New additivity properties of the relative entropy of entanglement and its generalizations
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929333040316416 |
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| author | Rubboli, Roberto Tomamichel, Marco |
| author_facet | Rubboli, Roberto Tomamichel, Marco |
| contents | We prove that the relative entropy of entanglement is additive when \emph{at least one of the two states} belongs to some specific class. We show that these classes include bipartite pure, maximally correlated, GHZ, Bell diagonal, isotropic, and generalized Dicke states. Previously, additivity was established only if \textit{both} states belong to the same class. Moreover, we extend these results to entanglement monotones based on the $α$-$z$ Rényi relative entropy. Notably, this family of monotones includes also the generalized robustness of entanglement and the geometric measure of entanglement. In addition, we prove that any monotone based on a quantum relative entropy is not additive for general states. We also compute closed-form expressions of the monotones for bipartite pure, Bell diagonal, isotropic, generalized Werner, generalized Dicke, and maximally correlated Bell diagonal states. Our results rely on developing a method that allows us to recast the initial convex optimization problem into a simpler linear one. Even though we mostly focus on entanglement theory, we expect that some of our technical results could be useful in investigating more general convex optimization problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_12804 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | New additivity properties of the relative entropy of entanglement and its generalizations Rubboli, Roberto Tomamichel, Marco Quantum Physics We prove that the relative entropy of entanglement is additive when \emph{at least one of the two states} belongs to some specific class. We show that these classes include bipartite pure, maximally correlated, GHZ, Bell diagonal, isotropic, and generalized Dicke states. Previously, additivity was established only if \textit{both} states belong to the same class. Moreover, we extend these results to entanglement monotones based on the $α$-$z$ Rényi relative entropy. Notably, this family of monotones includes also the generalized robustness of entanglement and the geometric measure of entanglement. In addition, we prove that any monotone based on a quantum relative entropy is not additive for general states. We also compute closed-form expressions of the monotones for bipartite pure, Bell diagonal, isotropic, generalized Werner, generalized Dicke, and maximally correlated Bell diagonal states. Our results rely on developing a method that allows us to recast the initial convex optimization problem into a simpler linear one. Even though we mostly focus on entanglement theory, we expect that some of our technical results could be useful in investigating more general convex optimization problems. |
| title | New additivity properties of the relative entropy of entanglement and its generalizations |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2211.12804 |