Quantum nonlocality

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cryptography, where high-quality random numbers are essential for ensuring the security of cryptographic protocols. Randomness certification is the process of verifying that the output of a random number generator is truly random and has not been tampered with by an adversary. DI randomness certification does this verification without making assumptions about the underlying devices that generate random numbers. Instead, randomness is certified by observing correlations between the outputs of different devices that are generated using the same physical process. Recent research has demonstrated the feasibility of DI randomness certification using entangled quantum systems, such as photons or electrons. Randomness expansion is taking a small amount of initial random seed and expanding it into a much larger sequence of random numbers. In DI randomness expansion, the expansion is done using measurements of quantum systems that are prepared in a highly entangled state. The security of the expansion is guaranteed by the laws of quantum mechanics, which make it impossible for an adversary to predict the expansion output. Recent research has shown that DI randomness expansion can be achieved using entangled photon pairs and measurement devices that violate a Bell inequality. Randomness amplification is the process of taking a small amount of initial random seed and increasing its randomness by using a cryptographic algorithm. In DI randomness amplification, this process is done using entanglement properties and quantum mechanics. The security of the amplification is guaranteed by the fact that any attempt by an adversary to manipulate the algorithm's output will inevitably introduce errors that can be detected and corrected. Recent research has demonstrated the feasibility of DI randomness amplification using quantum entanglement and the violation of a Bell inequality.

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happens to be a hard task, and current general methods only provide very low estimates. However, a Bell scenario with five inputs and three outputs suffices to provide arbitrarily high lower bounds on the underlying Hilbert space dimension. Quantum communication protocols which assume a knowledge of the local dimension of Alice and Bob's systems, but otherwise do not make claims on the mathematical description of the preparation and measuring devices involved are termed semi-device independent protocols. Currently, there exist semi-device independent protocols for quantum key distribution and randomness expansion.

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influence hidden variables at another party's distant location, if there is enough time for a superluminal influence (of finite, but otherwise unknown speed) to propagate from one point to the other. In this scenario, any bipartite experiment revealing Bell nonlocality can just provide lower bounds on the hidden influence's propagation speed. Quantum experiments with three or more parties can, nonetheless, disprove all such non-local hidden variable models.

4072:{\displaystyle \left|\psi \right\rangle ={\frac {1}{\sqrt {3}}}\left(\left|00\right\rangle +\left|01\right\rangle +\left|10\right\rangle \right)={\frac {1}{\sqrt {3}}}\left({\sqrt {2}}\left|+0\right\rangle +{\frac {1}{\sqrt {2}}}\left(\left|+1\right\rangle +\left|-1\right\rangle \right)\right)={\frac {1}{\sqrt {3}}}\left({\sqrt {2}}\left|0+\right\rangle +{\frac {1}{\sqrt {2}}}\left(\left|1+\right\rangle +\left|1-\right\rangle \right)\right)} 10112:. Remarkably, all boxes within the almost quantum set are shown to be compatible with the principles of NTCC, NANLC, ML and LO. There is also numerical evidence that almost-quantum boxes also comply with IC. It seems, therefore, that, even when the above principles are taken together, they do not suffice to single out the quantum set in the simplest Bell scenario of two parties, two inputs and two outputs. 976: 906:. Quantum nonlocality has been experimentally verified under a variety of physical assumptions. Any physical theory that aims at superseding or replacing quantum theory should account for such experiments and therefore cannot fulfill local realism; quantum nonlocality is a property of the universe that is independent of our description of nature. 10301:

likely to become increasingly crucial as quantum computing technology advances. In addition, a milder approach called semi-DI exists where random numbers can be generated with some assumptions on the working principle of the devices, environment, dimension, energy, etc., in which it benefits from ease-of-implementation and high generation rate.

1296:. The rules of quantum theory give predictions for the outcomes of measurements performed by the experimentalists. Alice, for example, will measure her particle to be spin-up in an average of fifty percent of measurements. However, according to the Copenhagen interpretation, Alice's measurement causes the state of the two particles to 10284:

pad of which Eve has no information whatsoever. The one-time pad can then be used to transmit a secret message through a public channel. Although the first security analyses on device-independent QKD relied on Eve carrying out a specific family of attacks, all such protocols have been recently proven unconditionally secure.

1204:{\displaystyle \left|\psi _{AB}\right\rangle ={\frac {1}{\sqrt {2}}}\left(\left|0\right\rangle _{A}\left|1\right\rangle _{B}-\left|1\right\rangle _{A}\left|0\right\rangle _{B}\right)={\frac {1}{\sqrt {2}}}\left(\left|-\right\rangle _{A}\left|+\right\rangle _{B}-\left|+\right\rangle _{A}\left|-\right\rangle _{B}\right)} 10292:

Nonlocality can be used to certify that the outcomes of one of the parties in a Bell experiment are partially unknown to an external adversary. By feeding a partially random seed to several non-local boxes, and, after processing the outputs, one can end up with a longer (potentially unbounded) string

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happens to be, Alice and Bob estimate how much knowledge an external quantum adversary Eve (the eavesdropper) could possess on the value of Alice and Bob's outputs. This estimation allows them to devise a reconciliation protocol at the end of which Alice and Bob share a perfectly correlated one-time

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for entanglement, there are none for nonlocality. Finally, reasonably simple examples of Bell inequalities have been found for which the quantum state giving the largest violation is never a maximally entangled state, showing that entanglement is, in some sense, not even proportional to nonlocality.

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The work of Bancal et al. generalizes Bell's result by proving that correlations achievable in quantum theory are also incompatible with a large class of superluminal hidden variable models. In this framework, faster-than-light signaling is precluded. However, the choice of settings of one party can

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Bell's demonstration is probabilistic in the sense that it shows that the precise probabilities predicted by quantum mechanics for some entangled scenarios cannot be met by a local hidden variable theory. (For short, here and henceforth "local theory" means "local hidden variables theory".) However,

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Macroscopic Locality (ML). In the considered setup, two separate parties conduct extensive low-resolution measurements over a large number of independently prepared pairs of correlated particles. ML states that any such “macroscopic” experiment must admit a local hidden variable model. It is proven

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would allow building a device capable of transmitting information faster than the speed of light. At the level of correlations between two parties, Einstein's causality translates in the requirement that Alice's measurement choice should not affect Bob's statistics, and vice versa. Otherwise, Alice

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The works above rely on the implicit assumption that any physical set of correlations must be closed under wirings. This means that any effective box built by combining the inputs and outputs of a number of boxes within the considered set must also belong to the set. Closure under wirings does not

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provides such a completion of quantum mechanics, with the introduction of hidden variables; however the theory is explicitly nonlocal. The interpretation therefore does not give an answer to Einstein's question, which was whether or not a complete description of quantum mechanics could be given in

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In the Copenhagen view of this experiment, Alice's measurement—and particularly her measurement choice—has a direct effect on Bob's state. However, under the assumption of locality, actions on Alice's system do not affect the "true", or "ontic" state of Bob's system. We see that the ontic state of

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DI randomness certification, expansion, and amplification are powerful techniques for generating high-quality random numbers that are secure against any potential attacks on the underlying devices used to generate random numbers. These techniques have critical applications in cryptography and are

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answered Einstein's question by showing that such local hidden variables can never reproduce the full range of statistical outcomes predicted by quantum theory. Bell showed that a local hidden variable hypothesis leads to restrictions on the strength of correlations of measurement results. If the

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It is true that for any pure entangled state there exists a choice of measurements that produce Bell nonlocal correlations, but the situation is more complex for mixed states. While any Bell nonlocal state must be entangled, there exist (mixed) entangled states which do not produce Bell nonlocal

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for the same reason. Therefore, the ontic state of Bob's system must be compatible with at least two quantum states; the quantum state is therefore not a complete descriptor of his system. Einstein, Podolsky and Rosen saw this as evidence of the incompleteness of the Copenhagen interpretation of

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can also provide lower bounds on the Hilbert space dimension of the local systems accessible to Alice and Bob. This problem is equivalent to deciding the existence of a matrix with low completely positive semidefinite rank. Finding lower bounds on the Hilbert space dimension based on statistics

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Nonlocality can be exploited to conduct quantum information tasks which do not rely on the knowledge of the inner workings of the prepare-and-measurement apparatuses involved in the experiment. The security or reliability of any such protocol just depends on the strength of the experimentally

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Quantum nonlocality is sometimes understood as being equivalent to entanglement. However, this is not the case. Quantum entanglement can be defined only within the formalism of quantum mechanics, i.e., it is a model-dependent property. In contrast, nonlocality refers to the impossibility of a

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The above considerations apply to model a quantum experiment. Consider two parties conducting local polarization measurements on a bipartite photonic state. The measurement result for the polarization of a photon can take one of two values (informally, whether the photon is polarized in that

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Device-independent (DI) randomness certification, expansion, and amplification are techniques used to generate high-quality random numbers that are secure against any potential attacks on the underlying devices used to generate random numbers. These techniques have critical applications in

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As shown, the statistics achievable by two or more parties conducting experiments in a classical system are constrained in a non-trivial way. Analogously, the statistics achievable by separate observers in a quantum theory also happen to be restricted. The first derivation of a non-trivial

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direction, or in the orthogonal direction). If each party is allowed to choose between just two different polarization directions, the experiment fits within the CHSH scenario. As noted by CHSH, there exist a quantum state and polarization directions which generate a box

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Tebyanian, H., Zahidy, M., Avesani, M., Stanco, A., Villoresi, P., & Vallone, G. (2021). Semi-device independent randomness generation based on quantum state's indistinguishability. Quantum Science and Technology, 6(4), 045026. doi: 10.1088/2058-9565/ac2047. URL:

5142:. Similar separations have been established in other types of causal structures. The characterization of the boundaries for classical correlations in such extended Bell scenarios is challenging, but there exist complete practical computational methods to achieve it. 3720:

quantum mechanics permits an even stronger violation of local theories: a possibilistic one, in which local theories cannot even agree with quantum mechanics on which events are possible or impossible in an entangled scenario. The first proof of this kind was due to

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that any microscopic experiment capable of violating Tsirelson's bound would also violate standard Bell nonlocality when brought to the macroscopic scale. Besides Tsirelson's bound, the principle of ML fully recovers the set of all two-point quantum correlators.

3706:. This demonstrates an explicit way in which a theory with ontological states that are local, with local measurements and only local actions cannot match the probabilistic predictions of quantum theory, disproving Einstein's hypothesis. Experimentalists such as 1287: 2137:

to locally characterize measurement results on each system: "It is a matter of indifference ... whether λ denotes a single variable or a set ... and whether the variables are discrete or continuous". However, it is equivalent (and more intuitive) to think of

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Bell inequalities are violated experimentally as predicted by quantum mechanics, then reality cannot be described by local hidden variables and the mystery of quantum nonlocal causation remains. However, Bell notes that the non-local hidden variable model of

4962:: directed acyclic graphs where each node represents a variable and an edge from a variable to another signifies that the former influences the latter and not otherwise, see the figure. In a standard bipartite Bell experiment, Alice's (Bob's) setting 10184:

The first device-independent protocol proposed was device-independent quantum key distribution (QKD). In this primitive, two distant parties, Alice and Bob, are distributed an entangled quantum state, that they probe, thus obtaining the statistics

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The works listed above describe what the quantum set of correlations looks like, but they do not explain why. Are quantum correlations unavoidable, even in post-quantum physical theories, or on the contrary, could there exist correlations outside

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when Alice and Bob reboot their experimental setup. Bell's assumption of local causality then stipulates that each local strategy defines the distributions of independent outcomes if Alice conducts experiment x and Bob conducts experiment

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All these principles can be experimentally falsified under the assumption that we can decide if two or more events are space-like separated. This sets this research program aside from the axiomatic reconstruction of quantum mechanics via

6540: 10574: 9885:. The principle of LO states that, for any multipartite box, the sum of the probabilities of any set of pair-wise locally orthogonal events cannot exceed 1. It is proven that any bipartite box violating Tsirelson's bound by an amount of 1705:

While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.

2833: 1552:". This steering occurs in such a way that no signal can be sent by performing such a state update; quantum nonlocality cannot be used to send messages instantaneously and is therefore not in direct conflict with causality concerns in 7230:

admits a quantum realization is a complicated problem. In fact, the dual problem of establishing whether a quantum box can have a perfect score at a non-local game is known to be undecidable. Moreover, the problem of deciding whether

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seem to enforce any limit on the maximum value of CHSH. However, it is not a void principle: on the contrary, in it is shown that many simple, intuitive families of sets of correlations in probability space happen to violate it.

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In view of this mismatch, Popescu and Rohrlich pose the problem of identifying a physical principle, stronger than the no-signalling conditions, that allows deriving the set of quantum correlations. Several proposals followed:

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have verified the quantum violation of the CHSH inequality as well as other formulations of Bell's inequality, to invalidate the local hidden variables hypothesis and confirm that reality is indeed nonlocal in the EPR sense.

9109: 5183:. Consider the CHSH Bell scenario detailed before, but this time assume that, in their experiments, Alice and Bob are preparing and measuring quantum systems. In that case, the CHSH parameter can be shown to be bounded by 964:

to challenge the idea that the quantum wavefunction was a complete description of reality, but instead they sparked a debate on the nature of reality. Afterwards, Einstein presented a variant of these ideas in a letter to

8726: 8583: 106: 4646: 3453: 1546: 1484: 1422: 1360: 12876:"Amazing: Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen proved that MIP* = RE and thus disproved Connes 1976 Embedding Conjecture, and provided a negative answer to Tsirelson's problem" 1714:) criticised the ambiguous terminology of the EPR paper, the thought experiment nevertheless generated a great deal of interest. Their notion of a "complete description" was later formalised by the suggestion of 11961:

Bancal, Jean-Daniel; Pironio, Stefano; Acin, Antonio; Liang, Yeong-Cherng; Scarani, Valerio; Gisin, Nicolas (2012). "Quantum nonlocality based on finite-speed causal influences leads to superluminal signaling".

5247: 5500: 3581: 8871: 5398: 4903: 4727: 4576: 4424: 4376: 4300: 4252: 4204: 10687: 9291:. The principle of NANLC states that non-local boxes should not give the two parties any advantage to play this game. It is proven that any box violating Tsirelson's bound would provide such an advantage. 9773: 7694: 1630:, since Alice can make a measurement that concludes with one of those states being the quantum description of his system. At the same time, it must also be compatible with one of the quantum states 2737: 1698: 6344: 5606: 1663: 1628: 6238: 3201: 1593: 14380:

Li, Hong-Wei; Yin, Zhen-Qiang; Wu, Yu-Chun; Zou, Xu-Bo; Wang, Shuang; Chen, Wei; Guo, Guang-Can; Han, Zheng-Fu (2011). "Semi-device-independent random-number expansion without entanglement".

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In the above definition, the space-like separation of the two parties conducting the Bell experiment was modeled by imposing that their associated operator algebras act on different factors

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Colbeck, R. & Kent, A. (2011). Private randomness expansion with untrusted devices. Journal of Physics A: Mathematical and Theoretical, 44(9), 095305. doi: 10.1088/1751-8113/44/9/095305

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correlations (although, operating on several copies of some of such states, or carrying out local post-selections, it is possible to witness nonlocal effects). Moreover, while there are

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Here, subscripts “A” and “B” distinguish the two particles, though it is more convenient and usual to refer to these particles as being in the possession of two experimentalists called

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described "two spatially separated particles which have both perfectly correlated positions and momenta" as a direct consequence of quantum theory. They intended to use the classical

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In the scenario proposed by Bell (a Bell scenario), two experimentalists, Alice and Bob, conduct experiments in separate labs. At each run, Alice (Bob) conducts an experiment

9603: 9551: 2185: 9016:(NTCC). This principle stipulates that nonlocal correlations should not be so strong as to allow two parties to solve all 1-way communication problems with some probability 5717: 3671: 4958:

The random variables measured in a general experiment can depend on each other in complicated ways. In the field of causal inference, such dependencies are represented via

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and named self-testing by Mayers and Yao. Self-testing is known to be robust against systematic noise, i.e., if the experimentally measured statistics are close enough to

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that, while complying with the no-signalling conditions, violates Tsirelson's bound, and is thus unrealizable in quantum physics. Dubbed the PR-box, it can be written as:

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Fritz, T.; A. B. Sainz; R. Augusiak; J. B. Brask; R. Chaves; A. Leverrier; A. Acín (2013). "Local orthogonality as a multipartite principle for quantum correlations".

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describing the experiment. Alternatively, one could model space-like separation by imposing that these two algebras commute. This leads to a different definition:

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is NP-hard. Characterizing quantum boxes is equivalent to characterizing the cone of completely positive semidefinite matrices under a set of linear constraints.

6268: 5530: 2574: 2529: 1935: 1860: 1815: 6642: 5611: 5096:). Bell's theorem can thus be interpreted as a separation between the quantum and classical predictions in a type of causal structures with just one hidden node 4158:

The experiment consists of this entangled state being shared between two experimenters, each of whom has the ability to measure either with respect to the basis

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in the strict sense defined by special relativity and, as such, the term "quantum nonlocality" is sometimes considered a misnomer. Still, it prompts many of the

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Brassard, G; Buhrman, H; Linden, N; Methot, AA; Tapp, A; Unger, F (2006). "Limit on Nonlocality in Any World in Which Communication Complexity Is Not Trivial".

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To prove unrealizability, the most known method is the Navascués–Pironio–Acín (NPA) hierarchy. This is an infinite decreasing sequence of sets of correlations

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Ghirardi, G.C.; Rimini, A.; Weber, T. (March 1980). "A general argument against superluminal transmission through the quantum mechanical measurement process".

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via local unitary transformations. This phenomenon, that can be interpreted as an instance of device-independent quantum tomography, was first pointed out by

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quantum theory, since the wavefunction is explicitly not a complete description of a quantum system under this assumption of locality. Their paper concludes:

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description of observed statistics in terms of a local hidden variable model, so it is independent of the physical model used to describe the experiment.

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Brunner, Nicolas; Pironio, Stefano; Acín, Antonio; Gisin, Nicolas; Methot, Andre Allan; Scarani, Valerio (2008). "Testing the Hilbert space dimension".

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and Rohrlich explore whether quantum correlations can be explained by appealing to relativistic causality alone. Namely, whether any hypothetical box

3023:{\displaystyle P(a,b|x,y)=\sum _{\lambda _{A},\lambda _{B}\in \Lambda }\rho (\lambda _{A},\lambda _{B})P_{A}(a|x,\lambda _{A})P_{B}(b|y,\lambda _{B})} 13529:

Pawlowski, M.; Paterek, T.; Kaszlikowski, D.; Scarani, V.; Winter, A.; Zukowski, M. (October 2009). "Information Causality as a Physical Principle".

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demonstrated a logical proof of quantum nonlocality that, like the GHZ proof is a possibilistic proof. It starts with the observation that the state

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Braun, D.; Choi, M.-S. (2008). "Hardy's test versus the Clauser-Horne-Shimony-Holt test of quantum nonlocality: Fundamental and practical aspects".

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Barrett, J.; Linden, N.; Massar, S.; Pironio, S.; Popescu, S.; Roberts, D. (2005). "Non-local correlations as an information theoretic resource".

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A simple Bayesian network. Rain influences whether the sprinkler is activated, and both rain and the sprinkler influence whether the grass is wet.

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Allcock, Jonathan; Nicolas Brunner; Noah Linden; Sandu Popescu; Paul Skrzypczyk; Tamás Vértesi (2009). "Closed sets of non-local correlations".

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is a set of correlations that is closed under wirings and can be characterized via semidefinite programming. It contains all correlations in

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Junge, M; Navascués, M; Palazuelos, C; Pérez-García, D; Scholz, VB; Werner, RF (2011). "Connes' embedding problem and Tsirelson's problem".

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Originally, it was unknown whether any of these principles (or a subset thereof) was strong enough to derive all the constraints defining

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can be correlated), then one can average over this distribution to obtain a formula for the joint probability of each measurement result:

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Ito, T.; Kobayashi, H.; Matsumoto, K. (2008). "Oracularization and two-prover one-round interactive proofs against nonlocal strategies".

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Wiseman, H.M.; Jones, S.J.; Doherty, A.C. (April 2007). "Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox".

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of comparable randomness or with a shorter but more random string. This last primitive can be proven impossible in a classical setting.

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Reid, M. D.; Drummond, P. D.; Bowen, W. P.; Cavalcanti, E. G.; Lam, P. K.; Bachor, H. A.; Andersen, U. L.; Leuchs, G. (2009-12-10).

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shared by Alice and Bob is such that it only admits a unique quantum realization. This means that there exist measurement operators

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Sikora, Jamie; Varvitsiotis, Antonios (2017). "Linear conic formulations for two-party correlations and values of nonlocal games".

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Hensen, B, et al. (October 2015). "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres".

10628: 8173:, (as claimed by Ji, Natarajan, Vidick, Wright, and Yuen) then a new method to detect the non-realizability of the correlations in 124: 15268: 10179: 9712: 945: 580: 7634: 14952: 14510: 11634: 11549:, Springer, Berlin, translated into English by Beyer, R.T., Princeton University Press, Princeton, cited by Baggott, J. (2004) 353: 14306:

Coladangelo, Andrea; Stark, Jalex (2018). "Unconditional separation of finite and infinite-dimensional quantum correlations".

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Linden, N.; Popescu, S.; Short, A. J.; Winter, A. (2007). "Quantum Nonlocality and Beyond: Limits from Nonlocal Computation".

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A box admitting such a decomposition is called a Bell local or a classical box. Fixing the number of possible values which

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denotes the sum modulo two. It can be verified that the CHSH value of this box is 4 (as opposed to the Tsirelson bound of

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of bits transmitted by Alice. It is shown that any box violating Tsirelson's bound would allow two parties to violate IC.

