Abstract
In this paper, I attempt a personal account of my understanding of the measurement problem in quantum mechanics, which has been largely in the tradition of the Copenhagen interpretation. I assume that (i) the quantum state is a representation of knowledge of a (real or hypothetical) observer relative to her experimental capabilities; (ii) measurements have definite outcomes in the sense that only one outcome occurs; (iii) quantum theory is universal and the irreversibility of the measurement process is only “for all practical purposes”. These assumptions are analyzed within quantum theory and their consistency is tested in Deutsch’s version of the Wigner’s friend gedanken experiment, where the friend reveals to Wigner whether she observes a definite outcome without revealing which outcome she observes. The view that holds the coexistence of the “facts of the world” common both for Wigner and his friend runs into the problem of the hidden variable program. The solution lies in understanding that “facts” can only exist relative to the observer.
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Notes
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The notion of irreducibility can be weakened to the requirement that the predictions conditioned on the variables are not more informative about the outcomes of future measurements than the predictions of quantum theory [7]. Formally, for every measurement, the probability distribution conditioned on the variable cannot have lower (Shannon) entropy than the quantum probability distribution.
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The so-called “non-local” features of quantum theory are not a subject of the present article. I should, however, like to mention that once one accepts the notion that probabilities can be irreducible, there is no reason to restrict them to be locally causal [11], i.e. to be decomposable as: \(p(a,b|x,y) = \int d\lambda \rho (\lambda ) P(a|x,\lambda ) P(b|y,\lambda )\), where x and y are choices of measurement settings in two separated laboratories, a and b are respective outcomes and \(\rho (\lambda )\) is a probability distribution. It appears that the main misunderstanding associated with Bell’s theorem stems from a failure to acknowledge the irreducibility of quantum probabilities irrespectively of the relative experimental space-time arrangements [12]. Bell’s local causality accepts that probabilities for local outcomes can be irreducibly probabilistic, but requires those for correlations to be factorized into (a convex mixture of) probabilities for local outcomes. There is no need for imposing such a constraint on a probabilistic theory, where probabilities are considered to be fundamental. Rather, the notion of locality should be based on a operationally well-defined no-signaling condition, and it is this condition whose violation is at odds with special relativity.
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Peres correctly notes that considering hypothetical observers is not a prerogative of quantum theory [20]. They are also used in thermodynamics, when we say that a perpetual-motion machine of the second kind cannot be built, or in the theory of special relativity, when we say that no signal can be transferred faster than the speed of light.
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In the same paper, Heisenberg concludes: “... the quantum mechanical predictions about the outcome of an arbitrary experiment are independent of the location of the cut ...” This can be seen as a consequence of “purification” in quantum theory, which states that every mixed state of system A can always be seen as a state belonging to a part of a composite system AB that itself is in a pure state. This state is unique up to a reversible transformation on B. The assumption of purification is one of the central features of quantum theory, which, taken as an axiom together with a few other axioms, makes it possible to explain why the theory has the very mathematical structure it does [23].
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When we introduce coarse-grained observables, we need to define the states that are “close” to each other to conflate them into coarse-grained outcomes. However, the terms “close” or “distant” make sense in a classical context only. There, “close” states correspond to neighboring outcomes in the real configuration space. For example, the coherent states conflated in the single outcome \(\Omega _0\) of the POVM all correspond to approximately the same direction \(\Omega _0\) in real space. Therefore, certain features of classicality need to be presumed before macroscopic states can be defined. An alternative would be the attempt to reconstruct the notions of closeness, distance, and space—and consequently, also the theories referring to these notions, such as quantum field theory—from within the formalism of the Hilbert space only. Useful tools for this attempt might be preferred tensor factorizations, coarse-grained observables, and symmetries. The results of Refs. [24, 25] present the first progress towards this goal. The most elementary quantum system, the qubit, resides in an abstract state space with SU(2) symmetry. This is locally isomorphic to the group SO(3) of rotations in three-dimensional space. Considering directional degrees of freedom (spin), this symmetry is found to be operationally justified in the symmetry of the configuration of macroscopic instruments used for transforming the spin state. Hereby one assumes that quantum theory is “closed”: the macroscopic instruments do not lie outside of the theory, but are described from within it in the limit of a large number of its constituents (as coherent states or “classical fields”) [25].
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The classical world arises from within quantum theory when neighboring outcomes are not distinguished but bunched together into slots in the measurements of limited precision. What would the classical world look like if non-neighboring outcomes were conflated to slots? To address this question, one could imagine an experiment on a person whose nerve fibers behind the retina are disconnected and again reconnected at different, randomly chosen, nerve extensions connecting to the brain. It seems reasonable to assume that the neighboring points of the object that is illuminated with light and observed by the person’s eye will no more be perceived by the person as neighboring points. One may wonder if, in the course of further interaction with the environment, the person’s brain will start to make sense out of the seen “disordered classical world”, or if it will post-process the signals to search for more “ordered” structures as a prerequisite for making sense out of them. The latter may eventually nullify the effect of the random reconnection of nerves, and the person will again perceive the ordinary classical world.
