Counterfactual definiteness

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Counterfactual definiteness

In some interpretations of quantum mechanics, counterfactual definiteness (CFD) is the ability to speak with meaning 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). 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 counter-factual 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 you cannot simultaneously know 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 not important: the point is that they cannot all be designated factual simultaneously.) 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. It is not present in interpretations such as the Copenhagen interpretation and its modern refinements, which regard the measured values as resulting from both the system being measured and the measuring apparatus, and regard the system as being indefinable in the absence of an interaction between the two. To give a macroscopic analogy using the idea of a laptop screen: The former type of interpretation sees quantum mechanical measurements as akin to measuring say the diagonal length of a laptop screen - this quantity is well defined regardless of whether one holds a ruler across the screen or not. The latter type of interpretation that rejects CFD sees quantum mechanical measurements as akin to asking how much your finger is warping the screen at the point where it is poking it - a question which is meaningless unless your finger is actually poking the screen.

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