Unequal Coherent Ergotropy at Equal Interferometric Visibility and Energy in Quantum-Controlled Interferometers
The ergotropy, or unitarily extractable work, of a quantum system naturally splits into incoherent and coherent contributions, respectively set by population inversion and coherences in the energy eigenbasis. We show that this split makes the coherent ergotropy of a Mach--Zehnder qubit sensitive to path predictability that the output interference fringes do not reveal. We send the path qubit, prepared as a thermal quantum battery, through two quantum-controlled interferometers built from the same elements, a fixed beam splitter and a beam splitter coherently controlled to be present or absent, differing only in the order in which these two act. The two arrangements share identical visibility, stored energy, and incoherent ergotropy at every point in the parameter space. Yet, for work extracted from the output battery alone, placing the controlled beam splitter first never yields less coherent ergotropy, and at fixed visibility and interferometric phase the difference is governed by the path predictability that this ordering creates. The same expression reappears as a purity deficit on the beam-splitter controller, so the work advantage comes with a Landauer bound on the cost of locally resetting the controller that is never larger at the same temperature. We further examine the effects of decoherence on the battery and consider a relativistic realization in which a Wigner rotation correlates the particle's spin with its path. Tracing out the spin decoheres the battery and allows the two arrangements to exhibit different coherent ergotropies even in regimes where their non-relativistic counterparts coincide. These results show that interference visibility is not a complete measure of the coherence available as work.