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Bohr on EPR, the Quantum Postulate, Determinism, and Contextuality

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Abstract

The famous EPR article of 1935 challenged the completeness of quantum mechanics and spurred decades of theoretical and experimental research into the foundations of quantum theory. A crowning achievement of this research is the demonstration that nature cannot in general consist in noncontextual pre-measurement properties that uniquely determine possible measurement outcomes, through experimental violations of Bell inequalities and Kochen-Specker theorems. In this article, I reconstruct an argument from Niels Bohr’s writings that the reality of the Einstein-Planck-de Broglie relations alone implies that no such properties can exist for momentum and position measurements, show how this argument responds to the challenge of EPR on general physical grounds, and advance that this reconstruction shows that and how Bohr’s “complementarity” is a view of the objective content and logic of quantum theory.

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Notes

  1. Mara Beller and Arthur Fine articulate this dismissal perhaps most influentially when they assert that only the acceptance of an “extreme positivist attitude” that “identif[ies] measurability and meaning” can salvage Bohr’s reply to EPR [1, pp. 12, 14] Similarly, Fine calls Bohr’s response to EPR “virtually textbook neo-positivism” while voicing complaints about positivism as a theory of meaning [2, pp. 8, 34-35] and Richard Healey characterizes Bohr as offering a theory of meaning and complains that it rests on too vague and anthropocentric of foundations [3, pp. 1536, 1553]. See also e.g. [1, p. 9], [2, pp. 4–5, 29, 31, 34–35], [4, p. 196], [5, p. 152], and [6, pp. 25, 29].

  2. See also [10].

  3. One such claim, following an argument that the quantum of action “sets a limit.. to the meaning we may ascribe to [information attainable by measurements]” is that “[w]e meet here in a new light the old truth that in our description of nature the purpose is not to disclose the real essence of the phenomena but only to track down, so far as it is possible, relations between the manifold aspects of our experience” [14, p. 296].

  4. While I agree with Folse at this level of description, I disagree with his suggestion that the quantum postulate challenges the principle of Separability per se [7, p. 264]: on my reading, the quantum postulate can only challenge Separability indirectly, in that either non-Separability or the existence of superluminal causality follows from the failure of Pre-Existing Properties (implied by the quantum postulate) and the reality of EPR correlations. See fn 21 and Sect. 6.

  5. There are three main novelties of my elaboration of Bohr’s response. First and primarily, Bohr’s protracted concern with experimental arrangements does not reflect an appeal to instrumentalism, pace Beller and Fine, but rather reflects physical consequences of the quantum postulate. Second, Bohr’s concession of the point that we can predict the position or momentum of the unmeasured particle in an EPR state with certainty is thinner than concession of the cogency of bare property ascription and is to be elaborated counterfactually. Third, the number of distinct physical situations / experimental arrangements at issue in considering the meaning of the EPR wavefunction is at least four.

  6. Thank you to an anonymous reviewer for suggesting that I explicitly situate this reconstruction of Bohr with respect to such no-go theorems.

  7. By “compatible operators” I mean operators representing measurements of compatible observables.

  8. See [17, pp. 8–9, 11–19] for a more accessible presentation of their reasoning.

  9. For example, Asher Peres and David Mermin show that the obtaining of such properties for all possible spin measurement outcomes for the singlet state of two electrons at all times is inconsistent with canonical commutation relations on spin angular momentum operators [19, 20]. See [17] for an overview of such theorems.

  10. There are also attempts to generalize the Kochen-Specker theorem for noncontextual properties that do not uniquely determine measurement outcomes, e.g. [21].

  11. See [22] for the inaugural paper and [23] for an overview of these inequalities, various philosophical disputes concerning the reasoning involved, and an overview of some of the corresponding experimental work.

