# Relativistic Limits on the Discretization and Temporal Resolution of a Quantum Clock

## TL;DR

The paper explores how relativity affects the precision and resolution of a quantum clock's time measurements.

## Contribution

The paper introduces relativistic limits on quantum clock discretization and temporal resolution using POVMs.

## Key findings

- A limit on time eigenvalue discretization is found when the Hamiltonian has an equally spaced spectrum.
- Increasing time states reduces the bound on minimum time quanta and allows treating clock values as continuous.
- A temporal resolution limit arises when using the clock in a relativistic spacetime measurement framework.

## Abstract

We provide a brief discussion regarding relativistic limits on the discretization and temporal resolution of time values in a quantum clock. Our clock is characterized by a time observable chosen to be the complement of a bounded and discrete Hamiltonian that can have an equally spaced or a generic spectrum. In the first case, the time observable can be described by a Hermitian operator, and we find a limit in the discretization for the time eigenvalues. Nevertheless, in both cases, the time observable can be described by a POVM, and, by increasing the number of time states, we show how the bound on the minimum time quantum can be reduced and identify the conditions under which the clock values can be treated as continuous. Finally, we find a limit for the temporal resolution of our time observable when the clock is used (together with light signals) in a relativistic framework for the measurement of spacetime distances.