4372:, a sender wishes to transmit an arbitrary quantum state of a particle to a possibly distant receiver. Consequently, the teleportation process is a quantum channel. The apparatus for the process itself requires a quantum channel for the transmission of one particle of an entangled-state to the receiver. Teleportation occurs by a joint measurement of the sent particle and the remaining entangled particle. This measurement results in classical information which must be sent to the receiver to complete the teleportation. Importantly, the classical information can be sent after the quantum channel has ceased to exist.
7972:
6834:
8779:
7962:
9081:
6590:
1018:
So far we have only defined quantum channel that transmits only quantum information. As stated in the introduction, the input and output of a channel can include classical information as well. To describe this, the formulation given so far needs to be generalized somewhat. A purely quantum channel,
3758:
2465:
2872:
5913:
3095:
6414:
3569:
5146:
914:. This relationship is same as that between the Schrödinger and Heisenberg descriptions of dynamics. The measurement statistics remain unchanged whether the observables are considered fixed while the states undergo operation or vice versa.
5289:
2579:
5951:
can be viewed as representing a message consisting of possibly infinite number of words. The limit supremum condition in the definition says that, in the limit, faithful transmission can be achieved by invoking the channel no more than
3873:
5385:
4732:
4173:
4502:
of the state is maintained during transmission through the channel. Contrast this with the transmission of electrical pulses through wires (a classical channel), where only classical information (e.g. 0s and 1s) can be sent.
846:
3400:
2966:
2223:
65:
of a system but as a pipeline intended to carry quantum information. (Some authors use the term "quantum operation" to include trace-decreasing maps while reserving "quantum channel" for strictly trace-preserving maps)
5062:
5567:
4998:
2589:
The observable map, in the Schrödinger picture, has a purely classical output algebra and therefore only describes measurement statistics. To take the state change into account as well, we define what is called a
5707:
4059:. Actually, all the states that result from the partial action of a measure-and-prepare channel are separable, and for this reason measure-and-prepare channels are also known as entanglement-breaking channels.
5807:
6400:
4929:
1387:
1141:
3950:
3618:
1906:
4284:
1547:
is sometimes given as the generalized mathematical framework for conveying both quantum and classical information. In axiomatizations of quantum mechanics, the classical information is carried in a
4035:
2364:
6781:
1084:
745:
2295:
1616:
5740:
1146:
Classical information can then be included in this formulation. The observables of a classical system can be assumed to be a commutative C*-algebra, i.e. the space of continuous functions
2751:
1954:
1829:
6686:
5489:
3233:
6819:
6332:
6236:
6201:
5816:
4800:
4360:. The time evolution is one example of a pure channel. This terminology again comes from the channel-state duality. A channel is pure if and only if its dual state is a pure state.
2644:
6060:
4832:
4496:
2008:
1726:
2977:
1441:
1777:
426:
468:
6585:{\displaystyle C(\mathbb {C} ^{m},\mathbb {C} ^{n})=C(\mathbb {C} ^{m\times m},\mathbb {C} ^{n\times n})=C(\mathbb {C} ^{m\times m},\mathbb {C} ^{n})={\frac {\log n}{\log m}}.}
3291:
1669:
6718:
6297:
6265:
6154:
2743:
1275:
5949:
3170:
6987:
Weedbrook, Christian; Pirandola, Stefano; GarcĂa-PatrĂłn, RaĂșl; Cerf, Nicolas J.; Ralph, Timothy C.; Shapiro, Jeffrey H.; Lloyd, Seth (2012). "Gaussian quantum information".
6095:
1485:
1302:
1008:
2699:
2108:
6907:
3466:
6946:
6125:
6013:
5641:
5442:
5070:
4454:
4426:
3125:
2356:
962:
904:
624:
5408:
4630:
4582:
4562:
1521:
192:
5177:
2484:
222:
5983:
5611:
4852:
4755:
4610:
4542:
4354:
4330:
4200:
4105:
4085:
3607:
3451:
3253:
2664:
2326:
2135:
2065:
2038:
935:
869:
765:
644:
597:
577:
531:
499:
375:
351:
328:
304:
277:
246:
152:
125:
1242:
1173:
1541:
1461:
1213:
1193:
3769:
1911:
The partial trace operation is a CPTP map, therefore a quantum channel in the Schrödinger picture. In the
Heisenberg picture, the dual map of this channel is
17:
5311:
4638:
4113:
6595:
The above says, for instance, an ideal quantum channel is no more efficient at transmitting classical information than an ideal classical channel. When
773:
8008:
5064:
be a chosen ideal channel. To make the comparison possible, one needs to encode and decode Ί via appropriate devices, i.e. we consider the composition
3299:
2883:
2143:
8815:
8576:
7161:
5003:
5500:
7039:
John A. Holbrook, David W. Kribs, and
Raymond Laflamme. "Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction."
579:
between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend
8209:
7123:
4179:
1683:
is the time. Clearly this gives a CPTP map in the Schrödinger picture and is therefore a channel. The dual map in the
Heisenberg picture is
378:
4949:
7853:
5652:
5745:
7754:
7415:
4388:. Single photons can be transmitted up to 100 km in standard fiber optics before losses dominate. The photon's time-of-arrival (
8747:
1010:. Physically speaking, this means that, in the Heisenberg picture, the trivial observable remains trivial after applying the channel.
6340:
4860:
1310:
1103:
8058:
7316:
3753:{\displaystyle \Phi _{2}\left({\begin{bmatrix}\rho (F_{1})\\\vdots \\\rho (F_{n})\end{bmatrix}}\right)=\sum _{i}\rho (F_{i})R_{i}.}
8759:
7641:
3892:
8443:
8001:
1864:
6405:
This is equivalent to the no-teleportation theorem: it is impossible to transmit quantum information via a classical channel.
8377:
4208:
2460:{\displaystyle \Psi (\rho )={\begin{bmatrix}\langle F_{1},\rho \rangle \\\vdots \\\langle F_{n},\rho \rangle \end{bmatrix}},}
74:
We will assume for the moment that all state spaces of the systems considered, classical or quantum, are finite-dimensional.
3958:
8300:
7965:
7151:
9107:
8808:
7923:
6733:
1025:
686:
7499:
8031:
7994:
7975:
7863:
7116:
2867:{\displaystyle \Phi (\rho )={\begin{bmatrix}\rho (F_{1})\cdot F_{1}\\\vdots \\\rho (F_{n})\cdot F_{n}\end{bmatrix}}.}
2234:
1578:
5712:
8141:
7791:
7449:
2471:
1917:
1785:
8714:
8423:
8418:
8126:
7786:
7514:
7494:
1676:
6644:
5908:{\displaystyle \lim _{\alpha }\Delta ({\hat {\Psi }}^{\otimes m_{\alpha }},\Psi _{id}^{\otimes n_{\alpha }})=0.}
5455:
3175:
8801:
8726:
8398:
7781:
1280:
Therefore, in the
Heisenberg picture, if the classical information is part of, say, the input, we would define
85:: the output of a channel at a given time depends only upon the corresponding input and not any previous ones.
7293:
6789:
6302:
6206:
6171:
9085:
8704:
8481:
8403:
8275:
8171:
8041:
7814:
7636:
7539:
7200:
6267:
will have an ideal channel denoted by the same symbol. We can now state some fundamental channel capacities.
3433:
prepares his (quantum) system in a specific state. In the Schrödinger picture, the first part of the channel
3090:{\displaystyle \Psi (f\otimes A)={\begin{bmatrix}f_{1}\Psi _{1}(A)\\\vdots \\f_{n}\Psi _{n}(A)\end{bmatrix}}}
2301:
533:
is only required to be not trace-increasing. In this article, it will be assumed that all channels are CPTP.
4766:
2597:
8438:
8372:
8367:
8338:
8036:
7819:
7687:
7278:
7109:
6967:
6019:
4808:
4459:
2475:
1978:
1689:
680:
1395:
9047:
8506:
8413:
7843:
7599:
7459:
7233:
7188:
7132:
1739:
395:
385:
4588:
between channels. Since a channel can be viewed as a linear operator, it is tempting to use the natural
2470:
where the inner product is the
HilbertâSchmidt inner product. Furthermore, viewing states as normalized
440:
9062:
8862:
8782:
8544:
8352:
8323:
7715:
7587:
7484:
7360:
7288:
7195:
6869:
6610:
It is relevant to note here that both of the above bounds on capacities can be broken, with the aid of
3261:
1624:
31:
6845:
6694:
6273:
6241:
6130:
2704:
2646:
be the effects (POVM) associated to an observable. In the Schrödinger picture, an instrument is a map
1543:. Positivity of a map is defined accordingly. This characterization is not universally accepted; the
1251:
8892:
8877:
8566:
8433:
8357:
8318:
8255:
8234:
8191:
8084:
7524:
7489:
7385:
7328:
6962:
5921:
3564:{\displaystyle \;\Phi _{1}(\rho )={\begin{bmatrix}\rho (F_{1})\\\vdots \\\rho (F_{n})\end{bmatrix}}.}
3130:
57:(CP) trace-preserving maps between spaces of operators. In other words, a quantum channel is just a
6076:
1466:
1283:
971:
42:, as well as classical information. An example of quantum information is the general dynamics of a
9035:
9030:
8959:
8934:
8919:
8914:
8909:
8633:
8613:
8603:
8593:
8549:
8109:
7697:
7670:
7646:
7609:
7400:
7333:
7268:
7253:
7223:
7146:
6957:
6691:
that is, it is the capacity of Κ with respect to the ideal channel on the classical one-bit system
5141:{\displaystyle {\hat {\Psi }}=D\circ \Phi \circ E:{\mathcal {B}}_{2}\rightarrow {\mathcal {A}}_{2}}
4041:
2669:
6883:
4040:
A measure-and-prepare channel can not be the identity map. This is precisely the statement of the
2070:
8328:
8239:
7848:
7582:
7474:
7444:
7243:
8742:
8290:
6916:
6100:
5988:
5616:
5417:
5284:{\displaystyle \Delta ({\hat {\Psi }},\Psi _{id})=\inf _{E,D}\|{\hat {\Psi }}-\Psi _{id}\|_{cb}}
4431:
4403:
9052:
8974:
8653:
8428:
8408:
8333:
8186:
7918:
7682:
7675:
7422:
6626:. These results indicate the significant role played by entanglement in quantum communication.
4333:
3103:
2574:{\displaystyle \Psi (\rho )={\begin{bmatrix}\rho (F_{1})\\\vdots \\\rho (F_{n})\end{bmatrix}}.}
2334:
940:
882:
602:
8229:
5393:
4615:
4567:
4547:
1490:
225:
161:
9025:
8999:
8824:
8678:
8176:
8156:
7838:
7464:
7390:
7355:
6615:
4369:
4052:
4045:
354:
197:
8964:
8887:
8709:
8638:
8583:
8313:
8121:
8079:
7628:
7377:
7228:
7074:
7006:
6611:
5968:
5596:
4837:
4740:
4595:
4585:
4527:
4397:
4393:
4339:
4315:
4185:
4090:
4070:
3592:
3436:
3238:
2649:
2311:
2120:
2043:
2016:
920:
854:
750:
629:
582:
562:
516:
484:
360:
336:
313:
282:
255:
231:
130:
103:
4336:, who introduced them). The minimum number of Kraus operators is called the Kraus rank of
1218:
1149:
437:
is the identity map on the ancilla, must also be positive. Therefore, it is required that
8:
9057:
9015:
8984:
8944:
8929:
8904:
8694:
8663:
8608:
8588:
8496:
8453:
8308:
8219:
8146:
8136:
8048:
7947:
7900:
7730:
7504:
7258:
7238:
7173:
7168:
475:
54:
39:
7078:
7010:
6065:
From the definition, it is vacuously true that 0 is an achievable rate for any channel.
3868:{\displaystyle \Phi (\rho )=\Phi _{2}\circ \Phi _{1}(\rho )=\sum _{i}\rho (F_{i})R_{i}.}
8949:
8839:
8754:
8623:
8521:
8214:
8161:
8053:
7663:
7509:
7311:
7248:
7090:
7064:
7022:
6996:
5960:
is the number of letters per invocation of the channel that can be sent without error.
2591:
1552:
1544:
1526:
1446:
1198:
1178:
647:
93:
Consider quantum channels that transmit only quantum information. This is precisely a
82:
4087:
in the
Heisenberg picture. With the assumption that everything is finite-dimensional,
3425:
performs the measurement of an observable and communicates the measurement outcome to
8994:
8882:
8852:
8847:
8764:
8673:
8643:
8571:
8534:
8529:
8511:
8476:
8466:
8270:
8166:
8131:
8114:
8017:
7927:
7572:
7479:
7436:
7367:
7283:
7263:
7218:
7178:
7156:
7026:
6619:
4499:
1548:
1304:
to include the relevant classical observables. An example of this would be a channel
94:
58:
6203:. As a slight abuse of notation, this ideal quantum channel will be also denoted by
8899:
8491:
8486:
8343:
8224:
7594:
7544:
7321:
7094:
7082:
7014:
5380:{\displaystyle \Psi _{id}^{\otimes n}=\Psi _{id}\otimes \cdots \otimes \Psi _{id}.}
4940:
4727:{\displaystyle \|\Phi -\Lambda \|=\sup\{\|(\Phi -\Lambda )(A)\|\;|\;\|A\|\leq 1\}.}
4168:{\displaystyle \Psi :\mathbb {C} ^{n\times n}\rightarrow \mathbb {C} ^{m\times m}.}
3883:
62:
6160:
dimensional quantum system, the ideal channel is the identity map on the space of
841:{\displaystyle \langle A,\Phi (\rho )\rangle =\langle \Phi ^{*}(A),\rho \rangle .}
8969:
8954:
8924:
8872:
8721:
8648:
8628:
8598:
8561:
8461:
8285:
7720:
7658:
7348:
5572:
is therefore a measure of the ability of the channel to transmit words of length
4056:
4564:. For instance, when the input and output algebras are identical, we can choose
4516:
Before giving the definition of channel capacity, the preliminary notion of the
8989:
8979:
8699:
8668:
8658:
8280:
8260:
8089:
7829:
7806:
7773:
7577:
7454:
4524:
of a channel needs to be discussed. When considering the capacity of a channel
4308:
4048:) is impossible. In other words, a quantum state can not be measured reliably.