  12. Outcome determinism appears to be what Guy Blaylock calls “counterfactual definiteness” and advances as a premise of Bell theorems [24, p. 115] This principle is the “assumption... that we are allowed to postulate a single definite result [that would result from] an individual measurement even when the measurement is not performed.” Blaylock does not mean, trivially, that any measurement outcome, were it to obtain, would obtain, but rather that there is only “one possibility for the potential result of a measurement.” This is just to advance that some property of the system to be measured uniquely determines what result a corresponding measurement upon that system would give. However, I agree with Myrvold et. al. in holding that outcome determinism holds only for a special case of the sorts of local properties at issue, and so is not a premise from which the inequalities are derived überhaupt [23, p. 30]: the inequalities do not assume that \(p^{1}_{a}(s| \lambda )\) and \(p^{2}_{b}(t| \lambda )\) are exclusively either 1 or 0 for every a, b, s, t, and \(\lambda\), which is precisely what Blaylock calls the principle of counterfactual definiteness and what I call Pre-Existing Properties.

  13. See [20] for a consideration of some logical relations between Bell theorems, the GHZ theorem, and theorems of the form of the Kochen-Specker theorem.

  14. As Arthur Fine clarifies [2, pp. 32–33], the logical form of the actual argument given in the article is much more and needlessly complex, in that the more complex reasoning crucially relies on a direct argument for the existence of simultaneous sharp values of position and momentum of some particle, and the success of this argument would suffice for establishing the incompleteness of quantum theory in light of the simple observation that quantum theory does not permit assignment of such values for any particle.

  15. I take (ii) to be both commonsensical and unobjectionable (and entirely compatible with Bohr’s reasoning), so I will simply tacitly assume it from now on. I flag it here just to make it explicit.

  16. Thank you to Guido Bacciagaluppi for helping me reach this reconstruction of EPR's reasoning.

  17. We can explain why there is no interference effect in the second arrangement if the HUR characterize the predictive significance of the initial state of the electron when localized in the first slit, as then the convolution of the spread in location of the first slit with the fringing pattern on the screen washes out the fringing pattern to give a non-fringed pattern on the screen [31, p. 357]. See [32, pp. 938–939] and [33, pp. 86–88] for mathematical details. As I exposit from Bohr’s pre-EPR writings in Sect. 5, the quantum postulate implies that the HUR indeed characterize the predictive significance of any initial conditions of momentum and position of any atomic entity.

  18. This language is also a continuation of pre-EPR language Bohr uses when discussing measurement in quantum theory [1, pp. 10–13].

  19. Beller and Fine also make this complaint in [2, pp. 4–5, 29, 31, 34–35] [4, p. 196], and [5, p. 152]. See also [6, p. 29], [34, p. 51], and [35, Ch. 13] for similar complaints. Cf. [36, p. 1329].

  20. Thus simply pointing out that what is “predictable with certainty” is relative to an experimental context and suggesting on this basis that so, too, are the “candidates for real status,” as Howard does on Bohr’s behalf [43, p. 256], does not engage with any of EPR’s premises. For the difference in experimental contexts is not a difference in the two physical situations in the neighborhood of particle 2, and hence, by Separability and Locality, should not make any difference as to what properties particle 2 has. The suggestion of Halvorson and Clifton that the elements of reality are those that are left invariant under some “relevant” group of symmetry transformations on the EPR state [44, pp. 11–16], while formally illuminating, also does not engage with EPR’s premises for the same reason. For the “relevant” group of symmetries is none other than that group of symmetries that (i) is applied to both particles and (ii) leaves the measured observable of particle 1 in a given experimental arrangement invariant. This amounts substantively to Howard’s suggestion that the only “candidates for real status” are set by the experimental context of making one sort of measurement or other on particle 1.

