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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.02434 |
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Table of Contents:
- Let $Φ$ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that $Φ$ is $η$-idempotent, namely, $\|Φ^2-Φ\|_{\mathrm{cb}} \leη$, and construct an associated $\varepsilon$-$C^*$ algebra (of almost-invariant observables) for $\varepsilon=O(η)$. 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 $η$-idempotent UCP map $Φ$, the $O(η)$-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel $Φ^*$ into a decoding channel, producing a state on $B$, and an encoding channel.