# Single-shot work extraction in quantum thermodynamics revisited

## TL;DR

This paper revisits single-shot quantum thermodynamics, providing a simple, intuitive approach to work extraction and formation, clarifying conceptual inconsistencies, and challenging traditional views on work storage systems.

## Contribution

It introduces a new elementary majorization-based method, offers a graphical interpretation of work quantities, and resolves conceptual issues regarding work storage in quantum thermodynamics.

## Key findings

- Maximum extractable work and free energies are graphically interpreted.
- An additional heat contribution is identified in general work extraction scenarios.
- The concept of work storage as a simple weight is challenged and refined.

## Abstract

We revisit the problem of work extraction from a system in contact with a heat bath to a work storage system, and the reverse problem of state formation from a thermal system state in single-shot quantum thermodynamics. A physically intuitive and mathematically simple approach using only elementary majorization theory and matrix analysis is developed, and a graphical interpretation of the maximum extractable work, minimum work cost of formation, and corresponding single-shot free energies is presented. This approach provides a bridge between two previous methods based respectively on the concept of thermomajorization and a comparison of subspace dimensions. In addition, a conceptual inconsistency with regard to general work extraction involving transitions between multiple energy levels of the work storage system is clarified and resolved. It is shown that an additional contribution to the maximum extractable work in those general cases should be interpreted not as work extracted from the system, but as heat transferred from the heat bath. Indeed, the additional contribution is an artifact of a work storage system (essentially a suspended "weight" that can be raised or lowered) that does not truly distinguish work from heat. The result calls into question the common concept that a work storage system in quantum thermodynamics is simply the quantum version of a suspended weight in classical thermodynamics.