Structure and positivity of linear maps preserving covariance under unitary evolution

TL;DR

This work characterizes maps that are covariant under unitary evolution from trace-class operators to bounded operators on tensorized Hilbert spaces. The central result (Theorem 1) gives a concrete six-parameter structure for such maps, with uniqueness for $ ext{dim}oldsymbol{ H} obreak oldsymbol{ H} obreak obreak o obreak dim oldsymbol{ H} obreak eq 2$, and sets the stage for analyzing self-adjointness and positivity (Theorem 2). The paper then specializes to the virtual broadcasting map, proving its unique characterization under covariance, permutation invariance, and consistency with classical broadcasting, and extends the analysis to maps continuous in the $W^*$-topology. Further sections extend the characterization to compact-operator domains, establish equivalences between positivity and complete positivity for covariant maps, and connect to completely bounded norms. Theorem 4 generalizes the covariance structure to $oldsymbol{ H}^{ obreak obreak m}$ via Schur–Weyl duality, giving an explicit decomposition in terms of permutation operators, which provides a comprehensive framework for unitary-covariant, multi-copy linear maps in quantum information settings.

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 $Φ:\mathcal{T(H)}\rightarrow \mathcal{B(H\otimes H)}$ that are continuous relative to the norm topology and covariance under unitary evolution (i.e., $Φ(UXU^*)=(U\otimes U)Φ(X)(U^*\otimes U^*)$ 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 $Ψ:\mathcal{B(H)}\rightarrow \mathcal{B(H\otimes H)}$ that are continuous relative to the $W^*$-topology and covariance under unitary evolution are also characterized.