 
Summary: PARTIAL TRANSPOSITION OF RANDOM STATES AND NONCENTERED
SEMICIRCULAR DISTRIBUTIONS
GUILLAUME AUBRUN
Abstract. Let W be a Wishart random matrix of size d2
× d2
, considered as a block matrix with d × 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 noncentered semicircular distribution when d . 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 trans
position 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 Cd
Cd
obtained after
partial tracing a random pure state over some ancilla of dimension d2
is typically PPT when > 4 and
typically nonPPT when < 4.
1. Introduction
