Operational Relation Between One-Time and Two-Time Work Protocols and Measurement-Resolution-Induced Transitions in Quantum Work Statistics

Abstract

We establish a direct operational connection between the one-time measurement (OTM) and two-time measurement (TTM) protocols for quantum work statistics, showing that OTM work values, including coherence signatures, can be reconstructed from standard TTM data through classical post-processing without any modification of the experimental setup. This reveals that coherence information commonly attributed to OTM schemes is already latent within TTM data and becomes explicit upon a natural post-processing step. For a driven two-level system subject to finite-resolution energy measurements, this reconstruction unveils a resolution-driven statistical transition in the work distribution: whereas projective and weak measurements yield smooth distributions, intermediate resolutions produce non-analytic structures whose character is determined by the dynamics and a critical meter resolution. We show that the non-analyticity of the OTM work distribution takes the form of square-root divergent peaks at specific work values, and that the separation between these singular points is an experimentally accessible quantity that vanishes at the critical resolution with a universal exponent. The same dynamical quantities that govern the transition also encode the relative entropy of coherence of the driven state, enabling coherence quantification from energy measurements alone. We demonstrate experimental accessibility in a nitrogen-vacancy center platform with parameters drawn from current experiments, where both sides of the transition and the coherence signatures are within reach of existing technology.