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).[1][2] 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.[3] In such discussions "meaningfully" means the ability to treat these unmeasured results on an equal footing with measured results in statistical calculations. It is this (sometimes assumed but unstated) aspect of counterfactual definiteness that is of direct relevance to physics and mathematical models of physical systems and not philosophical concerns regarding the meaning of unmeasured results.

Overview

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"), no-conspiracy (called also "asymmetry of time"),[4][5] and counterfactual definiteness (or "non-contextuality").

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]

As a noun, "counterfactual" may refer to an inferred effect or consequence of an unobserved macroscopic event. An example is counterfactual quantum computation.[9]

Theoretical considerations

An interpretation of quantum mechanics can be said to involve the use of counterfactual definiteness if it includes in the mathematical modelling outcomes of measurements that are counterfactual; in particular, those that are excluded according to quantum mechanics by the fact that quantum mechanics does not contain a description of simultaneous measurement of conjugate pairs of properties.[10]

For example, the 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 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 ⊥ (called "up tack" or "eet") 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 "time asymmetry" 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 classical 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] Bell's theorem proves that every type of quantum theory must necessarily violate locality or reject the possibility of extending the mathematical description with outcomes of measurements which were not actually made.[15][16]

Counterfactual definiteness is present in any interpretation of quantum mechanics that allows quantum mechanical measurement outcomes to be seen as deterministic functions of a system's state or of the state of the combined system and measurement apparatus. Cramer's (1986) transactional interpretation does not make that interpretation.[16]

Examples of interpretations rejecting counterfactual definiteness

Copenhagen interpretation

The traditional Copenhagen interpretation of quantum mechanics rejects counterfactual definiteness as it does not ascribe any value at all to a measurement that was not performed. When measurements are performed, values result, but these are not considered to be revelations of pre-existing values. In the words of Asher Peres "unperformed experiments have no results".[17]

Many worlds

The many-worlds interpretation rejects counterfactual definiteness in a different sense; instead of not assigning a value to measurements that were not performed, it ascribes many values. When measurements are performed each of these values gets realized as the resulting value in a different world of a branching reality. As Prof. Guy Blaylock of the University of Massachusetts Amherst puts it, "The many-worlds interpretation is not only counterfactually indefinite, it is factually indefinite as well."[18]

Consistent histories

The consistent histories approach rejects counterfactual definiteness in yet another manner; it ascribes single but hidden values to unperformed measurements and disallows combining values of incompatible measurements (counterfactual or factual) as such combinations do not produce results that would match any obtained purely from performed compatible measurements. When a measurement is performed the hidden value is nevertheless realized as the resulting value. Robert Griffiths likens these to "slips of paper" placed in "opaque envelopes".[19] Thus Consistent Histories does not reject counterfactual results per se, it rejects them only when they are being combined with incompatible results.[20] Whereas in the Copenhagen interpretation or the Many Worlds interpretation, the algebraic operations to derive Bell's inequality cannot proceed due to having no value or many values where a single value is required, in Consistent Histories, they can be performed but the resulting correlation coefficients can not be equated with those that would be obtained by actual measurements (which are instead given by the rules of quantum mechanical formalism). The derivation combines incompatible results only some of which could be factual for a given experiment and the rest counterfactual.

References

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").
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 Archived 2013-04-12 at the Wayback Machine
Enrique J. Galvez, "Undergraduate Laboratories Using Correlated Photons: Experiments on the Fundamentals of Quantum Mechanics," p. 2ff., 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."
Gábor Hofer-Szabó, Miklós Rédei, László E. Szabó, "The principle of the common cause" (Cambridge 2013), Sect. 9.2 "Local and nonconspiratorial common cause systems".
T.N. Palmer "Bell's conspiracy, Schrödinger's black cat and global invariant sets", Philosophical Transactions of the Royal Society A, 2015, vol. 373, issue 2047.
Christoph Saulder, "Contextuality and the Kochen-Specker Theorem", p. 11. Available from the author at: http://www.equinoxomega.net/files/studies/quantenphysik_Handout.pdf
Angel G. Valdenebro, "Assumptions Underlying Bell's Inequalities," p. 6.
Internet Encyclopedia of Philosophy, "The Einstein-Podolsky-Rosen Argument and the Bell Inequalities," section 3.
Rick Bradford, "The Observability of Counterfactuals" p.1. "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." http://www.rickbradford.co.uk/QM13Counterfactuals.pdf
Henry P Stapp S-matrix interpretation of quantum-theory Physical Review D Vol 3 #6 1303 (1971)
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 https://arxiv.org/abs/quant-ph/0104062v1 arXiv:quant-ph/0104062v1
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."
Inge S. Helland, "A new foundation of quantum mechanics," p. 3.
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
David Z Albert, Bohm's Alternative to Quantum Mechanics Scientific American (May 1994)
Cramer, John G. (1986-07-01). "The transactional interpretation of quantum mechanics". Reviews of Modern Physics. American Physical Society (APS). 58 (3): 647–687. Bibcode:1986RvMP...58..647C. doi:10.1103/revmodphys.58.647. ISSN 0034-6861.
Peres, Asher (1978). "Unperformed experiments have no results". American Journal of Physics. American Association of Physics Teachers (AAPT). 46 (7): 745–747. Bibcode:1978AmJPh..46..745P. doi:10.1119/1.11393. ISSN 0002-9505.
Blaylock, Guy (2010). "The EPR paradox, Bell's inequality, and the question of locality". American Journal of Physics. 78 (1): 111–120. arXiv:0902.3827. Bibcode:2010AmJPh..78..111B. doi:10.1119/1.3243279. ISSN 0002-9505. S2CID 118520639.
Griffiths, Robert B. (2010-10-21). "Quantum Locality". Foundations of Physics. Springer Nature. 41 (4): 705–733. arXiv:0908.2914. Bibcode:2011FoPh...41..705G. doi:10.1007/s10701-010-9512-5. ISSN 0015-9018. S2CID 15312828.
Griffiths, Robert B. (2012-03-16). "Quantum Counterfactuals and Locality". Foundations of Physics. Springer Nature. 42 (5): 674–684. arXiv:1201.0255. Bibcode:2012FoPh...42..674G. doi:10.1007/s10701-012-9637-9. ISSN 0015-9018. S2CID 118796867.

