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arXiv:2406.07219

On the Bures metric, C*-norm, and the quantum metric

Konrad Aguilar, KARINA BEHERA, Tron Omland, NICOLE WU

Published Jun 11, 2024• 0 citations
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Abstract

We prove that the topology on the density space with respect to a unital C*-algebra and a faithful induced by the C*-norm is finer than the Bures metric topology. We also provide an example when this containment is strict. Next, we provide a metric on the density space induced by a quantum metric in the sense of Rieffel and prove that the induced topology is the same as the topology induced by the Bures metric and C*-norm when the C*-algebra is assumed to be finite dimensional. Finally, we provide an example of when the Bures metric and induced quantum metric are not metric equivalent. Thus, we provide a bridge between these aspects of quantum information theory and noncommutative metric geometry.

DOI
10.48550/arxiv.2406.07219

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We study methods of inducing metrics on unital completely positive maps by employing seminorms arising in noncommutative geometry. Our main approach relies on the development of an infinite-dimensional C∗C^*C∗-algebraic analogue of the Choi-Jamiołkowski isomorphism. Under suitable conditions, we show that the induced metrics satisfy the quantum information theoretic properties of stability and chaining. Moreover, we show how to generate such metrics using constructions native to noncommutative geometry, by for example using external Kasparov products of spectral triples.
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