Self-testing maximally-dimensional genuinely entangled subspaces within the stabilizer formalism

TL;DR

This paper investigates the maximum dimension of genuinely entangled subspaces within the stabilizer formalism that can be self-tested, providing a framework, explicit constructions, and Bell inequalities for their certification.

Contribution

It introduces an efficient method to check genuine entanglement in stabilizer subspaces, determines their maximal dimension, and constructs Bell inequalities for self-testing these subspaces.

Findings

Maximal dimension of self-testable genuinely entangled stabilizer subspaces is established.

Explicit constructions of maximally-dimensional entangled subspaces for any number of qubits are provided.

Bell inequalities are developed that are maximally violated by states in these subspaces, aiding self-testing.

Abstract

Self-testing was originally introduced as a device-independent method of certification of entangled quantum states and local measurements performed on them. Recently, in [F. Baccari \textit{et al.}, arXiv:2003.02285] the notion of state self-testing has been generalized to entangled subspaces and the first self-testing strategies for exemplary genuinely entangled subspaces have been given. The main aim of our work is to pursue this line of research and to address the question how "large" (in terms of dimension) are genuinely entangled subspaces that can be self-tested, concentrating on the multiqubit stabilizer formalism. To this end, we first introduce a framework allowing to efficiently check whether a given stabilizer subspace is genuinely entangled. Building on it, we then determine the maximal dimension of genuinely entangled subspaces that can be constructed within the stabilizer subspaces and provide an exemplary construction of such maximally-dimensional subspaces for any number of qubits. Third, we construct Bell inequalities that are maximally violated by any entangled state from those subspaces and thus also any mixed states supported on them, and we show these inequalities to be useful for self-testing. Interestingly, our Bell inequalities allow for identification of higher-dimensional face structures in the boundaries of the sets of quantum correlations in the simplest multipartite Bell scenarios in which every observer performs two dichotomic measurements.