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## Capacities of quantum channels and how to find them (2003)

Venue: | Mathematical Programming |

Citations: | 9 - 1 self |

### Citations

12168 |
Elements of Information Theory
- Cover, Thomas
- 1991
(Show Context)
Citation Context ...apacity of a classical channel is essentially unique, and is representable as a single numerical quantity, which gives the amount of information that can be transmitted asymptotically per channel use =-=[46, 15]-=-. Quantum channels, unlike classical channels, do not have a single numerical quantity which can be defined as their capacity for transmitting information. Rather, quantum channels appear to have at l... |

10533 |
A mathematical theory of communication
- Shannon
- 1948
(Show Context)
Citation Context ...apacity of a classical channel is essentially unique, and is representable as a single numerical quantity, which gives the amount of information that can be transmitted asymptotically per channel use =-=[46, 15]-=-. Quantum channels, unlike classical channels, do not have a single numerical quantity which can be defined as their capacity for transmitting information. Rather, quantum channels appear to have at l... |

982 |
On the Einstein–Podolsky–Rosen Paradox
- Bell
- 1964
(Show Context)
Citation Context ...ite state, but which has a definite state when considered as a joint system of two qubits. In this state, the two photons have orthogonal 5spolarizations in whichever basis they are measured in. Bell =-=[6]-=- showed that the outcomes of measurements on the photons of this state cannot be reproduced by joint probability distributions which give probabilities for the outcomes of all possible measurements, a... |

642 | Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen
- Bennett, Brassard, et al.
- 1993
(Show Context)
Citation Context ...n unassisted classical channel. However, quantum teleportation lets a sender and a receiver who share an EPR pair of qubits communicate one qubit by sending two classical bits and using this EPR pair =-=[8]-=-. (See Figure 3.) In quantum teleportation, the sender measures the unknown quantum state state and the EPR pair in the Bell basis, and sends the receiver the two classical bits which are the results ... |

278 |
Communication via one- and two-particle operators on Einstein-Podolsky-Rosen
- Bennett, Wiesner
- 1992
(Show Context)
Citation Context ...two surprising phenomena in quantum communication: superdense coding and quantum teleportation. The process of superdense coding uses a shared EPR pair and a single qubit to encode two classical bits =-=[11]-=-. This is an improvement on the capacity of a noiseless, unassisted, quantum channel, which takes one quantum bit to send a classical bit. We will assume that the shared EPR pair is in the state 1 √ 2... |

215 |
A single quantum cannot be cloned
- Wootters, Zurek
- 1982
(Show Context)
Citation Context ...ible to increase the capacity of a classical channel by encoding information in a teleported qubit. Finally, there is a theorem of quantum mechanics that an unknown quantum state cannot be duplicated =-=[52]-=-. However, this no-cloning theorem (as it is known) is not violated; the original state is necessarily destroyed by the measurement, so teleportation cannot be used to clone a quantum state. We now gi... |

196 | The capacity of the quantum channel with general signal states
- Holevo
- 1998
(Show Context)
Citation Context ...ed between two or more channel uses, but do allow joint quantum measurements over arbitrarily many channel uses. A generalization of Shannon’s giving the C1,∞ capacity has been proven. Theorem (Holevo=-=[24]-=-, Schumacher–Westmoreland[45]): The C1,∞ capacity of a quantum channel, i.e., that capacity obtainable using codewords composed of signal states σi, where the probability of using σi is pi, is χ = HvN... |

181 |
Sending classical information via noisy quantum channels, Phys
- Schumacher, Westmoreland
- 1996
(Show Context)
Citation Context ...l uses, but do allow joint quantum measurements over arbitrarily many channel uses. A generalization of Shannon’s giving the C1,∞ capacity has been proven. Theorem (Holevo[24], Schumacher–Westmoreland=-=[45]-=-): The C1,∞ capacity of a quantum channel, i.e., that capacity obtainable using codewords composed of signal states σi, where the probability of using σi is pi, is χ = HvN( � piσi) − � piHvN(σi). (20)... |

177 | Concentrating partial entanglement by local operations - Bennett, Bernstein, et al. - 1996 |

168 | Unconditional Security Of Quantum Key Distribution Over Arbitrarily - Lo, Chau - 1999 |

