Abstract
This article considers the relationships between the character of physical law in quantum theory and Bohr’s concept of complementarity, under the assumption of the unrepresentable and possibly inconceivable nature of quantum objects and processes, an assumption that may be seen as the most radical departure from realism currently available. Complementarity, the article argues, is a reflection of the fact that, as against classical physics or relativity, the behavior of quantum objects of the same type, say, all electrons, is not governed by the same physical law in all contexts, specifically in complementary contexts. On the other hand, the mathematical formalism of quantum mechanics offers correct probabilistic or statistical predictions (no other predictions are possible on experimental grounds) in all contexts, here, again, under the assumption that quantum objects themselves and their behavior are beyond representation or even conception. Bohr, in this connection, spoke of “an entirely new situation as regards the description of physical phenomena that, the notion of complementarity aims at characterizing.” The article also considers the relationships among complementarity, entanglement, and quantum information, by basing these relationships on this understanding of complementarity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
I distinguish “the spirit of Copenhagen” from “the Copenhagen interpretation,” because there is no single such interpretation, as even Bohr had different versions of his interpretation. Many (but not all) of these interpretations do share some of the key features of Bohr’s interpretation(s). The interpretation adopted here follows that of Bohr (in its ultimate version, in place by 1930s), but with certain possible differences, which I indicate as I proceed.
Eventually, Bohr came to use the term “complementarity” to designate his overall interpretation of, importantly, both quantum phenomena and quantum mechanics. For clarity, however, I shall speak of Bohr’s interpretation, and by complementarity refer to the concept or principle of complementarity.
This and the following sections in part builds on an earlier study [8], but only in part: although there is some repetition, the present argument is different, especially by virtue of rethinking complementarity in terms of the character of quantum law.
Schrödinger’s derivation of quantum mechanics was primarily based on different, classical-like, principles, but could not avoid bringing in quantum ones, against his own grain, as discussed in [8, pp. 84–98].
It is not possible to survey these interpretations here. Just as does the Copenhagen interpretation, each rubric, on by now a long list (e.g., the many-worlds, consistent-histories, modal, relational, transcendental-pragmatist, and so forth) contains different versions. The literature dealing with the subject is immense. Standard reference sources, such as Wikipedia (“Interpretations of Quantum Mechanics” [11]), would list the most prominent such rubrics.
While the present argument advocates an overall nonrealist position in the spirit of Copenhagen, it is not a critique of realism, still a generally preferred philosophical view. One must also be mindful of additional complexities involved. One might, for example, argue against a realist interpretation of the wave function, specifically, as a continuous entity, and maintain a realist view of quantum mechanics, as referring to a discrete ultimate ontology. I refer here to Rovelli’s elegant article [13]. In the present view, following Bohr, no ontology, either continuous or discontinuous, can again be assigned at the ultimate level (of quantum objects and processes), but only an ultimately discrete ontology at the level of quantum phenomena, defined by what is observed in measuring instruments.
See, Jaeger’s helpful analysis of the concept of quantum objects from a realist perspective [16].
I distinguish causality, which is an ontological category, describing reality, from determinism, which is an epistemological category, describing our ability to predict the state of a system (ideally) exactly at any moment of time once we know its state at a given moment of time. Determinism is sometimes used in the same sense as causality is used here, and in the case of classical mechanics (which deals with single objects or a small number of objects), causality and determinism coincide. Once a classical system is large, one can no longer predict its causal behavior exactly, which is one of my reasons for distinguishing causality and determinism. Nonrealist interpretations of quantum mechanics automatically preclude not only determinism but also causality, and only allow for probabilistic or statistical predictions even in dealing with individual quantum objects. I shall explain the sense in which such predictions concerning individual quantum events could be statistical below.
This requirement is strengthened by special relativity, which restricts causes to those occurring in the backward (past) light cone of the event that is seen as an effect of this cause, while no event can be a cause of any event outside the forward (future) light cone of that event. These restrictions follow from the assumption that causal influences cannot travel faster than the speed of light in a vacuum, c. Principle theories do not require classical causality, which becomes difficult to assume in quantum physics, without violating special relativity or more generally locality, defined by the assumption that all physical influences are local. What may be called “the relativistic causality,” the prohibition of physical influences towards the past, may be maintained, as it is in quantum mechanics or quantum field theory, in the absence of classical causality. Relativity itself is, again, classically causal.
Although they do capture quantum discreteness as a defining principle of quantum theory, these postulates should not be confused with Bohr’s quantum postulate, introduced, along with complementarity, in the so-called Como lecture, in 1927. This postulate “attributes to any atomic process [rather than only to quantum jumps] an essential discontinuity, or rather individuality, ... and is symbolized by Planck’s quantum of action” [5, v. 1, p. 53]; emphasis added). For an illuminating discussion of Bohr’s 1913 postulates, see Folse’s article [17].
