Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.02434 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915144972369920 |
|---|---|
| author | Kitaev, Alexei |
| author_facet | Kitaev, Alexei |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02434 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost-idempotent quantum channels and approximate $C^*$-algebras Kitaev, Alexei Operator Algebras Mathematical Physics Quantum Physics 46L99 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. |
| title | Almost-idempotent quantum channels and approximate $C^*$-algebras |
| topic | Operator Algebras Mathematical Physics Quantum Physics 46L99 |
| url | https://arxiv.org/abs/2405.02434 |