  21. In a more recent article, Howard advances that Bohr’s response to Einstein consisted in his “[embrace] of entanglement, seeing in it the roots of complementarity” and a rejection of Separability accordingly, and that Bohr has been vindicated through quantum theory being “well-confirmed” and taking entanglement to be “fundamental” [45, pp. 59, 69–70, 79–80, 84]. Specifically, Howard advances that Bohr holds “[t]hat instrument and object form an entangled pair,” adducing as textual evidence Bohr’s pre-EPR statement that “an independent reality in the ordinary physical sense can neither be ascribed to the phenomena nor to the agencies of observation” because “...the quantum postulate implies that any observation of atomic phenomena will involve an interaction with the agency of observation not to be neglected” [46, p. 114]. Folse made essentially the same suggestion a few years prior [7, pp. 264ff]. This proposed reconstruction of Bohr’s response, however, does not directly engage with Separability in the sense at issue, for the simple reason that the difference in the two measuring devices for the two possible measurements on particle 1 of the EPR state is not a difference that obtains upon preparation of either of the two particles; rather, both particles were identically prepared by a preparing device, namely in an EPR state. Further, the possible interactions at issue when measuring either particle 1 or particle 2 are localized to the spatially separated neighborhoods of each respective particle, and so a simple appeal to these possible interactions does not give any reason to think the two-particle system should be non-Separable. Hence any “entanglement” between measuring device and system measured cannot directly speak for the non-Separability of the two particles in the EPR state. Moreover, as will become clear in Sect. 5, the quantum postulate does not even imply the reality of EPR correlations, so it cannot directly imply the non-Separability that would follow from them and the completeness of quantum theory. However, as we will see (Sect. 6), the quantum postulate does imply the failure of determinism for EPR measurements, which in turn implies either that the EPR state is non-Separable or that there is superluminal causality involved in EPR measurements if their outcomes are as predicted by wavefunction (1). If superluminal causality is a non-starter on relativistic grounds, the quantum postulate would thus imply that the EPR state is non-Separable indirectly, via the failure of determinism.

  22. Karen Barad also reads Bohr as challenging this principle [48, p. 107]. However, my reconstruction will make more explicit how he challenges it and how this serves to defuse the challenge from EPR than Barad’s discussion [48, pp. 269–275], which advances that the key issue is simply that there are distinct experimental arrangements [48, pp. 274–275]. As it stands, this is unsatisfactory for the reasons rehearsed above.

  23. Thank you to Noah Stemeroff for pushing me on this point.

  24. See [50] for an overview of different ways of understanding the HUR.

  25. To Beller’s credit, she also recognizes that the Como lecture primarily contains physical arguments as opposed to instrumentalist philosophy of science [5, pp. 117–134, esp. p. 123].

  26. Thank you to Sam Fletcher for pushing me to clarify this point. See [52] for an intuitive presentation of this issue.

  27. Within the formalism of quantum theory, we could say that only non-eigenstates of momentum can be spatially localized to any degree.

  28. To be precise, that the wave fields associated with massive particles spread out is a consequence both of this observation and the dispersion relation \(\omega \propto k^{2}\) postulated to hold for such particles in order to ensure correspondence with the Newtonian relation \(E_{\text {kinetic}} = \frac{p^{2}}{2m}\), given the relations \(E = \hbar \omega\) and \(p = \hbar k\).

  29. Bohr does not mark any approximation at work in the reasoning he references, but he appears to be referencing heuristic arguments from optics that are seen as approximate reasoning schemes when compared to the analysis of Fourier transforms from which the modern relation \(\Delta x \Delta p_{x} \ge \frac{\hbar }{2}\) is derived. This accounts for the difference of a factor 2 in the relation Bohr writes and the modern relation. See [32, pp. 925–928].

  30. The only way to get an empirically determined path from slit to subsequent screen is therefore by placing a series of intermediate slits, or by replacing the slit with e.g. a metal block with a small path drilled through it and putting a screen on the opposite end of the block, such that we know that any particle that makes it to the screen must have done so by passing through known intermediate positions. But these would just be totally different physical situations, where interactions of any particle with these other physical systems forces any through-passing particle to be so constrained. See [30, pp. 697–698] and cf. [54, pp. 311–312], [55, p. 313], [56, p. 391], and [57, p. 419].

  31. We can accordingly understand why Bohr advances that the HUR characterize the predictive significance of measurement results on quantum systems [46, pp. 122–123] [15, p. 151], [14, p. 296]. Cf. [54, p. 315].

  32. See also [49, p. 92], [58, p. 243], and [14, p. 295]. Cf. [54, pp. 303–305, 311], [55, pp. 313–314], [31, pp. 355–356], [56, p. 391], and [57, p. 419].

