Asymptotic Eigenvalue Distributions of Block-Transposed Wishart Matrices

Abstract

Abstract. We study the partial transposition W Γ = (id ⊗ t)W ∈ M dn (C) of a Wishart matrix W ∈ M dn (C) of parameters (dn, dm). Our main result is that, with d → ∞, the law of mW Γ is a free difference of free Poisson laws of parameters m(n ± 1)/2. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half line. IntroductionThe partial transposition of aΓ obtained by transposing each of the n × n blocks of W . That is, we let W Γ = (id ⊗ t)W , where id is the identity of M d (C), and t is the transposition of M n (C). The partial transposition operation can be defined using coordinates, as follows. A particular decompositionThe entries of a matrix W ∈ M d (C) ⊗ M n (C) can be indexed by four indices, i, j ∈ {1, . . . , d} identifying the block of A the matrix entry belongs to, and a, b ∈ {1, . . . , n} fixing the position of the matrix entry inside the block. Then, the partial transposed matrix W Γ has coordinates W Γ ia,jb = W ib,ja . Motivated by questions in quantum information theory, the partial transposition operation for Wishart matrices was studied by Aubrun [1], who showed that, for certain special values of the parameters, the empirical spectral distribution of W Γ converges in moments to a non-centered semicircular distribution. In this paper we discuss the general case, our main result being as follows.Theorem A. Let W be a complex Wishart matrix of parameters (dn, dm). Then, with d → ∞, the empirical spectral distribution of mW Γ converges in moments to a free difference of free Poisson distributions of respective parameters m(n ± 1)/2.Observe that if one applies the transposition map on the first factor of the tensor product M d (C)⊗M n (C) instead of the second one, the spectral distribution of W Γ remains2000 Mathematics Subject Classification. 60B20 (46L54, 81P45).

On the asymptotic distribution of block-modified random matrices

“…He computed the range of parameters of the Wishart distribution for which the partial transposition is still positive. Similar computations, in different asymptotic regimes, or for different maps acting on the blocks, have been performed in [BN12a,FŚ13,JLN14,JLN15]. In this paper, we unify these results by showing that operatorvalued free probability theory provides the right framework to study such questions.…”

“…In the case where the measure µ is a free Poisson measure of parameter λ > 0, we recover Theorem 4.1 from [BN12a]:…”

“…Historically, the study linear modifications of block-random matrices was initiated by Aubrun in [Aub12], where he studied partial transpositions of large Wishart random matrices. The same random matrix model, in the asymptotical regime where one dimension is kept fixed, while the other grows, was analyzed in [BN12a], and later generalized in [BN12b]. Let us consider the transpose map, transp : M n (C) → M n (C), transp(X) = X .…”

Random matrix techniques in quantum information theory

“…The PPT criterion for random quantum states has first been studied numerically in [ŽPBC07]. The analytic results in the following proposition are from [Aub12] (in the balanced case) and from [BN13] (in the unbalanced case); see also [FŚ13] for some improvements in the balanced case and the relation to meanders.…”

On the Separability of Unitarily Invariant Random Quantum States: The Unbalanced Regime

“…This case has been studied in [8,Theorem 6.2], where it has been shown that the measure MP Γ has positive support iff…”