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In a quantum mechanical experiment, the “observer” could be simulated by a qutrit with the following encoding [38]: \(|0\rangle \) for “knowing spin up”, \(|1\rangle \) for “knowing spin down” and \(|2\rangle \) for “I see no definite outcome”. The message is then encoded either in \(|2\rangle \) or in a state with the two-dimensional support spanned by vectors \(|0\rangle \) and \(|1\rangle \) (for example \(\frac{1}{2}(|0\rangle \langle 0|+|1\rangle \langle 1|\)). The superobserver applies the measurement with the projectors \(\hat{P}_1=|0\rangle \langle 0| + |1\rangle \langle 1|\) and \(\hat{P}_2=|2\rangle \langle 2|\).
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It seems to me that Deutsch had this particular view in mind when he claimed that the Copenhagen interpretation predicts the occurrence of the collapse. I see this view at most as a variant of the interpretation and (to my knowledge) not widely spread.
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A stronger argument against the possibility of seeing “blurred reality” has been brought to my attention by Jacques Pienaar. Consider a different experimental scenario where a referee prepares a state of the system either in a definite state \(|z + \rangle \) or \(|x + \rangle \) and the observer still performs the spin measurement along z-axis. In the first case the observer sees spin up and informs the superobserver about his outcome (for example by writing a message: “I see spin up along z-axis”). If the observer in the second case instead sees no definite outcome, e.g. a “blurred reality”, he informs the superobserver about this too, e.g. through the message “I observe no definite outcome”. The protocol would allow the superobserver to distinguish between nonorthogonal states perfectly. This is in disagreement with the laws of quantum mechanics.
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Here “locality” means that, for example, value \(A_1\) depends only on the local setting of Alice and not on the distant one of Bob. In a non-local hidden variable theory, we would need to distinguish between \(A_{11}\) and \(A_{12}\), depending on whether Bob’s setting is 1 or 2, respectively. It is not necessary to assume local deterministic values to derive Bell’ inequalities. Bell’s local causality is sufficient [11]. This however does not change the conclusions [12].
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One might be tempted to assume that the “facts” of the superobserver are the “real” ones, as he definitely has more reliable measurement instruments than the observer. This view cannot withstand the objection that the superobserver himself might be an object observed by yet another observer, the supersuperobserver, who describes the interference experiment of the superobserver quantum mechanically. The regression of increasingly more powerful observers might eventually find its end in a universe with a finite amount of resources.
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References
J. Bub, I. Pitowsky, Two dogmas about quantum mechanics, in Many Worlds? Everett, Quantum Theory, and Reality, ed. by S. Saunders, J. Barrett, A. Kent, D. Wallace (Oxford University Press, 2010), pp. 431–456
I. Pitowsky, Quantum mechanics as a theory of probability, in Festschrift in honor of Jeffrey Bub, ed. by W. Demopoulos, I. Pitowsky (Springer, Western Ontario Series in Philosophy of Science, New York, 2007)
T. Maudlin, Three measurement problems. Topoi 14(1), 7–15 (1995)
C. Brukner, A. Zeilinger, Information and fundamental elements of the structure of quantum theory, in Time, Quantum, Information, ed. by L. Castell, O. Ischebeck (Springer, 2003)
C.A. Fuch, R. Schack, Quantum-Bayesian coherence. Rev. Mod. Phys. 85, 1693 (2007)
C. Rovelli, Relational quantum mechanics. Int. J. Theor. Phys. 35, 1637–1678 (1996)
R. Colbeck, R. Renner, No extension of quantum theory can have improved predictive power. Nat. Commun. 2, 411 (2011)
G.C. Ghirardi, A. Rimini, T. Weber, Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D 34, 470 (1986)
L. Diosi, Models for universal reduction of macroscopic quantum fluctuations. Phys. Rev. A 40, 1165–1174 (1989)
R. Penrose, On gravity’s role in quantum state reduction. Gen. Relat. Gravit. 28, 581–600 (1996)
J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, Collected Papers on Quantum Philosophy (Cambridge University Press, 2004)
M. Zukowski, Č. Brukner, Quantum non-locality—it ain’t necessarily so.., Special issue on 50 years of Bell’s theorem. J. Phys. A: Math. Theor. 47, 424009 (2014)
C.A. Fuchs, R. Schack, QBism and the Greeks: why a quantum state does not represent an element of physical reality. arXiv:1412.4211 (2014)
S. Osnaghi, F. Freitas, O. Freire Jr., The origin of the Everettian heresy. Stud. Hist. Philos. Mod. Phys. 40(2), 97–123 (2009)
