We develop a framework to extend resource measures from one domain to a larger one. We find that all extensions of resource measures are bounded between two quantities that we call the minimal and maximal extensions. We discuss various applications of our framework. We show that any relative entropy (i.e., an additive function on pairs of quantum states that satisfies the data processing inequality) must be bounded by the min and max relative entropies. We prove that the generalized trace distance, the generalized fidelity, and the purified distance are optimal extensions. And in entanglement theory we introduce a technique to extend pure-state entanglement measures to mixed bipartite states.
We develop a framework to extend resource measures from one domain to a larger one. We find that all extensions of resource measures are bounded between two quantities that we call the minimal and maximal extensions. We discuss various applications of our framework. We show that any relative entropy (i.e., an additive function on pairs of quantum states that satisfies the data processing inequality) must be bounded by the min and max relative entropies. We prove that the generalized trace distance, the generalized fidelity, and the purified distance are optimal extensions. And in entanglement theory we introduce a technique to extend pure-state entanglement measures to mixed bipartite states.
The measures of distance frequently used in quantum information, such as trace distance or fidelity-based purified distance, are the smallest extensions of the corresponding distance measures in classical information theory. However, this way of extending classical distances is not unique, and a whole spectrum of other extensions can be defined. Here we study the question of how quantum information processing tasks are affected by a change of the error metrics to the maximal extensions of the classical distance measures, imposing more demanding error requirements on operational tasks. We find that the performance of some tasks, such as channel coding or quantum resource distillation, is not affected whatsoever by this change. However, for other tasks such as channel simulation and quantum resource dilution, we show that the rates are now given not by functions of the standard (Umegaki) quantum relative entropy, but ones based on the Belavkin-Staszewski relative entropy. Our results connect the Belavkin-Staszewski relative entropy with operational tasks and reveal that known reversibility results in quantum information, including the reverse Shannon theorem and the asymptotic reversibility of quantum resource transformations, no longer hold under maximal distance measures.
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.
A fundamental approach for the characterization and quantification of all kinds of resources is to study the conversion between different resource objects under certain constraints. Here we analyze, from a resource-nonspecific standpoint, the optimal efficiency of resource formation and distillation tasks with only a single copy of the given quantum state, thereby establishing a unified framework of one-shot quantum resource manipulation. We find general bounds on the optimal rates characterized by resource measures based on the smooth max- or min-relative entropies and hypothesis testing relative entropy, as well as the free robustness measure, providing them with general operational meanings in terms of optimal state conversion. Our results encompass a wide class of resource theories via the theory-dependent coefficients we introduce, and the discussions are solidified by important examples, such as entanglement, coherence, superposition, magic states, asymmetry, and thermal nonequilibrium.
The fidelity-based smooth min-relative entropy is a distinguishability measure that has appeared in a variety of contexts in prior work on quantum information, including resource theories like thermodynamics and coherence. Here we provide a comprehensive study of this quantity. First we prove that it satisfies several basic properties, including the dataprocessing inequality. We also establish connections between the fidelity-based smooth min-relative entropy and other widely used information-theoretic quantities, including smooth min-relative entropy and smooth sandwiched Rényi relative entropy, of which the sandwiched Rényi relative entropy and smooth max-relative entropy are special cases. After that, we use these connections to establish the second-order asymptotics of the fidelity-based smooth min-relative entropy and all smooth sandwiched Rényi relative entropies, finding that the first-order term is the quantum relative entropy and the second-order term involves the quantum relative entropy variance. Utilizing the properties derived, we also show how the fidelity-based smooth min-relative entropy provides one-shot bounds for operational tasks in general resource theories in which the target state is mixed, with a particular example being randomness distillation. The above observations then lead to second-order expansions of the upper bounds on distillable randomness, as well as the precise second-order asymptotics of the distillable randomness of particular classical–quantum states. Finally, we establish semi-definite programs for smooth maxrelative entropy and smooth conditional min-entropy, as well as a bilinear program for the fidelity-based smooth min-relative entropy, which we subsequently use to explore the tightness of a bound relating the last to the first.
The R\'enyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of R\'enyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory.
We propose a new quantum generalization of the family of R\'enyi entropies and the corresponding relative entropies that contains the min-entropy, collision entropy, von Neumann entropy as well as the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities and a limited duality relation.
We conclude that the treatment of these entropies is technically challenging and requires sophisticated tools from linear algebra. We share several conjectured properties which we were unable to prove.