Non-unique decompositions of mixed states and deterministic energy transfers

Abstract

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.

Figures (5)

• Figure 1: Non-unique decomposition and deterministic energy transfer. Illustration of the relationship between different decompositions of a fixed mixed state $\rho_B$ in system B and the induced trajectories in system A under the energy harvesting protocol. The two decompositions— $\{\rho_{B,1}, \rho_{B,2}\}$ represented by solid blue lines and $\{\rho_{B,1}', \rho_{B,2}'\}$ represented by dashed red lines—correspond to different pure-state ensembles that realize the same $\rho_B$. Each pure state in a decomposition drives a different trajectory of the qubit in system A at a designated time, shown in the matching colors.
• Figure 2: Robustness under source state perturbation for the quantum case. When two input states to system B differ by a small amount (i.e., $D(\rho_{B1}, \rho_{B2}) = \epsilon$), the corresponding output states $\rho_{A1}, \rho_{A2}$ in system A will not differ more: $D(\rho_{A1}, \rho_{A2})\leq \epsilon$
• Figure 3: Deterministic and approximate deterministic energy harvesting (DEH) dynamics in the Jaynes Cummings model. Each panel displays the fidelity between the time-evolved qubit state $\rho_A(t)$ and the excited state $|1\rangle$, with fidelity unity corresponding to perfect energy absorption. The time axis is shown in units of $g t$, where $g$ is the qubit--field coupling strength.
• Figure S4: Wigner function distributions for four quantum states of system B, evaluated at coherent state amplitude $\alpha = 1$. Panel (a) displays the Wigner function of the incoherent mixture $\rho_{B1} = \tfrac{1}{2}(|\alpha\rangle\langle\alpha| + |{-\alpha}\rangle\langle{-\alpha}|)$. Panel (b) corresponds to the even Schrödinger cat state $\rho_{B2} = \tfrac{1}{2}|+\rangle_{\alpha}\langle +|_{\alpha}$, and panel (c) to the odd cat state $\rho_{B3} = \tfrac{1}{2} |-\rangle_{\alpha}\langle -|_{\alpha}$. Panel (d) shows $\rho_{B4} = \tfrac{1}{2}(\rho_{B2} + \rho_{B3})$, a convex combination of even and odd cat states yielding the same density matrix as $\rho_{B1}$ but with a different pure-state decomposition. The vertical colorbar represents the values of the Wigner function.
• Figure S5: Entropy dynamics of each subsystem during the energy harvesting process governed by the Jaynes--Cummings interaction. The plot shows that, by stopping the interaction at an appropriate time, one can achieve energy transfer with almost no net entropy transfer, highlighting the reversibility of the process under optimal timing conditions.

Theorems & Definitions (22)

• Definition 1: DEH in the phase-space formalism
• Proposition 1: Decomposition-Irrelevance of DEH
• Proposition 2: Distinguishability of initial source states bounds distinguishability of final harvester states
• Definition 2: DEH in Density Matrix Formalism
• Proposition 3: Convex Closure of DEH States
• Corollary 3.1: Superposition-generated alternative DEH ensembles
• Proposition 1: Decomposition-Irrelevance of DEH
• Proposition 3: Convex Closure of DEH States
• Corollary 3.1: Superposition-generated alternative DEH ensembles
• Definition S1: Measurement Effects