K. Camilleri, M. Schlosshauer, Niels Bohr as philosopher of experiment: does decoherence theory challenge Bohr’s doctrine of classical concepts? Stud. Hist. Philos. Mod. Phys. 49, 73–83 (2015)
P. Heelan, Heisenberg and radical theoretical change. Z. Allgemeine Wissenschaftstheorie 6, 113–138 (1975)
AHQP, Archives for the History of Quantum Physics—Bohr’s Scientific Correspondence, 301 microfilm reels (American Philosophical Society, Philadelphia, 1986)
N. Harrigan, R.W. Spekkens, Einstein, incompleteness, and the epistemic view of quantum states. Found. Phys. 40, 125 (2010)
S. Malin, What are quantum states? Quantum Inf. Process. 5, 233–237 (2006)
A. Peres, When is a quantum measurement?. Ann. New York Acad. Sci. 480, New Tech. Ideas Quantum Meas. Theory 438 (1986)
E. Wigner, Symmetries and Reflections (Indiana University Press, 1967), p. 164
W. Pauli, Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg, vol. 2, ed. by K. von Meyenn, A. Hermann, V. F. Weisskopf (Springer, Berlin, 1985), pp. 1930–1939. For the English translation of Heisenberg’s manuscript with an introduction and bibliography see E. Crull, G. Bacciagaluppi. http://philsci-archive.pitt.edu/8590/
G. Chiribella, G. D’Ariano, P. Perinotti, Informational derivation of quantum theory. Phys. Rev. A 84, 012311 (2011)
M.P. Mueller, L. Masanes, Three-dimensionality of space and the quantum bit: an information-theoretic approach. New J. Phys. 15, 053040 (2013)
B. Dakic, Č. Brukner, The classical limit of a physical theory and the dimensionality of space, in Quantum Theory: Informational Foundations and Foils, ed. by G. Chiribella, R. Spekkens. (Springer, 2016) pp. 249–282. arXiv:1307.3984
J. Kofler, Č. Brukner, Classical world arising out of quantum physics under the restriction of coarse-grained measurements. Phys. Rev. Lett. 99, 180403 (2007)
W.H. Zurek, Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715 (2003)
J. Kofler, Č. Brukner, Conditions for quantum violation of macroscopic realism. Phys. Rev. Lett. 101, 090403 (2008)
A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic Publishers, New York, 1995)
A. Peres, Stability of quantum motion in chaotic and regular systems. Phys. Rev. A 30, 1610 (1984)
H.-J. Stöckmann, Q. Chaos, An Introduction (Cambridge University Press, Cambridge, 1999)
P. Jacquod, I. Adagideli, C.W.J. Beenakker, Decay of the Loschmidt echo for quantum states with sub-Planck-scale structures. Phys. Rev. Lett. 89, 154103 (2012)
A. Peres, Recurrence phenomena in quantum dynamics. Phys. Rev. Lett. 49, 1118 (1982)
A.E. Allahverdyan, R. Balian, T.M. Nieuwenhuizen, Understanding quantum measurement from the solution of dynamical models. Phys. Rep. 525, 1 (2013)
D. Deutsch, Quantum theory as a universal physical theory. Int. J. Theor. Phys. 24, I (1985)
E.P. Wigner, Remarks on the mind-body question, in The Scientist Speculates, ed. by I.J. Good (London, Heinemann, 1961)
X.Y. Zou, T.P. Grayson, L. Mandel, Observation of quantum interference effects in the frequency domain. Phys. Rev. Lett. 69, 3041 (1992)
C. Bennett, Private Communication
N. Bohr, Discussion with Einstein on epistemological problems in atomic physics, in Albert Einstein: Philosopher-Scientist, ed. by P.A. Schilpp (The Library of Living Philosophers, Evanston, Illinois, 1949)
L. Hardy, Quantum theory from five reasonable axioms. arXiv:quant-ph/0101012 (2001)
B. Dakic and C. Brukner, Quantum theory and beyond: is entanglement special?, in Deep Beauty: Understanding the Quantum World through Mathematical Innovation, ed. by H. Halvorson (Cambridge University Press, 2011), pp. 365–392
L. Masanes, M. Müller, A derivation of quantum theory from physical requirements. New J. Phys. 13, 063001 (2011)
A. Aufféves, P. Grangier, Contexts, systems and modalities: a new ontology for quantum mechanics. arXiv:1409.2120 (2014)
N. Bohr, The Philosophical Writings of Niels Bohr 3 (Ox Bow Press, Woodbridge, Conn., 1987)
N. Bohr, On the notions of causality and complementarity. Dialectica 2, 312–319 (1948)
As quoted in “The philosophy of Niels Bohr” by Aage Petersen, in the Bull. Atom. Sci. 19(7) (1963); “The Genius of Science: A Portrait Gallery” (2000) by Abraham Pais, p. 24, and “Niels Bohr: Reflections on Subject and Object” (2001) by Paul. McEvoy, p. 291
Acknowledgments
This work has been supported by the Austrian Science Fund (FWF) through CoQuS, SFB FoQuS, and Individual Project 2462. I would like to acknowledge discussions with Mateus Araujo, Borivoje Dakić, Philippe Grangier, Richard Healey, Johannes Kofler, Luis Masanes, Jaques Pienaar and Anton Zeilinger.
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Brukner, Č. (2017). On the Quantum Measurement Problem. In: Bertlmann, R., Zeilinger, A. (eds) Quantum [Un]Speakables II. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-38987-5_5
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