6273: 5535: 1633: 1598: 14044:. Proceedings of the 25th IEEE Symposium on Foundations of Computer Science. University of California. pp. 434–440. 11309:

Einstein, Albert. "Letter to E. Schrödinger" . Einstein Archives, ID: Call Number 22-47. Hebrew University of Jerusalem.

9297:(IC). The starting point is a bipartite communication scenario where one of the parts (Alice) is handed a random string 6169: 3122: 1563: 1282:{\textstyle \left|\pm \right\rangle ={\frac {1}{\sqrt {2}}}\left(\left|0\right\rangle \pm \left|1\right\rangle \right)} 12900: 2742: 14540: 14503: 14200:

Prakash, Anupam; Sikora, Jamie; Varvitsiotis, Antonios; Wei Zhaohui (2018). "Completely positive semidefinite rank".

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Navascués, Miguel; Wolfe, Elie (2020). "The Inflation Technique Completely Solves the Causal Compatibility Problem".

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Giustina, M, et al. (December 2015). "Significant-Loophole-Free Test of Bell's Theorem with Entangled Photons".

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Pawlowski, Marcin; Brunner, Nicolas (2011). "Semi-device-independent security of one-way quantum key distribution".

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Werner, R.F. (1989). "Quantum States with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model".

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Chapter 5. Quantum And Relativistic Protocols For Secure Multi-Party Computation (Thesis), University of Cambridge

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and Holt (CHSH) reformulated these inequalities in a manner that was more conducive to experimental testing (see

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Jevtic, S.; Rudolph, T (2015). "How Einstein and/or Schrödinger should have discovered Bell's theorem in 1936".

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Rowe MA, et al. (February 2001). "Experimental violation of a Bell's Inequality with efficient detection".

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Like the set of classical boxes, when represented in probability space, the set of no-signalling boxes forms a

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Acín, Antonio; Nicolas Gisin; Lluis Masanes (2006). "From Bell's Theorem to Secure Quantum Key Distribution".

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Hardy, Lucien (1993). "Nonlocality for two particles without inequalities for almost all entangled states".

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using just one bit of communication. It can be proven that any box violating Tsirelson's bound by more than

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This is characteristic ... of any such theory which reproduces exactly the quantum mechanical predictions.

1424:. Likewise, if Alice performs a measurement of spin in the x-direction, that is, with respect to the basis 531: 526: 497: 348: 129: 12754: 7073: 6018: 15015: 14922: 14245:

Navascués, Miguel; Vértesi, Tamás (2015). "Bounding the set of finite dimensional quantum correlations".

10361: 8017: 7765: 7703: 565: 310: 300: 7438: 1300:, so that if Alice performs a measurement of spin in the z-direction, that is with respect to the basis 15291: 15053: 14861: 14832: 11381: 7530: 7478: 5814:{\displaystyle P(a,b|x,y)=\left\langle \psi \right|E_{a}^{x}\otimes F_{b}^{y}\left|\psi \right\rangle } 1719: 910: 859: 511: 482: 9340: 9117: 9114:

No Advantage for Nonlocal Computation (NANLC). The following scenario is considered: given a function

8218: 8176: 6948: 15075: 14942: 14866: 14827: 14764: 14743: 14700: 14593: 9556: 9504: 7067: 4148:{\displaystyle |\pm \rangle ={\tfrac {1}{\sqrt {2}}}(\left|0\right\rangle \pm \left|1\right\rangle )} 3725: 2161: 1937:

when they respectively conduct the experiments x,y. In the following, each such set of probabilities

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that determine the statistics of measurement results, but to which an observer does not have access.

516: 477: 430: 405: 328: 188: 13326: 12811: 3638: 1862:. If Alice and Bob repeat their experiments several times, then they can estimate the probabilities 15142: 15122: 15112: 15102: 15058: 14618: 12473:

Junge, Marius; Palazuelos, C (2011). "Large violation of Bell inequalities with low entanglement".

12266:

Popescu, Sandu (1995). "Bell's Inequalities and Density Matrices: Revealing "Hidden" Nonlocality".

10819: 9013: 8077: 7116: 4796: 4732: 2363:{\displaystyle P(a,b|x,y,\lambda _{A},\lambda _{B})=P_{A}(a|x,\lambda _{A})P_{B}(b|y,\lambda _{B})} 600: 9982: 9953: 9950:. This state of affairs continued for some years until the construction of the almost quantum set 9259: 5903:

is not a polytope. On the contrary, it contains both straight and curved boundaries. In addition,

5032: 5005: 2606: 2579: 14837: 14748: 11801: 11669: 7349: 5978: 5309: 4829: 4765: 4653: 769: 487: 395: 15251: 14799: 12975: 10753: 10696: 10579: 10439: 10312: 10237: 10188: 10124: 9924: 8737: 8298: 8260: 7582: 7389: 7234: 7184: 7037: 6857: 6716: 6066: 5926: 5260: 3589: 3276: 3073: 2068: 2019: 1865: 445: 343: 109: 15162: 14937: 14917: 14842: 14695: 13321: 13304: 10814: 9233: 8944: 6535:{\displaystyle P(a,b|x,y)=\left\langle \psi \right|E_{a}^{x}F_{b}^{y}\left|\psi \right\rangle } 1715: 1297: 774: 492: 320: 290: 253: 14738: 14126: 12775:

Ji, Zhengfeng; Natarajan, Anand; Vidick, Thomas; Wright, John; Yuen, Henry (2020). "MIP*=RE".

11597: 10569:{\displaystyle {\tilde {E}}_{a}^{x},{\tilde {F}}_{b}^{y},\left|{\tilde {\psi }}\right\rangle } 9019: 5972:

which can be arbitrarily well approximated by quantum systems but are themselves not quantum.

2676: 2141: 2120: 973:'s take on EPR. The quantum state of the two particles prior to measurement can be written as 400: 15187: 14685: 14665: 13897:

Vazirani, Umesh; Vidick, Thomas (2014). "Fully Device-Independent Quantum Key Distribution".

12798: 10824: 9294: 8878: 6647: 6545: 5824: 3210: 3035: 1736: 961: 918: 243: 228: 15218: 15147: 15092: 14822: 14630: 14588: 14399: 14346: 14264: 14158: 14083: 14006: 13916: 13855: 13774: 13718: 13654: 13550: 13495: 13434: 13366: 13313: 13260: 13241:

Navascués, Miguel; Pironio, S; Acín, A (2007). "Bounding the Set of Quantum Correlations".

13130:

Ozawa, Narutaka (2013). "About the Connes Embedding Conjecture---Algebraic approaches---".

13096: 13043: 12838: 12786: 12645: 12602: 12545: 12492: 12423: 12350: 12285: 12224: 12171: 12034: 11981: 11924: 11871: 11810: 11782: 11733: 11678: 11520: 11481: 11424: 11339: 11252: 11193: 11098: 11033: 10972: 10901: 10864: 10809: 10742:, one can still determine the underlying state and measurement operators up to error bars. 10287: 9888: 9390: 8916: 8107: 7157: 7130: 7010: 6788: 6689: 5705:{\displaystyle \sum _{a}E_{a}^{x}={\mathbb {I} }_{A},\sum _{b}F_{b}^{y}={\mathbb {I} }_{B}} 5180: 5151: 3248: 560: 472: 198: 155: 12401: 966: 804: 8: 15203: 15172: 15117: 15097: 15005: 14962: 14817: 14728: 14645: 14557: 13755:

Navascués, M.; Y. Guryanova; M. J. Hoban; A. Acín (2015). "Almost Quantum Correlations".

12684:"Quantum analogues of the Bell inequalities. The case of two spatially separated domains" 12072:; Fritz, T (2019). "The Inflation Technique for Causal Inference with Latent Variables". 12046: 10829: 9207: 9181: 8418: 8396:(Bob) could signal Bob (Alice) instantaneously by choosing her (his) measurement setting 6247: 5509: 2553: 2508: 1914: 1839: 1794: 922: 887: 659: 467: 385: 193: 145: 14403: 14350: 14268: 14162: 14087: 14010: 13920: 13859: 13778: 13722: 13658: 13554: 13499: 13438: 13370: 13317: 13264: 13100: 13047: 12842: 12790: 12649: 12606: 12549: 12496: 12427: 12354: 12289: 12228: 12175: 12038: 11985: 11928: 11875: 11814: 11786: 11737: 11682: 11648: 11629: 11524: 11485: 11428: 11343: 11256: 11197: 11102: 11037: 10976: 10905: 10868: 6585: 15263: 15132: 15030: 14723: 14670: 14562: 14415: 14389: 14362: 14336: 14307: 14288: 14254: 14227: 14209: 14182: 14148: 14107: 14073: 14022: 13996: 13963: 13940: 13906: 13879: 13845: 13798: 13764: 13734: 13708: 13678: 13644: 13605: 13574: 13540: 13485: 13458: 13424: 13382: 13339: 13284: 13250: 13223: 13205: 13176: 13157: 13112: 13086: 13059: 13033: 12776: 12727: 12705: 12661: 12618: 12592: 12561: 12535: 12508: 12482: 12455: 12413: 12382: 12340: 12328: 12309: 12275: 12248: 12214: 12144: 12126: 12099: 12081: 12050: 12024: 11997: 11971: 11940: 11914: 11887: 11861: 11834: 11772: 11749: 11723: 11696: 11448: 11414: 11355: 11329: 11242: 11159: 11119: 11088: 11076: 11057: 11023: 10996: 10962: 10935: 9778: 9484: 9481:

Local Orthogonality (LO). This principle applies to multipartite Bell scenarios, where

9457: 9454:

of the mutual information between Alice's bit and Bob's guess cannot exceed the number

9437: 9417: 9320: 9300: 9161: 8399: 8087: 7997: 7912: 7827: 6990: 6886: 6768: 6115: 5906: 5886: 5866: 5079: 5059: 4985: 4965: 3721: 2534: 2489: 2192: 1820: 1775: 1553: 914: 380: 305: 238: 150: 11960: 11663:

Clauser, John F.; Horne, Michael A.; Shimony, Abner; Holt, Richard A. (October 1969).

11231:"Colloquium : The Einstein-Podolsky-Rosen paradox: From concepts to applications" 11230: 15273: 15182: 15152: 15080: 15043: 15038: 15020: 14985: 14975: 14779: 14675: 14640: 14623: 14526: 14482: 14460: 14441: 14419: 14366: 14280: 14199: 14186: 14174: 14099: 14026: 13944: 13932: 13871: 13790: 13738: 13670: 13566: 13511: 13450: 13386: 13343: 13276: 13161: 12856: 12709: 12665: 12459: 12447: 12439: 12374: 12301: 12240: 12187: 12148: 11891: 11826: 11576: 11573:

Quantum Non-Locality and Relativity : Metaphysical Intimations of Modern Physics

11554: 11440: 11385: 11278: 11211: 11163: 11124: 11049: 11000: 10988: 10939: 10927: 7034:

is constrained to be finite-dimensional, the closure of the corresponding set equals

1747: 1742: 1732: 814: 789: 729: 724: 624: 590: 570: 168: 27: 14292: 14231: 13802: 13682: 13404:. Symposium on the Foundations of Modern Physics. World Sci. Publ. pp. 441–460. 13288: 13116: 13063: 12622: 12526:

Thomas Vidick; Stephanie Wehner (2011). "More Non-locality with less Entanglement".

12313: 12103: 12001: 11838: 11766: 11753: 11700: 11452: 11359: 11061: 8441:

appropriately. Mathematically, Popescu and Rohrlich's no-signalling conditions are:

15000: 14995: 14852: 14733: 14407: 14354: 14276: 14272: 14219: 14170: 14166: 14111: 14091: 14014: 13928: 13924: 13883: 13863: 13782: 13726: 13662: 13615: 13578: 13558: 13531: 13503: 13462: 13442: 13374: 13331: 13268: 13227: 13215: 13147: 13139: 13104: 13051: 12846: 12695: 12653: 12610: 12565: 12553: 12500: 12435: 12431: 12366: 12358: 12293: 12252: 12236: 12232: 12179: 12136: 12091: 12054: 12042: 11989: 11944: 11932: 11879: 11818: 11741: 11686: 11643: 11528: 11489: 11432: 11347: 11268: 11260: 11201: 11151: 11114: 11110: 11106: 11045: 11041: 10980: 10917: 10909: 10872: 6429:{\displaystyle \sum _{a}E_{a}^{x}={\mathbb {I} },\sum _{b}F_{b}^{y}={\mathbb {I} }} 5883:. Contrary to the classical set of correlations, when viewed in probability space, 5162: 4959: 1549: 819: 809: 799: 699: 679: 664: 634: 502: 390: 14138: 13867: 13507: 13446: 13272: 12512: 12386: 11436: 5306:

admits a quantum realization if and only if there exists a pair of Hilbert spaces

4945: 4936: 15230: 15157: 15137: 15107: 15070: 15065: 14970: 14794: 12069: 11511: 11472: 10690: 9001: 5176: 3729: 3204: 1766: 1723:

terms of local hidden variables in keeping with the "Principle of Local Action".

949: 844: 714: 694: 440: 280: 12362: 12297: 11822: 11468:"Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" 11182:"Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" 10877: 10852: 15208: 15177: 15167: 14789: 14769: 14598: 14411: 14358: 13730: 12851: 12826: 12614: 12557: 11883: 11745: 11691: 11664: 11264: 9104:{\displaystyle 2{\sqrt {2}}\left({\frac {2}{\sqrt {3}}}-1\right)\approx 0.4377} 6112:

admits a field quantum realization if and only if there exists a Hilbert space

953: 899: 779: 739: 719: 689: 669: 619: 585: 435: 425: 218: 14223: 14064:

Pironio, S, et al. (2010). "Random numbers certified by Bell's theorem".

13698: 13219: 13143: 13108: 12580: 12578: 12504: 11936: 8721:{\displaystyle \sum _{b}P(a,b|x,y)=\sum _{b}P(a,b|x,y^{\prime })=:P_{A}(a|x).} 8578:{\displaystyle \sum _{a}P(a,b|x,y)=\sum _{a}P(a,b|x^{\prime },y)=:P_{B}(b|y),} 4681:

we conclude that if one of the experimenters had measured with respect to the

1911:, namely, the probability that Alice and Bob respectively observe the results 15306: 15127: 14980: 14871: 14774: 14690: 14660: 14613: 13984: 13023: 12925: 12443: 12378: 12183: 11551:

Beyond Measure: Modern physics, philosophy, and the meaning of quantum theory

11282: 11215: 8328: 7066:

In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen claimed a result in

1762: 1293: 903: 839: 834: 764: 734: 704: 575: 521: 248: 223: 101:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle } 11351: 15010: 14608: 14603: 14474: 14284: 14178: 14103: 13936: 13875: 13794: 13674: 13620: 13593: 13570: 13515: 13454: 13280: 12860: 12726:

Slofstra, William (2017). "The set of quantum correlations is not closed".

12451: 12305: 12244: 11830: 11532: 11494: 11444: 11206: 11181: 11128: 11053: 10992: 10931: 8257:, then the above two methods would provide a practical characterization of 6987:. This problem only appears in infinite dimensions: when the Hilbert space 5252: 4641:{\displaystyle \langle --|\psi \rangle =-{\tfrac {1}{2{\sqrt {3}}}}\neq 0.} 3740: 3707: 2117:

Bell formalized the idea of a hidden variable by introducing the parameter

1758: 957: 829: 824: 759: 744: 709: 203: 13357:

Rastall, Peter (1985). "Locality, Bell's theorem, and quantum mechanics".

12191: 12140: 12095: 6903:. Tsirelson's problem consists in deciding whether the inclusion relation 15048: 13850: 13490: 13429: 13255: 12370: 12345: 12280: 11919: 11728: 11419: 10922: 8215:

is needed. If Tsirelson's problem was solved in the affirmative, namely,

8084:

The NPA hierarchy thus provides a computational characterization, not of

939: 794: 749: 684: 639: 14095: 13562: 13077:

Fritz, Tobias (2012). "Tsirelson's problem and Kirchberg's conjecture".

10984: 10853:"Experimental Test of Bell's Inequalities Using Time- Varying Analyzers" 10288:

Device-independent randomness certification, expansion and amplification

4857:

are impossible. But then, if they had both measured with respect to the

13786: 13754: 13666: 13475: 13378: 13335: 13152: 12700: 12683: 12657: 12205:

Palazuelos, Carlos (2012). "Super-activation of quantum non-locality".

11509:

Furry, W.H. (March 1936). "Remarks on Measurements in Quantum Theory".

11273: 11155: 7579:. That method, however, can just be used to prove the realizability of 3448:{\displaystyle S_{\rm {CHSH}}\equiv E(0,0)+E(1,0)+E(0,1)-E(1,1)\leq 2,} 1711: 1541:{\displaystyle \{\left|+\right\rangle _{B},\left|-\right\rangle _{B}\}} 1479:{\displaystyle \{\left|+\right\rangle _{A},\left|-\right\rangle _{A}\}} 1417:{\displaystyle \{\left|0\right\rangle _{B},\left|1\right\rangle _{B}\}} 1355:{\displaystyle \{\left|0\right\rangle _{A},\left|1\right\rangle _{A}\}} 970: 784: 754: 674: 649: 644: 629: 14379: 14018: 13634: 13055: 12636:

Cirel'son, BS (1980). "Quantum generalizations of Bell's inequality".

12015:

Fritz, Tobias (2012). "Beyond Bell's Theorem: Correlation Scenarios".

11993: 11713: 8324:

which nonetheless do not lead to any unphysical operational behavior?

4950: 14902: 14583: 13414: 11467: 10913: 2158:

as a local "strategy" or "message" that occurs with some probability

275: 14495: 13302:

Popescu, Sandu; Rohrlich, Daniel (1994). "Nonlocality as an axiom".

9000:). This box had been identified earlier, by Rastall and Khalfin and 14312: 14214: 13528: 13210: 13001:"Landmark Computer Science Proof Cascades Through Physics and Math" 12827:"How 'spooky' is quantum physics? The answer could be incalculable" 12781: 12732: 12418: 12131: 12086: 11093: 11028: 10967: 8731: 5242:{\displaystyle -2{\sqrt {2}}\leq \mathrm {CHSH} \leq 2{\sqrt {2}}.} 654: 14394: 14341: 14259: 14153: 14078: 14001: 13968: 13911: 13769: 13713: 13649: 13610: 13545: 13400:

Khalfin, L.A.; Tsirelson, B.S. (1985). Lahti; et al. (eds.).

13181: 13091: 13038: 12597: 12540: 12487: 12219: 12029: 11976: 11866: 11777: 11334: 11247: 917:

and its universal speed limit of objects. Thus, quantum theory is

14457:

The Quantum Handshake: Entanglement, Nonlocality and Transactions

12329:"Entanglement-Assisted Local Manipulation of Pure Quantum States" 11767:

Daniel M. Greenberger; Michael A. Horne; Anton Zeilinger (2007),

10850: 5495:{\displaystyle E_{a}^{x}:H_{A}\to H_{A},F_{b}^{y}:H_{B}\to H_{B}} 3576:{\displaystyle E(x,y)\equiv \sum _{a,b=0,1}(-1)^{a+b}P(a,b|x,y).} 3203:. In that representation, the set of all classical boxes forms a 2576:

and the local variable describing her (his) experiment has value

8866:{\displaystyle P(a,b|x,y)={\frac {1}{2}}\delta _{xy,a\oplus b}.} 5393:{\displaystyle \left|\psi \right\rangle \in H_{A}\otimes H_{B}} 4946:

Analogs of Bell’s theorem in more complicated causal structures

4937:

Nonlocal hidden variable models with a finite propagation speed

4898:{\displaystyle \{\left|0\right\rangle ,\left|1\right\rangle \}} 4722:{\displaystyle \{\left|0\right\rangle ,\left|1\right\rangle \}} 4571:{\displaystyle \{\left|+\right\rangle ,\left|-\right\rangle \}} 4419:{\displaystyle \{\left|+\right\rangle ,\left|-\right\rangle \}} 4371:{\displaystyle \{\left|0\right\rangle ,\left|1\right\rangle \}} 4295:{\displaystyle \{\left|0\right\rangle ,\left|1\right\rangle \}} 4247:{\displaystyle \{\left|+\right\rangle ,\left|-\right\rangle \}} 4199:{\displaystyle \{\left|0\right\rangle ,\left|1\right\rangle \}} 1560:

Bob's system must be compatible with one of the quantum states

14042:

Generating quasi-random sequences from slightly-random sources

13835: 11905:

Nikolić, Hrvoje (2007). "Quantum Mechanics: Myths and Facts".

10173: 7181:

are, in principle, unbounded, determining whether a given box

2486:) denotes the probability that Alice (Bob) obtains the result 13174: 12525: 10682:{\displaystyle E_{a}^{x},F_{b}^{y},\left|\psi \right\rangle } 5175:

statistical limit on the set of quantum correlations, due to

13399: 13195: 12067: 11665:"Proposed Experiment to Test Local Hidden-Variable Theories" 8289: 14127:

https://iopscience.iop.org/article/10.1088/2058-9565/ac2047

13825:. IEEE Symposium on Foundations of Computer Science (FOCS). 12579:

Yeong-Cherng Liang; Tamás Vértesi; Nicolas Brunner (2010).

11228: 11141: 10851:

Aspect, Alain; Dalibard, Jean; Roger, Gérard (1982-12-20).

