Structure and positivity of linear maps preserving covariance under unitary evolution

Abstract

Let $\mathcal{H}$ be a complex finite-dimensional or infinite-dimensional separable Hilbert space, $\mathcal{B}(\mathcal{H})$ and $\mathcal{T}(\mathcal{H})$ be the Banach spaces of all bounded linear operators and of all trace class operators on $\mathcal{H},$ respectively. In this paper, we give a concrete description of the linear maps $\Phi :\mathcal{T}(\mathcal{H})\to \mathcal{B}(\mathcal{H}\otimes \mathcal{H})$ that are continuous relative to the norm topology and covariance under unitary evolution (i.e., $\Phi (UX{U}^{\ast })=(U\otimes U)\Phi (X)(U^{\ast }\otimes U^{\ast })$ for all $X\in \mathcal{T}(\mathcal{H})$ and unitary operators $U\in \mathcal{B}(\mathcal{H})).$ Using this, we obtain the equivalent conditions for this class of maps to be self-adjoint or positive. As a corollary, we get that the virtual broadcasting map ${\mathcal{B}}_{vb}:\mathcal{T}(\mathcal{H})\to \mathcal{B}(\mathcal{H}\otimes \mathcal{H})$ with the form ${\mathcal{B}}_{vb}(X)=\frac{1}{2}[S(I\otimes X)+S(X\otimes I)]$ is uniquely determined by three conditions: covariance under unitary evolution, invariance under permutation of the copies and consistency with classical broadcasting, where $S\in \mathcal{B}(\mathcal{H}\otimes \mathcal{H})$ is the swap operator. Moreover, the linear maps $\Psi :\mathcal{B}(\mathcal{H})\to \mathcal{B}(\mathcal{H}\otimes \mathcal{H})$ that are continuous relative to the $W^{\ast }$-topology and covariance under unitary evolution are also characterized.