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Journal of the Australian Mathematical Society

LINEAR MAPS PRESERVING TENSOR PRODUCTS OF RANK-ONE HERMITIAN MATRICES

Jinli Xu, Baodong Zheng, Ajda Fošner

Published Nov 21, 2014• 4 citations
Publisher

Abstract

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.

DOI
10.1017/s1446788714000603

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For a positive integer n≥2n\geq 2n≥2 , let MnM_{n}Mn​ be the set of n×nn\times nn×n complex matrices and HnH_{n}Hn​ the set of Hermitian matrices in MnM_{n}Mn​ . We characterize injective linear maps ϕ:Hm1⋯ml→Hn{\it\phi}:H_{m_{1}\cdots m_{l}}\rightarrow H_{n}ϕ:Hm1​⋯ml​​→Hn​ 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 Ak∈HmkA_{k}\in H_{m_{k}}Ak​∈Hmk​​ , k=1,…,lk=1,\dots ,lk=1,…,l , where l,m1,…,ml≥2l,m_{1},\dots ,m_{l}\geq 2l,m1​,…,ml​≥2 are positive integers. The necessity of the injectivity assumption is shown. Moreover, the connection of the problem to quantum information science is mentioned.
Let m,n≥2m,n\ge 2m,n≥2 be positive integers, MmM_mMm​ the set of m×mm\times mm×m complex matrices and MnM_nMn​ the set of n×nn\times nn×n complex matrices. Regard MmnM_{mn}Mmn​ as the tensor space Mm⊗MnM_m\otimes M_nMm​⊗Mn​. Suppose ∣⋅∣|\cdot|∣⋅∣ is the Ky Fan kkk-norm with 1≤k≤mn1 \le k \le mn1≤k≤mn, or the Schatten ppp-norm with 1≤p≤∞1 \le p \le \infty1≤p≤∞ (p≠2p\ne 2p=2) on MmnM_{mn}Mmn​. It is shown that a linear map ϕ:Mmn→Mmnϕ: M_{mn} \rightarrow M_{mn}ϕ:Mmn​→Mmn​ satisfying ∣A⊗B∣=∣ϕ(A⊗B)∣|A\otimes B| = |ϕ(A\otimes B)|∣A⊗B∣=∣ϕ(A⊗B)∣ for all A∈MmA \in M_mA∈Mm​ and B∈MnB \in M_nB∈Mn​ if and only if there are unitary U,V∈MmnU, V \in M_{mn}U,V∈Mmn​ such that ϕϕϕ has the form A⊗B↦U(φ1(A)⊗φ2(B))VA\otimes B \mapsto U(φ_1(A) \otimes φ_2(B))VA⊗B↦U(φ1​(A)⊗φ2​(B))V, where φi(X)φ_i(X)φi​(X) is either the identity map X↦XX \mapsto XX↦X or the transposition map X↦XtX \mapsto X^tX↦Xt. The results are extended to tensor space Mn1⊗...⊗MnmM_{n_1} \otimes ... \otimes M_{n_m}Mn1​​⊗...⊗Mnm​​ of higher level. The connection of the problem to quantum information science is mentioned.
We show the linearity and injectivity of two maps ϕ1\phi_1ϕ1​ and ϕ2\phi_2ϕ2​ between certain subsets of matrix spaces that satisfy tr⁡(ϕ1(A)ϕ2(B))=tr⁡(AB)\operatorname{tr}(\phi_1(A)\phi_2(B))=\operatorname{tr}(AB)tr(ϕ1​(A)ϕ2​(B))=tr(AB). We apply it to characterize maps ϕi\phi_iϕi​ (i=1,…,mi=1, \ldots, mi=1,…,m) satisfying tr⁡(ϕ1(A1)⋯ϕm(Am))=tr⁡(A1⋯Am)\operatorname{tr} (\phi_1(A_1)\cdots \phi_m(A_m))=\operatorname{tr} (A_1\cdots A_m)tr(ϕ1​(A1​)⋯ϕm​(Am​))=tr(A1​⋯Am​) on maps S→S\mathcal{S}\to \mathcal{S}S→S in which S\mathcal{S}S is the set of nnn-by-nnn general or Hermitian matrices for m≥3m\ge 3m≥3 and positive definite matrices for m≥2m\ge 2m≥2, and on completely positive maps Mn→MnM_n\to M_nMn​→Mn​ for m≥2m\ge 2m≥2. Then we give several equivalent characterizations of invertible completely positive maps, that is, ϕ:Mn→Mk\phi: M_n\to M_kϕ:Mn​→Mk​ and ψ:Mk→Mn\psi:M_k\to M_nψ:Mk​→Mn​ such that