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

TL;DR

This paper establishes a fundamental three-way trade-off relation between time, cost, and error in thermodynamic operations, providing a unified framework that extends the third law of thermodynamics to quantum and classical regimes.

Contribution

It introduces the concept of separated states and formulates a universal trade-off relation applicable to various thermodynamic processes, advancing understanding of fundamental limits in thermodynamics.

Findings

Derives a universal time-cost-error trade-off relation for thermodynamic operations.

Quantifies the unattainability principle of the third law in a general framework.

Extends the trade-off relation to quantum dynamics, including non-Markovian processes.

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 $\tau\mathcal{C}\varepsilon_{\tau}\ge 1-\eta$. 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.