Partial transposition of random states and non-centered semicircular distributions

TL;DR

This paper studies the eigenvalue distribution of a partially transposed Wishart matrix, revealing a non-centered semicircular law and identifying a threshold for positivity of partial transpose in high dimensions.

Contribution

It establishes the limiting eigenvalue distribution of the partial transpose of Wishart matrices and determines a sharp threshold for the positivity of partial transpose in large dimensions.

Findings

Eigenvalues follow a non-centered semicircular distribution as dimension grows.

Extreme eigenvalues converge to the spectrum edge.

Threshold for PPT states at parameter value 4.

Abstract

Let W be a Wishart random matrix of size d^2 times d^2, considered as a block matrix with d times d blocks. Let Y be the matrix obtained by transposing each block of W. We prove that the empirical eigenvalue distribution of Y approaches a non-centered semicircular distribution when d tends to infinity. We also show the convergence of extreme eigenvalues towards the edge of the expected spectrum. The proofs are based on the moments method. This matrix model is relevant to Quantum Information Theory and corresponds to the partial transposition of a random induced state. A natural question is: "When does a random state have a positive partial transpose (PPT)?". We answer this question and exhibit a strong threshold when the parameter from the Wishart distribution equals 4. When d gets large, a random state on C^d tensor C^d obtained after partial tracing a random pure state over some ancilla of dimension alpha.d^2 is typically PPT when alpha>4 and typically non-PPT when alpha<4.