Coherent Ergotropy and Quantum Fisher Information: An Exact Two-Level Identity, a Non-Universal Fisher Bound, and an Anisotropic Coherence Horizon

Abstract

The coherent ergotropy of a quantum system—the work extractable by unitary control that is stored in energy-basis coherences—is usually treated as a purely thermodynamic quantity. This preprint establishes its exact geometric content for a two-level system and precisely delineates the limits of that connection. First, it proves the exact identity C = ε(√(z² + F_Q) − |z|), where z is the population imbalance and F_Q is the quantum Fisher information conjugate to the level phase, equivalently the squared Ramsey visibility. Coherent work is therefore a joint function of the population imbalance and the phase QFI, rather than a function of the QFI alone. Second, an exact two-level bound C ≤ ½√(F_Q^H) is established for the Hamiltonian-generated quantum Fisher information, with saturation on the Bloch equator. An explicit qutrit counterexample, extended by embedding, demonstrates that this bound fails for every dimension d ≥ 3. The work further corrects the target state entering the quantum speed-limit formulation of work extraction, showing that unitary extraction must drive the system toward its passive state rather than its dephased state. Finally, the exact identity predicts an anisotropic coherence horizon under pure dephasing: the 1/e lifetime of coherent work changes continuously from T2 on the Bloch equator to T2/2 near the poles. Coherent ergotropy can consequently be reconstructed experimentally from three calibrated quantities—the population imbalance, Ramsey visibility, and level splitting—without a work-extraction protocol, full state tomography, or an ancilla.