2 results
Linear maps preserving
$(p,k)$-norms of tensor products of matrices
- Part of
-
- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 15 December 2023, pp. 1-21
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
Let
$m,n\ge 2$ be integers. Denote by 
$M_n$ the set of 
$n\times n$ complex matrices and 
$\|\cdot \|_{(p,k)}$ the 
$(p,k)$ norm on 
$M_{mn}$ with a positive integer 
$k\leq mn$ and a real number 
$p>2$. We show that a linear map 
$\phi :M_{mn}\rightarrow M_{mn}$ satisfies 
$$ \begin{align*}\|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)} \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n\end{align*} $$if and only if there exist unitary matrices
$U,V\in M_{mn}$ such that 
$$ \begin{align*}\phi(A\otimes B)=U(\varphi_1(A)\otimes \varphi_2(B))V \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n,\end{align*} $$where
$\varphi _s$ is the identity map or the transposition map 
$X\mapsto X^T$ for 
$s=1,2$. The result is also extended to multipartite systems.
Maps on Quantum States in
$C^{\ast }$-algebras Preserving von Neumann Entropy or Schatten 
$p$-norm of Convex Combinations
- Part of
-
- Journal:
- Canadian Mathematical Bulletin / Volume 62 / Issue 1 / March 2019
- Published online by Cambridge University Press:
- 09 January 2019, pp. 75-80
- Print publication:
- March 2019
-
- Article
-
- You have access
- Export citation
-
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
$p$-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 
$C^{\ast }$-algebra.
