Counterfactual definiteness

In some interpretations of quantum mechanics, counterfactual definiteness (CFD) is the ability to speak meaningfully of the definiteness of the results of measurements that have not been performed (i.e. the ability to assume the existence of objects, and properties of objects, even when they have not been measured), which is at odds with superdeterminism. A macroscopic example of CFD would be the assumption—without measurement—that a ball, thrown into the air, will return to the Earth due to gravity. CFD says that if a phenomenon (the return of an airborne ball to the Earth) has been reproducibly measured in the past, one can safely assume its presence in the future without having to refer to additional measurement events for proof of its existence. More rigorously, an interpretation of quantum mechanics satisfies CFD if it includes in the statistical population of measurement results, those measurements which are counterfactual by virtue of their being excluded by the quantum mechanical prohibition on simultaneous measurement of certain pairs of properties.

For example, the Heisenberg uncertainty principle states that one cannot simultaneously know, with arbitrarily high precision, both the position and momentum of a particle. Suppose one measures the position: this act destroys any information about its momentum. Is it then possible to talk about the outcome one would have obtained if one had measured its momentum instead of its position? In terms of mathematical formalism, is such a counterfactual momentum measurement to be included, together with the factual position measurement, in the statistical population of possible outcomes describing the particle? If the position was found to be r₀ then in an interpretation satisfying CFD, the statistical population describing position and momentum would contain all pairs (r₀,p) for every possible momentum value p, whereas an interpretation that rejects counterfactual values completely would only have the pair (r₀,⊥) where ⊥ denotes an undefined value. To use a macroscopic analogy, an interpretation which rejects CFD views measuring the position as akin to asking where in a room a person is standing, while measuring the momentum is akin to asking if the person's lap is empty or has something on it. If the person has been made to stand then that person has no lap and neither of the statements "the person's lap is empty" and "there is something on the person's lap" is true. Any statistical calculation based on values where the person is standing at some place in the room and simultaneously has a lap as if sitting would be meaningless.

Counterfactual definiteness is a basic assumption, which, together with locality, leads to Bell inequalities. In their derivation it is explicitly assumed that every possible measurement, even if not performed, can be included in statistical calculations. The calculation involves averaging over sets of outcomes which cannot all be simultaneously factual—if some are assumed to be factual outcomes of an experiment others have to be assumed counterfactual. (Which ones are designated as factual is determined by the experimenter: the outcomes of the measurements he actually performs become factual by virtue of his choice to do so, the outcomes of the measurements he doesn't perform are counterfactual.) Bell's Theorem actually proves that every type of quantum theory must necessarily violate either locality or CFD.

CFD is present in any interpretation of quantum mechanics that regards quantum mechanical measurements to be objective descriptions of a system's state independent of the measuring process, but also if regarded as an objective description of the system and the measurement apparatus.

See also

Counterfactual measurement in quantum mechanics. A Mach-Zehnder interferometer is adjusted so that 100% of the output impinges upon detector B. Having been adjusted in such a manner, 100% of the light will continue to reach B even if the light intensity is reduced so that only one photon at a time travels through the apparatus. It appears as if the wave function of each individual photon travels both paths and engages in interference at the last beam splitter, so that only the wave to B is constructive. In figure (a), although the photon is illustrated as having taken the "northern" branch of the interferometer, it interferes with itself so that only detector B detects the photon. The same holds for figure (b), although the photon is considered to have taken the "southern" branch of the interferometer.
Figure (c) illustrates the situation where an obstacle has been introduced on the "southern" branch of the interferometer. 50% of the photons are deflected by mirror M, and the remaining photons are split 25% to detector A, and 25% to detector B.
Figure (d) illustrates the fundamental paradox raised by this demonstration. An individual photon arriving at detector A must have traversed the "northern" path and could not have interacted with mirror M. The arrival of a photon at detector A constitutes proof that an obstacle exists on the "southern" path, but no exchange of energy has taken place between the photon and obstacle M. How could the photon possibly have acquired information about M without an exchange of energy?

• Superdeterminism
• Determinism
• Quantum indeterminacy
• Interaction-free measurement
• Elitzur–Vaidman bomb-tester
• Renninger negative-result experiment
• Scientific realism
• Naive realism

External links

• Quantum nonlocality without counterfactual definiteness?
• On Some Consequences of the Breakdown of Counterfactual Definiteness in the Quantum World by W. De Baere
• On Bell and CFD by W. M. de Muynck1, W. De Baere, and H. Martens
• CFD by Brian Skyrms
• CFD by Stapp (1988) and 1990