  33. See also [46, p. 123] and [14, p. 296].

  34. The statement that “the impossibility of neglecting the interaction with the agency of measurement means that every observation introduces a new uncontrollable element” for quantum systems echoes the claim that “...any observation of atomic phenomena will involve an interaction with the agency of observation not to be neglected” [15, p. 148]. Not only does this latter statement replace the statement that “...the quantum postulate implies that no observation of atomic phenomena is possible without their essential disturbance” in an earlier draft of the Como lecture [49, p. 91], it still serves to problematize “our usual description of physical phenomena[,]... based entirely on the idea that the phenomena concerned may be observed without disturbing them appreciably” [15, p. 148].

  35. Such inference is, for example, part of the logic of evidence of Newtonian gravitational theory, according to which the successive determinations of the positions as a function of time of the various gravitating bodies in the solar system allow us to compute their masses and hence uniquely infer their positions prior to the first set of measurements; these masses and relative positions are then a set of initial conditions from which the states determined in each successive measurement, as well as each state successive to those partially determined from the measurements taken, are forward-evolved. Thus the previous and future states can be inferred and predicted with ever-increasing accuracy in proportion to the precision and number of measurements of a succession of states, because each such state evolves continuously and deterministically out of the previous ones [59].

  36. This explains why the reasoning from the quantum postulate to this understanding of the HUR is presented in the section prior to one from which I have just quoted (titled “Measurements in Quantum Theory”) and why the section from which I have just quoted focuses on the HUR.

  37. For example, if one were to place a movable slit instead of a screen beyond a rigidly bolted, preparing slit through which a particle propagates with zero motion transverse to the preparing slit, there will nonetheless be a spread in the degree to which the subsequent movable slit moves from run to run.

  38. In the Como lecture, Bohr only makes this point about position measurements, but the same is true of momentum measurements.

  39. I take it that this is also what Bohr means when he says that the quantum postulate implies “a resignation as regards causal space-time coordination of atomic processes,” that a “rigorous definition of [a] system is no longer possible,” and “the impossibility of causal space-time description of [the] phenomena,” and so a limitation of “the classical physical ideas when applied to atomic phenomena” [49, p. 91] [15, p. 148].

  40. For instance, because the HUR represent the reciprocal relation between the spatial extent of some atomic entity and the spread in momentum that entity has, for there to be a determinate amount of total momentum transfer from the particle to the movable slit such that one has measured the momentum of the particle, this interaction must have taken place over a larger region than just the slit-width, from which region the wavefield of that particle must accordingly propagate.

  41. When Bohr concedes that a measurement of the position or momentum of particle 1 of the EPR state “automatically determine[s]” the position or momentum of particle 2, therefore, he cannot mean that this measurement determines a noncontextual property that exists prior to and uniquely explains the measurement outcome of any hypothetical measurement performed on particle 2 [30, p. 699]. This must instead be read counterfactually: if we had an EPR preparation procedure, and if we were to place a (position, momentum) measuring device that would allow the predicted measurement outcome for particle 2, it would register that outcome. This analysis therefore provides an argument for the meaninglessness of the inference of properties that obtain independently of any measuring context, just as Allen Stairs suggests [60, p. 236]. It is however unclear why the counterfactual indicated here should itself be considered meaningless: Stairs simply asserts that this is “the best answer” for an indeterminist [60, p. 236], but it is not clear why the counterfactual is meaningless just because no measurement reveals a noncontextual pre-existing property. Perhaps what Stairs has in mind is that the counterfactual situation would be a different total physical situation, and so the counterfactual is “meaningless” if its meaning is thought to be indicative of a noncontextual pre-existing property.

  42. This spread is wider in proportion to the sharpness of the momentum measurement on particle 2: the sharper the momentum of particle 2 during measurement, the larger the region of space throughout which particle 2 must be spread during measurement.

  43. Accordingly, I agree with Thomas Ryckman that Bohr takes the key mistake of Einstein / EPR to lie in an appeal to a “well-defined used of the concept of ‘state’ ” of an atomic entity independently of its relations to measuring devices [54, p. 313] [27, p. 152] and additionally urge that this mistake is more specifically the neglect of the general failure of Pre-Existing Properties for measurements of momentum and position implied by the quantum postulate.

