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We study the partial transposition WΓ=(id⊗t)W∈Mdn(ℂ) of a Wishart matrix W∈Mdn(ℂ) of parameters (dn,dm). Our main result is that, with d→∞, the law of mWΓ 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.
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