Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0809.0115 · Physical Review A

Unexpected reemergence of the von Neumann theorem

Marek Żukowski

Published Feb 23, 2009• 8 citations
arXivHTMLPDF

Abstract

It is shown here that the ``simple test of quantumness for a single system'' of Alicki and Van Ryn has exactly the same relation to the discussion of the problem of describing the quantum system via a classical probabilistic scheme (that is in terms of hidden variables or within a realistic theory) as the von Neumann theorem [Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932); Mathematical Foundations of Quantum Mechanics (Princeton University, Princeton, NJ, 1955)]. The latter one was shown by Bell [Rev. Mod. Phys. 38, 447 (1966)] to stem from an assumption that the hidden variable values for a sum of two noncommuting observables have to be, for each individual system, equal to sums of eigenvalues of the two operators. One cannot find a justification for such an assumption to hold for noncommeasurable variables. On the positive side, the criterion may be useful in rejecting models which are based on stochastic classical fields. Nevertheless, the example used by the authors has a classical optical realization.

DOI
10.1103/physreva.79.024103

Related papers

Commentaire de “Hilbert and von Neumann on the Mathematical Foundations of Quantum Mechanics”

Presses de l'Université Laval eBooks•Jun 15, 2020•0 citations

Homer Nodded: Von Neumann’s Surprising Oversight

N. David Mermin, Rüdiger Schack

We review the famous no-hidden-variables theorem in von Neumann’s 1932 book on the mathematical foundations of quantum mechanics (Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932 ). We describe the notorious gap in von Neumann’s argument, pointed out by Hermann (Abhandlungen der Fries’schen Schule 6:75–152, 1935 ) and, more famously, by Bell (Rev Modern Phys 38:447–452, 1966 ). We disagree with recent papers claiming that Hermann and Bell failed to understand what von Neumann was actually doing.

arXiv:1805.10311•Foundations of Physics•Jul 30, 2018•31 citations

Readers offer their own magic moments with John Bell

Robert B. Griffiths

Reinhold Bertlmann’s article “Magic moments with John Bell” was a pleasure to read. I add a note concerning some later developments that are significant for understanding the implications of Bell’s quantum foundations work.The crucial point is that under the Bell locality hypothesis for a joint measurement as given in the article,E(a,b)=∫dλρ(λ)A(a,λ)·B(b,λ)the quantity λ that represents the hidden variable(s) is a classical, not a quantum, object, and ρ(λ) is a classical probability distribution. Such expressions only make sense in the context of quantum theory when the operators of interest commute with each other, but that is not the case when this expression is used to derive a Bell inequality.Many years ago Arthur Fine pointed out that hidden variables and Bell inequalities were “imposing requirements to make well defined precisely those probability distributions for noncommuting observables whose rejection is the very essence of quantum mechanics.”11. A. Fine, Phys. Rev. Lett. 48, 291 (1982). https://doi.org/10.1103/PhysRevLett.48.291 Thus the real issue is not one of locality but, instead, the proper use of probabilities in quantum theory.Fine’s point has been confirmed by the later development of a fully consistent way of introducing probabilities in quantum mechanics.22. R. B. Griffiths, Stanford Encyclopedia of Philosophy (2014); http://plato.stanford.edu/entries/qm-consistent-histories. That approach has shown that when quantum mechanics is interpreted using subspaces of the Hilbert space rather than classical hidden variables to represent physical properties, it is local: There are no ghostly nonlocal influences.We cannot know how Bell would have responded to those new developments, which only came to full fruition after his untimely death. The reader willing to dig into my somewhat tedious discussion33. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press (2002), sec. 23.4. of the Einstein-Podolsky-Rosen and Bohm situation will find an explicit proof that a measurement by Alice has not the slightest effect on the spin on Bob’s side. More recent is a general argument that quantum mechanics satisfies a principle of Einstein locality: What is done to system A has no effect on system B as long as there is no interaction between the two.44. R. B. Griffiths, Found. Phys. 41, 705 (2011). https://doi.org/10.1007/s10701-010-9512-5Those proofs of quantum locality begin by taking very seriously John von Neumann’s proposal that quantum properties correspond to Hilbert subspaces, whose projectors do not, in general, commute with one another.55. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U. Press (1955), sec. 3.5. Bell, though, assumed that such properties can be represented by classical hidden variables, which always commute. Thus Bell’s theorem teaches that if one uses classical hidden variables in place of quantum properties, the result will contradict quantum theory and disagree with experiment. Indeed, while Bell started off by pointing out deficiencies in von Neumann’s argument against hidden variables, the end result of his work is an even better and much more conclusive argument, backed by experiment, for rejecting classical hidden variables.REFERENCESSection:ChooseTop of pageREFERENCES <<1. A. Fine, Phys. Rev. Lett. 48, 291 (1982). https://doi.org/10.1103/PhysRevLett.48.291, Google ScholarCrossref2. R. B. Griffiths, Stanford Encyclopedia of Philosophy (2014); http://plato.stanford.edu/entries/qm-consistent-histories. Google Scholar3. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press (2002), sec. 23.4. Google Scholar4. R. B. Griffiths, Found. Phys. 41, 705 (2011). https://doi.org/10.1007/s10701-010-9512-5, Google ScholarCrossref5. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U. Press (1955), sec. 3.5. Google Scholar© 2015 American Institute of Physics.