249:
46:. An example of classical information is a text document transmitted over the
7101:
7086:
7018:
6864:
Another measure of how well a quantum channel preserves information is called
3395:{\displaystyle {\tilde {\Psi }}(A)=\sum _{i}\Psi _{i}(A)=\sum _{i}M_{i}AM_{i}}
2961:{\displaystyle f={\begin{bmatrix}f_{1}\\\vdots \\f_{n}\end{bmatrix}}\in C(X).}
2067:'s are assumed to be positive operators acting on appropriate state space and
9101:
9040:
8939:
8618:
8471:
8362:
8265:
8181:
8151:
8104:
7651:
7469:
7395:
5445:
5167:
are unital CP maps with appropriate domains. The quantity of interest is the
4589:
4055:, a channel is measure-and-prepare if and only if the corresponding state is
2218:{\displaystyle f={\begin{bmatrix}f_{1}\\\vdots \\f_{n}\end{bmatrix}}\in C(X)}
1847:
155:
9020:
8501:
8099:
8094:
7871:
7796:
5057:{\displaystyle \Psi _{id}:{\mathcal {B}}_{2}\rightarrow {\mathcal {A}}_{2}}
2308:
to obtain the quantum mechanical observable. It can be easily checked that
5562:{\displaystyle \Delta ({\hat {\Psi }}^{\otimes m},\Psi _{id}^{\otimes n})}
4760:
To make the operator norm even a more undesirable candidate, the quantity
1097:. Therefore, we can say a channel is a unital CP map between C*-algebras:
1019:
in the
Heisenberg picture, is a linear map Κ between spaces of operators:
8857:
8539:
7881:
7735:
7273:
4381:
8793:
6833:
513:. In the literature, sometimes the fourth property is weakened so that
7942:
7876:
7740:
6910:
4396:
are used as a basis to encode quantum information for purposes such as
1094:
965:
599:
uniquely to the full space of operators. This leads to the adjoint map
6618:
allows one to transmit quantum information using a classical channel.
5294:
with the infimum being taken over all possible encoders and decoders.
4993:{\displaystyle \Psi :{\mathcal {B}}_{1}\rightarrow {\mathcal {A}}_{1}}
4400:. The channel is capable of transmitting not only basis states (e.g.
8393:
8074:
7725:
3172:(this can always be done since elements of a POVM are positive) then
7986:
7910:
7886:
7745:
7710:
6986:
5702:{\displaystyle \{n_{\alpha }\},\{m_{\alpha }\}\subset \mathbb {N} }
47:
7069:
7001:
5802:{\displaystyle \lim \sup _{\alpha }(n_{\alpha }/m_{\alpha })<r}
7937:
7554:
1568:
For a purely quantum system, the time evolution, at certain time
158:) of the sending and receiving ends, respectively, of a channel.
4380:
Experimentally, a simple implementation of a quantum channel is
7914:
7410:
6395:{\displaystyle C(\mathbb {C} ^{m},\mathbb {C} ^{n\times n})=0.}
4924:{\displaystyle \|\Phi \|_{cb}=\sup _{n}\|\Phi \otimes I_{n}\|.}
4385:
1382:{\displaystyle \Psi :L(H_{B})\otimes C(X)\rightarrow L(H_{A}).}
1136:{\displaystyle \Psi :{\mathcal {B}}\rightarrow {\mathcal {A}}.}
4044:, which says classical teleportation (not to be confused with
81:
in the section title carries the same meaning as in classical
7183:
4939:
The mathematical model of a channel used here is same as the
43:
7932:
7405:
7338:
6629:
5444:
independently and is the quantum mechanical counterpart of
2111:
1093:). The operator spaces can be viewed as finite-dimensional
3945:{\displaystyle \Phi ^{*}=\Phi _{1}^{*}\circ \Phi _{2}^{*}}
7549:
7534:
1901:{\displaystyle \rho ^{A}=\operatorname {Tr} _{B}\;\rho .}
6634:
Using the same notation as the previous subsection, the
6786:
where the reference system is now the one qubit system
6073:
As stated before, for a system with observable algebra
4737:
However, the operator norm may increase when we tensor
4384:(or free-space for that matter) transmission of single
4279:{\displaystyle \Psi (A)=\sum _{i=1}^{N}K_{i}AK_{i}^{*}}
747:
as a map between
Hilbert spaces, we obtain its adjoint
7055:, Lecture Notes in Physics Volume 611, Springer, 2002.
4030:{\displaystyle \;\Phi ^{*}(A)=\sum _{i}R_{i}(A)F_{i}.}
3641:
3612:
takes the above classical state to the density matrix
3498:
3007:
2898:
2775:
2508:
2388:
2158:
2073:
6919:
6886:
6792:
6736:
6697:
6647:
6417:
6343:
6305:
6276:
6244:
6209:
6174:
6133:
6103:
6079:
6022:
5991:
5971:
5924:
5819:
5748:
5715:
5655:
5619:
5599:
5503:
5458:
5420:
5396:
5314:
5180:
5073:
5006:
4952:
4863:
4840:
4811:
4769:
4743:
4641:
4618:
4598:
4584:
to be the identity map. Such a comparison requires a
4570:
4550:
4530:
4462:
4434:
4406:
4342:
4318:
4211:
4188:
4116:
4093:
4073:
3961:
3895:
3772:
3621:
3595:
3469:
3460:
making a measurement, i.e. it is the observable map:
3439:
3302:
3264:
3241:
3178:
3133:
3106:
2980:
2886:
2754:
2707:
2672:
2652:
2600:
2487:
2367:
2337:
2314:
2237:
2146:
2123:
2046:
2019:
1981:
1920:
1867:
1788:
1742:
1736:
Consider a composite quantum system with state space
1692:
1627:
1581:
1529:
1493:
1469:
1449:
1398:
1313:
1286:
1254:
1221:
1201:
1181:
1152:
1106:
1028:
974:
943:
923:
885:
857:
776:
753:
689:
632:
605:
585:
565:
519:
487:
443:
398:
363:
339:
316:
285:
258:
234:
200:
164:
133:
106:
6880:
A bistochastic quantum channel is a quantum channel
6270:
The channel capacity of the classical ideal channel
6238:. Similarly, a classical system with output algebra
548:
only constitute a proper subset of the operators on
7376:
3429:classically. According to the message he receives,
481:Density matrices are specified to have trace 1, so
27:
Foundational object in quantum communication theory
6940:
6901:
6813:
6776:{\displaystyle C(\Psi ,\mathbb {C} ^{2\times 2}),}
6775:
6712:
6680:
6584:
6394:
6326:
6291:
6259:
6230:
6195:
6148:
6119:
6089:
6054:
6007:
5977:
5956:times the length of a word. One can also say that
5943:
5907:
5801:
5734:
5701:
5635:
5605:
5561:
5483:
5436:
5402:
5379:
5283:
5140:
5056:
4992:
4923:
4846:
4826:
4794:
4749:
4726:
4624:
4604:
4576:
4556:
4536:
4490:
4448:
4420:
4348:
4324:
4278:
4194:
4167:
4099:
4079:
4029:
3944:
3867:
3752:
3601:
3563:
3445:
3394:
3285:
3247:
3227:
3164:
3119:
3089:
2960:
2866:
2737:
2693:
2658:
2638:
2573:
2459:
2350:
2320:
2289:
2217:
2129:
2102:
2059:
2032:
2002:
1948:
1900:
1823:
1771:
1720:
1663:
1610:
1535:
1515:
1479:
1455:
1435:
1381:
1296:
1269:
1236:
1207:
1187:
1167:
1135:
1079:{\displaystyle \Psi :L(H_{B})\rightarrow L(H_{A})}
1078:
1002:
956:
929:
898:
863:
840:
759:
740:{\displaystyle \Phi :L(H_{A})\rightarrow L(H_{B})}
739:
638:
618:
591:
571:
525:
493:
462:
420:
369:
345:
322:
298:
271:
240:
216:
186:
146:
119:
4934:
4834:The solution is to introduce, for any linear map
509:used to describe a map are sometimes abbreviated
9099:
5821:
5753:
5749:
5222:
4887:
4660:
4544:, we need to compare it with an "ideal channel"
2114:.) In the Heisenberg picture, the corresponding
310:As required by postulates of quantum mechanics,
7131:
6875:
2290:{\displaystyle \;\Psi (f)=\sum _{i}f_{i}F_{i}.}
1611:{\displaystyle \rho \rightarrow U\rho \;U^{*},}
392:is coupled to the system, then the induced map
5735:{\displaystyle m_{\alpha }\rightarrow \infty }
4375:
4107:is a unital CP map between spaces of matrices
4067:Consider the case of a purely quantum channel
3408:
38:is a communication channel which can transmit
8809:
8002:
7117:
3421:wish to communicate in the following manner:
69:
7043:. Volume 2, Number 5, p. 381-419. Oct 2003.
5938:
5925:
5688:
5675:
5669:
5656:
5269:
5237:
4915:
4896:
4871:
4864:
4789:
4770:
4718:
4709:
4703:
4693:
4666:
4663:
4654:
4642:
4485:
4471:
4443:
4415:
3293:gives precisely the observable map. The map
2633:
2601:
2443:
2424:
2410:
2391:
2358:takes density matrices to classical states:
1949:{\displaystyle A\rightarrow A\otimes I_{B},}
1824:{\displaystyle \rho \in H_{A}\otimes H_{B},}
832:
804:
798:
777:
5000:be a channel in the Heisenberg picture and
4511:
1975:An observable associates a numerical value
9080:
8816:
8802:
8009:
7995:
7124:
7110:
6681:{\displaystyle C(\Psi ,\mathbb {C} ^{2}),}
6616:entanglement-assisted teleportation scheme
6023:
5484:{\displaystyle {\hat {\Psi }}^{\otimes m}}
4702:
4696:
4180:Choi's theorem on completely positive maps
3962:
3470:
3228:{\displaystyle \;\Psi _{i}(A)=M_{i}AM_{i}}
3179:
2971:The dual map in the Heisenberg picture is
2238:
1891:
1594:
8823:
7068:
7000:
6795:
6751:
6700:
6662:
6537:
6516:
6486:
6465:
6441:
6426:
6408:Moreover, the following equalities hold:
6367:
6352:
6308:
6279:
6247:
6212:
6177:
6062:is the supremum of all achievable rates.
5695:
4146:
4125:
1996:
1257:
6980:
6814:{\displaystyle \mathbb {C} ^{2\times 2}}
6630:Classical and quantum channel capacities
6327:{\displaystyle \mathbb {C} ^{n\times n}}
6299:with respect to a quantum ideal channel
6231:{\displaystyle \mathbb {C} ^{n\times n}}
6196:{\displaystyle \mathbb {C} ^{n\times n}}
5583:This leads to the following definition:
4456:) but also superpositions of them (e.g.
4356:. A channel with Kraus rank 1 is called
3889:In the Heisenberg picture, the dual map
3127:is defined in the following way: Factor
1089:that is unital and completely positive (
1013:
507:completely positive and trace preserving
154:be the state spaces (finite-dimensional
7642:Continuous-variable quantum information
5305:times, so we consider the tensor power
4757:with the identity map on some ancilla.
3763:The total operation is the composition
357:of positive elements. In other words,
194:will denote the family of operators on
53:Terminologically, quantum channels are
14:
9100:
4795:{\displaystyle \|\Phi \otimes I_{n}\|}
2639:{\displaystyle \{F_{1},\dots ,F_{n}\}}
88:
8797:
8016:
7990:
7105:
7058:
6068:
6055:{\displaystyle \;C(\Psi ,\Psi _{id})}
5301:, the ideal channel is to be applied
4827:{\displaystyle n\rightarrow \infty .}
4491:{\displaystyle |0\rangle +|1\rangle }
2003:{\displaystyle f_{i}\in \mathbb {C} }
1721:{\displaystyle A\rightarrow U^{*}AU.}
536:
333:Since density matrices are positive,
97:, whose properties we now summarize.
6828:
3405:describes the overall state change.
1436:{\displaystyle L(H_{B})\otimes C(X)}
228:, a purely quantum channel is a map
18:Completely positive trace-preserving
7053:How to Correct Small Quantum Errors
6824:
4592:. In other words, the closeness of
4506:
4046:entanglement-assisted teleportation
1772:{\displaystyle H_{A}\otimes H_{B}.}
937:is assumed to be trace preserving,
917:It can be directly checked that if
421:{\displaystyle I_{n}\otimes \Phi ,}
24:
6920:
6887:
6743:
6654:
6140:
6127:is by definition the identity map
6105:
6082:
6037:
6030:
5993:
5972:
5869:
5840:
5830:
5729:
5621:
5600:
5536:
5514:
5504:
5463:
5422:
5362:
5340:
5316:
5256:
5243:
5203:
5190:
5181:
5127:
5110:
5095:
5077:
5043:
5026:
5008:
4979:
4962:
4953:
4899:
4867:
4841:
4818:
4773:
4744:
4678:
4672:
4651:
4645:
4619:
4599:
4571:
4551:
4531:
4343:
4319:
4212:
4189:
4117:
4094:
4074:
3964:
3928:
3910:
3897:
3802:
3789:
3773:
3623:
3596:
3472:
3440:
3338:
3306:
3265:
3242:
3181:
3108:
3061:
3021:
2981:
2755:
2653:
2488:
2368:
2339:
2331:The corresponding Schrödinger map
2315:
2239:
2124:
1472:
1443:is still a C*-algebra. An element
1314:
1289:
1125:
1115:
1107:
1029:
976:
945:
924:
887:
858:
808:
786:
754:
690:
683:inner product. Therefore, viewing
633:
607:
586:
566:
520:
488:
463:{\displaystyle I_{n}\otimes \Phi }
457:
412:
364:
340:
317:
235:
25:
9119:
4854:between C*-algebras, the cb-norm
3878:Channels of this form are called
3589:, the second part of the channel
3286:{\displaystyle \Psi (f\otimes I)}
2110:. (Such a collection is called a
1664:{\displaystyle U=e^{-iHt/\hbar }}
1656:
1563:
9079:
8778:
8777:
7971:
7970:
7961:
7960:
6832:
6713:{\displaystyle \mathbb {C} ^{2}}
6292:{\displaystyle \mathbb {C} ^{m}}
6260:{\displaystyle \mathbb {C} ^{m}}
6149:{\displaystyle I_{\mathcal {B}}}
5414:inputs undergoing the operation
4363:
2738:{\displaystyle C(X)\otimes L(H)}
1277:with entry-wise multiplication.
