Certifying ergotropy under partial information

Abstract

Ergotropy, the maximum work extractable from a quantum system, is a central resource in quantum physics. Computing ergotropy is well established when the system state is fully known, but its estimation under partial information remains an open problem. Here we introduce a general certification framework that lower bounds ergotropy using only the expectation values of a limited set of arbitrary observables. The method naturally applies in the finite-statistics regime, yielding confidence-certified bounds that explicitly incorporate shot noise. We benchmark our approach on both synthetic data and experimental measurements from an IBM quantum processor. This establishes a robust and experimentally accessible tool for certifying extractable work in realistic quantum settings.

Figures (5)

• Figure 1: Schematic illustrating the problem under consideration. We assume the Hamiltonian $H$ of the system is fully characterized and we want to find a lower bound $\mathcal{E}_{\rm LB}$ for the unknown ergotropy $\mathcal{E}$ under partial information on the true state $\rho$. This is obtained through an informationally incomplete set of measurements, defined by a set of expectation values $\{ \langle \mathcal{O}_i \rangle = {\rm tr}(\rho \mathcal{O}_i)\}_{i\,\in \mathcal{I}}$.
• Figure 2: Schematic representation of the two-step procedure for determining the ergotropy lower bound. In step (i), we select a state compatible with the measurement constraints and construct the corresponding optimal unitary for work extraction. Keeping this unitary fixed, step (ii) determines the worst-case scenario by minimizing the energy difference over the state. This procedure yields the desired lower bound on the ergotropy.
• Figure 3: (a) Certified ergotropy lower bounds as a function of the number $K$ of measured observables (random Pauli strings). We considered the true states \ref{['pure-states']}, GHZ (blue), product (green), and W (red), and XXZ Hamiltonian ($J_1=1$, $J_y=1$, $\Delta=0.5$, $B=0$). (b) Same as in (a) but for the negative-temperature state \ref{['negative-gibbs']} ($\beta=-1$) and MFI Hamiltonian ($B=0.5$, $G=0.5$, $\Delta=1$). For each value of $K$, each realization is generated according to the hierarchical sampling described in the text. The plotted curves show the median over 20 realizations, with shaded regions indicating the interquartile range. Dashed horizontal lines mark the true ergotropies. We considered a system of 5 qubits.
• Figure 4: Probabilistic ergotropy lower bounds as a function of the number $K$ of measured observables (confidence level $1-\delta=0.997$). (a) Effect of finite sampling on the ergotropy lower bound, considering a five-qubit ANNNI model ($J_1=1$, $J_2=-1$, $B=0.5$) and the true state \ref{['superposition']}, superposition of extremal Hamiltonian eigenstates. Curves correspond to different numbers of simulated measurement shots per Pauli observable (see legend). For each $K$, the point shown is the median over 20 realizations using hierarchical sampling. Shaded regions denote the interquartile range across realizations. Dashed horizontal lines indicate the true ergotropies. (b) Ergotropy lower bounds obtained from experimental data for a four-qubit GHZ state and considering XXZ Hamiltonian ($J_1=J_y=1$, $\Delta=0.5$, $B=0$). Here the Pauli constraints are incorporated in the fixed experimental order, without resampling; each expectation value is estimated from $2^{14}$ measurement shots. Each point corresponds to a single deterministic realization, and the optimal unitary is updated only when it yields a strictly tighter bound, ensuring monotonicity of the certified curve. Here, the reference ergotropy (dashed line) is computed on the target GHZ state.
• Figure 5: Ergotropy lower bound as a function of the mean energy using our two-step protocol (blue curve). The considered observables are the Pauli strings appearing in the Pauli decomposition of the Hamiltonian (associated to nonzero components). We also show the Hamiltonian-only ergotropy lower bound of Ref. canzio2025extracting (red) and the energy-basis measurement bound of Eq \ref{['theorem:IC']} (green). We considered a four-qubit system and XXZ Hamiltonian ($J_1=J_y=-1$, $\Delta=-0.5$, $B=0$). The true state is parametrized as a superposition of the lowest- and highest-energy eigenstates, $|\psi(s)\rangle \propto |\epsilon_1\rangle + s|\epsilon_d\rangle$. Notably, our approach certifies the presence of coherent ergotropy at low energies, as highlighted in the inset, where the blue curve lies above the green one.

Theorems & Definitions (4)

• Lemma 1
• proof
• Lemma 2
• proof