Physics Today•Nov 30, 2015•0 citations

On von Neumann and Bell Theorems Applied to Quantumness Tests

Robert Alicki

arXiv:0810.4708•Foundations of Physics•Feb 18, 2009•6 citations

Kochen-Specker Theorem and Games

Laura Mančinska

One of the statements one would hear pretty early in any quantum mechanics course is that for quantum particles there are properties which do not have definite values. For example, if we measure |+〉 in standard basis, the best we can do to describe the outcome is to say, that with probability 1/2 we will get 0 and with probability 1/2 we will get 1. Yet one could wonder, is this really the way things are or is it just because our description of this quantum particle is incomplete, but in fact there are some hidden variables inside every quantum system that contain enough information to tell the outcome of measurement (see Fig. 1). However, many people disdain the idea of hidden variables, so there has been rather long history of theorems that forbid theories involving hidden variables. Usually if one wants to prove some no hidden variables theorem he has to assume something about the hidden variables theories he is going to forbid. One of such assumptions is non-contextuality, which means that the outcome we get after measuring with an observable is the same regardless of what other observables we choose to measure at the same time. The first theorem forbidding hidden variables theories is due to von Neumann in 1932. Then for a third of a century everyone cited von Neumann’s work carelessly, till Bell in 1966 indicated that there was an unreasonable assumption in von Neumann’s proof and urged for further examination of the problem (see [6] and [1] for more historical details). Bell also constructed a hidden variables model for one qubit that could assign the outcome of any observable in a way consistent with quantum mechanics. He also proved two theorems that forbid non-contextual hidden variables theories using counterexamples with larger quantum systems. Yet surprisingly he himself thought that hidden variables do exists and considered these theorems only as characterizations of the hidden variables theory we should look for. In this essay we will discuss several versions of one of the two Bell’s theorems. The theorem we will consider was also independently proven by Kochen and Specker in 1967 (see [5]), hence it is called Bell-Kochen-Specker theorem. For shortness and in order not to mix this theorem with another Bell’s theorem from now on we will refer to it as Kochen-Specker theorem. We will be interested in proving several versions of this theorem and the following lemma will turn out to be of great use:

•Jan 1, 2007•0 citations

Generalization of von Neumann's theorem to irreproducible measurements within the framework of quantum field theory. V

