We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with n copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every n ∈ ℕ, there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all n. For higher dimensions, we reduce the existence question of such non-trivial “tensor-stable positive maps” to a one-parameter family of maps and show that an affirmative answer would imply the existence of non-positive partial transpose bound entanglement. As an application, we show that any tensor-stable positive map that is not completely positive yields an upper bound on the quantum channel capacity, which for the transposition map gives the well-known cb-norm bound. We, furthermore, show that the latter is an upper bound even for the local operations and classical communications-assisted quantum capacity, and that moreover it is a strong converse rate for this task.
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January 2016
Research Article|
July 23 2015
Positivity of linear maps under tensor powers
Alexander Müller-Hermes;
1Zentrum Mathematik, Technische
Universität München
, 85748 Garching, Germany
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David Reeb;
1Zentrum Mathematik, Technische
Universität München
, 85748 Garching, Germany
2Institute for Theoretical Physics,
Leibniz Universität Hannover
, 30167 Hannover, Germany
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Michael M. Wolf
1Zentrum Mathematik, Technische
Universität München
, 85748 Garching, Germany
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a)
Electronic address: [email protected]
b)
Electronic address: [email protected]
c)
Electronic address: [email protected]
J. Math. Phys. 57, 015202 (2016)
Article history
Received:
March 27 2015
Accepted:
July 08 2015
Citation
Alexander Müller-Hermes, David Reeb, Michael M. Wolf; Positivity of linear maps under tensor powers. J. Math. Phys. 1 January 2016; 57 (1): 015202. https://doi.org/10.1063/1.4927070
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