Quantum $U$-channels on $S$-spaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913333616050176 |
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| author | Bag, Priyabrata Rohilla, Azad Trivedi, Harsh |
| author_facet | Bag, Priyabrata Rohilla, Azad Trivedi, Harsh |
| contents | If the symmetry, (an operator $J$ satisfying $J=J^*=J^{-1}$) which defines the Krein space, is replaced by a (not necessarily self-adjoint) unitary, then we have the notion of an $S$-space which was introduced by Szafraniec. In this paper, we consider $S$-spaces and study the structure of completely $U$-positive maps between the algebras of bounded linear operators. We first give a Stinespring-type representation for a completely $U$-positive map. On the other hand, we introduce Choi $U$-matrix of a linear map and establish the equivalence of the Kraus $U$-decompositions and Choi $U$-matrices. Then we study properties of nilpotent completely $U$-positive maps. We develop the $U$-PPT criterion for separability of quantum $U$-states and discuss the entanglement breaking condition of quantum $U$-channels and explore $U$-PPT squared conjecture. Finally, we give concrete examples of completely $U$-positive maps and examples of $3 \otimes 3$ quantum $U$-states which are $U$-entangled and $U$-separable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18160 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum $U$-channels on $S$-spaces Bag, Priyabrata Rohilla, Azad Trivedi, Harsh Functional Analysis 46E22, ~46L05, ~46L08, ~47B50, ~81T05 If the symmetry, (an operator $J$ satisfying $J=J^*=J^{-1}$) which defines the Krein space, is replaced by a (not necessarily self-adjoint) unitary, then we have the notion of an $S$-space which was introduced by Szafraniec. In this paper, we consider $S$-spaces and study the structure of completely $U$-positive maps between the algebras of bounded linear operators. We first give a Stinespring-type representation for a completely $U$-positive map. On the other hand, we introduce Choi $U$-matrix of a linear map and establish the equivalence of the Kraus $U$-decompositions and Choi $U$-matrices. Then we study properties of nilpotent completely $U$-positive maps. We develop the $U$-PPT criterion for separability of quantum $U$-states and discuss the entanglement breaking condition of quantum $U$-channels and explore $U$-PPT squared conjecture. Finally, we give concrete examples of completely $U$-positive maps and examples of $3 \otimes 3$ quantum $U$-states which are $U$-entangled and $U$-separable. |
| title | Quantum $U$-channels on $S$-spaces |
| topic | Functional Analysis 46E22, ~46L05, ~46L08, ~47B50, ~81T05 |
| url | https://arxiv.org/abs/2404.18160 |