1270:{\displaystyle \mathbb {R} ^{n}}
626:, which describes the action of
555:and same can be said for system
5944:{\displaystyle \{n_{\alpha }\}}
5159:is a decoder. In this context,
4062:
3165:{\displaystyle F_{i}=M_{i}^{2}}
306:with the following properties:
8727:Relativistic quantum mechanics
7063:, Cambridge University Press,
7041:Quantum Information Processing
7033:
6929:
6923:
6896:
6890:
6767:
6740:
6672:
6651:
6603:, the best one can achieve is
6547:
6511:
6502:
6460:
6451:
6421:
6383:
6347:
6090:{\displaystyle {\mathcal {B}}}
6049:
6027:
5896:
5843:
5833:
5790:
5762:
5726:
5556:
5517:
5507:
5466:
5246:
5215:
5193:
5184:
5121:
5080:
5037:
4973:
4935:Definition of channel capacity
4815:
4805:may increase without bound as
4698:
4690:
4684:
4681:
4669:
4478:
4464:
4436:
4408:
4221:
4215:
4141:
4011:
4005:
3979:
3973:
3849:
3836:
3817:
3811:
3782:
3776:
3734:
3721:
3690:
3677:
3660:
3647:
3547:
3534:
3517:
3504:
3487:
3481:
3353:
3347:
3321:
3315:
3309:
3280:
3268:
3196:
3190:
3076:
3070:
3036:
3030:
2996:
2984:
2952:
2946:
2837:
2824:
2794:
2781:
2764:
2758:
2732:
2726:
2717:
2711:
2688:
2682:
2557:
2544:
2527:
2514:
2497:
2491:
2377:
2371:
2248:
2242:
2228:to the quantum mechanical one
2212:
2206:
1924:
1731:
1696:
1585:
1480:{\displaystyle {\mathcal {A}}}
1430:
1424:
1415:
1402:
1373:
1360:
1354:
1351:
1345:
1336:
1323:
1297:{\displaystyle {\mathcal {B}}}
1231:
1225:
1162:
1156:
1120:
1073:
1060:
1054:
1051:
1038:
1003:{\displaystyle \Phi ^{*}(I)=I}
991:
985:
823:
817:
795:
789:
734:
721:
715:
712:
699:
679:) are Hilbert spaces with the
559:. However, once a linear map
388:of arbitrary finite dimension
181:
168:
13:
1:
8705:Quantum statistical mechanics
8482:Quantum differential calculus
8404:Delayed-choice quantum eraser
8172:Symmetry in quantum mechanics
7637:Adiabatic quantum computation
6973:
3582:prepares his system in state
2694:{\displaystyle \rho \in L(H)}
2584:
2103:{\textstyle \sum _{i}F_{i}=I}
1970:
1248:-dimensional Euclidean space
7688:Topological quantum computer
6968:Quantum depolarizing channel
6902:{\displaystyle \Phi (\rho )}
6876:Bistochastic quantum channel
5576:faithfully by being invoked
5450:m invocations of the channel
5297:To transmit words of length
4518:norm of complete boundedness
4332:(after the German physicist
2476:Riesz representation theorem
2137:maps a classical observable
1846:, is obtained by taking the
7:
8507:Quantum stochastic calculus
8497:Quantum measurement problem
8419:MachâZehnder interferometer
7966:Quantum information science
7133:Quantum information science
6951:
5587:A non-negative real number
4376:In the experimental setting
3409:Measure-and-prepare channel
1963:is an observable of system
1558:
1244:can be identified with the
906:maps observables on system
541:Density matrices acting on
10:
9124:
9108:Quantum information theory
9063:Thermoacoustic heat engine
7361:quantum gate teleportation
7061:Quantum Information Theory
7051:M. Keyl and R. F. Werner,
6941:{\displaystyle \Phi (I)=I}
6870:fidelity of quantum states
6120:{\displaystyle \Psi _{id}}
6008:{\displaystyle \Psi _{id}}
5636:{\displaystyle \Psi _{id}}
5437:{\displaystyle \Psi _{id}}
4449:{\displaystyle |1\rangle }
4421:{\displaystyle |0\rangle }
501:has to preserve the trace.
70:Memoryless quantum channel
32:quantum information theory
9075:
9048:Immersive virtual reality
9008:
8838:
8831:
8773:
8735:
8687:
8567:Quantum complexity theory
8545:Quantum cellular automata
8520:
8452:
8386:
8299:
8248:
8235:Path integral formulation
8202:
8067:
8024:
7956:
7899:
7862:
7828:
7805:
7772:
7763:
7696:
7625:
7563:
7523:
7490:Quantum Fourier transform
7435:
7386:Post-quantum cryptography
7329:Entanglement distillation
7302:
7211:
7139:
7087:10.1017/9781316809976.001
7019:10.1103/RevModPhys.84.621
6989:Reviews of Modern Physics
6963:Amplitude damping channel
3578:-th measurement outcome,
3120:{\displaystyle \Psi _{i}}
2351:{\displaystyle \Psi ^{*}}
957:{\displaystyle \Phi ^{*}}
899:{\displaystyle \Phi ^{*}}
619:{\displaystyle \Phi ^{*}}
61:viewed not merely as the
9031:Digital scent technology
8634:Quantum machine learning
8614:Quantum key distribution
8604:Quantum image processing
8594:Quantum error correction
8444:Wheeler's delayed choice
7976:Quantum mechanics topics
7671:Quantum machine learning
7647:One-way quantum computer
7500:Quantum phase estimation
7401:Quantum key distribution
7334:Monogamy of entanglement
6958:No-communication theorem
5403:{\displaystyle \otimes }
4625:{\displaystyle \Lambda }
4577:{\displaystyle \Lambda }
4557:{\displaystyle \Lambda }
4512:The cb-norm of a channel
4042:no teleportation theorem
3574:If, in the event of the
2666:with pure quantum input
2010:to a quantum mechanical
1516:{\displaystyle a=x^{*}x}
653:The spaces of operators
474:. Such maps are called
187:{\displaystyle L(H_{A})}
8550:Quantum finite automata
7583:Randomized benchmarking
7445:Amplitude amplification
7059:Wilde, Mark M. (2017),
9053:Magnetic refrigeration
8654:Quantum neural network
7683:Quantum Turing machine
7676:quantum neural network
7423:Quantum secret sharing
6942:
6903:
6815:
6777:
6714:
6682:
6586:
6396:
6328:
6293:
6261:
6232:
6197:
6150:
6121:
6091:
6056:
6009:
5979:
5945:
5909:
5803:
5736:
5703:
5637:
5607:
5563:
5485:
5438:
5404:
5381:
5285:
5142:
5058:
4994:
4925:
4848:
4828:
4796:
4751:
4728:
4626:
4606:
4578:
4558:
4538:
4492:
4450:
4422:
4350:
4326:
4280:
4247:
4196:
4169:
4101:
4081:
4031:
3946:
3869:
3754:
3603:
3565:
3447:
3396:
3287:
3249:
3229:
3166:
3121:
3091:
2962:
2868:
2739:
2701:and with output space
2695:
2660:
2640:
2575:
2461:
2352:
2322:
2291:
2219:
2131:
2104:
2061:
2034:
2004:
1950:
1902:
1825:
1773:
1722:
1665:
1612:
1537:
1517:
1487:is called positive if
1481:
1457:
1437:
1383:
1298:
1271:
1238:
1209:
1189:
1169:
1137:
1080:
1004:
958:
931:
900:
865:
842:
761:
741:
640:
620:
593:
573:
527:
495:
464:
422:
371:
347:
324:
300:
273:
242:
218:
217:{\displaystyle H_{A}.}
188:
148:
121:
9026:Cloak of invisibility
8825:Emerging technologies
8679:Quantum teleportation
8192:Waveâparticle duality
7755:Entanglement-assisted
7716:quantum convolutional
7391:Quantum coin flipping
7356:Quantum teleportation
7317:entanglement-assisted
7147:DiVincenzo's criteria
6943:
6904:
6868:, and it arises from
6816:
6778:
6715:
6683:
6587:
6397:
6329:
6294:
6262:
6233:
6198:
6151:
6122:
6092:
6057:
6010:
5980:
5978:{\displaystyle \Psi }
5946:
5910:
5804:
5737:
5704:
5638:
5608:
5606:{\displaystyle \Psi }
5564:
5486:
5439:
5405:
5382:
5286:
5143:
5059:
4995:
4926:
4849:
4847:{\displaystyle \Phi }
4829:
4797:
4752:
4750:{\displaystyle \Phi }
4729:
4627:
4612:to the ideal channel
4607:
4605:{\displaystyle \Phi }
4579:
4559:
4539:
4537:{\displaystyle \Phi }
4493:
4451:
4423:
4390:time-bin entanglement
4370:quantum teleportation
4351:
4349:{\displaystyle \Psi }
4327:
4325:{\displaystyle \Psi }
4281:
4227:
4197:
4195:{\displaystyle \Psi }
4170:
4102:
4100:{\displaystyle \Psi }
4082:
4080:{\displaystyle \Psi }
4053:channel-state duality
4032:
3947:
3870:
3755:
3604:
3602:{\displaystyle \Phi }
3566:
3448:
3446:{\displaystyle \Phi }
3397:
3288:
3250:
3248:{\displaystyle \Psi }
3230:
3167:
3122:
3092:
2963:
2869:
2740:
2696:
2661:
2659:{\displaystyle \Phi }
2641:
2576:
2462:
2353:
2323:
2321:{\displaystyle \Psi }
2292:
2220:
2132:
2130:{\displaystyle \Psi }
2105:
2062:
2060:{\displaystyle F_{i}}
2035:
2033:{\displaystyle F_{i}}
2005:
1951:
1903:
1834:the reduced state of
1826:
1774:
1723:
1666:
1613:
1538:
1518:
1482:
1458:
1438:
1384:
1299:
1272:
1239:
1210:
1190:
1170:
1138:
1081:
1014:Classical information
1005:
959:
932:
930:{\displaystyle \Phi }
901:
866:
864:{\displaystyle \Phi }
843:
762:
760:{\displaystyle \Phi }
742:
641:
639:{\displaystyle \Phi }
621:
594:
592:{\displaystyle \Phi }
574:
572:{\displaystyle \Phi }
528:
526:{\displaystyle \Phi }
496:
494:{\displaystyle \Phi }
465:
423:
372:
370:{\displaystyle \Phi }
348:
346:{\displaystyle \Phi }
325:
323:{\displaystyle \Phi }
301:
299:{\displaystyle H_{B}}
274:
272:{\displaystyle H_{A}}
243:
241:{\displaystyle \Phi }
219:
189:
149:
147:{\displaystyle H_{B}}
122:
120:{\displaystyle H_{A}}
8710:Quantum field theory
8639:Quantum metamaterial
8584:Quantum cryptography
8314:Consistent histories
7566:processor benchmarks
7495:Quantum optimization
7378:Quantum cryptography
7189:physical vs. logical
6917:
6884:
6790:
6734:
6695:
6645:
6415:
6341:
6303:
6274:
6242:
6207:
6172:
6156:. Thus for a purely
6131:
6101:
6097:, the ideal channel
6077:
6020:
5989:
5969:
5965:channel capacity of
5922:
5817:
5746:
5713:
5653:
5617:
5597:
5501:
5456:
5418:
5410:operation describes
5394:
5312:
5178:
5071:
5004:
4950:
4861:
4838:
4809:
4767:
4741:
4639:
4616:
4596:
4568:
4548:
4528:
4460:
4432:
4404:
4398:quantum cryptography
4340:
4316:
4209:
4186:
4114:
4091:
4071:
3959:
3893:
3770:
3619:
3593:
3467:
3437:
3413:Suppose two parties
3300:
3262:
3239:
3176:
3131:
3104:
2978:
2884:
2752:
2705:
2670:
2650:
2598:
2485:
2365:
2335:
2312:
2300:In other words, one
2235:
2144:
2121:
2071:
2044:
2017:
1979:
1918:
1865:
1854:with respect to the
1786:
1740:
1690:
1625:
1579:
1527:
1491:
1467:
1447:
1396:
1311:
1284:
1252:
1237:{\displaystyle C(X)}
1219:
1199:
1179:
1168:{\displaystyle C(X)}
1150:
1104:
1026:
972:
941:
921:
883:
855:
774:
751:
687:
630:
603:
583:
563:
517:
485:
470:is positive for all
441:
396:
361:
337:
314:
283:
256:
232:
198:
162:
131:
104:
9058:Phased-array optics
9016:Acoustic levitation
8695:Quantum fluctuation
8664:Quantum programming
8624:Quantum logic gates
8609:Quantum information
8589:Quantum electronics
8049:Classical mechanics
7279:Quantum speed limit
7174:Quantum programming
7169:Quantum information
7079:2011arXiv1106.1445W
7011:2012RvMP...84..621W
5895:
5593:achievable rate of
5555:
5335:
4275:
4202:must take the form
3941:
3923:
3880:measure-and-prepare
3456:simply consists of
3161:
2474:, and invoking the
476:completely positive
330:needs to be linear.
226:Schrödinger picture
89:Schrödinger picture
55:completely positive
40:quantum information
8748:in popular culture
8530:Quantum algorithms
8378:Von NeumannâWigner
8358:Objective collapse
8054:Old quantum theory
7928:Forest/Rigetti QCS
7664:quantum logic gate
7450:BernsteinâVazirani
7437:Quantum algorithms
7312:Classical capacity
7196:Quantum processors
7179:Quantum simulation
6938:
6899:
6844:. You can help by
6811:
6773:
6710:
6678:
6638:of a channel Κ is
6636:classical capacity
6582:
6392:
6324:
6289:
6257:
6228:
6193:
6146:
6117:
6087:
6069:Important examples
6052:
6005:
5975:
5941:
5905:
5868:
5829:
5799:
5761:
5732:
5699:
5649:For all sequences
5633:
5603:
5559:
5535:
5481:
5434:
5400:
5377:
5315:
5281:
5236:
5169:best case scenario
5155:is an encoder and
5138:
5054:
4990:
4921:
4895:
4844:
4824:
4792:
4747:
4724:
4632:can be defined by
4622:
4602:
4574:
4554:
4534:
4488:
4446:
4418:
4346:
4322:
4276:
4261:
4192:
4165:
4097:
4077:
4027:
3994:
3942:
3927:
3909:
3865:
3832:
3750:
3717:
3695:
3599:
3561:
3552:
3443:
3392:
3368:
3336:
3283:
3255:is CP and unital.
3245:
3225:
3162:
3147:
3117:
3087:
3081:
2958:
2934:
2864:
2855:
2735:
2691:
2656:
2636:
2592:quantum instrument
2571:
2562:
2457:
2448:
2348:
2328:is CP and unital.
