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.
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.
We explore coherent ergotropy extraction in quantum batteries from a resource-geometric point of view. For an initial state ρ, we quantify the coherent extraction process by the coherent ergotropy Ec(ρ) and the coherent extraction distance Dcext(ρ) between the active state σρ and the passive state Pρ. This defines the geometric power capacity Πc(ρ)=Ec(ρ)/Dcext(ρ), which measures the coherent ergotropy released unit minimal unitary distance. We prove that, for any driving Hamiltonian satisfying ∥Vt∥≤ν, the actual coherent discharging power is bounded by Pcext(ρ;Vt)≤νΠc(ρ), showing that Πc(ρ) is a capacity under unit driving norm rather than the power of a particular protocol. General bounds on Πc(ρ) are derived by combining relative entropy bounds on coherent ergotropy with geometric bounds on the coherent extraction distance. We also formulate coherence measure induced bounds and protocol-corrected capacities involving the effective speed of a given Hamiltonian. Qubit and qutrit examples demonstrate that Πc(ρ) captures a resource-geometric feature of coherent discharging beyond coherent ergotropy or coherence measures alone.
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.
The amount of extractable work from a physical system is fundamentally connected to the information available about its state, as illustrated by Maxwell's demon and the Gibbs paradox. In standard thermodynamic protocols involving system--bath interactions, the maximum work is given by the free-energy difference between the initial state and the corresponding Gibbs state at the bath temperature. This motivates a natural question: does information also limit work extraction in closed quantum systems that do not involve a heat bath and where work is obtained through unitary operations generated by a time-dependent Hamiltonian? While ergotropy quantifies the maximum work extractable via unitary operations, it assumes complete knowledge of the quantum state, typically requiring full state tomography. In realistic scenarios, however, only partial information is accessible. In this case, the relevant figure of merit is observational ergotropy, which depends explicitly on the measurement used to probe the system. We show that observational ergotropy decreases under classical post-processing of measurement outcomes, implying that fine-grained measurements allow greater work extraction than coarse-grained ones. Moreover, maximizing observational ergotropy over all possible measurements recovers standard ergotropy, which decomposes into incoherent (classical) and coherent (quantum) contributions. Our results demonstrate that coherence in the measurement projectors constitutes the key resource, enabling work extraction beyond the incoherent limit and establishing coherence as the origin of quantum advantage in observational ergotropy extraction.
We investigate the decomposition of ergotropy into incoherent and coherent contributions for quantum systems subject to typical Markovian noise channels. The incoherent part originates from population inversion in the energy eigenbasis after dephasing, while the coherent part captures the role of quantum coherence in work extraction. For single-qubit systems, we derive explicit conditions for freezing and enhancement of coherent ergotropy and obtain an analytical upper bound, showing that it cannot exceed one half of the state's quantum coherence. We then study two classes of separable two-qubit states under local noise. For Bell-diagonal states, which are locally completely passive and possess no local coherence, we prove that the total extractable work equals the average of geometric quantum and classical correlations. In this case, coherent ergotropy cannot be enhanced, although freezing occurs under specific noise conditions. By contrast, for separable states with local coherence, coherent ergotropy can increase under all considered noise channels, including phase-flip and depolarizing noise. Extending the analysis to multipartite systems, we show that both the magnitude and range of noise-induced enhancement grow with the number of qubits, indicating collective reinforcement. Finally, we demonstrate through an explicit example that entanglement does not prevent this enhancement: coherent ergotropy may increase under noise even for entangled states. Our results reveal that noise can assist energy storage, challenging the conventional view of noise as purely detrimental and suggesting compatibility between noise-assisted enhancement and fast entanglement-based charging mechanisms in quantum batteries.
Ergotropy is defined as the maximum amount of work that can be extracted through a unitary cyclic evolution. It plays a crucial role in assessing the work capacity of a quantum system. Recently, the significance of quantum coherence in work extraction has been theoretically identified, revealing that quantum states with more coherence possess more ergotropy compared to their dephased counterparts. However, an experimental study of the coherent ergotropy remains absent. Here, we report an experimental investigation of the coherent ergotropy in a single spin system. Based on the method of measuring ergotropy with an ancilla qubit, both the coherent and incoherent components of the ergotropy for the nonequilibrium state were successfully extracted. The increase in ergotropy induced by the increase in the coherence of the system was observed by varying the coherence of the state. Our work reveals the interplay between quantum thermodynamics and quantum information theory, future investigations could further explore the role other quantum attributes play in thermodynamic protocols.
We analyze the geometric phase and dynamic phase acquired by a qubit coupled to an environment through pure dephasing, establishing a direct connection between phase accumulation and ergotropy. We show that the dynamic phase depends solely on the incoherent ergotropy, reflecting its purely energetic origin. In contrast, the geometric phase exhibits a nontrivial dependence on both the coherent and incoherent contributions to the total ergotropy, encoding the interplay between coherence, dissipation, and energy extraction. By performing a perturbative expansion in the qubit-environment coupling strength, we demonstrate that, in the weak-coupling and long-time regime, the geometric phase becomes determined exclusively by the incoherent ergotropy, which coincides with the asymptotic value of the total ergotropy reached under decoherence. These results provide a clear physical distinction between dynamic and geometric phases in open quantum systems and establish geometric phases as sensitive probes of energetic resources. Furthermore,~in superconducting circuit implementations, our findings suggest that the ergotropy of a two-level system could be inferred indirectly from geometric-phase measurements using standard techniques such as quantum state tomography.