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965:" it into one of its basis states with a given probability. Because of the entanglement, measurement of one qubit will "collapse" the other qubit to a state whose measurement will yield one of two possible values, where the value depends on which Bell's state the two qubits are in initially. Bell's states can be generalized to certain quantum states of multi-qubit systems, such as the
4324:
2399:. The mixed state implies that not all the information on this first qubit is known. Bell States are either symmetric or antisymmetric with respect to the subsystems. Bell states are maximally entangled in the sense that its reduced density operators are maximally mixed, the multipartite generalization of Bell states in this spirit is called the
2476:
particle is localized at a particular detector registering a "click" to indicate that a particle has been detected. Such devices can be constructed from, for example: mirrors, beam splitters, and wave plates – and are attractive from an experimental perspective because they are easy to use and have a high measurement
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After Alice applies the desired transformations to her qubit, she sends it to Bob. Bob then performs a measurement on the Bell state, which projects the entangled state onto one of the four two-qubit basis vectors, one of which will coincide with the original two bit message Alice was trying to send.
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was the first to prove that the measurement correlations in the Bell State are stronger than could ever exist between classical systems. This hints that quantum mechanics allows information processing beyond what is possible with classical mechanics. In addition, the Bell states form an orthonormal
1007:
The qubit held by Alice (subscript "A") can be in a superposition of 0 and 1. If Alice measured her qubit in the standard basis, the outcome would be either 0 or 1, each with probability 1/2; if Bob (subscript "B") also measured his qubit, the outcome would be the same as for Alice. Thus, Alice and
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allows two individuals to communicate two bits of classical information by only sending a single qubit. The basis of this phenomenon is the entangled states or Bell states of a two qubit system. In this example, Alice and Bob are very far from each other, and have each been given one qubit of the
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Experiments that utilize so-called "linear evolution, local measurement" techniques cannot realize a complete Bell state measurement. Linear evolution means that the detection apparatus acts on each particle independent of the state or evolution of the other, and local measurement means that each
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is the transfer of a quantum state over a distance. It is facilitated by entanglement between A, the giver, and B, the receiver of this quantum state. This process has become a fundamental research topic for quantum communication and computing. More recently, scientists have been testing its
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For entanglement in a single qubit variable, only three distinct classes out of four Bell states are distinguishable using such linear optical techniques. This means two Bell states cannot be distinguished from each other, limiting the efficiency of quantum communication protocols such as
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is the use of quantum mechanical properties in order to encode and send information safely. The theory behind this process is the fact that it is impossible to measure a quantum state of a system without disturbing the system. This can be used to detect eavesdropping within a system.
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on qubit A, a measurement can be made in the computational basis. The CNOT gate performs the act of un-entangling the two previously entangled qubits. This allows the information to be converted from quantum information to a measurement of classical information.
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states, allows the experimenter to trace over one variable and achieve a complete Bell state measurement in the other. Leveraging so-called hyper-entangled systems thus has an advantage for teleportation. It also has advantages for other protocols such as
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basis and can therefore be defined with an appropriate measurement. Because Bell states are entangled states, information on the entire system may be known, while withholding information on the individual subsystems. For example, the Bell state is a
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2472:. The result of a Bell state measurement is used by one's co-conspirator to reconstruct the original state of a teleported particle from half of an entangled pair (the "quantum channel") that was previously shared between the two ends.
2462:, states that any measurement can be moved to the end of the circuit. The second principle, the principle of implicit measurement, states that at the end of a quantum circuit, measurement can be assumed for any unterminated wires.
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Alice and Bob share an EPR pair and each took one qubit before they became separated. Alice must deliver a qubit of information to Bob, but she does not know the state of this qubit and can only send classical information to Bob.
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This perfect correlation at a distance is special: maybe the two particles "agreed" in advance, when the pair was created (before the qubits were separated), which outcome they would show in case of a measurement.
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1991:. This will then act as a control input to the CNOT gate, which only inverts the target (the second qubit) when the control (the first qubit) is 1. Thus, the CNOT gate transforms the second qubit as follows
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be made perfect by the use of any "pre-agreement" stored in some hidden variables – but that quantum mechanics predicts perfect correlations. In a more refined formulation known as the
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The steps below show the necessary quantum gate transformations, and resulting Bell states, that Alice needs to apply to her qubit for each possible two bit message she desires to send to Bob.
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4351:. It enables two parties to produce a shared random secret key that can be used to encrypt messages. Its private key is created between the two parties through a public channel.
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Bob would each seemingly have random outcome. Through communication they would discover that, although their outcomes separately seemed random, these were perfectly correlated.
4259:{\displaystyle 11:-iY=XZ={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}\longrightarrow |\psi \rangle ={\frac {|01\rangle -|10\rangle }{\sqrt {2}}}\equiv |{\Psi ^{-}}\rangle }
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3957:{\displaystyle 01:Z={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}\longrightarrow |\psi \rangle ={\frac {|00\rangle -|11\rangle }{\sqrt {2}}}\equiv |{\Phi ^{-}}\rangle }
1906:
1840:
4726:{\displaystyle ={\frac {1}{2{\sqrt {2}}}}((|+\rangle _{A}+|-\rangle _{A})(|+\rangle _{B}+|-\rangle _{B})-(|+\rangle _{A}-|-\rangle _{A})(|+\rangle _{B}-|-\rangle _{B}))}
4099:{\displaystyle 10:X={\begin{bmatrix}0&1\\1&0\end{bmatrix}}\longrightarrow |\psi \rangle ={\frac {|01\rangle +|10\rangle }{\sqrt {2}}}\equiv |{\Psi ^{+}}\rangle }
3812:{\displaystyle 00:I={\begin{bmatrix}1&0\\0&1\end{bmatrix}}\longrightarrow |\psi \rangle ={\frac {|00\rangle +|11\rangle }{\sqrt {2}}}\equiv |{\Phi ^{+}}\rangle }
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Independent measurements made on two qubits that are entangled in Bell states positively correlate perfectly if each qubit is measured in the relevant basis. For the
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1744:{\displaystyle |\Psi ^{-}\rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}{\big )}\qquad (4)}
1588:{\displaystyle |\Psi ^{+}\rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |1\rangle _{B}+|1\rangle _{A}\otimes |0\rangle _{B}{\big )}\qquad (3)}
1432:{\displaystyle |\Phi ^{-}\rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |0\rangle _{B}-|1\rangle _{A}\otimes |1\rangle _{B}{\big )}\qquad (2)}
1276:{\displaystyle |\Phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B}{\big )}\qquad (1)}
4428:
1994:
1055:(a sort of common-sense formulation of how information is conveyed), but certain systems permitted in quantum mechanics can attain values as high as
1031:
paper", there is something missing in the description of the qubit pair given above – namely this "agreement", called more formally a
1915:
36:
3304:
2430:
Quantum circuit that performs Bell decoding. Bell states are sometimes called EPR pairs. Notice that the circuit that decodes the Bell state
372:
1004:. They are in a superposition of 0 and 1 – a linear combination of the two states. Their entanglement means the following:
4903:{\displaystyle ={\frac {1}{2{\sqrt {2}}}}(|++\rangle +|+-\rangle +|-+\rangle +|--\rangle -|++\rangle +|+-\rangle +|-+\rangle -|--\rangle )}
2805:
correlations can be understood by measuring both qubits in the same basis and observing perfectly anti-correlated results. More generally,
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Given Alice's measurements, Bob performs one of four operations on his half of the EPR pair and recovers the original quantum state.
