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arXiv:1611.06352

An Uncertainty Principle For Completely Positive Maps

Jeremy Levick

Published Nov 19, 2016• 4 citations
arXivHTMLPDF

Abstract

We present an "uncertainty principle" for quantum channels, showing a relationship between the dimensions of the range of a channel and the range of its complement. We examine some interesting specific cases, and discuss consequences for privacy of quantum channels.
DOI
10.48550/arxiv.1611.06352

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We establish operator structure identities for quantum channels and their error-correcting and private codes, emphasizing the complementarity relationship between the two perspectives. Relevant structures include correctable and private operator algebras, and operator spaces such as multiplicative domains and nullspaces of quantum channels and their complementary maps. For the case of privatizing to quantum states, we also derive dimension inequalities on the associated operator algebras that further quantify the trade-off between correction and privacy.