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arXiv:0810.3478 · Journal of Mathematical Physics

Generalized relative entropies and the capacity of classical-quantum channels

Milán Mosonyi, Nilanjana Datta

Published Jul 1, 2009• 72 citations
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Abstract

We provide lower and upper bounds on the information transmission capacity of one single use of a classical-quantum channel. The lower bound is expressed in terms of the Hoeffding capacity, which we define similarly to the Holevo capacity but replacing the relative entropy with the Hoeffding distance. Similarly, our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max-relative entropy in the definition of the divergence radius of a channel.

DOI
10.1063/1.3167288

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We provide lower and upper bounds on the information transmission capacity of one single use of a classical-quantum channel. The lower bound is expressed in terms of the Hoeffding capacity, which we define similarly to the Holevo capacity but replacing the relative entropy with the Hoeffding distance. Similarly, our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max-relative entropy in the definition of the divergence radius of a channel.
Following Csiszár's approach in classical information theory, it is shown that the quantum α-relative entropies with parameter α ∈ (0,1) can be represented as generalized cutoff rates, and hence a direct operational interpretation of the quantum α-relative entropies are provided. It is also shown that various generalizations of the Holevo capacity, defined in terms of the α-relative entropies, coincide for the parameter range α ∈ (0,2], and an upper bound on the one-shot ε-capacity of a classical-quantum channel in terms of these capacities is given.
We investigate bounds in the transmission of classical information through quantum systems. Our focus lies in the generalized Holevo theorem, which provides a single-letter Holevo-like inequality from arbitrary quantum distance measures. Through the introduction of the [Formula: see text]-[Formula: see text]-Rényi relative entropies, which comprise known relevant quantities such as the Rényi relative entropy and the sandwiched Rényi relative entropy, we establish the Holevo–Rényi inequality. This result leads to a quantum bound for the [Formula: see text]-mutual information, suggesting new insights into communication channel performance and the fundamental limits for reliability functions in memoryless multi-letter communication channels.
We obtain lower and upper bounds on the one-shot strong converse capacity for the transmission of classical information through quantum channels, in terms of a generalized relative entropy quantity, namely, the smoothed max-relative entropy. For the case of a memoryless channel, these bounds converge independently to the Holevo capacity, in the limit of asymptotically many uses. We also employ our one-shot result to obtain an upper bound to the strong converse capacity of a channel with long-term memory which can be shown to violate the strong converse property. Finally, we show bounds on the strong converse rate of hypothesis testing in the one-shot scenario, again in terms of the smoothed max-relative entropies.
With the recent development of quantum information theory, some attempts have been made to construct information theory beyond quantum theory. Here, we consider hypothesis-testing relative entropy and one-shot classical capacity, that is, the optimal rate of classical information transmitted by using a single channel under the constraint of a certain error probability, in general physical theories where states and measurements are operationally defined. Then we obtain the upper bound of the one-shot classical capacity by generalizing the method given by Wang and Renner [Phys. Rev. Lett. 108, 200501 (2012)]. Also, we derive the lower bound of the capacity by showing the existence of a good code that can transmit classical information with a certain error probability. Applying the above two bounds, we prove the asymptotic equivalence between classical capacity and hypothesis-testing relative entropy in any general physical theory.
We define classical quantum multiway channels for transmission of classical information, after recent work by Allahverdyan and Saakian. Bounds on the capacity region are derived in a uniform way, which are analogous to the classically known ones, simply replacing Shannon entropy with von Neumann entropy. For the single receiver case (multiple-access channel) the exact capacity region is determined. These results are applied to the case of noisy channels, with arbitrary input signal states. A second issue of this work is the presentation of a calculus of quantum information quantities, based on the algebraic formulation of quantum theory.