Covariant decomposable maps on C*-algebras and quantum dynamics
Krzysztof Szczygielski
TL;DR
The paper extends the theory of covariant completely positive maps to decomposable maps between unital C*-algebras, providing dilation-based characterizations (extending Scutaru) and explicit operator-sum descriptions under covariance with respect to both general groups and the maximal commutative subgroup of U(n). It shows how covariant decomposable maps decompose into covariant CP and coCP parts, with concrete Frobenius-basis structures and Kraus-type representations, and applies these results to quantum dynamics, giving necessary and sufficient conditions for covariance of D-divisible, time-local generators on matrix algebras. The work connects algebraic dilation theory with dynamical semigroups, offering a framework for analyzing symmetry-constrained quantum evolutions beyond complete positivity. It hence provides foundational tools for characterizing covariant decomposable evolutions and their generators in finite dimensions, with potential extensions to broader C*-algebras and quantum information applications.
Abstract
We characterize covariant positive decomposable maps between unital C*-algebras in terms of a dilation theorem, which generalizes a seminal result by H. Scutaru from Rep. Math. Phys. 16 (1):79-87, 1979. As a case study, we provide a certain characterization of the operator sum representation of maps on $\mathbb{M}_{n}(\mathbb{C})$, covariant with respect to the maximal commutative subgroup of $\mathrm{U}(n)$. A connection to quantum dynamics is established by specifying sufficient and necessary conditions for covariance of D-divisible (decomposably divisible) quantum evolution families, recently introduced in J. Phys. A: Math. Theor. 56 (2023) 485202.