On the rank of extremal marginal states
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| Format: | Preprint |
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2024
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| author | Devendra, Repana Dey, Pankaj Dey, Santanu |
| author_facet | Devendra, Repana Dey, Pankaj Dey, Santanu |
| contents | Let $ρ_1$ and $ρ_2$ be two states on $\mathbb{C}^{d_1}$ and $\mathbb{C}^{d_2}$ respectively. The marginal state space, denoted by $\mathcal{C}(ρ_1,ρ_2)$, is the set of all states $ρ$ on $\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$ with partial traces $ρ_1, ρ_2$. K. R. Parthasarathy established that if $ρ$ is an extreme point of $\mathcal{C}(ρ_1,ρ_2)$, then the rank of $ρ$ does not exceed $\sqrt{d_1^2+d_2^2-1}$. Rudolph posed a question regarding the tightness of this bound. In 2010, Ohno gave an affirmative answer by providing examples in low-dimensional matrix algebras $\mathbb{M}_3$ and $\mathbb{M}_4$. This article aims to provide a positive answer to the Rudolph question in various matrix algebras. Our approaches, to obtain the extremal marginal states with tight upper bound, are based on Choi-Jamiołkowski isomorphism and tensor product of extreme points. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10041 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the rank of extremal marginal states Devendra, Repana Dey, Pankaj Dey, Santanu Operator Algebras Mathematical Physics Functional Analysis 46L30, 81P47, 47L07, 46M05, 46N50 Let $ρ_1$ and $ρ_2$ be two states on $\mathbb{C}^{d_1}$ and $\mathbb{C}^{d_2}$ respectively. The marginal state space, denoted by $\mathcal{C}(ρ_1,ρ_2)$, is the set of all states $ρ$ on $\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$ with partial traces $ρ_1, ρ_2$. K. R. Parthasarathy established that if $ρ$ is an extreme point of $\mathcal{C}(ρ_1,ρ_2)$, then the rank of $ρ$ does not exceed $\sqrt{d_1^2+d_2^2-1}$. Rudolph posed a question regarding the tightness of this bound. In 2010, Ohno gave an affirmative answer by providing examples in low-dimensional matrix algebras $\mathbb{M}_3$ and $\mathbb{M}_4$. This article aims to provide a positive answer to the Rudolph question in various matrix algebras. Our approaches, to obtain the extremal marginal states with tight upper bound, are based on Choi-Jamiołkowski isomorphism and tensor product of extreme points. |
| title | On the rank of extremal marginal states |
| topic | Operator Algebras Mathematical Physics Functional Analysis 46L30, 81P47, 47L07, 46M05, 46N50 |
| url | https://arxiv.org/abs/2412.10041 |