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Strong converses for classical information transmission and hypothesis testing

Nilanjana Datta, Milán Mosonyi, Min-Hsiu Hsieh, Fernando G. S. L. Brandão

Published Jun 15, 2011• 2 citations
OpenAlex

Abstract

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.

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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.
We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Then using a relative entropy version of the quantum asymptotic equipartition property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger ( ε-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot ε-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the ε-independent version of the relative entropy-QAEP, we can recover both the Holevo- Schumacher- Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.
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.
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.
The one-shot classical capacity of a quantum channel quantifies the amount of classical information that can be transmitted through a single use of the channel such that the error probability is below a certain threshold. In this work, we show that this capacity is well approximated by a relative-entropy-type measure defined via hypothesis testing. Combined with a quantum version of Stein's lemma, our results give a conceptually simple proof of the well-known Holevo-Schumacher-Westmoreland theorem for the capacity of memoryless channels. More generally, we obtain tight capacity formulas for arbitrary (not necessarily memoryless) channels.
We study the problem of transmission of information over classical and classical-quantum channels in the one-shot regime where the underlying codes are constrained to be group codes. In the achievability part, we introduce a new input probability distribution that incorporates the encoding homomorphism and the underlying channel law. Using a random coding argument, we characterize the performance of group codes in terms of hypothesis testing relative-entropic quantities. In the converse part, we establish bounds by leveraging a hypothesis testing-based approach. Furthermore, we apply the one-shot result to the asymptotic stationary memoryless setting, and establish a single-letter lower bound on group capacities for both classes of channels. Moreover, we derive a matching upper bound on the asymptotic group capacity.
We study entanglement-assisted quantum and classical communication over a single use of a quantum channel, which itself can correspond to a finite number of uses of a channel with arbitrarily correlated noise. We obtain characterizations of the corresponding one-shot capacities by establishing upper and lower bounds on them in terms of the difference of two smoothed entropic quantities. In the case of a memoryless channel, the upper and lower bounds converge to the known single-letter formulas for the corresponding capacities, in the limit of asymptotically many uses of it. Our results imply that the difference of two smoothed entropic quantities characterizing the one-shot entanglement-assisted capacities serves as a one-shot analog of the mutual information, since it reduces to the mutual information, between the output of the channel and a system purifying its input, in the asymptotic, memoryless scenario.