  44. In the context of Bell theorems, Bohr’s reasoning thus recommends viewing the obtaining of correlated measurement outcomes of the form predicted for the EPR state as indications of a genuine failure of what is called outcome independence for such measurements on such states. See [23]. Thank you to Juliusz Doboszewski for suggesting I spell this out.

  45. Thus, on my reading, differences between later Einstein’s argumentative strategy regarding the completeness of quantum theory vis-à-vis Separability and Locality and EPR’s (e.g. that later Einstein explicitly marks his argument as independent of the eigenstate-eigenvalue link [29, p. 181], [61]) do not change the basic picture of how Bohr’s response interfaces with such concerns. For, any interesting differences of detail aside, the basic form of the reasoning is still that from correlated measurement outcomes on spatially separated systems, one can infer an incompatibility of the completeness of quantum theory with the conjunction of Separability and Locality, and Bohr’s reasoning still interfaces with such reasoning precisely by showing that Separability or Locality must fail if there are correlated measurement outcomes of position and momentum on spatially separated particles and one accepts the reality of the quantum of action.

  46. There may well be other shortcomings of Bohr’s response beyond those obscurities. For example, there could be problems with Bohr’s discussion of experimental arrangements suited to study the EPR state, as Beller advances [5, p. 150]. These would not however affect the validity of the reasoning reconstructed here.

  47. See also [55, p. 317]. What remains to be specified in arriving at this view is why and in what sense Bohr was concerned here exclusively with “experimental conditions.”

  48. This suggestion finds support in the historical work of Makoto Katsumori [9, pp. 28–31, 50–53, 69–77], who shows that this change accompanied Bohr’s rejection of the language of “disturbance.” See also [7, pp. 263–265] and [8, pp. 135–137].

  49. Indeed, we can see Bohr oscillating between these two ways of thinking about the quantum of action within the same articles as early as 1929 and as late as 1939. In 1929, Bohr writes that “the discovery of the quantum of action” shows us that “...any observation necessitates an interference with the course of phenomena, which is of such a nature that it deprives us of the foundation underlying the causal mode of description” indicating the former point of view, but then one sentence later writes that “...this should not be regarded as a hindrance to further advance; we must only be prepared for the necessity of an ever extending abstraction from our customary demands for a directly visualizable description of nature,” indicating that our conceptualization of “the course of phenomena” as “interfered with” is only a demand of the classical worldview, in accordance with the latter point of view [58, p. 249]. In 1939, in the very article in which Bohr first introduces the above novel definition of “phenomenon,” Bohr still voices that measurements of quantum systems have an “essential influence on the phenomenon itself,” in contradistinction to the status of measurements vis-à-vis physical systems “within the scope of classical physics” [54, p. 311]. Thank you to Thomas Ryckman for drawing my attention to such passages.

  50. Thus, adducing the quantum postulate in response to EPR is not the assumption of the completeness of quantum mechanics per se, as Folse claims [7, p. 261]; rather, the postulate shows EPR’s conclusion to be false on general physical grounds and informs Bohr’s view of what quantum theory represents in the first place, the latter of which Folse urges in a later article [62].

  51. All four commitments can be found in e.g. the most recent textbook from Griffiths and Schroeter [64, pp. 17–19, 125–126, 133–136]. Dirac and von Neumann were the first to codify them in the quantum formalism [65, pp. 217, 351–357, 418], [66, pp. 34–36].

  52. von Neumann as well as Dirac codified this part of the textbook view [65, 66]; it appears that Pauli was the first to suggest it in print [67, p. 83].

  53. On the ideality of elementary plane waves in Bohr, see e.g. [51, p. 97], [49, p. 92], [46, pp. 116, 129, 132], [15, pp. 149, 156], [54, p. 304]. Bohr continually counterposes the ideality of such plane waves with “the idea of material particles” which are as such localized in space [51, pp. 76, 79], [49, p. 92], [46, p. 116], [15, p. 149], [54, p. 304].