133 |
Bell’s theorem without inequalities
- Greenberger, Horn, et al.
- 1990
(Show Context)
Citation Context ...measurements, and in which each of the single photons has a definite probability distribution for the outcome of measurements on it, independent of the measurements which are made on the other photon =-=[6, 20]-=-. In other words, there cannot be any set of hidden variables associated with each photon that determines the probability distribution obtained when this photon is measured in any particular basis. Tw... |

117 |
Fidelity for mixed quantum states
- Jozsa
- 1994
(Show Context)
Citation Context .... If both the output state ρout and the input state ρin are mixed states, the fidelity is defined � Tr ρ 1/2 1/2 outρinρout , an expression which, despite its appearance, is symmetric in ρin and ρout =-=[29]-=-. In the case where either ρout or ρin is pure, this is equivalent to the previous definition, and for mixed states it is a relatively simple expression which gives an upper bound on the probability o... |

111 | Entanglement-assisted capacity of a quantum channel and the reverse shannon theorem
- Bennett, Shor, et al.
- 2002
(Show Context)
Citation Context ...e a quantum state. We now give another capacity for quantum channels, one which has a capacity formula which can actually be completely proven, even in the case of infinite-dimensional Hilbert spaces =-=[9, 10, 25, 26]-=-. Recall that if N is a noiseless quantum channel, and if the sender and receiver possess shared EPR pairs, they can use superdense coding to double the classical information capacity of N . Similarly... |

108 |
Quantum coding, Phys
- Schumacher
- 1993
(Show Context)
Citation Context ... the above definition of entropy for thermodynamics. One can ask whether this is also the correct definition of entropy for information theory. We will first give the example of quantum source coding =-=[30, 44]-=-, also called Schumacher compression, for which we will see that it is indeed the right definition. We consider a memoryless quantum source that at each time step emits the pure state vi with probabil... |

81 | General formulas for capacity of classical-quantum channels
- Hayashi, Nagaoka
- 2003
(Show Context)
Citation Context ...where state σi has pi. We will sometimes write χ({σi, pi}i) so as to explicitly show this dependence. Another approach to proving this theorem, which also provides some additional results, appears in =-=[37, 39]-=- We later give a sketch of the proof of the C1,∞ capacity formula in the special case where the σi are pure states. We will first ask: Does this formula give the true capacity of a quantum channel N ?... |

74 | Classical information capacity of a quantum channel, Phys
- Hausladen, Jozsa, et al.
- 1996
(Show Context)
Citation Context ...e discussion of the proof of the Holevo-Schumacher-Westmoreland theorem in the special case where the σi are pure states. The proof of this case in fact appeared before the general theorem was proved =-=[21]-=-. The proof uses three ingredients. These are (1) random codes, (2) typical subspaces, and (3) the square root measurement. The square root measurement is also called the “pretty good” measurement, an... |

66 |
EntanglementAssisted Classical Capacity of Noisy Quantum Channels, Phys
- Bennett, Shor, et al.
- 1999
(Show Context)
Citation Context ...e a quantum state. We now give another capacity for quantum channels, one which has a capacity formula which can actually be completely proven, even in the case of infinite-dimensional Hilbert spaces =-=[9, 10, 25, 26]-=-. Recall that if N is a noiseless quantum channel, and if the sender and receiver possess shared EPR pairs, they can use superdense coding to double the classical information capacity of N . Similarly... |

60 | Smolin “Quantum-channel capacity of very noisy channels” Phys
- DiVincenzo, Shor, et al.
- 1998
(Show Context)
Citation Context ...the maximum over the tensor product of n uses of the channel, and let n go to infinity, because unlike the 22 isclassical (or the quantum) mutual information, the coherent information is not additive =-=[16]-=-. The quantity Q(N) (34) is the quantum capacity of a noisy quantum channel N [34, 5, 27, 50]. Even for maximizing the single-symbol expression (that is, taking n = 1 in Eq. (34)), the calculation of ... |

59 | Additivity of the classical capacity of entanglement-breaking quantum channels
- Shor
- 2002
(Show Context)
Citation Context ...en separate inputs to the channel helps to increase channel capacity, it would be possible to exceed this χmax. This can be addressed by answering a question that is simple to state: Is χmax additive =-=[1, 40, 31, 49, 36, 3]-=-? That is, if we have two quantum channels N1 and N2, is χmax(N1 ⊗ N2) = χmax(N1) + χmax(N2). (23) Proving superadditivity of the quantity χmax (i.e., the ≥ direction of Eq. (23)) is easy. The open qu... |

56 | Inequalities for quantum entropy: A review with conditions for equality
- Ruskai
- 2002
(Show Context)
Citation Context ...done using the estimate H(ρ + ǫ∆) = H(ρ − ǫTr∆log ρ + O(ǫ 2 ) 17swhich holds for matrices with Trρ = 1 and Tr∆ = 0. It can be derived from the integral expression for log(ρ + ǫδ) given in Eq. (20) of =-=[42]-=-. We thus propose to find the value of C1,∞ numerically by iterating the following steps 1. With a fixed set of vi, and the constraint � gram (26). i piviv † i = ρ, solve the linear pro2. Change ρ so ... |

50 | On some additivity problems in quantum information theory
- Amosov, Holevo, et al.
- 2000
(Show Context)
Citation Context ...en separate inputs to the channel helps to increase channel capacity, it would be possible to exceed this χmax. This can be addressed by answering a question that is simple to state: Is χmax additive =-=[1, 40, 31, 49, 36, 3]-=-? That is, if we have two quantum channels N1 and N2, is χmax(N1 ⊗ N2) = χmax(N1) + χmax(N2). (23) Proving superadditivity of the quantity χmax (i.e., the ≥ direction of Eq. (23)) is easy. The open qu... |

44 |
The capacity of the quantum depolarizing channel
- King
(Show Context)
Citation Context ...en separate inputs to the channel helps to increase channel capacity, it would be possible to exceed this χmax. This can be addressed by answering a question that is simple to state: Is χmax additive =-=[1, 40, 31, 49, 36, 3]-=-? That is, if we have two quantum channels N1 and N2, is χmax(N1 ⊗ N2) = χmax(N1) + χmax(N2). (23) Proving superadditivity of the quantity χmax (i.e., the ≥ direction of Eq. (23)) is easy. The open qu... |

41 |
The capacity of a noisy quantum channel” Phys
- Lloyd
- 1997
(Show Context)
Citation Context ...nity, because unlike the 22 isclassical (or the quantum) mutual information, the coherent information is not additive [16]. The quantity Q(N) (34) is the quantum capacity of a noisy quantum channel N =-=[34, 5, 27, 50]-=-. Even for maximizing the single-symbol expression (that is, taking n = 1 in Eq. (34)), the calculation of the coherent information appears to be a difficult optimization problem, as there may be mult... |

29 |
Remarks on additivity of the Holevo channel capacity and of the entanglement formation
- Matsumoto, Shimono, et al.
- 2004
(Show Context)
Citation Context ...l capacities have been fairly straightforward, often using gradient descent techniques [40]. More research has been done on the calculation of the entanglement of formation [51, 4], a related problem =-=[36]-=-. None of these programs have used combinatorial optimization techniques. For one of the capacities discussed in this paper—the entanglementassisted capacity—this technique may be fairly efficient, as... |

28 |
Private Communication
- Fuchs
- 2008
(Show Context)
Citation Context ...ble, it has been conjectured that the measurement optimizing accessible information is always a von Neumann measurement, in part because extensive computer experiments have not found a counterexample =-=[17]-=-. This conjecture has been proven for quantum states in two dimensions [33]. Our next example shows that this conjecture does not hold for ensembles composed of three or more states. 11sOur second exa... |

24 |
Information theoretical aspects of quantum measurement
- Holevo
- 1973
(Show Context)
Citation Context ... one might formulate the conjecture that Iacc ≤ HvN. This is true, as in fact is a somewhat stronger theorem which we will shortly state. The first published proof of this theorem was given by Holevo =-=[23]-=-. It was earlier conjectured by Gordon [18] and stated by Levitin with no proof [32]. Theorem (Holevo): Suppose that we have a memoryless source emitting an ensemble of (possibly mixed) states σi, whe... |

23 |
An algorithm for calculating the capacity of an arbitrary discrete memoryless channel
- Arimoto
- 1972
(Show Context)
Citation Context ... {Xi}, then the conditional entropy is H(Y |X) = � Pr(X = Xi)H(Y |X = Xi). (5) i There is an efficient algorithm, the Arimoto-Blahut algorithm, for calculating the capacity (2) of a classical channel =-=[2, 12, 15, 43]-=-. 3sWhen the formula for mutual information is extended to the quantum case, two generalizations have been found that both give capacities of a quantum channel, although these capacities differ in bot... |

23 |
Characterizing Entanglement
- Bruß
- 2002
(Show Context)
Citation Context ...mines the probability distribution obtained when this photon is measured in any particular basis. Two quantum systems such as an EPR pair which are non-classically correlated are said to be entangled =-=[13]-=-. The next fundamental principle of quantum mechanics we discuss is the linearity principle. This principle states that an isolated quantum system undergoes linear evolution. Because the quantum syste... |

20 | The early days of information theory - Pierce - 1973 |

19 | On entanglement-assisted classical capacity
- Holevo
- 2002
(Show Context)
Citation Context ...e a quantum state. We now give another capacity for quantum channels, one which has a capacity formula which can actually be completely proven, even in the case of infinite-dimensional Hilbert spaces =-=[9, 10, 25, 26]-=-. Recall that if N is a noiseless quantum channel, and if the sender and receiver possess shared EPR pairs, they can use superdense coding to double the classical information capacity of N . Similarly... |

19 | A new proof of the channel coding theorem via hypothesis testing in quantum information theory
- Ogawa, Nagaoka
(Show Context)
Citation Context ...where state σi has pi. We will sometimes write χ({σi, pi}i) so as to explicitly show this dependence. Another approach to proving this theorem, which also provides some additional results, appears in =-=[37, 39]-=- We later give a sketch of the proof of the C1,∞ capacity formula in the special case where the σi are pure states. We will first ask: Does this formula give the true capacity of a quantum channel N ?... |

15 | An introduction to quantum error correction,” in Quantum Computation: A Grand Mathematical Challenge for the Twenty-First - Gottesman |

13 |
On quantum measure of information
- Levitin
- 1969
(Show Context)
Citation Context ...omewhat stronger theorem which we will shortly state. The first published proof of this theorem was given by Holevo [23]. It was earlier conjectured by Gordon [18] and stated by Levitin with no proof =-=[32]-=-. Theorem (Holevo): Suppose that we have a memoryless source emitting an ensemble of (possibly mixed) states σi, where σi is emitted with probability pi. Let χ = HvN( � piσi) − � piHvN(σi). (18) Then ... |

12 |
Noise at optical frequencies; information theory
- Gordon
- 1964
(Show Context)
Citation Context ...cc ≤ HvN. This is true, as in fact is a somewhat stronger theorem which we will shortly state. The first published proof of this theorem was given by Holevo [23]. It was earlier conjectured by Gordon =-=[18]-=- and stated by Levitin with no proof [32]. Theorem (Holevo): Suppose that we have a memoryless source emitting an ensemble of (possibly mixed) states σi, where σi is emitted with probability pi. Let χ... |

11 | Unified approach to quantum capacities: towards quantum noisy coding theorem - Horodecki, Horodecki, et al. |

8 |
Blahut: Computation of Channel Capacity and Rate-Distortion Functions
- E
- 1972
(Show Context)
Citation Context ... {Xi}, then the conditional entropy is H(Y |X) = � Pr(X = Xi)H(Y |X = Xi). (5) i There is an efficient algorithm, the Arimoto-Blahut algorithm, for calculating the capacity (2) of a classical channel =-=[2, 12, 15, 43]-=-. 3sWhen the formula for mutual information is extended to the quantum case, two generalizations have been found that both give capacities of a quantum channel, although these capacities differ in bot... |

8 | The adaptive classical capacity of a quantum channel, or information capacities of three symmetric pure states in three dimensions
- Shor
- 2002
(Show Context)
Citation Context ...h hill climbing does not seem like it would be efficient, I do not have any alternative techniques to suggest. 1sThis paper originates in my research investigating the capacities of a quantum channel =-=[48]-=-. In order to show that a certain channel capacity (which I do not deal with in this paper; it is less natural than the capacities covered here) lies strictly between two other channel capacities, I n... |