1051:, it is shown that a certain correlation measure cannot exceed the value 2 if one assumes that physics respects the constraints of
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applications in information transfer through optical fibers. The process of quantum teleportation is defined as the following:
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The result of a measurement of a single qubit in a Bell state is indeterminate, but upon measuring the first qubit in the
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Quantum cryptography can be considered a state of entanglement between two multi-dimensional systems, also known as two-
5145:
5024:
917:. This normalization implies that the overall probability of the particle being in one of the mentioned states is 1:
119:
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208:
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2276:{\displaystyle |\beta (x,y)\rangle =\left({\frac {|0,y\rangle +(-1)^{x}|1,{\bar {y}}\rangle }{\sqrt {2}}}\right),}
2636:
state, this means selecting the same basis for both qubits. If an experimenter chose to measure both qubits in a
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In this example, Alice is trying to communicate two bits of classical information, one of four two bit strings:
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arguments that these correlations (the one for the 0, 1 basis and the one for the +, â basis) cannot
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Bell state using the same basis, the qubits would appear positively correlated when measuring in the
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2346:-basis, the result of measuring the second qubit is guaranteed to yield the same value (for the
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1082:. Thus, quantum theory violates the Bell inequality and the idea of local "hidden variables".
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5158:"Distinguishability of Hyper-Entangled Bell States by Linear Evolution and Local Measurement"
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Alice measures her qubits, obtaining one of four results, and sends this information to Bob.
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2488:. If a Bell state is measured from this ambiguous class, the teleportation event fails.
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and form a maximally entangled basis, known as the Bell basis, of the four-dimensional
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Understanding of Bell's states is useful in analysis of quantum communication, such as
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1814:(see picture). As an example, the pictured quantum circuit takes the two qubit input
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729:
724:
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27:
2148:). More generally, the circuit transforms the input in accordance with the equation
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4499:{\displaystyle |\Phi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(|00\rangle -|11\rangle )}
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Entangling particles in multiple qubit variables, such as (for photonic systems)
2064:{\displaystyle {\frac {(|00\rangle +|11\rangle )}{\sqrt {2}}}=|\Phi ^{+}\rangle }
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Although there are many possible ways to create entangled Bell states through
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1984:{\displaystyle (|0\rangle |0\rangle +|1\rangle |0\rangle ) \over {\sqrt {2}}}
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101:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
3609:
gate to her qubit; and finally, if Alice wanted to send the two bit message
5231:
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5195:
5131:
3642:
3361:{\displaystyle |\psi \rangle ={\frac {|00\rangle +|11\rangle }{\sqrt {2}}}}
1036:
1024:
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759:
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Quantum measurement obeys two key principles. The first, the principle of
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Bell states). This implies that the measurement outcomes are correlated.
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985:
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1806:, the simplest takes a computational basis as the input, and contains a
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that determines which of the four Bell states the two qubits are in.
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basis, and partially (probabilistically) correlated in other bases.
654:
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transformations locally, transforming the initial entangled state
4971:{\displaystyle ={\frac {1}{\sqrt {2}}}(|+-\rangle +|-+\rangle )}
2434:
to the circuit that encodes, or creates, Bell states (described
2920:
Relationship between the correlated bases of two qubits in the
2137:{\displaystyle |00\rangle ,|01\rangle ,|10\rangle ,|11\rangle }
903:
2504:, in which hyper-entanglement increases the channel capacity.
2435:
4355:
2420:
1001:
5171:"Deterministic Quantum Teleportation through Fiber Channels"
1754:
1117:
are designated as "Bell states". They are known as the four
961:. Due to this superposition, measurement of the qubit will "
2465:
The following are applications of Bell state measurements:
2395:, but the reduced density operator of the first qubit is a
3641:, she would do nothing to her qubit. Alice performs these
2446:
in the Bell basis can be seen in quantum computing. If a
2913:, and observing perfectly positively correlated results.
1090:
Four specific two-qubit states with the maximal value of
4323:
2840:
can be understood by measuring the first qubit in basis
4320:
The following quantum circuit describes teleportation:
4146:
3989:
3844:
3702:
2419:: It is a joint quantum-mechanical measurement of two
996:
The Bell states are four specific maximally entangled
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3554:, she would apply a NOT gate; if she wanted to send
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5086:Zaman, Fakhar; Jeong, Youngmin (2 October 2018).
5310:
3522:to her qubit. Similarly, if Alice wants to send
957:. Entanglement is a basis-independent result of
910:. The Bell's states are a form of entangled and
4295:It is performed step by step as the following:
3470:. If Alice chooses to send the two bit message
2527:variables, one can distinguish between at most
2468:Bell state measurement is the crucial step in
5226:
5047:"A Complete Basis of Generalized Bell States"
2593:Bell states using linear optical techniques.
1726:
1638:
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950:{\displaystyle \langle \Phi |\Phi \rangle =1}
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2703:
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2450:is applied to qubits A and B, followed by a
2366:Bell states) or the opposite value (for the
2337:
2255:
2207:
2180:
2144:, the circuit outputs the four Bell states (
2131:
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2103:
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2012:
1967:
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5236:Quantum computation and quantum information
5017:Quantum Computation and Quantum Information
4306:Alice then sends the first qubit through a
5297:On the Einstein Podolsky and Rosen paradox
5085:
2596:
2401:absolutely maximally entangled (AME) state
1842:and transforms it to the first Bell state
874:
860:
5204:
5194:
5121:
5111:
5070:
2760:{\displaystyle \{|+\rangle ,|-\rangle \}}
2712:{\displaystyle \{|0\rangle ,|1\rangle \}}
2406:
1880:Explicitly, the Hadamard gate transforms
1755:Creating Bell states via quantum circuits
1119:maximally entangled two-qubit Bell states
984:prevents this behavior from transmitting
4322:
4273:
2915:
2425:
1758:
906:that represent the simplest examples of
5014:
4331:
4327:Quantum circuit for teleporting a qubit
5311:
2507:In general, for hyper-entanglement in
2074:For the four basic two-qubit inputs,
1763:Quantum circuit to create Bell state
5271:An introduction to quantum computing
5088:"Counterfactual Bell-State Analysis"
5044:
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5038:
5036:
5010:
5008:
5006:
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5002:
5000:
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5168:
3502:, she would perform the phase flip
1873:{\displaystyle |\Phi ^{+}\rangle .}
13:
4438:
4243:
4083:
3941:
3796:
3673:into one of the four Bell states.
3233:{\displaystyle |\Psi ^{-}\rangle }
3218:
3161:{\displaystyle |\Psi ^{+}\rangle }
3146:
3089:{\displaystyle |\Phi ^{-}\rangle }
3074:
3023:{\displaystyle |\Phi ^{+}\rangle }
3008:
2948:{\displaystyle |\Phi ^{-}\rangle }
2933:
2833:{\displaystyle |\Psi ^{+}\rangle }
2818:
2798:{\displaystyle |\Psi ^{+}\rangle }
2783:
2664:{\displaystyle |\Phi ^{-}\rangle }
2649:
2629:{\displaystyle |\Phi ^{+}\rangle }
2614:
2373:
2353:
2049:
1855:
1791:{\displaystyle |\Phi ^{+}\rangle }
1776:
1609:
1453:
1297:
1141:
935:
927:
406:Sum-over-histories (path integral)
92:
66:
22:Part of a series of articles about
14:
5335:
5033:
4991:
4299:Alice sends her qubits through a
43:
988:faster than the speed of light.
