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Linear transformations between multipartite quantum systems that map the set of tensor product of idempotent matrices into idempotent matrix set. (English) Zbl 06203142
Summary: Let $H_n$ be the set of $n \times n$ complex Hermitian matrices and $\cal P_n$ (resp., $\cal T_n$) be the set of all idempotent (resp., tripotent) matrices in $H_n$. In $l$-partite quantum system $H_{m_{1}\cdots m_{l}} = \otimes^l_1 H_{m_i}$, $\otimes^l_1 \cal P_{m_i}$ (resp., $\otimes^l_1 \cal T_{m_i}$) denotes the set of all decomposable elements $\otimes^l_1 A_i$ such that $A_i \in \cal P_{m_i}$ (resp., $A_i \in \cal T_{m_i}$). In this paper, linear maps $\phi$ from $H_{m_{1}\cdots m_{l}}$ to $H_n$ with $n \leq m_1 \cdots m_l$ such that $\phi(\otimes^l_1 \cal P_{m_i}) \in \cal P_n$ are characterized. As its application, the structure of linear maps $\phi$ from $H_{m_{1}\cdots m_{l}}$ to $H_n$ with $n \leq m_1 \cdots m_l$ such that $\phi(\otimes^l_1 \cal T_{m_i}) \in \cal T_n$ is also obtained.
MSC:
46Functional analysis
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References:
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