From the Choi Formalism in Infinite Dimensions to Unique Decompositions of Generators of Completely Positive Dynamical Semigroups

Abstract

Given any separable complex Hilbert space, any trace-class operator $B$ which does not have purely imaginary trace, and any generator $L$ of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a unique bounded operator $K$ and a unique completely positive map $\Phi$ such that (i) $L=K(\cdot )+(\cdot )K^{\ast }+\Phi$, (ii) the superoperator $\Phi (B^{\ast }(\cdot )B)$ is trace class and has vanishing trace, and (iii) $\mathrm{tr}(B^{\ast }K)$ is a real number. Central to our proof is a modified version of the Choi formalism which relates completely positive maps to positive semi-definite operators. We characterize when this correspondence is injective and surjective, respectively, which in turn explains why the proof idea of our main result cannot extend to non-separable Hilbert spaces. In particular, we find examples of positive semi-definite operators which have empty pre-image under the Choi formalism as soon as the underlying Hilbert space is infinite-dimensional.

19 pages main. Generalizes arXiv:2310.04037 to infinite dimensions

References in corpus (4)

• Unbounded generators of dynamical semigroups
• A canonical Hamiltonian for open quantum systems
• Work, Heat and Internal Energy in Open Quantum Systems: A Comparison of Four Approaches from the Autonomous System Framework
• Understanding and Generalizing Unique Decompositions of Generators of Dynamical Semigroups