3431:{\displaystyle '00','01','10',}
3285:
2145:
1731:
1575:
1419:
1263:
1035:. In his famous paper of 1964,
5169:Huo, Meiru (19 October 2018).
5162:
5150:
5146:"Embedded Bell State Analysis"
5138:
5079:
5045:Sych, Denis (7 January 2009).
5019:. Cambridge University Press.
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4358:(quantum digit) entanglement.
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3666:{\displaystyle |\psi \rangle }
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2719:basis, anti-correlated in the
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1053:local "hidden-variable" theory
991:
969:for three or more subsystems.
931:
556:Relativistic quantum mechanics
88:
81:
62:
1:
5301:Bell System Technical Journal
5072:10.1088/1367-2630/11/1/013006
4984:
2906:{\displaystyle b_{2}=X.b_{1}}
1085:
596:Quantum statistical mechanics
2867:, the second qubit in basis
2495:and a two-element subset of
1110:{\displaystyle 2{\sqrt {2}}}
1075:{\displaystyle 2{\sqrt {2}}}
16:Quantum states of two qubits
7:
5319:Quantum information science
4361:
2417:quantum information science
2415:is an important concept in
1901:{\displaystyle |00\rangle }
1835:{\displaystyle |00\rangle }
888:quantum information science
566:Quantum information science
10:
5340:
5240:Cambridge University Press
5113:10.1038/s41598-018-32928-8
4277:
2307:{\displaystyle {\bar {y}}}
5015:Nielsen, Michael (2010).
3273:{\displaystyle X.Z.b_{1}}
2559:{\displaystyle 2^{n+1}-1}
2338:Properties of Bell states
5156:Pisenti, Gaebler, Lynn.
5075:– via IOP Science.
4413:
4349:quantum key distribution
4343:The most common form of
2497:orbital angular momentum
982:no-communication theorem
601:Quantum machine learning
354:Wheeler's delayed-choice
5275:Oxford University Press
3195:{\displaystyle X.b_{1}}
3123:{\displaystyle Z.b_{1}}
2597:Bell state correlations
311:LeggettâGarg inequality
5196:10.1126/sciadv.aas9401
5051:New Journal of Physics
4972:
4904:
4727:
4500:
4328:
4260:
4100:
3958:
3813:
3667:
3635:
3603:
3586:, she would apply the
3580:
3548:
3516:
3496:
3464:
3432:
3362:
3274:
3234:
3196:
3162:
3124:
3090:
3052:
3024:
2986:
2956:
2949:
2907:
2861:
2834:
2799:
2761:
2713:
2665:
2630:
2587:
2560:
2521:
2439:
2407:Bell state measurement
2380:
2360:
2328:
2308:
2277:
2138:
2065:
1985:
1902:
1874:
1836:
1799:
1792:
1745:
1589:
1433:
1277:
1111:
1076:
1027:in their famous 1935 "
951:
102:
4973:
4905:
4728:
4501:
4393:Quantum teleportation
4368:Bell test experiments
4326:
4285:Quantum teleportation
4280:Quantum teleportation
4274:Quantum teleportation
4261:
4101:
3959:
3814:
3668:
3636:
3604:
3581:
3549:
3517:
3497:
3465:
3433:
3363:
3275:
3235:
3197:
3163:
3125:
3091:
3053:
3051:{\displaystyle b_{1}}
3025:
2987:
2985:{\displaystyle b_{2}}
2950:
2919:
2908:
2862:
2860:{\displaystyle b_{1}}
2835:
2800:
2762:
2714:
2666:
2631:
2588:
2586:{\displaystyle 4^{n}}
2561:
2522:
2470:quantum teleportation
2442:A helpful example of
2429:
2381:
2379:{\displaystyle \Psi }
2361:
2359:{\displaystyle \Phi }
2329:
2309:
2278:
2139:
2066:
1986:
1903:
1875:
1837:
1793:
1762:
1746:
1590:
1434:
1278:
1112:
1077:
978:quantum teleportation
952:
296:Elitzur–Vaidman
286:Davisson–Germer
103:
4913:
4736:
4509:
4429:
4398:Quantum cryptography
4345:quantum cryptography
4337:Quantum cryptography
4332:Quantum cryptography
4111:
3969:
3824:
3682:
3649:
3634:{\displaystyle '00'}
3613:
3590:
3579:{\displaystyle '11'}
3558:
3547:{\displaystyle '10'}
3526:
3506:
3495:{\displaystyle '01'}
3474:
3463:{\displaystyle '11'}
3442:
3375:
3305:
3245:
3209:
3173:
3137:
3101:
3065:
3035:
2999:
2969:
2924:
2871:
2844:
2809:
2774:
2723:
2675:
2640:
2605:
2570:
2531:
2511:
2460:deferred measurement
2370:
2350:
2318:
2289:
2154:
2078:
1995:
1916:
1884:
1846:
1818:
1767:
1600:
1444:
1288:
1132:
1094:
1059:
1049:BellâCHSH inequality
921:
908:quantum entanglement
561:Quantum field theory
473:Consistent histories
110:Schrödinger equation
37:
5228:Nielsen, Michael A.
5187:2018SciA....4.9401H
5144:Kwiat, Weinfurter.
5104:2018NatSR...814641Z
5063:2009NJPh...11a3006S
4408:Bell diagonal state
2444:quantum measurement
2314:is the negation of
349:Stern–Gerlach
146:Classical mechanics
5092:Scientific Reports
4968:
4900:
4723:
4496:
4329:
4256:
4174:
4096:
4014:
3954:
3872:
3809:
3727:
3663:
3631:
3602:{\displaystyle iY}
3599:
3576:
3544:
3512:
3492:
3460:
3428:
3358:
3270:
3230:
3192:
3158:
3120:
3086:
3048:
3020:
2982:
2957:
2945:
2903:
2857:
2830:
2795:
2757:
2709:
2661:
2626:
2583:
2556:
2517:
2440:
2376:
2356:
2324:
2304:
2273:
2134:
2061:
1973:
1898:
1870:
1832:
1800:
1788:
1741:
1585:
1429:
1273:
1107:
1072:
1041:probability theory
947:
537:Von NeumannâWigner
517:Objective-collapse
316:Mach–Zehnder
306:Leggett inequality
301:Franck–Hertz
151:Old quantum theory
98:
5284:978-0-19-857049-3
5263:Laflamme, Raymond
5249:978-0-521-63503-5
4929:
4928:
4759:
4756:
4532:
4529:
4463:
4462:
4388:Superdense coding
4373:Bell's inequality
4231:
4230:
4071:
4070:
3929:
3928:
3784:
3783:
3515:{\displaystyle Z}
3356:
3355:
3299:entangled state.
3296:Superdense coding
3291:Superdense coding
3283:
3282:
2520:{\displaystyle n}
2502:superdense coding
2327:{\displaystyle y}
2301:
2264:
2263:
2252:
2038:
2037:
1981:
1979:
1634:
1633:
1478:
1477:
1322:
1321:
1166:
1165:
1105:
1070:
1039:showed by simple
1037:John S. Bell
1015:Hence, following
974:superdense coding
884:
883:
591:Scattering theory
571:Quantum computing
344:Schrödinger's cat
276:Bell's inequality
84:
59:
28:Quantum mechanics
5331:
5304:
5287:
5252:
5232:Chuang, Isaac L.