2318:
2287:
2263:
2215:
2194:
2127:
2100:
2083:
2057:
2030:
2000:
1946:
1898:
1821:
1769:
1718:
1661:
1608:
1553:Frobenius category
1545:quantum instrument
1533:
1513:
1477:
1453:
1433:
1379:
1294:
1267:
1234:
1205:
1185:
1165:
1133:
1076:
1000:
954:
927:
910:to observables on
896:
861:
838:
757:
737:
648:Heisenberg picture
636:
616:
589:
569:
537:Heisenberg picture
523:
491:
460:
418:
367:
353:must preserve the
343:
320:
296:
269:
238:
214:
184:
144:
117:
83:information theory
9095:
9094:
9071:
9070:
8878:complexity theory
8863:cellular automata
8791:
8790:
8765:Quantum mysticism
8743:Schrödinger's cat
8674:Quantum simulator
8644:Quantum metrology
8572:Quantum computing
8535:Quantum amplifier
8512:Quantum spacetime
8477:Quantum cosmology
8467:Quantum chemistry
8167:Scattering theory
8115:Zero-point energy
8110:Degenerate levels
8018:Quantum mechanics
7984:
7983:
7895:
7894:
7792:Linear optical QC
7573:Quantum supremacy
7527:complexity theory
7480:Quantum annealing
7431:
7430:
7368:Superdense coding
7157:Quantum computing
6862:
6861:
6624:two bit per qubit
6620:Superdense coding
6605:one bit per qubit
6577:
5846:
5820:
5752:
5520:
5469:
5249:
5221:
5196:
5083:
4886:
3985:
3823:
3708:
3359:
3327:
3312:
2254:
2074:
1549:Frobenius algebra
1536:{\displaystyle x}
1456:{\displaystyle a}
1208:{\displaystyle X}
1188:{\displaystyle X}
95:quantum operation
59:quantum operation
16:(Redirected from
9115:
9083:
9082:
8960:machine learning
8935:key distribution
8920:image processing
8910:error correction
8836:
8835:
8818:
8811:
8804:
8795:
8794:
8781:
8780:
8492:Quantum geometry
8487:Quantum dynamics
8344:Superdeterminism
8276:RaritaâSchwinger
8225:Matrix mechanics
8080:Braâket notation
8011:
8004:
7997:
7988:
7987:
7974:
7973:
7964:
7963:
7770:
7769:
7700:error correction
7629:computing models
7595:Relaxation times
7485:Quantum counting
7374:
7373:
7322:quantum capacity
7269:No-teleportation
7254:No-communication
7126:
7119:
7112:
7103:
7102:
7097:
7072:
7044:
7037:
7031:
7030:
7004:
6984:
6947:
6945:
6944:
6939:
6908:
6906:
6905:
6900:
6866:channel fidelity
6857:
6854:
6836:
6829:
6825:Channel fidelity
6820:
6818:
6817:
6812:
6810:
6809:
6798:
6782:
6780:
6779:
6774:
6766:
6765:
6754:
6725:quantum capacity
6719:
6717:
6716:
6711:
6709:
6708:
6703:
6687:
6685:
6684:
6679:
6671:
6670:
6665:
6591:
6589:
6588:
6583:
6578:
6576:
6565:
6554:
6546:
6545:
6540:
6531:
6530:
6519:
6501:
6500:
6489:
6480:
6479:
6468:
6450:
6449:
6444:
6435:
6434:
6429:
6401:
6399:
6398:
6393:
6382:
6381:
6370:
6361:
6360:
6355:
6333:
6331:
6330:
6325:
6323:
6322:
6311:
6298:
6296:
6295:
6290:
6288:
6287:
6282:
6266:
6264:
6263:
6258:
6256:
6255:
6250:
6237:
6235:
6234:
6229:
6227:
6226:
6215:
6202:
6200:
6199:
6194:
6192:
6191:
6180:
6155:
6153:
6152:
6147:
6145:
6144:
6143:
6126:
6124:
6123:
6118:
6116:
6115:
6096:
6094:
6093:
6088:
6086:
6085:
6061:
6059:
6058:
6053:
6048:
6047:
6014:
6012:
6011:
6006:
6004:
6003:
5985:with respect to
5984:
5982:
5981:
5976:
5950:
5948:
5947:
5942:
5937:
5936:
5914:
5912:
5911:
5906:
5894:
5893:
5892:
5879:
5864:
5863:
5862:
5861:
5848:
5847:
5839:
5828:
5808:
5806:
5805:
5800:
5789:
5788:
5779:
5774:
5773:
5760:
5741:
5739:
5738:
5733:
5725:
5724:
5708:
5706:
5705:
5700:
5698:
5687:
5686:
5668:
5667:
5642:
5640:
5639:
5634:
5632:
5631:
5613:with respect to
5612:
5610:
5609:
5604:
5568:
5566:
5565:
5560:
5554:
5546:
5531:
5530:
5522:
5521:
5513:
5490:
5488:
5487:
5482:
5480:
5479:
5471:
5470:
5462:
5443:
5441:
5440:
5435:
5433:
5432:
5409:
5407:
5406:
5401:
5386:
5384:
5383:
5378:
5373:
5372:
5351:
5350:
5334:
5326:
5290:
5288:
5287:
5282:
5280:
5279:
5267:
5266:
5251:
5250:
5242:
5235:
5214:
5213:
5198:
5197:
5189:
5147:
5145:
5144:
5139:
5137:
5136:
5131:
5130:
5120:
5119:
5114:
5113:
5085:
5084:
5076:
5063:
5061:
5060:
5055:
5053:
5052:
5047:
5046:
5036:
5035:
5030:
5029:
5019:
5018:
4999:
4997:
4996:
4991:
4989:
4988:
4983:
4982:
4972:
4971:
4966:
4965:
4930:
4928:
4927:
4922:
4914:
4913:
4894:
4882:
4881:
4853:
4851:
4850:
4845:
4833:
4831:
4830:
4825:
4801:
4799:
4798:
4793:
4788:
4787:
4756:
4754:
4753:
4748:
4733:
4731:
4730:
4725:
4701:
4631:
4629:
4628:
4623:
4611:
4609:
4608:
4603:
4583:
4581:
4580:
4575:
4563:
4561:
4560:
4555:
4543:
4541:
4540:
4535:
4507:Channel capacity
4497:
4495:
4494:
4489:
4481:
4467:
4455:
4453:
4452:
4447:
4439:
4427:
4425:
4424:
4419:
4411:
4355:
4353:
4352:
4347:
4331:
4329:
4328:
4323:
4285:
4283:
4282:
4277:
4274:
4269:
4257:
4256:
4246:
4241:
4201:
4199:
4198:
4193:
4174:
4172:
4171:
4166:
4161:
4160:
4149:
4140:
4139:
4128:
4106:
4104:
4103:
4098:
4086:
4084:
4083:
4078:
4036:
4034:
4033:
4028:
4023:
4022:
4004:
4003:
3993:
3972:
3971:
3951:
3949:
3948:
3943:
3940:
3935:
3922:
3917:
3905:
3904:
3874:
3872:
3871:
3866:
3861:
3860:
3848:
3847:
3831:
3810:
3809:
3797:
3796:
3759:
3757:
3756:
3751:
3746:
3745:
3733:
3732:
3716:
3704:
3700:
3699:
3689:
3688:
3659:
3658:
3631:
3630:
3608:
3606:
3605:
3600:
3570:
3568:
3567:
3562:
3557:
3556:
3546:
3545:
3516:
3515:
3480:
3479:
3452:
3450:
3449:
3444:
3401:
3399:
3398:
3393:
3391:
3390:
3378:
3377:
3367:
3346:
3345:
3335:
3314:
3313:
3305:
3292:
3290:
3289:
3284:
3254:
3252:
3251:
3246:
3234:
3232:
3231:
3226:
3224:
3223:
3211:
3210:
3189:
3188:
3171:
3169:
3168:
3163:
3160:
3155:
3143:
3142:
3126:
3124:
3123:
3118:
3116:
3115:
3096:
3094:
3093:
3088:
3086:
3085:
3069:
3068:
3059:
3058:
3029:
3028:
3019:
3018:
2967:
2965:
2964:
2959:
2939:
2938:
2931:
2930:
2910:
2909:
2873:
2871:
2870:
2865:
2860:
2859:
2852:
2851:
2836:
2835:
2809:
2808:
2793:
2792:
2744:
2742:
2741:
2736:
2700:
2698:
2697:
2692:
2665:
2663:
2662:
2657:
2645:
2643:
2642:
2637:
2632:
2631:
2613:
2612:
2580:
2578:
2577:
2572:
2567:
2566:
2556:
2555:
2526:
2525:
2466:
2464:
2463:
2458:
2453:
2452:
2436:
2435:
2403:
2402:
2357:
2355:
2354:
2349:
2347:
2346:
2327:
2325:
2324:
2319:
2306:against the POVM
2296:
2294:
2293:
2288:
2283:
2282:
2273:
2272:
2262:
2224:
2222:
2221:
2216:
2199:
2198:
2191:
2190:
2170:
2169:
2136:
2134:
2133:
2128:
2109:
2107:
2106:
2101:
2093:
2092:
2082:
2066:
2064:
2063:
2058:
2056:
2055:
2039:
2037:
2036:
2031:
2029:
2028:
2009:
2007:
2006:
2001:
1999:
1991:
1990:
1955:
1953:
1952:
1947:
1942:
1941:
1907:
1905:
1904:
1899:
1890:
1889:
1877:
1876:
1830:
1828:
1827:
1822:
1817:
1816:
1804:
1803:
1778:
1776:
1775:
1770:
1765:
1764:
1752:
1751:
1727:
1725:
1724:
1719:
1708:
1707:
1670:
1668:
1667:
1662:
1660:
1659:
1655:
1617:
1615:
1614:
1609:
1604:
1603:
1542:
1540:
1539:
1534:
1522:
1520:
1519:
1514:
1509:
1508:
1486:
1484:
1483:
1478:
1476:
1475:
1463:of a C*-algebra
1462:
1460:
1459:
1454:
1442:
1440:
1439:
1434:
1414:
1413:
1388:
1386:
1385:
1380:
1372:
1371:
1335:
1334:
1303:
1301:
1300:
1295:
1293:
1292:
1276:
1274:
1273:
1268:
1266:
1265:
1260:
1243:
1241:
1240:
1235:
1214:
1212:
1211:
1206:
1194:
1192:
1191:
1186:
1174:
1172:
1171:
1166:
1142:
1140:
1139:
1134:
1129:
1128:
1119:
1118:
1085:
1083:
1082:
1077:
1072:
1071:
1050:
1049:
1009:
1007:
1006:
1001:
984:
983:
963:
961:
960:
955:
953:
952:
936:
934:
933:
928:
905:
903:
902:
897:
895:
894:
871:takes states on
870:
868:
867:
862:
847:
845:
844:
839:
816:
815:
766:
764:
763:
758:
746:
744:
743:
738:
733:
732:
711:
710:
645:
643:
642:
637:
625:
623:
622:
617:
615:
614:
598:
596:
595:
590:
578:
576:
575:
570:
532:
530:
529:
524:
500:
498:
497:
492:
469:
467:
466:
461:
453:
452:
427:
425:
424:
419:
408:
407:
376:
374:
373:
368:
352:
350:
349:
344:
329:
327:
326:
321:
305:
303:
302:
297:
295:
294:
278:
276:
275:
270:
268:
267:
250:density matrices
247:
245:
244:
239:
223:
221:
220:
215:
210:
209:
193:
191:
190:
185:
180:
179:
153:
151:
150:
145:
143:
142:
126:
124:
123:
118:
116:
115:
63:reduced dynamics
21:
9123:
9122:
9118:
9117:
9116:
9114:
9113:
9112:
9098:
9097:
9096:
9091:
9067:
9004:
8915:finite automata
8827:
8822:
8792:
8787:
8769:
8755:Wigner's friend
8731:
8722:Quantum gravity
8683:
8669:Quantum sensing
8649:Quantum network
8629:Quantum machine
8599:Quantum imaging
8562:Quantum circuit
8557:Quantum channel
8516:
8462:Quantum biology
8448:
8424:ElitzurâVaidman
8399:DavissonâGermer
8382:
8334:Hidden-variable
8324:de BroglieâBohm
8301:Interpretations
8295:
8244:
8198:
8085:Complementarity
8063:
8020:
8015:
7985:
7980:
7952:
7902:
7891:
7864:Superconducting
7858:
7824:
7815:Neutral atom QC
7807:Ultracold atoms
7801:
7766:implementations
7765:
7759:
7699:
7692:
7659:Quantum circuit
7627:
7621:
7615:
7605:
7565:
7559:
7526:
7519:
7475:Hidden subgroup
7427:
7416:other protocols
7372:
7349:quantum network
