Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Time-cost-error trade-off relation in thermodynamics: The third law and beyond

Tan Van Vu, Keiji Saito

TL;DR

This work establishes a universal, information-theoretic three-way trade-off between time, cost, and error for thermodynamic tasks aimed at suppressing undesired state probabilities, encapsulated by the bound $\tau\mathcal{C}\varepsilon_\tau \ge 1-\eta$. By introducing separated states, the authors unify classical and quantum settings across erasure, cooling, copying, and kinetic proofreading, and show that exact zero error is unattainable with finite resources, generalizing the unattainability aspect of the third law. The framework combines a kinetic contribution (max escape rate) with a thermodynamic contribution (entropy production), yielding a thermokinetic cost $\mathcal{C}=\omega\Phi(\overline{\sigma})$ that governs feasible state transformations in finite time. They extend the bound to quantum regimes, deriving analogous results for Markovian Lindblad dynamics and for non-Markovian dynamics with finite-size reservoirs, including a no-go for exact classical copying and a quantitative bound on cooling temperatures. Demonstrations on classical bits, qubits, and finite reservoirs illustrate the bound’s tightness and its broad applicability to fundamental limits in nonequilibrium thermodynamics, with implications for future exploration of continuous-variable systems, measurement, and reaction networks.

Abstract

Elucidating fundamental limitations inherent in physical systems is a central subject in physics. For important thermodynamic operations such as information erasure, cooling, and copying, resources like time and energetic cost must be expended to achieve the desired outcome within a predetermined error margin. In the context of cooling, the unattainability principle of the third law of thermodynamics asserts that infinite "resources" are needed to reach absolute zero. However, the precise identification of relevant resources and how they jointly constrain achievable error remains unclear within the frameworks of stochastic and quantum thermodynamics. In this work, we introduce the concept of separated states, which consist of fully unoccupied and occupied states, and formulate the corresponding thermokinetic cost and error, thereby establishing a unifying framework for a broad class of thermodynamic operations. We then uncover a three-way trade-off relation between time, cost, and error for thermodynamic operations aimed at creating separated states, simply expressed as $τ\mathcal{C}\varepsilon_τ\ge 1-η$. This fundamental relation is applicable to diverse thermodynamic operations, including information erasure, cooling, and copying. It provides a profound quantification of the unattainability principle in the third law of thermodynamics in a general form. Building upon this relation, we explore the quantitative limitations governing cooling operations, the preparation of separated states, and a no-go theorem for exact classical copying. Furthermore, we extend these findings to the quantum regime, encompassing both Markovian and non-Markovian dynamics. Specifically, within Lindblad dynamics, we derive a similar three-way trade-off relation that quantifies the cost of achieving a pure state with a given error.

Time-cost-error trade-off relation in thermodynamics: The third law and beyond

TL;DR

This work establishes a universal, information-theoretic three-way trade-off between time, cost, and error for thermodynamic tasks aimed at suppressing undesired state probabilities, encapsulated by the bound . By introducing separated states, the authors unify classical and quantum settings across erasure, cooling, copying, and kinetic proofreading, and show that exact zero error is unattainable with finite resources, generalizing the unattainability aspect of the third law. The framework combines a kinetic contribution (max escape rate) with a thermodynamic contribution (entropy production), yielding a thermokinetic cost that governs feasible state transformations in finite time. They extend the bound to quantum regimes, deriving analogous results for Markovian Lindblad dynamics and for non-Markovian dynamics with finite-size reservoirs, including a no-go for exact classical copying and a quantitative bound on cooling temperatures. Demonstrations on classical bits, qubits, and finite reservoirs illustrate the bound’s tightness and its broad applicability to fundamental limits in nonequilibrium thermodynamics, with implications for future exploration of continuous-variable systems, measurement, and reaction networks.

