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, $\mathcal{B(H)}$ and $\mathcal{T(H)}$ 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 $\Phi:\mathcal{T(H)}\rightarrow \mathcal{B(H\otimes H)}$ that are continuous relative to the norm topology and covariance under unitary evolution (i.e., $\Phi(UXU^*)=(U\otimes U)\Phi(X)(U^*\otimes U^*)$ for all $X\in\mathcal{T(H)}$ and unitary operators $U\in\mathcal{B(H)}).$ 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 $\mathcal{B}_{vb}:\mathcal{T(H)}\rightarrow \mathcal{B(H\otimes H)}$ 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 $S\in\mathcal{B(H\otimes H)}$ is the swap operator. Moreover, the linear maps $\Psi:\mathcal{B(H)}\rightarrow \mathcal{B(H\otimes H)}$ that are continuous relative to the $W^*$-topology and covariance under unitary evolution are also characterized.