7344:Quantum channel
7304:
7298:
7244:No-broadcasting
7234:GottesmanâKnill
7207:
7135:
7130:
7048:
7047:
7038:
7034:
6985:
6981:
6976:
6954:
6918:
6915:
6914:
6885:
6882:
6881:
6878:
6858:
6852:
6849:
6842:needs expansion
6827:
6799:
6794:
6793:
6791:
6788:
6787:
6755:
6750:
6749:
6735:
6732:
6731:
6704:
6699:
6698:
6696:
6693:
6692:
6666:
6661:
6660:
6646:
6643:
6642:
6632:
6566:
6555:
6553:
6541:
6536:
6535:
6520:
6515:
6514:
6490:
6485:
6484:
6469:
6464:
6463:
6445:
6440:
6439:
6430:
6425:
6424:
6416:
6413:
6412:
6371:
6366:
6365:
6356:
6351:
6350:
6342:
6339:
6338:
6312:
6307:
6306:
6304:
6301:
6300:
6283:
6278:
6277:
6275:
6272:
6271:
6251:
6246:
6245:
6243:
6240:
6239:
6216:
6211:
6210:
6208:
6205:
6204:
6181:
6176:
6175:
6173:
6170:
6169:
6139:
6138:
6134:
6132:
6129:
6128:
6108:
6104:
6102:
6099:
6098:
6081:
6080:
6078:
6075:
6074:
6071:
6040:
6036:
6021:
6018:
6017:
5996:
5992:
5990:
5987:
5986:
5970:
5967:
5966:
5932:
5928:
5923:
5920:
5919:
5888:
5884:
5880:
5872:
5857:
5853:
5849:
5838:
5837:
5836:
5824:
5818:
5815:
5814:
5784:
5780:
5775:
5769:
5765:
5756:
5747:
5744:
5743:
5720:
5716:
5714:
5711:
5710:
5694:
5682:
5678:
5663:
5659:
5654:
5651:
5650:
5624:
5620:
5618:
5615:
5614:
5598:
5595:
5594:
5547:
5539:
5523:
5512:
5511:
5510:
5502:
5499:
5498:
5472:
5461:
5460:
5459:
5457:
5454:
5453:
5452:corresponds to
5425:
5421:
5419:
5416:
5415:
5395:
5392:
5391:
5365:
5361:
5343:
5339:
5327:
5319:
5313:
5310:
5309:
5272:
5268:
5259:
5255:
5241:
5240:
5225:
5206:
5202:
5188:
5187:
5179:
5176:
5175:
5132:
5126:
5125:
5124:
5115:
5109:
5108:
5107:
5075:
5074:
5072:
5069:
5068:
5048:
5042:
5041:
5040:
5031:
5025:
5024:
5023:
5011:
5007:
5005:
5002:
5001:
4984:
4978:
4977:
4976:
4967:
4961:
4960:
4959:
4951:
4948:
4947:
4937:
4909:
4905:
4890:
4874:
4870:
4862:
4859:
4858:
4839:
4836:
4835:
4810:
4807:
4806:
4783:
4779:
4768:
4765:
4764:
4742:
4739:
4738:
4697:
4640:
4637:
4636:
4617:
4614:
4613:
4597:
4594:
4593:
4569:
4566:
4565:
4549:
4546:
4545:
4529:
4526:
4525:
4514:
4509:
4477:
4463:
4461:
4458:
4457:
4435:
4433:
4430:
4429:
4407:
4405:
4402:
4401:
4378:
4366:
4341:
4338:
4337:
4317:
4314:
4313:
4309:Kraus operators
4305:
4297:. The matrices
4270:
4265:
4252:
4248:
4242:
4231:
4210:
4207:
4206:
4187:
4184:
4183:
4150:
4145:
4144:
4129:
4124:
4123:
4115:
4112:
4111:
4092:
4089:
4088:
4072:
4069:
4068:
4065:
4018:
4014:
3999:
3995:
3989:
3967:
3963:
3960:
3957:
3956:
3936:
3931:
3918:
3913:
3900:
3896:
3894:
3891:
3890:
3856:
3852:
3843:
3839:
3827:
3805:
3801:
3792:
3788:
3771:
3768:
3767:
3741:
3737:
3728:
3724:
3712:
3694:
3693:
3684:
3680:
3671:
3670:
3664:
3663:
3654:
3650:
3637:
3636:
3632:
3626:
3622:
3620:
3617:
3616:
3611:
3594:
3591:
3590:
3587:
3551:
3550:
3541:
3537:
3528:
3527:
3521:
3520:
3511:
3507:
3494:
3493:
3475:
3471:
3468:
3465:
3464:
3455:
3438:
3435:
3434:
3411:
3386:
3382:
3373:
3369:
3363:
3341:
3337:
3331:
3304:
3303:
3301:
3298:
3297:
3263:
3260:
3259:
3240:
3237:
3236:
3219:
3215:
3206:
3202:
3184:
3180:
3177:
3174:
3173:
3156:
3151:
3138:
3134:
3132:
3129:
3128:
3111:
3107:
3105:
3102:
3101:
3080:
3079:
3064:
3060:
3054:
3050:
3047:
3046:
3040:
3039:
3024:
3020:
3014:
3010:
3003:
3002:
2979:
2976:
2975:
2933:
2932:
2926:
2922:
2919:
2918:
2912:
2911:
2905:
2901:
2894:
2893:
2885:
2882:
2881:
2854:
2853:
2847:
2843:
2831:
2827:
2818:
2817:
2811:
2810:
2804:
2800:
2788:
2784:
2771:
2770:
2753:
2750:
2749:
2706:
2703:
2702:
2671:
2668:
2667:
2651:
2648:
2647:
2627:
2623:
2608:
2604:
2599:
2596:
2595:
2587:
2561:
2560:
2551:
2547:
2538:
2537:
2531:
2530:
2521:
2517:
2504:
2503:
2486:
2483:
2482:
2447:
2446:
2431:
2427:
2421:
2420:
2414:
2413:
2398:
2394:
2384:
2383:
2366:
2363:
2362:
2342:
2338:
2336:
2333:
2332:
2313:
2310:
2309:
2278:
2274:
2268:
2264:
2258:
2236:
2233:
2232:
2193:
2192:
2186:
2182:
2179:
2178:
2172:
2171:
2165:
2161:
2154:
2153:
2145:
2142:
2141:
2122:
2119:
2118:
2088:
2084:
2078:
2072:
2069:
2068:
2051:
2047:
2045:
2042:
2041:
2024:
2020:
2018:
2015:
2014:
1995:
1986:
1982:
1980:
1977:
1976:
1973:
1937:
1933:
1919:
1916:
1915:
1885:
1881:
1872:
1868:
1866:
1863:
1862:
1812:
1808:
1799:
1795:
1787:
1784:
1783:
1760:
1756:
1747:
1743:
1741:
1738:
1737:
1734:
1703:
1699:
1691:
1688:
1687:
1651:
1638:
1634:
1626:
1623:
1622:
1599:
1595:
1580:
1577:
1576:
1566:
1561:
1528:
1525:
1524:
1504:
1500:
1492:
1489:
1488:
1471:
1470:
1468:
1465:
1464:
1448:
1445:
1444:
1409:
1405:
1397:
1394:
1393:
1367:
1363:
1330:
1326:
1312:
1309:
1308:
1288:
1287:
1285:
1282:
1281:
1261:
1256:
1255:
1253:
1250:
1249:
1220:
1217:
1216:
1200:
1197:
1196:
1180:
1177:
1176:
1151:
1148:
1147:
1124:
1123:
1114:
1113:
1105:
1102:
1101:
1067:
1063:
1045:
1041:
1027:
1024:
1023:
1016:
979:
975:
973:
970:
969:
948:
944:
942:
939:
938:
922:
919:
918:
890:
886:
884:
881:
880:
856:
853:
852:
811:
807:
775:
772:
771:
752:
749:
748:
728:
724:
706:
702:
688:
685:
684:
681:HilbertâSchmidt
678:
665:
631:
628:
627:
610:
606:
604:
601:
600:
584:
581:
580:
564:
561:
560:
553:
546:
539:
518:
515:
514:
505:The adjectives
486:
483:
482:
448:
444:
442:
439:
438:
436:
403:
399:
397:
394:
393:
362:
359:
358:
338:
335:
334:
315:
312:
311:
290:
286:
284:
281:
280:
263:
259:
257:
254:
253:
233:
230:
229:
205:
201:
199:
196:
195:
175:
171:
163:
160:
159:
138:
134:
132:
129:
128:
111:
107:
105:
102:
101:
91:
72:
36:quantum channel
28:
23:
22:
15:
12:
11:
5:
9121:
9111:
9110:
9093:
9092:
9090:
9089:
9076:
9073:
9072:
9069:
9068:
9066:
9065:
9060:
9055:
9050:
9045:
9044:
9043:
9033:
9028:
9023:
9018:
9012:
9010:
9006:
9005:
9003:
9002:
8997:
8992:
8987:
8982:
8977:
8975:neural network
8972:
8967:
8962:
8957:
8952:
8947:
8942:
8937:
8932:
8927:
8922:
8917:
8912:
8907:
8902:
8897:
8896:
8895:
8885:
8880:
8875:
8870:
8865:
8860:
8855:
8850:
8844:
8842:
8833:
8829:
8828:
8821:
8820:
8813:
8806:
8798:
8789:
8788:
8786:
8785:
8774:
8771:
8770:
8768:
8767:
8762:
8757:
8752:
8751:
8750:
8739:
8737:
8733:
8732:
8730:
8729:
8724:
8719:
8718:
8717:
8707:
8702:
8700:Casimir effect
8697:
8691:
8689:
8685:
8684:
8682:
8681:
8676:
8671:
8666:
8661:
8659:Quantum optics
8656:
8651:
8646:
8641:
8636:
8631:
8626:
8621:
8616:
8611:
8606:
8601:
8596:
8591:
8586:
8581:
8580:
8579:
8569:
8564:
8559:
8554:
8553:
8552:
8542:
8537:
8532:
8526:
8524:
8518:
8517:
8515:
8514:
8509:
8504:
8499:
8494:
8489:
8484:
8479:
8474:
8469:
8464:
8458:
8456:
8450:
8449:
8447:
8446:
8441:
8436:
8434:Quantum eraser
8431:
8426:
8421:
8416:
8411:
8406:
8401:
8396:
8390:
8388:
8384:
8383:
8381:
8380:
8375:
8370:
8365:
8360:
8355:
8350:
8349:
8348:
8347:
8346:
8331:
8326:
8321:
8316:
8311:
8305:
8303:
8297:
8296:
8294:
8293:
8288:
8283:
8278:
8273:
8268:
8263:
8258:
8252:
8250:
8246:
8245:
8243:
8242:
8237:
8232:
8227:
8222:
8217:
8212:
8206:
8204:
8200:
8199:
8197:
8196:
8195:
8194:
8189:
8179:
8174:
8169:
8164:
8159:
8154:
8149:
8144:
8139:
8134:
8129:
8124:
8119:
8118:
8117:
8112:
8107:
8102:
8092:
8090:Density matrix
8087:
8082:
8077:
8071:
8069:
8065:
8064:
8062:
8061:
8056:
8051:
8046:
8045:
8044:
8034:
8028:
8026:
8022:
8021:
8014:
8013:
8006:
7999:
7991:
7982:
7981:
7979:
7978:
7968:
7957:
7954:
7953:
7951:
7950:
7948:many others...
7945:
7940:
7935:
7930:
7921:
7907:
7905:
7897:
7896:
7893:
7892:
7890:
7889:
7884:
7879:
7874:
7868:
7866:
7860:
7859:
7857:
7856:
7851:
7846:
7841:
7835:
7833:
7826:
7825:
7823:
7822:
7820:Trapped-ion QC
7817:
7811:
7809:
7803:
7802:
7800:
7799:
7794:
7789:
7784:
7778:
7776:
7774:Quantum optics
7767:
7761:
7760:
7758:
7757:
7752:
7751:
7750:
7743:
7738:
7733:
7728:
7723:
7718:
7713:
7704:
7702:
7694:
7693:
7691:
7690:
7685:
7680:
7679:
7678:
7668:
7667:
7666:
7656:
7655:
7654:
7644:
7639:
7633:
7631:
7623:
7622:
7620:
7619:
7618:
7617:
7613:
7607:
7603:
7592:
7591:
7590:
7580:
7578:Quantum volume
7575:
7569:
7567:
7561:
7560:
7558:
7557:
7552:
7547:
7542:
7537:
7531:
7529:
7521:
7520:
7518:
7517:
7512:
7507:
7502:
7497:
7492:
7487:
7482:
7477:
7472:
7467:
7462:
7457:
7455:Boson sampling
7452:
7447:
7441:
7439:
7433:
7432:
7429:
7428:
7426:
7425:
7420:
7419:
7418:
7413:
7408:
7398:
7393:
7388:
7382:
7380:
7371:
7370:
7365:
7364:
7363:
7353:
7352:
7351:
7341:
7336:
7331:
7326:
7325:
7324:
7319:
7308:
7306:
7300:
7299:
7297:
7296:
7291:
7289:SolovayâKitaev
7286:
7281:
7276:
7271:
7266:
7261:
7256:
7251:
7246:
7241:
7236:
7231:
7226:
7221:
7215:
7213:
7209:
7208:
7206:
7205:
7204:
7203:
7193:
7192:
7191:
7181:
7176:
7171:
7166:
7165:
7164:
7154:
7149:
7143:
7141:
7137:
7136:
7129:
7128:
7121:
7114:
7106:
7100:
7099:
7056:
7046:
7045:
7032:
6995:(2): 621â669.
