Sample Complexity of Black Box Work Extraction

TL;DR

This work analyzes how many copies of an unknown quantum state are required to estimate extractable work, quantified by ergotropy. It proves that a single-copy random state yields vanishing ergotropy in the large-environment limit, highlighting a fundamental limitation of black-box work extraction. When multiple copies are available, it introduces a sample-efficient randomized protocol to estimate observational ergotropy with additive accuracy and high probability, with favorable $O(\|H\|^2/\varepsilon^2)$ scaling and only logarithmic dependence on system dimension. The authors also establish robustness bounds for estimation under imperfect information and present a qubit-efficient quantum algorithm for work extraction from local Hamiltonians, advancing practical thermodynamic assessments on near-term quantum devices.

Abstract

Extracting work from a physical system is one of the cornerstones of quantum thermodynamics. The extractable work, as quantified by ergotropy, necessitates a complete description of the quantum system. This is significantly more challenging when the state of the underlying system is unknown, as quantum tomography is extremely inefficient. In this article, we analyze the number of samples of the unknown state required to extract work. With only a single copy of an unknown state, we prove that ergotropy approaches zero in the asymptotic limit, rendering work extraction nearly impossible. In contrast, when multiple copies are available, we quantify the sample complexity required to estimate extractable work, establishing a scaling relationship that balances the desired accuracy with success probability. Our work develops a sample-efficient protocol to assess the utility of unknown states as quantum batteries and opens avenues for estimating thermodynamic quantities using near-term quantum computers.

Figures (6)

• Figure 1: Schematic of the possibilities for extracting work when a quantum system is prepared in some unknown state $\rho$. We show that it is nearly impossible to extract any work with only a single copy of $\rho$. On the other hand, when multiple copies are available, we provide a sample and qubit-efficient randomized quantum algorithm for work extraction that requires only a few samples of $\rho$.
• Figure 2: Ergotropy of random states. We plot the ergotropy of random quantum states (generated by partial tracing $NN'$-dimensional Haar random pure states) for $N=2$ (Fig. \ref{['fg.N2']}), $N=3$ (Fig. \ref{['fg.N3']}), $N=4$ (Fig. \ref{['fg.N4']}), and $N=5$ (Fig. \ref{['fg.N5']}), as a function of $N'$. The Hamiltonian is considered to be $H=\sum_{i=1}^N h_i/N$, where $h_i=\ket{1}\bra{1}$, and $\norm{H}=1$. For each $N$ and $N'$, we sample several random density matrices and compute the ergotropy for each of them (indicated by the cluster of points for each $N, N'$). The red curve in each figure is a plot of the function $\sqrt{N/N'}$, which is an upper bound on ergotropy of random states with high probability as established in Theorem \ref{['thm:impossibility-work']}. We find that our upper bound becomes increasingly tighter as the ratio $N/N'$ decreases.
• Figure 3: Work extraction from translationally invariant Heisenberg XXX model. The Hamiltonian for $n$ spins is given by $H_{n} = - \sum_{j=1}^n\sigma_z^{(j)} \otimes \sigma_z^{(j+1)} - \sum_{j=1}^n\sigma_x^{(j)}$ (with the periodic boundary condition), where $\sigma_z^{(j)}$ and $\sigma_x^{(j)}$ for Pauli-$Z$ and Pauli-$X$ matrices for the $j$th spin, respectively. For $n=3$, let the spins be in the state $\rho = \ket{11}\bra{11}\otimes \mathbb{I}_2/4+ \frac{1}{2}\ket{\psi}\bra{\psi}$, where $\ket{\psi} = \frac{1}{\sqrt{3}}\left(-\ket{011} + \ket{100} + \ket{101}\right)$. We have $\mathrm{Erg}\left(\rho\right)\approx 2.53$ while $\mathrm{Erg}_{\mathrm{obs}}\left(\rho\right)\approx 1.37$. To estimate the latter quantity, we perform projective measurement $\{L_i\}_{i=1}^4$ with $L_1=\ket{00}\bra{00}\otimes \mathbb{I}_4$, $L_2=\ket{01}\bra{01}\otimes \mathbb{I}_4$, $L_3=\ket{10}\bra{10}\otimes \mathbb{I}_4$, and $L_4=\ket{11}\bra{11}\otimes \mathbb{I}_4$. The figure shows how Protocol \ref{['prot1']} approximates $\mathrm{Erg}_{\mathrm{obs}}\left(\rho\right)$ as a function of the number of samples $T$. For $T$ as in Eq. \ref{['eq:samp-work-prot']}, we obtain an $\varepsilon$-accurate estimate of $\mathrm{Erg}_{\mathrm{obs}}\left(\rho\right)$ for $\varepsilon=10^{-2}$, and failure probability $\delta=10^{-4}$.
• Figure 4: The schematic above shows the action of global unitary on measured state to yield a corresponding passive state for $n=3~ (N=8)$ and Hamiltonian $H^{(8)}$ given by Eq. \ref{['eq:toy-ham']}. Let the projective measurement be given by $\{P_i\}_{i=1}^4$, where $P_i=\sum_{b=1}^2 \ket{\psi_i^{(b)}}\bra{\psi_i^{(b)}}$ and $i=1,\cdots,4$. Let $\rho$ be the unknown state and $p_i=\mathrm{Tr}\left[P_i\rho\right]$. Let $\pi$ be the permutation on $4$ objects such that $p_{\pi(i)}\geq p_{\pi(i+1)}$ for all $i\in[3]$. Then we define $U_{\mathrm{gl}}= \ket{\phi_1}\bra{\psi_{\pi(1)}^{(1)}} + \ket{\phi_2}\bra{\psi_{\pi(1)}^{(2)}} + \ket{\phi_3}\bra{\psi_{\pi(2)}^{(1)}}+\ket{\phi_4}\bra{\psi_{\pi(2)}^{(2)}} + \ket{\phi_5}\bra{\psi_{\pi(3)}^{(1)}}+ \ket{\phi_6}\bra{\psi_{\pi(3)}^{(2)}} + \ket{\phi_7}\bra{\psi_{\pi(4)}^{(1)}} + \ket{\phi_8}\bra{\psi_{\pi(4)}^{(2)}}$.
• Figure 5: Observational ergotropy and sample complexity. Consider a state $\rho=(1/3)\sum_{i=1}^3\rho_i$, where $\rho_1$, $\rho_2$, and $\rho_3$ are given in Eq. \ref{['eq:first-ex-rho']}. Figs. (\ref{['fg.samp2']}) and (\ref{['fg.samp1']}) show the convergence of our protocol from main text to the exact value of observational ergotropy, as we increase number of samples, with respect to measurements in entangled (Sec. \ref{['app:case1']}) and product (Sec. \ref{['app:case2']}) bases, respectively.

Theorems & Definitions (15)

• Theorem 1: Impossibility of work extraction from a random state
• Lemma 1
• Lemma 2
• Lemma 3: Concentration of measure phenomenon
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma A6
• proof