Jensen-Shannon divergence as a measure of distinguishability between mixed quantum states

Abstract

We discuss an alternative to relative entropy as a measure of distance between mixed quantum states. The proposed quantity is an extension to the realm of quantum theory of the Jensen-Shannon divergence ͑JSD͒ between probability distributions. The JSD has several interesting properties. It arises in information theory and, unlike the Kullback-Leibler divergence, it is symmetric, always well-defined, and bounded. We show that the quantum JSD shares with the relative entropy most of the physically relevant properties, in particular those required for a "good" quantum distinguishability measure. We relate it to other known quantum distances and we suggest possible applications in the field of the quantum information theory.

Structural reducibility of multilayer networks

“…To overcome this problem, we adopt a greedy agglomerative hierarchical clustering algorithm 34 to explore the space of partitions, based on a concept of distance similar to the one adopted in quantum physics to quantify the distance between mixed quantum states 28 . More specifically, capitalizing on the concept of Von Neumann entropy of a graph, we use the quantum Jensen-Shannon divergence to quantify the (dis-) similarity between all pairs of layers of a multilayer network (see Methods).…”

“…Inspired by a similar question arising in quantum physics when one needs to quantify the distance between mixed quantum states 28 , we propose here a method to aggregate some of the layers of a multilayer system while maximizing its distinguishability from the aggregated network. The method is based on a purely information theoretic perspective, which makes use of the definition of Von Neumann entropy of a graph.…”

Distance between quantum states in the presence of initial qubit-environment correlations: A comparative study

“…The Jensen-Shannon divergence is a tool which allows one to overcome this sort of problem. It is defined in terms of a symmetrized relative entropy between states; here, however, we use instead the following expression [3]:…”

“…Examples of such distance measures comprise the trace distance, HilbertSchmidt distance, Bures distance, Hellinger distance and Jensen-Shannon divergence, to mention a few, see also in Refs. [1][2][3][4][5]. These metrics possess distinct properties like being Riemannian, monotone (contractive), with bounds and relations among them [6].…”

Purification-based metric to measure the distance between quantum states and processes

“…Therefore, in order to provide a way to determine how well a quantum protocol is working, distance measures need to be devised to allow us to determine how close two quantum states or two quantum processes are to each other. A variety of distance measures have been developed for this purposes, like trace distance, Fidelity, Bures distance, Hilbert-Schmidt distance, Hellinger distance and Quantum Jensen-Shannon divergence, just to name a few [12,[21][22][23][24][25][26][27]. Quantum processes can be represented by means of positive and trace-preserving maps E defined on the set of density operators belonging to B(H) + 1 , that is, the set of positive trace one operators ρ on a Hilbert space H. We say that the map E is monotonous under quantum operations with respect to a given distance D(ρ, σ), or contractive for short, if…”