9768:{\displaystyle ({\bar {a}}^{\prime }|{\bar {x}}^{\prime })} 9553:

in their local labs. They respectively obtain the outcomes

7122: 13240: 7689:{\displaystyle Q^{1}\supset Q^{2}\supset Q^{3}\supset ...} 7119:, a famous conjecture in the theory of operator algebras. 14326: 14039: 13402:

Quantum and quasi-classical analogs of Bell inequalities

12402:"Neither Contextuality nor Nonlocality Admits Catalysts" 11598:"The Einstein-Podolsky-Rosen Argument in Quantum Theory" 5253:

The sets of quantum correlations and Tsirelson’s problem

14305: 14244: 13820: 13301: 12774: 11662: 11375: 7280:

can be approximated by a quantum system with precision

5863:

In the following, the set of such boxes will be called

3771:

defined below can be written in a few suggestive ways:

2788:

of being selected (shared randomness is allowed, i.e.,

13896: 13591: 7386:, one can explore, using variational methods, whether 6765:

How does this new set relate to the more conventional

4612: 4101: 1486:, then Bob's system will be left in one of the states 1362:, then Bob's system will be left in one of the states 1217: 14435: 13982: 11179: 10756: 10699: 10631: 10582: 10491: 10442: 10414: 10364: 10315: 10240: 10191: 10127: 10056: 10014: 9985: 9956: 9927: 9891: 9846: 9801: 9781: 9715: 9663: 9611: 9559: 9507: 9487: 9460: 9440: 9420: 9414:, for which purpose Alice is allowed to transmit Bob 9393: 9343: 9337:

bits. The second part, Bob, receives a random number

9323: 9303: 9262: 9236: 9210: 9184: 9164: 9120: 9056: 9022: 8973: 8947: 8919: 8881: 8789: 8740: 8590: 8447: 8421: 8402: 8337: 8301: 8263: 8221: 8179: 8137: 8110: 8090: 8020: 8000: 7935: 7915: 7853: 7830: 7768: 7706: 7637: 7585: 7533: 7481: 7441: 7392: 7352: 7286: 7237: 7187: 7160: 7133: 7076: 7040: 7013: 6993: 6951: 6909: 6889: 6860: 6818: 6791: 6771: 6719: 6692: 6650: 6588: 6548: 6444: 6352: 6276: 6250: 6172: 6138: 6118: 6069: 6021: 5981: 5929: 5909: 5889: 5869: 5827: 5720: 5614: 5538: 5512: 5406: 5352: 5312: 5263: 5192: 5102: 5082: 5062: 5035: 5008: 4988: 4968: 4911: 4863: 4832: 4799: 4768: 4735: 4687: 4656: 4584: 4536: 4503: 4467: 4432: 4384: 4336: 4308: 4260: 4212: 4164: 4085: 3777: 3749: 3679: 3641: 3592: 3463: 3330: 3279: 3251: 3213: 3125: 3076: 3038: 2836: 2794: 2745: 2732:{\displaystyle \lambda _{A},\lambda _{B}\in \Lambda } 2699: 2679: 2639: 2609: 2582: 2556: 2537: 2511: 2492: 2435: 2378: 2218: 2195: 2164: 2144: 2123: 2071: 2022: 1943: 1917: 1868: 1842: 1823: 1797: 1778: 1671: 1636: 1601: 1566: 1492: 1430: 1368: 1306: 979: 39: 12116: 11851: 11575:(3rd ed.). John Wiley & Sons. p. 111. 11319: 11299:" Reports on Progress in Physics 41.12 (1978): 1881. 11297:

Bell's theorem. Experimental tests and implications.

11180:

Einstein, A.; Podolsky, B.; Rosen, N. (1935-05-15).