6978:
6977:
6975:
6972:
6971:
6970:
6965:
6960:
6953:
6950:
6937:
6934:
6931:
6928:
6925:
6922:
6898:
6895:
6892:
6889:
6877:
6874:
6860:
6859:
6839:
6837:
6826:
6823:
6808:
6805:
6802:
6797:
6784:
6783:
6772:
6769:
6764:
6761:
6758:
6753:
6748:
6745:
6742:
6739:
6723:Similarly the
6707:
6702:
6689:
6688:
6677:
6674:
6669:
6664:
6659:
6656:
6653:
6650:
6631:
6628:
6593:
6592:
6581:
6575:
6572:
6569:
6564:
6561:
6558:
6552:
6549:
6544:
6539:
6534:
6529:
6526:
6523:
6518:
6513:
6510:
6507:
6504:
6499:
6496:
6493:
6488:
6483:
6478:
6475:
6472:
6467:
6462:
6459:
6456:
6453:
6448:
6443:
6438:
6433:
6428:
6423:
6420:
6403:
6402:
6391:
6388:
6385:
6380:
6377:
6374:
6369:
6364:
6359:
6354:
6349:
6346:
6321:
6318:
6315:
6310:
6286:
6281:
6254:
6249:
6225:
6222:
6219:
6214:
6190:
6187:
6184:
6179:
6142:
6137:
6114:
6111:
6107:
6084:
6070:
6067:
6051:
6046:
6043:
6039:
6035:
6032:
6029:
6026:
6002:
5999:
5995:
5974:
5940:
5935:
5931:
5927:
5916:
5915:
5904:
5901:
5898:
5891:
5887:
5883:
5878:
5875:
5871:
5867:
5860:
5856:
5852:
5845:
5842:
5835:
5832:
5827:
5823:
5811:
5810:
5798:
5795:
5792:
5787:
5783:
5778:
5772:
5768:
5764:
5759:
5755:
5751:
5731:
5728:
5723:
5719:
5697:
5693:
5690:
5685:
5681:
5677:
5674:
5671:
5666:
5662:
5658:
5646:
5645:
5630:
5627:
5623:
5602:
5570:
5569:
5558:
5553:
5550:
5545:
5542:
5538:
5534:
5529:
5526:
5519:
5516:
5509:
5506:
5478:
5475:
5468:
5465:
5431:
5428:
5424:
5399:
5388:
5387:
5376:
5371:
5368:
5364:
5360:
5357:
5354:
5349:
5346:
5342:
5338:
5333:
5330:
5325:
5322:
5318:
5292:
5291:
5278:
5275:
5271:
5265:
5262:
5258:
5254:
5248:
5245:
5239:
5234:
5231:
5228:
5224:
5220:
5217:
5212:
5209:
5205:
5201:
5195:
5192:
5186:
5183:
5149:
5148:
5135:
5129:
5123:
5118:
5112:
5106:
5103:
5100:
5097:
5094:
5091:
5088:
5082:
5079:
5051:
5045:
5039:
5034:
5028:
5022:
5017:
5014:
5010:
4987:
4981:
4975:
4970:
4964:
4958:
4955:
4936:
4933:
4932:
4931:
4920:
4917:
4912:
4908:
4904:
4901:
4898:
4893:
4889:
4885:
4880:
4877:
4873:
4869:
4866:
4843:
4823:
4820:
4817:
4814:
4803:
4802:
4791:
4786:
4782:
4778:
4775:
4772:
4746:
4735:
4734:
4723:
4720:
4717:
4714:
4711:
4708:
4705:
4700:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4621:
4601:
4573:
4553:
4533:
4513:
4510:
4508:
4505:
4487:
4484:
4480:
4476:
4473:
4470:
4466:
4445:
4442:
4438:
4417:
4414:
4410:
4377:
4374:
4365:
4362:
4345:
4321:
4301:
4287:
4286:
4273:
4268:
4264:
4260:
4255:
4251:
4245:
4240:
4237:
4234:
4230:
4226:
4223:
4220:
4217:
4214:
4191:
4176:
4175:
4164:
4159:
4156:
4153:
4148:
4143:
4138:
4135:
4132:
4127:
4122:
4119:
4096:
4076:
4064:
4061:
4038:
4037:
4026:
4021:
4017:
4013:
4010:
4007:
4002:
3998:
3992:
3988:
3984:
3981:
3978:
3975:
3970:
3966:
3952:is defined by
3939:
3934:
3930:
3926:
3921:
3916:
3912:
3908:
3903:
3899:
3876:
3875:
3864:
3859:
3855:
3851:
3846:
3842:
3838:
3835:
3830:
3826:
3822:
3819:
3816:
3813:
3808:
3804:
3800:
3795:
3791:
3787:
3784:
3781:
3778:
3775:
3761:
3760:
3749:
3744:
3740:
3736:
3731:
3727:
3723:
3720:
3715:
3711:
3707:
3703:
3698:
3692:
3687:
3683:
3679:
3676:
3673:
3672:
3669:
3666:
3665:
3662:
3657:
3653:
3649:
3646:
3643:
3642:
3640:
3635:
3629:
3625:
3609:
3598:
3585:
3572:
3571:
3560:
3555:
3549:
3544:
3540:
3536:
3533:
3530:
3529:
3526:
3523:
3522:
3519:
3514:
3510:
3506:
3503:
3500:
3499:
3497:
3492:
3489:
3486:
3483:
3478:
3474:
3453:
3442:
3410:
3407:
3403:
3402:
3389:
3385:
3381:
3376:
3372:
3366:
3362:
3358:
3355:
3352:
3349:
3344:
3340:
3334:
3330:
3326:
3323:
3320:
3317:
3311:
3308:
3282:
3279:
3276:
3273:
3270:
3267:
3244:
3235:. We see that
3222:
3218:
3214:
3209:
3205:
3201:
3198:
3195:
3192:
3187:
3183:
3159:
3154:
3150:
3146:
3141:
3137:
3114:
3110:
3098:
3097:
3084:
3078:
3075:
3072:
3067:
3063:
3057:
3053:
3049:
3048:
3045:
3042:
3041:
3038:
3035:
3032:
3027:
3023:
3017:
3013:
3009:
3008:
3006:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2969:
2968:
2957:
2954:
2951:
2948:
2945:
2942:
2937:
2929:
2925:
2921:
2920:
2917:
2914:
2913:
2908:
2904:
2900:
2899:
2897:
2892:
2889:
2875:
2874:
2863:
2858:
2850:
2846:
2842:
2839:
2834:
2830:
2826:
2823:
2820:
2819:
2816:
2813:
2812:
2807:
2803:
2799:
2796:
2791:
2787:
2783:
2780:
2777:
2776:
2774:
2769:
2766:
2763:
2760:
2757:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2690:
2687:
2684:
2681:
2678:
2675:
2655:
2635:
2630:
2626:
2622:
2619:
2616:
2611:
2607:
2603:
2586:
2583:
2582:
2581:
2570:
2565:
2559:
2554:
2550:
2546:
2543:
2540:
2539:
2536:
2533:
2532:
2529:
2524:
2520:
2516:
2513:
2510:
2509:
2507:
2502:
2499:
2496:
2493:
2490:
2468:
2467:
2456:
2451:
2445:
2442:
2439:
2434:
2430:
2426:
2423:
2422:
2419:
2416:
2415:
2412:
2409:
2406:
2401:
2397:
2393:
2390:
2389:
2387:
2382:
2379:
2376:
2373:
2370:
2345:
2341:
2317:
2298:
2297:
2286:
2281:
2277:
2271:
2267:
2261:
2257:
2253:
2250:
2247:
2244:
2241:
2226:
2225:
2214:
2211:
2208:
2205:
2202:
2197:
2189:
2185:
2181:
2180:
2177:
2174:
2173:
2168:
2164:
2160:
2159:
2157:
2152:
2149:
2126:
2116:observable map
2099:
2096:
2091:
2087:
2081:
2077:
2054:
2050:
2027:
2023:
1998:
1994:
1989:
1985:
1972:
1969:
1957:
1956:
1945:
1940:
1936:
1932:
1929:
1926:
1923:
1909:
1908:
1897:
1894:
1888:
1884:
1880:
1875:
1871:
1832:
1831:
1820:
1815:
1811:
1807:
1802:
1798:
1794:
1791:
1768:
1763:
1759:
1755:
1750:
1746:
1733:
1730:
1729:
1728:
1717:
1714:
1711:
1706:
1702:
1698:
1695:
1658:
1654:
1650:
1647:
1644:
1641:
1637:
1633:
1630:
1619:
1618:
1607:
1602:
1598:
1593:
1590:
1587:
1584:
1572:, is given by
1565:
1564:Time evolution
1562:
1560:
1557:
1532:
1512:
1507:
1503:
1499:
1496:
1474:
1452:
1432:
1429:
1426:
1423:
1420:
1417:
1412:
1408:
1404:
1401:
1390:
1389:
1378:
1375:
1370:
1366:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1333:
1329:
1325:
1322:
1319:
1316:
1291:
1264:
1259:
1233:
1230:
1227:
1224:
1204:
1184:
1164:
1161:
1158:
1155:
1144:
1143:
1132:
1127:
1122:
1117:
1112:
1109:
1087:
1086:
1075:
1070:
1066:
1062:
1059:
1056:
1053:
1048:
1044:
1040:
1037:
1034:
1031:
1015:
1012:
999:
996:
993:
990:
987:
982:
978:
951:
947:
926:
893:
889:
860:
849:
848:
837:
834:
831:
828:
825:
822:
819:
814:
810:
806:
803:
800:
797:
794:
791:
788:
785:
782:
779:
756:
736:
731:
727:
723:
720:
717:
714:
709:
705:
701:
698:
695:
692:
674:
661:
635:
613:
609:
588:
568:
551:
544:
538:
535:
522:
503:
502:
490:
479:
459:
456:
451:
447:
432:
417:
414:
411:
406:
402:
382:
366:
342:
331:
319:
293:
289:
266:
262:
237:
213:
208:
204:
183:
178:
174:
170:
167:
156:Hilbert spaces
141:
137:
114:
110:
90:
87:
71:
68:
26:
9:
6:
4:
3:
2:
9120:
9109:
9106:
9105:
9103:
9088:
9087:
9078:
9077:
9074:
9064:
9061:
9059:
9056:
9054:
9051:
9049:
9046:
9042:
9041:Plasma window
9039:
9038:
9037:
9034:
9032:
9029:
9027:
9024:
9022:
9019:
9017:
9014:
9013:
9011:
9007:
9001:
9000:teleportation
8998:
8996:
8993:
8991:
8988:
8986:
8983:
8981:
8978:
8976:
8973:
8971:
8968:
8966:
8963:
8961:
8958:
8956:
8953:
8951:
8948:
8946:
8943:
8941:
8938:
8936:
8933:
8931:
8928:
8926:
8923:
8921:
8918:
8916:
8913:
8911:
8908:
8906:
8903:
8901:
8898:
8894:
8891:
8890:
8889:
8886:
8884:
8881:
8879:
8876:
8874:
8871:
8869:
8866:
8864:
8861:
8859:
8856:
8854:
8851:
8849:
8846:
8845:
8843:
8841:
8837:
8834:
8830:
8826:
8819:
8814:
8812:
8807:
8805:
8800:
8799:
8796:
8784:
8776:
8775:
8772:
8766:
8763:
8761:
8758:
8756:
8753:
8749:
8746:
8745:
8744:
8741:
8740:
8738:
8734:
8728:
8725:
8723:
8720:
8716:
8713:
8712:
8711:
8708:
8706:
8703:
8701:
8698:
8696:
8693:
8692:
8690:
8686:
8680:
8677:
8675:
8672:
8670:
8667:
8665:
8662:
8660:
8657:
8655:
8652:
8650:
8647:
8645:
8642:
8640:
8637:
8635:
8632:
8630:
8627:
8625:
8622:
8620:
8619:Quantum logic
8617:
8615:
8612:
8610:
8607:
8605:
8602:
8600:
8597:
8595:
8592:
8590:
8587:
8585:
8582:
8578:
8575:
8574:
8573:
8570:
8568:
8565:
8563:
8560:
8558:
8555:
8551:
8548:
8547:
8546:
8543:
8541:
8538:
8536:
8533:
8531:
8528:
8527:
8525:
8523:
8519:
8513:
8510:
8508:
8505:
8503:
8500:
8498:
8495:
8493:
8490:
8488:
8485:
8483:
8480:
8478:
8475:
8473:
8472:Quantum chaos
8470:
8468:
8465:
8463:
8460:
8459:
8457:
8455:
8451:
8445:
8442:
8440:
8439:SternâGerlach
8437:
8435:
8432:
8430:
8427:
8425:
8422:
8420:
8417:
8415:
8412:
8410:
8407:
8405:
8402:
8400:
8397:
8395:
8392:
8391:
8389:
8385:
8379:
8376:
8374:
8373:Transactional
8371:
8369:
8366:
8364:
8363:Quantum logic
8361:
8359:
8356:
8354:
8351:
8345:
8342:
8341:
8340:
8337:
8336:
8335:
8332:
8330:
8327:
8325:
8322:
8320:
8317:
8315:
8312:
8310:
8307:
8306:
8304:
8302:
8298:
8292:
8289:
8287:
8284:
8282:
8279:
8277:
8274:
8272:
8269:
8267:
8264:
8262:
8259:
8257:
8254:
8253:
8251:
8247:
8241:
8238:
8236:
8233:
8231:
8228:
8226:
8223:
8221:
8218:
8216:
8213:
8211:
8208:
8207:
8205:
8201:
8193:
8190:
8188:
8185:
8184:
8183:
8182:Wave function
8180:
8178:
8175:
8173:
8170:
8168:
8165:
8163:
8160:
8158:
8157:Superposition
8155:
8153:
8152:Quantum state
8150:
8148:
8145:
8143:
8140:
8138:
8135:
8133:
8130:
8128:
8125:
8123:
8120:
8116:
8113:
8111:
8108:
8106:
8105:Excited state
8103:
8101:
8098:
8097:
8096:
8093:
8091:
8088:
8086:
8083:
8081:
8078:
8076:
8073:
8072:
8070:
8066:
8060:
8057:
8055:
8052:
8050:
8047:
8043:
8040:
8039:
8038:
8035:
8033:
8030:
8029:
8027:
8023:
8019:
8012:
8007:
8005:
8000:
7998:
7993:
7992:
7989:
7977:
7969:
7967:
7959:
7958:
7955:
7949:
7946:
7944:
7941:
7939:
7936:
7934:
7931:
7929:
7925:
7922:
7920:
7916:
7912:
7909:
7908:
7906:
7904:
7898:
7888:
7885:
7883:
7880:
7878:
7875:
7873:
7870:
7869:
7867:
7865:
7861:
7855:
7852:
7850:
7847:
7845:
7844:Spin qubit QC
7842:
7840:
7837:
7836:
7834:
7831:
7827:
7821:
7818:
7816:
7813:
7812:
7810:
7808:
7804:
7798:
7795:
7793:
7790:
7788:
7785:
7783:
7780:
7779:
7777:
7775:
7771:
7768:
7762:
7756:
7753:
7749:
7748:
7744:
7742:
7739:
7737:
7734:
7732:
7729:
7727:
7724:
7722:
7719:
7717:
7714:
7712:
7709:
7708:
7706:
7705:
7703:
7701:
7695:
7689:
7686:
7684:
7681:
7677:
7674:
7673:
7672:
7669:
7665:
7662:
7661:
7660:
7657:
7653:
7652:cluster state
7650:
7649:
7648:
7645:
7643:
7640:
7638:
7635:
7634:
7632:
7630:
7624:
7616:
7612:
7608:
7606:
7602:
7598:
7597:
7596:
7593:
7589:
7586:
7585:
7584:
7581:
7579:
7576:
7574:
7571:
7570:
7568:
7562:
7556:
7553:
7551:
7548:
7546:
7543:
7541:
7538:
7536:
7533:
7532:
7530:
7528:
7522:
7516:
7513:
7511:
7508:
7506:
7503:
7501:
7498:
7496:
7493:
7491:
7488:
7486:
7483:
7481:
7478:
7476:
7473:
7471:
7468:
7466:
7463:
7461:
7460:DeutschâJozsa
7458:
7456:
7453:
7451:
7448:
7446:
7443:
7442:
7440:
7438:
7434:
7424:
7421:
7417:
7414:
7412:
7409:
7407:
7404:
7403:
7402:
7399:
7397:
7396:Quantum money
7394:
7392:
7389:
7387:
7384:
7383:
7381:
7379:
7375:
7369:
7366:
7362:
7359:
7358:
7357:
7354:
7350:
7347:
7346:
7345:
7342:
7340:
7337:
7335:
7332:
7330:
7327:
7323:
7320:
7318:
7315:
7314:
7313:
7310:
7309:
7307:
7305:communication
7301:
7295:
7292:
7290:
7287:
7285:
7282:
7280:
7277:
7275:
7272:
7270:
7267:
7265:
7262:
7260:
7257:
7255:
7252:
7250:
7247:
7245:
7242:
7240:
7237:
7235:
7232:
7230:
7227:
7225:
7222:
7220:
7217:
7216:
7214:
