The Quantum Entropy Cone of Stabiliser States

TL;DR

The paper addresses the problem of characterising quantum entropy cones for multi-party systems, focusing on stabiliser states. It develops a framework showing that stabiliser-state entropy vectors satisfy linear rank inequalities, notably the Ingleton inequality in the 4-party case, and that SSA together with weak monotonicity suffice to characterise the 4-party stabiliser cone when combined with Ingleton. The authors derive the stabiliser-entropy formula $S(J) = \log \frac{|\widehat{G}|}{|\widehat{G_J}|} - \log d_J$ and use it to prove that pure stabiliser states on $(N{+}1)$ parties yield $N$-party entropy vectors obeying Ingleton and related inequalities, with explicit extremal rays for the 4-party case realised by stabiliser states (including a 5-qubit code construction). They show that the 4-party Ingleton cone’s extremal rays are attainable by stabiliser constructions and discuss open questions about additional non-Shannon-type inequalities in larger-party settings, linking group-theoretic structures to quantum entropy cones.

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.

Theorems & Definitions (20)

• Proposition 1
• proof
• remark 1
• definition 1
• Proposition 3
• proof
• lemma 1
• proof
• theorem 4
• proof