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LINEAR MAPS PRESERVING TENSOR PRODUCTS OF RANK-ONE HERMITIAN MATRICES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 98 / Issue 3 / June 2015
- Published online by Cambridge University Press:
- 21 November 2014, pp. 407-428
- Print publication:
- June 2015
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- Article
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For a positive integer
$n\geq 2$, let 
$M_{n}$ be the set of 
$n\times n$ complex matrices and 
$H_{n}$ the set of Hermitian matrices in 
$M_{n}$. We characterize injective linear maps 
${\it\phi}:H_{m_{1}\cdots m_{l}}\rightarrow H_{n}$ satisfying for all
$$\begin{eqnarray}\text{rank}(A_{1}\otimes \cdots \otimes A_{l})=1\Longrightarrow \text{rank}({\it\phi}(A_{1}\otimes \cdots \otimes A_{l}))=1\end{eqnarray}$$
$A_{k}\in H_{m_{k}}$, 
$k=1,\dots ,l$, where 
$l,m_{1},\dots ,m_{l}\geq 2$ are positive integers. The necessity of the injectivity assumption is shown. Moreover, the connection of the problem to quantum information science is mentioned.
