Measurement events relative to temporal quantum reference frames

Ladina Hausmann; Alexander Schmidhuber; Esteban Castro-Ruiz

doi:10.22331/q-2025-01-30-1616

Measurement events relative to temporal quantum reference frames

Ladina Hausmann¹, Alexander Schmidhuber¹, and Esteban Castro-Ruiz^1,2,3

¹Institute for Theoretical Physics, ETH Zurich, Switzerland
²Université Paris-Saclay, Inria, CNRS, LMF, 91190 Gif-sur-Yvette, France
³Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria

Published:	2025-01-30, volume 9, page 1616
Editor:	Maximilian Lock
Eprint:	arXiv:2308.10967v4
Doi:	https://doi.org/10.22331/q-2025-01-30-1616
Citation:	Quantum 9, 1616 (2025).

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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.

Popular summary

In quantum mechanics, time enters as an external parameter in the Schrödinger equation. In general relativity, time is only meaningful relative to a physical system — a clock. In 1983, Page and Wootters proposed a framework that includes the clock as a quantum system as well. In their framework, time arises relationally, from the correlations between the clock and the system. Despite its success, the physical interpretation of the framework is still heavily debated. In particular, understanding measurements performed at different times has proved a conceptually thorny issue. Recently, there has been progress. Two independent groups have managed to model consecutive measurements within the Page-Wootters approach. In the "twirled observable" approach, measurements consist of joint observations of the clock and the system that are independent of any external time (see left figure). In the "purified measurement" approach, the interaction between the system, the measurement apparatus, and the clock is modelled explicitly (see right figure). Are the twirled observable and the purified measurement approaches physically equivalent? Here, we show that while the two approaches give the same results when the used clocks are ideal, the two approaches result in radically different pictures when the clocks have finite resources, yielding them no longer ideal. We resolve the reason for the discrepancy, showing that the notion of a "measurement event" differs between one approach and the other.

Along the way, we explore what physics "looks like" in the Page-Wootters framework from the point of view of a clock with limited resources. We point out fundamental limits to the unitarity of time evolution and the sharp definition of time-ordering between events, and argue that these features can be re-established in a world where time is fundamentally discrete.

► BibTeX data

@article{Hausmann2025measurementevents,
  doi = {10.22331/q-2025-01-30-1616},
  url = {https://doi.org/10.22331/q-2025-01-30-1616},
  title = {Measurement events relative to temporal quantum reference frames},
  author = {Hausmann, Ladina and Schmidhuber, Alexander and Castro-Ruiz, Esteban},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1616},
  month = jan,
  year = {2025}
}

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Cited by

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[4] Julian De Vuyst, Stefan Eccles, Philipp A. Höhn, and Josh Kirklin, "Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy", Journal of High Energy Physics 2025 7, 63 (2025).

[5] Максим Чурилов, "Quantum Reference Frames Beyond Ideality: Complete-Order Quotients, Resource Lattices, and Certified Transformations", (2026).

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[9] Carlo Cepollaro and Flaminia Giacomini, "Quantum generalisation of Einstein's equivalence principle can be verified with entangled clocks as quantum reference frames", Classical and Quantum Gravity 41 18, 185009 (2024).

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