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arXiv:2505.07510 · Physical review. D/Physical review. D.

Toward a test of the Born rule in high-energy collisions

Antony Valentini, Mira Varma

Published Dec 5, 2025• 0 citations
arXivHTMLPDF

Abstract

We consider how the Born rule, a fundamental principle of quantum mechanics, can be tested for particles created on the shortest timescales ( ∼ 10 − 25 s ) currently accessible at high-energy colliders. We focus on targeted tests of the Born rule for spin or polarization probabilities, which offer a particularly clean experimental signal, and which can be described by a simple hidden-variables model of two-state systems proposed by Bell. These probabilities test a remarkable feature of the quantum formalism, whereby expectation values for incompatible experiments are linearly related. Born-rule violations can be parametrized by nonlinear expectation values for quantum measurements of spin or polarization, together with anomalies in ensemble averages, which may then be constrained by experiments. Notable experiments considered here include the recent detection of single photons from top-quark decay, and the indirect measurement of tau-lepton polarization. Repurposing these experiments as tests of the Born rule, however, presents several challenges, which are discussed in this paper.

DOI
10.1103/4llk-pt1j

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De Gruyter eBooks•Oct 24, 2019•0 citations
We consider how the Born rule, a fundamental principle of quantum mechanics, can be tested for particles created on the shortest timescales ( ∼ 10 − 25 s ) currently accessible at high-energy colliders. We focus on targeted tests of the Born rule for spin or polarization probabilities, which offer a particularly clean experimental signal, and which can be described by a simple hidden-variables model of two-state systems proposed by Bell. These probabilities test a remarkable feature of the quantum formalism, whereby expectation values for incompatible experiments are linearly related. Born-rule violations can be parametrized by nonlinear expectation values for quantum measurements of spin or polarization, together with anomalies in ensemble averages, which may then be constrained by experiments. Notable experiments considered here include the recent detection of single photons from top-quark decay, and the indirect measurement of tau-lepton polarization. Repurposing these experiments as tests of the Born rule, however, presents several challenges, which are discussed in this paper.
Born's rule relates detection probabilities to the modulus square of the wave function.It is one of the central principles of quantum mechanics together with that of linear superposition.Single-particle interference is accordingly limited to pairs of quantum paths and higher-order interferences are prohibited.Deviations from Born's law have been quantified via the Sorkin parameter which is proportional to the third-order term.However, while the linearity of quantum theory has been experimentally tested to the level of 10 -20 eV, the Sorkin parameter has only been measured to an accuracy of 2 × 10 -3 in the quantum regime.We here investigate Born's law using many-particle interferences and demonstrate that all interference terms of order (2M + 1) and greater vanish for M particles.We further introduce a family of many-particle Sorkin parameters and show that they are exponentially more sensitive to deviations from Born's rule than their single-particle counterpart.
In this thesis, we experimentally test fundamental properties of quantum mechanics, namely \nnon-locality (in the form of three new families of Bell's inequalities) and the symmetry of \nenvariance. To accomplish these we use a Sagnac source of polarization entangled photon \npairs. \nIn chapters one and two we discuss the relevant background information in quantum \ninformation theory, nonlinear optics, experimental realization of polarization entangled \nphotons and a trouble-shooting and maintenance guide for a Saganc source. \nIn chapter three we experiment with a set of three newly derived families of Bell's \ninequalities. These three families are predicted to yield the largest volume of violation of \nthe local hidden variable models (LHVM). Our experimental results are in good agreement \nwith those predictions and therefore, represent the largest volume of experimental violation \nof LHVM to date. We showed a violation of up to 30 sigma from what is predicted by LHVM, \nand our results followed closely to the predictions of quantum mechanics. \nIn chapter four we experimentally test envariance, an assisted-symmetry exhibited by \nspecifi c quantum systems. Envariance is a fundamental property in the quantum world that \nhas lacked, until now, extensive experimental study. The symmetry has ramifi cations in the \nfoundations of quantum mechanics, and plays an integral role in a proof of Born's rule [1]. \nOur results serve as a benchmark the property of envariance. We show that experimental \nquantum states can be (99.66+/- 0.04)% envariant over a wide range of transformations, as \nmeasured using the average quantum fi delity [2], and (99.963 +/- 0.005)% as measured using \na modifi ed average Bhattacharya Coeffi cient [3], a measure of the overlap of two probability \ndistributions.