7628:, and not its unrealizability with quantum systems. 1693:{\displaystyle \left|\rightarrow \right\rangle _{B}} 12677: 12675: 11404: 9434:bits. The principle of IC states that the sum over 6339:{\displaystyle \{E_{a}^{x}\}_{a},\{F_{b}^{y}\}_{b}} 5601:{\displaystyle \{E_{a}^{x}\}_{a},\{F_{b}^{y}\}_{b}} 4254:. We see that if they each measure with respect to 1658:{\displaystyle \left|\leftarrow \right\rangle _{B}} 1623:{\displaystyle \left|\downarrow \right\rangle _{B}} 12472: 12327:Jonathan, Daniel; Plenio, Martin B. (1999-10-25). 10791: 10734: 10681: 10617: 10568: 10477: 10428: 10400: 10350: 10275: 10226: 10162: 10104: 10042: 10000: 9971: 9942: 9897: 9877: 9832: 9787: 9775:are said to be locally orthogonal if there exists 9767: 9701: 9649: 9597: 9545: 9493: 9466: 9446: 9426: 9406: 9379: 9329: 9309: 9283: 9248: 9222: 9196: 9170: 9150: 9103: 9042: 8992: 8959: 8933: 8905: 8865: 8775: 8720: 8577: 8433: 8408: 8387: 8316: 8278: 8249: 8207: 8165: 8123: 8096: 8068: 8006: 7983: 7921: 7901: 7836: 7816: 7754: 7688: 7620: 7571: 7519: 7467: 7427: 7378: 7335: 7272: 7222: 7173: 7146: 7104: 7055: 7026: 6999: 6979: 6937: 6895: 6875: 6846: 6804: 6777: 6754: 6705: 6674: 6636: 6572: 6534: 6428: 6338: 6262: 6232: 6158: 6124: 6104: 6053: 6007: 5964: 5915: 5895: 5875: 5851: 5813: 5704: 5600: 5524: 5494: 5392: 5338: 5298: 5241: 5134: 5088: 5068: 5048: 5021: 4994: 4974: 4925: 4897: 4849: 4818: 4785: 4754: 4721: 4673: 4640: 4570: 4522: 4489: 4454: 4418: 4370: 4322: 4294: 4246: 4198: 4147: 4071: 3763: 3698: 3665: 3627: 3575: 3447: 3314: 3265: 3237: 3195: 3111: 3062: 3022: 2820: 2780: 2731: 2685: 2665: 2622: 2595: 2568: 2543: 2523: 2498: 2478: 2421: 2362: 2201: 2179: 2150: 2129: 2106: 2057: 2008: 1929: 1903: 1854: 1829: 1809: 1784: 1692: 1657: 1622: 1587: 1540: 1478: 1416: 1354: 1281: 1203: 100: 14040:Santha, Miklos; Vazirani, Umesh V. (1984-10-24). 12901:"MIP*=RE, disproving Connes embedding conjecture" 11289: 10170:. These protocols are termed device-independent. 6233:{\displaystyle E_{a}^{x}:H\to H,F_{b}^{y}:H\to H} 5923:is not closed: this means that there exist boxes 3196:{\displaystyle \left(P(a,b|x,y)\right)_{a,b,x,y}} 1588:{\displaystyle \left|\uparrow \right\rangle _{B}} 15304: 12672: 12581:"Semi-device-independent bounds on entanglement" 2781:{\displaystyle \rho (\lambda _{A},\lambda _{B})} 11604:. Metaphysics Research Lab, Stanford University 10115: 7336:{\displaystyle 1/\operatorname {poly} (|X||Y|)} 7115:Tsirelson's problem can be shown equivalent to 5145: 3732:in 1993 The state involved is often called the 933: 12681: 12326: 11376:Nielsen, Michael A.; Chuang, Isaac L. (2000). 7435:can be realized in a bipartite quantum system 6159:{\displaystyle \left|\psi \right\rangle \in H} 4650:This leads to the paradox: having the outcome 3207:. In the Bell scenario studied by CHSH, where 1548:. Schrödinger referred to this phenomenon as " 14511: 13976: 13823:Quantum Cryptography with Imperfect Apparatus 12635: 9158:, two parties are distributed the strings of 8388:{\displaystyle P(a,b|x,y)\not \in {\bar {Q}}} 4905:basis, by locality the result must have been 867: 13987:(2012). "Free randomness can be amplified". 12824: 11547:Mathematische Grundlagen der Quantenmechanik 11077:"Strong Loophole-Free Test of Local Realism" 10750:The degree of non-locality of a quantum box 9374: 9350: 6327: 6308: 6296: 6277: 5589: 5570: 5558: 5539: 4892: 4864: 4844: 4813: 4780: 4749: 4716: 4688: 4668: 4602: 4585: 4565: 4537: 4413: 4385: 4365: 4337: 4289: 4261: 4241: 4213: 4193: 4165: 4094: 3714: 2003: 1944: 1535: 1493: 1473: 1431: 1411: 1369: 1349: 1307: 95: 69: 13821:Mayers, Dominic; Yao, Andrew C.-C. (1998). 13295: 12721: 12719: 12629: 11378:Quantum Computation and Quantum Information 11322:Journal of the Optical Society of America B 10174:Device-independent quantum key distribution 5135:{\displaystyle (\lambda _{A},\lambda _{B})} 14518: 14504: 14473: 14454: 13694: 13692: 12204: 12198: 12110: 11371: 11369: 10485:such that any other physical realization 9878:{\displaystyle a_{k}\not =a_{k}^{\prime }} 5002:), together with her (his) local variable 4729:basis instead, the outcome must have been 3070:can each take, one can represent each box 874: 860: 14393: 14340: 14311: 14258: 14213: 14152: 14077: 14000: 13967: 13951: 13910: 13849: 13768: 13712: 13648: 13619: 13609: 13544: 13489: 13428: 13325: 13254: 13209: 13180: 13151: 13090: 13037: 12850: 12780: 12751:"Bell inequalities and operator algebras" 12731: 12699: 12596: 12539: 12486: 12417: 12344: 12279: 12218: 12130: 12085: 12061: 12028: 11975: 11956: 11954: 11918: 11865: 11776: 11727: 11690: 11647: 11493: 11418: 11333: 11272: 11246: 11222: 11205: 11118: 11092: 11027: 11013: 10966: 10921: 10876: 10429:{\displaystyle \left|\psi \right\rangle } 9501:parties respectively conduct experiments 8993:{\displaystyle 2{\sqrt {2}}\approx 2.828} 8290:The physics of supra-quantum correlations 6938:{\displaystyle {\bar {Q}}\subseteq Q_{c}} 6847:{\displaystyle {\bar {Q}}\subseteq Q_{c}} 6421: 6383: 6346:represent complete measurements. Namely, 5691: 5646: 5608:represent complete measurements. Namely, 3764:{\displaystyle \left|\psi \right\rangle } 3699:{\displaystyle 2{\sqrt {2}}\approx 2.828} 2821:{\displaystyle \lambda _{A},\lambda _{B}} 2666:{\displaystyle \lambda _{A},\lambda _{B}} 14063: 13890: 12998: 12923: 12725: 12716: 12399: 11630:"On the Einstein Podolsky Rosen paradox" 11302: 11075:Shalm, LK, et al. (December 2015). 10846: 10844: 10105:{\displaystyle P(a,b|x,y)\not \in Q_{c}} 9387:. Their goal is to transmit Bob the bit 8734:. Popescu and Rohrlich identified a box 7984:{\displaystyle P(a,b|x,y)\not \in Q^{k}} 7902:{\displaystyle P(a,b|x,y)\not \in Q_{c}} 7123:Characterization of quantum correlations 4949: 4497:However, sometimes they see the outcome 14373: 14320: 14132: 13957: 13689: 13356: 12951:"Halting Is Poly-Time Quantum Provable" 12753:. Open quantum problems. Archived from 12265: 12259: 11904: 11623: 11621: 11619: 11602:The Stanford Encyclopedia of Philosophy 11570: 11539: 11398: 11366: 11135: 11074: 11007: 10952: 10180:Device-independent quantum cryptography 10043:{\displaystyle Q_{c}\supset {\bar {Q}}} 9702:{\displaystyle ({\bar {a}}|{\bar {x}})} 9650:{\displaystyle ({\bar {a}}|{\bar {x}})} 5169: 4490:{\displaystyle \left|0-\right\rangle .} 4455:{\displaystyle \left|-0\right\rangle ,} 2479:{\displaystyle P_{B}(b|y,\lambda _{B})} 2422:{\displaystyle P_{A}(a|x,\lambda _{A})} 1710:Although various authors (most notably 925:discussions concerning quantum theory. 909:Quantum nonlocality does not allow for 15305: 14238: 14193: 14033: 13816: 13814: 13812: 13189: 12973: 12768: 12745: 12743: 12519: 12475:Communications in Mathematical Physics 12161: 12155: 12008: 11951: 11295:Clauser, John F., and Abner Shimony. " 10745: 4926:{\displaystyle \left|11\right\rangle } 4523:{\displaystyle \left|--\right\rangle } 4323:{\displaystyle \left|11\right\rangle } 2009:{\displaystyle \{P(a,b|x,y):a,b,x,y\}} 894:refers to the phenomenon by which the 14525: 14499: 13829: 13750: 13748: 13592:Navascués, M.; H. Wunderlich (2009). 13469: 13129: 13076: 12948: 12898: 12873: 12572: 12466: 12014: 11798: 11792: 11707: 11508: 11502: 11175: 11173: 10946: 10841: 9833:{\displaystyle x_{k}=x_{k}^{\prime }} 8166:{\displaystyle {\bar {Q}}\not =Q_{c}} 6785:defined above? It can be proven that 5056:), influence her (his) local outcome 14479:Fundamentals of Quantum Entanglement 14299: 13408: 11656: 11627: 11616: 11595: 11589: 11465: 11459: 11308: 11068: 10891: 7112:, thus solving Tsirelson's problem. 7105:{\displaystyle {\bar {Q}}\neq Q_{c}} 6054:{\displaystyle H=H_{A}\otimes H_{B}} 2065:. In the quantum nonlocality slang, 1817:in her (his) lab, obtaining outcome 902:do not allow an interpretation with 13809: 13594:"A Glance Beyond the Quantum Model" 13234: 12924:Aaronson, Scott (16 January 2020). 12740: 11760: 11649:10.1103/PhysicsPhysiqueFizika.1.195 11553:, Oxford University Press, Oxford, 10885: 10401:{\displaystyle E_{a}^{x},F_{b}^{y}} 8069:{\displaystyle P(a,b|x,y)\in Q^{k}} 7817:{\displaystyle P(a,b|x,y)\in Q^{k}} 7755:{\displaystyle P(a,b|x,y)\in Q_{c}} 13: 14429: 13745: 13628: 13585: 13522: 11170: 10050:, but also some non-quantum boxes 9912:Generalized Probabilistic Theories 9870: 9825: 9757: 9733: 8677: 8528: 7468:{\displaystyle H_{A}\otimes H_{B}} 5219: 5216: 5213: 5210: 4330:. If one measures with respect to 4302:, then they never see the outcome 3657: 3654: 3651: 3648: 3346: 3343: 3340: 3337: 3322:must satisfy the CHSH inequality: 2903: 2726: 2680: 2531:when she (he) conducts experiment 1726: 406:Sum-over-histories (path integral) 92: 66: 22:Part of a series of articles about 14: 15329: 10234:. Based on how non-local the box 7572:{\displaystyle \dim(H_{B})=d_{B}} 7520:{\displaystyle \dim(H_{A})=d_{A}} 6713:the set of all such correlations 43: 15287: 15286: 14436:Grib, AA; Rodrigues, WA (1999). 13958:Colbeck, Roger (December 2006). 12999:Hartnett, Kevin (4 March 2020). 12949:Regan, Kenneth W. (2020-01-15). 11545:von Neumann, J. (1932/1955). In 9657:is called an event. Two events 9380:{\displaystyle k\in \{1,...,n\}} 9151:{\displaystyle f_{0,1}^{n}\to 1} 8250:{\displaystyle {\bar {Q}}=Q_{c}} 8208:{\displaystyle Q_{c}-{\bar {Q}}} 6980:{\displaystyle {\bar {Q}}=Q_{c}} 6945:is strict, i.e., whether or not 3119:as a finite vector with entries 14118: 14057: 14048: 13393: 13350: 13168: 13123: 13070: 13017: 12992: 12967: 12942: 12917: 12892: 12867: 12818: 12638:Letters in Mathematical Physics 12400:Karvonen, Martti (2021-10-13). 12393: 12320: 11898: 11845: 11564: 11313: 10304: 9598:{\displaystyle a_{1},...,a_{n}} 9546:{\displaystyle x_{1},...,x_{n}} 4530:when measuring with respect to 2180:{\displaystyle \rho (\lambda )} 913:, and hence is compatible with 911:faster-than-light communication 15236:Relativistic quantum mechanics 14438:Nonlocality in Quantum Physics 14277:10.1103/PhysRevLett.115.020501 14171:10.1103/PhysRevLett.100.210503 13929:10.1103/physrevlett.113.140501 12436:10.1103/PhysRevLett.127.160402 12237:10.1103/PhysRevLett.109.190401 12047:10.1088/1367-2630/14/10/103001 11111:10.1103/PhysRevLett.115.250402 11046:10.1103/PhysRevLett.115.250401 10786: 10773: 10760: 10729: 10716: 10703: 10612: 10599: 10586: 10556: 10526: 10499: 10472: 10459: 10446: 10345: 10332: 10319: 10270: 10257: 10244: 10221: 10208: 10195: 10157: 10144: 10131: 10086: 10073: 10060: 10034: 9992: 9963: 9934: 9762: 9750: 9739: 9726: 9716: 9696: 9690: 9680: 9673: 9664: 9644: 9638: 9628: 9621: 9612: 9278: 9266: 9142: 8819: 8806: 8793: 8770: 8757: 8744: 8712: 8705: 8698: 8682: 8662: 8649: 8630: 8617: 8604: 8569: 8562: 8555: 8539: 8519: 8506: 8487: 8474: 8461: 8428: 8422: 8379: 8367: 8354: 8341: 8308: 8270: 8228: 8199: 8144: 8050: 8037: 8024: 7965: 7952: 7939: 7883: 7870: 7857: 7798: 7785: 7772: 7736: 7723: 7710: 7615: 7602: 7589: 7553: 7540: 7501: 7488: 7422: 7409: 7396: 7330: 7326: 7318: 7313: 7305: 7301: 7267: 7254: 7241: 7217: 7204: 7191: 7083: 7047: 6958: 6916: 6867: 6825: 6749: 6736: 6723: 6625: 6589: 6474: 6461: 6448: 6224: 6194: 6099: 6086: 6073: 5959: 5946: 5933: 5750: 5737: 5724: 5479: 5435: 5293: 5280: 5267: 5129: 5103: 4834: 4801: 4770: 4737: 4658: 4595: 4426:, they never see the outcomes 4142: 4114: 4087: 3666:{\displaystyle S_{\rm {CHSH}}} 3622: 3609: 3596: 3567: 3554: 3541: 3523: 3513: 3479: 3467: 3433: 3421: 3412: 3400: 3391: 3379: 3370: 3358: 3309: 3296: 3283: 3161: 3148: 3135: 3106: 3093: 3080: 3017: 2997: 2990: 2977: 2957: 2950: 2937: 2911: 2866: 2853: 2840: 2775: 2749: 2739:has an associated probability 2673:can take values from some set 2563: 2557: 2518: 2512: 2473: 2453: 2446: 2416: 2396: 2389: 2357: 2337: 2330: 2317: 2297: 2290: 2274: 2235: 2222: 2174: 2168: 2101: 2088: 2075: 2052: 2039: 2026: 1976: 1963: 1950: 1898: 1885: 1872: 1849: 1843: 1804: 1798: 1677: 1642: 1607: 1572: 556:Relativistic quantum mechanics 88: 81: 62: 1: 15214:Quantum statistical mechanics 14991:Quantum differential calculus 14913:Delayed-choice quantum eraser 14681:Symmetry in quantum mechanics 14481:. Institute of Physics (UK). 13868:10.1103/PhysRevLett.97.120405 13508:10.1103/PhysRevLett.99.180502 13447:10.1103/PhysRevLett.96.250401 13273:10.1103/physrevlett.98.010401 12974:Vidick, Thomas (2020-01-14). 12825:Castelvecchi, Davide (2020). 12688:Journal of Soviet Mathematics 11600:. In Zalta, Edward N. (ed.). 11437:10.1103/physrevlett.98.140402 10835: 9204:and asked to output the bits 8327:In their seminal 1994 paper, 6015:of the overall Hilbert space 4819:{\displaystyle |{-}0\rangle } 4755:{\displaystyle |{-}1\rangle } 898:statistics of a multipartite 596:Quantum statistical mechanics 16:Deviations from local realism 12955:Gödel's Lost Letter and P=NP 11596:Fine, Arthur (Winter 2017). 10116:Device independent protocols 10001:{\displaystyle {\tilde {Q}}} 9972:{\displaystyle {\tilde {Q}}} 9284:{\displaystyle f(x\oplus y)} 5146:Entanglement and nonlocality 5049:{\displaystyle \lambda _{B}} 5022:{\displaystyle \lambda _{A}} 4933:, which is also impossible. 2623:{\displaystyle \lambda _{B}} 2596:{\displaystyle \lambda _{A}} 934:Einstein, Podolsky and Rosen 7: 15016:Quantum stochastic calculus 15006:Quantum measurement problem 14928:Mach–Zehnder interferometer 12363:10.1103/PhysRevLett.83.3566 12298:10.1103/PhysRevLett.74.2619 12119:Journal of Causal Inference 11823:10.1103/PhysRevLett.71.1665 11769:Going beyond Bell's Theorem 10878:10.1103/PhysRevLett.49.1804 10803: 7379:{\displaystyle d_{A},d_{B}} 7346:For small fixed dimensions 6008:{\displaystyle H_{A},H_{B}} 5339:{\displaystyle H_{A},H_{B}} 4850:{\displaystyle |0-\rangle } 4786:{\displaystyle |1-\rangle } 4674:{\displaystyle |--\rangle } 566:Quantum information science 10: 15334: 14412:10.1103/PhysRevA.84.034301 14359:10.1103/PhysRevA.84.010302 13731:10.1103/PhysRevA.80.062107 12899:Barak, Boaz (2020-01-14). 12852:10.1038/d41586-020-00120-6 12615:10.1103/PhysRevA.83.022108 12558:10.1103/PhysRevA.83.052310 11884:10.1103/physreva.78.032114 11746:10.1103/PhysRevA.71.022101 11692:10.1103/PhysRevLett.23.880 11382:Cambridge University Press 11265:10.1103/RevModPhys.81.1727 10792:{\displaystyle P(a,b|x,y)} 10735:{\displaystyle P(a,b|x,y)} 10618:{\displaystyle P(a,b|x,y)} 10478:{\displaystyle P(a,b|x,y)} 10351:{\displaystyle P(a,b|x,y)} 10276:{\displaystyle P(a,b|x,y)} 10227:{\displaystyle P(a,b|x,y)} 10177: 10163:{\displaystyle P(a,b|x,y)} 9943:{\displaystyle {\bar {Q}}} 9111:is incompatible with NTCC. 8776:{\displaystyle P(a,b|x,y)} 8317:{\displaystyle {\bar {Q}}} 8279:{\displaystyle {\bar {Q}}} 7621:{\displaystyle P(a,b|x,y)} 7428:{\displaystyle P(a,b|x,y)} 7273:{\displaystyle P(a,b|x,y)} 7223:{\displaystyle P(a,b|x,y)} 7056:{\displaystyle {\bar {Q}}} 6876:{\displaystyle {\bar {Q}}} 6755:{\displaystyle P(a,b|x,y)} 6105:{\displaystyle P(a,b|x,y)} 5965:{\displaystyle P(a,b|x,y)} 5299:{\displaystyle P(a,b|x,y)} 5149: 3628:{\displaystyle P(a,b|x,y)} 3315:{\displaystyle P(a,b|x,y)} 3112:{\displaystyle P(a,b|x,y)} 2107:{\displaystyle P(a,b|x,y)} 2058:{\displaystyle P(a,b|x,y)} 1904:{\displaystyle P(a,b|x,y)} 1730: 937: 928: 15282: 15244: 15196: 15076:Quantum complexity theory 15054:Quantum cellular automata 15029: 14961: 14895: 14808: 14757: 14744:Path integral formulation 14711: 14576: 14533: 14224:10.1007/s10107-017-1198-4 13220:10.1007/s10107-016-1049-8 13144:10.1007/s11537-013-1280-5 13109:10.1142/S0129055X12500122 12874:Kalai, Gil (2020-01-17). 12682:Tsirel'son, B.S. (1987). 12505:10.1007/s00220-011-1296-8 11937:10.1007/s10701-007-9176-y 11235:Reviews of Modern Physics 9249:{\displaystyle a\oplus b} 8960:{\displaystyle a\oplus b} 7068:quantum complexity theory 6166:and projection operators 5400:and projection operators 3715:Possibilistic nonlocality 2693:. If each pair of values 15143:Quantum machine learning 15123:Quantum key distribution 15113:Quantum image processing 15103:Quantum error correction 14953:Wheeler's delayed choice 14202:Mathematical Programming 13198:Mathematical Programming 12184:10.1103/PhysRevA.40.4277 11144:Lettere al Nuovo Cimento 10820:Quantum pseudo-telepathy 9043:{\displaystyle p>1/2} 9014:communication complexity 7127:Since the dimensions of 7117:Connes embedding problem 2686:{\displaystyle \Lambda } 2151:{\displaystyle \lambda } 2130:{\displaystyle \lambda } 2016:will be denoted by just 601:Quantum machine learning 354:Wheeler's delayed-choice 15059:Quantum finite automata 14247:Physical Review Letters 14141:Physical Review Letters 13899:Physical Review Letters 13838:Physical Review Letters 13478:Physical Review Letters 13417:Physical Review Letters 13243:Physical Review Letters 12406:Physical Review Letters 12333:Physical Review Letters 12268:Physical Review Letters 12207:Physical Review Letters 11802:Physical Review Letters 11670:Physical Review Letters 11635:Physics Physique Физика 11407:Physical Review Letters 11352:10.1364/JOSAB.32.000A50 11081:Physical Review Letters 11016:Physical Review Letters 10857:Physical Review Letters 8906:{\displaystyle a,b,x,y} 6883:denotes the closure of 6675:{\displaystyle a,b,x,y} 6573:{\displaystyle a,b,x,y} 5852:{\displaystyle a,b,x,y} 3245:can take values within 3238:{\displaystyle a,b,x,y} 3063:{\displaystyle a,b,x,y} 311:Leggett–Garg inequality 15163:Quantum neural network 13621:10.1098/rspa.2009.0453 13359:Foundations of Physics 13305:Foundations of Physics 12880:Combinatorics and more 12806:Cite journal requires 11907:Foundations of Physics 11533:10.1103/PhysRev.49.476 11495:10.1103/PhysRev.48.696 11207:10.1103/PhysRev.47.777 10793: 10736: 10683: 10619: 10570: 10479: 10430: 10402: 10352: 10277: 10228: 10164: 10121:measured correlations 10106: 10044: 10002: 9973: 9944: 9899: 9879: 9834: 9789: 9769: 9703: 9651: 9605:. The pair of vectors 9599: 9547: 9495: 9468: 9448: 9428: 9408: 9381: 9331: 9311: 9285: 9250: 9224: 9198: 9172: 9152: 9105: 9044: 8994: 8961: 8935: 8907: 8867: 8777: 8722: 8579: 8435: 8410: 8389: 8318: 8280: 8251: 8209: 8167: 8125: 8098: 8070: 8008: 7985: 7923: 7903: 7838: 7818: 7756: 7690: 7622: 7573: 7521: 7469: 7429: 7380: 7337: 7274: 7224: 7175: 7148: 7106: 7070:that would imply that 7057: 7028: 7001: 6981: 6939: 6897: 6877: 6848: 6806: 6779: 6756: 6707: 6676: 6638: 6574: 6536: 6430: 6340: 6264: 6234: 6160: 6132:, a normalized vector 6126: 6106: 6055: 6009: 5966: 5917: 5897: 5877: 5853: 5815: 5706: 5602: 5526: 5496: 5394: 5346:, a normalized vector 5340: 5300: 5257:Mathematically, a box 5243: 5136: 5090: 5070: 5050: 5023: 4996: 4976: 4955: 4927: 4899: 4851: 4820: 4787: 4756: 4723: 4675: 4642: 4572: 4524: 4491: 4456: 4420: 4372: 4324: 4296: 4248: 4200: 4149: 4073: 3765: 3700: 3667: 3629: 3577: 3449: 3316: 3267: 3239: 3197: 3113: 3064: 3024: 2822: 2782: 2733: 2687: 2667: 2624: 2597: 2570: 2545: 2525: 2500: 2480: 2423: 2364: 2203: 2181: 2152: 2131: 2108: 2059: 2010: 1931: 1905: 1856: 1831: 1811: 1786: 1756: 1708: 1694: 1659: 1624: 1589: 1542: 1480: 1418: 1356: 1283: 1205: 102: 15188:Quantum teleportation 14701:Wave–particle duality 13757:Nature Communications 13637:Nature Communications 12141:10.1515/jci-2018-0008 12096:10.1515/jci-2017-0020 11571:Maudlin, Tim (2011). 