Probing the geometry of correlation matrices with randomized measurements

TL;DR

This paper explores the geometric structure of correlation matrices derived from bipartite quantum states, linking their singular values to entanglement properties and measurement frameworks, with implications for quantum information theory.

Contribution

It introduces a novel connection between the geometry of correlation matrices, randomized measurements, and quantum measurement structures like mutually unbiased bases and SIC-POVMs.

Findings

Characterized the boundary of moments for separable states.

Linked correlation matrix geometry to measurement frameworks.

Provided explicit constructions for extremal points.

Abstract

The generalized Bloch decomposition of a bipartite quantum state gives rise to a correlation matrix whose singular values provide rich information about non-local properties of the state, such as the dimensionality of entanglement. While some entanglement criteria based on the singular values exist, a complete understanding of the geometry of admissible correlation matrices is lacking. We provide a deeper insight into the geometry of the singular values of the correlation matrices of limited Schmidt number. First, we provide a link to the framework of randomized measurements and show how to obtain knowledge about the singular values in this framework by constructing observables that yield the same moments as one obtains from orthogonal averages over the Bloch sphere. We then focus on the case of separable states and characterize the boundary of the set of the first two non-vanishing moments by giving explicit constructions for some of the faces and extremal points. These constructions yield a connection between the geometry of the correlation matrices and the existence problems of maximal sets of mutually unbiased bases, as well as SIC-POVMs.