I have considered it on previous occasions, most recently in [8, pp. 68–83].
While Bohr did not expressly state his position on this point, it appears to accord with this view [8, pp. 180–184].
This concept of randomness is not ontological, because one cannot ascertain the reality of this randomness, but epistemological. It is ultimately a matter of assumption or belief, practically justified in a given interpretation.
It is true that the concept of complementarity, for example, as defined here, is very general and allows for applications beyond physics. Bohr (tentatively) and others, such as W. Pauli, K. G. Jung, M. Delbrück, and others proposed using it in philosophy, biology, and psychology. I shall, however, not consider these extensions here.
Part of the genealogy of the concept of complementarity was a conception, developed by Bohr in early psychological studies (before he began to study physics), that human cognitions must, under certain circumstances, be positioned in incompatible planes. Bohr saw this situation as analogous to the way Riemann surfaces work in the theory of functions of complex variables. A Riemann surface allow one to remove ambiguity and properly define functions of complex variables, such as \(f(z)= \sqrt{z}\). It is ambiguous, and hence not properly definable, when considered on the complex plane, as \(\sqrt{z}\) has two meanings, but is well defined on the corresponding Riemann surface, because it has a single meaning on each of the two mutually exclusive, “complementary,” sheets of the surface. Bohr reflected on this analogy, specifically using the Riemann surface for \(\sqrt{z}\) as an example, in his final interview (he died shortly thereafter) [30]. Of course, the Riemann surface for \(\sqrt{z}\) rigorously defines two mutually exclusive mathematical domains without involving probability. In quantum physics, while the physical behavior of quantum objects of the same type is different and indeed incompatible in each complementary experimental setup, the quantum-mechanical formalism, while equally capably of predicting this behavior in both setups, does so only in probabilistic or statistical terms, which is, as I explained, in accord with what is observed. Nevertheless, the analogy is telling as concerns the character of Bohr’s thinking and the architecture of concepts, as considered above.
In part correlatively, complementarity was also considered differently in the Como lecture, by using, as the main example of the concept, the complementarity of “the space-time coordination and the claim of causality” [5, v. 1, pp. 54–55]. Bohr abandoned this complementarity, along with the view in question, because he abandoned the possibility of applying the concept of causality in quantum physics in any circumstances, rather than allowing it in certain circumstances, as complementary to the space-time coordination. As noted earlier, in Bohr’s later works, the concept of complementarity became primarily associated with two complementarities—that of the position and momentum measurements and that of the time and the energy measurements, correlative to the uncertainty relations.
It is conceivable that a nonrepresentational model would provide exact predictions of the outcomes of the individual processes considered. Einstein, however, always, including in EPR’s article and related communications, understood completeness in terms of representational realist models (e.g., [29, p. 138]).
I only cite some of the key earlier experiments. There have been numerous experiments performed since, some in order to find loopholes in these and other experiments, seen as confirming Bell’s theorem.
The thought experiment (dealing with continuous variables), proposed by EPR cannot be physically realized because the EPR-entangled quantum state is not normalizable, which does not affect the fundamentals of the case. There are experiments (e.g., those involving photon pairs produced in parametric down conversion) that statistically approximate the idealized entangled state constructed by EPR for continuous variables. I cannot consider these experiments here, but they can be shown to be consistent with the present argument.
See also the remainder of Mermin’s excellent discussion, which contains an elegant proof of Bell’s theorem [47, pp. 110–176].
As Schrödinger was the first to note, one could simultaneously make alternative (complementary) measurements on \(S_{1 }\), say, the position measurement, which determines its position, and \(S_{2}\), the momentum measurement, which determines its momentum, and thus simultaneously predict (ideally exactly) the alternative second variable for each system, the momentum for \(S_{1}\) and the position for \(S_{2}\). This joint determination, however, is not simultaneous in the same location, and, thus, is in accord within EPR’s initial criterion of reality, without the restriction in question.
Einstein does note on the same occasion (and elsewhere) that the paradox is eliminated if quantum mechanics is only a statistical theory of ensembles and not of individual events, because, in this case, no single measurement of a given variable on \(S_{1}\) or, more accurately, \(S_{1n}\) determines the value of the corresponding variable on \(S_{2n }\) [37, p. 205].
There are realist views of quantum entanglement, either in realist interpretations of quantum mechanics (for example, the many worlds interpretation) or in alternative theories, such as Bohmian mechanics, or those in which the level of reality handled by quantum mechanics is underlain by a deeper reality (even within the proper, low-energy, scope of quantum mechanics), such as that of classical random fields, recently proposed by Khrennikov [48]. The so-called superdeterminism is another realist view of entanglement and quantum phenomena that might be mentioned here, as offering a particularly striking contrast to the present view, because it explains away the complexities discussed here by denying a free choice of performing one or the other EPR measurement, a free choice central to Bohr’s position and defining complementarity (e.g., [49]).
