Probing moments of multipartite density matrices with randomized measurements

Hui Zhao, Jing-Yun Zhao, Naihuan Jing

Nikolai Wyderka, Andreas Ketterer

The generalized Bloch decomposition of a bipartite quantum state gives rise to a correlation matrix whose singular values provide rich information about nonlocal 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 nonvanishing 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 symmetric, informationally complete, positive operator-valued measures.

Gerardo A. Paz-Silva, John Henry Reina

Nikolai Wyderka, Andreas Ketterer

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.

Matteo G. A. Paris

Vladimir Al. Osipov, H.-J. Sommers, Karol Życzkowski

Ensembles of random density matrices determined by various probability measures are analysed. A simple and efficient algorithm to generate at random density matrices distributed according to the Bures measure is proposed. This procedure may serve as an initial step in performing the Bayesian approach to quantum state estimation based on the Bures prior. We study the distribution of purity of random mixed states. The moments of the distribution of purity are determined for quantum states generated with respect to the Bures measure. This calculation serves as an exemplary application of the ‘deform-and-study’ approach in the theory of integrable systems leading to one of Painlevés transcendents.