Quantum Entropy Prover

TL;DR

This paper introduces a framework and a Python package for proving quantum information inequalities, extending classical methods to quantum systems and enabling the discovery of non-Shannon-type inequalities.

Contribution

It develops a quantum analog of classical linear programming methods for information inequalities and provides a practical tool for researchers in quantum information theory.

Findings

The qITIP package successfully proves quantum information inequalities.

The framework handles constrained inequalities, aiding in discovering non-Shannon-type inequalities.

Demonstrations show the method's effectiveness on various quantum states.

Abstract

Information inequalities govern the ultimate limitations in information theory and as such play an pivotal role in characterizing what values the entropy of multipartite states can take. Proving an information inequality, however, quickly becomes arduous when the number of involved parties increases. For classical systems, [Yeung, IEEE Trans. Inf. Theory (1997)] proposed a framework to prove Shannon-type inequalities via linear programming. Here, we derive an analogous framework for quantum systems, based on the strong sub-additivity and weak monotonicity inequalities for the von-Neumann entropy. Importantly, this also allows us to handle constrained inequalities, which - in the classical case - served as a crucial tool in proving the existence of non-standard, so-called non-Shannon-type inequalities [Zhang & Yeung, IEEE Trans. Inf. Theory (1998)]. Our main contribution is the Python package qITIP, for which we present the theory and demonstrate its capabilities with several illustrative examples