Convex set of quantum states with positive partial transpose analysed by hit and run algorithm

TL;DR

This paper analyzes the convex set of quantum states with positive partial transpose using a hit and run algorithm, providing insights into the distribution and convergence properties of PPT states in high-dimensional quantum systems.

Contribution

It introduces an efficient hit and run sampling method for PPT states and derives their level density distribution, improving over previous sampling techniques.

Findings

The hit and run algorithm converges faster for K≥3.

PPT states' level density differs from Marchenko-Pastur distribution.

Explicit probability distribution for PPT states' spectrum is derived.

Abstract

The convex set of quantum states of a composite $K \times K$ system with positive partial transpose is analysed. A version of the hit and run algorithm is used to generate a sequence of random points covering this set uniformly and an estimation for the convergence speed of the algorithm is derived. For $K\ge 3$ this algorithm works faster than sampling over the entire set of states and verifying whether the partial transpose is positive. The level density of the PPT states is shown to differ from the Marchenko-Pastur distribution, supported in [0,4] and corresponding asymptotically to the entire set of quantum states. Based on the shifted semi--circle law, describing asymptotic level density of partially transposed states, and on the level density for the Gaussian unitary ensemble with constraints for the spectrum we find an explicit form of the probability distribution supported in [0,3], which describes well the level density obtained numerically for PPT states.