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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2512.12319 |
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| _version_ | 1866914200536743936 |
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| author | Li, Yuan Wang, Shuaijie Xu, Xiaoming |
| author_facet | Li, Yuan Wang, Shuaijie Xu, Xiaoming |
| contents | Let $\mathcal{H}$ be a complex finite-dimensional or infinite-dimensional separable Hilbert space, $\mathcal{B(H)}$ and $\mathcal{T(H)}$ be the Banach spaces of all bounded linear operators and of all trace class operators on $\mathcal{H},$ respectively. In this paper, we give a concrete description of the linear maps $Φ:\mathcal{T(H)}\rightarrow \mathcal{B(H\otimes H)}$ that are continuous relative to the norm topology and covariance under unitary evolution (i.e., $Φ(UXU^*)=(U\otimes U)Φ(X)(U^*\otimes U^*)$ for all $X\in\mathcal{T(H)}$ and unitary operators $U\in\mathcal{B(H)}).$ Using this, we obtain the equivalent conditions for this class of maps to be self-adjoint or positive. As a corollary, we get that the virtual broadcasting map $\mathcal{B}_{vb}:\mathcal{T(H)}\rightarrow \mathcal{B(H\otimes H)}$ with the form $\mathcal{B}_{vb}(X)=\frac{ 1}{2}[S(I\otimes X)+S(X\otimes I)]$ is uniquely determined by three conditions: covariance under unitary evolution, invariance under permutation of the copies and consistency with classical broadcasting, where $S\in\mathcal{B(H\otimes H)}$ is the swap operator. Moreover, the linear maps $Ψ:\mathcal{B(H)}\rightarrow \mathcal{B(H\otimes H)}$ that are continuous relative to the $W^*$-topology and covariance under unitary evolution are also characterized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure and positivity of linear maps preserving covariance under unitary evolution Li, Yuan Wang, Shuaijie Xu, Xiaoming Functional Analysis Mathematical Physics Let $\mathcal{H}$ be a complex finite-dimensional or infinite-dimensional separable Hilbert space, $\mathcal{B(H)}$ and $\mathcal{T(H)}$ be the Banach spaces of all bounded linear operators and of all trace class operators on $\mathcal{H},$ respectively. In this paper, we give a concrete description of the linear maps $Φ:\mathcal{T(H)}\rightarrow \mathcal{B(H\otimes H)}$ that are continuous relative to the norm topology and covariance under unitary evolution (i.e., $Φ(UXU^*)=(U\otimes U)Φ(X)(U^*\otimes U^*)$ for all $X\in\mathcal{T(H)}$ and unitary operators $U\in\mathcal{B(H)}).$ Using this, we obtain the equivalent conditions for this class of maps to be self-adjoint or positive. As a corollary, we get that the virtual broadcasting map $\mathcal{B}_{vb}:\mathcal{T(H)}\rightarrow \mathcal{B(H\otimes H)}$ with the form $\mathcal{B}_{vb}(X)=\frac{ 1}{2}[S(I\otimes X)+S(X\otimes I)]$ is uniquely determined by three conditions: covariance under unitary evolution, invariance under permutation of the copies and consistency with classical broadcasting, where $S\in\mathcal{B(H\otimes H)}$ is the swap operator. Moreover, the linear maps $Ψ:\mathcal{B(H)}\rightarrow \mathcal{B(H\otimes H)}$ that are continuous relative to the $W^*$-topology and covariance under unitary evolution are also characterized. |
| title | Structure and positivity of linear maps preserving covariance under unitary evolution |
| topic | Functional Analysis Mathematical Physics |
| url | https://arxiv.org/abs/2512.12319 |