Scalable Reconstruction of Density Matrices

Scalable reconstruction of density matrices.

Recent contributions in the field of quantum state tomography have shown that, despite the exponential growth of Hilbert space with the number of subsystems, tomography of one-dimensional quantum systems may still be performed efficiently by tailored reconstruction schemes. Here, we discuss a scalable method to reconstruct mixed states that are well approximated by matrix product operators. The...

Reconstruction of matrices from submatrices

For an arbitrary matrix A of n×n symbols, consider its submatrices of size k×k, obtained by deleting n−k rows and n−k columns. Optionally, the deleted rows and columns can be selected symmetrically or independently. We consider the problem of whether these multisets determine matrix A. Following the ideas of Krasikov and Roditty in the reconstruction of sequences from subsequences, we replace t...

Scalable geometric density estimation

It is standard to assume a low-dimensional structure in estimating a high-dimensional density. However, popular methods, such as probabilistic principal component analysis, scale poorly computationally. We introduce a novel empirical Bayes method that we term geometric density estimation (GEODE) and show that, with mild conditions and among all d-dimensional linear subspaces, the span of the d ...

Scalable multiscale density estimation

Asymptotics of random density matrices

We investigate random density matrices obtained by partial tracing larger random pure states. We show that there is a strong connection between these random density matrices and the Wishart ensemble of random matrix theory. We provide asymptotic results on the behavior of the eigenvalues of random density matrices, including convergence of the empirical spectral measure. We also study the large...