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Register a new userRelating different quantum generalizations of the conditional Renyi entropy
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Recently a new quantum generalization of the Renyi divergence and the corresponding conditional Renyi entropies was proposed. Here we report on a surprising relation between conditional Renyi entropies based on this new generalization and conditional Renyi entropies based on the quantum relative Renyi entropy that was used in previous literature. Our result generalizes the well-known duality relation H(A|B) + H(A|C) = 0 of the conditional von Neumann entropy for tripartite pure states to Renyi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional Renyi entropies and derive a new entropic uncertainty relation.
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The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-$sigma$-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.
In this short note, I show how a recent result of Alhejji and Smith [arXiv:1909.00787] regarding an optimal uniform continuity bound for classical conditional entropy leads to an optimal uniform continuity bound for quantum conditional entropy of classical--quantum states. The bound is optimal in the sense that there always exists a pair of classical--quantum states saturating the bound, and so no further improvements are possible. An immediate application is a uniform continuity bound for entanglement of formation that improves upon the one previously given by Winter in [arXiv:1507.07775]. Two intriguing open questions are raised regarding other possible uniform continuity bounds for conditional entropy, one about quantum--classical states and another about fully quantum bipartite states.
We establish a discrete analog of the Renyi entropy comparison due to Bobkov and Madiman. For log-concave variables on the integers, the min entropy is within log e of the usual Shannon entropy. Additionally we investigate the entropic Rogers-Shephard inequality studied by Madiman and Kontoyannis, and establish a sharp Renyi version for certain parameters in both the continuous and discrete cases
Feature selection, in the context of machine learning, is the process of separating the highly predictive feature from those that might be irrelevant or redundant. Information theory has been recognized as a useful concept for this task, as the prediction power stems from the correlation, i.e., the mutual information, between features and labels. Many algorithms for feature selection in the literature have adopted the Shannon-entropy-based mutual information. In this paper, we explore the possibility of using Renyi min-entropy instead. In particular, we propose an algorithm based on a notion of conditional Renyi min-entropy that has been recently adopted in the field of security and privacy, and which is strictly related to the Bayes error. We prove that in general the two approaches are incomparable, in the sense that we show that we can construct datasets on which the Renyi-based algorithm performs better than the corresponding Shannon-based one, and datasets on which the situation is reversed. In practice, however, when considering datasets of real data, it seems that the Renyi-based algorithm tends to outperform the other one. We have effectuate several experiments on the BASEHOCK, SEMEION, and GISETTE datasets, and in all of them we have indeed observed that the Renyi-based algorithm gives better results.
The scaled Renyi information plays a significant role in evaluating the performance of information processing tasks by virtue of its connection to the error exponent analysis. In quantum information theory, there are three generalizations of the classical Renyi divergence---the Petzs, sandwiched, and log-Euclide
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