Abstract
We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions and . Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary , a general relation between the -point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite . Then, we focus on the moments of the Schmidt number , the reciprocal of the purity. This is a random variable supported on , which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for for and 3 and arbitrary , and also for square systems by spotting for the latter a connection with the probability that the smallest eigenvalue of an matrix belonging to the Gaussian unitary ensemble is larger than . As a by-product, we present an exact asymptotic expansion for for finite as . Our results are corroborated by numerical simulations whenever possible, with excellent agreement.
1 More- Received 5 February 2016
DOI:https://doi.org/10.1103/PhysRevE.93.052106
©2016 American Physical Society