Measurement events relative to temporal quantum reference frames

TL;DR

This paper compares two approaches to the Page-Wootters formalism for quantum time, revealing differences in their operational meaning, especially for non-ideal clocks, and discusses implications for unitarity and temporal order.

Contribution

It clarifies the operational distinctions between twirled observable and purified measurement approaches in quantum reference frames, especially for finite-resource clocks.

Findings

Both approaches agree for ideal clocks.

Purified measurement yields non-unitary, time non-local evolution for non-ideal clocks.

Discrepancies suggest fundamental limitations in defining temporal order operationally.

Abstract

The Page-Wootters formalism is a proposal for reconciling the background-dependent, quantum-mechanical notion of time with the background independence of general relativity. However, the physical meaning of this framework remains debated. In this work, we compare two consistent approaches to the Page-Wootters formalism to clarify the operational meaning of evolution and measurements with respect to a temporal quantum reference frame. The so-called "twirled observable" approach implements measurements as operators that are invariant with respect to the Hamiltonian constraint. The "purified measurement" approach instead models measurements dynamically by modifying the constraint itself. While both approaches agree in the limit of ideal clocks, a natural generalization of the purified measurement approach to the case of non-ideal, finite-resource clocks yields a radically different picture. We discuss the physical origin of this discrepancy and argue that these approaches describe operationally distinct situations. Moreover, we show that, for non-ideal clocks, the purified measurement approach yields a time non-local evolution equation, which can lead to non-unitary evolution. Moreover, it implies a fundamental limitation to the operational definition of the temporal order of events. Nevertheless, unitarity and definite temporal order can be restored if we assume that time is discrete.