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arXiv:0709.2953 · Physical Review A

Quantum entanglement capacity with classical feedback

Alan W. Leung

Published Jan 18, 2008• 7 citations
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Abstract

For any quantum discrete memoryless channel, we define a quantity called quantum entanglement capacity with classical feedback $({E}_{B})$, and we show that this quantity lies between two other well-studied quantities. These two quantities---namely the quantum capacity assisted by two-way classical communication $({Q}_{2})$ and the quantum capacity with classical feedback $({Q}_{B})$---are widely conjectured to be different: There exists a quantum discrete memoryless channel for which ${Q}_{2}>{Q}_{B}$. We then present a general scheme to convert any quantum error-correcting codes into adaptive protocols for this newly defined quantity of the quantum depolarizing channel, and illustrate with the repetition code and Shor code. We contrast the present notion with entanglement purification protocols by showing that, whilst the Leung-Shor protocol can be applied directly, recurrence methods need to be supplemented with other techniques but at the same time offer a way to improve the aforementioned repetition code. For the quantum depolarizing channel, we prove a formula that gives lower bounds on the quantum capacity with classical feedback from any ${E}_{B}$ protocols. We then apply this formula to the ${E}_{B}$ protocols that we discuss to obtain lower bounds on the quantum capacity with classical feedback of the quantum depolarizing channel.

DOI
10.1103/physreva.77.012322

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For any quantum discrete memoryless channel, we define a quantity called quantum entanglement capacity with classical feedback (EB)({E}_{B})(EB​), and we show that this quantity lies between two other well-studied quantities. These two quantities---namely the quantum capacity assisted by two-way classical communication (Q2)({Q}_{2})(Q2​) and the quantum capacity with classical feedback (QB)({Q}_{B})(QB​)---are widely conjectured to be different: There exists a quantum discrete memoryless channel for which Q2>QB{Q}_{2}>{Q}_{B}Q2​>QB​. We then present a general scheme to convert any quantum error-correcting codes into adaptive protocols for this newly defined quantity of the quantum depolarizing channel, and illustrate with the repetition code and Shor code. We contrast the present notion with entanglement purification protocols by showing that, whilst the Leung-Shor protocol can be applied directly, recurrence methods need to be supplemented with other techniques but at the same time offer a way to improve the aforementioned repetition code. For the quantum depolarizing channel, we prove a formula that gives lower bounds on the quantum capacity with classical feedback from any EB{E}_{B}EB​ protocols. We then apply this formula to the EB{E}_{B}EB​ protocols that we discuss to obtain lower bounds on the quantum capacity with classical feedback of the quantum depolarizing channel.
We initiate the study of zero-error communication via quantum channels when the receiver and the sender have at their disposal a noiseless feedback channel of unlimited quantum capacity, generalizing Shannon's zero-error communication theory with instantaneous feedback. We first show that this capacity is only a function of the linear span of Choi-Kraus operators of the channel, which generalizes the bipartite equivocation graph of a classical channel, and which we dub non-commutative bipartite graph. Then, we go on to show that the feedback-assisted capacity is non-zero (allowing for a constant amount of activating noiseless communication) if and only if the non-commutative bipartite graph is non-trivial, and give a number of equivalent characterizations. This result involves a far-reaching extension of the conclusive exclusion of quantum states. We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede. We demonstrate that this bound to have many good properties, including being additive and given by a minimax formula. We also prove a coding theorem showing that this quantity is the entanglement-assisted capacity against an adversarially chosen channel from the set of all channels with the same Choi-Kraus span, which can also be interpreted as the feedback-assisted unambiguous capacity. The proof relies on a generalization of the Postselection Lemma (de Finetti reduction) that allows to reflect additional constraints, and which we believe to be of independent interest. This capacity is a relaxation of the feedback-assisted zero-error capacity; however, we have to leave open the question of whether they coincide in general. We illustrate our ideas with a number of examples, including classical-quantum channels and Weyl diagonal channels, and close with an extensive discussion of open questions.
We present a family of entanglement purification protocols that generalize four previous methods, namely the recurrence method, the modified recurrence method, and the two methods proposed by Maneva-Smolin and Leung-Shor. We will show that this family of protocols have improved yields over a wide range of initial fidelities F, and hence imply new lower bounds on the quantum capacity assisted by two-way classical communication of the quantum depolarizing channel. In particular, the yields of these protocols are higher than the yield of universal hashing for F less than 0.993 and as F goes to 1.
The entanglement-assisted classical capacity of a noisy quantum channel (C/sub E/) is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entanglement, which may be used up by the communication protocol. We show that the capacity C/sub E/ is given by an expression parallel to that for the capacity of a purely classical channel: i.e., the maximum, over channel inputs /spl rho/, of the entropy of the channel input plus the entropy of the channel output minus their joint entropy, the latter being defined as the entropy of an entangled purification of /spl rho/ after half of it has passed through the channel. We calculate entanglement-assisted capacities for two interesting quantum channels, the qubit amplitude damping channel and the bosonic channel with amplification/attenuation and Gaussian noise. We discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel, alone or in the presence of ancillary resources such as prior entanglement. In the classical analog of entanglement-assisted communication - communication over a discrete memoryless channel (DMC) between parties who share prior random information - we show that one parameter is sufficient, i.e., that in the presence of prior shared random information, all DMCs of equal capacity can simulate one another with unit asymptotic efficiency.
We generalize the random coding argument of stabilizer codes and derive a lower bound on the quantum capacity of an arbitrary discrete memoryless quantum channel. For the depolarizing channel, our lower bound coincides with that obtained by Bennett et al. We also slightly improve the quantum Gilbert–Varshamov bound for general stabilizer codes, and establish an analog of the quantum Gilbert–Varshamov bound for linear stabilizer codes. Our proof is restricted to the binary quantum channels, but its extension of to l-adic channels is straight forward.
We prove that the classical capacity of an arbitrary quantum channel assisted by a free classical feedback channel is bounded from above by the maximum average output entropy of the quantum channel. As a consequence of this bound, we conclude that a classical feedback channel does not improve the classical capacity of a quantum erasure channel, and by taking into account energy constraints, we conclude the same for a pure-loss bosonic channel. The method for establishing the aforementioned entropy bound involves identifying an information measure having two key properties: 1) it does not increase under a one-way local operations and classical communication channel from the receiver to the sender and 2) a quantum channel from sender to receiver cannot increase the information measure by more than the maximum output entropy of the channel. This information measure can be understood as the sum of two terms, with one corresponding to classical correlation and the other to entanglement.