11466:Bohr, N (July 1935). 10825:Quantum contextuality 10794: 10737: 10684: 10620: 10571: 10480: 10431: 10403: 10353: 10278: 10229: 10165: 10107: 10045: 10003: 9974: 9945: 9900: 9898:{\displaystyle 0.052} 9880: 9835: 9790: 9770: 9704: 9652: 9600: 9548: 9496: 9469: 9449: 9429: 9409: 9407:{\displaystyle x_{k}} 9382: 9332: 9312: 9295:Information Causality 9286: 9251: 9225: 9199: 9173: 9153: 9106: 9045: 8995: 8962: 8936: 8934:{\displaystyle {0,1}} 8908: 8868: 8778: 8723: 8580: 8436: 8411: 8390: 8319: 8281: 8252: 8210: 8168: 8126: 8124:{\displaystyle Q_{c}} 8099: 8071: 8009: 7986: 7924: 7904: 7839: 7819: 7757: 7696:with the properties: 7691: 7623: 7574: 7522: 7470: 7430: 7381: 7338: 7275: 7225: 7176: 7174:{\displaystyle H_{B}} 7149: 7147:{\displaystyle H_{A}} 7107: 7058: 7029: 7027:{\displaystyle Q_{c}} 7007:in the definition of 7002: 6982: 6940: 6898: 6878: 6849: 6812:is closed. Moreover, 6807: 6805:{\displaystyle Q_{c}} 6780: 6757: 6708: 6706:{\displaystyle Q_{c}} 6677: 6639: 6575: 6537: 6431: 6341: 6265: 6235: 6161: 6127: 6107: 6056: 6010: 5967: 5918: 5898: 5878: 5854: 5816: 5707: 5603: 5527: 5497: 5395: 5341: 5301: 5244: 5137: 5091: 5071: 5051: 5024: 4997: 4977: 4953: 4928: 4900: 4852: 4821: 4788: 4757: 4724: 4676: 4643: 4573: 4525: 4492: 4457: 4421: 4373: 4325: 4297: 4249: 4201: 4150: 4074: 3766: 3701: 3668: 3630: 3578: 3450: 3317: 3273:, any Bell local box 3268: 3266:{\displaystyle {0,1}} 3240: 3198: 3114: 3065: 3025: 2823: 2783: 2734: 2688: 2668: 2625: 2598: 2571: 2546: 2526: 2501: 2481: 2424: 2365: 2204: 2182: 2153: 2132: 2109: 2060: 2011: 1932: 1906: 1857: 1832: 1812: 1787: 1752: 1737:Bell test experiments 1703: 1695: 1660: 1625: 1590: 1543: 1481: 1419: 1357: 1284: 1206: 962:principle of locality 296:Elitzur–Vaidman 286:Davisson–Germer 103: 15318:Quantum field theory 15219:Quantum field theory 15148:Quantum metamaterial 15093:Quantum cryptography 14823:Consistent histories 11384:. pp. 112–113. 10810:Action at a distance 10754: 10697: 10629: 10580: 10489: 10440: 10412: 10408:and a quantum state 10362: 10313: 10238: 10189: 10125: 10054: 10012: 9983: 9954: 9925: 9889: 9844: 9799: 9779: 9713: 9661: 9609: 9557: 9505: 9485: 9458: 9438: 9418: 9391: 9341: 9321: 9301: 9260: 9256:is a good guess for 9234: 9208: 9182: 9162: 9118: 9054: 9020: 8971: 8945: 8917: 8879: 8787: 8738: 8588: 8445: 8419: 8400: 8335: 8299: 8261: 8219: 8177: 8135: 8108: 8088: 8078:semidefinite program 8018: 7998: 7933: 7913: 7909:, then there exists 7851: 7828: 7766: 7704: 7635: 7583: 7531: 7479: 7439: 7390: 7350: 7284: 7235: 7185: 7158: 7131: 7074: 7038: 7011: 6991: 6949: 6907: 6887: 6858: 6816: 6789: 6769: 6717: 6690: 6648: 6586: 6546: 6442: 6350: 6274: 6248: 6170: 6136: 6116: 6067: 6019: 5979: 5927: 5907: 5887: 5867: 5825: 5718: 5612: 5536: 5510: 5404: 5350: 5310: 5261: 5190: 5170:Quantum correlations 5152:Quantum entanglement 5100: 5080: 5060: 5033: 5006: 4986: 4966: 4909: 4861: 4830: 4797: 4766: 4733: 4685: 4654: 4582: 4534: 4501: 4465: 4430: 4382: 4334: 4306: 4258: 4210: 4162: 4083: 3775: 3747: 3677: 3639: 3590: 3461: 3328: 3277: 3249: 3211: 3123: 3074: 3036: 2834: 2792: 2743: 2697: 2677: 2637: 2607: 2580: 2554: 2535: 2509: 2490: 2433: 2376: 2216: 2193: 2162: 2142: 2121: 2069: 2020: 1941: 1915: 1866: 1840: 1821: 1795: 1776: 1669: 1634: 1599: 1564: 1490: 1428: 1366: 1304: 1215: 977: 561:Quantum field theory 473:Consistent histories 110:Schrödinger equation 37: 15313:Quantum measurement 15204:Quantum fluctuation 15173:Quantum programming 15133:Quantum logic gates 15118:Quantum information 15098:Quantum electronics 14558:Classical mechanics 14459:. Springer Verlag. 14455:Cramer, JG (2015). 14440:. Springer Verlag. 14404:2011PhRvA..84c4301L 14351:2011PhRvA..84a0302P 14269:2015PhRvL.115b0501N 14163:2008arXiv0802.0760B 14096:10.1038/nature09008 14088:2010Natur.464.1021P 14072:(7291): 1021–1024. 14011:2012NatPh...8..450C 13921:2014PhRvL.113n0501V 13860:2006PhRvL..97l0405A 13779:2015NatCo...6.6288N 13723:2009PhRvA..80f2107A 13659:2013NatCo...4.2263F 13563:10.1038/nature08400 13555:2009Natur.461.1101P 13539:(7267): 1101–1104. 13500:2007PhRvL..99r0502L 13439:2006PhRvL..96y0401B 13371:1985FoPh...15..963R 13318:1994FoPh...24..379P 13265:2007PhRvL..98a0401N 13101:2012RvMaP..2450012F 13048:2011JMP....52a2102J 12976:"A Masters project" 12843:2020Natur.577..461C 12791:2020arXiv200104383J 12650:1980LMaPh...4...93C 12607:2011PhRvA..83b2108L 12550:2011PhRvA..83e2310V 12497:2011CMaPh.306..695J 12428:2021PhRvL.127p0402K 12355:1999PhRvL..83.3566J 12290:1995PhRvL..74.2619P 12229:2012PhRvL.109s0401P 12176:1989PhRvA..40.4277W 12039:2012NJPh...14j3001F 11986:2012NatPh...8..867B 11929:2007FoPh...37.1563N 11876:2008PhRvA..78c2114B 11815:1993PhRvL..71.1665H 11787:2007arXiv0712.0921G 11738:2005PhRvA..71b2101B 11683:1969PhRvL..23..880C 11628:Bell, John (1964). 11525:1936PhRv...49..476F 11486:1935PhRv...48..696B 11429:2007PhRvL..98n0402W 11344:2015JOSAB..32A..50J 11257:2009RvMP...81.1727R 11198:1935PhRv...47..777E 11103:2015PhRvL.115y0402S 11038:2015PhRvL.115y0401G 10985:10.1038/nature15759 10977:2015Natur.526..682H 10906:2001Natur.409..791R 10869:1982PhRvL..49.1804A 10830:Quantum foundations 10815:Popper's experiment 10746:Dimension witnesses 10664: 10646: 10542: 10515: 10397: 10379: 10309:Sometimes, the box 9874: 9829: 9223:{\displaystyle a,b} 9197:{\displaystyle x,y} 9141: 8434:{\displaystyle (y)} 8014:, deciding whether 6624: 6606: 6520: 6505: 6415: 6377: 6325: 6294: 6263:{\displaystyle x,y} 6217: 6187: 5799: 5781: 5684: 5639: 5587: 5556: 5525:{\displaystyle x,y} 5465: 5421: 2569:{\displaystyle (y)} 2524:{\displaystyle (b)} 1930:{\displaystyle a,b} 1855:{\displaystyle (b)} 1810:{\displaystyle (y)} 892:quantum nonlocality 888:theoretical physics 349:Stern–Gerlach 146:Classical mechanics 15257:in popular culture 15039:Quantum algorithms 14887:Von Neumann–Wigner 14867:Objective collapse 14563:Old quantum theory 13787:10.1038/ncomms7288 13667:10.1038/ncomms3263 13379:10.1007/bf00739036 13336:10.1007/BF02058098 12701:10.1007/BF01663472 12658:10.1007/bf00417500 11156:10.1007/BF02817189 10789: 10732: 10679: 10650: 10632: 10615: 10566: 10519: 10492: 10475: 10426: 10398: 10383: 10365: 10348: 10273: 10224: 10160: 10102: 10040: 9998: 9969: 9940: 9895: 9875: 9860: 9830: 9815: 9785: 9765: 9699: 9647: 9595: 9543: 9491: 9464: 9444: 9424: 9404: 9377: 9327: 9307: 9281: 9246: 9220: 9194: 9168: 9148: 9121: 9101: 9040: 8990: 8957: 8931: 8903: 8863: 8773: 8718: 8645: 8600: 8575: 8502: 8457: 8431: 8406: 8385: 8314: 8276: 8247: 8205: 8163: 8121: 8094: 8066: 8004: 7981: 7919: 7899: 7834: 7814: 7752: 7686: 7618: 7569: 7517: 7465: 7425: 7376: 7333: 7270: 7220: 7171: 7144: 7102: 7053: 7024: 6997: 6977: 6935: 6893: 6873: 6844: 6802: 6775: 6752: 6703: 6672: 6637:{\displaystyle =0} 6634: 6610: 6592: 6570: 6532: 6506: 6491: 6426: 6401: 6400: 6363: 6362: 6336: 6311: 6280: 6260: 6230: 6203: 6173: 6156: 6122: 6102: 6051: 6005: 5962: 5913: 5893: 5873: 5849: 5811: 5785: 5767: 5702: 5670: 5669: 5625: 5624: 5598: 5573: 5542: 5522: 5492: 5451: 5407: 5390: 5336: 5296: 5239: 5132: 5086: 5066: 5046: 5019: 4992: 4972: 4956: 4923: 4895: 4847: 4816: 4783: 4752: 4719: 4671: 4638: 4630: 4568: 4520: 4487: 4452: 4416: 4368: 4320: 4292: 4244: 4196: 4145: 4112: 4069: 3761: 3722:Daniel Greenberger 3696: 3663: 3625: 3573: 3512: 3445: 3312: 3263: 3235: 3193: 3109: 3060: 3020: 2907: 2818: 2778: 2729: 2683: 2663: 2620: 2593: 2566: 2541: 2521: 2496: 2476: 2419: 2360: 2199: 2177: 2148: 2127: 2104: 2055: 2006: 1927: 1901: 1852: 1827: 1807: 1782: 1690: 1655: 1620: 1585: 1554:special relativity 1538: 1476: 1414: 1352: 1279: 1201: 915:special relativity 537:Von Neumann–Wigner 517:Objective-collapse 316:Mach–Zehnder 306:Leggett inequality 301:Franck–Hertz 151:Old quantum theory 98: 15300: 15299: 15274:Quantum mysticism 15252:Schrödinger's cat 15183:Quantum simulator 15153:Quantum metrology 15081:Quantum computing 15044:Quantum amplifier 15021:Quantum spacetime 14986:Quantum cosmology 14976:Quantum chemistry 14676:Scattering theory 14624:Zero-point energy 14619:Degenerate levels 14527:Quantum mechanics 14488:978-0-7503-2226-3 14466:978-3-319-24642-0 14447:978-0-306-46182-8 14382:Physical Review A 14329:Physical Review A 14019:10.1038/nphys2300 13701:Physical Review A 13604:(2115): 881–890. 13056:10.1063/1.3514538 12905:Windows On Theory 12837:(7791): 461–462. 12585:Physical Review A 12528:Physical Review A 12339:(17): 3566–3569. 12274:(14): 2619–2622. 12164:Physical Review A 11994:10.1038/nphys2460 11913:(11): 1563–1611. 11854:Physical Review A 11809:(11): 1665–1668. 11716:Physical Review A 11391:978-0-521-63503-5 10961:(7575): 682–686. 10900:(6822): 791–794. 10863:(25): 1804–1807. 10559: 10529: 10502: 10037: 9995: 9966: 9937: 9788:{\displaystyle k} 9753: 9729: 9693: 9676: 9641: 9624: 9494:{\displaystyle n} 9467:{\displaystyle s} 9447:{\displaystyle k} 9427:{\displaystyle s} 9330:{\displaystyle n} 9310:{\displaystyle x} 9171:{\displaystyle n} 9082: 9081: 9065: 8982: 8833: 8636: 8591: 8493: 8448: 8409:{\displaystyle x} 8382: 8311: 8273: 8231: 8202: 8147: 8097:{\displaystyle Q} 8076:can be cast as a 8007:{\displaystyle k} 7922:{\displaystyle k} 7837:{\displaystyle k} 7086: 7050: 7000:{\displaystyle H} 6961: 6919: 6896:{\displaystyle Q} 6870: 6828: 6778:{\displaystyle Q} 6391: 6353: 6125:{\displaystyle H} 5916:{\displaystyle Q} 5896:{\displaystyle Q} 5876:{\displaystyle Q} 5660: 5615: 5234: 5204: 5181:Tsirelson's bound 5089:{\displaystyle b} 5069:{\displaystyle a} 4995:{\displaystyle y} 4975:{\displaystyle x} 4960:Bayesian networks 4629: 4626: 4111: 4110: 4079:where, as above, 4017: 4016: 3986: 3974: 3973: 3909: 3908: 3878: 3866: 3865: 3802: 3801: 3688: 3485: 2872: 2544:{\displaystyle x} 2499:{\displaystyle a} 2202:{\displaystyle y} 2114:is termed a box. 1830:{\displaystyle a} 1785:{\displaystyle x} 1720:Bohmian mechanics 1242: 1241: 1114: 1113: 1014: 1013: 967:Erwin Schrödinger 884: 883: 591:Scattering theory 571:Quantum computing 344:Schrödinger's cat 276:Bell's inequality 84: 59: 28:Quantum mechanics 15325: 15290: 15289: 15001:Quantum geometry 14996:Quantum dynamics 14853:Superdeterminism 14785:Rarita–Schwinger 14734:Matrix mechanics 14589:Bra–ket notation 14520: 14513: 14506: 14497: 14496: 14492: 14470: 14451: 14424: 14423: 14397: 14377: 14371: 14370: 14344: 14335:(1): 010302(R). 14324: 14318: 14317: 14315: 14303: 14297: 14296: 14262: 14242: 14236: 14235: 14217: 14208:(1–2): 397–431. 14197: 14191: 14190: 14156: 14136: 14130: 14122: 14116: 14115: 14081: 14061: 14055: 14052: 14046: 14045: 14037: 14031: 14030: 14004: 13983:Colbeck, Roger; 13980: 13974: 13973: 13971: 13955: 13949: 13948: 13914: 13894: 13888: 13887: 13853: 13851:quant-ph/0510094 13833: 13827: 13826: 13818: 13807: 13806: 13772: 13752: 13743: 13742: 13716: 13696: 13687: 13686: 13652: 13632: 13626: 13625: 13623: 13613: 13589: 13583: 13582: 13548: 13526: 13520: 13519: 13493: 13491:quant-ph/0610097 13473: 13467: 13466: 13432: 13430:quant-ph/0508042 13412: 13406: 13405: 13397: 13391: 13390: 13354: 13348: 13347: 13329: 13299: 13293: 13292: 13258: 13256:quant-ph/0607119 13238: 13232: 13231: 13213: 13204:(1–2): 431–463. 13193: 13187: 13186: 13184: 13172: 13166: 13165: 13155: 13127: 13121: 13120: 13094: 13074: 13068: 13067: 13041: 13021: 13015: 13014: 13012: 13011: 12996: 12990: 12989: 12987: 12986: 12971: 12965: 12964: 12962: 12961: 12946: 12940: 12939: 12937: 12936: 12930:Shtetl-Optimized 12921: 12915: 12914: 12912: 12911: 12896: 12890: 12889: 12887: 12886: 12871: 12865: 12864: 12854: 12822: 12816: 12815: 12809: 12804: 12802: 12794: 12784: 12772: 12766: 12765: 12763: 12762: 12747: 12738: 12737: 12735: 12723: 12714: 12713: 12703: 12679: 12670: 12669: 12633: 12627: 12626: 12600: 12576: 12570: 12569: 12543: 12523: 12517: 12516: 12490: 12470: 12464: 12463: 12421: 12397: 12391: 12390: 12348: 12346:quant-ph/9905071 12324: 12318: 12317: 12283: 12281:quant-ph/9502005 12263: 12257: 12256: 12222: 12202: 12196: 12195: 12170:(8): 4277–4281. 12159: 12153: 12152: 12134: 12114: 12108: 12107: 12089: 12074:Causal Inference 12065: 12059: 12058: 12032: 12012: 12006: 12005: 11979: 11970:(867): 867–870. 11958: 11949: 11948: 11922: 11920:quant-ph/0609163 11902: 11896: 11895: 11869: 11849: 11843: 11842: 11796: 11790: 11789: 11780: 11764: 11758: 11757: 11731: 11729:quant-ph/0404097 11711: 11705: 11704: 11694: 11660: 11654: 11653: 11651: 11625: 11614: 11613: 11611: 11609: 11593: 11587: 11586: 11568: 11562: 11561:, pages 144–145. 11543: 11537: 11536: 11506: 11500: 11499: 11497: 11463: 11457: 11456: 11422: 11420:quant-ph/0612147 11402: 11396: 11395: 11373: 11364: 11363: 11337: 11317: 11311: 11310: 11306: 11300: 11293: 11287: 11286: 11276: 11250: 11241:(4): 1727–1751. 11226: 11220: 11219: 11209: 11177: 11168: 11167: 11139: 11133: 11132: 11122: 11096: 11072: 11066: 11065: 11031: 11011: 11005: 11004: 10970: 10950: 10944: 10943: 10925: 10914:10.1038/35057215 10889: 10883: 10882: 10880: 10848: 10798: 10796: 10795: 10790: 10776: 10741: 10739: 10738: 10733: 10719: 10688: 10686: 10685: 10680: 10678: 10663: 10658: 10645: 10640: 10625:is connected to 10624: 10622: 10621: 10616: 10602: 10575: 10573: 10572: 10567: 10565: 10561: 10560: 10552: 10541: 10536: 10531: 10530: 10522: 10514: 10509: 10504: 10503: 10495: 10484: 10482: 10481: 10476: 10462: 10435: 10433: 10432: 10427: 10425: 10407: 10405: 10404: 10399: 10396: 10391: 10378: 10373: 10357: 10355: 10354: 10349: 10335: 10282: 10280: 10279: 10274: 10260: 10233: 10231: 10230: 10225: 10211: 10169: 10167: 10166: 10161: 10147: 10111: 10109: 10108: 10103: 10101: 10100: 10076: 10049: 10047: 10046: 10041: 10039: 10038: 10030: 10024: 10023: 10007: 10005: 10004: 9999: 9997: 9996: 9988: 9978: 9976: 9975: 9970: 9968: 9967: 9959: 9949: 9947: 9946: 9941: 9939: 9938: 9930: 9904: 9902: 9901: 9896: 9884: 9882: 9881: 9876: 9873: 9868: 9856: 9855: 9839: 9837: 9836: 9831: 9828: 9823: 9811: 9810: 9794: 9792: 9791: 9786: 9774: 9772: 9771: 9766: 9761: 9760: 9755: 9754: 9746: 9742: 9737: 9736: 9731: 9730: 9722: 9708: 9706: 9705: 9700: 9695: 9694: 9686: 9683: 9678: 9677: 9669: 9656: 9654: 9653: 9648: 9643: 9642: 9634: 9631: 9626: 9625: 9617: 9604: 9602: 9601: 9596: 9594: 9593: 9569: 9568: 9552: 9550: 9549: 9544: 9542: 9541: 9517: 9516: 9500: 9498: 9497: 9492: 9473: 9471: 9470: 9465: 9453: 9451: 9450: 9445: 9433: 9431: 9430: 9425: 9413: 9411: 9410: 9405: 9403: 9402: 9386: 9384: 9383: 9378: 9336: 9334: 9333: 9328: 9316: 9314: 9313: 9308: 9290: 9288: 9287: 9282: 9255: 9253: 9252: 9247: 9229: 9227: 9226: 9221: 9203: 9201: 9200: 9195: 9177: 9175: 9174: 9169: 9157: 9155: 9154: 9149: 9140: 9135: 9110: 9108: 9107: 9102: 9094: 9090: 9083: 9077: 9073: 9066: 9061: 9049: 9047: 9046: 9041: 9036: 8999: 8997: 8996: 8991: 8983: 8978: 8966: 8964: 8963: 8958: 8940: 8938: 8937: 8932: 8930: 8912: 8910: 8909: 8904: 8872: 8870: 8869: 8864: 8859: 8858: 8834: 8826: 8809: 8782: 8780: 8779: 8774: 8760: 8727: 8725: 8724: 8719: 8708: 8697: 8696: 8681: 8680: 8665: 8644: 8620: 8599: 8584: 8582: 8581: 8576: 8565: 8554: 8553: 8532: 8531: 8522: 8501: 8477: 8456: 8440: 8438: 8437: 8432: 8415: 8413: 8412: 8407: 8394: 8392: 8391: 8386: 8384: 8383: 8375: 8357: 8323: 8321: 8320: 8315: 8313: 8312: 8304: 8285: 8283: 8282: 8277: 8275: 8274: 8266: 8256: 8254: 8253: 8248: 8246: 8245: 8233: 8232: 8224: 8214: 8212: 8211: 8206: 8204: 8203: 8195: 8189: 8188: 8172: 8170: 8169: 8164: 8162: 8161: 8149: 8148: 8140: 8130: 8128: 8127: 8122: 8120: 8119: 8103: 8101: 8100: 8095: 8075: 8073: 8072: 8067: 8065: 8064: 8040: 8013: 8011: 8010: 8005: 7990: 7988: 7987: 7982: 7980: 7979: 7955: 7928: 7926: 7925: 7920: 7908: 7906: 7905: 7900: 7898: 7897: 7873: 7843: 7841: 7840: 7835: 7823: 7821: 7820: 7815: 7813: 7812: 7788: 