# Second law of thermodynamics with quantum memory

## TL;DR

This paper introduces a quantum thermodynamic framework involving multiple heat reservoirs, ancillary systems, and quantum memory, deriving new bounds on work extraction influenced by quantum correlations and the second law.

## Contribution

It presents a novel inequality linking quantum mutual information and entropy changes, establishing a lower bound on work gain that incorporates quantum entanglement effects.

## Key findings

- Maximum work from two engines exceeds a bound related to quantum entanglement.
- Derived a new inequality constraining work gain based on quantum information.
- Identified conditions under which the second law's traditional bounds are modified.

## Abstract

We design a heat engine with multi-heat-reservoir, ancillary system and quantum memory. We then derive an inequality related with the second law of thermodynamics, and give a new limitation about the work gain from the engine by analyzing the entropy change and quantum mutual information change during the process. In addition and remarkably, by combination of two independent engines and with the help of the entropic uncertainty relation with quantum memory, we find that the total maximum work gained from those two heat engines should be larger than a quantity related with quantum entanglement between the ancillary state and the quantum memory. This result provides a lower bound for the maximum work extracted, in contrast with the upper bound in the conventional second law of thermodynamics. However, the validity of this inequality depends on whether the maximum work can achieve the upper bound.