630 research outputs found
Partial transpose of random quantum states: exact formulas and meanders
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.Comment: v2: change of title, change of some methods of proof
Simplifying additivity problems using direct sum constructions
We study the additivity problems for the classical capacity of quantum
channels, the minimal output entropy and its convex closure. We show for each
of them that additivity for arbitrary pairs of channels holds iff it holds for
arbitrary equal pairs, which in turn can be taken to be unital. In a similar
sense, weak additivity is shown to imply strong additivity for any convex
entanglement monotone. The implications are obtained by considering direct sums
of channels (or states) for which we show how to obtain several information
theoretic quantities from their values on the summands. This provides a simple
and general tool for lifting additivity results.Comment: 5 page
Low entropy output states for products of random unitary channels
In this paper, we study the behaviour of the output of pure entangled states
after being transformed by a product of conjugate random unitary channels. This
study is motivated by the counterexamples by Hastings and Hayden-Winter to the
additivity problems. In particular, we study in depth the difference of
behaviour between random unitary channels and generic random channels. In the
case where the number of unitary operators is fixed, we compute the limiting
eigenvalues of the output states. In the case where the number of unitary
operators grows linearly with the dimension of the input space, we show that
the eigenvalue distribution converges to a limiting shape that we characterize
with free probability tools. In order to perform the required computations, we
need a systematic way of dealing with moment problems for random matrices whose
blocks are i.i.d. Haar distributed unitary operators. This is achieved by
extending the graphical Weingarten calculus introduced in Collins and Nechita
(2010)
Commentary: Birth, Growth and Recreation of Higher Education Research Institute
æžć: Higher Education Research: Challenges and Prospects. Report of RIHEâs 50th Anniversary International Symposium, 202
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