Non-unique decompositions of mixed states and deterministic energy transfers

We investigate the impact of non-unique decompositions of mixed states on energy transfer. Mixed states generally have non-unique decompositions into pure states in quantum theory and, by definition, in other non-classical probabilistic theories. We consider energy transfers constituting deterministic energy harvesting, wherein the source transfers energy to the harvester but not entropy. We use the possibility of non-unique decompositions to derive that if source states in a set jointly lead to deterministic energy harvesting for the given harvesting system and interaction, then that set can be expanded to include both mixtures and superpositions of the original states in the set. As a paradigmatic example, we model the source as an EM mode transferring energy to a 2-level system harvester via the Jaynes-Cummings model. We show that the set of coherent EM mode states with fixed $|α|$ that jointly achieve deterministic energy transfer can be expanded to include all mixtures and superpositions of those states. More generally, the results link the defining feature of a non-classical probability theory with the ability to achieve energy transfer without entropy transfer.

💡 Research Summary

The paper investigates how the non‑uniqueness of mixed‑state decompositions—a hallmark of quantum theory and any non‑classical probabilistic framework—affects the possibility of deterministic energy harvesting (DEH), i.e., transferring energy from a source to a load without accompanying entropy flow. After a brief motivation, the authors formalize DEH within the generalized probabilistic theories (GPT) setting, where each subsystem is described by a normalized quasiprobability distribution on phase space and dynamics are given by an affine (linear) map Φτ generated by a theory‑specific Hamiltonian kernel.

Definition 1 (DEH in phase‑space) states that a set of source states {Wi B} achieves DEH if, for a fixed interaction and evolution time τ, the marginal state of the harvester A is driven from a pure low‑energy distribution WA(0) to a pure high‑energy distribution WA(1) for every Wi B in the set.

Proposition 1 (Decomposition‑Irrelevance) follows directly from the linearity of Φτ: (i) any convex mixture of source states that individually achieve DEH also achieves DEH, and (ii) any pure state appearing in an alternative decomposition of the same mixed source must itself achieve DEH. This shows that the specific way a mixed source is written as a sum of pure states is irrelevant for the harvesting outcome.

The authors then address robustness. Proposition 2 establishes a bound on the distinguishability of final harvester states in terms of the distinguishability of the initial source states, using a generic statistical distance D that reduces to the trace distance or relative entropy in the quantum case. Because quantum pure states are not perfectly distinguishable, the bound guarantees that small perturbations of the source lead to only small changes in the harvested state, a property illustrated in Fig. 2.

Moving to the density‑matrix formalism, Definition 2 recasts DEH as the requirement that TrB