Related papers: Almost-idempotent quantum channels and approximate $C^*$-algebras

Almost-idempotent quantum channels and approximate $C^*$-algebras

URL: http://arxiv.org/abs/2405.02434v1
Date: Fri, 3 May 2024 18:59:50 GMT
Title: Almost-idempotent quantum channels and approximate $C^*$-algebras
Authors: Alexei Kitaev,
Abstract summary: We prove that any finite-dimensional $varepsilon$-$C*$ algebra is $O(varepsilon)$-isomorphic to a genuine $C*$ algebra.
Score: 0.03922370499388702
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: Let $\Phi$ be a unital completely positive map on the space of operators on some Hilbert space. We assume that $\Phi$ is almost idempotent, namely, $\|\Phi^2-\Phi\|_{\mathrm{cb}} \le\eta$, and construct a corresponding "$\varepsilon$-$C^*$ algebra" for $\varepsilon=O(\eta)$. This type of structure has the axioms of a unital $C^*$ algebra but the associativity and other axioms involving the multiplication and the unit hold up to $\varepsilon$. We further prove that any finite-dimensional $\varepsilon$-$C^*$ algebra is $O(\varepsilon)$-isomorphic to a genuine $C^*$ algebra. These bounds are universal, i.e.\ do not depend on the dimensionality or other parameters.

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