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

• URL: http://arxiv.org/abs/2405.02434v2
• Date: Tue, 11 Feb 2025 01:26:18 GMT
• Title: Almost-idempotent quantum channels and approximate $C^*$-algebras
• Authors: Alexei Kitaev,
• Abstract summary: We prove that any finite-dimensional $varepsilon$-$C*$ algebra $A$ is $O(varepsilon)$-isomorphic to some genuine $C*$ algebra $B$. When $A$ comes from a finite-dimensional $eta$-idempotent UCP map $Phi$, the $O(eta)$-isomorphism and its inverse can be realized by UCP maps.
• Score: 0.03922370499388702
• License:
• Abstract: Let $\Phi$ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that $\Phi$ is $\eta$-idempotent, namely, $\|\Phi^2-\Phi\|_{\mathrm{cb}} \le\eta$, and construct an associated $\varepsilon$-$C^*$ algebra (of almost-invariant observables) 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 prove that any finite-dimensional $\varepsilon$-$C^*$ algebra $A$ is $O(\varepsilon)$-isomorphic to some genuine $C^*$ algebra $B$. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When $A$ comes from a finite-dimensional $\eta$-idempotent UCP map $\Phi$, the $O(\eta)$-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel $\Phi^*$ into a decoding channel, producing a state on $B$, and an encoding channel.

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