The price of keeping time
In plain wordsA device that runs a quantum operation not respecting time translation, with a clock that must come back, pays per use over many runs a work that can exceed the known thermodynamic capacity; one operation preserving thermal equilibrium costs work while its reverse is free.
The result
As the paper states it, in briefWhat a clock costs a device that runs an operation which does not respect time translation.
Questionson the index
- Partly answered Do thermal operations reach the thermodynamic capacity of a channel that is not time-covariant? posed by others
- Partly answered Can thermal operations with a time reference achieve what Gibbs-preserving maps achieve? posed by others
How this was checked
- Blind checks
- Every result passed a blind refutation-first check
- Machine checks
- Machine checks in Python
- In Lean
- Finite and algebraic cores in Lean with Mathlib.
- Lean
- Selected parts.
- Peer review
- Not yet.
The verification label is the one shared by the four companion papers, given in full on The price of time order.
Every check here is internal. How we check results.
Novelty and credit
- Novelty
- Extends named prior work: the capital bound uses the reuse argument of Chiribella, Yang and Renner, and Tajima, Yamaguchi, Takagi and Kuramochi, Chiribella, Yang and Renner, and Tajima, Shiraishi and Saito already bound the resource from errors alone. New, in the paper’s words: the return error enters as a separate parameter, and the bounds fall on the controller and on the heat baths.
- Correction
- Version 1.1.1 corrects one sentence of the batch-law appendix: version 1.1.0 said that a minimizing need not exist, but the infimum is attained. No result depended on it. Corrections
Versionsfive, each with its own DOI
| Version | Date | DOI | Notes |
|---|---|---|---|
| 1.1.1 | 10.5281/zenodo.23790746 | Correction: Version 1.1.0 said, in the batch-law appendix, that a minimizing Y need not exist. The infimum is attained (Rockafellar, Convex Analysis, Theorem 28.2). No result depended on the sentence. PDF hosted here. | |
| 1.1.0 | 10.5281/zenodo.23325723 | ||
| 1.0.2 | 10.5281/zenodo.23323046 | ||
| 1.0.1 | 10.5281/zenodo.23310740 | ||
| 1.0.0 | 10.5281/zenodo.23310633 |
Paper, code and links
papers/keeping-time-v1.1.1.pdf, version 1.1.1, 831,092 bytes, 48 pages.SHA-256 7898df281f88b35aebde231053de8e272f096e503066acbc581716eafca6bd5e
- DOI
10.5281/zenodo.23790746; all versions 10.5281/zenodo.23310632
Corrections
- Version 1.1.1. Version 1.1.0 said, in the batch-law appendix, that a minimizing Y need not exist. The infimum is attained (Rockafellar, Convex Analysis, Theorem 28.2). No result depended on the sentence.doi:10.5281/zenodo.23790746
- The four thermodynamics papers of 11 October cited Thompson et al. with a DOI that did not resolve. Version 1.0.2 of each paper gives the correct DOI, 10.1103/dryx-l5rt.
All corrections, and what counts as one.
Cite
@misc{douglas2026keepingtime,
author = {Douglas, Seth and Mghirbi, Nidhal},
title = {The price of keeping time},
year = {2026},
version = {1.1.1},
doi = {10.5281/zenodo.23790746},
url = {https://doi.org/10.5281/zenodo.23790746}
}