5219:
5218:
5208:
5198:
5181:(10): eaas9401.
5175:Science Advances
5166:
5160:
5154:
5148:
5142:
5136:
5135:
5125:
5115:
5083:
5077:
5076:
5074:
5042:
5031:
5030:
5012:
4978:
4977:
4975:
4974:
4969:
4955:
4938:
4930:
4924:
4920:
4909:
4907:
4906:
4901:
4887:
4870:
4853:
4836:
4819:
4802:
4785:
4768:
4760:
4758:
4757:
4752:
4743:
4732:
4730:
4729:
4724:
4716:
4715:
4703:
4695:
4694:
4682:
4671:
4670:
4658:
4650:
4649:
4637:
4623:
4622:
4610:
4602:
4601:
4589:
4578:
4577:
4565:
4557:
4556:
4544:
4533:
4531:
4530:
4525:
4516:
4505:
4503:
4502:
4497:
4486:
4472:
4464:
4458:
4454:
4446:
4445:
4436:
4424:
4403:Quantum circuits
4265:
4263:
4262:
4257:
4252:
4251:
4250:
4240:
4232:
4226:
4225:
4218:
4204:
4198:
4187:
4179:
4178:
4105:
4103:
4102:
4097:
4092:
4091:
4090:
4080:
4072:
4066:
4065:
4058:
4044:
4038:
4027:
4019:
4018:
3963:
3961:
3960:
3955:
3950:
3949:
3948:
3938:
3930:
3924:
3923:
3916:
3902:
3896:
3885:
3877:
3876:
3818:
3816:
3815:
3810:
3805:
3804:
3803:
3793:
3785:
3779:
3778:
3771:
3757:
3751:
3740:
3732:
3731:
3672:
3670:
3669:
3664:
3656:
3640:
3638:
3637:
3632:
3630:
3622:
3608:
3606:
3605:
3600:
3585:
3583:
3582:
3577:
3575:
3567:
3553:
3551:
3550:
3545:
3543:
3535:
3521:
3519:
3518:
3513:
3501:
3499:
3498:
3493:
3491:
3483:
3469:
3467:
3466:
3461:
3459:
3451:
3437:
3435:
3434:
3429:
3424:
3416:
3408:
3400:
3392:
3384:
3367:
3365:
3364:
3359:
3357:
3351:
3350:
3343:
3329:
3323:
3312:
3279:
3277:
3276:
3271:
3269:
3268:
3239:
3237:
3236:
3231:
3226:
3225:
3216:
3201:
3199:
3198:
3193:
3191:
3190:
3167:
3165:
3164:
3159:
3154:
3153:
3144:
3129:
3127:
3126:
3121:
3119:
3118:
3095:
3093:
3092:
3087:
3082:
3081:
3072:
3057:
3055:
3054:
3049:
3047:
3046:
3029:
3027:
3026:
3021:
3016:
3015:
3006:
2991:
2989:
2988:
2983:
2981:
2980:
2959:
2958:
2954:
2952:
2951:
2946:
2941:
2940:
2931:
2912:
2910:
2909:
2904:
2902:
2901:
2883:
2882:
2866:
2864:
2863:
2858:
2856:
2855:
2839:
2837:
2836:
2831:
2826:
2825:
2816:
2804:
2802:
2801:
2796:
2791:
2790:
2781:
2766:
2764:
2763:
2758:
2747:
2733:
2718:
2716:
2715:
2710:
2699:
2685:
2670:
2668:
2667:
2662:
2657:
2656:
2647:
2635:
2633:
2632:
2627:
2622:
2621:
2612:
2592:
2590:
2589:
2584:
2582:
2581:
2565:
2563:
2562:
2557:
2549:
2548:
2526:
2524:
2523:
2518:
2413:Bell measurement
2385:
2383:
2382:
2377:
2365:
2363:
2362:
2357:
2333:
2331:
2330:
2325:
2313:
2311:
2310:
2305:
2303:
2302:
2294:
2282:
2280:
2279:
2274:
2269:
2265:
2259:
2258:
2254:
2253:
2245:
2236:
2231:
2230:
2197:
2191:
2161:
2143:
2141:
2140:
2135:
2127:
2113:
2099:
2085:
2070:
2068:
2067:
2062:
2057:
2056:
2047:
2039:
2033:
2032:
2022:
2008:
1999:
1990:
1988:
1987:
1982:
1980:
1975:
1963:
1952:
1938:
1927:
1918:
1907:
1905:
1904:
1899:
1891:
1879:
1877:
1876:
1871:
1863:
1862:
1853:
1841:
1839:
1838:
1833:
1825:
1804:quantum circuits
1797:
1795:
1794:
1789:
1784:
1783:
1774:
1750:
1748:
1747:
1742:
1730:
1729:
1723:
1722:
1710:
1702:
1701:
1689:
1681:
1680:
1668:
1660:
1659:
1647:
1642:
1641:
1635:
1629:
1625:
1617:
1616:
1607:
1594:
1592:
1591:
1586:
1574:
1573:
1567:
1566:
1554:
1546:
1545:
1533:
1525:
1524:
1512:
1504:
1503:
1491:
1486:
1485:
1479:
1473:
1469:
1461:
1460:
1451:
1438:
1436:
1435:
1430:
1418:
1417:
1411:
1410:
1398:
1390:
1389:
1377:
1369:
1368:
1356:
1348:
1347:
1335:
1330:
1329:
1323:
1317:
1313:
1305:
1304:
1295:
1282:
1280:
1279:
1274:
1262:
1261:
1255:
1254:
1242:
1234:
1233:
1221:
1213:
1212:
1200:
1192:
1191:
1179:
1174:
1173:
1167:
1161:
1157:
1149:
1148:
1139:
1125:for two qubits:
1116:
1114:
1113:
1108:
1106:
1101:
1081:
1079:
1078:
1073:
1071:
1066:
956:
954:
953:
948:
934:
876:
869:
862:
503:Superdeterminism
156:Braâket notation
107:
105:
104:
99:
91:
86:
85:
77:
65:
60:
58:
47:
19:
18:
5339:
5338:
5334:
5333:
5332:
5330:
5329:
5328:
5309:
5308:
5295:
5285:
5261:Kaye, Phillip;
5250:
5223:
5222:
5167:
5163:
5155:
5151:
5143:
5139:
5084:
5080:
5043:
5034:
5027:
5013:
4992:
4987:
4982:
4981:
4951:
4934:
4919:
4914:
4911:
4910:
4883:
4866:
4849:
4832:
4815:
4798:
4781:
4764:
4751:
4747:
4742:
4737:
4734:
4733:
4711:
4707:
4699:
4690:
4686:
4678:
4666:
4662:
4654:
4645:
4641:
4633:
4618:
4614:
4606:
4597:
4593:
4585:
4573:
4569:
4561:
4552:
4548:
4540:
4524:
4520:
4515:
4510:
4507:
4506:
4482:
4468:
4453:
4441:
4437:
4432:
4430:
4427:
4426:
4425:
4421:
4416:
4364:
4334:
4282:
4276:
4246:
4242:
4241:
4236:
4214:
4200:
4199:
4197:
4183:
4173:
4172:
4167:
4161:
4160:
4152:
4142:
4141:
4112:
4109:
4108:
4086:
4082:
4081:
4076:
4054:
4040:
4039:
4037:
4023:
4013:
4012:
4007:
4001:
4000:
3995:
3985:
3984:
3970:
3967:
3966:
3944:
3940:
3939:
3934:
3912:
3898:
3897:
3895:
3881:
3871:
3870:
3862:
3856:
3855:
3850:
3840:
3839:
3825:
3822:
3821:
3799:
3795:
3794:
3789:
3767:
3753:
3752:
3750:
3736:
3726:
3725:
3720:
3714:
3713:
3708:
3698:
3697:
3683:
3680:
3679:
3652:
3650:
3647:
3646:
3623:
3616:
3614:
3611:
3610:
3591:
3588:
3587:
3568:
3561:
3559:
3556:
3555:
3536:
3529:
3527:
3524:
3523:
3507:
3504:
3503:
3484:
3477:
3475:
3472:
3471:
3452:
3445:
3443:
3440:
3439:
3417:
3409:
3401:
3393:
3385:
3378:
3376:
3373:
3372:
3339:
3325:
3324:
3322:
3308:
3306:
3303:
3302:
3293:
3288:
3264:
3260:
3246:
3243:
3242:
3221:
3217:
3212:
3210:
3207:
3206:
3186:
3182:
3174:
3171:
3170:
3149:
3145:
3140:
3138:
3135:
3134:
3114:
3110:
3102:
3099:
3098:
3077:
3073:
3068:
3066:
3063:
3062:
3042:
3038:
3036:
3033:
3032:
3011:
3007:
3002:
3000:
2997:
2996:
2976:
2972:
2970:
2967:
2966:
2936:
2932:
2927:
2925:
2922:
2921:
2897:
2893:
2878:
2874:
2872:
2869:
2868:
2851:
2847:
2845:
2842:
2841:
2821:
2817:
2812:
2810:
2807:
2806:
2786:
2782:
2777:
2775:
2772:
2771:
2743:
2729:
2724:
2721:
2720:
2695:
2681:
2676:
2673:
2672:
2652:
2648:
2643:
2641:
2638:
2637:
2617:
2613:
2608:
2606:
2603:
2602:
2599:
2577:
2573:
2571:
2568:
2567:
2566:classes out of
2538:
2534:
2532:
2529:
2528:
2512:
2509:
2508:
2409:
2371:
2368:
2367:
2351:
2348:
2347:
2340:
2319:
2316:
2315:
2293:
2292:
2290:
2287:
2286:
2244:
2243:
2232:
2226:
2222:
2193:
2192:
2190:
2186:
2157:
2155:
2152:
2151:
2123:
2109:
2095:
2081:
2079:
2076:
2075:
2052:
2048:
2043:
2018:
2004:
2000:
1998:
1996:
1993:
1992:
1974:
1959:
1948:
1934:
1923:
1917:
1914:
1913:
1887:
1885:
1882:
1881:
1858:
1854:
1849:
1847:
1844:
1843:
1821:
1819:
1816:
1815:
1779:
1775:
1770:
1768:
1765:
1764:
1757:
1725:
1724:
1718:
1714:
1706:
1697:
1693:
1685:
1676:
1672:
1664:
1655:
1651:
1643:
1637:
1636:
1624:
1612:
1608:
1603:
1601:
1598:
1597:
1569:
1568:
1562:
1558:
1550:
1541:
1537:
1529:
1520:
1516:
1508:
1499:
1495:
1487:
1481:
1480:
1468:
1456:
1452:
1447:
1445:
1442:
1441:
1413:
1412:
1406:
1402:
1394:
1385:
1381:
1373:
1364:
1360:
1352:
1343:
1339:
1331:
1325:
1324:
1312:
1300:
1296:
1291:
1289:
1286:
1285:
1257:
1256:
1250:
1246:
1238:
1229:
1225:
1217:
1208:
1204:
1196:
1187:
1183:
1175:
1169:
1168:
1156:
1144:
1140:
1135:
1133:
1130:
1129:
1100:
1095:
1092:
1091:
1088:
1065:
1060:
1057:
1056:
1033:hidden variable
994:
930:
922:
919:
918:
880:
851:
850:
849:
614:
606:
605:
551:
550:Advanced topics
543:
542:
541:
493:Hidden-variable
483:de BroglieâBohm
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:
5337:
5327:
5326:
5324:Quantum states
5321:
5307:
5306:
5293:
5283:
5267:Mosca, Michele
5258:
5248:
5221:
5220:
5161:
5149:
5137:
5078:
5032:
5025:
4989:
4988:
4986:
4983:
4980:
4979:
4967:
4964:
4961:
4958:
4954:
4950:
4947:
4944:
4941:
4937:
4933:
4927:
4923:
4918:
4899:
4896:
4893:
4890:
4886:
4882:
4879:
4876:
4873:
4869:
4865:
4862:
4859:
4856:
4852:
4848:
4845:
4842:
4839:
4835:
4831:
4828:
4825:
4822:
4818:
4814:
4811:
4808:
4805:
4801:
4797:
4794:
4791:
4788:
4784:
4780:
4777:
4774:
4771:
4767:
4763:
4755:
4750:
4746:
4741:
4722:
4719:
4714:
4710:
4706:
4702:
4698:
4693:
4689:
4685:
4681:
4677:
4674:
4669:
4665:
4661:
4657:
4653:
4648:
4644:
4640:
4636:
4632:
4629:
4626:
4621:
4617:
4613:
4609:
4605:
4600:
4596:
4592:
4588:
4584:
4581:
4576:
4572:
4568:
4564:
4560:
4555:
4551:
4547:
4543:
4539:
4536:
4528:
4523:
4519:
4514:
4495:
4492:
4489:
4485:
4481:
4478:
4475:
4471:
4467:
4461:
4457:
4452:
4449:
4444:
4440:
4435:
4418:
4417:
4415:
4412:
4411:
4410:
4405:
4400:
4395:
4390:
4385:
4380:
4375:
4370:
4363:
4360:
4333:
4330:
4318:
4317:
4314:
4311:
4304:
4278:Main article:
4275:
4272:
4255:
4249:
4245:
4239:
4235:
4229:
4224:
4221:
4217:
4213:
4210:
4207:
4203:
4196:
4193:
4190:
4186:
4182:
4177:
4171:
4168:
4166:
4163:
4162:
4159:
4156:
4153:
4151:
4148:
4147:
4145:
4140:
4137:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4095:
4089:
4085:
4079:
4075:
4069:
4064:
4061:
4057:
4053:
4050:
4047:
4043:
4036:
4033:
4030:
4026:
4022:
4017:
4011:
4008:
4006:
4003:
4002:
3999:
3996:
3994:
3991:
3990:
3988:
3983:
3980:
3977:
3974:
3953:
3947:
3943:
3937:
3933:
3927:
3922:
3919:
3915:
3911:
3908:
3905:
3901:
3894:
3891:
3888:
3884:
3880:
3875:
3869:
3866:
3863:
3861:
3858:
3857:
3854:
3851:
3849:
3846:
3845:
3843:
3838:
3835:
3832:
3829:
3808:
3802:
3798:
3792:
3788:
3782:
3777:
3774:
3770:
3766:
3763:
3760:
3756:
3749:
3746:
3743:
3739:
3735:
3730:
3724:
3721:
3719:
3716:
3715:
3712:
3709:
3707:
3704:
3703:
3701:
3696:
3693:
3690:
3687:
3662:
3659:
3655:
3629:
3626:
3621:
3618:
3598:
3595:
3574:
3571:
3566:
3563:
3542:
3539:
3534:
3531:
3511:
3490:
3487:
3482:
3479:
3458:
3455:
3450:
