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Concentration estimates for random subspaces of a tensor product, and application to Quantum Information Theory

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Abstract

Given a random subspace H_n chosen uniformly in a tensor product of Hilbert spaces V_n ⊗ W , we consider the collection K_n of all singular values of all norm one elements of H_n with respect to the tensor structure. A law of large numbers has been obtained for this random set in the context of W fixed and the dimension of H_n, V_n tending to infinity at the same speed in 2012 in a paper from S. Belinschi, B. Collins and I. Nechita. In this paper, we provide measure concentration estimates in this context. The probabilistic study of K_n was motivated by important questions in Quantum Information Theory, and allowed to provide the smallest known dimension (184) for the dimension an an ancilla space allowing Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where violation of MOE occurs.
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Dates and versions

ensl-03053098 , version 1 (10-12-2020)

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  • HAL Id : ensl-03053098 , version 1

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Benoît Collins, Félix Parraud. Concentration estimates for random subspaces of a tensor product, and application to Quantum Information Theory. 2020. ⟨ensl-03053098⟩
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