Precision Tests of the Born Rule

Precision tests of the Born rule search for a specified departure from quadratic quantum probabilities. A three-path interference experiment offers a useful null quantity: ordinary complex amplitudes with the Born rule have pairwise interference but no irreducible third-order term. Turning this identity into an experimental bound requires controlling changes in the apparatus when paths are opened or blocked.

The third-order interference combination

For fixed path amplitudes $a,b,c$, write $P_A=|a|^2$, $P_{AB}=|a+b|^2$, and similarly for the other subsets. Then

$$I_3={}P_{ABC}-P_{AB}-P_{AC}-P_{BC} +P_A+P_B+P_C-P_\varnothing$$

vanishes, with $P_\varnothing=0$ for the ideal model. Expanding each modulus squared shows that every diagonal term and every pairwise cross term cancels. There is no product involving all three amplitudes in a quadratic probability.

For $a=b=c=1$ in arbitrary common intensity units, the combination is $9-4-4-4+1+1+1=0$. These intensities must have a common normalization. Renormalizing each aperture setting to unit total probability would compare different quantities and destroy the test.

In a real count-rate experiment, $P_\varnothing$ estimates background under a compatible acquisition protocol. One may divide $I_3$ by a stated sum of pairwise interference magnitudes to form a dimensionless statistic; the denominator and treatment of its uncertainty must be reported.

What a three-slit apparatus actually changes

Blocking a slit changes boundary conditions. The field with two open apertures is not exactly the sum of fields obtained by independently opening each one, because propagation and scattering depend on the full apparatus. Sawant and collaborators explicitly analyzed this limitation of the naive path-sum model.

Consequently, a nonzero measured aperture combination can arise within standard wave theory and the Born rule. It must be compared with the correct prediction for the actual boundary conditions. Detector nonlinearity, source drift, imperfect extinction, near-field coupling, and unequal acquisition times can also mimic a signal.

Sinha and collaborators’ 2010 photonic experiment is an early quantitative example. Its quoted sensitivity belongs to that experiment and normalization; it is not a universal or current limit on every possible Born-rule modification.

What is being tested?

A candidate rule such as $P\propto|\psi|^{2+\epsilon}$ needs a normalization prescription and a definition for composite systems and measurements. A bound on $I_3$ constrains modifications that generate the corresponding higher-order interference, under the apparatus model. It does not automatically bound every alteration of probabilities or dynamics.

Likewise, observing $I_3=0$ within uncertainty is consistent with the Born rule but does not derive it. Born Rule as a Postulate concerns its logical role; Tests of Linearity concerns time evolution. These are related but distinct empirical questions.

Exercise

Suppose a detector returns $F(P)=P+\epsilon P^2$ instead of a linear intensity response. For $a=b=c=1$ and zero background, what apparent $I_3$ results?

Solution

The ordinary term cancels. The quadratic detector correction gives $\epsilon(9^2-3\times4^2+3\times1^2)=36\epsilon$. Thus an apparently nonzero higher-order interference signal can be caused entirely by detector response even when the incident intensities obey the Born rule.

References

• Sawant, R., Samuel, J., Sinha, A., Sinha, S., and Sinha, U., “Non-classical paths in interference experiments,” Physical Review Letters 113, 120406 (2014), doi:10.1103/PhysRevLett.113.120406, arXiv:1308.2022 — changing boundary conditions and path-sum corrections.
• Sinha, U., Couteau, C., Jennewein, T., Laflamme, R., and Weihs, G., “Ruling out multi-order interference in quantum mechanics,” Science 329, 418–421 (2010), doi:10.1126/science.1190545 — a three-slit null test.