The Quantum Entropy Cone of Stabiliser States

TL;DR

This paper characterizes the entropy cone of stabiliser states in multi-party quantum systems, revealing that classical inequalities combined with a specific linear rank inequality fully describe their von Neumann entropy vectors.

Contribution

It identifies the complete set of inequalities, including the Ingleton inequality, that define the entropy cone for stabiliser states, extending understanding of quantum entropy constraints.

Findings

Entropy vectors for stabiliser states satisfy classical and Ingleton inequalities.

Strong subadditivity, weak monotonicity, and Ingleton inequality characterize the entropy cone.

The Ingleton inequality is the unique additional inequality for four-party stabiliser states.

Abstract

We investigate the universal linear inequalities that hold for the von Neumann entropies in a multi-party system, prepared in a stabiliser state. We demonstrate here that entropy vectors for stabiliser states satisfy, in addition to the classic inequalities, a type of linear rank inequalities associated with the combinatorial structure of normal subgroups of certain matrix groups. In the 4-party case, there is only one such inequality, the so-called Ingleton inequality. For these systems we show that strong subadditivity, weak monotonicity and Ingleton inequality exactly characterize the entropy cone for stabiliser states.