Conservation Laws For Every Quantum Measurement Outcome

Daniel Collins; Sandu Popescu

doi:10.22331/q-2025-07-29-1815

Conservation Laws For Every Quantum Measurement Outcome

H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL

Published:	2025-07-29, volume 9, page 1815
Editor:	Leon Loveridge
Eprint:	arXiv:2404.18621v3
Doi:	https://doi.org/10.22331/q-2025-07-29-1815
Citation:	Quantum 9, 1815 (2025).

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Abstract

In the paradigmatic example of quantum measurements, whenever one measures a system which starts in a superposition of two states of a conserved quantity, it jumps to one of the two states, implying different final values for the quantity that should have been conserved. The standard law of conservation for quantum mechanics handles this jump by stating only that the total distribution of the conserved quantity over repeated measurements is unchanged, but states nothing about individual cases. Here however we show that one can go beyond this and have conservation in each individual instance. We made our arguments in the case of angular momentum of a particle on a circle, where many technicalities simplify, and bring arguments to show that this holds in full generality. Hence we argue that the conservation law in quantum mechanics should be rewritten, to go beyond its hitherto statistical formulation, to state that the total of a conserved quantity is unchanged in every individual measurement outcome. As a further crucial element, we show that conservation can be localised at the level of the system of interest and its relevant frame of reference, and is independent on any assumptions on the distribution of the conserved quantity over the entire universe.

Featured image: When a superposition of angular momentum being clockwise and anti-clockwise is measured, and the outcome is that angular momentum is anti-clockwise, is angular momentum conserved for this individual run of the experiment?

Popular summary

Conservation laws are some of the most important laws in physics, so much so that conservation of energy is taught in high school physics. However quantum mechanics is fundamentally probabilistic, and so quantum mechanical conservation laws are statistical, saying that the distribution of e.g. energy, generated by many repeated experiments, is conserved, but they say nothing about what happens in each individual experiment. Indeed, this was generally thought to be impossible. Whilst these statistical conservation laws have been extremely successful, we argue, and prove for a large class of cases, that we can do better and that conservation laws hold even for individual measurement outcomes. Should this prove to be true in general, it will completely modify the basic formulation and understanding of these laws, give us stronger quantum conservation laws, and allow us to give more powerful and intuitive descriptions of fundamental quantum-mechanical experiments.

► BibTeX data

@article{Collins2025conservationlaws,
  doi = {10.22331/q-2025-07-29-1815},
  url = {https://doi.org/10.22331/q-2025-07-29-1815},
  title = {Conservation {L}aws {F}or {E}very {Q}uantum {M}easurement {O}utcome},
  author = {Collins, Daniel and Popescu, Sandu},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1815},
  month = jul,
  year = {2025}
}

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Cited by

[1] Lev Vaidman, Open Systems: Physics, Metaphysics, and Methodology 283 (2026) ISBN:9780198929246.

[2] Ovidiu Cristinel Stoica, "Observation as Physication. A single-world unitary no-conspiracy interpretation of quantum mechanics", arXiv:2412.09669, (2024).

[3] Eloi Descamps, Nicolas Fabre, Astghik Saharyan, Arne Keller, and Pérola Milman, "Superselection Rules and Bosonic Quantum Computational Resources", Physical Review Letters 133 26, 260605 (2024).

[4] Edward J. Gillis, "Wave Function Collapse, Lorentz Invariance, and the Third Postulate of Relativity", arXiv:2405.05335, (2024).

[5] Ovidiu Cristinel Stoica, "Freedom in the Many-Worlds Interpretation", Foundations of Physics 54 5, 68 (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 16:47:07) and SAO/NASA ADS (last updated successfully 2026-08-09 16:47:08). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.