Counterfactual definiteness

In 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). The term "Counterfactual definiteness" is used in discussions of physics calculations, especially those related to the phenomenon called quantum entanglement and those related to the Bell inequalities.[1]

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.[2][3]

Introduction

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 (or "non-contextuality??), and no conspiracy.[4]

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.[5]

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."[6]

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 < #ai > which unambiguously determine the measurement outcome < ai >, given that the corresponding measurement a is performed."[7]

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.[8] These phenomena are not directly germane to the subject under consideration here.

The main theoretical considerations

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.[9]

For example, the Heisenberg uncertainty principle states that one cannot simultaneously know, with arbitrarily high precision, both the position and momentum of a particle.[10] 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 r0 then in an interpretation that permits counterfactual definiteness, 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.[11] 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.[12]

The dependability of counterfactually definite values is a basic assumption, which, together with "no conspiracy"and "local causality" 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.[13] 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.[14][15]

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.

Related topics

References

  1. ↑ Enrique J. Galvez, "Undergraduate Laboratories Using Correlated Photons: Experiments on the Fundamentals of Quantum Mechanics," p. 2f, says, "Bell formulated a set of inequalities, now known as 'Bell’s inequalities,' that would test non-locality. Should an experiment verify these inequalities, then nature would be demonstrated to be local and quantum mechanics incorrect. Conversely, a measurement of a violation of the inequalities would vindicate quantum mechanics’ non-local properties."
  2. ↑ Inge S. Helland, "A new foundation of quantum mechanics," p. 386: "Counterfactual definiteness is defined as the ability to speak with results of measurements that have not been performed (i.e., the ability to assure the existence of objects, and properties of objects, even when they have not been measured").
  3. ↑ W. M. de Muynck, W. De Baere, and H. Martens, "Interpretations of Quantum Mechanics, Joint Measurement of Incompatible Observables, and Counterfactual Definiteness" p. 54 says: "Counterfactual reasoning deals with nonactual physical processes and events and plays an important role in physical argumentations. In such reasonings it is assumed that, if some set of manipulations were carried out, then the resulting physical processes would give rise to effects which are determined by the formal laws of the theory applying in the envisaged domain of experimentation.
    The physical justification of counterfactual reasoning depends on the context in which it is used. Rigorously speaking, given some theoretical framework, such reasoning is always allowed and justified as soon as one is sure of the possibility of at least one realization of the pre-assumed set of manipulations. In general, in counterfactual reasoning it is even understood that the physical situations to which the reasoning applies can be reproduced at will, and hence may be realized more than once."Text was downloaded from: http://www.phys.tue.nl/ktn/Wim/i1.pdf
  4. ↑ http://en.citizendium.org/wiki/Entanglement_%28physics%29#Counterfactual_definiteness
  5. ↑ Christoph Saulder, "Contextuality and the Kochen-Specker Theorem", p. 11.
  6. ↑ Angel G. Valdenebro, "Assumptions Underlying Bell's Inequalities," p. 6.
  7. ↑ Internet Encyclopedia of Philosophy, "The Einstein-Podolsky-Rosen Argument and the Bell Inequalities," section 3.
  8. ↑ Rick Bradford,"The Observability of Counterfactuals" p. 1 says: "Suppose something could have happened, but actually did not happen. In classical physics the fact that an event could have happened but didn't can make no difference to any future outcome. Only those things which actually happen can influence the future evolution of the world. But in quantum mechanics it is otherwise. The potential for an event to happen can influence future outcomes even if the event does not happen. Something that could happen but actually does not is called as counterfactual. In quantum mechanics counterfactuals are observable they have measurable consequences. The Elitzur-Vaidman bomb test provides a striking illustration of this. "See: http://www.rickbradford.co.uk/QM13Counterfactuals.pdf
  9. ↑ Henry P Stapp S-matrix interpretation of quantum-theory Physical Review D Vol 3 #6 1303 (1971)
  10. ↑ Yakir Aharonov et al., "Revisiting Hardy's Paradox: Counterfactual Statements, Real Measurements, Entanglement and Weak Values, p. 1, says, "For example, according to Heisenberg’s uncertainty relations, an absolutely precise measurement of position reduces the uncertainty in position to zero Δx = 0 but produces an infinite uncertainty in momentum Δp = ∞." See http://arxiv.org/abs/quant-ph/0104062v1 arXiv:quant-ph/0104062v1
  11. ↑ Yakir Aharonov, et al, "Revisiting Hardy's Paradox: Counterfactual Statements, Real Measurements, Entanglement and Weak Values," p.1 says, "The main argument against counterfactual statements is that if we actually perform measurements to test them, we disturb the system significantly, and in such disturbed conditions no paradoxes arise."
  12. ↑ Inge S. Helland, "A new foundation of quantum mechanics," p. 3.
  13. ↑ Yakir Aharonov, et al, "Revisiting Hardy's Paradox: Counterfactual Statements, Real Measurements, Entanglement and Weak Values," says, "In 1964 Bell published a proof that any deterministic hidden variable theory that reproduces the quantum mechanical statistics must be nonlocal (in a precise sense of non-locality there in defined) , Subsequently, Bell' s theorem has been generalized to cover stochastic hidden variable theories. Commenting on Bell' s earlier paper. Stapp (1971) suggests that the proof rests on the assumption of "counterfactual definiteness" : essentially the assumption that subjunctive conditionals of the form: " If measurement M had been performed, result R would have been obtained" always have a definite truth value (even for measurements that were not carried out because incompatible measurements were being made) and that the quantum mechanical statistics are the probabilities of such conditionals." p. 1 arXiv:quant-ph/0104062v1
  14. ↑ David Z Albert, Bohm's Alternative to Quantum Mechanics Scientific American (May 1994)
  15. ↑ John G. Cramer The transactional interpretation of quantum mechanics Reviews of Modern Physics Vol 58, #3 pp.647-687 (1986)

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