3447:
3427:
3423:
3420:
3415:
3412:
3407:
3404:
3399:
3396:
3391:
3388:
3383:
3380:
3354:
3349:
3346:
3342:
3338:
3335:
3332:
3328:
3321:
3318:
3315:
3311:
3292:
3289:
3287:
3284:
3281:
3280:
3267:
3263:
3259:
3256:
3253:
3250:
3240:
3229:
3224:
3220:
3215:
3203:
3202:
3189:
3185:
3181:
3178:
3168:
3157:
3152:
3148:
3143:
3131:
3130:
3117:
3113:
3109:
3106:
3096:
3085:
3080:
3076:
3071:
3059:
3058:
3045:
3041:
3030:
3019:
3014:
3010:
3005:
2993:
2992:
2979:
2975:
2963:
2944:
2939:
2935:
2930:
2900:
2896:
2892:
2889:
2886:
2881:
2877:
2854:
2850:
2829:
2824:
2820:
2815:
2794:
2789:
2785:
2780:
2756:
2753:
2750:
2746:
2742:
2739:
2736:
2732:
2728:
2708:
2705:
2702:
2698:
2694:
2691:
2688:
2684:
2680:
2660:
2655:
2651:
2646:
2625:
2620:
2616:
2611:
2598:
2595:
2580:
2576:
2555:
2552:
2547:
2544:
2541:
2537:
2516:
2432:is the adjoint
2408:
2405:
2375:
2355:
2339:
2336:
2323:
2300:
2297:
2272:
2268:
2262:
2257:
2251:
2248:
2242:
2239:
2235:
2229:
2225:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2196:
2189:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2160:
2133:
2130:
2126:
2122:
2119:
2116:
2112:
2108:
2105:
2102:
2098:
2094:
2091:
2088:
2084:
2060:
2055:
2051:
2046:
2042:
2036:
2031:
2028:
2025:
2021:
2017:
2014:
2011:
2007:
2003:
1978:
1972:
1969:
1966:
1962:
1958:
1955:
1951:
1947:
1944:
1941:
1937:
1933:
1930:
1926:
1922:
1897:
1894:
1890:
1869:
1866:
1861:
1857:
1852:
1831:
1828:
1824:
1787:
1782:
1778:
1773:
1756:
1753:
1752:
1751:
1740:
1737:
1734:
1728:
1721:
1717:
1713:
1709:
1705:
1700:
1696:
1692:
1688:
1684:
1679:
1675:
1671:
1667:
1663:
1658:
1654:
1650:
1646:
1640:
1632:
1628:
1623:
1620:
1615:
1611:
1606:
1595:
1584:
1581:
1578:
1572:
1565:
1561:
1557:
1553:
1549:
1544:
1540:
1536:
1532:
1528:
1523:
1519:
1515:
1511:
1507:
1502:
1498:
1494:
1490:
1484:
1476:
1472:
1467:
1464:
1459:
1455:
1450:
1439:
1428:
1425:
1422:
1416:
1409:
1405:
1401:
1397:
1393:
1388:
1384:
1380:
1376:
1372:
1367:
1363:
1359:
1355:
1351:
1346:
1342:
1338:
1334:
1328:
1320:
1316:
1311:
1308:
1303:
1299:
1294:
1283:
1272:
1269:
1266:
1260:
1253:
1249:
1245:
1241:
1237:
1232:
1228:
1224:
1220:
1216:
1211:
1207:
1203:
1199:
1195:
1190:
1186:
1182:
1178:
1172:
1164:
1160:
1155:
1152:
1147:
1143:
1138:
1104:
1099:
1087:
1084:
1069:
1064:
998:quantum states
993:
990:
946:
943:
940:
937:
933:
929:
926:
900:quantum states
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:
5336:
5325:
5322:
5320:
5317:
5316:
5314:
5302:
5298:
5294:
5291:
5286:
5280:
5276:
5272:
5268:
5264:
5259:
5256:
5251:
5245:
5241:
5237:
5233:
5229:
5225:
5224:
5216:
5212:
5207:
5202:
5197:
5192:
5188:
5184:
5180:
5176:
5172:
5165:
5159:
5153:
5147:
5141:
5133:
5129:
5124:
5119:
5114:
5109:
5105:
5101:
5097:
5093:
5089:
5082:
5073:
5068:
5064:
5060:
5057:(1): 013006.
5056:
5052:
5048:
5041:
5039:
5037:
5028:
5026:9781139495486
5022:
5018:
5011:
5009:
5007:
5005:
5003:
5001:
4999:
4997:
4995:
4990:
4959:
4956:
4948:
4942:
4939:
4925:
4921:
4916:
4891:
4888:
4880:
4874:
4871:
4863:
4857:
4854:
4846:
4840:
4837:
4829:
4823:
4820:
4812:
4806:
4803:
4795:
4789:
4786:
4778:
4772:
4769:
4753:
4748:
4744:
4739:
4712:
4704:
4696:
4691:
4683:
4667:
4659:
4651:
4646:
4638:
4627:
4619:
4611:
4603:
4598:
4590:
4574:
4566:
4558:
4553:
4545:
4526:
4521:
4517:
4512:
4487:
4479:
4473:
4459:
4455:
4450:
4442:
4423:
4419:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4365:
4359:
4357:
4352:
4350:
4346:
4341:
4338:
4325:
4321:
4315:
4312:
4309:
4308:Hadamard gate
4305:
4302:
4298:
4297:
4296:
4293:
4289:
4286:
4281:
4271:
4267:
4247:
4233:
4227:
4219:
4211:
4205:
4194:
4188:
4175:
4169:
4164:
4157:
4154:
4149:
4143:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4106:
4087:
4073:
4067:
4059:
4051:
4045:
4034:
4028:
4015:
4009:
4004:
3997:
3992:
3986:
3981:
3978:
3975:
3972:
3964:
3945:
3931:
3925:
3917:
3909:
3903:
3892:
3886:
3873:
3867:
3864:
3859:
3852:
3847:
3841:
3836:
3833:
3830:
3827:
3819:
3800:
3786:
3780:
3772:
3764:
3758:
3747:
3741:
3728:
3722:
3717:
3710:
3705:
3699:
3694:
3691:
3688:
3685:
3677:
3674:
3657:
3644:
3627:
3624:
3619:
3617:
3596:
3593:
3572:
3569:
3564:
3562:
3540:
3537:
3532:
3530:
3509:
3488:
3485:
3480:
3478:
3456:
3453:
3448:
3446:
3425:
3421:
3418:
3413:
3410:
3405:
3402:
3397:
3394:
3389:
3386:
3381:
3379:
3369:
3352:
3344:
3336:
3330:
3319:
3313:
3300:
3297:
3265:
3261:
3257:
3254:
3251:
3248:
3241:
3222:
3205:
3204:
3187:
3183:
3179:
3176:
3169:
3150:
3133:
3132:
3115:
3111:
3107:
3104:
3097:
3078:
3061:
3060:
3043:
3039:
3031:
3012:
2995:
2994:
2977:
2973:
2964:
2961:
2960:
2937:
2918:
2914:
2898:
2894:
2890:
2887:
2884:
2879:
