Work Extraction from Unknown Quantum Sources

Energy extraction is a central task in thermodynamics. In quantum physics, ergotropy measures the amount of work extractable under cyclic Hamiltonian control. As its full extraction requires perfect knowledge of the initial state, however, it does not characterize the work value of unknown or untrusted quantum sources. Fully characterizing such sources would require quantum tomography, which is prohibitively costly in experiments due to the exponential growth of required measurements and operational limitations. Here, we therefore derive a new notion of ergotropy applicable when nothing is known about the quantum states produced by the source, apart from what can be learned by performing only a single type of coarse-grained measurement. We find that in this case the extracted work is defined by the Boltzmann and observational entropy in cases where the measurement outcomes are, or are not, used in the work extraction, respectively. This notion of ergotropy represents a realistic measure of extractable work, which can be used as the relevant figure of merit to characterize a quantum battery.

Figure 1

Scheme of work extraction from unknown sources. In the limit of the large number of iterations $N$, the ergotropy per state is larger in Stage 1 at the cost of having to store the measurement outcomes. It is given by $W=\mathrm{tr}[H(\rho -{\rho }_{\beta })]$, where ${\rho }_{\beta }$ is a thermal state with temperature $\beta$ implicitly defined by the mean Boltzmann entropy through $S_{\mathrm{vN}}({\rho }_{\beta })=S_{\mathcal{C}}^B$ in Stage 1, and by observational entropy through $S_{\mathrm{vN}}({\rho }_{\beta })=S_{\mathcal{C}}$ in Stage 2, respectively. See Eqs. (11) and (14). The energy $\mathrm{tr}[H\rho ]$ is unknown, but it can be estimated from Eq. (17).

Figure 2

The work extraction protocol in Stage 1 (left) and Stage 2 (right) for $N=1$ (single state; top) and $N=2$ (two states; bottom). $\mathcal{C}$ denotes the measurement.

Figure 3

Observational ergotropy and observational entropy as a function of time for a thermalizing system of four particles. Here, $\mathcal{C}$ is chosen to be a local energy coarse graining [Eq. (16)]; the initial state $|{\psi }_1⟩\otimes |{\psi }_2⟩$, with $|{\psi }_1⟩=|000000⟩$ and $|{\psi }_2⟩=|111100⟩$, evolves with the Hamiltonian given by Eq. (18), with $T=V=1$ and $T^{\prime }=V^{\prime }=0.96$. We choose the energy resolution $\mathrm{\Delta }E=(E_1-E_0)/2$, where $E_0$ and $E_1$ are the ground and first excited state energy, respectively. True ergotropy $W_{\mathcal{C}}^{\infty }(\rho )$ (solid purple) is unknown to experimenters. However, it is possible to estimate the value of $W_{\mathcal{C}}^{\infty }({\rho }_{\mathrm{cg}})$ (red dotted), ${\rho }_{\mathrm{cg}}={\sum }_{E_1,E_2}[p_{E_1{E}_2}(t)/V_{E_1}{V}_{E_2}]P_{E_1}\otimes P_{E_2}$, and be sure that the true value lies within the light-red shaded region, between the red-dashed lines representing $W_{\mathcal{C}}^{\infty }({\rho }_{\mathrm{cg}})\pm 2(\parallel H_{\mathrm{int}}\parallel +\mathrm{\Delta }E)$. We compare this with observational entropy $S_{\mathcal{C}}(\rho )$ (blue long-dashed) called “nonequilibrium thermodynamic entropy” for this particular coarse graining [52]. It is bounded by the sum of initial thermodynamic entropies (dark green dot-dot-dashed) and by the final thermodynamic entropy (green dot-dashed), $S_{\mathrm{th}}\equiv S_{C_E}$, defined as observational entropy with global energy coarse graining with the same resolution $\mathrm{\Delta }E$ [50, 52]. Observational ergotropy is directly defined by observational entropy, and inversely related to it, as per Eqs. (14) and (15).