Existence of a time–cost–error trade-off for non-Markovian dynamics with infinite-dimensional reservoirs

Establish whether a three-way trade-off relation between operation time, dissipative/kinetic cost, and error holds for general non-Markovian quantum dynamics in which the system is coupled to an infinite-dimensional reservoir; specifically, construct a rigorous inequality of the form τ C ετ ≥ 1 − η (with εt defined from the smallest eigenvalue of the system’s density matrix, εt = −[ln λS(t)]−1, and η = ετ/ε0) that remains meaningful in the infinite-dimensional limit by developing a cost functional that does not depend on reservoir dimensionality and yields a nontrivial bound as the reservoir dimension tends to infinity.

Background

The paper establishes a unified three-way trade-off relation among time, cost, and error for thermodynamic operations that aim to create separated states, with extensions to quantum settings. For non-Markovian dynamics with a finite-dimensional reservoir, the authors derive a trade-off of the form τ C ετ ≥ 1 − η, where the cost is defined via entropy production as C = τ−1 Ψ(λ−1 Στ), with λ the smallest eigenvalue of the initial composite state and Στ the total entropy production.

However, when the reservoir is infinite-dimensional, the dimensional factor implicit in λ renders the bound trivial in the limit, preventing a meaningful inequality. The authors therefore highlight as an open question whether a comparable time–cost–error trade-off can be formulated for non-Markovian quantum dynamics with infinite-dimensional reservoirs, and how to define a cost functional that leads to a nontrivial, informative bound in this regime.