Quantum Reactivity: An Indicator of Quantum Correlation

Abstract

Defining a robust measure of quantum correlation for multipartite states is an unresolved and challenging problem. Existing measures of quantum correlation are either not scalable or do not satisfy all the accepted properties of a measure of quantum correlation. We introduce a novel geometric measure of quantum correlation that we refer to as quantum reactivity. This measure is extendable to an arbitrary large number of qubits and satisfies the required properties of monotonicity and invariance under unitary operations. Our approach is based on generalization of Schumacher's singlet state triangle inequality that used an information geometry-based entropic distance. We define quantum reactivity as the familiar ratio of surface area to volume. To accomplish this, we use a generalization of information distance to area, volume and higher-dimensional volumes. We examine a spectrum of multipartite states (Werner, W, GHZ etc.) and demonstrate that the quantum reactivity measure is a monotonic function for quantum correlation which satisfies all the properties of a measure for quantum correlation, and provides an ordering of these quantum states as to their degree of correlation.

Geometric measures of information for quantum state characterization

“…[22] We developed these entropic geometric measures to explore the geometric structure of quantum states, and to find a measure of quantum entanglement. [1,10] Entanglement measures based on relative entropy have been introduced earlier and perhaps it is worth revisiting these using the geometric insight we provided. [21,4] Geometry is often a valuable guide for scientific discovery.…”

“…Nevertheless, we have recently shown the volume formulae introduced in this paper appear to yield a reasonable measure for quantum correlation satisfying accepted conditions. [1,2] In particular, we have found recently that quantum reactivity appears to be a viable measure of quantum correlations for a multipartite state. [1] The term reactivity, as commonly used for chemical reactions, is a ratio of surface area to volume.…”

“…[1,2] In particular, we have found recently that quantum reactivity appears to be a viable measure of quantum correlations for a multipartite state. [1] The term reactivity, as commonly used for chemical reactions, is a ratio of surface area to volume. This can be defined for an d-partite quantum system using the entropic volume formulae introduced in this manuscript as a weighted sum over the "space of measurements,"…”

“…The integral in the weighted sum is over all measurements. [1] This geometric approach based on Shannon's entropy of the quantum measurement outcomes appears to provide a viable measure of quantum correlation, and is a generalization of Schumacher's approach. [18] It is scalable in that it can be generalized to an arbitrary large multipartite state in general position.…”

“…[1,2,10] It is well accepted that quantum entanglement is the key resource for quantum computing, [12] and this has led to many studies on this topic. [11,23,19,17,8,9,16,7] We recently proposed a novel information-geometric measure of quantum correlation, that we refer to as quantum reactivity R. [1,2] It is similar to the reactivity in chemistry in that it is a ratio of surface area to volume. However, our areas and volumes are entropic in nature and are defined on the outcomes of measurements (a "space of measurements") from an ensemble of identically-prepared quantum systems.…”