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The goal of this note is to show that Hastings' counterexample to the additivity of minimal output von Neumann entropy can be readily deduced from a sharp version of Dvoretzky's theorem on almost spherical sections of convex bodies.
Lévy, P.: Problémes concrets d’analyse fonctionnelle
1951
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Dvoretzky, A.: Some Results on Convex Bodies and Banach Spaces
1961
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Marchenko, V. A., Pastur, L. A.: The distribution of eigenvalues in certain sets of random matrices
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Dudley, R. M.: The sizes of compact subsets of Hilbert space and continuity of Gaussian processes
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Milman, V.: A new proof of the theorem of A. Dvoretzky on sections of convex bodies
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Figiel, T., Lindenstrauss, J., Milman, V. D.: The dimension of almost spherical sections of convex bodies
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Jain, N. C., Marcus, M. B.: Continuity of subgaussian processes
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Geman, S.: A limit theorem for the norm of random matrices
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Silverstein, J. W.: The smallest eigenvalue of a large-dimensional Wishart matrix
1985
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Gordon, Y.: On Milman’s inequality and random subspaces which escape through a mesh in 𝐑 n \mathbf{R}^{n}
1988
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Pisier, G.: The volume of convex bodies and Banach space geometry
1989
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Schechtman, G.: A remark concerning the dependence on ε \varepsilon in Dvoretzky’s theorem
1989
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Nielsen, M. A., Chuang, I. L.: Quantum computation and quantum information
2000
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Haagerup, U., Thorbjørnsen, S.: Random matrices with complex Gaussian entries
2003
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Holevo, A. S.: The additivity problem in quantum information theory
2006
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Hayden, P., Leung, D., Winter, A.: Aspects of generic entanglement
2006
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2008
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Hastings, M. B.: Superadditivity of communication capacity using entangled inputs
2009
Later among the works it cites.
2010
Closest in time.
Fukuda, M., King, C., Moser, D.: Comments on Hastings’ Additivity Counterexamples
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Shor, P. W.: Equivalence of additivity questions in quantum information theory
2004
Cited alongside, same era.
Talagrand, M.: The generic chaining. Upper and Lower bounds of Stochastic Processes
2005
Cited alongside, same era.
Cited in the paper.
2010
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Aubrun, G., Szarek, S., Werner, E.: Non-additivity of Rényi entropy and Dvoretzky’s theorem
2010
Closest in time.