The price of keeping time

Abstract

A device that runs a quantum operation which does not respect time translation needs its own clock. This paper asks what owning that clock costs. Gates. The clock is paid for in capital, in the energy range of the heat baths, and in an energy that comes back only as a lottery, each growing faster than the number of uses as the error vanishes. The batch law. With a returned clock, the batch work per use of every channel is a single-letter quantity B. It equals the thermodynamic capacity when the channel's environment can carry its energy change, and can exceed it otherwise, even for channels of capacity zero. How large the clock must be. At fixed error, the same condition sets how the clock's capital spread must grow: not at all for covariant channels, like the square root of the number of uses when the environment can carry the energy change (by a construction that does not attain the optimal work), and at least linearly when it cannot. A controller that comes back only in state. Counting the controller's free energy, no time reference does better in batch than a returned clock. A controller that comes back close in state but does not refund its free energy, as an embezzling state does, can lower the battery's work to the capacity or below it, but only by paying the difference from its own free energy, like a second battery; to pay a fixed amount per use within return error η, its energy range plus kT times the logarithm of its dimension must be at least of order the number of uses divided by η, and a classical ladder of that size suffices. Time reversal. In batch with a returned clock, a Gibbs-preserving qutrit channel costs 0.1547 kT per use, while its Petz reverse costs nothing. Noise horizon. Any fixed amount of full-rank thermal noise makes both directions free, but for a finite run what matters is the expected number of noisy uses. If it tends to zero, the original work rate remains; if it diverges, sublinear work is possible; if it tends to a positive finite value, the work rate stays positive, between two envelopes whose equality is open. All rates here are optimal asymptotic work rates at vanishing error over the complete interaction, with hardware allowed to grow; where a controller comes back only approximately, its actual mean energy and entropy are also refunded up to sublinear changes, except in the free-energy ledger of a controller that comes back only in state, which counts its unrefunded free energy instead. This record archives the manuscript, its TeX source and its machine checks (Python, and Lean with Mathlib). The manuscript is the authority for exact statements and hypotheses; the verification notes list what is checked by machine and what is proved only in the text. How the work was done, including the use of AI models, is described in the manuscript. This paper is one of four on the thermodynamics of time order: Only quantum channels save work by waiting (doi:10.5281/zenodo.23310624; https://github.com/Apsiape/save-work-by-waiting); The work cost of answering in real time (doi:10.5281/zenodo.23310628; https://github.com/Apsiape/answering-in-real-time); The price of time order (doi:10.5281/zenodo.23310622; https://github.com/Apsiape/price-of-time-order). Scoped licenses: the manuscript, PDF and TeX source are licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0). The software and machine-readable verification artifacts are additionally available under the MIT License. See RIGHTS.md in the archive. The aggregate Other (Open) category represents this mixed-license package.