Statistics of leading digits leads to unification of quantum correlations

TL;DR

This paper reveals a universal pattern in the distribution of first significant digits of quantum correlation measures, following Benford's law in some cases and signaling quantum phase transitions through violations.

Contribution

It demonstrates the universal behavior of first-digit distributions in quantum correlation measures and links violations to quantum phase transitions in specific models.

Findings

Quantum correlation measures follow Benford's law in Haar-random states.

Violations of Benford's law signal quantum phase transitions.

Universal digit distribution patterns extend to multipartite and higher-dimensional systems.

Abstract

We show that the frequency distribution of the first significant digits of the numbers in the data sets generated from a large class of measures of quantum correlations, which are either entanglement measures, or belong to the information-theoretic paradigm, exhibit a universal behaviour. In particular, for Haar uniformly simulated arbitrary two-qubit states, we find that the first-digit distribution corresponding to a collection of chosen computable quantum correlation quantifiers tend to follow the first-digit law, known as the Benford's law, when the rank of the states increases. Considering a two-qubit state which is obtained from a system governed by paradigmatic spin Hamiltonians, namely, the XY model in a transverse field, and the XXZ model, we show that entanglement as well as information theoretic measures violate the Benford's law. We quantitatively discuss the violation of the Benford's law by using a violation parameter, and demonstrate that the violation parameter can signal quantum phase transitions occurring in these models. We also comment on the universality of the statistics of first significant digits corresponding to appropriate measures of quantum correlations in the case of multipartite systems as well as systems in higher dimensions.