2025/12/13 by Yuan Li, Li, Yuan, Shuaijie Wang +3
Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2512.12319
openalex publication_date 2025/12/13 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28
Let H be a complex finite-dimensional or infinite-dimensional separable Hilbert space, B(H) and T(H) be the Banach spaces of all bounded linear operators and of all trace class operators on H, respectively. In this paper, we give a concrete description of the linear maps Φ:T(H)→ B(H⊗ H) that are continuous relative to the norm topology and covariance under unitary evolution (i.e., Φ(UXU^*)=(U⊗ U)Φ(X)(U^*⊗ U^*) for all X\inT(H) and unitary operators U\inB(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 Bvb:T(H)→ B(H⊗ H) with the form Bvb(X)=( 1)/(2)[S(I⊗ X)+S(X⊗ I)] is uniquely determined by three conditions: covariance under unitary evolution, invariance under permutation of the copies and consistency with classical broadcasting, where S\inB(H⊗ H) is the swap operator. Moreover, the linear maps Ψ:B(H)→ B(H⊗ H) that are continuous relative to the W^*-topology and covariance under unitary evolution are also characterized.