References
Feynman, R.: The Character of Physical Law. MIT Press, Cambridge, MA (1965) rpt (1994)
Bohr, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696–702 (1935)
Schrödinger, E.: The present situation in quantum mechanics (1935). In: Wheeler, J.A., Zurek, W.H. (eds.) Quantum Theory and Measurement, pp. 152–167. Princeton University Press, Princeton (1983)
Heisenberg, W.: The Physical Principles of the Quantum Theory. Dover, New York (1930). Translated by Eckhart, K., and Hoyt, F. C. rpt (1949)
Bohr, N.: The Philosophical Writings of Niels Bohr, vol. 3. Ox Bow Press, Woodbridge (1987)
Wheeler, J.A.: Geons, Black Holes, and Quantum Foam: A Life in Physics. W. W. Norton, New York (1998)
Von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1932). R. T. rpt, Translated by Beyer (1983)
Plotnitsky, A.: The Principles of Quantum Theory, from Planck’s Quanta to the Higgs Boson: The Nature of Quantum Reality and the Spirit of Copenhagen. Springer/Nature, New York (2016)
Deleuze, G., Guattari, F.: What is Philosophy? Columbia University Press, New York (1994). Translated by Tomlinson, H., and Burchell, G
Bohr, N.: Causality and complementarity (1937). In: Faye, J., Folse, H.J. (eds.) The Philosophical Writings of Niels Bohr. Causality and Complementarity, Supplementary Papers, vol. 4, pp. 83–91. Ox Bow Press, Woodbridge, CT (1994)
Interpretations of quantum mechanics. Wikipedia. http://en.wikipedia.org/wiki/Interpretationsofquantummechanics
Einstein, A.: 1919, What is the Theory of Relativity? (1919). Einstein, A. Ideas and Opinions, pp. 227–231. Bonanza Books, New York (1954)
Rovelli, C.: An argument against a realistic interpretation of the wave function. Found. Phys. 46, 1229–1237 (2016)
Plotnitsky, A., Khrennikov, A.: Reality without realism: on the ontological and epistemological architecture of quantum mechanics. Found. Phys. 25(10), 1269–1300 (2015)
Kant, I.: Critique of Pure Reason. Cambridge University Press, Cambridge (1997). Translated by Guyer. P., and Wood, A. W
Jaeger, G.: Quantum Objects: Non-local Correlation, Causality and Objective Indefiniteness in the Quantum World. Springer, New York (2013)
Folse, H.J.: The methodological lesson of complementarity: Bohr’s naturalistic epistemology. Phys. Scr. T163, 014001 (2014). doi:10.1088/0031-8949/2014/T163/014001
Heisenberg, W.: Quantum-theoretical re-interpretation of kinematical and mechanical relations. In: Van der Waerden, B.L. (ed.) Sources of Quantum Mechanics, pp. 261–277. Dover, New York, Reprint 1968 (1925)
Mehra, J., Rechenberg, H.: The Historical Development of Quantum Theory, 6 vols. Springer, Berlin (2001)
Chiribella, G., D’Ariano, G.M., Perinotti, P.: Informational derivation of quantum theory. Phys. Rev. A 84, 012311-1–012311-39 (2011)
D’Ariano, G.M., Chribella, G., Perinotti, P.: Quantum Theory from First Principles: An Informational Approach. Cambridge University Press, Cambridge (2017)
Hardy, L.: Foliable operational structures for general probabilistic theory. In: Halvorson, H. (ed.) Deep Beauty: Understanding the Quantum World Through Mathematical Innovation, pp. 409–442. Cambridge University Press, Cambridge (2011)
Wheeler, J.A.: Information, physics, quantum: the search for links. In: Zurek, W.H. (ed.) Complexity, Entropy, and the Physics of Information. Addison-Wesley, Redwood City, CA (1990)
Zeilinger, A.: A foundational principle for quantum mechanics. Found. Phys. 29(4), 631–643 (1999)
Hardy, L.: Quantum mechanics from five reasonable axioms. arXiv:quant-ph/0101012v4 (2001)
Fuchs, C.A.: Quantum mechanics as quantum information, mostly. J. Mod. Opt. 50, 987–223 (2003)
Fuchs, C.A., Mermin, N.D., Schack, R.: An introduction to QBism with an application to the locality of quantum mechanics. Am. J. Phys. 82, 749 (2014). doi:10.1119/1.4874855
Pauli, W.: Writings on Physics and Philosophy. Springer, Berlin (1994)
Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? In: Wheeler, J.A., Zurek, W.H. (ed.) Quantum Theory and Measurement, pp. 138–141. Princeton University Press, Princeton, Reprint 1983 (1935)