7210:
7202:
7199:
7198:
7197:
7194:
7190:
7187:
7186:
7185:
7182:
7180:
7177:
7175:
7172:
7170:
7167:
7163:
7160:
7159:
7158:
7155:
7153:
7150:
7148:
7145:
7144:
7142:
7138:
7134:
7127:
7122:
7120:
7115:
7113:
7108:
7107:
7104:
7096:
7092:
7088:
7084:
7080:
7076:
7071:
7066:
7062:
7057:
7054:
7050:
7049:
7042:
7036:
7028:
7024:
7020:
7016:
7012:
7008:
7003:
6998:
6994:
6990:
6983:
6979:
6969:
6966:
6964:
6961:
6959:
6956:
6955:
6949:
6935:
6932:
6926:
6912:
6893:
6873:
6871:
6867:
6856:
6847:
6843:
6840:This section
6838:
6835:
6831:
6830:
6822:
6806:
6803:
6800:
6770:
6762:
6759:
6756:
6746:
6737:
6730:
6729:
6728:
6726:
6721:
6705:
6675:
6667:
6657:
6648:
6641:
6640:
6639:
6637:
6627:
6625:
6621:
6617:
6613:
6608:
6606:
6602:
6598:
6579:
6573:
6570:
6567:
6562:
6559:
6556:
6550:
6542:
6532:
6527:
6524:
6521:
6508:
6505:
6497:
6494:
6491:
6481:
6476:
6473:
6470:
6457:
6454:
6446:
6436:
6431:
6418:
6411:
6410:
6409:
6406:
6389:
6386:
6378:
6375:
6372:
6362:
6357:
6344:
6337:
6336:
6335:
6319:
6316:
6313:
6284:
6268:
6252:
6223:
6220:
6217:
6188:
6185:
6182:
6167:
6164: Ă
6163:
6159:
6135:
6112:
6109:
6066:
6063:
6044:
6041:
6033:
6024:
6016:, denoted by
6015:
6000:
5997:
5961:
5959:
5955:
5933:
5929:
5902:
5899:
5889:
5885:
5881:
5876:
5873:
5865:
5858:
5854:
5850:
5825:
5813:
5812:
5796:
5793:
5785:
5781:
5776:
5770:
5766:
5757:
5721:
5717:
5691:
5683:
5679:
5672:
5664:
5660:
5648:
5647:
5643:
5628:
5625:
5590:
5586:
5585:
5584:
5581:
5579:
5575:
5551:
5548:
5543:
5540:
5532:
5527:
5524:
5497:
5496:
5495:
5494:The quantity
5492:
5476:
5473:
5451:
5448:. Similarly,
5447:
5446:concatenation
5429:
5426:
5413:
5397:
5374:
5369:
5366:
5358:
5355:
5352:
5347:
5344:
5336:
5331:
5328:
5323:
5320:
5308:
5307:
5306:
5304:
5300:
5295:
5276:
5273:
5263:
5260:
5252:
5232:
5229:
5226:
5218:
5210:
5207:
5199:
5174:
5173:
5172:
5170:
5166:
5162:
5158:
5154:
5133:
5116:
5104:
5101:
5098:
5092:
5089:
5086:
5067:
5066:
5065:
5049:
5032:
5020:
5015:
5012:
4985:
4968:
4956:
4944:
4942:
4941:classical one
4918:
4910:
4906:
4902:
4891:
4883:
4878:
4875:
4857:
4856:
4855:
4821:
4812:
4784:
4780:
4776:
4763:
4762:
4761:
4758:
4721:
4715:
4712:
4706:
4687:
4675:
4657:
4648:
4635:
4634:
4633:
4591:
4590:operator norm
4587:
4523:
4519:
4504:
4501:
4482:
4474:
4468:
4440:
4412:
4399:
4395:
4391:
4387:
4383:
4373:
4371:
4364:Teleportation
4361:
4359:
4335:
4311:
4310:
4304:
4300:
4296:
4292:
4271:
4266:
4262:
4258:
4253:
4249:
4243:
4238:
4235:
4232:
4228:
4224:
4218:
4205:
4204:
4203:
4181:
4162:
4157:
4154:
4151:
4136:
4133:
4130:
4120:
4110:
4109:
4108:
4060:
4058:
4054:
4049:
4047:
4043:
4024:
4019:
4015:
4008:
4000:
3996:
3990:
3986:
3982:
3976:
3968:
3955:
3954:
3953:
3937:
3932:
3924:
3919:
3914:
3906:
3901:
3887:
3885:
3881:
3862:
3857:
3853:
3844:
3840:
3833:
3828:
3824:
3820:
3814:
3806:
3798:
3793:
3785:
3779:
3766:
3765:
3764:
3747:
3742:
3738:
3729:
3725:
3718:
3713:
3709:
3705:
3701:
3696:
3685:
3681:
3674:
3667:
3655:
3651:
3644:
3638:
3633:
3627:
3615:
3614:
3613:
3588:
3581:
3577:
3558:
3553:
3542:
3538:
3531:
3524:
3512:
3508:
3501:
3495:
3490:
3484:
3476:
3463:
3462:
3461:
3459:
3432:
3428:
3424:
3420:
3416:
3406:
3387:
3383:
3379:
3374:
3370:
3364:
3360:
3356:
3350:
3342:
3332:
3328:
3324:
3318:
3296:
3295:
3294:
3277:
3274:
3271:
3256:
3220:
3216:
3212:
3207:
3203:
3199:
3193:
3185:
3157:
3152:
3148:
3144:
3139:
3135:
3112:
3082:
3073:
3065:
3055:
3051:
3043:
3033:
3025:
3015:
3011:
3004:
2999:
2993:
2990:
2987:
2974:
2973:
2972:
2955:
2949:
2943:
2940:
2935:
2927:
2923:
2915:
2906:
2902:
2895:
2890:
2887:
2880:
2879:
2878:
2861:
2856:
2848:
2844:
2840:
2832:
2828:
2821:
2814:
2805:
2801:
2797:
2789:
2785:
2778:
2772:
2767:
2761:
2748:
2747:
2746:
2729:
2723:
2720:
2714:
2708:
2685:
2679:
2676:
2673:
2628:
2624:
2620:
2617:
2614:
2609:
2605:
2593:
2568:
2563:
2552:
2548:
2541:
2534:
2522:
2518:
2511:
2505:
2500:
2494:
2481:
2480:
2479:
2478:, we can put
2477:
2473:
2454:
2449:
2440:
2437:
2432:
2428:
2417:
2407:
2404:
2399:
2395:
2385:
2380:
2374:
2361:
2360:
2359:
2343:
2329:
2307:
2305:
2284:
2279:
2275:
2269:
2265:
2259:
2255:
2251:
2245:
2231:
2230:
2229:
2209:
2203:
2200:
2195:
2187:
2183:
2175:
2166:
2162:
2155:
2150:
2147:
2140:
2139:
2138:
2117:
2113:
2097:
2094:
2089:
2085:
2079:
2075:
2052:
2048:
2025:
2021:
2013:
1992:
1987:
1983:
1968:
1966:
1962:
1943:
1938:
1934:
1930:
1927:
1921:
1914:
1913:
1912:
1895:
1892:
1886:
1882:
1878:
1873:
1869:
1861:
1860:
1859:
1857:
1853:
1849:
1848:partial trace
1845:
1841:
1837:
1818:
1813:
1809:
1805:
1800:
1796:
1792:
1789:
1782:
1781:
1780:
1766:
1761:
1757:
1753:
1748:
1744:
1715:
1712:
1709:
1704:
1700:
1693:
1686:
1685:
1684:
1682:
1678:
1674:
1652:
1648:
1645:
1642:
1639:
1635:
1631:
1628:
1605:
1600:
1596:
1591:
1588:
1582:
1575:
1574:
1573:
1571:
1556:
1554:
1550:
1546:
1530:
1510:
1505:
1501:
1497:
1494:
1450:
1427:
1421:
1418:
1410:
1406:
1399:
1376:
1368:
1364:
1357:
1348:
1342:
1339:
1331:
1327:
1320:
1317:
1307:
1306:
1305:
1278:
1262:
1247:
1228:
1222:
1215:is finite so
1202:
1182:
1159:
1153:
1130:
1110:
1100:
1099:
1098:
1096:
1092:
1068:
1064:
1057:
1046:
1042:
1035:
1032:
1022:
1021:
1020:
1011:
997:
994:
988:
980:
967:
949:
915:
913:
909:
891:
878:
874:
835:
829:
826:
820:
812:
801:
792:
783:
780:
770:
769:
768:
729:
725:
718:
707:
703:
696:
693:
682:
677:
673:
669:
664:
660:
656:
651:
649:
611:
558:
554:
547:
534:
512:
508:
480:
477:
473:
454:
449:
445:
435:
431:
415:
409:
404:
400:
391:
387:
383:
380:
356:
332:
309:
308:
307:
291:
287:
264:
260:
251:
227:
211:
206:
202:
176:
172:
165:
157:
139:
135:
112:
108:
98:
96:
86:
84:
80:
75:
67:
64:
60:
56:
51:
49:
45:
41:
37:
33:
19:
9084:
9021:Anti-gravity
8965:metamaterial
8893:post-quantum
8888:cryptography
8867:
8556:
8502:Quantum mind
8414:FranckâHertz
8256:KleinâGordon
8210:Formulations
8203:Formulations
8132:Interference
8122:Entanglement
8100:Ground state
8095:Energy level
8068:Fundamentals
8032:Introduction
7872:Charge qubit
7797:KLM protocol
7746:
7610:
7600:
7343:
7294:Purification
7224:EastinâKnill
7060:
7052:
7040:
7035:
6992:
6988:
6982:
6879:
6865:
6863:
6850:
6846:adding to it
6841:
6785:
6724:
6722:
6690:
6635:
6633:
6623:
6612:entanglement
6609:
6604:
6600:
6596:
6594:
6407:
6404:
6269:
6165:
6161:
6157:
6072:
6064:
5964:
5962:
5957:
5953:
5917:
5592:
5588:
5582:
5577:
5573:
5571:
5493:
5449:
5411:
5389:
5302:
5298:
5296:
5293:
5168:
5164:
5160:
5156:
5152:
5150:
4945:
4938:
4804:
4759:
4736:
4521:
4517:
4515:
4394:polarization
4389:
4379:
4367:
4357:
4307:
4302:
4298:
4294:
4290:
4288:
4177:
4066:
4063:Pure channel
4050:
4039:
3888:
3879:
3877:
3762:
3583:
3579:
3575:
3573:
3457:
3430:
3426:
3422:
3418:
3414:
3412:
3404:
3258:Notice that
3257:
3099:
2970:
2876:
2588:
2469:
2330:
2303:
2299:
2227:
2115:
2011:
1974:
1964:
1960:
1958:
1910:
1855:
1851:
1843:
1839:
1835:
1833:
1779:For a state
1735:
1680:
1672:
1620:
1569:
1567:
1391:
1279:
1245:
1195:. We assume
1175:on some set
1145:
1090:
1088:
1017:
916:
911:
907:
876:
875:to those on
872:
850:
675:
671:
667:
662:
658:
654:
652:
556:
549:
542:
540:
510:
506:
504:
471:
433:
429:
389:
379:positive map
99:
92:
78:
76:
73:
52:
35:
29:
9036:Force field
8985:programming
8945:logic clock
8930:information
8905:electronics
8760:EPR paradox
8540:Quantum bus
8409:Double-slit
8387:Experiments
8353:Many-worlds
8291:Schrödinger
8240:Phase space
8230:Schrödinger
8220:Interaction
8177:Uncertainty
8147:Nonlocality
8142:Measurement
8137:Decoherence
8127:Hamiltonian
7903:programming
7882:Phase qubit
7787:Circuit QED
7259:No-deleting
7201:cloud-based
6622:. achieves
5918:A sequence
4382:fiber optic
4306:are called
2472:functionals
2302:integrates
1732:Restriction
1677:Hamiltonian
1095:C*-algebras
8950:logic gate
8848:algorithms
8688:Extensions
8522:Technology
8368:Relational
8319:Copenhagen
8215:Heisenberg
8162:Tunnelling
8025:Background
7943:libquantum
7877:Flux qubit
7782:Cavity QED
7731:BaconâShor
7721:stabilizer
7249:No-cloning
6974:References
4334:Karl Kraus
2585:Instrument
1971:Observable
1838:on system
968:, that is,
252:acting on
79:memoryless
8995:simulator
8883:computing
8853:amplifier
8394:Bell test
8249:Equations
8075:Born rule
7849:NV center
7284:Threshold
7264:No-hiding
7229:Gleason's
7070:1106.1445
7027:119250535
7002:1110.3234
6921:Φ
6909:which is
6894:ρ
6888:Φ
6853:June 2008
6804:×
6760:×
6744:Ψ
6655:Ψ
6571:
6560:
6525:×
6495:×
6474:×
6376:×
6317:×
6221:×
6186:×
6168:matrices
6106:Ψ
6038:Ψ
6031:Ψ
5994:Ψ
5973:Ψ
5934:α
5890:α
5882:⊗
5870:Ψ
5859:α
5851:⊗
5844:^
5841:Ψ
5831:Δ
5826:α
5809:, we have
5786:α
5771:α
5758:α
5730:∞
5727:→
5722:α
5692:⊂
5684:α
5665:α
5622:Ψ
5601:Ψ
5549:⊗
5537:Ψ
5525:⊗
5518:^
5515:Ψ
5505:Δ
5474:⊗
5467:^
5464:Ψ
5423:Ψ
5398:⊗
5363:Ψ
5359:⊗
5356:⋯
5353:⊗
5341:Ψ
5329:⊗
5317:Ψ
5270:‖
5257:Ψ
5253:−
5247:^
5244:Ψ
5238:‖
5204:Ψ
5194:^
5191:Ψ
5182:Δ
5122:→
5099:∘
5096:Φ
5093:∘
5081:^
5078:Ψ
5038:→
5009:Ψ
4974:→
4954:Ψ
4916:‖
4903:⊗
4900:Φ
4897:‖
4872:‖
4868:Φ
4865:‖
4842:Φ
4819:∞
4816:→
4790:‖
4777:⊗
4774:Φ
4771:‖
4745:Φ
4713:≤
4710:‖
4704:‖
4694:‖
4679:Λ
4676:−
4673:Φ
4667:‖
4655:‖
4652:Λ
4649:−
4646:Φ
4643:‖
4620:Λ
4600:Φ
4572:Λ
4552:Λ
4532:Φ
4500:coherence
4486:⟩
4472:⟩
4444:⟩
4416:⟩
4344:Ψ
4320:Ψ
4272:∗
4229:∑
4213:Ψ
4190:Ψ
4155:×
4142:→
4134:×
4118:Ψ
4095:Ψ
4075:Ψ
4057:separable
3987:∑
3969:∗
3965:Φ
3938:∗
3929:Φ
3925:∘
3920:∗
3911:Φ
3902:∗
3898:Φ
3834:ρ
3825:∑
3815:ρ
3803:Φ
3799:∘
3790:Φ
3780:ρ
3774:Φ
3719:ρ
3710:∑
3675:ρ
3668:⋮
3645:ρ
3624:Φ
3597:Φ
3532:ρ
3525:⋮
3502:ρ
3485:ρ
3473:Φ
3441:Φ
3361:∑
3339:Ψ
3329:∑
3310:~
3307:Ψ
3275:⊗
3266:Ψ
3243:Ψ
3182:Ψ
3109:Ψ
3062:Ψ
3044:⋮
3022:Ψ
2991:⊗
2982:Ψ
2941:∈
2916:⋮
2841:⋅
2822:ρ
2815:⋮
2798:⋅
2779:ρ
2762:ρ
2756:Φ
2721:⊗
2677:∈
2674:ρ
2654:Φ
2618:…
2542:ρ
2535:⋮
2512:ρ
2495:ρ
2489:Ψ
2444:⟩
2441:ρ
2425:⟨
2418:⋮
2411:⟩
2408:ρ
2392:⟨
2375:ρ
2369:Ψ
2344:∗
2340:Ψ
2316:Ψ
2256:∑
2240:Ψ
2201:∈
2176:⋮
2125:Ψ
2076:∑
1993:∈
1931:⊗
1925:→
1893:ρ
1870:ρ
1806:⊗
1793:∈
1790:ρ
1754:⊗
1705:∗
1697:→
1657:ℏ
1640:−
1601:∗
1592:ρ
1586:→
1583:ρ
1523:for some
1506:∗
1419:⊗
1355:→
1340:⊗
1315:Ψ
1121:→
1108:Ψ
1055:→
1030:Ψ
981:∗
977:Φ
950:∗
946:Φ
925:Φ
892:∗
888:Φ
859:Φ
833:⟩
830:ρ
813:∗
809:Φ
805:⟨
799:⟩
793:ρ
787:Φ
778:⟨
767:given by
755:Φ
716:→
691:Φ
634:Φ
612:∗
608:Φ
587:Φ
567:Φ
521:Φ
489:Φ
458:Φ
455:⊗
413:Φ
410:⊗
365:Φ
341:Φ
318:Φ
236:Φ
9102:Category
8900:dynamics
8783:Category
8577:Timeline
8329:Ensemble
8309:Bayesian
8271:Majorana
8187:Collapse
8059:Glossary
8042:Timeline
7911:OpenQASM
7887:Transmon
7764:Physical
7564:Quantum
7465:Grover's
7239:Holevo's
7212:Theorems
7162:timeline
7152:NISQ era
6952:See also
6727:of Κ is
4498:). The
1858:system:
1559:Examples
248:between
48:Internet
8990:sensing
8970:network
8955:machine
8925:imaging
8873:circuit
8868:channel
8840:Quantum
8736:Related
8715:History
8454:Science
8286:Rydberg
8037:History
7901:Quantum
7839:Kane QC
7698:Quantum
7626:Quantum
7555:PostBQP
7525:Quantum
7510:Simon's
7303:Quantum
7140:General
7095:2515538
7075:Bibcode
7007:Bibcode
6913:, i.e.
