Divisibility of Non-Invertible Quantum Dynamics in the Heisenberg Picture

Abstract:We study divisibility of finite-dimensional quantum dynamics in the Heisenberg picture when the dynamical maps need not be invertible. Heisenberg divisible dynamics determines unique reduced propagators on the accessible operator systems, and Heisenberg CP-divisibility is shown to be equivalent to an operator-norm monotonicity. Although each reduced propagator admits a unital completely positive extension, independently chosen extensions need not satisfy a composition law. Generalized inverses restore this law algebraically, but do not, in general, preserve complete positivity. We characterize the resulting obstruction and give a sufficient construction based on unital completely positive projections onto the accessible operator systems. It is shown that for qubit dynamics Heisenberg CP-divisibility always guarantees a completely positive generalized-inverse propagator. Finally, we analyze time-local generators at singular times and identify the associated gauge freedom on dynamically inaccessible directions.

arXiv:2610.07986 [quant-ph]

(or arXiv:2610.07986v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.07986 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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