264 research outputs found
Converse bounds for private communication over quantum channels
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 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 several converse bounds
on the private transmission capabilities of all phase-insensitive bosonic
channels.Comment: v3: 53 pages, 3 figures, final version accepted for publication in
IEEE Transactions on Information Theor
Strong converse rates for quantum communication
We revisit a fundamental open problem in quantum information theory, namely
whether it is possible to transmit quantum information at a rate exceeding the
channel capacity if we allow for a non-vanishing probability of decoding error.
Here we establish that the Rains information of any quantum channel is a strong
converse rate for quantum communication: For any sequence of codes with rate
exceeding the Rains information of the channel, we show that the fidelity
vanishes exponentially fast as the number of channel uses increases. This
remains true even if we consider codes that perform classical post-processing
on the transmitted quantum data. As an application of this result, for
generalized dephasing channels we show that the Rains information is also
achievable, and thereby establish the strong converse property for quantum
communication over such channels. Thus we conclusively settle the strong
converse question for a class of quantum channels that have a non-trivial
quantum capacity.Comment: v4: 13 pages, accepted for publication in IEEE Transactions on
Information Theor
On the Second-Order Asymptotics for Entanglement-Assisted Communication
The entanglement-assisted classical capacity of a quantum channel is known to
provide the formal quantum generalization of Shannon's classical channel
capacity theorem, in the sense that it admits a single-letter characterization
in terms of the quantum mutual information and does not increase in the
presence of a noiseless quantum feedback channel from receiver to sender. In
this work, we investigate second-order asymptotics of the entanglement-assisted
classical communication task. That is, we consider how quickly the rates of
entanglement-assisted codes converge to the entanglement-assisted classical
capacity of a channel as a function of the number of channel uses and the error
tolerance. We define a quantum generalization of the mutual information
variance of a channel in the entanglement-assisted setting. For covariant
channels, we show that this quantity is equal to the channel dispersion, and
thus completely characterize the convergence towards the entanglement-assisted
classical capacity when the number of channel uses increases. Our results also
apply to entanglement-assisted quantum communication, due to the equivalence
between entanglement-assisted classical and quantum communication established
by the teleportation and super-dense coding protocols.Comment: v2: Accepted for publication in Quantum Information Processin
Correlation detection and an operational interpretation of the R�nyi mutual information
A variety of new measures of quantum R�nyi mutual information and quantum R�nyi conditional entropy have recently been proposed, and some of their mathematical properties explored. Here, we show that the R�nyi mutual information attains operational meaning in the context of composite hypothesis testing, when the null hypothesis is a fixed bipartite state and the alternative hypothesis consists of all product states that share one marginal with the null hypothesis. This hypothesis testing problem occurs naturally in channel coding, where it corresponds to testing whether a state is the output of a given quantum channel or of a "useless" channel whose output is decoupled from the environment. Similarly, we establish an operational interpretation of R�nyi conditional entropy by choosing an alternative hypothesis that consists of product states that are maximally mixed on one system. Specialized to classical probability distributions, our results also establish an operational interpretation of R�nyi mutual information and R�nyi conditional entropy
The fidelity of recovery is multiplicative
© 1963-2012 IEEE. Fawzi and Renner recently established a lower bound on the conditional quantum mutual information (CQMI) of tripartite quantum states in terms of the fidelity of recovery (FoR), i.e., the maximal fidelity of the state with a state reconstructed from its marginal by acting only on the system. The FoR measures quantum correlations by the local recoverability of global states and has many properties similar to the CQMI. Here, we generalize the FoR and show that the resulting measure is multiplicative by utilizing semi-definite programming duality. This allows us to simplify an operational proof by Brandão et al. of the above-mentioned lower bound that is based on quantum state redistribution. In particular, in contrast to the previous approaches, our proof does not rely on de Finetti reductions
Operational interpretation of Rényi information measures via composite hypothesis testing against product and markov distributions
© 1963-2012 IEEE. We revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis. We introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms. In this case, we find the threshold rate, the optimal error and strong converse exponents (at large deviations from the threshold), and the second order asymptotics (at small deviations from the threshold). We apply our results to find the operational interpretations of various Rényi information measures. In case the alternative hypothesis is comprised of bipartite product distributions, we find that the optimal error and strong converse exponents are determined by the variations of Rényi mutual information. In case the alternative hypothesis consists of tripartite distributions satisfying the Markov property, we find that the optimal exponents are determined by the variations of Rényi conditional mutual information. In either case, the relevant notion of Rényi mutual information depends on the precise choice of the alternative hypothesis. As such, this paper also strengthens the view that different definitions of Rényi mutual information, conditional entropy, and conditional mutual information are adequate depending on the context in which the measures are used
Converse bounds for private communication over quantum channels
© 1963-2012 IEEE. 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
Quantum coding with finite resources
The quantum capacity of a memoryless channel determines the maximal rate at which we can communicate reliably over asymptotically many uses of the channel. Here we illustrate that this asymptotic characterization is insufficient in practical scenarios where decoherence severely limits our ability to manipulate large quantum systems in the encoder and decoder. In practical settings, we should instead focus on the optimal trade-off between three parameters: the rate of the code, the size of the quantum devices at the encoder and decoder, and the fidelity of the transmission. We find approximate and exact characterizations of this trade-off for various channels of interest, including dephasing, depolarizing and erasure channels. In each case, the trade-off is parameterized by the capacity and a second channel parameter, the quantum channel dispersion. In the process, we develop several bounds that are valid for general quantum channels and can be computed for small instances
Decomposition Rules for Quantum Rényi Mutual Information with an Application to Information Exclusion Relations
We prove decomposition rules for quantum R\'enyi mutual information, generalising the relation to inequalities between R\'enyi mutual information and R\'enyi entropy of different orders. The proof uses Beigi's generalisation of Reisz-Thorin interpolation to operator norms, and a variation of the argument employed by Dupuis which was used to show chain rules for conditional R\'enyi entropies. The resulting decomposition rule is then applied to establish an information exclusion relation for R\'enyi mutual information, generalising the original relation by Hall
Moderate Deviation Analysis for Classical Communication over Quantum Channels
© 2017, Springer-Verlag GmbH Germany. We analyse families of codes for classical data transmission over quantum channels that have both a vanishing probability of error and a code rate approaching capacity as the code length increases. To characterise the fundamental tradeoff between decoding error, code rate and code length for such codes we introduce a quantum generalisation of the moderate deviation analysis proposed by AltÅg and Wagner as well as Polyanskiy and Verdú. We derive such a tradeoff for classical-quantum (as well as image-additive) channels in terms of the channel capacity and the channel dispersion, giving further evidence that the latter quantity characterises the necessary backoff from capacity when transmitting finite blocks of classical data. To derive these results we also study asymmetric binary quantum hypothesis testing in the moderate deviations regime. Due to the central importance of the latter task, we expect that our techniques will find further applications in the analysis of other quantum information processing tasks
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