C*-extreme entanglement breaking maps on operator systems
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910294412886016 |
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| author | Balasubramanian, Sriram Hotwani, Neha |
| author_facet | Balasubramanian, Sriram Hotwani, Neha |
| contents | Let $\mathcal E$ denote the set of all unital entanglement breaking (UEB) linear maps defined on an operator system $\mathcal S \subset M_d$ and, mapping into $M_n$. As it turns out, the set $\mathcal E$ is not only convex in the classical sense but also in a quantum sense, namely it is $C^*$-convex. The main objective of this article is to describe the $C^*$-extreme points of this set $\mathcal E$. By observing that every EB map defined on the operator system $\mathcal S$ dilates to a positive map with commutative range and also extends to an EB map on $M_d$, we show that the $C^*$-extreme points of the set $\mathcal E$ are precisely the UEB maps that are maximal in the sense of Arveson (\cite{A} and \cite{A69}) and that they are also exactly the linear extreme points of the set $\mathcal E$ with commutative range. We also determine their explicit structure, thereby obtaining operator system generalizations of the analogous structure theorem and the Krein-Milman type theorem given in \cite{BDMS}. As a consequence, we show that $C^*$-extreme (UEB) maps in $\mathcal E$ extend to $C^*$-extreme UEB maps on the full algebra. Finally, we obtain an improved version of the main result in \cite{BDMS}, which contains various characterizations of $C^*$-extreme UEB maps between the algebras $M_d$ and $M_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07642 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | C*-extreme entanglement breaking maps on operator systems Balasubramanian, Sriram Hotwani, Neha Operator Algebras Mathematical Physics Functional Analysis 81P40, 47L07 (primary), 15B48, 81R15, 81P42, 81P45 (Secondary) Let $\mathcal E$ denote the set of all unital entanglement breaking (UEB) linear maps defined on an operator system $\mathcal S \subset M_d$ and, mapping into $M_n$. As it turns out, the set $\mathcal E$ is not only convex in the classical sense but also in a quantum sense, namely it is $C^*$-convex. The main objective of this article is to describe the $C^*$-extreme points of this set $\mathcal E$. By observing that every EB map defined on the operator system $\mathcal S$ dilates to a positive map with commutative range and also extends to an EB map on $M_d$, we show that the $C^*$-extreme points of the set $\mathcal E$ are precisely the UEB maps that are maximal in the sense of Arveson (\cite{A} and \cite{A69}) and that they are also exactly the linear extreme points of the set $\mathcal E$ with commutative range. We also determine their explicit structure, thereby obtaining operator system generalizations of the analogous structure theorem and the Krein-Milman type theorem given in \cite{BDMS}. As a consequence, we show that $C^*$-extreme (UEB) maps in $\mathcal E$ extend to $C^*$-extreme UEB maps on the full algebra. Finally, we obtain an improved version of the main result in \cite{BDMS}, which contains various characterizations of $C^*$-extreme UEB maps between the algebras $M_d$ and $M_n$. |
| title | C*-extreme entanglement breaking maps on operator systems |
| topic | Operator Algebras Mathematical Physics Functional Analysis 81P40, 47L07 (primary), 15B48, 81R15, 81P42, 81P45 (Secondary) |
| url | https://arxiv.org/abs/2306.07642 |