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Optimal measurement-based quantum thermal machines in a finite-size system

Chinonso Onah, Obinna Uzoh, Obinna Abah

Abstract

We present a measurement-based quantum thermal machine that extracts work from the back-action of generalized quantum measurements whose working medium is a coupled two-level quantum system. Specifically, we derive universal optimization criteria for a three-stroke measurement-based engine cycle with coupled two-level system of Ising-like interaction as a working medium. Furthermore, we present two numerical algorithms to optimize the engine work extraction and enhance its performance. Our numerical results demonstrate: (i) efficiency peaks in the projective-measurement limit; (ii) symmetry breaking (detuning or weak coupling) enlarges the exploitable energy gap; and (iii) performance remains robust ($>50\%$ of optimum) under $\sim\!10^\circ$ feedback-pulse errors. The framework is platform-agnostic and directly implementable with current superconducting, trapped-ion, or NMR technologies, providing a concrete route to scalable, measurement-powered quantum thermal machines.

Optimal measurement-based quantum thermal machines in a finite-size system

Abstract

We present a measurement-based quantum thermal machine that extracts work from the back-action of generalized quantum measurements whose working medium is a coupled two-level quantum system. Specifically, we derive universal optimization criteria for a three-stroke measurement-based engine cycle with coupled two-level system of Ising-like interaction as a working medium. Furthermore, we present two numerical algorithms to optimize the engine work extraction and enhance its performance. Our numerical results demonstrate: (i) efficiency peaks in the projective-measurement limit; (ii) symmetry breaking (detuning or weak coupling) enlarges the exploitable energy gap; and (iii) performance remains robust ( of optimum) under feedback-pulse errors. The framework is platform-agnostic and directly implementable with current superconducting, trapped-ion, or NMR technologies, providing a concrete route to scalable, measurement-powered quantum thermal machines.

Paper Structure

This paper contains 20 sections, 1 theorem, 91 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Measurement-based quantum engine
  3. Constructing the feedback Hamiltonian
  4. Optimization problem: Optimal feedback
  5. Examples
  6. Single qubit measurement-based engine
  7. Two-body measurement-based engine
  8. Algorithms for the angles optimisation
  9. Hybrid local–global search (Alg. \ref{['alg:extended_optimal_feedback']}).
  10. Pure grid-sweep baseline (Alg. \ref{['alg:gridsearch_feedback']}).
  11. Conclusions
  12. Optimal feedback of a measurement-based engine
  13. Local transformation of single-body Pauli matrices
  14. Local transformation of two-body Pauli matrices
  15. Global transformation of Pauli matrices
  16. ...and 5 more sections

Key Result

Theorem 1

Consider a quantum measurement-based machine of finite size consisting of $N$ coupled two-level systems described by the Ising-like Hamiltonian of the form Padurariu2010Pita-vidal2024: where $\mathbf{I}_{\otimes N}$ is the identity matrix of the $N$-space, $\epsilon_j$ is the energy of qubit $j$, $\sigma_z^{j}$ is the Pauli-$Z$ operator on qubit $j$, and $\Delta_{zz}$ denotes the coupling between

Figures (6)

  • Figure 1: Left panel: Schematic illustration of feedback driven measurement-based quantum machine with a two-level system. Right panel: Thermal device working substance composing of a six coupled two-level system. Each two-level system (skyblue circle) is initially aligned along the $z$-axis and the gray dashed lines represent couplings between the two-level systems. Red arrows indicate weak $\sigma_x$ measurements by detectors $D_i$, and green arcs labeled $\theta_i$ represent local feedback (rotations around $y$-axis) applied based on measurement outcomes.
  • Figure 2: Left to right: Optimal feedback angle, maximum work extraction, and efficiency as a function of measurement strength parameter for a single qubit measurement-based engine. The inset is the energy functional as a function of optimal angle with the red-cross corresponding to the optimal angle deduced from the grid search algorithm.
  • Figure 3: Energy spectrum and gap of the two-coupled quantum spin system with $\epsilon_1\!=\!\epsilon_2\!=\!0.5$ as a function of the interaction strength $\Delta_z$. Energy gap $\Delta E$ as a function of $\Delta_z$. For $\Delta_z\le-0.25$ (ferromagnetic branch) the gap saturates at $\Delta E=1$; inside the window $-0.25\le\Delta_z\le0.25$ it varies linearly; for $\Delta_z\ge0.25$ (antiferromagnetic branch) it grows unbounded as $\Delta E\simeq 2\Delta_z$.
  • Figure 4: Efficiency as a function of measurement strength for two qubits measurement-based engine at optimal feedback angles for three engine configuration, (a) $\Delta\!=\!0.0$ and (b) $\Delta\!=\!-0.2$. A single-qubit engine ($n=1$) is represented by purple dotted-dashed curve, a two-qubit engine with a single detector ($n=2:D_1$) is teal dashed curve, and a two-qubit engine with double detectors ($n=2:D_1$ and $D_2$) is denoted with pink solid curve. Insets: Output work as function of measurement strength. Parameters are $\epsilon_1\!=\!-0.05$ and $\epsilon_2\!=\!-0.10$.
  • Figure 5: Efficiency $\eta$ as a function of detuning $\xi$ for the two coupled qubits measurement-based engine at fixed qubit coupling $\Delta\!=\!-0.20$ and measurement strength $\kappa\!=\!0.10$. The red curve shows the case of one detector while the green curve correspond to the case of using two detectors ($D_1$ and $D_2$).
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1: Stationarity Conditions for the Optimal Feedback Angles
  • proof