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[1] 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").

[2] 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 Archived 2013-04-12 at the Wayback Machine

[3] Enrique J. Galvez, "Undergraduate Laboratories Using Correlated Photons: Experiments on the Fundamentals of Quantum Mechanics," p. 2ff., 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."

[4] Gábor Hofer-Szabó, Miklós Rédei, László E. Szabó, "The principle of the common cause" (Cambridge 2013), Sect. 9.2 "Local and nonconspiratorial common cause systems".

[5] T.N. Palmer "Bell's conspiracy, Schrödinger's black cat and global invariant sets", Philosophical Transactions of the Royal Society A, 2015, vol. 373, issue 2047.

[6] Christoph Saulder, "Contextuality and the Kochen-Specker Theorem", p. 11. Available from the author at: http://www.equinoxomega.net/files/studies/quantenphysik_Handout.pdf

[7] Angel G. Valdenebro, "Assumptions Underlying Bell's Inequalities," p. 6.

[8] Internet Encyclopedia of Philosophy, "The Einstein-Podolsky-Rosen Argument and the Bell Inequalities," section 3.

[Bradford-9] Rick Bradford, "The Observability of Counterfactuals" p.1. "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." http://www.rickbradford.co.uk/QM13Counterfactuals.pdf

[10] Henry P Stapp S-matrix interpretation of quantum-theory Physical Review D Vol 3 #6 1303 (1971)

[11] 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 https://arxiv.org/abs/quant-ph/0104062v1 arXiv:quant-ph/0104062v1

[12] 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."

[13] Inge S. Helland, "A new foundation of quantum mechanics," p. 3.

[14] 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

[15] David Z Albert, Bohm's Alternative to Quantum Mechanics Scientific American (May 1994)

[Cramer1986-16] Cramer, John G. (1986-07-01). "The transactional interpretation of quantum mechanics". Reviews of Modern Physics. American Physical Society (APS). 58 (3): 647–687. Bibcode:1986RvMP...58..647C. doi:10.1103/revmodphys.58.647. ISSN 0034-6861.

[Peres_1978_pp._745–747-17] Peres, Asher (1978). "Unperformed experiments have no results". American Journal of Physics. American Association of Physics Teachers (AAPT). 46 (7): 745–747. Bibcode:1978AmJPh..46..745P. doi:10.1119/1.11393. ISSN 0002-9505.

[Blaylock_2010_pp._111–120-18] Blaylock, Guy (2010). "The EPR paradox, Bell's inequality, and the question of locality". American Journal of Physics. 78 (1): 111–120. arXiv:0902.3827. Bibcode:2010AmJPh..78..111B. doi:10.1119/1.3243279. ISSN 0002-9505. S2CID 118520639.

[Griffiths_pp._705–733-19] Griffiths, Robert B. (2010-10-21). "Quantum Locality". Foundations of Physics. Springer Nature. 41 (4): 705–733. arXiv:0908.2914. Bibcode:2011FoPh...41..705G. doi:10.1007/s10701-010-9512-5. ISSN 0015-9018. S2CID 15312828.

[Griffiths_pp._674–684-20] Griffiths, Robert B. (2012-03-16). "Quantum Counterfactuals and Locality". Foundations of Physics. Springer Nature. 42 (5): 674–684. arXiv:1201.0255. Bibcode:2012FoPh...42..674G. doi:10.1007/s10701-012-9637-9. ISSN 0015-9018. S2CID 118796867.