2875:
2852:
2848:
2822:
2787:
2768:
2748:
2740:
2734:
2700:
2692:
2686:
2653:
2618:
2594:
2578:
2574:
2553:
2550:
2545:
2542:
2539:
2535:
2514:
2505:
2503:
2498:
2494:
2489:
2487:
2486:teleportation
2481:
2479:
2478:cross-section
2473:
2471:
2466:
2463:
2461:
2456:
2453:
2452:Hadamard gate
2449:
2445:
2437:
2433:
2428:
2424:
2422:
2418:
2414:
2404:
2402:
2398:
2394:
2389:
2345:
2335:
2321:
2295:
2283:
2270:
2266:
2260:
2246:
2240:
2237:
2227:
2219:
2216:
2210:
2204:
2201:
2198:
2187:
2183:
2174:
2171:
2168:
2162:
2149:
2147:
2128:
2120:
2114:
2106:
2100:
2092:
2086:
2072:
2053:
2040:
2034:
2023:
2015:
2009:
1976:
1964:
1953:
1945:
1939:
1928:
1911:
1910:superposition
1892:
1867:
1859:
1826:
1813:
1809:
1808:Hadamard gate
1805:
1780:
1761:
1735:
1719:
1711:
1703:
1698:
1690:
1682:
1677:
1669:
1661:
1656:
1648:
1630:
1626:
1621:
1613:
1596:
1579:
1563:
1555:
1547:
1542:
1534:
1526:
1521:
1513:
1505:
1500:
1492:
1474:
1470:
1465:
1457:
1440:
1423:
1407:
1399:
1391:
1386:
1378:
1370:
1365:
1357:
1349:
1344:
1336:
1318:
1314:
1309:
1301:
1284:
1267:
1251:
1243:
1235:
1230:
1222:
1214:
1209:
1201:
1193:
1188:
1180:
1162:
1158:
1153:
1145:
1128:
1127:
1126:
1124:
1123:Hilbert space
1120:
1102:
1097:
1083:
1067:
1062:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1013:
1009:
1005:
1003:
999:
989:
987:
983:
979:
975:
970:
968:
964:
960:
959:superposition
944:
941:
916:
915:basis vectors
913:
909:
905:
901:
898:are specific
897:
893:
892:Bell's states
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:
5296:
5270:
5235:
5178:
5174:
5164:
5152:
5140:
5098:(1): 14641.
5095:
5091:
5081:
5054:
5050:
5016:
4422:
4353:
4342:
4335:
4319:
4294:
4290:
4283:
4268:
4107:
3965:
3820:
3678:
3675:
3643:quantum gate
3370:
3301:
3294:
3286:Applications
2769:
2600:
2506:
2493:polarization
2490:
2482:
2474:
2467:
2464:
2457:
2441:
2412:
2410:
2343:
2341:
2284:
2150:
2146:listed above
2073:
1801:
1118:
1089:
1044:
1014:
1010:
1006:
995:
971:
895:
891:
885:
431:KleinâGordon
367:Formulations
204:Energy level
199:Entanglement
182:Fundamentals
169:Interference
120:Introduction
4378:EPR paradox
2397:mixed state
992:Bell states
986:information
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
5313:Categories
4985:References
2962:Bell state
2393:pure state
1086:Bell basis
912:normalized
815:Sommerfeld
730:Heisenberg
725:Gutzwiller
665:de Broglie
613:Scientists
527:Relational
478:Copenhagen
381:Heisenberg
239:Tunnelling
140:Background
4963:⟩
4957:−
4946:⟩
4943:−
4895:⟩
4892:−
4889:−
4881:−
4878:⟩
4872:−
4861:⟩
4858:−
4844:⟩
4830:−
4827:⟩
4824:−
4821:−
4810:⟩
4804:−
4793:⟩
4790:−
4776:⟩
4709:⟩
4705:−
4697:−
4688:⟩
4664:⟩
4660:−
4652:−
4643:⟩
4628:−
4616:⟩
4612:−
4595:⟩
4571:⟩
4567:−
4550:⟩
4491:⟩
4480:−
4477:⟩
4448:⟩
4443:−
4439:Φ
4383:GHZ state
4301:CNOT gate
4254:⟩
4248:−
4244:Ψ
4234:≡
4223:⟩
4212:−
4209:⟩
4192:⟩
4189:ψ
4181:⟶
4155:−
4121:−
4094:⟩
4084:Ψ
4074:≡
4063:⟩
4049:⟩
4032:⟩
4029:ψ
4021:⟶
3952:⟩
3946:−
3942:Φ
3932:≡
3921:⟩
3910:−
3907:⟩
3890:⟩
3887:ψ
3879:⟶
3865:−
3807:⟩
3797:Φ
3787:≡
3776:⟩
3762:⟩
3745:⟩
3742:ψ
3734:⟶
3661:⟩
3658:ψ
3348:⟩
3334:⟩
3317:⟩
3314:ψ
3228:⟩
3223:−
3219:Ψ
3156:⟩
3147:Ψ
3084:⟩
3079:−
3075:Φ
3018:⟩
3009:Φ
2943:⟩
2938:−
2934:Φ
2828:⟩
2819:Ψ
2793:⟩
2784:Ψ
2752:⟩
2749:−
2738:⟩
2704:⟩
2690:⟩
2659:⟩
2654:−
2650:Φ
2624:⟩
2615:Φ
2551:−
2448:CNOT gate
2388:John Bell
2374:Ψ
2354:Φ
2299:¯
2256:⟩
2250:¯
2217:−
2208:⟩
2181:⟩
2163:β
2132:⟩
2118:⟩
2104:⟩
2090:⟩
2059:⟩
2050:Φ
2027:⟩
2013:⟩
1968:⟩
1957:⟩
1943:⟩
1932:⟩
1896:⟩
1865:⟩
1856:Φ
1830:⟩
1812:CNOT gate
1786:⟩
1777:Φ
1716:⟩
1704:⊗
1695:⟩
1683:−
1674:⟩
1662:⊗
1653:⟩
1619:⟩
1614:−
1610:Ψ
1560:⟩
1548:⊗
1539:⟩
1518:⟩
1506:⊗
1497:⟩
1463:⟩
1454:Ψ
1404:⟩
1392:⊗
1383:⟩
1371:−
1362:⟩
1350:⊗
1341:⟩
1307:⟩
1302:−
1298:Φ
1248:⟩
1236:⊗
1227:⟩
1206:⟩
1194:⊗
1185:⟩
1151:⟩
1142:Φ
967:GHZ state
939:⟩
936:Φ
928:Φ
925:⟨
896:EPR pairs
845:Zeilinger
690:Ehrenfest
419:Equations
96:⟩
93:Ψ
82:^
70:⟩
67:Ψ
44:ℏ
5269:(2007),
5234:(2000),
5215:30345350
5132:30279547
4362:See also
3628:′
3620:′
3573:′
3565:′
3541:′
3533:′
3489:′
3481:′
3457:′
3449:′
3422:′
3414:′
3406:′
3398:′
3390:′
3382:′
1021:Podolsky
1017:Einstein
963:collapse
770:Millikan
695:Einstein
680:Davisson
635:Blackett
620:Aharonov
488:Ensemble
468:Bayesian
373:Overview
254:Collapse
234:Symmetry
125:Glossary
5206:6195333
5183:Bibcode
5123:6168595
5100:Bibcode
5059:Bibcode
1908:into a
1000:of two
902:of two
810:Simmons
800:Rydberg
765:Moseley
745:Kramers
735:Hilbert
720:Glauber
715:Feynman
700:Everett
670:Compton
441:Rydberg
130:History
5303:, 1964
5290:pp. 75
5281:
5255:pp. 25
5246:
5213:
5203:
5130:
5120:
5023:
2965:Basis
2955:state.