7761: 7759: 7758: 7753: 7751: 7750: 7726: 7695: 7693: 7692: 7687: 7673: 7672: 7660: 7659: 7647: 7646: 7627: 7625: 7624: 7619: 7605: 7578: 7576: 7575: 7570: 7568: 7567: 7552: 7551: 7526: 7524: 7523: 7518: 7516: 7515: 7500: 7499: 7474: 7472: 7471: 7466: 7464: 7463: 7451: 7450: 7434: 7432: 7431: 7426: 7412: 7385: 7383: 7382: 7377: 7375: 7374: 7362: 7361: 7342: 7340: 7339: 7334: 7329: 7321: 7316: 7308: 7294: 7279: 7277: 7276: 7271: 7257: 7229: 7227: 7226: 7221: 7207: 7180: 7178: 7177: 7172: 7170: 7169: 7153: 7151: 7150: 7145: 7143: 7142: 7111: 7109: 7108: 7103: 7101: 7100: 7088: 7087: 7079: 7062: 7060: 7059: 7054: 7052: 7051: 7043: 7033: 7031: 7030: 7025: 7023: 7022: 7006: 7004: 7003: 6998: 6986: 6984: 6983: 6978: 6976: 6975: 6963: 6962: 6954: 6944: 6942: 6941: 6936: 6934: 6933: 6921: 6920: 6912: 6902: 6900: 6899: 6894: 6882: 6880: 6879: 6874: 6872: 6871: 6863: 6853: 6851: 6850: 6845: 6843: 6842: 6830: 6829: 6821: 6811: 6809: 6808: 6803: 6801: 6800: 6784: 6782: 6781: 6776: 6761: 6759: 6758: 6753: 6739: 6712: 6710: 6709: 6704: 6702: 6701: 6681: 6679: 6678: 6673: 6643: 6641: 6640: 6635: 6623: 6618: 6605: 6600: 6579: 6577: 6576: 6571: 6541: 6539: 6538: 6533: 6531: 6519: 6514: 6504: 6499: 6490: 6464: 6435: 6433: 6432: 6427: 6425: 6424: 6414: 6409: 6399: 6387: 6386: 6376: 6371: 6361: 6345: 6343: 6342: 6337: 6335: 6334: 6324: 6319: 6304: 6303: 6293: 6288: 6269: 6267: 6266: 6261: 6239: 6237: 6236: 6231: 6216: 6211: 6186: 6181: 6165: 6163: 6162: 6157: 6149: 6131: 6129: 6128: 6123: 6111: 6109: 6108: 6103: 6089: 6060: 6058: 6057: 6052: 6050: 6049: 6037: 6036: 6014: 6012: 6011: 6006: 6004: 6003: 5991: 5990: 5971: 5969: 5968: 5963: 5949: 5922: 5920: 5919: 5914: 5902: 5900: 5899: 5894: 5882: 5880: 5879: 5874: 5858: 5856: 5855: 5850: 5820: 5818: 5817: 5812: 5810: 5798: 5793: 5780: 5775: 5766: 5740: 5711: 5709: 5708: 5703: 5701: 5700: 5695: 5694: 5683: 5678: 5668: 5656: 5655: 5650: 5649: 5638: 5633: 5623: 5607: 5605: 5604: 5599: 5597: 5596: 5586: 5581: 5566: 5565: 5555: 5550: 5531: 5529: 5528: 5523: 5501: 5499: 5498: 5493: 5491: 5490: 5478: 5477: 5464: 5459: 5447: 5446: 5434: 5433: 5420: 5415: 5399: 5397: 5396: 5391: 5389: 5388: 5376: 5375: 5363: 5345: 5343: 5342: 5337: 5335: 5334: 5322: 5321: 5305: 5303: 5302: 5297: 5283: 5248: 5246: 5245: 5240: 5235: 5230: 5222: 5205: 5200: 5141: 5139: 5138: 5133: 5128: 5127: 5115: 5114: 5095: 5093: 5092: 5087: 5075: 5073: 5072: 5067: 5055: 5053: 5052: 5047: 5045: 5044: 5028: 5026: 5025: 5020: 5018: 5017: 5001: 4999: 4998: 4993: 4981: 4979: 4978: 4973: 4932: 4930: 4929: 4924: 4922: 4904: 4902: 4901: 4896: 4891: 4877: 4856: 4854: 4853: 4848: 4837: 4825: 4823: 4822: 4817: 4809: 4804: 4792: 4790: 4789: 4784: 4773: 4761: 4759: 4758: 4753: 4745: 4740: 4728: 4726: 4725: 4720: 4715: 4701: 4680: 4678: 4677: 4672: 4661: 4647: 4645: 4644: 4639: 4631: 4628: 4627: 4622: 4613: 4598: 4577: 4575: 4574: 4569: 4564: 4550: 4529: 4527: 4526: 4521: 4519: 4515: 4496: 4494: 4493: 4488: 4483: 4479: 4461: 4459: 4458: 4453: 4448: 4444: 4425: 4423: 4422: 4417: 4412: 4398: 4377: 4375: 4374: 4369: 4364: 4350: 4329: 4327: 4326: 4321: 4319: 4301: 4299: 4298: 4293: 4288: 4274: 4253: 4251: 4250: 4245: 4240: 4226: 4205: 4203: 4202: 4197: 4192: 4178: 4154: 4152: 4151: 4146: 4141: 4127: 4113: 4106: 4102: 4090: 4078: 4076: 4075: 4070: 4068: 4064: 4063: 4059: 4058: 4054: 4039: 4035: 4018: 4012: 4008: 4003: 3999: 3987: 3982: 3975: 3969: 3965: 3960: 3956: 3955: 3951: 3950: 3946: 3931: 3927: 3910: 3904: 3900: 3895: 3891: 3879: 3874: 3867: 3861: 3857: 3852: 3848: 3847: 3833: 3819: 3803: 3797: 3793: 3788: 3770: 3768: 3767: 3762: 3760: 3705: 3703: 3702: 3697: 3689: 3684: 3672: 3670: 3669: 3664: 3662: 3661: 3660: 3634: 3632: 3631: 3626: 3612: 3582: 3580: 3579: 3574: 3557: 3537: 3536: 3511: 3454: 3452: 3451: 3446: 3351: 3350: 3349: 3321: 3319: 3318: 3313: 3299: 3272: 3270: 3269: 3264: 3262: 3244: 3242: 3241: 3236: 3202: 3200: 3199: 3194: 3192: 3191: 3168: 3164: 3151: 3118: 3116: 3115: 3110: 3096: 3069: 3067: 3066: 3061: 3029: 3027: 3026: 3021: 3016: 3015: 3000: 2989: 2988: 2976: 2975: 2960: 2949: 2948: 2936: 2935: 2923: 2922: 2906: 2899: 2898: 2886: 2885: 2856: 2827: 2825: 2824: 2819: 2817: 2816: 2804: 2803: 2787: 2785: 2784: 2779: 2774: 2773: 2761: 2760: 2738: 2736: 2735: 2730: 2722: 2721: 2709: 2708: 2692: 2690: 2689: 2684: 2672: 2670: 2669: 2664: 2662: 2661: 2649: 2648: 2629: 2627: 2626: 2621: 2619: 2618: 2602: 2600: 2599: 2594: 2592: 2591: 2575: 2573: 2572: 2567: 2550: 2548: 2547: 2542: 2530: 2528: 2527: 2522: 2505: 2503: 2502: 2497: 2485: 2483: 2482: 2477: 2472: 2471: 2456: 2445: 2444: 2428: 2426: 2425: 2420: 2415: 2414: 2399: 2388: 2387: 2369: 2367: 2366: 2361: 2356: 2355: 2340: 2329: 2328: 2316: 2315: 2300: 2289: 2288: 2273: 2272: 2260: 2259: 2238: 2210: 2208: 2206: 2205: 2200: 2186: 2184: 2183: 2178: 2157: 2155: 2154: 2149: 2136: 2134: 2133: 2128: 2113: 2111: 2110: 2105: 2091: 2064: 2062: 2061: 2056: 2042: 2015: 2013: 2012: 2007: 1966: 1936: 1934: 1933: 1928: 1910: 1908: 1907: 1902: 1888: 1861: 1859: 1858: 1853: 1836: 1834: 1833: 1828: 1816: 1814: 1813: 1808: 1791: 1789: 1788: 1783: 1716:hidden variables 1699: 1697: 1696: 1691: 1689: 1688: 1683: 1664: 1662: 1661: 1656: 1654: 1653: 1648: 1629: 1627: 1626: 1621: 1619: 1618: 1613: 1594: 1592: 1591: 1586: 1584: 1583: 1578: 1547: 1545: 1544: 1539: 1534: 1533: 1528: 1513: 1512: 1507: 1485: 1483: 1482: 1477: 1472: 1471: 1466: 1451: 1450: 1445: 1423: 1421: 1420: 1415: 1410: 1409: 1404: 1389: 1388: 1383: 1361: 1359: 1358: 1353: 1348: 1347: 1342: 1327: 1326: 1321: 1288: 1286: 1285: 1280: 1278: 1274: 1273: 1259: 1243: 1237: 1233: 1228: 1210: 1208: 1207: 1202: 1200: 1196: 1195: 1194: 1189: 1177: 1176: 1171: 1156: 1155: 1150: 1138: 1137: 1132: 1115: 1109: 1105: 1100: 1096: 1095: 1094: 1089: 1077: 1076: 1071: 1056: 1055: 1050: 1038: 1037: 1032: 1015: 1009: 1005: 1000: 996: 995: 876: 869: 862: 503:Superdeterminism 156:Bra–ket notation 107: 105: 104: 99: 91: 86: 85: 77: 65: 60: 58: 47: 19: 18: 15333: 15332: 15328: 15327: 15326: 15324: 15323: 15322: 15303: 15302: 15301: 15296: 15278: 15264:Wigner's friend 15240: 15231:Quantum gravity 15192: 15178:Quantum sensing 15158:Quantum network 15138:Quantum machine 15108:Quantum imaging 15071:Quantum circuit 15066:Quantum channel 15025: 14971:Quantum biology 14957: 14933:Elitzur–Vaidman 14908:Davisson–Germer 14891: 14843:Hidden-variable 14833:de Broglie–Bohm 14810:Interpretations 14804: 14753: 14707: 14594:Complementarity 14572: 14529: 14524: 14489: 14467: 14448: 14432: 14430:Further reading 14427: 14378: 14374: 14325: 14321: 14304: 14300: 14243: 14239: 14198: 14194: 14137: 14133: 14123: 14119: 14062: 14058: 14053: 14049: 14038: 14034: 13981: 13977: 13956: 13952: 13895: 13891: 13834: 13830: 13819: 13810: 13753: 13746: 13697: 13690: 13633: 13629: 13598:Proc. R. Soc. A 13590: 13586: 13527: 13523: 13474: 13470: 13413: 13409: 13398: 13394: 13355: 13351: 13327:10.1.1.508.4193 13300: 13296: 13239: 13235: 13194: 13190: 13173: 13169: 13128: 13124: 13079:Rev. Math. Phys 13075: 13071: 13022: 13018: 13009: 13007: 13005:Quanta Magazine 12997: 12993: 12984: 12982: 12972: 12968: 12959: 12957: 12947: 12943: 12934: 12932: 12922: 12918: 12909: 12907: 12897: 12893: 12884: 12882: 12872: 12868: 12823: 12819: 12807: 12805: 12796: 12795: 12773: 12769: 12760: 12758: 12749: 12748: 12741: 12724: 12717: 12680: 12673: 12634: 12630: 12577: 12573: 12524: 12520: 12471: 12467: 12398: 12394: 12325: 12321: 12264: 12260: 12203: 12199: 12160: 12156: 12115: 12111: 12070:Spekkens, R. W. 12066: 12062: 12013: 12009: 11959: 11952: 11903: 11899: 11850: 11846: 11797: 11793: 11765: 11761: 11712: 11708: 11677:(15): 880–884. 11661: 11657: 11626: 11617: 11607: 11605: 11594: 11590: 11583: 11569: 11565: 11544: 11540: 11512:Physical Review 11507: 11503: 11473:Physical Review 11464: 11460: 11403: 11399: 11392: 11374: 11367: 11318: 11314: 11307: 11303: 11294: 11290: 11227: 11223: 11192:(10): 777–780. 11186:Physical Review 11178: 11171: 11150:(10): 293–298. 11140: 11136: 11073: 11069: 11012: 11008: 10951: 10947: 10890: 10886: 10849: 10842: 10838: 10806: 10772: 10755: 10752: 10751: 10748: 10715: 10698: 10695: 10694: 10668: 10659: 10654: 10641: 10636: 10630: 10627: 10626: 10598: 10581: 10578: 10577: 10551: 10550: 10546: 10537: 10532: 10521: 10520: 10510: 10505: 10494: 10493: 10490: 10487: 10486: 10458: 10441: 10438: 10437: 10436:giving rise to 10415: 10413: 10410: 10409: 10392: 10387: 10374: 10369: 10363: 10360: 10359: 10331: 10314: 10311: 10310: 10307: 10290: 10256: 10239: 10236: 10235: 10207: 10190: 10187: 10186: 10182: 10176: 10143: 10126: 10123: 10122: 10118: 10096: 10092: 10072: 10055: 10052: 10051: 10029: 10028: 10019: 10015: 10013: 10010: 10009: 9987: 9986: 9984: 9981: 9980: 9958: 9957: 9955: 9952: 9951: 9929: 9928: 9926: 9923: 9922: 9890: 9887: 9886: 9869: 9864: 9851: 9847: 9845: 9842: 9841: 9824: 9819: 9806: 9802: 9800: 9797: 9796: 9780: 9777: 9776: 9756: 9745: 9744: 9743: 9738: 9732: 9721: 9720: 9719: 9714: 9711: 9710: 9685: 9684: 9679: 9668: 9667: 9662: 9659: 9658: 9633: 9632: 9627: 9616: 9615: 9610: 9607: 9606: 9589: 9585: 9564: 9560: 9558: 9555: 9554: 9537: 9533: 9512: 9508: 9506: 9503: 9502: 9486: 9483: 9482: 9459: 9456: 9455: 9439: 9436: 9435: 9419: 9416: 9415: 9398: 9394: 9392: 9389: 9388: 9342: 9339: 9338: 9322: 9319: 9318: 9302: 9299: 9298: 9261: 9258: 9257: 9235: 9232: 9231: 9209: 9206: 9205: 9183: 9180: 9179: 9163: 9160: 9159: 9136: 9125: 9119: 9116: 9115: 9072: 9071: 9067: 9060: 9055: 9052: 9051: 9032: 9021: 9018: 9017: 8977: 8972: 8969: 8968: 8946: 8943: 8942: 8920: 8918: 8915: 8914: 8913:take values in 8880: 8877: 8876: 8839: 8835: 8825: 8805: 8788: 8785: 8784: 8756: 8739: 8736: 8735: 8704: 8692: 8688: 8676: 8672: 8661: 8640: 8616: 8595: 8589: 8586: 8585: 8561: 8549: 8545: 8527: 8523: 8518: 8497: 8473: 8452: 8446: 8443: 8442: 8420: 8417: 8416: 8401: 8398: 8397: 8374: 8373: 8353: 8336: 8333: 8332: 8303: 8302: 8300: 8297: 8296: 8292: 8265: 8264: 8262: 8259: 8258: 8241: 8237: 8223: 8222: 8220: 8217: 8216: 8194: 8193: 8184: 8180: 8178: 8175: 8174: 8157: 8153: 8139: 8138: 8136: 8133: 8132: 8115: 8111: 8109: 8106: 8105: 8089: 8086: 8085: 8060: 8056: 8036: 8019: 8016: 8015: 7999: 7996: 7995: 7975: 7971: 7951: 7934: 7931: 7930: 7914: 7911: 7910: 7893: 7889: 7869: 7852: 7849: 7848: 7829: 7826: 7825: 7808: 7804: 7784: 7767: 7764: 7763: 7746: 7742: 7722: 7705: 7702: 7701: 7668: 7664: 7655: 7651: 7642: 7638: 7636: 7633: 7632: 7601: 7584: 7581: 7580: 7563: 7559: 7547: 7543: 7532: 7529: 7528: 7511: 7507: 7495: 7491: 7480: 7477: 7476: 7459: 7455: 7446: 7442: 7440: 7437: 7436: 7408: 7391: 7388: 7387: 7370: 7366: 7357: 7353: 7351: 7348: 7347: 7325: 7317: 7312: 7304: 7290: 7285: 7282: 7281: 7253: 7236: 7233: 7232: 7203: 7186: 7183: 7182: 7165: 7161: 7159: 7156: 7155: 7138: 7134: 7132: 7129: 7128: 7125: 7096: 7092: 7078: 7077: 7075: 7072: 7071: 7042: 7041: 7039: 7036: 7035: 7018: 7014: 7012: 7009: 7008: 6992: 6989: 6988: 6971: 6967: 6953: 6952: 6950: 6947: 6946: 6929: 6925: 6911: 6910: 6908: 6905: 6904: 6888: 6885: 6884: 6862: 6861: 6859: 6856: 6855: 6838: 6834: 6820: 6819: 6817: 6814: 6813: 6796: 6792: 6790: 6787: 6786: 6770: 6767: 6766: 6735: 6718: 6715: 6714: 6697: 6693: 6691: 6688: 6687: 6649: 6646: 6645: 6619: 6614: 6601: 6596: 6587: 6584: 6583: 6547: 6544: 6543: 6521: 6515: 6510: 6500: 6495: 6480: 6460: 6443: 6440: 6439: 6420: 6419: 6410: 6405: 6395: 6382: 6381: 6372: 6367: 6357: 6351: 6348: 6347: 6330: 6326: 6320: 6315: 6299: 6295: 6289: 6284: 6275: 6272: 6271: 6249: 6246: 6245: 6212: 6207: 6182: 6177: 6171: 6168: 6167: 6139: 6137: 6134: 6133: 6117: 6114: 6113: 6085: 6068: 6065: 6064: 6045: 6041: 6032: 6028: 6020: 6017: 6016: 5999: 5995: 5986: 5982: 5980: 5977: 5976: 5945: 5928: 5925: 5924: 5908: 5905: 5904: 5888: 5885: 5884: 5868: 5865: 5864: 5826: 5823: 5822: 5800: 5794: 5789: 5776: 5771: 5756: 5736: 5719: 5716: 5715: 5696: 5690: 5689: 5688: 5679: 5674: 5664: 5651: 5645: 5644: 5643: 5634: 5629: 5619: 5613: 5610: 5609: 5592: 5588: 5582: 5577: 5561: 5557: 5551: 5546: 5537: 5534: 5533: 5511: 5508: 5507: 5486: 5482: 5473: 5469: 5460: 5455: 5442: 5438: 5429: 5425: 5416: 5411: 5405: 5402: 5401: 5384: 5380: 5371: 5367: 5353: 5351: 5348: 5347: 5330: 5326: 5317: 5313: 5311: 5308: 5307: 5279: 5262: 5259: 5258: 5255: 5229: 5209: 5199: 5191: 5188: 5187: 5172: 5154: 5148: 5123: 5119: 5110: 5106: 5101: 5098: 5097: 5081: 5078: 5077: 5061: 5058: 5057: 5040: 5036: 5034: 5031: 5030: 5013: 5009: 5007: 5004: 5003: 4987: 4984: 4983: 4967: 4964: 4963: 4948: 4939: 4912: 4910: 4907: 4906: 4881: 4867: 4862: 4859: 4858: 4833: 4831: 4828: 4827: 4805: 4800: 4798: 4795: 4794: 4769: 4767: 4764: 4763: 4741: 4736: 4734: 4731: 4730: 4705: 4691: 4686: 4683: 4682: 4657: 4655: 4652: 4651: 4621: 4617: 4611: 4594: 4583: 4580: 4579: 4554: 4540: 4535: 4532: 4531: 4508: 4504: 4502: 4499: 4498: 4472: 4468: 4466: 4463: 4462: 4437: 4433: 4431: 4428: 4427: 4402: 4388: 4383: 4380: 4379: 4354: 4340: 4335: 4332: 4331: 4309: 4307: 4304: 4303: 4278: 4264: 4259: 4256: 4255: 4230: 4216: 4211: 4208: 4207: 4182: 4168: 4163: 4160: 4159: 4131: 4117: 4100: 4086: 4084: 4081: 4080: 4047: 4043: 4028: 4024: 4023: 4019: 4007: 3992: 3988: 3981: 3980: 3976: 3964: 3939: 3935: 3920: 3916: 3915: 3911: 3899: 3884: 3880: 3873: 3872: 3868: 3856: 3837: 3823: 3809: 3808: 3804: 3792: 3778: 3776: 3773: 3772: 3750: 3748: 3745: 3744: 3730:Anton Zeilinger 3717: 3683: 3678: 3675: 3674: 3647: 3646: 3642: 3640: 3637: 3636: 3608: 3591: 3588: 3587: 3553: 3526: 3522: 3489: 3462: 3459: 3458: 3336: 3335: 3331: 3329: 3326: 3325: 3295: 3278: 3275: 3274: 3252: 3250: 3247: 3246: 3212: 3209: 3208: 3205:convex polytope 3169: 3147: 3131: 3127: 3126: 3124: 3121: 3120: 3092: 3075: 3072: 3071: 3037: 3034: 3033: 3011: 3007: 2996: 2984: 2980: 2971: 2967: 2956: 2944: 2940: 2931: 2927: 2918: 2914: 2894: 2890: 2881: 2877: 2876: 2852: 2835: 2832: 2831: 2812: 2808: 2799: 2795: 2793: 2790: 2789: 2769: 2765: 2756: 2752: 2744: 2741: 2740: 2717: 2713: 2704: 2700: 2698: 2695: 2694: 2678: 2675: 2674: 2657: 2653: 2644: 2640: 2638: 2635: 2634: 2614: 2610: 2608: 2605: 2604: 2587: 2583: 2581: 2578: 2577: 2555: 2552: 2551: 2536: 2533: 2532: 2510: 2507: 2506: 2491: 2488: 2487: 2467: 2463: 2452: 2440: 2436: 2434: 2431: 2430: 2410: 2406: 2395: 2383: 2379: 2377: 2374: 2373: 2351: 2347: 2336: 2324: 2320: 2311: 2307: 2296: 2284: 2280: 2268: 2264: 2255: 2251: 2234: 2217: 2214: 2213: 2194: 2191: 2190: 2189: 2163: 2160: 2159: 2143: 2140: 2139: 2122: 2119: 2118: 2087: 2070: 2067: 2066: 2038: 2021: 2018: 2017: 1962: 1942: 1939: 1938: 1916: 1913: 1912: 1884: 1867: 1864: 1863: 1841: 1838: 1837: 1822: 1819: 1818: 1796: 1793: 1792: 1777: 1774: 1773: 1767:CHSH inequality 1750:are different: 1739: 1729: 1727:Bell inequality 1684: 1673: 1672: 1670: 1667: 1666: 1649: 1638: 1637: 1635: 1632: 1631: 1614: 1603: 1602: 1600: 1597: 1596: 1579: 1568: 1567: 1565: 1562: 1561: 1529: 1518: 1517: 1508: 1497: 1496: 1491: 1488: 1487: 1467: 1456: 1455: 1446: 1435: 1434: 1429: 1426: 1425: 1405: 1394: 1393: 1384: 1373: 1372: 1367: 1364: 1363: 1343: 1332: 1331: 1322: 1311: 1310: 1305: 1302: 1301: 1263: 1249: 1248: 1244: 1232: 1218: 1216: 1213: 1212: 1190: 1179: 1178: 1172: 1161: 1160: 1151: 1140: 1139: 1133: 1122: 1121: 1120: 1116: 1104: 1090: 1079: 1078: 1072: 1061: 1060: 1051: 1040: 1039: 1033: 1022: 1021: 1020: 1016: 1004: 988: 984: 980: 978: 975: 974: 950:Albert Einstein 942: 936: 931: 880: 851: 850: 849: 614: 606: 605: 551: 550:Advanced topics 543: 542: 541: 493:Hidden-variable 483:de Broglie–Bohm 462: 460:Interpretations 452: 451: 450: 420: 412: 411: 410: 368: 360: 359: 358: 325: 281:CHSH inequality 270: 262: 261: 260: 189:Complementarity 183: 175: 174: 173: 141: 112: 87: 76: 75: 61: 51: 46: 38: 35: 34: 17: 12: 11: 5: 15331: 15321: 15320: 15315: 15298: 15297: 15295: 15294: 15283: 15280: 15279: 15277: 15276: 15271: 15266: 15261: 15260: 15259: 15248: 15246: 15242: 15241: 15239: 15238: 15233: 15228: 15227: 15226: 15216: 15211: 15209:Casimir effect 15206: 15200: 15198: 15194: 15193: 15191: 15190: 15185: 15180: 15175: 15170: 15168:Quantum optics 15165: 15160: 15155: 15150: 15145: 15140: 15135: 15130: 15125: 15120: 15115: 15110: 15105: 15100: 15095: 15090: 15089: 15088: 15078: 15073: 15068: 15063: 15062: 15061: 15051: 15046: 15041: 15035: 15033: 15027: 15026: 15024: 15023: 15018: 15013: 15008: 15003: 14998: 14993: 14988: 14983: 14978: 14973: 14967: 14965: 14959: 14958: 14956: 14955: 14950: 14945: 14943:Quantum eraser 14940: 14935: 14930: 14925: 14920: 14915: 14910: 14905: 14899: 14897: 14893: 14892: 14890: 14889: 14884: 14879: 14874: 14869: 14864: 14859: 14858: 14857: 14856: 14855: 14840: 14835: 14830: 14825: 14820: 14814: 14812: 14806: 14805: 14803: 14802: 14797: 14792: 14787: 14782: 14777: 14772: 14767: 14761: 14759: 14755: 14754: 14752: 14751: 14746: 14741: 14736: 14731: 14726: 14721: 14715: 14713: 14709: 14708: 14706: 14705: 14704: 14703: 14698: 14688: 14683: 14678: 14673: 14668: 14663: 14658: 14653: 14648: 14643: 14638: 14633: 14628: 14627: 14626: 14621: 14616: 14611: 14601: 14599:Density matrix 14596: 14591: 14586: 14580: 14578: 14574: 14573: 14571: 14570: 14565: 14560: 14555: 14554: 14553: 14543: 14537: 14535: 14531: 14530: 14523: 14522: 14515: 14508: 14500: 14494: 14493: 14487: 14471: 14465: 14452: 14446: 14431: 14428: 14426: 14425: 14372: 14319: 14298: 14237: 14192: 14147:(21): 210503. 14131: 14117: 14056: 14047: 14032: 13995:(6): 450–453. 13989:Nature Physics 13985:Renner, Renato 13975: 13950: 13905:(14): 140501. 