# Efficiency versus Speed in Quantum Heat Engines: Rigorous Constraint from Lieb-Robinson Bound

## TL;DR

This paper establishes a rigorous fundamental limit on the efficiency of quantum heat engines operating at finite power, showing that achieving Carnot efficiency at finite speed is impossible, even with non-Markovian dynamics.

## Contribution

It introduces a novel inequality based on Lieb-Robinson bounds that constrains quantum heat engine efficiency and speed, extending the analysis to non-Markovian processes.

## Key findings

- Finite power engines cannot reach Carnot efficiency.
- Most energy remains localized near the engine after operation.
- The inequality applies to non-Markovian quantum dynamics.

## Abstract

A long standing open problem whether a heat engine with finite power achieves the Carnot efficiency is investigated. We rigorously prove a general trade-off inequality on thermodynamic efficiency and time interval of a cyclic process with quantum heat engines. In a first step, employing the Lieb-Robinson bound we establish an inequality on the change in a local observable caused by an operation far from support of the local observable. This inequality provides a rigorous characterization of the following intuitive picture that most of the energy released from the engine to the cold bath remains near the engine when the cyclic process is finished. Using the above description, we finally prove an upper bound on efficiency with the aid of quantum information geometry. In addition, since our inequality falls down into the conventional second law of thermodynamics in Markovian limit, we adopt a completely different treatment which is developed in the context of classical stochastic processes. Our result generally excludes the possibility of a process with finite speed at the Carnot efficiency in quantum heat engines. In particular, the obtained constraint covers engines evolving with non-Markovian dynamics, which almost all previous studies on this topics fail to address.