7 | Quantum capacity is properly defined without encodings”, Phys
- Barnum, Smolin, et al.
- 1998
(Show Context)
Citation Context ...nity, because unlike the 22 isclassical (or the quantum) mutual information, the coherent information is not additive [16]. The quantity Q(N) (34) is the quantum capacity of a noisy quantum channel N =-=[34, 5, 27, 50]-=-. Even for maximizing the single-symbol expression (that is, taking n = 1 in Eq. (34)), the calculation of the coherent information appears to be a difficult optimization problem, as there may be mult... |

5 | Entanglement-assisted capacities of constrained quantum channels
- Holevo
(Show Context)
Citation Context |

5 | Numerical Experiments on the Capacity of Quantum Channel with Entangled Input
- Osawa, Nagaoka
- 2001
(Show Context)
Citation Context ...to test these techniques in the near future. To date, the means used for numerical computations of quantum channel capacities have been fairly straightforward, often using gradient descent techniques =-=[40]-=-. More research has been done on the calculation of the entanglement of formation [51, 4], a related problem [36]. None of these programs have used combinatorial optimization techniques. For one of th... |

4 | 2000 Quantum information theory: results and open problems Geom
- Shor
(Show Context)
Citation Context ...ent; this extrapolates between the C1,∞ capacity and the entanglement-assisted capacity. The description of quantum information theory and capacities contained in here is largely taken from the paper =-=[47]-=-. 2 Quantum Information Theory The discipline of information theory was founded by Claude Shannon in a truly remarkable paper [46] which laid down the foundations of the subject. We begin with a quote... |

3 |
De Moor, Variational characterizations of separability and entanglement of formation, Phys
- Audenaert, Verstraete, et al.
(Show Context)
Citation Context ...putations of quantum channel capacities have been fairly straightforward, often using gradient descent techniques [40]. More research has been done on the calculation of the entanglement of formation =-=[51, 4]-=-, a related problem [36]. None of these programs have used combinatorial optimization techniques. For one of the capacities discussed in this paper—the entanglementassisted capacity—this technique may... |

3 |
Relaxation Method For Calculating Quantum Entanglement. eprint arXiv:quant-ph/0101123
- Tucci
- 2001
(Show Context)
Citation Context ...putations of quantum channel capacities have been fairly straightforward, often using gradient descent techniques [40]. More research has been done on the calculation of the entanglement of formation =-=[51, 4]-=-, a related problem [36]. None of these programs have used combinatorial optimization techniques. For one of the capacities discussed in this paper—the entanglementassisted capacity—this technique may... |

2 |
Optimal quantum measurements for pure and mixed states
- Levitin
- 1995
(Show Context)
Citation Context ...ormation is always a von Neumann measurement, in part because extensive computer experiments have not found a counterexample [17]. This conjecture has been proven for quantum states in two dimensions =-=[33]-=-. Our next example shows that this conjecture does not hold for ensembles composed of three or more states. 11sOur second example is three photons with polarizations that differ by 60 ◦ each. These ar... |

2 |
Iterating the Arimoto-Blahut algorithm for faster convergence
- Sayir
- 2000
(Show Context)
Citation Context ... {Xi}, then the conditional entropy is H(Y |X) = � Pr(X = Xi)H(Y |X = Xi). (5) i There is an efficient algorithm, the Arimoto-Blahut algorithm, for calculating the capacity (2) of a classical channel =-=[2, 12, 15, 43]-=-. 3sWhen the formula for mutual information is extended to the quantum case, two generalizations have been found that both give capacities of a quantum channel, although these capacities differ in bot... |

1 |
On strong superadditivity of the entanglement of formation,” LANL e-print quant-ph/0303045, available at http://xxx.lanl.gov
- Audenaert, Braunstein
(Show Context)
Citation Context |

1 |
A new proof of the quantum noiseless coding the
- Jozsa, Schumacher
- 1994
(Show Context)
Citation Context ... the above definition of entropy for thermodynamics. One can ask whether this is also the correct definition of entropy for information theory. We will first give the example of quantum source coding =-=[30, 44]-=-, also called Schumacher compression, for which we will see that it is indeed the right definition. We consider a memoryless quantum source that at each time step emits the pure state vi with probabil... |