13889: 13844:(12): 120405. 13828: 13808: 13744: 13688: 13627: 13584: 13521: 13484:(18): 180502. 13468: 13423:(25): 250401. 13407: 13392: 13365:(9): 963–972. 13349: 13312:(3): 379–385. 13294: 13233: 13188: 13167: 13122: 13085:(5): 1250012. 13069: 13016: 12991: 12966: 12941: 12916: 12891: 12866: 12817: 12808:|journal= 12767: 12739: 12715: 12694:(4): 557–570. 12671: 12628: 12571: 12518: 12481:(3): 695–746. 12465: 12412:(16): 160402. 12392: 12319: 12258: 12213:(19): 190401. 12197: 12154: 12109: 12060: 12023:(10): 103001. 12007: 11964:Nature Physics 11950: 11897: 11844: 11791: 11759: 11706: 11655: 11642:(3): 195–200. 11615: 11588: 11581: 11563: 11538: 11501: 11480:(8): 696–702. 11458: 11413:(14): 140402. 11397: 11390: 11365: 11312: 11301: 11288: 11221: 11169: 11134: 11087:(25): 250402. 11067: 11022:(25): 250401. 11006: 10945: 10884: 10839: 10837: 10834: 10833: 10832: 10827: 10822: 10817: 10812: 10805: 10802: 10788: 10785: 10782: 10779: 10775: 10771: 10768: 10765: 10762: 10759: 10747: 10744: 10731: 10728: 10725: 10722: 10718: 10714: 10711: 10708: 10705: 10702: 10677: 10674: 10671: 10667: 10662: 10657: 10653: 10649: 10644: 10639: 10635: 10614: 10611: 10608: 10605: 10601: 10597: 10594: 10591: 10588: 10585: 10564: 10558: 10555: 10549: 10545: 10540: 10535: 10528: 10525: 10518: 10513: 10508: 10501: 10498: 10474: 10471: 10468: 10465: 10461: 10457: 10454: 10451: 10448: 10445: 10424: 10421: 10418: 10395: 10390: 10386: 10382: 10377: 10372: 10368: 10347: 10344: 10341: 10338: 10334: 10330: 10327: 10324: 10321: 10318: 10306: 10303: 10289: 10286: 10272: 10269: 10266: 10263: 10259: 10255: 10252: 10249: 10246: 10243: 10223: 10220: 10217: 10214: 10210: 10206: 10203: 10200: 10197: 10194: 10178:Main article: 10175: 10172: 10159: 10156: 10153: 10150: 10146: 10142: 10139: 10136: 10133: 10130: 10117: 10114: 10099: 10095: 10091: 10088: 10085: 10082: 10079: 10075: 10071: 10068: 10065: 10062: 10059: 10036: 10033: 10027: 10022: 10018: 9994: 9991: 9965: 9962: 9936: 9933: 9907: 9906: 9894: 9872: 9867: 9863: 9859: 9854: 9850: 9827: 9822: 9818: 9814: 9809: 9805: 9784: 9764: 9759: 9752: 9749: 9741: 9735: 9728: 9725: 9718: 9698: 9692: 9689: 9682: 9675: 9672: 9666: 9646: 9640: 9637: 9630: 9623: 9620: 9614: 9592: 9588: 9584: 9581: 9578: 9575: 9572: 9567: 9563: 9540: 9536: 9532: 9529: 9526: 9523: 9520: 9515: 9511: 9490: 9479: 9475: 9463: 9443: 9423: 9401: 9397: 9376: 9373: 9370: 9367: 9364: 9361: 9358: 9355: 9352: 9349: 9346: 9326: 9306: 9292: 9280: 9277: 9274: 9271: 9268: 9265: 9245: 9242: 9239: 9219: 9216: 9213: 9193: 9190: 9187: 9167: 9147: 9144: 9139: 9134: 9131: 9128: 9124: 9112: 9100: 9097: 9093: 9089: 9086: 9080: 9076: 9070: 9064: 9059: 9039: 9035: 9031: 9028: 9025: 8989: 8986: 8981: 8976: 8956: 8953: 8950: 8929: 8926: 8923: 8902: 8899: 8896: 8893: 8890: 8887: 8884: 8862: 8857: 8854: 8851: 8848: 8845: 8842: 8838: 8832: 8829: 8824: 8821: 8818: 8815: 8812: 8808: 8804: 8801: 8798: 8795: 8792: 8772: 8769: 8766: 8763: 8759: 8755: 8752: 8749: 8746: 8743: 8717: 8714: 8711: 8707: 8703: 8700: 8695: 8691: 8687: 8684: 8679: 8675: 8671: 8668: 8664: 8660: 8657: 8654: 8651: 8648: 8643: 8639: 8635: 8632: 8629: 8626: 8623: 8619: 8615: 8612: 8609: 8606: 8603: 8598: 8594: 8574: 8571: 8568: 8564: 8560: 8557: 8552: 8548: 8544: 8541: 8538: 8535: 8530: 8526: 8521: 8517: 8514: 8511: 8508: 8505: 8500: 8496: 8492: 8489: 8486: 8483: 8480: 8476: 8472: 8469: 8466: 8463: 8460: 8455: 8451: 8430: 8427: 8424: 8405: 8381: 8378: 8372: 8369: 8366: 8363: 8360: 8356: 8352: 8349: 8346: 8343: 8340: 8310: 8307: 8291: 8288: 8272: 8269: 8244: 8240: 8236: 8230: 8227: 8201: 8198: 8192: 8187: 8183: 8160: 8156: 8152: 8146: 8143: 8118: 8114: 8093: 8082: 8081: 8063: 8059: 8055: 8052: 8049: 8046: 8043: 8039: 8035: 8032: 8029: 8026: 8023: 8003: 7992: 7978: 7974: 7970: 7967: 7964: 7961: 7958: 7954: 7950: 7947: 7944: 7941: 7938: 7918: 7896: 7892: 7888: 7885: 7882: 7879: 7876: 7872: 7868: 7865: 7862: 7859: 7856: 7845: 7833: 7811: 7807: 7803: 7800: 7797: 7794: 7791: 7787: 7783: 7780: 7777: 7774: 7771: 7749: 7745: 7741: 7738: 7735: 7732: 7729: 7725: 7721: 7718: 7715: 7712: 7709: 7685: 7682: 7679: 7676: 7671: 7667: 7663: 7658: 7654: 7650: 7645: 7641: 7617: 7614: 7611: 7608: 7604: 7600: 7597: 7594: 7591: 7588: 7566: 7562: 7558: 7555: 7550: 7546: 7542: 7539: 7536: 7514: 7510: 7506: 7503: 7498: 7494: 7490: 7487: 7484: 7462: 7458: 7454: 7449: 7445: 7424: 7421: 7418: 7415: 7411: 7407: 7404: 7401: 7398: 7395: 7373: 7369: 7365: 7360: 7356: 7332: 7328: 7324: 7320: 7315: 7311: 7307: 7303: 7300: 7297: 7293: 7289: 7269: 7266: 7263: 7260: 7256: 7252: 7249: 7246: 7243: 7240: 7219: 7216: 7213: 7210: 7206: 7202: 7199: 7196: 7193: 7190: 7168: 7164: 7141: 7137: 7124: 7121: 7099: 7095: 7091: 7085: 7082: 7049: 7046: 7021: 7017: 6996: 6974: 6970: 6966: 6960: 6957: 6932: 6928: 6924: 6918: 6915: 6892: 6869: 6866: 6841: 6837: 6833: 6827: 6824: 6799: 6795: 6774: 6751: 6748: 6745: 6742: 6738: 6734: 6731: 6728: 6725: 6722: 6700: 6696: 6684: 6683: 6671: 6668: 6665: 6662: 6659: 6656: 6653: 6633: 6630: 6627: 6622: 6617: 6613: 6609: 6604: 6599: 6595: 6591: 6581: 6569: 6566: 6563: 6560: 6557: 6554: 6551: 6530: 6527: 6524: 6518: 6513: 6509: 6503: 6498: 6494: 6489: 6486: 6483: 6479: 6476: 6473: 6470: 6467: 6463: 6459: 6456: 6453: 6450: 6447: 6437: 6423: 6418: 6413: 6408: 6404: 6398: 6394: 6390: 6385: 6380: 6375: 6370: 6366: 6360: 6356: 6333: 6329: 6323: 6318: 6314: 6310: 6307: 6302: 6298: 6292: 6287: 6283: 6279: 6259: 6256: 6253: 6229: 6226: 6223: 6220: 6215: 6210: 6206: 6202: 6199: 6196: 6193: 6190: 6185: 6180: 6176: 6155: 6152: 6148: 6145: 6142: 6121: 6101: 6098: 6095: 6092: 6088: 6084: 6081: 6078: 6075: 6072: 6048: 6044: 6040: 6035: 6031: 6027: 6024: 6002: 5998: 5994: 5989: 5985: 5961: 5958: 5955: 5952: 5948: 5944: 5941: 5938: 5935: 5932: 5912: 5892: 5872: 5861: 5860: 5848: 5845: 5842: 5839: 5836: 5833: 5830: 5809: 5806: 5803: 5797: 5792: 5788: 5784: 5779: 5774: 5770: 5765: 5762: 5759: 5755: 5752: 5749: 5746: 5743: 5739: 5735: 5732: 5729: 5726: 5723: 5713: 5699: 5693: 5687: 5682: 5677: 5673: 5667: 5663: 5659: 5654: 5648: 5642: 5637: 5632: 5628: 5622: 5618: 5595: 5591: 5585: 5580: 5576: 5572: 5569: 5564: 5560: 5554: 5549: 5545: 5541: 5521: 5518: 5515: 5489: 5485: 5481: 5476: 5472: 5468: 5463: 5458: 5454: 5450: 5445: 5441: 5437: 5432: 5428: 5424: 5419: 5414: 5410: 5387: 5383: 5379: 5374: 5370: 5366: 5362: 5359: 5356: 5333: 5329: 5325: 5320: 5316: 5295: 5292: 5289: 5286: 5282: 5278: 5275: 5272: 5269: 5266: 5254: 5251: 5250: 5249: 5238: 5233: 5228: 5225: 5221: 5218: 5215: 5212: 5208: 5203: 5198: 5195: 5179:, is known as 5171: 5168: 5147: 5144: 5131: 5126: 5122: 5118: 5113: 5109: 5105: 5085: 5065: 5043: 5039: 5016: 5012: 4991: 4971: 4947: 4944: 4938: 4935: 4921: 4918: 4915: 4894: 4890: 4887: 4884: 4880: 4876: 4873: 4870: 4866: 4846: 4843: 4840: 4836: 4815: 4812: 4808: 4803: 4782: 4779: 4776: 4772: 4751: 4748: 4744: 4739: 4718: 4714: 4711: 4708: 4704: 4700: 4697: 4694: 4690: 4670: 4667: 4664: 4660: 4637: 4634: 4625: 4620: 4616: 4610: 4607: 4604: 4601: 4597: 4593: 4590: 4587: 4567: 4563: 4560: 4557: 4553: 4549: 4546: 4543: 4539: 4518: 4514: 4511: 4507: 4486: 4482: 4478: 4475: 4471: 4451: 4447: 4443: 4440: 4436: 4415: 4411: 4408: 4405: 4401: 4397: 4394: 4391: 4387: 4378:and the other 4367: 4363: 4360: 4357: 4353: 4349: 4346: 4343: 4339: 4318: 4315: 4312: 4291: 4287: 4284: 4281: 4277: 4273: 4270: 4267: 4263: 4243: 4239: 4236: 4233: 4229: 4225: 4222: 4219: 4215: 4195: 4191: 4188: 4185: 4181: 4177: 4174: 4171: 4167: 4144: 4140: 4137: 4134: 4130: 4126: 4123: 4120: 4116: 4109: 4105: 4099: 4096: 4093: 4089: 4067: 4062: 4057: 4053: 4050: 4046: 4042: 4038: 4034: 4031: 4027: 4022: 4015: 4011: 4006: 4002: 3998: 3995: 3991: 3985: 3979: 3972: 3968: 3963: 3959: 3954: 3949: 3945: 3942: 3938: 3934: 3930: 3926: 3923: 3919: 3914: 3907: 3903: 3898: 3894: 3890: 3887: 3883: 3877: 3871: 3864: 3860: 3855: 3851: 3846: 3843: 3840: 3836: 3832: 3829: 3826: 3822: 3818: 3815: 3812: 3807: 3800: 3796: 3791: 3787: 3784: 3781: 3759: 3756: 3753: 3716: 3713: 3695: 3692: 3687: 3682: 3659: 3656: 3653: 3650: 3645: 3624: 3621: 3618: 3615: 3611: 3607: 3604: 3601: 3598: 3595: 3572: 3569: 3566: 3563: 3560: 3556: 3552: 3549: 3546: 3543: 3540: 3535: 3532: 3529: 3525: 3521: 3518: 3515: 3510: 3507: 3504: 3501: 3498: 3495: 3492: 3488: 3484: 3481: 3478: 3475: 3472: 3469: 3466: 3444: 3441: 3438: 3435: 3432: 3429: 3426: 3423: 3420: 3417: 3414: 3411: 3408: 3405: 3402: 3399: 3396: 3393: 3390: 3387: 3384: 3381: 3378: 3375: 3372: 3369: 3366: 3363: 3360: 3357: 3354: 3348: 3345: 3342: 3339: 3334: 3311: 3308: 3305: 3302: 3298: 3294: 3291: 3288: 3285: 3282: 3261: 3258: 3255: 3234: 3231: 3228: 3225: 3222: 3219: 3216: 3190: 3187: 3184: 3181: 3178: 3175: 3172: 3167: 3163: 3160: 3157: 3154: 3150: 3146: 3143: 3140: 3137: 3134: 3130: 3108: 3105: 3102: 3099: 3095: 3091: 3088: 3085: 3082: 3079: 3059: 3056: 3053: 3050: 3047: 3044: 3041: 3019: 3014: 3010: 3006: 3003: 2999: 2995: 2992: 2987: 2983: 2979: 2974: 2970: 2966: 2963: 2959: 2955: 2952: 2947: 2943: 2939: 2934: 2930: 2926: 2921: 2917: 2913: 2910: 2905: 2902: 2897: 2893: 2889: 2884: 2880: 2875: 2871: 2868: 2865: 2862: 2859: 2855: 2851: 2848: 2845: 2842: 2839: 2815: 2811: 2807: 2802: 2798: 2777: 2772: 2768: 2764: 2759: 2755: 2751: 2748: 2728: 2725: 2720: 2716: 2712: 2707: 2703: 2682: 2660: 2656: 2652: 2647: 2643: 2633:Suppose that 2617: 2613: 2590: 2586: 2565: 2562: 2559: 2540: 2520: 2517: 2514: 2495: 2475: 2470: 2466: 2462: 2459: 2455: 2451: 2448: 2443: 2439: 2418: 2413: 2409: 2405: 2402: 2398: 2394: 2391: 2386: 2382: 2359: 2354: 2350: 2346: 2343: 2339: 2335: 2332: 2327: 2323: 2319: 2314: 2310: 2306: 2303: 2299: 2295: 2292: 2287: 2283: 2279: 2276: 2271: 2267: 2263: 2258: 2254: 2250: 2247: 2244: 2241: 2237: 2233: 2230: 2227: 2224: 2221: 2198: 2176: 2173: 2170: 2167: 2147: 2126: 2103: 2100: 2097: 2094: 2090: 2086: 2083: 2080: 2077: 2074: 2054: 2051: 2048: 2045: 2041: 2037: 2034: 2031: 2028: 2025: 2005: 2002: 1999: 1996: 1993: 1990: 1987: 1984: 1981: 1978: 1975: 1972: 1969: 1965: 1961: 1958: 1955: 1952: 1949: 1946: 1926: 1923: 1920: 1900: 1897: 1894: 1891: 1887: 1883: 1880: 1877: 1874: 1871: 1851: 1848: 1845: 1826: 1806: 1803: 1800: 1781: 1733:Bell's theorem 1728: 1725: 1687: 1682: 1679: 1676: 1652: 1647: 1644: 1641: 1617: 1612: 1609: 1606: 1582: 1577: 1574: 1571: 1537: 1532: 1527: 1524: 1521: 1516: 1511: 1506: 1503: 1500: 1495: 1475: 1470: 1465: 1462: 1459: 1454: 1449: 1444: 1441: 1438: 1433: 1413: 1408: 1403: 1400: 1397: 1392: 1387: 1382: 1379: 1376: 1371: 1351: 1346: 1341: 1338: 1335: 1330: 1325: 1320: 1317: 1314: 1309: 1277: 1272: 1269: 1266: 1262: 1258: 1255: 1252: 1247: 1240: 1236: 1231: 1227: 1224: 1221: 1199: 1193: 1188: 1185: 1182: 1175: 1170: 1167: 1164: 1159: 1154: 1149: 1146: 1143: 1136: 1131: 1128: 1125: 1119: 1112: 1108: 1103: 1099: 1093: 1088: 1085: 1082: 1075: 1070: 1067: 1064: 1059: 1054: 1049: 1046: 1043: 1036: 1031: 1028: 1025: 1019: 1012: 1008: 1003: 999: 994: 991: 987: 983: 954:Boris Podolsky 938:Main article: 935: 932: 930: 927: 900:quantum system 882: 881: 879: 878: 871: 864: 856: 853: 852: 848: 847: 842: 837: 832: 827: 822: 817: 812: 807: 802: 797: 792: 787: 782: 777: 772: 767: 762: 757: 752: 747: 742: 737: 732: 727: 722: 717: 712: 707: 702: 697: 692: 687: 682: 677: 672: 667: 662: 657: 652: 647: 642: 637: 632: 627: 622: 616: 615: 612: 611: 608: 607: 604: 603: 598: 593: 588: 586:Density matrix 583: 578: 573: 568: 563: 558: 552: 549: 548: 545: 544: 540: 539: 534: 529: 524: 519: 514: 509: 508: 507: 506: 505: 490: 485: 480: 475: 470: 464: 463: 458: 457: 454: 453: 449: 448: 443: 438: 433: 428: 422: 421: 418: 417: 414: 413: 409: 408: 403: 398: 393: 388: 383: 377: 376: 375: 369: 366: 365: 362: 361: 357: 356: 351: 346: 340: 339: 338: 337: 336: 334:Delayed-choice 329:Quantum eraser 324: 323: 318: 313: 308: 303: 298: 293: 288: 283: 278: 272: 271: 268: 267: 264: 263: 259: 258: 257: 256: 246: 241: 236: 231: 226: 221: 219:Quantum number 216: 211: 206: 201: 196: 191: 185: 184: 181: 180: 177: 176: 172: 171: 166: 160: 159: 158: 153: 148: 142: 139: 138: 135: 134: 133: 132: 127: 122: 114: 113: 108: 97: 94: 90: 83: 80: 74: 71: 68: 64: 57: 54: 50: 45: 42: 31: 30: 24: 23: 15: 9: 6: 4: 3: 2: 15330: 15319: 15316: 15314: 15311: 15310: 15308: 15293: 15285: 15284: 15281: 15275: 15272: 15270: 15267: 15265: 15262: 15258: 15255: 15254: 15253: 15250: 15249: 15247: 15243: 15237: 15234: 15232: 15229: 15225: 15222: 15221: 15220: 15217: 15215: 15212: 15210: 15207: 15205: 15202: 15201: 15199: 15195: 15189: 15186: 15184: 15181: 15179: 15176: 15174: 15171: 15169: 15166: 15164: 15161: 15159: 15156: 15154: 15151: 15149: 15146: 15144: 15141: 15139: 15136: 15134: 15131: 15129: 15128:Quantum logic 15126: 15124: 15121: 15119: 15116: 15114: 15111: 15109: 15106: 15104: 15101: 15099: 15096: 15094: 15091: 15087: 15084: 15083: 15082: 15079: 15077: 15074: 15072: 15069: 15067: 15064: 15060: 15057: 15056: 15055: 15052: 15050: 15047: 15045: 15042: 15040: 15037: 15036: 15034: 15032: 15028: 15022: 15019: 15017: 15014: 15012: 15009: 15007: 15004: 15002: 14999: 14997: 14994: 14992: 14989: 14987: 14984: 14982: 14981:Quantum chaos 14979: 14977: 14974: 14972: 14969: 14968: 14966: 14964: 14960: 14954: 14951: 14949: 14948:Stern–Gerlach 14946: 14944: 14941: 14939: 14936: 14934: 14931: 14929: 14926: 14924: 14921: 14919: 14916: 14914: 14911: 14909: 14906: 14904: 14901: 14900: 14898: 14894: 14888: 14885: 14883: 14882:Transactional 14880: 14878: 14875: 14873: 14872:Quantum logic 14870: 14868: 14865: 14863: 14860: 14854: 14851: 14850: 14849: 14846: 14845: 14844: 14841: 14839: 14836: 14834: 14831: 14829: 14826: 14824: 14821: 14819: 14816: 14815: 14813: 14811: 14807: 14801: 14798: 14796: 14793: 14791: 14788: 14786: 14783: 14781: 14778: 14776: 14773: 14771: 14768: 14766: 14763: 14762: 14760: 14756: 14750: 14747: 14745: 14742: 14740: 14737: 14735: 14732: 14730: 14727: 14725: 14722: 14720: 14717: 14716: 14714: 14710: 14702: 14699: 14697: 14694: 14693: 14692: 14691:Wave function 14689: 14687: 14684: 14682: 14679: 14677: 14674: 14672: 14669: 14667: 14666:Superposition 14664: 14662: 14661:Quantum state 14659: 14657: 14654: 14652: 14649: 14647: 14644: 14642: 14639: 14637: 14634: 14632: 14629: 14625: 14622: 14620: 14617: 14615: 14614:Excited state 14612: 14610: 14607: 14606: 14605: 14602: 14600: 14597: 14595: 14592: 14590: 14587: 14585: 14582: 14581: 14579: 14575: 14569: 14566: 14564: 14561: 14559: 14556: 14552: 14549: 14548: 14547: 14544: 14542: 14539: 14538: 14536: 14532: 14528: 14521: 14516: 14514: 14509: 14507: 14502: 14501: 14498: 14490: 14484: 14480: 14476: 14472: 14468: 14462: 14458: 14453: 14449: 14443: 14439: 14434: 14433: 14421: 14417: 14413: 14409: 14405: 14401: 14396: 14391: 14388:(3): 034301. 14387: 14383: 14376: 14368: 14364: 14360: 14356: 14352: 14348: 14343: 14338: 14334: 14330: 14323: 14314: 14309: 14302: 14294: 14290: 14286: 14282: 14278: 14274: 14270: 14266: 14261: 14256: 14253:(2): 020501. 14252: 14248: 14241: 14233: 14229: 14225: 14221: 14216: 14211: 14207: 14203: 14196: 14188: 14184: 14180: 14176: 14172: 14168: 14164: 14160: 14155: 14150: 14146: 14142: 14135: 14128: 14121: 14113: 14109: 14105: 14101: 14097: 14093: 14089: 14085: 14080: 14075: 14071: 14067: 14060: 14051: 14043: 14036: 14028: 14024: 14020: 14016: 14012: 14008: 14003: 13998: 13994: 13990: 13986: 13979: 13970: 13965: 13961: 13954: 13946: 13942: 13938: 13934: 13930: 13926: 13922: 13918: 13913: 13908: 13904: 13900: 13893: 13885: 13881: 13877: 13873: 13869: 13865: 13861: 13857: 13852: 13847: 13843: 13839: 13832: 13824: 13817: 13815: 13813: 13804: 13800: 13796: 13792: 13788: 13784: 13780: 13776: 13771: 13766: 13762: 13758: 13751: 13749: 13740: 13736: 13732: 13728: 13724: 13720: 13715: 13710: 13707:(6): 062107. 13706: 13702: 13695: 13693: 13684: 13680: 13676: 13672: 13668: 13664: 13660: 13656: 13651: 13646: 13642: 13638: 13631: 13622: 13617: 13612: 13607: 13603: 13599: 13595: 13588: 13580: 13576: 13572: 13568: 13564: 13560: 13556: 13552: 13547: 13542: 13538: 13534: 13533: 13525: 13517: 13513: 13509: 13505: 13501: 13497: 13492: 13487: 13483: 13479: 13472: 13464: 13460: 13456: 13452: 13448: 13444: 13440: 13436: 13431: 13426: 13422: 13418: 13411: 13403: 13396: 13388: 13384: 13380: 13376: 13372: 13368: 13364: 13360: 13353: 13345: 13341: 13337: 13333: 13328: 13323: 13319: 13315: 13311: 13307: 13306: 13298: 13290: 13286: 13282: 13278: 13274: 13270: 13266: 13262: 13257: 13252: 13249:(1): 010401. 13248: 13244: 13237: 13229: 13225: 13221: 13217: 13212: 13207: 13203: 13199: 13192: 13183: 13178: 13171: 13163: 13159: 13154: 13149: 13145: 13141: 13137: 13133: 13126: 13118: 13114: 13110: 13106: 13102: 13098: 13093: 13088: 13084: 13080: 13073: 13065: 13061: 13057: 13053: 13049: 13045: 13040: 13035: 13032:(1): 012102. 13031: 13027: 13026:J. Math. Phys 13020: 13006: 13002: 12995: 12981: 12977: 12970: 12956: 12952: 12945: 12931: 12927: 12920: 12906: 12902: 12895: 12881: 12877: 12870: 12862: 12858: 12853: 12848: 12844: 12840: 12836: 12832: 12828: 12821: 12813: 12800: 12792: 12788: 12783: 12778: 12771: 12757:on 2019-12-06 12756: 12752: 12746: 12744: 12734: 12729: 12722: 12720: 12711: 12707: 12702: 12697: 12693: 12689: 12685: 12678: 12676: 12667: 12663: 12659: 12655: 12651: 12647: 12644:(2): 93–100. 12643: 12639: 12632: 12624: 12620: 12616: 12612: 12608: 12604: 12599: 12594: 12591:(2): 022108. 12590: 12586: 12582: 12575: 12567: 12563: 12559: 12555: 12551: 12547: 12542: 12537: 12534:(5): 052310. 