  54. One of Bohr’s observations that invites confusion in this connection is that in certain measurements of position we “cut ourselves off from” any knowledge of the momentum transfer between the particle and the rigidly bolted diaphragm through which it passes, which can appear to suggest that no simultaneous measurements of position and momentum are possible at all [30, p. 697]; this observation is correct as far as it goes, but, as I have just argued, does not imply that measurements of momentum in general cannot also serve as measurements of position. With such examples, I take it that Bohr means to illustrate what approaching the classical ideal of any (known) complete localization of an atomic entity demands, namely rigidly bolting the diaphragm to a laboratory frame (otherwise, the measuring slit is liable to move during measurement, which means that the region in which the atomic entity is interacting with the diaphragm is larger than it would be if the same diaphragm were bolted, and thus further from the classical ideal), and pointing out that in such situations, there is, owing to the quantum postulate, a spread in how the atomic entity exchanges momentum with the rigidly bolted diaphragm in proportion to how small the slit is. For some reason (plausibly simply as a consequence of additional epistemic sensibilities), Bohr is also concerned to observe that such momentum transfer is itself unknown in such cases.

  55. See [68] for details on POVMs vis-à-vis simultaneous unsharp measurements.

  56. It is worth noting in this connection that not all historical instances of Born’s rule were advanced with the eigenstate-eigenvalue link. See for example [69, pp. 865–6].

  57. To be clear, there is no reason to think that subjective acts of ascertainment (such as visually attending to an experimental arrangement) “cause” said quantum jumps, in the way that such acts are thought to cause “collapse” of the wavefunction on the orthodox or pop-science view of quantum measurement: the most natural thing to say here is that we ascertain the measurement record, which is itself a perceivable amplification of an instance of such a jump. The only thing we as experimenting subjects can do to make a causal difference to the quantum phenomena is change the experimental arrangement and thus consider a different experimental context—itself a different physical situation—in which different jumps can occur and be amplified such that we can perceive them with our senses.

  58. That this was Bohr’s general point of view finds strong textual support in “On the Notions of Causality and Complementarity":

    Incidentally, it may be remarked that the construction and the functioning of all apparatus like diaphragms and shutters, serving to define geometry and timing of the experimental arrangements, or photographic plates used for recording the localization of atomic objects, will depend on properties of materials which are themselves essentially determined by the quantum of action. Still, this circumstance is irrelevant for the study of simple atomic phenomena where, in the specification of the experimental conditions, we may to a very high degree of approximation disregard the molecular constitution of the measuring instruments. If only the instruments are sufficiently heavy compared with the atomic objects under investigation, we can in particular neglect the requirements of [the HUR] as regards the control of the localization in space and time of the single pieces of apparatus relative to each other [55, pp. 315–6].

    Here, the quantum of action is presented as necessary for experimental arrangements even to have the properties they have that make them suitable as experimental arrangements in the first place, and the representation of these arrangements with classical concepts is presented as some pragmatically justifiable coarse-grained representation of configurations of essentially quantum-mechanical constituents.

  59. Thank you to an anonymous reviewer for pushing me on this point.

  60. For one example of such a neo-Bohrian attempt, see [71]. For an objection that such an approach makes quantum theory non-universal, see [72].

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Acknowledgements

Thank you to Thomas Ryckman for a comprehensive, penetrating, and historically sensitive introduction to the various issues involved in interpreting Bohr’s thought and in making sense of the Bohr-Einstein debate. Thank you to participants in the Graduate Student Workshop of Stanford’s Philosophy Department in Autumn 2022 for helping me talk through the relations between Pre-Existing Properties, Separability, and Locality. Thank you to participants in the Lichtenberg Group’s Work in Progress session in May 2023 at the University of Bonn for a helpful discussion of a draft of this article. Thank you to three anonymous reviewers for helpful comments and suggestions on an earlier draft of this article and to Robert Engelman for helpful comments on a more mature draft of this article. Thank you to Guido Bacciagaluppi for helpful comments on a mature draft of this article and in particular for correcting my understanding of the EPR argument.

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Hall, Z. Bohr on EPR, the Quantum Postulate, Determinism, and Contextuality. Found Phys 54, 31 (2024). https://doi.org/10.1007/s10701-024-00764-8

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