5580:times.
4522:cb-norm
4386:photons
4051:In the
1675:is the
1392:Notice
646:in the
386:ancilla
224:In the
8980:optics
8832:Fields
8429:Popper
7919:IBM QX
7915:Qiskit
7854:NMR QC
7832:-based
7736:Steane
7707:Codes
7505:Shor's
7411:SARG04
7219:Bell's
7093:
7025:
6911:unital
6614:. The
5709:where
5591:is an
5151:where
4586:metric
4289:where
3886:form.
3884:Holevo
3882:or in
3100:where
2594:. Let
2012:effect
1959:where
1621:where
966:unital
851:While
666:) and
428:where
384:If an
9009:Other
8940:logic
8339:Local
8281:Pauli
8261:Dirac
7741:Toric
7184:Qubit
7091:S2CID
7065:arXiv
7023:S2CID
6997:arXiv
4520:, or
4392:) or
377:is a
44:qubit
9086:List
8266:Weyl
7933:Cirq
7924:Quil
7830:Spin
7726:Shor
7406:BB84
7339:LOCC
5963:The
5794:<
5742:and
5390:The
5163:and
4946:Let
4358:pure
3417:and
2877:Let
2112:POVM
1679:and
1671:and
511:CPTP
355:cone
279:and
127:and
100:Let
77:The
34:, a
8858:bus
7747:gnu
7711:CSS
7588:XEB
7550:QMA
7545:QIP
7540:EQP
7535:BQP
7515:VQE
7470:HHL
7274:PBR
7083:doi
7015:doi
6848:.
6568:log
6557:log
6334:is
5822:lim
5754:sup
5750:lim
5223:inf
4888:sup
4661:sup
4368:In
4312:of
4178:By
1850:of
1551:or
964:is
30:In
9104::
7938:Q#
7089:,
7081:,
7073:,
7021:.
7013:.
7005:.
6993:84
6991:.
6948:.
6872:.
6821:.
6720:.
6607:.
6599:=
6390:0.
5903:0.
5644:if
5491:.
5171::
4943:.
4428:,
4295:nm
4293:â€
4182:,
2745::
2040:.
1967:.
1883:Tr
1842:,
1555:.
1091:CP
879:,
650::
50:.
8817:e
8810:t
8803:v
8010:e
8003:t
7996:v
7926:â
7917:â
7913:â
7614:2
7611:T
7604:1
7601:T
7125:e
7118:t
7111:v
7098:.
7085::
7077::
7067::
7029:.
7017::
7009::
6999::
6936:I
6933:=
6930:)
6927:I
6924:(
6897:)
6891:(
6855:)
6851:(
6807:2
6801:2
6796:C
6771:,
6768:)
6763:2
6757:2
6752:C
6747:,
6741:(
6738:C
6706:2
6701:C
6676:,
6673:)
6668:2
6663:C
6658:,
6652:(
6649:C
6601:m
6597:n
6580:.
6574:m
6563:n
6551:=
6548:)
6543:n
6538:C
6533:,
6528:m
6522:m
6517:C
6512:(
6509:C
6506:=
6503:)
6498:n
6492:n
6487:C
6482:,
6477:m
6471:m
6466:C
6461:(
6458:C
6455:=
6452:)
6447:n
6442:C
6437:,
6432:m
6427:C
6422:(
6419:C
6387:=
6384:)
6379:n
6373:n
6368:C
6363:,
6358:m
6353:C
6348:(
6345:C
6320:n
6314:n
6309:C
6285:m
6280:C
6253:m
6248:C
6224:n
6218:n
6213:C
6189:n
6183:n
6178:C
6166:n
6162:n
6158:n
6141:B
6136:I
6113:d
6110:i
6083:B
6050:)
6045:d
6042:i
6034:,
6028:(
6025:C
6001:d
5998:i
5958:r
5954:r
5939:}
5930:n
5926:{
5900:=
5897:)
5886:n
5877:d
5874:i
5866:,
5855:m
5834:(
5797:r
5791:)
5782:m
5777:/
5767:n
5763:(
5718:m
5696:N
5689:}
5680:m
5676:{
5673:,
5670:}
5661:n
5657:{
5629:d
5626:i
5589:r
5578:m
5574:n
5557:)
5552:n
5544:d
5541:i
5533:,
5528:m
5508:(
5477:m
5430:d
5427:i
5412:n
5375:.
5370:d
5367:i
5348:d
5345:i
5337:=
5332:n
5324:d
5321:i
5303:n
5299:n
5277:b
5274:c
5264:d
5261:i
5233:D
5230:,
5227:E
5219:=
5216:)
5211:d
5208:i
5200:,
5185:(
5165:D
5161:E
5157:D
5153:E
5134:2
5128:A
5117:2
5111:B
5105::
5102:E
5090:D
5087:=
5050:2
5044:A
5033:2
5027:B
5021::
5016:d
5013:i
4986:1
4980:A
4969:1
4963:B
4957::
4919:.
4911:n
4907:I
4892:n
4884:=
4879:b
4876:c
4822:.
4813:n
4785:n
4781:I
4722:.
4719:}
4716:1
4707:A
4699:|
4691:)
4688:A
4685:(
4682:)
4670:(
4664:{
4658:=
4483:1
4479:|
4475:+
4469:0
4465:|
4441:1
4437:|
4413:0
4409:|
4303:i
4299:K
4291:N
4267:i
4263:K
4259:A
4254:i
4250:K
4244:N
4239:1
4236:=
4233:i
4225:=
4222:)
4219:A
4216:(
4163:.
4158:m
4152:m
4147:C
4137:n
4131:n
4126:C
4121::
4025:.
4020:i
4016:F
4012:)
4009:A
4006:(
4001:i
3997:R
3991:i
3983:=
3980:)
3977:A
3974:(
3933:2
3915:1
3907:=
3863:.
3858:i
3854:R
3850:)
3845:i
3841:F
3837:(
3829:i
3821:=
3818:)
3812:(
3807:1
3794:2
3786:=
3783:)
3777:(
3748:.
3743:i
3739:R
3735:)
3730:i
3726:F
3722:(
3714:i
3706:=
3702:)
3697:]
3691:)
3686:n
3682:F
3678:(
3661:)
3656:1
3652:F
3648:(
3639:[
3634:(
3628:2
3610:2
3586:i
3584:R
3580:B
3576:i
3559:.
3554:]
3548:)
3543:n
3539:F
3535:(
3518:)
3513:1
3509:F
3505:(
3496:[
3491:=
3488:)
3482:(
3477:1
3458:A
3454:1
3431:B
3427:B
3423:A
3419:B
3415:A
3388:i
3384:M
3380:A
3375:i
3371:M
3365:i
3357:=
3354:)
3351:A
3348:(
3343:i
3333:i
3325:=
3322:)
3319:A
3316:(
3281:)
3278:I
3272:f
3269:(
3221:i
3217:M
3213:A
3208:i
3204:M
3200:=
3197:)
3194:A
3191:(
3186:i
3158:2
3153:i
3149:M
3145:=
3140:i
3136:F
3113:i
3083:]
3077:)
3074:A
3071:(
3066:n
3056:n
3052:f
3037:)
3034:A
3031:(
3026:1
3016:1
3012:f
3005:[
3000:=
2997:)
2994:A
2988:f
2985:(
2956:.
2953:)
2950:X
2947:(
2944:C
2936:]
2928:n
2924:f
2907:1
2903:f
2896:[
2891:=
2888:f
2862:.
2857:]
2849:n
2845:F
2838:)
2833:n
2829:F
2825:(
2806:1
2802:F
2795:)
2790:1
2786:F
2782:(
2773:[
2768:=
2765:)
2759:(
2733:)
2730:H
2727:(
2724:L
2718:)
2715:X
2712:(
2709:C
2689:)
2686:H
2683:(
2680:L
2634:}
2629:n
2625:F
2621:,
2615:,
2610:1
2606:F
2602:{
2569:.
2564:]
2558:)
2553:n
2549:F
2545:(
2528:)
2523:1
2519:F
2515:(
2506:[
2501:=
2498:)
2492:(
2455:,
2450:]
2438:,
2433:n
2429:F
2405:,
2400:1
2396:F
2386:[
2381:=
2378:)
2372:(
2304:f
2285:.
2280:i
2276:F
2270:i
2266:f
2260:i
2252:=
2249:)
2246:f
2243:(
2213:)
2210:X
2207:(
2204:C
2196:]
2188:n
2184:f
2167:1
2163:f
2156:[
2151:=
2148:f
2098:I
2095:=
2090:i
2086:F
2080:i
2053:i
2049:F
2026:i
2022:F
1997:C
1988:i
1984:f
1965:A
1961:A
1944:,
1939:B
1935:I
1928:A
1922:A
1896:.
1887:B
1879:=
1874:A
1856:B
1852:Ï
1844:Ï
1840:A
1836:Ï
1819:,
1814:B
1810:H
1801:A
1797:H
1767:.
1762:B
1758:H
1749:A
1745:H
1716:.
1713:U
1710:A
1701:U
1694:A
1681:t
1673:H
1653:/
1649:t
1646:H
1643:i
1636:e
1632:=
1629:U
1606:,
1597:U
1589:U
1570:t
1531:x
1511:x
1502:x
1498:=
1495:a
1473:A
1451:a
1431:)
1428:X
1425:(
1422:C
1416:)
1411:B
1407:H
1403:(
1400:L
1377:.
1374:)
1369:A
1365:H
1361:(
1358:L
1352:)
1349:X
1346:(
1343:C
1337:)
1332:B
1328:H
1324:(
1321:L
1318::
1290:B
1263:n
1258:R
1246:n
1232:)
1229:X
1226:(
1223:C
1203:X
1183:X
1163:)
1160:X
1157:(
1154:C
1131:.
1126:A
1116:B
1111::
1074:)
1069:A
1065:H
1061:(
1058:L
1052:)
1047:B
1043:H
1039:(
1036:L
1033::
998:I
995:=
992:)
989:I
986:(
912:A
908:B
877:B
873:A
836:.
827:,
824:)
821:A
818:(
802:=
796:)
790:(
784:,
781:A
735:)
730:B
726:H
722:(
719:L
713:)
708:A
704:H
700:(
697:L
694::
676:B
672:H
670:(
668:L
663:A
659:H
657:(
655:L
557:B
552:A
550:H
545:A
543:H
478:.
472:n
450:n
446:I
434:n
430:I
416:,
405:n
401:I
390:n
381:.
292:B
288:H
265:A
261:H
212:.
207:A
203:H
182:)
177:A
173:H
169:(
166:L
140:B
136:H
113:A
109:H
20:)
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