Are quantum measurements universal?
Abstract
With D. Aerts, we showed in 2014, within the extended Bloch representation of quantum mechanics, that the Born rule can be understood as resulting from a universal average over all the possible ways in which the measurement-interactions can be selected. The cellular construction used there was since questioned, in the context of Bertrand's paradox, by N. Shackel, who observed that the result depends on the discretization adopted, that the sequence of averages is constant, and that the sense in which the average is taken over "all" densities remains unclear. Taking these criticisms as a stimulus, we re-examine the notion. The key is to recognize that a universal measurement is a random way of selecting a measurement which does not privilege any portion of the structure describing the available interactions, a property we call indifference. We prove that every indifferent universal measurement, however constructed, produces exactly the Born probabilities: the universal average is unique, and the cellular construction of 2014 is one among infinitely many ways of obtaining it, exact at every finite resolution. Non-trivial indifferent universal measurements exist, but must involve deterministic measurements: Dirichlet processes are indifferent universal measurements reaching every way of selecting an interaction, whereas an indifferent universal measurement free of deterministic components necessarily selects the uniform density, so that the Born rule holds not only on average but with certainty. Deviations from the quantum statistics in sequences of measurements are governed by a single parameter, in principle accessible to experiment. The only geometric inputs are the Euclidean length of the membrane and the orthogonal landing of the point-particle on it; the latter is where the content of Gleason's theorem lies. Generalizations to N outcomes and to Bertrand's paradox are discussed.
Related
- Progress toward quantum-noise-limited interferometric measurements of optical nonlinearity in vacuum
- Realization of a Primary Irradiance Scale for Solar irradiance Measurements based on a Predictable Quantum Efficient Detector and a Quasi-Monochromatic Portable Tunable Source.
- Estimating the concurrence for quantum states via symmetric measurements
- Harnessing Post-selected von Neumann Measurements to Strengthen Entanglement and Quantum Signatures in Coherent States
- Beyond Shor's Conjecture: Optimal Quantum Measurements at Fixed Error
BibTeX
@misc{bianchi2026quantum,
title = {{Are quantum measurements universal?}},
author = {Massimiliano Sassoli de Bianchi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37873},
eprint = {2610.37873},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37873},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Are quantum measurements universal?
AU - Massimiliano Sassoli de Bianchi
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37873
UR - https://arxiv.org/abs/2610.37873
AB - With D. Aerts, we showed in 2014, within the extended Bloch representation of quantum mechanics, that the Born rule can be understood as resulting from a universal average over all the possible ways in which the measurement-interactions can be selected. The cellular construction used there was since questioned, in the context of Bertrand's paradox, by N. Shackel, who observed that the result depends on the discretization adopted, that the sequence of averages is constant, and that the sense in which the average is taken over "all" densities remains unclear. Taking these criticisms as a stimulus, we re-examine the notion. The key is to recognize that a universal measurement is a random way of selecting a measurement which does not privilege any portion of the structure describing the available interactions, a property we call indifference. We prove that every indifferent universal measurement, however constructed, produces exactly the Born probabilities: the universal average is unique, and the cellular construction of 2014 is one among infinitely many ways of obtaining it, exact at every finite resolution. Non-trivial indifferent universal measurements exist, but must involve deterministic measurements: Dirichlet processes are indifferent universal measurements reaching every way of selecting an interaction, whereas an indifferent universal measurement free of deterministic components necessarily selects the uniform density, so that the Born rule holds not only on average but with certainty. Deviations from the quantum statistics in sequences of measurements are governed by a single parameter, in principle accessible to experiment. The only geometric inputs are the Euclidean length of the membrane and the orthogonal landing of the point-particle on it; the latter is where the content of Gleason's theorem lies. Generalizations to N outcomes and to Bertrand's paradox are discussed.
ER -