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arXiv:1211.0396 · SIAM Journal on Matrix Analysis and Applications

Linear Maps Preserving Ky Fan Norms and Schatten Norms of Tensor Products of Matrices

Ajda Fošner, Zejun Huang, Chi-Kwong Li, Nung-Sing Sze

Published Jan 1, 2013• 15 citations
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Abstract

For a positive integer $n$, let $M_n$ be the set of $n\times n$ complex matrices. Suppose $\|\cdot\|$ is the Ky Fan $k$-norm with $1 \le k \le mn$ or the Schatten $p$-norm with $1 \le p \le \infty$ ($p\ne 2$) on $M_{mn}$, where $m,n\ge 2$ are positive integers. It is shown that a linear map $\phi: M_{mn} \rightarrow M_{mn}$ satisfying $\|A\otimes B\| = \|\phi(A\otimes B)\| for all A \in M_m and B \in M_n$ if and only if there are unitary $U, V \in M_{mn}$ such that $\phi$ has the form $A\otimes B \mapsto U(\varphi_1(A) \otimes \varphi_2(B))V$, where $\varphi_s(X)$ is either the identity map $X \mapsto X$ or the transposition map $X \mapsto X^t$. The results are extended to tensor space $M_{n_1} \otimes \cdots \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.

DOI
10.1137/120891794

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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.
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.
We define 2-indexed (q,p)(q,p)(q,p)-Schatten quasi-norms for any q,p>0q,p > 0q,p>0 on operators on a tensor product of Hilbert spaces, naturally extending the norms defined by Pisier's theory of operator-valued Schatten spaces. We establish several desirable properties of these quasi-norms, such as relational consistency and the behavior on block diagonal operators, assuming that ∣1q−1p∣≤1|\frac{1}{q} - \frac{1}{p}| \leq 1∣q1​−p1​∣≤1. In fact, we show that this condition is essentially necessary for natural properties to hold. Furthermore, for linear maps between spaces of such quasi-norms, we introduce completely bounded quasi-norms and co-quasi-norms. We prove that the q→pq \to pq→p completely bounded co-quasi-norm is super-multiplicative for tensor products of quantum channels for q≥p>0q \geq p>0q≥p>0, extending an influential result of [Devetak, Junge, King, Ruskai, 2006]. Our proofs rely on elementary matrix analysis and operator convexity tools and do not require operator space theory. On the applications side, we demonstrate that these quasi-norms can be used to express relevant quantum information measures such as Rényi conditional entropies for α≥12α\geq \frac{1}{2}α≥21​ or the Sandwiched Rényi Umlaut information for α<1α< 1α<1. Our multiplicativity results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output Rényi-ααα-entropy for α≥12α\geq\frac{1}{2}α≥21​ extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output Rényi-ααα entropy for α≥12α\geq \frac{1}{2}α≥21​.
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.
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.