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arXiv:1706.00390 · Journal of Physics A Mathematical and Theoretical

Coherence and entanglement measures based on Rényi relative entropies

Huangjun Zhu, Masahito Hayashi, Lin Chen

Published Sep 29, 2017• 51 citations
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Abstract

Abstract We study systematically resource measures of coherence and entanglement based on Rényi relative entropies, which include the logarithmic robustness of coherence, geometric coherence, and conventional relative entropy of coherence together with their entanglement analogues. First, we show that each Rényi relative entropy of coherence is equal to the corresponding Rényi relative entropy of entanglement for any maximally correlated state. By virtue of this observation, we establish a simple operational connection between entanglement measures and coherence measures based on Rényi relative entropies. We then prove that all these coherence measures, including the logarithmic robustness of coherence, are additive. Accordingly, all these entanglement measures are additive for maximally correlated states. In addition, we derive analytical formulas for Rényi relative entropies of entanglement of maximally correlated states and bipartite pure states, which reproduce a number of classic results on the relative entropy of entanglement and logarithmic robustness of entanglement in a unified framework. Several nontrivial bounds for Rényi relative entropies of coherence (entanglement) are further derived, which improve over results known previously. Moreover, we determine all states whose relative entropy of coherence is equal to the logarithmic robustness of coherence. As an application, we provide an upper bound for the exact coherence distillation rate, which is saturated for pure states.

DOI
10.1088/1751-8121/aa8ffc

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Abstract We study systematically resource measures of coherence and entanglement based on Rényi relative entropies, which include the logarithmic robustness of coherence, geometric coherence, and conventional relative entropy of coherence together with their entanglement analogues. First, we show that each Rényi relative entropy of coherence is equal to the corresponding Rényi relative entropy of entanglement for any maximally correlated state. By virtue of this observation, we establish a simple operational connection between entanglement measures and coherence measures based on Rényi relative entropies. We then prove that all these coherence measures, including the logarithmic robustness of coherence, are additive. Accordingly, all these entanglement measures are additive for maximally correlated states. In addition, we derive analytical formulas for Rényi relative entropies of entanglement of maximally correlated states and bipartite pure states, which reproduce a number of classic results on the relative entropy of entanglement and logarithmic robustness of entanglement in a unified framework. Several nontrivial bounds for Rényi relative entropies of coherence (entanglement) are further derived, which improve over results known previously. Moreover, we determine all states whose relative entropy of coherence is equal to the logarithmic robustness of coherence. As an application, we provide an upper bound for the exact coherence distillation rate, which is saturated for pure states.
We show that the distillable coherence---which is equal to the relative entropy of coherence---is, up to a constant factor, always bounded by the ${\ensuremath{\ell}}_{1}$-norm measure of coherence (defined as the sum of absolute values of off diagonals). Thus the latter plays a similar role as logarithmic negativity plays in entanglement theory and this is the best operational interpretation from a resource-theoretic viewpoint. Consequently the two measures are intimately connected to another operational measure, the robustness of coherence. We find also relationships between these measures, which are tight for general states, and the tightest possible for pure and qubit states. For a given robustness, we construct a state having minimum distillable coherence.
Abstract We establish a rigorous equivalence between quantum coherence measures defined by the sandwiched Rényi relative entropy and generalized conditional entropies associated with auxiliary quantum systems, providing a direct operational interpretation of sandwiched Rényi relative entropy of coherence as the conditional uncertainty of projective measurement outcomes with quantum memory. Motivated by this connection, we develop a coherence-based approach to quantum uncertainty relations. Within this approach, we introduce a coherence measure induced by the two-parameter α - z -Rényi relative entropy and derive corresponding uncertainty relations for both pure and mixed quantum states. The resulting state-independent bounds recover known results in appropriate limits and yield tighter bounds over a broad parameter regime. Our results provide a systematic connection between quantum coherence, generalized conditional entropies, and uncertainty relations, thereby offering a unified perspective on quantum uncertainty in terms of Rényi-type information measures.
Maximal Rényi Relative Entropy for α>2α>2α>2
Quantum relative entropies play a fundamental role in quantum information theory. In the classical setting, Rényi relative entropies constitute, up to linear combinations, the most general class of relative entropies, naturally motivating the search for their minimal and maximal quantum extensions. The minimal extension is known to be the reverse sandwiched Rényi relative entropy for α∈[0,1/2)α\in[0,1/2)α∈[0,1/2) and the sandwiched Rényi relative entropy for α≥1/2α\geq 1/2α≥1/2. In contrast, the maximal extension had previously been identified only for α∈[0,2]α\in[0,2]α∈[0,2], where it is given by the geometric Rényi relative entropy. In this work, we complete this characterization by proving that for α>2α>2α>2, the maximal extension is given by the ααα-zzz Rényi relative entropy with z=α−1z=α-1z=α−1. As an application, we determine when an energy-incoherent state can be transformed into an energy-coherent state by a Gibbs-preserving operation assisted by an uncorrelated catalyst, thereby fully characterizing the coherence-generating power of this class of operations in the catalytic setting.
Quantum coherence and quantum correlations are important quantum resources for quantum computation and quantum information. In this paper, using entropy-based measures, we investigate the relationships between quantum correlated coherence, which is the coherence between subsystems, and two main kinds of quantum correlations as defined by quantum discord as well as quantum entanglement. In particular, we show that quantum discord and quantum entanglement can be well characterized by quantum correlated coherence. Moreover, we prove that the entanglement measure formulated by quantum correlated coherence is lower and upper bounded by the relative entropy of entanglement and the entanglement of formation, respectively, and equal to the relative entropy of entanglement for all the maximally correlated states.
The uncertainty relation reveals the intrinsic difference that distinguishes the quantum world from the classical world. In this paper, the quantum uncertainty relations of the sandwiched R\'enyi relative entropy of coherence (SRREOC) and unified $(\ensuremath{\alpha},\ensuremath{\beta})$-relative entropy of coherence (UREOC) are studied. First, the uncertainty relation of the SRREOC for pure states, with extension to mixed states by convex-roof construction, is given in a ddd-dimensional quantum system. Second, the analytical formula of the UREOC is presented; it is shown that the UREOC is a good candidate for a quantum coherence measure. Furthermore, the quantum uncertainty relation of the UREOC for pure states is obtained in a ddd-dimensional quantum system, then it is generalized to mixed states. In addition, our results cover the quantum uncertainty relations of the other three coherence measures. Third, two methods are given to compare the lower bounds of the SRREOC and UREOC for single-qubit pure states. By comparison, we find that the results in this paper not only can be applied to single-qubit states but also can be extended to a ddd-dimensional quantum system.
Abstract It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the distance is measured by the quasi-relative entropy. First, we consider a maximally entangled state, and show that the closest separable state is the same for any quasi-relative entropy as for the relative entropy of entanglement. Then, we show that this also holds for a certain class of functions and any pure state. And at last, we consider any pure state on two qubit systems and a large class of operator convex function. For these, we find the closest separable state, which may not be the same one as for the relative entropy of entanglement.