Volume monogamy of quantum steering ellipsoids for multi-qubit systems

TL;DR

This paper explores the volume monogamy relations of quantum steering ellipsoids in multi-qubit systems, establishing new inequalities for pure and mixed states, and analyzing the effects of noise on these quantum correlations.

Contribution

It derives new volume monogamy relations for pure 3- and 4-qubit states, extends these to n-qubits, and examines the impact of noise on quantum steering ellipsoids.

Findings

Volume monogamy relation holds for pure 3-qubit states and is stronger than CKW inequality.

Weaker monogamy relation applicable to mixed 3-qubit states.

Volume of two-qubit states does not increase under local noise.

Abstract

The quantum steering ellipsoid can be used to visualise two-qubit states, and thus provides a generalisation of the Bloch picture for the single qubit. Recently, a monogamy relation for the volumes of steering ellipsoids has been derived for pure 3-qubit states and shown to be stronger than the celebrated Coffman-Kundu-Wootters (CKW) inequality. We first demonstrate the close connection between this volume monogamy relation and the classification of pure 3-qubit states under stochastic local operations and classical communication (SLOCC). We then show that this monogamy relation does not hold for general mixed 3-qubit states and derive a weaker monogamy relation that does hold for such states. We also prove a volume monogamy relation for pure 4-qubit states, and generalize our 3-qubit inequality to n qubits. Finally, we study the effect of noise on the quantum steering ellipsoid and find that the volume of any two-qubit state is non-increasing when the state is exposed to arbitrary local noise. This implies that any volume monogamy relation for a given class of multi-qubit states remains valid under the addition of local noise. We investigate this quantitatively for the experimentally relevant example of isotropic noise.