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arXiv · 2204.08819

Counterexamples to the extendibility of positive unital norm-one maps

Giulio Chiribella·Kenneth R. Davidson·Vern I. Paulsen·Mizanur Rahaman
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Abstract

Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C*-algebra containing it. An analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a positive map on the full matrix algebra. The first counterexample is an unextendible positive unital map with unit norm, the second counterexample is an unextendible positive unital isometry on a real operator space, and the third counterexample is an unextendible positive unital isometry on a complex operator space.

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Giulio Chiribella, Kenneth R. Davidson, Vern I. Paulsen, Mizanur Rahaman. 2022-04-19. Counterexamples to the extendibility of positive unital norm-one maps. https://doi.org/10.1016/j.laa.2023.01.003

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