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Counterfactual definiteness (CFD) means that the output of a system can, in principle, be calculated by using an explicit formula y = f(x). Although the aspects of the universe that can adequately be described by classical physics are compatible with counterfactual definiteness, those aspects of the universe that can only be adequately modelled or described by quantum mechanics are not in all cases compatible with counterfactual definiteness. ^{[1]}
Counterfactual definiteness is a term used in discussions of physics calculations, especially those related to the phenomenon called quantum entanglement and those related to the Bell inequalities.^{[2]} The single adjective "counterfactual" may also appear in physics discussions where it is frequently treated as a noun. The word "counterfactual" does not mean "characterized by being opposed to fact." Instead, it is used to characterize values that could have been measured but, for one reason or another, were not.^{[3]}^{[4]} Counterfactual measurement values are values derived by means other than direct observation or measure, such as by calculation on the basis of well-substantiated theory. If one knows an equation that permits deriving reliably expected values from a list of inputs to the physical system under investigation, then one has "counterfactual definiteness" in the knowledge of that system. Much of what humans regularly do in the field of engineering to design complex machines such as submarines or the Hubble Telescope depend on the use of physics theories to calculate desired specifications for structures, lenses, etc. The values used to grind a lens or fortify a bulkhead are not the ad hoc results of empirical measurements made to substantiate the theories of optics or the conclusions of materials scientists, but that does not make them in any sense unreliable. They are as reliable as all of the research done to create them, and all the instances of use in the real world in which they have not been disproven.
The subject of counterfactual definiteness receives attention in the study of quantum mechanics because it is argued that, when challenged by the findings of quantum mechanics, classical physics must give up its claim to one of three assumptions: locality (no "spooky action at a distance"), counterfactual definiteness, and no conspiracy.^{[5]} If physics gives up the claim to locality, it brings into question our ordinary ideas about causality and suggests that events may transpire at faster-than-light speeds.^{[6]} If physics gives up the "no conspiracy" condition, it becomes possible for "nature to force experimenters to measure what she wants, and when she wants, hiding whatever she does not like physicists to see."^{[7]} If physics rejects the possibility that, in all cases, there can be "counterfactual definiteness," then it rejects some features that humans are very much accustomed to regarding as enduring features of the universe. "The elements of reality the EPR paper is talking about are nothing but what the property interpretation calls properties existing independently of the measurements. In each run of the experiment, there exist some elements of reality, the system has particular properties < #a_{i} > which unambiguously determine the measurement outcome < a_{i} >, given that the corresponding measurement a is performed."^{[8]} If physics gives up locality, that amounts to admitting the possibility of sudden "magical" intervention in human affairs from unknown and possibly unknowable sources of action.
Something else, something that may be called "counterfactuality," permits inferring effects that have immediate and observable consequences in the macro world even though there is no empirical knowledge of them. One such example is the Elitzur-Vaidman bomb tester.^{[9]} These phenomena are not directly germane to the subject under consideration here.
An interpretation of quantum mechanics can be said to involve the use of counterfactual definiteness if it includes in the statistical population of measurement results, any measurements that are counterfactual because they are excluded by the quantum mechanical impossibility of simultaneous measurement of conjugate pairs of properties.^{[10]}
For example, the Heisenberg uncertainty principle states that one cannot simultaneously know, with arbitrarily high precision, both the position and momentum of a particle.^{[11]} Suppose one measures the position of a particle. This act destroys any information about its momentum. Is it then possible to talk about the outcome that 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 were found to be r_{0} then in an interpretation that permits counterfactual definiteness, the statistical population describing position and momentum would contain all pairs (r_{0},p) for every possible momentum value p, whereas an interpretation that rejects counterfactual values completely would only have the pair (r_{0},⊥) where ⊥ denotes an undefined value.^{[12]} To use a macroscopic analogy, an interpretation which rejects counterfactual definiteness views measuring the position as akin to asking where in a room a person is located, while measuring the momentum is akin to asking whether the person's lap is empty or has something on it. If the person's position has changed by making him or her stand rather than sit, then that person has no lap and neither the statement "the person's lap is empty" nor "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.^{[13]}
The dependability of counterfactually definite values is a basic assumption, which, together with "no conspiracy"and "local realism" led to the Bell inequalities. Bell showed that the results of experiments intended to test the idea of hidden variables would be predicted to fall within certain limits based on all three of these assumptions, which are considered principles fundamental to classic physics, but that the results found within those limits would be inconsistent with the predictions of quantum mechanical theory. Experiments have shown that quantum mechanical results predictably exceed those classical limits. Calculating expectations based on Bell's work implies that for quantum physics the assumption of "local realism" must be abandoned.^{[14]} In Bell's 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 that 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 proves that every type of quantum theory must necessarily violate locality or reject the possibility of reliable measurements of the counterfactual and definite kind.^{[15]}^{[16]}
Counterfactual definiteness 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.
Hidden variable theory, Alice and Bob, Quantum mechanics, EPR paradox, Quantum entanglement