2421:qubits
2285:where
1810:and a
1023:, and
1002:qubits
980:. The
904:qubits
890:, the
840:Zeeman
835:Wigner
785:Planck
755:Landau
740:Jordan
391:Matrix
321:Popper
4414:Notes
4356:qudit
2436:above
1025:Rosen
795:Raman
780:Pauli
775:Onnes
710:Fermi
685:Debye
675:Dirac
640:Bloch
630:Bethe
498:Local
436:Pauli
426:Dirac
224:State
5279:ISBN
5244:ISBN
5211:PMID
5128:PMID
5021:ISBN
2770:The
2411:The
1045:both
976:and
830:Wien
825:Weyl
790:Rabi
760:Laue
750:Lamb
705:Fock
660:Bose
655:Born
650:Bohr
645:Bohm
625:Bell
5201:PMC
5191:doi
5118:PMC
5108:doi
5067:doi
4347:is
3438:or
1912:of
1029:EPR
894:or
886:In
5315::
5299:,
5288:,
5277:,
5273:,
5265:;
5253:,
5242:,
5238:,
5230:;
5209:.
5199:.
5189:.
5177:.
5173:.
5126:.
5116:.
5106:.
5094:.
5090:.
5065:.
5055:11
5053:.
5049:.
5035:^
4993:^
4488:11
4474:00
4266:.
4220:10
4206:01
4115:11
4060:10
4046:01
3973:10
3918:11
3904:00
3828:01
3773:11
3759:00
3686:00
3625:00
3570:11
3538:10
3486:01
3454:11
3419:10
3403:01
3387:00
3368:.
3345:11
3331:00
2480:.
2438:).
2403:.
2334:.
2129:11
2115:10
2101:01
2087:00
2071:.
2024:11
2010:00
1893:00
1827:00
1019:,
5305:.
5292:.
5257:.
5217:.
5193::
5185::
5179:4
5134:.
5110::
5102::
5096:8
5069::
5061::
5029:.
4966:)
4960:+
4953:|
4949:+
4940:+
4936:|
4932:(
4926:2
4922:1
4917:=
4898:)
4885:|
4875:+
4868:|
4864:+
4855:+
4851:|
4847:+
4841:+
4838:+
4834:|
4817:|
4813:+
4807:+
4800:|
4796:+
4787:+
4783:|
4779:+
4773:+
4770:+
4766:|
4762:(
4754:2
4749:2
4745:1
4740:=
4721:)
4718:)
4713:B
4701:|
4692:B
4684:+
4680:|
4676:(
4673:)
4668:A
4656:|
4647:A
4639:+
4635:|
4631:(
4625:)
4620:B
4608:|
4604:+
4599:B
4591:+
4587:|
4583:(
4580:)
4575:A
4563:|
4559:+
4554:A
4546:+
4542:|
4538:(
4535:(
4527:2
4522:2
4518:1
4513:=
4494:)
4484:|
4470:|
4466:(
4460:2
4456:1
4451:=
4434:|
4310:.
4303:.
4238:|
4228:2
4216:|
4202:|
4195:=
4185:|
4176:]
4170:0
4165:1
4158:1
4150:0
4144:[
4139:=
4136:Z
4133:X
4130:=
4127:Y
4124:i
4118::
4088:+
4078:|
4068:2
4056:|
4052:+
4042:|
4035:=
4025:|
4016:]
4010:0
4005:1
3998:1
3993:0
3987:[
3982:=
3979:X
3976::
3936:|
3926:2
3914:|
3900:|
3893:=
3883:|
3874:]
3868:1
3860:0
3853:0
3848:1
3842:[
3837:=
3834:Z
3831::
3801:+
3791:|
3781:2
3769:|
3765:+
3755:|
3748:=
3738:|
3729:]
3723:1
3718:0
3711:0
3706:1
3700:[
3695:=
3692:I
3689::
3654:|
3597:Y
3594:i
3510:Z
3426:,
3411:,
3395:,
3353:2
3341:|
3337:+
3327:|
3320:=
3310:|
3266:1
3262:b
3258:.
3255:Z
3252:.
3249:X
3214:|
3188:1
3184:b
3180:.
3177:X
3151:+
3142:|
3116:1
3112:b
3108:.
3105:Z
3070:|
3044:1
3040:b
3013:+
3004:|
2978:2
2974:b
2929:|
2899:1
2895:b
2891:.
2888:X
2885:=
2880:2
2876:b
2853:1
2849:b
2823:+
2814:|
2788:+
2779:|
2755:}
2745:|
2741:,
2735:+
2731:|
2727:{
2707:}
2701:1
2697:|
2693:,
2687:0
2683:|
2679:{
2645:|
2619:+
2610:|
2579:n
2575:4
2554:1
2546:1
2543:+
2540:n
2536:2
2515:n
2344:z
2322:y
2296:y
2271:,
2267:)
2261:2
2247:y
2241:,
2238:1
2234:|
2228:x
2224:)
2220:1
2214:(
2211:+
2205:y
2202:,
2199:0
2195:|
2188:(
2184:=
2178:)
2175:y
2172:,
2169:x
2166:(
2159:|
2125:|
2121:,
2111:|
2107:,
2097:|
2093:,
2083:|
2054:+
2045:|
2041:=
2035:2
2030:)
2020:|
2016:+
2006:|
2002:(
1977:2
1971:)
1965:0
1961:|
1954:1
1950:|
1946:+
1940:0
1936:|
1929:0
1925:|
1921:(
1889:|
1868:.
1860:+
1851:|
1823:|
1798:.
1781:+
1772:|
1739:)
1736:4
1733:(
1727:)
1720:B
1712:0
1708:|
1699:A
1691:1
1687:|
1678:B
1670:1
1666:|
1657:A
1649:0
1645:|
1639:(
1631:2
1627:1
1622:=
1605:|
1583:)
1580:3
1577:(
1571:)
1564:B
1556:0
1552:|
1543:A
1535:1
1531:|
1527:+
1522:B
1514:1
1510:|
1501:A
1493:0
1489:|
1483:(
1475:2
1471:1
1466:=
1458:+
1449:|
1427:)
1424:2
1421:(
1415:)
1408:B
1400:1
1396:|
1387:A
1379:1
1375:|
1366:B
1358:0
1354:|
1345:A
1337:0
1333:|
1327:(
1319:2
1315:1
1310:=
1293:|
1271:)
1268:1
1265:(
1259:)
1252:B
1244:1
1240:|
1231:A
1223:1
1219:|
1215:+
1210:B
1202:0
1198:|
1189:A
1181:0
1177:|
1171:(
1163:2
1159:1
1154:=
1146:+
1137:|
1103:2
1098:2
1068:2
1063:2
945:1
942:=
932:|
875:e
868:t
861:v
89:|
79:H
73:=
63:|
56:t
53:d
49:d
41:i
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