12533: 12529: 12522: 12514: 12510: 12506: 12502: 12498: 12494: 12489: 12484: 12480: 12476: 12469: 12461: 12457: 12453: 12449: 12445: 12441: 12437: 12433: 12429: 12425: 12420: 12415: 12411: 12407: 12403: 12396: 12388: 12384: 12380: 12376: 12372: 12368: 12364: 12360: 12356: 12352: 12347: 12342: 12338: 12334: 12330: 12323: 12315: 12311: 12307: 12303: 12299: 12295: 12291: 12287: 12282: 12277: 12273: 12269: 12262: 12254: 12250: 12246: 12242: 12238: 12234: 12230: 12226: 12221: 12216: 12212: 12208: 12201: 12193: 12189: 12185: 12181: 12177: 12173: 12169: 12165: 12158: 12150: 12146: 12142: 12138: 12133: 12128: 12124: 12120: 12113: 12105: 12101: 12097: 12093: 12088: 12083: 12079: 12075: 12071: 12068:Wolfe, Elie; 12064: 12056: 12052: 12048: 12044: 12040: 12036: 12031: 12026: 12022: 12018: 12011: 12003: 11999: 11995: 11991: 11987: 11983: 11978: 11973: 11969: 11965: 11957: 11955: 11946: 11942: 11938: 11934: 11930: 11926: 11921: 11916: 11912: 11908: 11901: 11893: 11889: 11885: 11881: 11877: 11873: 11868: 11863: 11860:(3): 032114. 11859: 11855: 11848: 11840: 11836: 11832: 11828: 11824: 11820: 11816: 11812: 11808: 11804: 11803: 11795: 11788: 11784: 11779: 11774: 11770: 11763: 11755: 11751: 11747: 11743: 11739: 11735: 11730: 11725: 11722:(2): 022101. 11721: 11717: 11710: 11702: 11698: 11693: 11688: 11684: 11680: 11676: 11672: 11671: 11666: 11659: 11650: 11645: 11641: 11637: 11636: 11631: 11624: 11622: 11620: 11603: 11599: 11592: 11584: 11582:9781444331264 11578: 11574: 11567: 11560: 11559:0-19-852927-9 11556: 11552: 11548: 11542: 11534: 11530: 11526: 11522: 11518: 11514: 11513: 11505: 11496: 11491: 11487: 11483: 11479: 11475: 11474: 11469: 11462: 11454: 11450: 11446: 11442: 11438: 11434: 11430: 11426: 11421: 11416: 11412: 11408: 11401: 11393: 11387: 11383: 11379: 11372: 11370: 11361: 11357: 11353: 11349: 11345: 11341: 11336: 11331: 11327: 11323: 11316: 11305: 11298: 11292: 11284: 11280: 11275: 11270: 11266: 11262: 11258: 11254: 11249: 11244: 11240: 11236: 11232: 11225: 11217: 11213: 11208: 11203: 11199: 11195: 11191: 11187: 11183: 11176: 11174: 11165: 11161: 11157: 11153: 11149: 11145: 11138: 11130: 11126: 11121: 11116: 11112: 11108: 11104: 11100: 11095: 11090: 11086: 11082: 11078: 11071: 11063: 11059: 11055: 11051: 11047: 11043: 11039: 11035: 11030: 11025: 11021: 11017: 11010: 11002: 10998: 10994: 10990: 10986: 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3869: 3862: 3858: 3853: 3849: 3844: 3841: 3838: 3834: 3830: 3827: 3824: 3820: 3816: 3813: 3810: 3805: 3798: 3794: 3789: 3785: 3782: 3779: 3757: 3754: 3751: 3742: 3737: 3735: 3731: 3727: 3726:Michael Horne 3723: 3712: 3709: 3693: 3690: 3685: 3680: 3643: 3619: 3616: 3613: 3605: 3602: 3599: 3593: 3583: 3570: 3564: 3561: 3558: 3550: 3547: 3544: 3538: 3533: 3530: 3527: 3519: 3516: 3508: 3505: 3502: 3499: 3496: 3493: 3490: 3486: 3482: 3476: 3473: 3470: 3464: 3455: 3442: 3439: 3436: 3430: 3427: 3424: 3418: 3415: 3409: 3406: 3403: 3397: 3394: 3388: 3385: 3382: 3376: 3373: 3367: 3364: 3361: 3355: 3352: 3332: 3323: 3306: 3303: 3300: 3292: 3289: 3286: 3280: 3259: 3256: 3253: 3232: 3229: 3226: 3223: 3220: 3217: 3214: 3206: 3188: 3185: 3182: 3179: 3176: 3173: 3170: 3165: 3158: 3155: 3152: 3144: 3141: 3138: 3132: 3128: 3103: 3100: 3097: 3089: 3086: 3083: 3077: 3057: 3054: 3051: 3048: 3045: 3042: 3039: 3030: 3012: 3008: 3004: 3001: 2993: 2985: 2981: 2972: 2968: 2964: 2961: 2953: 2945: 2941: 2932: 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1639: 1615: 1610: 1604: 1580: 1575: 1569: 1557: 1555: 1551: 1530: 1525: 1522: 1519: 1514: 1509: 1504: 1501: 1498: 1468: 1463: 1460: 1457: 1452: 1447: 1442: 1439: 1436: 1406: 1401: 1398: 1395: 1390: 1385: 1380: 1377: 1374: 1344: 1339: 1336: 1333: 1328: 1323: 1318: 1315: 1312: 1299: 1295: 1294:Alice and Bob 1290: 1275: 1270: 1267: 1264: 1260: 1256: 1253: 1250: 1245: 1238: 1234: 1229: 1225: 1222: 1219: 1197: 1191: 1186: 1183: 1180: 1173: 1168: 1165: 1162: 1157: 1152: 1147: 1144: 1141: 1134: 1129: 1126: 1123: 1117: 1110: 1106: 1101: 1097: 1091: 1086: 1083: 1080: 1073: 1068: 1065: 1062: 1057: 1052: 1047: 1044: 1041: 1034: 1029: 1026: 1023: 1017: 1010: 1006: 1001: 997: 992: 989: 985: 981: 972: 968: 963: 959: 955: 951: 947: 941: 926: 924: 920: 916: 912: 907: 905: 904:local realism 901: 897: 893: 889: 877: 872: 870: 865: 863: 858: 857: 855: 854: 846: 843: 841: 838: 836: 833: 831: 828: 826: 823: 821: 818: 816: 813: 811: 808: 806: 803: 801: 798: 796: 793: 791: 788: 786: 783: 781: 778: 776: 773: 771: 768: 766: 763: 761: 758: 756: 753: 751: 748: 746: 743: 741: 738: 736: 733: 731: 728: 726: 723: 721: 718: 716: 713: 711: 708: 706: 703: 701: 698: 696: 693: 691: 688: 686: 683: 681: 678: 676: 673: 671: 668: 666: 663: 661: 658: 656: 653: 651: 648: 646: 643: 641: 638: 636: 633: 631: 628: 626: 623: 621: 618: 617: 610: 609: 602: 599: 597: 594: 592: 589: 587: 584: 582: 579: 577: 576:Quantum chaos 574: 572: 569: 567: 564: 562: 559: 557: 554: 553: 547: 546: 538: 535: 533: 532:Transactional 530: 528: 525: 523: 522:Quantum logic 520: 518: 515: 513: 510: 504: 501: 500: 499: 496: 495: 494: 491: 489: 486: 484: 481: 479: 476: 474: 471: 469: 466: 465: 461: 456: 455: 447: 444: 442: 439: 437: 434: 432: 429: 427: 424: 423: 416: 415: 407: 404: 402: 399: 397: 394: 392: 389: 387: 384: 382: 379: 378: 374: 371: 370: 364: 363: 355: 352: 350: 347: 345: 342: 341: 335: 332: 331: 330: 327: 326: 322: 319: 317: 314: 312: 309: 307: 304: 302: 299: 297: 294: 292: 289: 287: 284: 282: 279: 277: 274: 273: 266: 265: 255: 252: 251: 250: 249:Wave function 247: 245: 242: 240: 237: 235: 232: 230: 229:Superposition 227: 225: 222: 220: 217: 215: 212: 210: 207: 205: 202: 200: 197: 195: 192: 190: 187: 186: 179: 178: 170: 167: 165: 162: 161: 157: 154: 152: 149: 147: 144: 143: 137: 136: 131: 128: 126: 123: 121: 118: 117: 116: 115: 111: 78: 72: 55: 52: 48: 40: 33: 32: 29: 26: 25: 21: 20: 15011:Quantum mind 14923:Franck–Hertz 14765:Klein–Gordon 14719:Formulations 14712:Formulations 14655: 14641:Interference 14631:Entanglement 14609:Ground state 14604:Energy level 14577:Fundamentals 14541:Introduction 14478: 14456: 14437: 14385: 14381: 14375: 14332: 14328: 14322: 14301: 14250: 14246: 14240: 14205: 14201: 14195: 14144: 14140: 14134: 14120: 14069: 14065: 14059: 14050: 14041: 14035: 13992: 13988: 13978: 13959: 13953: 13902: 13898: 13892: 13841: 13837: 13831: 13822: 13760: 13756: 13704: 13700: 13640: 13636: 13630: 13601: 13597: 13587: 13536: 13530: 13524: 13481: 13477: 13471: 13420: 13416: 13410: 13401: 13395: 13362: 13358: 13352: 13309: 13303: 13297: 13246: 13242: 13236: 13201: 13197: 13191: 13170: 13135: 13132:Jpn. J. Math 13131: 13125: 13082: 13078: 13072: 13029: 13025: 13019: 13008:. Retrieved 13004: 12994: 12983:. Retrieved 12979: 12969: 12958:. Retrieved 12954: 12944: 12933:. Retrieved 12929: 12919: 12908:. Retrieved 12904: 12894: 12883:. Retrieved 12879: 12869: 12834: 12830: 12820: 12799:cite journal 12770: 12759:. Retrieved 12755:the original 12691: 12687: 12641: 12637: 12631: 12588: 12584: 12574: 12531: 12527: 12521: 12478: 12474: 12468: 12409: 12405: 12395: 12336: 12332: 12322: 12271: 12267: 12261: 12210: 12206: 12200: 12167: 12163: 12157: 12122: 12118: 12112: 12077: 12073: 12063: 12020: 12016: 12010: 11967: 11963: 11910: 11906: 11900: 11857: 11853: 11847: 11806: 11800: 11794: 11768: 11762: 11719: 11715: 11709: 11674: 11668: 11658: 11639: 11633: 11606:. Retrieved 11601: 11591: 11572: 11566: 11550: 11546: 11541: 11516: 11510: 11504: 11477: 11471: 11461: 11410: 11406: 11400: 11377: 11328:(4): 50–55. 11325: 11321: 11315: 11304: 11291: 11238: 11234: 11224: 11189: 11185: 11147: 11143: 11137: 11084: 11080: 11070: 11019: 11015: 11009: 10958: 10954: 10948: 10897: 10893: 10887: 10860: 10856: 10749: 10308: 10305:Self-testing 10299: 10295: 10291: 10183: 10119: 9920: 9916: 9908: 9905:violates LO. 9012:Non-trivial 9006: 8874: 8729: 8326: 8293: 8083: 7630: 7345: 7126: 7114: 7065: 6764: 6685: 6063: 5974: 5862: 5256: 5177:B. Tsirelson 5173: 5159: 5155: 4957: 4940: 4649: 4157: 3741:Lucien Hardy 3738: 3718: 3708:Alain Aspect 3584: 3456: 3324: 3031: 2830: 2632: 2371: 2212: 2116: 1771: 1757: 1753: 1740: 1709: 1704: 1558: 1291: 958:Nathan Rosen 944:In the 1935 943: 923:foundational 908: 891: 885: 431:Klein–Gordon 367:Formulations 213: 204:Energy level 199:Entanglement 182:Fundamentals 169:Interference 120:Introduction 15269:EPR paradox 15049:Quantum bus 14918:Double-slit 14896:Experiments 14862:Many-worlds 14800:Schrödinger 14749:Phase space 14739:Schrödinger 14729:Interaction 14686:Uncertainty 14656:Nonlocality 14651:Measurement 14646:Decoherence 14636:Hamiltonian 13153:2433/173118 13138:: 147–183. 12371:10044/1/245 12017:New J. Phys 11274:10072/37941 6270:, the sets 5532:, the sets 940:EPR paradox 896:measurement 820:von Neumann 805:Schrödinger 581:EPR paradox 512:Many-worlds 446:Schrödinger 401:Schrödinger 396:Phase-space 386:Interaction 291:Double-slit 269:Experiments 244:Uncertainty 214:Nonlocality 209:Measurement 194:Decoherence 164:Hamiltonian 15307:Categories 15197:Extensions 15031:Technology 14877:Relational 14828:Copenhagen 14724:Heisenberg 14671:Tunnelling 14534:Background 14475:Duarte, FJ 14313:1804.05116 14215:1604.07199 13211:1506.07297 13010:2020-03-09 12985:2020-03-06 12960:2020-03-06 12935:2020-03-06 12910:2020-03-06 12885:2020-03-06 12782:2001.04383 12761:2019-12-05 12733:1703.08618 12419:2102.07637 12132:1707.06476 12087:1609.00672 11608:6 December 11519:(6): 476. 11094:1511.03189 11029:1511.03190 10968:1508.05949 10836:References 9795:such that 7929:such that 6644:, for all 6542:, for all 6240:such that 5821:, for all 5502:such that 5150:See also: 1731:See also: 1712:Niels Bohr 971:David Bohm 815:Sommerfeld 730:Heisenberg 725:Gutzwiller 665:de Broglie 613:Scientists 527:Relational 478:Copenhagen 381:Heisenberg 239:Tunnelling 140:Background 14903:Bell test 14758:Equations 14584:Born rule 14420:118407749 14395:1108.1480 14367:119300029 14342:1103.4105 14260:1412.0924 14187:119256543 14154:0802.0760 14079:0911.3427 14027:118309394 14002:1105.3195 13969:0911.3814 13945:119299119 13912:1210.1810 13770:1403.4621 13739:118677048 13714:0908.1496 13650:1210.3018 13611:0907.0372 13546:0905.2292 13387:122298281 13344:120333148 13322:CiteSeerX 13182:0810.0693 13162:121154563 13092:1008.1168 13039:1008.1142 12980:MyCQstate 12926:"MIP*=RE" 12710:119363229 12666:120680226 12598:1012.1513 12541:1011.5206 12488:1007.3043 12460:231924967 12444:0031-9007 12379:0031-9007 12220:1205.3118 12149:155100141 12125:: 70–91. 12030:1206.5115 11977:1110.3795 11892:119267461 11867:0808.0052 11778:0712.0921 11335:1411.4387 11283:0034-6861 11248:0806.0270 11216:0031-899X 11164:121145494 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7793:, 7790:x 7786:| 7782:b 7779:, 7776:a 7773:( 7770:P 7748:c 7744:Q 7737:) 7734:y 7731:, 7728:x 7724:| 7720:b 7717:, 7714:a 7711:( 7708:P 7684:. 7681:. 7678:. 7670:3 7666:Q 7657:2 7653:Q 7644:1 7640:Q 7616:) 7613:y 7610:, 7607:x 7603:| 7599:b 7596:, 7593:a 7590:( 7587:P 7565:B 7561:d 7557:= 7554:) 7549:B 7545:H 7541:( 7513:A 7509:d 7505:= 7502:) 7497:A 7493:H 7489:( 7461:B 7457:H 7448:A 7444:H 7423:) 7420:y 7417:, 7414:x 7410:| 7406:b 7403:, 7400:a 7397:( 7394:P 7372:B 7368:d 7364:, 7359:A 7355:d 7331:) 7327:| 7323:Y 7319:| 7314:| 7310:X 7306:| 7302:( 7292:/ 7288:1 7268:) 7265:y 7262:, 7259:x 7255:| 7251:b 7248:, 7245:a 7242:( 7239:P 7218:) 7215:y 7212:, 7209:x 7205:| 7201:b 7198:, 7195:a 7192:( 7189:P 7167:B 7163:H 7140:A 7136:H 7098:c 7094:Q 7081:Q 7045:Q 7020:c 7016:Q 6995:H 6973:c 6969:Q 6965:= 6956:Q 6931:c 6927:Q 6914:Q 6891:Q 6865:Q 6840:c 6836:Q 6823:Q 6798:c 6794:Q 6773:Q 6750:) 6747:y 6744:, 6741:x 6737:| 6733:b 6730:, 6727:a 6724:( 6721:P 6699:c 6695:Q 6682:. 6670:y 6667:, 6664:x 6661:, 6658:b 6655:, 6652:a 6632:0 6629:= 6626:] 6621:y 6616:b 6612:F 6608:, 6603:x 6598:a 6594:E 6590:[ 6580:. 6568:y 6565:, 6562:x 6559:, 6556:b 6553:, 6550:a 6523:| 6517:y 6512:b 6508:F 6502:x 6497:a 6493:E 6488:| 6478:= 6475:) 6472:y 6469:, 6466:x 6462:| 6458:b 6455:, 6452:a 6449:( 6446:P 6436:. 6422:I 6417:= 6412:y 6407:b 6403:F 6397:b 6389:, 6384:I 6379:= 6374:x 6369:a 6365:E 6359:a 6332:b 6328:} 6322:y 6317:b 6313:F 6309:{ 6306:, 6301:a 6297:} 6291:x 6286:a 6282:E 6278:{ 6258:y 6255:, 6252:x 6228:H 6222:H 6219:: 6214:y 6209:b 6205:F 6201:, 6198:H 6192:H 6189:: 6184:x 6179:a 6175:E 6154:H 6141:| 6120:H 6100:) 6097:y 6094:, 6091:x 6087:| 6083:b 6080:, 6077:a 6074:( 6071:P 6047:B 6043:H 6034:A 6030:H 6026:= 6023:H 6001:B 5997:H 5993:, 5988:A 5984:H 5960:) 5957:y 5954:, 5951:x 5947:| 5943:b 5940:, 5937:a 5934:( 5931:P 5911:Q 5891:Q 5871:Q 5859:. 5847:y 5844:, 5841:x 5838:, 5835:b 5832:, 5829:a 5802:| 5796:y 5791:b 5787:F 5778:x 5773:a 5769:E 5764:| 5754:= 5751:) 5748:y 5745:, 5742:x 5738:| 5734:b 5731:, 5728:a 5725:( 5722:P 5712:. 5698:B 5692:I 5686:= 5681:y 5676:b 5672:F 5666:b 5658:, 5653:A 5647:I 5641:= 5636:x 5631:a 5627:E 5621:a 5594:b 5590:} 5584:y 5579:b 5575:F 5571:{ 5568:, 5563:a 5559:} 5553:x 5548:a 5544:E 5540:{ 5520:y 5517:, 5514:x 5488:B 5484:H 5475:B 5471:H 5467:: 5462:y 5457:b 5453:F 5449:, 5444:A 5440:H 5431:A 5427:H 5423:: 5418:x 5413:a 5409:E 5386:B 5382:H 5373:A 5369:H 5355:| 5332:B 5328:H 5324:, 5319:A 5315:H 5294:) 5291:y 5288:, 5285:x 5281:| 5277:b 5274:, 5271:a 5268:( 5265:P 5237:. 5232:2 5227:2 5220:H 5217:S 5214:H 5211:C 5202:2 5197:2 5130:) 5125:B 5117:, 5112:A 5104:( 5084:b 5076:( 5064:a 5042:B 5029:( 5015:A 4990:y 4982:( 4970:x 4914:| 4893:} 4886:1 4883:| 4879:, 4872:0 4869:| 4865:{ 4839:0 4835:| 4811:0 4802:| 4775:1 4771:| 4747:1 4738:| 4717:} 4710:1 4707:| 4703:, 4696:0 4693:| 4689:{ 4659:| 4624:3 4619:2 4615:1 4606:= 4596:| 4566:} 4556:| 4552:, 4545:+ 4542:| 4538:{ 4506:| 4485:. 4474:0 4470:| 4450:, 4442:0 4435:| 4414:} 4404:| 4400:, 4393:+ 4390:| 4386:{ 4366:} 4359:1 4356:| 4352:, 4345:0 4342:| 4338:{ 4311:| 4290:} 4283:1 4280:| 4276:, 4269:0 4266:| 4262:{ 4242:} 4232:| 4228:, 4221:+ 4218:| 4214:{ 4194:} 4187:1 4184:| 4180:, 4173:0 4170:| 4166:{ 4143:) 4136:1 4133:| 4122:0 4119:| 4115:( 4108:2 4104:1 4098:= 4088:| 4066:) 4061:) 4049:1 4045:| 4041:+ 4033:+ 4030:1 4026:| 4021:( 4014:2 4010:1 4005:+ 3997:+ 3994:0 3990:| 3984:2 3978:( 3971:3 3967:1 3962:= 3958:) 3953:) 3944:1 3937:| 3933:+ 3925:1 3922:+ 3918:| 3913:( 3906:2 3902:1 3897:+ 3889:0 3886:+ 3882:| 3876:2 3870:( 3863:3 3859:1 3854:= 3850:) 3839:| 3835:+ 3825:| 3821:+ 3811:| 3806:( 3799:3 3795:1 3790:= 3780:| 3752:| 3686:2 3681:2 3658:H 3655:S 3652:H 3649:C 3644:S 3623:) 3620:y 3617:, 3614:x 3610:| 3606:b 3603:, 3600:a 3597:( 3594:P 3571:. 3568:) 3565:y 3562:, 3559:x 3555:| 3551:b 3548:, 3545:a 3542:( 3539:P 3534:b 3531:+ 3528:a 3524:) 3520:1 3514:( 3509:1 3506:, 3503:0 3500:= 3497:b 3494:, 3491:a 3480:) 3477:y 3474:, 3471:x 3468:( 3465:E 3443:, 3440:2 3434:) 3431:1 3428:, 3425:1 3422:( 3419:E 3413:) 3410:1 3407:, 3404:0 3401:( 3398:E 3395:+ 3392:) 3389:0 3386:, 3383:1 3380:( 3377:E 3374:+ 3371:) 3368:0 3365:, 3362:0 3359:( 3356:E 3347:H 3344:S 3341:H 3338:C 3333:S 3310:) 3307:y 3304:, 3301:x 3297:| 3293:b 3290:, 3287:a 3284:( 3281:P 3260:1 3257:, 3254:0 3233:y 3230:, 3227:x 3224:, 3221:b 3218:, 3215:a 3189:y 3186:, 3183:x 3180:, 3177:b 3174:, 3171:a 3166:) 3162:) 3159:y 3156:, 3153:x 3149:| 3145:b 3142:, 3139:a 3136:( 3133:P 3129:( 3107:) 3104:y 3101:, 3098:x 3094:| 3090:b 3087:, 3084:a 3081:( 3078:P 3058:y 3055:, 3052:x 3049:, 3046:b 3043:, 3040:a 3018:) 3013:B 3005:, 3002:y 2998:| 2994:b 2991:( 2986:B 2982:P 2978:) 2973:A 2965:, 2962:x 2958:| 2954:a 2951:( 2946:A 2942:P 2938:) 2933:B 2925:, 2920:A 2912:( 2896:B 2888:, 2883:A 2870:= 2867:) 2864:y 2861:, 2858:x 2854:| 2850:b 2847:, 2844:a 2841:( 2838:P 2814:B 2806:, 2801:A 2776:) 2771:B 2763:, 2758:A 2750:( 2719:B 2711:, 2706:A 2659:B 2651:, 2646:A 2616:B 2603:( 2589:A 2564:) 2561:y 2558:( 2539:x 2519:) 2516:b 2513:( 2494:a 2474:) 2469:B 2461:, 2458:y 2454:| 2450:b 2447:( 2442:B 2438:P 2429:( 2417:) 2412:A 2404:, 2401:x 2397:| 2393:a 2390:( 2385:A 2381:P 2358:) 2353:B 2345:, 2342:y 2338:| 2334:b 2331:( 2326:B 2322:P 2318:) 2313:A 2305:, 2302:x 2298:| 2294:a 2291:( 2286:A 2282:P 2278:= 2275:) 2270:B 2262:, 2257:A 2249:, 2246:y 2243:, 2240:x 2236:| 2232:b 2229:, 2226:a 2223:( 2220:P 2209:: 2197:y 2175:) 2169:( 2102:) 2099:y 2096:, 2093:x 2089:| 2085:b 2082:, 2079:a 2076:( 2073:P 2053:) 2050:y 2047:, 2044:x 2040:| 2036:b 2033:, 2030:a 2027:( 2024:P 2004:} 2001:y 1998:, 1995:x 1992:, 1989:b 1986:, 1983:a 1980:: 1977:) 1974:y 1971:, 1968:x 1964:| 1960:b 1957:, 1954:a 1951:( 1948:P 1945:{ 1925:b 1922:, 1919:a 1899:) 1896:y 1893:, 1890:x 1886:| 1882:b 1879:, 1876:a 1873:( 1870:P 1850:) 1847:b 1844:( 1825:a 1805:) 1802:y 1799:( 1780:x 1686:B 1675:| 1651:B 1640:| 1616:B 1605:| 1581:B 1570:| 1536:} 1531:B 1520:| 1515:, 1510:B 1502:+ 1499:| 1494:{ 1474:} 1469:A 1458:| 1453:, 1448:A 1440:+ 1437:| 1432:{ 1412:} 1407:B 1399:1 1396:| 1391:, 1386:B 1378:0 1375:| 1370:{ 1350:} 1345:A 1337:1 1334:| 1329:, 1324:A 1316:0 1313:| 1308:{ 1276:) 1268:1 1265:| 1254:0 1251:| 1246:( 1239:2 1235:1 1230:= 1220:| 1198:) 1192:B 1181:| 1174:A 1166:+ 1163:| 1153:B 1145:+ 1142:| 1135:A 1124:| 1118:( 1111:2 1107:1 1102:= 1098:) 1092:B 1084:0 1081:| 1074:A 1066:1 1063:| 1053:B 1045:1 1042:| 1035:A 1027:0 1024:| 1018:( 1011:2 1007:1 1002:= 993:B 990:A 982:| 875:e 868:t 861:v 89:| 79:H 73:= 63:| 56:t 53:d 49:d 41:i

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Knowledge

Quantum nonlocality

Index