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A Demon that remembers: An agential approach towards quantum thermodynamics of temporal correlations

Ruo Cheng Huang

Abstract

This thesis develops a decision-theoretic framework for extracting thermodynamic work from temporal correlations in quantum systems. We model a classical agent -- lacking quantum memory -- performing adaptive work extraction through continuous inference and decision-making under uncertainty. By introducing $ρ^*$-ideal protocols, we demonstrate that exploiting memory effects allows adaptive strategies to surpass non-adaptive bounds. We formalize this via the Time-Ordered Free Energy (TOFE), a novel upper bound for causal, adaptive operations that reveals a thermodynamic gap linked to adaptive ordered discord. Additionally, we tackle work extraction from unknown sources using reinforcement learning. By adapting multi-armed bandit algorithms, we show an agent can simultaneously learn an unknown i.i.d. quantum state and extract work, achieving polylogarithmic cumulative dissipation that significantly outperforms standard tomography. Overall, this work lays the foundation for predictive and learning-based quantum thermodynamics.

A Demon that remembers: An agential approach towards quantum thermodynamics of temporal correlations

Abstract

This thesis develops a decision-theoretic framework for extracting thermodynamic work from temporal correlations in quantum systems. We model a classical agent -- lacking quantum memory -- performing adaptive work extraction through continuous inference and decision-making under uncertainty. By introducing -ideal protocols, we demonstrate that exploiting memory effects allows adaptive strategies to surpass non-adaptive bounds. We formalize this via the Time-Ordered Free Energy (TOFE), a novel upper bound for causal, adaptive operations that reveals a thermodynamic gap linked to adaptive ordered discord. Additionally, we tackle work extraction from unknown sources using reinforcement learning. By adapting multi-armed bandit algorithms, we show an agent can simultaneously learn an unknown i.i.d. quantum state and extract work, achieving polylogarithmic cumulative dissipation that significantly outperforms standard tomography. Overall, this work lays the foundation for predictive and learning-based quantum thermodynamics.

Paper Structure

This paper contains 90 sections, 15 theorems, 312 equations, 25 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Maxwell's Demon
  3. Time-varying resources
  4. Agential approach
  5. Learning and Update of Belief
  6. Belief about the future distribution
  7. Belief about process structure
  8. Major Contributions
  9. Outline of the Thesis
  10. Theoretical background
  11. Quantum Information
  12. Multi-partite states
  13. Information-theoretic quantities
  14. Quantum Thermodynamics
  15. Thermal operation
  16. ...and 75 more sections

Key Result

Theorem 3.1

Each $\rho^*$-ideal work-extraction protocol that thermalizes all $d$-dimensional quantum states and exhibits at most $d$ distinct extracted-work values. These extracted-work values can be expressed, in terms of the ideal input's spectral decomposition $\rho^* = \sum_{n} \lambda_n \ket{\lambda_n} \b where $\mathcal{F}_{\text{eq}}$ is the equilibrium free energy and $H$ is the Hamiltonian of the sy

Figures (25)

  • Figure 1: Illustration of Szilard’s Engine. The box has an initial volume $V=\alpha L$. After isothermal expansion of the single particle within the box, the attached weight gains energy $\Delta U=k_BT\ln2$. However, in order for the demon to acquire knowledge of the particle’s position again, a minimum energy cost of $\Delta W=k_BT\ln2$ must be expended. The net energy change is at most zero, thereby preserving the second law of thermodynamics.
  • Figure 2: Depiction of temporally correlated systems. Panel (a) shows a sequence of boxes that remains invariant over time; no feedback control is required to extract work from such a sequence. Panel (b) illustrates a sequence with an alternating pattern, where the agent must retain information about the preceding box to extract work effectively. Panel (c) depicts a system with more complex temporal correlations, requiring a larger memory or inference mechanism for optimal work extraction.
  • Figure 3: Diagrammatic representation of qubits on a Bloch sphere. $\theta$ is the angle from the $Z$-axis and $\phi$ is the angle measured from the $X$-axis. Pure states reside on the surface while mixed states occupy the interior of the sphere.
  • Figure 4: Venn diagram of entropic quantities. The blue and red circles represent the entropies of random variables $X$ and $Y$, respectively. Their union corresponds to the joint entropy of $H(X,Y)$, and the intersection represents the mutual information $I(X;Y)$.
  • Figure 5: A circuit diagram representation of the work extraction protocol. The system $Q$ represents the system where free energy is drawn, $B$ is a battery, and $R$ represents a thermal reservoir as an ancillary system. The protocol aims to transform $\rho_Q$ to a thermal state $\gamma_Q$ with the help of the thermal states from the reservoir; the free energy lost in system $Q$ will be balanced by the increase in energy of the battery $B$.
  • ...and 20 more figures

Theorems & Definitions (29)

  • Definition 2.1: Quantum State
  • Definition 2.2: Hidden Markov Model
  • Definition 2.3: Causal State
  • Definition 2.4: Unifilarity
  • Definition 2.5: $\epsilon$-Machine
  • Definition 3.1: $\rho^*$-ideal work extraction protocol
  • Theorem 3.1: The Work Distribution of $\rho^*$-Ideal Protocols
  • proof
  • Theorem 4.1: Optimal Recursive Belief Update
  • Proposition 4.1
  • ...and 19 more