Partial transposition of random states and non-centered semicircular\n distributions

Abstract

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