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arXiv:0806.3525 · Physical Review A

Secret-key-assisted private classical communication capacity over quantum channels

Min-Hsiu Hsieh, Zhicheng Luo, Todd A. Brun

Published Oct 8, 2008• 17 citations
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Abstract

We prove a regularized formula for the secret-key-assisted capacity region of a quantum channel for transmitting private classical information. This result parallels the work of Devetak et al. (e-print arXiv:quant-ph/0512015) on entanglement-assisted quantum communication capacity . This formula provides a family protocol, the private father protocol, under the resource inequality framework that includes private classical communication without secret-key assistance as a child protocol.

DOI
10.1103/physreva.78.042306

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We prove a regularized formula for the secret-key-assisted capacity region of a quantum channel for transmitting private classical information. This result parallels the work of Devetak et al. (e-print arXiv:quant-ph/0512015) on entanglement-assisted quantum communication capacity . This formula provides a family protocol, the private father protocol, under the resource inequality framework that includes private classical communication without secret-key assistance as a child protocol.
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In this paper, we consider the problem of transmitting classical and quantum information reliably over an entanglement-assisted quantum (EAQ) channel. Our main result is a capacity theorem that gives a 3-D achievable rate region. Points in the region are rate triples, consisting of the classical communication rate, the quantum communication rate, and the entanglement consumption rate of a particular coding scheme. The crucial protocol in achieving the boundary points of the capacity region is a protocol that we name the classically enhanced father(CEF) protocol. The CEF protocol is more general than other protocols in the family tree of quantum Shannon theoretic protocols, in the sense that several previously known quantum protocols are now child protocols of it. The CEF protocol also shows an improvement over a timesharing strategy for the case of a qubit dephasing channel-this result justifies the need for simultaneous coding of classical and quantum information over an EAQ channel. Our capacity theorem is of a multiletter nature (requiring a limit over many uses of the channel), but it reduces to a single-letter characterization for at least three channels: the completely depolarizing channel, the quantum erasure channel, and the qubit dephasing channel.
We consider using a secret key and a noisy quantum channel to generate noiseless public communication and noiseless private communication. The optimal protocol for this setting is the publicly enhanced private father protocol. This protocol exploits random coding techniques and ``piggybacking'' of public information along with secret-key-assisted private codes. The publicly enhanced private father protocol is a generalization of the secret-key-assisted protocol of Hsieh, Luo, and Brun and a generalization of a protocol for simultaneous communication of public and private information suggested by Devetak and Shor.
A formula for the capacity of a quantum channel for transmitting private classical information is derived. This is shown to be equal to the capacity of the channel for generating a secret key, and neither capacity is enhanced by forward public classical communication. Motivated by the work of Schumacher and Westmoreland on quantum privacy and quantum coherence, parallels between private classical information and quantum information are exploited to obtain an expression for the capacity of a quantum channel for generating pure bipartite entanglement. The latter implies a new proof of the quantum channel coding theorem and a simple proof of the converse. The coherent information plays a role in all of the above mentioned capacities
Shared entanglement is a resource available to parties communicating over a quantum channel, much akin to public coins in classical communication protocols. Whereas shared randomness does not help in the transmission of information, or significantly reduce the classical complexity of computing functions (as compared to private-coin protocols), shared entanglement leads to startling phenomena such as ``quantum teleportation'' and ``superdense coding.'' The problem of characterising the power of prior entanglement has puzzled many researchers. In this paper, we revisit the problem of transmitting classical bits over an entanglement-assisted quantum channel. We derive a new, optimal bound on the number of quantum bits required for this task, for any given probability of error. All known lower bounds in the setting of bounded error entanglement-assisted communication are based on sophisticated information theoretic arguments. In contrast, our result is derived from first principles, using a simple linear algebraic technique.
We study the symmetric-side-channel-assisted private capacity of a quantum channel, for which we provide a single- letter formula. This capacity is additive, convex, and, for degradable channels, equal to the unassisted private capacity. While a channel's (unassisted) capacity for private classical communication may be strictly larger than its quantum capacity, we will show that these capacities are equal for degradable channels, thus demonstrating the equivalence of privacy and quantum coherence in this context. We use these ideas to find new bounds on the key rate of quantum key distribution protocols with one-way classical post-processing. For the Bennett-Brassard-84 (BB84) protocol, our results demonstrate that collective attacks are strictly stronger than individual attacks.