Multiple Quantum Hypothesis Testing Expressions and Classical-Quantum Channel Converse Bounds

Alternative exact expressions are derived for the minimum error probability of a hypothesis test discriminating among $M$ quantum states. The first expression corresponds to the error probability of a binary hypothesis test with certain parameters; the second involves the optimization of a given information-spectrum measure. Particularized in the classical-quantum channel coding setting, this characterization implies the tightness of two existing converse bounds; one derived by Matthews and Wehner using hypothesis-testing, and one obtained by Hayashi and Nagaoka via an information-spectrum approach.Comment: Presented at the 2016 IEEE International Symposium on Information Theory, July 10-15, 2016, Barcelona, Spai

Strong Converse and Second-Order Asymptotics of Channel Resolvability

We study the problem of channel resolvability for fixed i.i.d. input distributions and discrete memoryless channels (DMCs), and derive the strong converse theorem for any DMCs that are not necessarily full rank. We also derive the optimal second-order rate under a condition. Furthermore, under the condition that a DMC has the unique capacity achieving input distribution, we derive the optimal second-order rate of channel resolvability for the worst input distribution.Comment: 7 pages, a shorter version will appear in ISIT 2014, this version includes the proofs of technical lemmas in appendice

One-shot rates for entanglement manipulation under non-entangling maps

We obtain expressions for the optimal rates of one- shot entanglement manipulation under operations which generate a negligible amount of entanglement. As the optimal rates for entanglement distillation and dilution in this paradigm, we obtain the max- and min-relative entropies of entanglement, the two logarithmic robustnesses of entanglement, and smoothed versions thereof. This gives a new operational meaning to these entanglement measures. Moreover, by considering the limit of many identical copies of the shared entangled state, we partially recover the recently found reversibility of entanglement manipu- lation under the class of operations which asymptotically do not generate entanglement.Comment: 7 pages; no figure

Smooth Renyi Entropies and the Quantum Information Spectrum

Many of the traditional results in information theory, such as the channel coding theorem or the source coding theorem, are restricted to scenarios where the underlying resources are independent and identically distributed (i.i.d.) over a large number of uses. To overcome this limitation, two different techniques, the information spectrum method and the smooth entropy framework, have been developed independently. They are based on new entropy measures, called spectral entropy rates and smooth entropies, respectively, that generalize Shannon entropy (in the classical case) and von Neumann entropy (in the more general quantum case). Here, we show that the two techniques are closely related. More precisely, the spectral entropy rate can be seen as the asymptotic limit of the smooth entropy. Our results apply to the quantum setting and thus include the classical setting as a special case

Generalized Entropies

We study an entropy measure for quantum systems that generalizes the von Neumann entropy as well as its classical counterpart, the Gibbs or Shannon entropy. The entropy measure is based on hypothesis testing and has an elegant formulation as a semidefinite program, a type of convex optimization. After establishing a few basic properties, we prove upper and lower bounds in terms of the smooth entropies, a family of entropy measures that is used to characterize a wide range of operational quantities. From the formulation as a semidefinite program, we also prove a result on decomposition of hypothesis tests, which leads to a chain rule for the entropy.Comment: 21 page