Abstract

Elucidating fundamental limitations inherent in physical systems is a central subject in physics. For important thermodynamic operations such as information erasure, cooling, and copying, resources like time and energetic cost must be expended to achieve the desired outcome within a predetermined error margin. In the context of cooling, the unattainability principle of the third law of thermodynamics asserts that infinite "resources" are needed to reach absolute zero. However, the precise identification of relevant resources and how they jointly constrain achievable error remains unclear within the frameworks of stochastic and quantum thermodynamics. In this work, we introduce the concept of separated states, which consist of fully unoccupied and occupied states, and formulate the corresponding thermokinetic cost and error, thereby establishing a unifying framework for a broad class of thermodynamic operations. We then uncover a three-way trade-off relation between time, cost, and error for thermodynamic operations aimed at creating separated states, simply expressed as . This fundamental relation is applicable to diverse thermodynamic operations, including information erasure, cooling, and copying. It provides a profound quantification of the unattainability principle in the third law of thermodynamics in a general form. Building upon this relation, we explore the quantitative limitations governing cooling operations, the preparation of separated states, and a no-go theorem for exact classical copying. Furthermore, we extend these findings to the quantum regime, encompassing both Markovian and non-Markovian dynamics. Specifically, within Lindblad dynamics, we derive a similar three-way trade-off relation that quantifies the cost of achieving a pure state with a given error.
Paper Structure (32 sections, 4 theorems, 151 equations, 7 figures)

This paper contains 32 sections, 4 theorems, 151 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Summary of results
  4. Relevant literature
  5. Setup
  6. Separated states
  7. Dynamics, error, and cost
  8. Results
  9. Time-cost-error trade-off relation
  10. More on the unattainability principle in the third law of thermodynamics
  11. Limitation in preparation of separated states
  12. No-go theorem of exact classical copying
  13. Hardness of copying
  14. Generalization to quantum cases
  15. Markovian dynamics
  16. ...and 17 more sections

Key Result

Proposition 1

Given two probability distributions $\ket{p}=[p_1,\dots,p_d]^\top$ and $\ket{q}=[q_1,\dots,q_d]^\top$, let $\ket{p^{\le}}$ and $\ket{q^{\le}}$ be their corresponding sorted distributions (i.e., $p_1^\le\le\dots\le p_d^\le$ and $q_1^\le\le\dots\le q_d^\le$). Then, the following inequality holds true:

Figures (7)

  • Figure 1: In thermodynamic processes that reduce the observation probability to zero, a trade-off relation between time, cost, and error naturally arises. Examples include: (i) erasing information from bits (where the probability of all microstates corresponding to the logical state '$1$’ is reduced to zero), (ii) cooling toward the ground state (where the probability of excited states is reduced to zero), (iii) copying (where the probability of states differing between the source and target is reduced to zero), and (iv) kinetic proofreading (where the probability of pathways producing incorrect products is reduced to zero).
  • Figure 2: The schematic illustrates the preparation of a separated state in Markov jump processes. After finite-time operations, the final distribution is approximately separated, i.e., the total probability of finding the system in the undesired states belonging to the set $\mathfrak{u}$ is nearly zero.
  • Figure 3: (a) Schematic illustration of the kinetic and thermodynamic contributions $\omega_t$ and $\overline{\sigma}_t$. (b) The behavior of the cost term as the function of the kinetic ($\omega$) and thermodynamic ($\overline{\sigma}$) contributions. The cost term vanishes when either contribution is zero and increases as a function of the other contribution when one is fixed. (c) Visualization of the function $\Phi(x)$ and its upper bound $\Theta(x)$, which are both monotonically increasing functions.
  • Figure 4: Numerical illustration of the bound \ref{['eq:temp.lb']} in cooling a classical bit. The energy gap is modulated as $\Delta_g(t)=0.1+2.9t/\tau$, with transition rates satisfying $w_{12}(t)+w_{21}(t)=5$. The duration $\tau$ is varied, while the bath inverse temperature is fixed at $\beta=1$. The effective temperature $T$, shown as the solid line, is consistently bounded from below by both $T_{\rm lb}$ and $T_{\rm{lb},1}$, represented by the dashed and dash-dotted lines, respectively.
  • Figure 5: An example of preparing a separated state using a particle in a box. This scheme reduces the accessible state space during the operation, and therefore, it is not included in our setup.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof