Non-classicality criteria from phase-space representations and information-theoretical constraints are maximally inequivalent

TL;DR

This paper demonstrates that two major criteria for quantum non-classicality, based on phase-space and information theory, are fundamentally incompatible, revealing different aspects of quantum correlations beyond entanglement and discord.

Contribution

The study proves that the sets of classical states defined by phase-space and information-theoretic criteria are nearly disjoint, highlighting their maximal inequivalence and different notions of classicality.

Findings

C and P sets of states are almost disjoint.

Most CC states exhibit quantumness in P-representation.

Almost all P-classical states have positive quantum discord.

Abstract

We consider two celebrated criteria for defining the non-classicality of bipartite bosonic quantum systems, the first stemming from information theoretic concepts and the second from physical constraints on the quantum phase-space. Consequently, two sets of allegedly classical states are singled out: i) the set C composed of the so called classical-classical (CC) states---separable states that are locally distinguishable and do not possess quantum discord; ii) the set P of states endowed with a positive P-representation (P-classical states)---mixture of Glauber coherent states that, e.g., fail to show negativity of their Wigner function. By showing that C and P are almost disjoint, we prove that the two defining criteria are maximally inequivalent. Thus, the notions of classicality that they put forward are radically different. In particular, generic CC states show quantumness in their P-representation and, viceversa, almost all P-classical states have positive quantum discord, hence are not CC. This inequivalence is further elucidated considering different applications of P-classical and CC states. Our results suggest that there are other quantum correlations in nature than those revealed by entanglement and quantum discord.