On the preservers of maximally entangled states (Q2332410)

Let $\mathcal{X}$ and $\mathcal{Y}$ be finite-dimensional complex Hilbert spaces with $\dim \mathcal{X}=n$ and $\dim \mathcal{Y}=m$. Let $L(\mathcal{X})$ denote the space of all linear transformations on $\mathcal{X}$ and $\mathcal{X} \otimes \mathcal{Y}$ the tensor product of $\mathcal{X}$ and $\mathcal{Y}$ (which is itself is a finite dimensional Hilbert space). A unit vector $u\in \mathcal{X}\otimes \mathcal{Y}$ is said to be maximally entangled if it can be written in the form \[ u=\sum\limits_{j=1}^{r}\frac{1}{\sqrt{r}}x_{j}\otimes y_{j} \] for othonormal sets of vectors $\{x_{1},\dots ,x_{r}\}\subset \mathcal{X}$, $\{y_{1},\dots ,y_{r}\}\subset \mathcal{Y}$ and $r=\min \{m,n\}$. The projection operator $uu^{\ast }\in L(\mathcal{X}\otimes \mathcal{Y})$ (induced by the above $u$) is called a maximally entangled state. We denote by $\mathrm{MES}_{\mathcal{X},\mathcal{Y}}$ the set of maximally entagled states over $\mathcal{X}$ and $\mathcal{Y}$, i.e., \[ \mathrm{MES}_{\mathcal{X},\mathcal{Y}}=\{uu^{\ast }: u\text{ is maximally entangled}\}\subset L(\mathcal{X}\otimes \mathcal{Y}). \] We say that the map $\Phi : \operatorname{span}(\mathrm{MES}_{\mathcal{X},\mathcal{Y}})\rightarrow \operatorname{span}(\mathrm{MES}_{\mathcal{X},\mathcal{Y}})$ preserves the maximal entanglement (or preserves $\mathrm{MES}_{\mathcal{X},\mathcal{Y}}$) if $\Phi (\mathrm{MES}_{\mathcal{X}, \mathcal{Y}})\subseteq \mathrm{MES}_{\mathcal{X},\mathcal{Y}}$. Motivated by applications in quantum information theory, where the concept of entanglement is very important, the authors prove the following result. Let $\mathcal{Y}=\mathcal{X}^{k}=\mathcal{X\oplus }$ $\mathcal{\cdots \oplus X}$, $k>1$. If \[ \Phi :\operatorname{span}(\mathrm{MES}_{\mathcal{X},\mathcal{Y}})\rightarrow \operatorname{span}(\mathrm{MES}_{ \mathcal{X},\mathcal{Y}}) \] is an invertible linear map that preserves $\mathrm{MES}_{\mathcal{X},\mathcal{Y}}$, then $\Phi $ is of the form \[ \Phi (X)=(U\otimes V)X^{\sigma }(U\otimes V)^{\ast },\qquad X\in \operatorname{span}(\mathrm{MES}_{\mathcal{X},\mathcal{Y}}), \] where $U\in L(\mathcal{X})$, $V\in L(\mathcal{Y})$ are unitary and $A\mapsto A^{\sigma }$ is either the identity or the transpose map. Using this result, the authors generalize the invertible case of Theorem 1.1 from [\textit{E. Poon}, Linear Algebra Appl. 468, 122--144 (2015; Zbl 1307.15045)].