Stabilizer information inequalities from phase space distributions

TL;DR

This paper shows that the von Neumann entropy of stabilizer states obeys all classical balanced information inequalities, using their classical phase-space model via the discrete Wigner function.

Contribution

It demonstrates that stabilizer states satisfy classical information inequalities through their phase-space representation, bridging quantum and classical entropy properties.

Findings

Von Neumann entropy of stabilizer states obeys all classical balanced inequalities.

Stabilizer states have a classical model via the discrete Wigner function.

Results extend to multi-mode Gaussian states.

Abstract

The Shannon entropy of a collection of random variables is subject to a number of constraints, the best-known examples being monotonicity and strong subadditivity. It remains an open question to decide which of these "laws of information theory" are also respected by the von Neumann entropy of many-body quantum states. In this article, we consider a toy version of this difficult problem by analyzing the von Neumann entropy of stabilizer states. We find that the von Neumann entropy of stabilizer states satisfies all balanced information inequalities that hold in the classical case. Our argument is built on the fact that stabilizer states have a classical model, provided by the discrete Wigner function: The phase-space entropy of the Wigner function corresponds directly to the von Neumann entropy of the state, which allows us to reduce to the classical case. Our result has a natural counterpart for multi-mode Gaussian states, which sheds some light on the general properties of the construction. We also discuss the relation of our results to recent work by Linden, Ruskai, and Winter.