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Maps preserving trace of products and invertible completely positive maps

Huajun Huang, Ming-Cheng Tsai

Published Mar 22, 2021• 0 citations
OpenAlex

Abstract

We show the linearity and injectivity of two maps $\phi_1$ and $\phi_2$ between certain subsets of matrix spaces that satisfy $\operatorname{tr}(\phi_1(A)\phi_2(B))=\operatorname{tr}(AB)$. We apply it to characterize maps $\phi_i$ ($i=1, \ldots, m$) satisfying $$\operatorname{tr} (\phi_1(A_1)\cdots \phi_m(A_m))=\operatorname{tr} (A_1\cdots A_m)$$ on maps $\mathcal{S}\to \mathcal{S}$ in which $\mathcal{S}$ is the set of $n$-by-$n$ general or Hermitian matrices for $m\ge 3$ and positive definite matrices for $m\ge 2$, and on completely positive maps $M_n\to M_n$ for $m\ge 2$. Then we give several equivalent characterizations of invertible completely positive maps, that is, $\phi: M_n\to M_k$ and $\psi:M_k\to M_n$ such that $\psi(\phi(A))=A$ for $A\in M_n$, 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 $\phi$ between matrix spaces satisfies $\phi(I)=I$ and $\phi(A^k)=\phi(A)^k$ for a fixed integer $k\ge 2$ if and only if $\phi(A^r)=\phi(A)^r$ for all $r\in {\mathbb Z}_+$. We characterize maps $\phi, \psi: \mathcal{S} \to \mathcal{S}$ satisfying $\operatorname{tr} (\phi(A)\psi(B))=\operatorname{tr} (AB)$ and $\operatorname{tr} (\phi(A)\psi(B)^k)=\operatorname{tr} (AB^k)$ for a fixed $k\ge 2$, where $\mathcal{S}$ is the set of general or Hermitian matrices.

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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 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.
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.
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.
Abstract Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten ppp -norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a C∗C^{\ast }C∗ -algebra.
Uhlmann showed that there exists a positive, unital and trace-preserving map transforming a Hermitian matrix AAA into another BBB if and only if the vector of eigenvalues of AAA majorizes that of BBB. In this work I characterize the existence of such a transformation when one of the conditions of unitality or trace preservation is dropped. This induces two possible preorders in the set of Hermitian matrices and I argue how this can be used to construct measures quantifying the lack of positive semidefiniteness of any given Hermitian matrix with relevant monotonicity properties. It turns out that the measures in each of the two formalisms are essentially unique.