As a fundamental postulate of quantum mechanics, Born's rule assigns probabilities to the measurement outcomes of quantum systems and excludes multiorder quantum interference. Here we report an experiment on a single spin in diamond to test Born's rule by inspecting the third-order quantum interference. The ratio of the third-order quantum interference to the second order in our experiment is bounded to the scale of $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, which provides a stringent constraint on the potential breakdown of Born's rule.
Recently a new model with hidden variables of the wave type was elaborated, so called prequantum classical statistical field theory (PCSFT). Roughly speaking PCSFT is a classical signal theory applied to a special class of signals—“quantum systems”. PCSFT reproduces successfully all probabilistic predictions of QM, including correlations for entangled systems. This model peacefully coexists with all known no‐go theorems, including Bell’s theorem. In our approach QM is an approximate model. All probabilistic predictions of QM are only (quite good) approximations of “real physical averages”. The latter are averages with respect to fluctuations of prequantum fields. In particular, Born’s rule is only an approximate rule. More precise experiments should demonstrate its violation. We present a simple experiment which has to produce statistical data violating Born’s rule. Since the PCSFT‐presentation of this experiment may be difficult for experimenters, we reformulate consequences of PCSFT in terms of the conventional wave function. In general, deviation from Born’s rule is rather small. We found an experiment amplifying this deviation. We start with a toy example. Then we present a more realistic example based on Gaussian states with very small dispersion.
Recently a new model with hidden variables of the wave type was elaborated, so called prequantum classical statistical field theory (PCSFT). Roughly speaking PCSFT is a classical signal theory applied to a special class of signals—“quantum systems”. PCSFT reproduces successfully all probabilistic predictions of QM, including correlations for entangled systems. This model peacefully coexists with all known no‐go theorems, including Bell’s theorem. In our approach QM is an approximate model. All probabilistic predictions of QM are only (quite good) approximations of “real physical averages”. The latter are averages with respect to fluctuations of prequantum fields. In particular, Born’s rule is only an approximate rule. More precise experiments should demonstrate its violation. We present a simple experiment which has to produce statistical data violating Born’s rule. Since the PCSFT‐presentation of this experiment may be difficult for experimenters, we reformulate consequences of PCSFT in terms of the conventional wave function. In general, deviation from Born’s rule is rather small. We found an experiment amplifying this deviation. We start with a toy example. Then we present a more realistic example based on Gaussian states with very small dispersion.
Traditionally, some version of Born’s rule is always considered to be an indispensable part of any interpretation of quantum mechanics, either as a postulate, or as a result derived-not always on the basis of convincing reasoning-from other postulates. In this section, we have a close look at the possible forms of Born’s rule and discuss the limits of its validity. All traditional foundations of quantum mechanics heavily depend on the concept of (hypothetical, idealized) experiments-far too heavily. This is one of the reasons why these foundations are still unsettled, over 90 years after the discovery of the basic equations for modern quantum mechanics. No other theory has such controversial foundations. The main reason is that the starting point of the usual interpretations is an idealization of the measurement process that is taken too seriously, namely as the indisputable truth about everything measured. But in reality, this idealization is only a didactical trick for the newcomer to make the formal definitions of quantum mechanics a bit easier to swallow. Except in a few very simple cases, it is too far removed from experimental practice to tell much about real measurements, and hence about how quantum physics is used in real applications. In experimental physics, measurement is a very complex thing, far more complex than Born’s rule (the usual starting point) suggests. To measure the distance between two galaxies, the mass of the top quark, or the Lamb shift, just to mention three basic examples, cannot be captured by the idealistic measurement concept used there, nor by any of the refinements of it discussed in the literature. In each of the three cases mentioned, one assembles a lot of auxiliary information and ultimately calculates the measurement result from a best fit of a model to the data. Clearly, the theory must already be in place in order to do that. We do not even know what a top quark should be, whose mass we are measuring, unless we have a theory that tells us this! The two most accurately determined observables in the history of quantum physics, namely the anomalous magnetic moment of the electron and Lamb shift, are not even q-observables! To present the stage for the criticism of Born’s rule in Section 14.3, we first need to clarify the meaning of the term “Born’s rule”. To distinguish different useful meanings, we look in Sections 14.1-14.2 at the early history of Born’s rule.