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arXiv:0903.3920 · Physical Review A

Public and private communication with a quantum channel and a secret key

Min-Hsiu Hsieh, Mark M. Wilde

Published Aug 5, 2009• 20 citations
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Abstract

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.

DOI
10.1103/physreva.80.022306

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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.
We consider the problem of transmitting classical and quantum information reliably over an entanglement-assisted quantum channel. Our main result is a capacity theorem that gives a three-dimensional 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 protocol. The classically-enhanced father 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 classically-enhanced father protocol also shows an improvement over a time-sharing strategy for the case of a qubit dephasing channel--this result justifies the need for simultaneous coding of classical and quantum information over an entanglement-assisted quantum channel. Our capacity theorem is of a multi-letter 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.
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
We introduce a new protocol, the channel-state coding protocol, to quantum Shannon theory. This protocol generates entanglement between a sender and receiver by coding for a noisy quantum channel with the aid of a noisy shared state. The mother and father protocols arise as special cases of the channel-state coding protocol, where the channel is noiseless or the state is a noiseless maximally entangled state, respectively. The channel-state coding protocol paves the way for formulating entanglement-assisted quantum error-correcting codes that are robust to noise in shared entanglement. Finally, the channel-state coding protocol leads to a Smith-Yard superactivation, where we can generate entanglement using a zero-capacity erasure channel and a non-distillable bound entangled state.
This paper presents a simple way for an eavesdropper to eavesdrop freely the secret message in the experimental realization of quantum communication protocol proposed by Beige et al (2002 Acta Phys. Pol. A 101 357). Moreover, it introduces an efficient quantum secure communication protocol based on a publicly known key with decoy photons and two biased bases by modifying the original protocol. The total efficiency of this new protocol is double that of the original one. With a low noise quantum channel, this protocol can be used for transmitting a secret message. At present, this protocol is good for generating a private key efficiently.
In this paper, we give tradeoffs between classical communication, quantum communication, and entanglement for processing information in the Shannon-theoretic setting. We first prove a “unit-resource” capacity theorem that applies to the scenario where only the above three noiseless resources are available for consumption or generation. The optimal strategy mixes the three fundamental protocols of teleportation, superdense coding, and entanglement distribution. We then provide an achievable rate region and a matching multiletter converse for the “direct-static” capacity theorem. This theorem applies to the scenario where a large number of copies of a noisy bipartite state are available (in addition to consumption or generation of the above three noiseless resources). Our coding strategy involves a protocol that we name theclassically assisted state redistribution protocoland the three fundamental protocols. We finally provide an achievable rate region and a matching multiletter converse for the “direct-dynamic” capacity theorem. This theorem applies to the scenario where a large number of uses of a noisy quantum channel are available in addition to the consumption or generation of the three noiseless resources. Our coding strategy combines theclassically enhanced father protocolwith the three fundamental unit protocols.