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arXiv:1602.08898 · IEEE Transactions on Information Theory

Converse Bounds for Private Communication Over Quantum Channels

Mark M. Wilde, Marco Tomamichel, Mario Berta

Published Jan 5, 2017• 154 citations
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Abstract

This paper establishes several converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state, which is a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here, we use this approach along with a “privacy test” to establish a general meta-converse bound for private communication, which has a number of applications. The meta-converse allows for proving that any quantum channel's relative entropy of entanglement is a strong converse rate for private communication. For covariant channels, the meta-converse also leads to second-order expansions of relative entropy of entanglement bounds for private communication rates. For such channels, the bounds also apply to the private communication setting in which the sender and the receiver are assisted by unlimited public classical communication, and as such, they are relevant for establishing various converse bounds for quantum key distribution protocols conducted over these channels. We find precise characterizations for several channels of interest and apply the methods to establish converse bounds on the private transmission capabilities of all phase-insensitive bosonic channels.

DOI
10.1109/tit.2017.2648825

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This paper establishes several converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state, which is a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here, we use this approach along with a “privacy test” to establish a general meta-converse bound for private communication, which has a number of applications. The meta-converse allows for proving that any quantum channel's relative entropy of entanglement is a strong converse rate for private communication. For covariant channels, the meta-converse also leads to second-order expansions of relative entropy of entanglement bounds for private communication rates. For such channels, the bounds also apply to the private communication setting in which the sender and the receiver are assisted by unlimited public classical communication, and as such, they are relevant for establishing various converse bounds for quantum key distribution protocols conducted over these channels. We find precise characterizations for several channels of interest and apply the methods to establish converse bounds on the private transmission capabilities of all phase-insensitive bosonic channels.
We establish a converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state, which is a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here we use this approach along with a “privacy test” to establish a general meta-converse bound for private communication.
We consider private communication over bosonic Gaussian channels via the most general adaptive protocols based on local operations and two-way classical communication. These protocols include all the possible strategies allowed by quantum mechanics where two remote parties have local quantum computers but do not share prior quantum entanglement. In this context, Pirandola-Laurenza-Ottaviani-Banchi (PLOB) [Nat. Commun. 8, 15043 (2017)] established weak converse upper bounds for the secret key capacity of these channels. These bounds were computed by combining teleportation stretching, able to simplify any adaptive protocol into a block form, and the channel’s relative entropy of entanglement, so that data-processing properties allow one to write simple single-letter quantities. Here we discuss an extension of these bounds to repeater-assisted quantum communications. Then, using an energy-constrained version of the diamond norm and the Braunstein-Kimble teleportation protocol, we can rigorously show the strong converse property of the bounds discovered by PLOB. Our analysis provides a full mathematical justification of recent claims appeared in the literature.
Abstract Werner states have a host of interesting properties, which often serve to illuminate the unusual properties of quantum information. Starting from these states, one may define a family of quantum channels, known as the Holevo–Werner channels, which themselves afford several unusual properties. In this paper we use the teleportation covariance of these channels to upper bound their two-way assisted quantum and secret-key capacities. This bound may be expressed in terms of relative entropy distances, such as the relative entropy of entanglement, and also in terms of the squashed entanglement. Most interestingly, we show that the relative entropy bounds are strictly subadditive for a sub-class of the Holevo–Werner channels, so that their regularization provides a tighter performance. These information-theoretic results are first found for point-to-point communication and then extended to repeater chains and quantum networks, under different types of routing strategies.
We explore several new converse bounds for classical communication over quantum channels in both the one-shot and asymptotic regimes. First, we show that the Matthews-Wehner meta-converse bound for entanglementassisted classical communication can be achieved by activated, no-signaling assisted codes, suitably generalizing a result for classical channels. Second, we derive a new efficiently computable meta-converse on the amount of classical information unassisted codes can transmit over a single use of a quantum channel. As applications, we provide a finite resource analysis of classical communication over quantum erasure channels, including the second-order and moderate deviation asymptotics. Third, we explore the asymptotic analogue of our new meta-converse, the Υ-information of the channel. We show that its regularization is an upper bound on the classical capacity, which is generally tighter than the entanglement-assisted capacity and other known efficiently computable strong converse bounds. For covariant channels, we show that the Υ-information is a strong converse bound.
Abstract We review recent results on the simulation of quantum channels, the reduction of adaptive protocols (teleportation stretching), and the derivation of converse bounds for quantum and private communication, as established in PLOB (Pirandola et al 2017 Nat. Commun. 8 15043). We start by introducing a general weak converse bound for private communication based on the relative entropy of entanglement. We discuss how combining this bound with channel simulation and teleportation stretching, PLOB established the two-way quantum and private capacities of several fundamental channels, including the bosonic lossy channel. We then provide a rigorous proof of the strong converse property of these bounds by adopting a correct use of the Braunstein–Kimble teleportation protocol for the simulation of bosonic Gaussian channels. This analysis provides a full justification of claims presented in the follow-up paper WTB (Wilde et al 2017 IEEE Trans. Inf. Theory 63 1792–817) whose upper bounds for Gaussian channels would be otherwise infinitely large. Besides clarifying contributions in the area of channel simulation and protocol reduction, we also present some generalizations of the tools to other entanglement measures and novel results on the maximum excess noise which is tolerable in quantum key distribution.
With the rapid growth of quantum technologies, knowing the fundamental characteristics of quantum systems and protocols is essential for their effective implementation. A particular communication setting that has received increased focus is related to quantum key distribution and distributed quantum computation. In this setting, a quantum channel connects a sender to a receiver, and their goal is to distill either a secret key or entanglement, along with the help of arbitrary local operations and classical communication (LOCC). In this work, we establish a general theory of energy-constrained, LOCC-assisted private and quantum capacities of quantum channels, which are the maximum rates at which an LOCC-assisted quantum channel can reliably establish a secret key or entanglement, respectively, subject to an energy constraint on the channel input states. We prove that the energy-constrained squashed entanglement of a channel is an upper bound on these capacities. We also explicitly prove that a thermal state maximizes a relaxation of the squashed entanglement of all phase-insensitive, single-mode input bosonic Gaussian channels, generalizing results from prior work. After doing so, we prove that a variation of the method introduced by Goodenough et al. [New J. Phys. 18, 063005 (2016)] leads to improved upper bounds on the energy-constrained secret-key-agreement capacity of a bosonic thermal channel. We then consider a multipartite setting and prove that two known multipartite generalizations of the squashed entanglement are in fact equal. We finally show that the energy-constrained, multipartite squashed entanglement plays a role in bounding the energy-constrained LOCC-assisted private and quantum capacity regions of quantum broadcast channels.