Characterization of several kinds of quantum analogues of relative entropy

We review the properties of the quantum relative entropy function and discuss its application to problems of classical and quantum information transfer and to quantum data compression. We then outline further uses of relative entropy to quantify quantum entanglement and analyze its manipulation.

We argue from the point of view of statistical inference that the quantum relative entropy is a good measure for distinguishing between two quantum states (or two classes of quantum states) described by density matrices. We extend this notion to describe the amount of entanglement between two quantum systems from a statistical point of view. Our measure is independent of the number of entangled systems and their dimensionality.

The aim of the present paper is to give axiomatic characterization of quantum relative entropy utilizing resource conversion scenario. We consider two sets of axioms: non-asymptotic and asymptotic. In the former setting, we prove that the upperbound and the lowerbund of $\mathrm{D}^{Q}(\rho||\sigma) $ is $\mathrm{D}^{R}(\rho||\sigma) :=\mathrm{tr}% \,\rho\ln\sqrt{\rho}\sigma^{-1}\sqrt{\rho}$ and $\mathrm{D}(\rho||\sigma) :=$ $\mathrm{tr}\,\rho(\ln\rho-\ln\sigma) $, respective...

In this paper it was proved that the quantum relative entropy $D(\sigma \| \rho)$ can be asymptotically attained by Kullback Leibler divergences of probabilities given by a certain sequence of POVMs. The sequence of POVMs depends on $\rho$, but is independent of the choice of $\sigma$.

Quantum mechanics and information theory are among the most important scientific discoveries of the last century. Although these two areas initially developed separately it has emerged that they are in fact intimately related. In this review I will show how quantum information theory extends traditional information theory by exploring the limits imposed by quantum, rather than classical mechanics on information storage and transmission. The derivation of many key results uniq...

We give a pedagogical introduction to quantum discord. We the discuss the problem of separation of total correlations in a given quantum state into entanglement, dissonance, and classical correlations using the concept of relative entropy as a distance measure of correlations. This allows us to put all correlations on an equal footing. Entanglement and dissonance, whose definition is introduced here, jointly belong to what is known as quantum discord. Our methods are complete...

In this paper, we introduce and study unified $(r,s)$-relative entropy and quantum unified $(r,s)$-relative entropy, in particular, our main results of quantum unified $(r,s)$-relative entropy are established on the separable complex Hilbert spaces. Moreover, the entanglement-measure of states due to the quantum unified $(r,s)$-relative entropy is considered, too. Our results improved a uncorrect statement on the monotone property of entanglement-measure function.

The aim of the present paper is to give axiomatic characterization of quantum relative entropy utilizing resource conversion scenario. We consider two sets of axioms: non-asymptotic and asymptotic. In the former setting, we prove that the upperbound and the lowerbund of D^{Q}({\rho}||{\sigma}) is D^{R}({\rho}||{\sigma}):=tr{\rho}ln{\sigma}^{1/2}{\rho}^{-1}{\sigma}^{1/2} and D({\rho}||{\sigma}):= tr{\rho}(ln{\rho}-ln{\sigma}), respectively. In the latter setting, we prove uniq...

In this article we propose a quantum version of Shannon's conditional entropy. Given two density matrices $\rho$ and $\sigma$ on a finite dimensional Hilbert space and with $S(\rho)=-\tr\rho\ln\rho$ being the usual von Neumann entropy, this quantity $S(\rho|\sigma)$ is concave in $\rho$ and satisfies $0\le S(\rho|\sigma)\le S(\rho)$, a quantum analogue of Shannon's famous inequality. Thus we view $S(\rho|\sigma)$ as the entropy of $\rho$ conditioned by $\sigma$.

We prove that the quantum relative entropy is a rate function in large deviation principle. Next, we define information criteria for quantum states and estimate the accuracy of the use of them. Most of the results in this paper are essentially based on Hiai-Ohya-Tsukada theorem.