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Thermalization of finite complexity and its application to heat bath algorithmic cooling

Xueyuan Hu, Valerio Scarani

TL;DR

The paper develops a finite-complexity, collision-model framework for thermal operations and uses it to study cooling below bath temperature. It shows that single collisions cannot cool below the bath if the initial system state has a well-defined effective temperature, but cooling is possible when the state lacks such an effective temperature; it provides explicit examples and an iterative protocol for sub-bath cooling without a machine in certain finite-dimensional settings. By introducing a single-qubit machine, the authors demonstrate improved cooling limits and positive energy efficiency, with asymptotic cooling reaching $\beta^*=4\beta$ under unrestricted CoP, and they connect these results to a unified HBAC perspective under finite reservoirs. Overall, the work extends HBAC concepts to finite-resource thermodynamics, clarifying the role of resource constraints, effective temperature, and minimal auxiliary systems in achieving sub-bath cooling and efficient entropy management.

Abstract

We introduce a class of thermal operations based on the collision model, where the system sequentially interacts with uncorrelated bath molecules via energy-preserving unitaries. To ensure finite complexity, each molecule is constrained to be no larger than the system. We identify a necessary condition for cooling below the bath temperature via a single collision: the system must initially lack a well-defined effective temperature, even a negative one. By constructing a iterative protocol, we demonstrate that sub-bath cooling is achievable without a machine under these restricted thermal operations. Moreover, introducing a qubit machine further enhances both the cooling limit and energy efficiency. These findings contribute to the broader study of cooling with finite resources.

Thermalization of finite complexity and its application to heat bath algorithmic cooling

TL;DR

The paper develops a finite-complexity, collision-model framework for thermal operations and uses it to study cooling below bath temperature. It shows that single collisions cannot cool below the bath if the initial system state has a well-defined effective temperature, but cooling is possible when the state lacks such an effective temperature; it provides explicit examples and an iterative protocol for sub-bath cooling without a machine in certain finite-dimensional settings. By introducing a single-qubit machine, the authors demonstrate improved cooling limits and positive energy efficiency, with asymptotic cooling reaching under unrestricted CoP, and they connect these results to a unified HBAC perspective under finite reservoirs. Overall, the work extends HBAC concepts to finite-resource thermodynamics, clarifying the role of resource constraints, effective temperature, and minimal auxiliary systems in achieving sub-bath cooling and efficient entropy management.

Abstract

We introduce a class of thermal operations based on the collision model, where the system sequentially interacts with uncorrelated bath molecules via energy-preserving unitaries. To ensure finite complexity, each molecule is constrained to be no larger than the system. We identify a necessary condition for cooling below the bath temperature via a single collision: the system must initially lack a well-defined effective temperature, even a negative one. By constructing a iterative protocol, we demonstrate that sub-bath cooling is achievable without a machine under these restricted thermal operations. Moreover, introducing a qubit machine further enhances both the cooling limit and energy efficiency. These findings contribute to the broader study of cooling with finite resources.
Paper Structure (28 sections, 3 theorems, 92 equations, 4 figures, 1 table)

This paper contains 28 sections, 3 theorems, 92 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. thermal operations based on the collision model
  3. definition
  4. Comparison with Markov thermal processes
  5. Both the system and the molecule are a single qubit
  6. Beyond two qubits
  7. Single-collision cooling
  8. No-go theorems
  9. Possibility of cooling below the bath's temperature
  10. Iterative cooling protocols
  11. A unified view of Heat-bath Algorithmic Cooling (HBAC)
  12. Protocols in the absence of a machine
  13. Protocols with a qubit machine
  14. Improved energy efficiency
  15. Improved cooling limit
  16. ...and 13 more sections

Key Result

Theorem 1

For any qubit state $\rho$ and number of collisions $N$, and Besides, the convex hull of $\mathbb{CO}_{\mathcal{C}_\beta(H^{(2)},H^{(2)},1)}(\rho)$ equals to $\mathbb{CO}_{\mathrm{MTO}_\beta}(\rho)$.

Figures (4)

  • Figure 1: The configuration of one round in HBAC under coherent control. The target system, the machine and the heat bath are labeled as $S$, $M$, and $R$, respectively. The energy-non-preserving unitary $V$ is the recharging routine, and $\Gamma$, realized by interacting with a heat bath via a energy-preserving unitary $U$, is the thermalizing routine.
  • Figure 2: The set of reachable states from initial state $\vec{p}=(\tau_1,\tau_0,\tau_2)$. Here we choose $q=0.5$.
  • Figure 3: The work consumption and the energy reduction for both Protocol I (Panel (a)) and Protocol II (Panel (b)). Here we choose $q=0.3$ and $E=1$.
  • Figure 4: Comparison between the cumulative CoP of Protocols I and II. Here we choose $q=0.3$.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1