Probing the geometry of correlation matrices with randomized measurements

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.

9 pages (+8 pages of Appendix), 5 figures

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• Analyzing quantum entanglement with the Schmidt decomposition in operator space
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• Detecting high-dimensional entanglement in cold-atom quantum simulators
• Experimental certification of high-dimensional entanglement with randomized measurements
• Bound entangled Bell diagonal states of unequal local dimensions, and their witnesses
• Real randomized measurements for analyzing properties of quantum states
• Quantum null-hypothesis device-independent Schmidt rank witness
• Detecting high-dimensional entanglement by randomized product projections
• Certifying entanglement dimensionality by $k$-reduction moments