Revision 1

Accepted · first revision · record

Title: Maximal precision of a quantum clock per unit entropy
Plain: A clock must waste energy to tick, and classical clocks become steadier only in proportion to the waste. A 2025 theory paper showed quantum clocks can become steadier exponentially faster, so the ultimate limit is open.
Precise: For an autonomous clock driven by thermal baths, let $N = \mu^2 / \sigma^2$ be the precision of the waiting time between ticks and $\Sigma$ the entropy produced per tick ($k_B = 1$); classical Markov clocks obey $N \le \Sigma / 2$. Meier et al. (Nature Physics 2025, arXiv:2407.07948) found a coherent many-body clock with $N$ growing exponentially in $\Sigma$. Determine the optimal $N(\Sigma, d)$ for clocks of Hilbert-space dimension $d$, and demonstrate $N > \Sigma / 2$ experimentally.
Settled by: A proved upper bound on $N$ matched by a construction, plus an implementation exceeding $N = \Sigma / 2$.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: open
Status note: Exponential scaling of $N$ with $\Sigma$ was shown theoretically (Meier et al., Nature Physics 2025); a September 2026 superconducting experiment reached a current TUR ratio $1.71 \pm 0.17$ below the classical value 2 (arXiv:2609.37149), but no clock with $N > \Sigma/2$ has been reported; the optimal scaling is unknown.
Posed since: 2017