Bohr, N.: Interview with Thomas Kuhn, Aage Petersen, and Eric Rüdinger, 17 November 1962, Niels Bohr Archive. Copenhagen and American Institute of Physics, College Park, MD (1962)
Born, M.: Quantenmechanik der Stoßvorgänge. Z. Phys. 38, 803–827 (1926)
Ozawa, M.: Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurements. Phys. Rev. A 67, 042105 (2003)
Busch, P., Shilladay, C.: Complementarity and uncertainty in Mach-Zehnder interferometry and beyond. Phys. Rep. 435, 1–31 (2006)
Bohr, N.: The Causality Problem in Atomic Physics (1938). In: Faye, J., Folse, H.J. (eds.) The Philosophical Writings of Niels Bohr. Causality and Complementarity, Supplementary Papers, vol. 4, pp. 94–121. Ox Bow Press, Woodbridge, CT (1987)
Schrödinger, E.: Discussion of probability relations between separated systems. Proc. Camb. Philos. Soc. 31, 555–563 (1935)
Schrödinger, E.: Discussion of probability relations between separated systems. Proc. Camb. Philos. Soc. 32, 446–452 (1936)
Born, M.: The Einstein-Born Letters. Walker, New York (2005). Translated by Born, I
Einstein, A.: Remarks to the essays appearing in this collective volume. In: Schillp, P. (ed.) Albert Einstein: Philosopher-Scientist, pp. 663–688. Tudor, New York, New York (1949)
Plotnitsky, A.: Epistemology and Probability: Bohr, Heisenberg. Schrödinger and the Nature of Quantum-Theoretical Thinking. Springer, New York (2009)
Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going beyond Bell’s theorem. In: Kafatos, M. (ed.) Bell’s Theorem, Quantum Theory and Conceptions of the Universe, pp. 69–72. Kluwer, Dordrecht (1989)
Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A.: Bell’s theorem without inequalities. Am. J. Phys. 58, 1131–1142 (1990)
Hardy, L.: Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett. 71, 1665–1668 (1993)
Aspect, A., Dalibard, J., Roger, G.: Experimental test of Bell’s inequalities using time varying analyzers. Phys. Rev. Lett. 49, 1804–1807 (1982)
D’Ariano, G.M., Jaeger, G.M., Khrennikov, G., Plotnitsky, A. (eds.): Quantum theory: advances and problems. Phys. Scr. T163, 014006 (2014)
Khrennikov, A., de Raedt, H., Plotnitsky, A., Polyakov, S. (eds.): Probing the Limits of Quantum Mechanics: Theory and Experiment. 1 and 2, Found. Phys. 1(45), 7 (2015) (Found. Phys. 2(45), 8)
D’Ariano, G.M., Khrennikov, A. (eds.): Quantum foundations: information approach. Philos. Trans. R. Soc. A 374, 20150244 (2016)
Mermin, N.D.: Boojums All the Way Through. Cambridge University Press, Cambridge (1990)
Khrennikov, A.: Demystification of Quantum Entanglement, arXiv:0905.4791v3 [physics.gen-ph] (2010)
‘t Hooft, G.: Determinism in free bosons. Int. J. Theor. Phys. 42, 355–361 (2003)
Penrose, R.: On gravity’s role in quantum state reduction. In: Callender, C., Huggett, N. (eds.) Physics Meets Philosophy at the Planck Scale: Contemporary Theories of Quantum Gravity, pp. 290–304. Cambridge University Press, Cambridge (2001)
Haroche, S.: Entanglement and decoherence in cavity quantum electrodynamics experiments. In: Gonis, T., Turchi, P.E.A. (eds.) Decoherence and Its Implications in Quantum Computation and Information Transfer, pp. 211–223. IOS Press, Amsterdam (2001)
Haroche, S., Raimond, J.-M.: Exploring the Quantum: Atoms, Cavities, and Photons. Oxford University Press, Oxford (2006)
Acknowledgements
I would like to thank Mauro D’Ariano, Jan Faye, Henry Folse, Laurent Freidel, Lucien Hardy, Gregg Jaeger, Andrei Khrennikov, and Paolo Perinotti for invaluable discussions concerning the subjects addressed in this article. I am also grateful to both anonymous readers of the article for helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
“The deep truth is imageless.” —P. B. Shelley, Prometheus Unbound (Act II.iv.116)
Rights and permissions
About this article
Cite this article
Plotnitsky, A. On the Character of Quantum Law: Complementarity, Entanglement, and Information. Found Phys 47, 1115–1154 (2017). https://doi.org/10.1007/s10701-017-0101-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-017-0101-8