Journal of Function Spaces and Applications

Linear Transformations between Multipartite Quantum Systems That Map the Set of Tensor Product of Idempotent Matrices into Idempotent Matrix Set

Jinli Xu, Baodong Zheng, Hongmei Yao

Published Jan 1, 2013• 6 citations

Abstract

Let H n be the set of n × n complex Hermitian matrices and 𝒫 n (resp., 𝒯 n ) be the set of all idempotent (resp., tripotent) matrices in H n . In l -partite quantum system H m 1 T m l = ⊗ 1 l H m i , ⊗ 1 l 𝒫 m i (resp., ⊗ 1 l 𝒯 m i ) denotes the set of all decomposable elements ⊗ 1 l A i such that A i ∈ 𝒫 m i (resp., A i ∈ 𝒯 m

DOI: 10.1155/2013/182569

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Journal of Function Spaces and Applications

Linear Transformations between Multipartite Quantum Systems That Map the Set of Tensor Product of Idempotent Matrices into Idempotent Matrix Set

Jinli Xu, Baodong Zheng, Hongmei Yao

Published Jan 1, 2013• 6 citations

Publisher

Abstract

DOI: 10.1155/2013/182569

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Jinli Xu, Baodong Zheng, Ajda Fošner

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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 $$\begin{eqnarray}\text{rank}(A_{1}\otimes \cdots \otimes A_{l})=1\Longrightarrow \text{rank}({\it\phi}(A_{1}\otimes \cdots \otimes A_{l}))=1\end{eqnarray}$$ for all

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.