ψ(ϕ(A))=A\psi(\phi(A))=Aψ(ϕ(A))=A for A∈MnA\in M_nA∈Mn​, and provide a comprehensive list of such examples. These maps are connected to the trace-of-product preserving property and are useful in quantum information theory. In addition, we show that a linear map ϕ\phiϕ between matrix spaces satisfies ϕ(I)=I\phi(I)=Iϕ(I)=I and ϕ(Ak)=ϕ(A)k\phi(A^k)=\phi(A)^kϕ(Ak)=ϕ(A)k for a fixed integer k≥2k\ge 2k≥2 if and only if ϕ(Ar)=ϕ(A)r\phi(A^r)=\phi(A)^rϕ(Ar)=ϕ(A)r for all r∈Z+r\in {\mathbb Z}_+r∈Z+​. We characterize maps ϕ,ψ:S→S\phi, \psi: \mathcal{S} \to \mathcal{S}ϕ,ψ:S→S satisfying tr⁡(ϕ(A)ψ(B))=tr⁡(AB)\operatorname{tr} (\phi(A)\psi(B))=\operatorname{tr} (AB)tr(ϕ(A)ψ(B))=tr(AB) and tr⁡(ϕ(A)ψ(B)k)=tr⁡(ABk)\operatorname{tr} (\phi(A)\psi(B)^k)=\operatorname{tr} (AB^k)tr(ϕ(A)ψ(B)k)=tr(ABk) for a fixed k≥2k\ge 2k≥2, where S\mathcal{S}S is the set of general or Hermitian matrices.
For a positive integer nnn, let MnM_nMn​ be the set of n×nn\times nn×n complex matrices. Suppose ∥⋅∥\|\cdot\|∥⋅∥ is the Ky Fan kkk-norm with 1≤k≤mn1 \le k \le mn1≤k≤mn or the Schatten ppp-norm with 1≤p≤∞1 \le p \le \infty1≤p≤∞ (p≠2p\ne 2p=2) on MmnM_{mn}Mmn​, where m,n≥2m,n\ge 2m,n≥2 are positive integers. It is shown that a linear map ϕ:Mmn→Mmn\phi: M_{mn} \rightarrow M_{mn}ϕ:Mmn​→Mmn​ satisfying ∥A⊗B∥=∥ϕ(A⊗B)∥forallA∈MmandB∈Mn\|A\otimes B\| = \|\phi(A\otimes B)\| for all A \in M_m and B \in M_n∥A⊗B∥=∥ϕ(A⊗B)∥forallA∈Mm​andB∈Mn​ if and only if there are unitary U,V∈MmnU, V \in M_{mn}U,V∈Mmn​ such that ϕ\phiϕ has the form A⊗B↦U(φ1(A)⊗φ2(B))VA\otimes B \mapsto U(\varphi_1(A) \otimes \varphi_2(B))VA⊗B↦U(φ1​(A)⊗φ2​(B))V, where φs(X)\varphi_s(X)φs​(X) is either the identity map X↦XX \mapsto XX↦X or the transposition map X↦XtX \mapsto X^tX↦Xt. The results are extended to tensor space Mn1⊗⋯⊗MnmM_{n_1} \otimes \cdots \otimes M_{n_m}Mn1​​⊗⋯⊗Mnm​​ of higher level. The connection of the problem to quantum information science is mentioned.
In this article, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ on mn × mn Hermitian matrices such that φ(A ⊗ B) and A ⊗ B have the same spectrum for any m × m Hermitian A and n × n Hermitian B. Such a map has the form A ⊗ B ↦ U(ϕ1(A) ⊗ ϕ2(B))U* for mn × mn Hermitian matrices in tensor form A ⊗ B, where U is a unitary matrix, and for j ∈ {1, 2}, ϕ j is the identity map X ↦ X or the transposition map X ↦ X t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A ⊗ B is also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems.
Let n>1 be a positive integer, {H1,…,Hn} be a finite collection of complex Hilbert spaces with dim⁡(Hk)≥2, and P1(Hk) be the set of all rank-1 self-adjoint projections on Hk, k=1,…,n. Set DP1⨂k=1nHk={A1⊗⋯⊗An:Ak∈P1(Hk),k=1,…,n}.We characterize the maps ϕ from D(P1(⨂k=1nHk)) to P1(⨂k=1nHk) preserving transition probability, i.e. tr(AB)=tr(ϕ(A)ϕ(B)),for all A,B∈DP1⨂k=1nHk.A particular case corresponding to n = 1 is well known as (non-surjective version) Wigner's theorem. Our result may be considered as a generalization of Wigner's theorem.