Partial transpose of random quantum states: exact formulas and meanders

TL;DR

This paper derives exact formulas and explores asymptotic regimes for the eigenvalue distribution of the partial transpose of random quantum states, revealing new connections to meander polynomials and providing convergence and deviation results.

Contribution

It introduces exact combinatorial formulas for moments and identifies a new matrix model for meander polynomials, expanding understanding of eigenvalue distributions in quantum information.

Findings

Exact combinatorial formulas for moments of partial transpose

Identification of three asymptotic regimes, including a new meander polynomial model

Convergence to semicircular distribution with bounds on extreme eigenvalues

Abstract

We investigate the asymptotic behavior of the empirical eigenvalues distribution of the partial transpose of a random quantum state. The limiting distribution was previously investigated via Wishart random matrices indirectly (by approximating the matrix of trace 1 by the Wishart matrix of random trace) and shown to be the semicircular distribution or the free difference of two free Poisson distributions, depending on how dimensions of the concerned spaces grow. Our use of Wishart matrices gives exact combinatorial formulas for the moments of the partial transpose of the random state. We find three natural asymptotic regimes in terms of geodesics on the permutation groups. Two of them correspond to the above two cases; the third one turns out to be a new matrix model for the meander polynomials. Moreover, we prove the convergence to the semicircular distribution together with its extreme eigenvalues under weaker assumptions, and show large deviation bound for the latter.