Thursday, November 8, 2012

1211.1525 (Motohisa Fukuda et al.)

Partial transpose of random quantum states, Weingarten calculus and

Motohisa Fukuda, Piotr Śniady
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 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. We use instead Weingarten calculus to study this problem and find three natural 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.
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