Appendix 4—figure 1. Mutual information $MI(\overrightarrow{c},t)$ between clock state $\overrightarrow{c}$ and time ${\displaystyle t}$ is only affected by the variance of the clock state distribution ${\displaystyle p(\overrightarrow{c}|t)}$ at a given time ${\displaystyle t}$ along the direction of motion and not orthogonal to it.

In this toy example, we assume the distribution $p(\overrightarrow{c}|t)$ to be supported on a rectangle of size ${\displaystyle a_x}$ and ${\displaystyle a_y}$ in a 2d clock state space. The clock state moves at a speed ${\displaystyle u}$ in the x-direction. Time telling quality is affected by how much the population at different times overlap with each other. Consequently, clocks with large ${\displaystyle a_x}$ and small ${\displaystyle a_y}$ (bottom) have lower mutual information ${\displaystyle MI(\overrightarrow{c},t)}$ relative to clocks with small ${\displaystyle a_x}$ and large ${\displaystyle a_y}$ (top). Consequently, we use the population variance along the direction of motion as an instantaneous measure of time-telling ability in the paper.