Yu.M. Lomsadze

Russian Physics Journal•Sep 1, 1979•0 citations
It is shown here that the ``simple test of quantumness for a single system'' of Alicki and Van Ryn has exactly the same relation to the discussion of the problem of describing the quantum system via a classical probabilistic scheme (that is in terms of hidden variables or within a realistic theory) as the von Neumann theorem [Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932); Mathematical Foundations of Quantum Mechanics (Princeton University, Princeton, NJ, 1955)]. The latter one was shown by Bell [Rev. Mod. Phys. 38, 447 (1966)] to stem from an assumption that the hidden variable values for a sum of two noncommuting observables have to be, for each individual system, equal to sums of eigenvalues of the two operators. One cannot find a justification for such an assumption to hold for noncommeasurable variables. On the positive side, the criterion may be useful in rejecting models which are based on stochastic classical fields. Nevertheless, the example used by the authors has a classical optical realization.
We review the famous no-hidden-variables theorem in von Neumann’s 1932 book on the mathematical foundations of quantum mechanics (Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932 ). We describe the notorious gap in von Neumann’s argument, pointed out by Hermann (Abhandlungen der Fries’schen Schule 6:75–152, 1935 ) and, more famously, by Bell (Rev Modern Phys 38:447–452, 1966 ). We disagree with recent papers claiming that Hermann and Bell failed to understand what von Neumann was actually doing.
Reinhold Bertlmann’s article “Magic moments with John Bell” was a pleasure to read. I add a note concerning some later developments that are significant for understanding the implications of Bell’s quantum foundations work.The crucial point is that under the Bell locality hypothesis for a joint measurement as given in the article,E(a,b)=∫dλρ(λ)A(a,λ)·B(b,λ)the quantity λ that represents the hidden variable(s) is a classical, not a quantum, object, and ρ(λ) is a classical probability distribution. Such expressions only make sense in the context of quantum theory when the operators of interest commute with each other, but that is not the case when this expression is used to derive a Bell inequality.Many years ago Arthur Fine pointed out that hidden variables and Bell inequalities were “imposing requirements to make well defined precisely those probability distributions for noncommuting observables whose rejection is the very essence of quantum mechanics.”11. A. Fine, Phys. Rev. Lett. 48, 291 (1982). https://doi.org/10.1103/PhysRevLett.48.291 Thus the real issue is not one of locality but, instead, the proper use of probabilities in quantum theory.Fine’s point has been confirmed by the later development of a fully consistent way of introducing probabilities in quantum mechanics.22. R. B. Griffiths, Stanford Encyclopedia of Philosophy (2014); http://plato.stanford.edu/entries/qm-consistent-histories. That approach has shown that when quantum mechanics is interpreted using subspaces of the Hilbert space rather than classical hidden variables to represent physical properties, it is local: There are no ghostly nonlocal influences.We cannot know how Bell would have responded to those new developments, which only came to full fruition after his untimely death. The reader willing to dig into my somewhat tedious discussion33. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press (2002), sec. 23.4. of the Einstein-Podolsky-Rosen and Bohm situation will find an explicit proof that a measurement by Alice has not the slightest effect on the spin on Bob’s side. More recent is a general argument that quantum mechanics satisfies a principle of Einstein locality: What is done to system A has no effect on system B as long as there is no interaction between the two.44. R. B. Griffiths, Found. Phys. 41, 705 (2011). https://doi.org/10.1007/s10701-010-9512-5Those proofs of quantum locality begin by taking very seriously John von Neumann’s proposal that quantum properties correspond to Hilbert subspaces, whose projectors do not, in general, commute with one another.55. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U. Press (1955), sec. 3.5. Bell, though, assumed that such properties can be represented by classical hidden variables, which always commute. Thus Bell’s theorem teaches that if one uses classical hidden variables in place of quantum properties, the result will contradict quantum theory and disagree with experiment. Indeed, while Bell started off by pointing out deficiencies in von Neumann’s argument against hidden variables, the end result of his work is an even better and much more conclusive argument, backed by experiment, for rejecting classical hidden variables.REFERENCESSection:ChooseTop of pageREFERENCES <<1. A. Fine, Phys. Rev. Lett. 48, 291 (1982). https://doi.org/10.1103/PhysRevLett.48.291, Google ScholarCrossref2. R. B. Griffiths, Stanford Encyclopedia of Philosophy (2014); http://plato.stanford.edu/entries/qm-consistent-histories. Google Scholar3. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press (2002), sec. 23.4. Google Scholar4. R. B. Griffiths, Found. Phys. 41, 705 (2011). https://doi.org/10.1007/s10701-010-9512-5, Google ScholarCrossref5. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U. Press (1955), sec. 3.5. Google Scholar© 2015 American Institute of Physics.
One of the statements one would hear pretty early in any quantum mechanics course is that for quantum particles there are properties which do not have definite values. For example, if we measure |+〉 in standard basis, the best we can do to describe the outcome is to say, that with probability 1/2 we will get 0 and with probability 1/2 we will get 1. Yet one could wonder, is this really the way things are or is it just because our description of this quantum particle is incomplete, but in fact there are some hidden variables inside every quantum system that contain enough information to tell the outcome of measurement (see Fig. 1). However, many people disdain the idea of hidden variables, so there has been rather long history of theorems that forbid theories involving hidden variables. Usually if one wants to prove some no hidden variables theorem he has to assume something about the hidden variables theories he is going to forbid. One of such assumptions is non-contextuality, which means that the outcome we get after measuring with an observable is the same regardless of what other observables we choose to measure at the same time. The first theorem forbidding hidden variables theories is due to von Neumann in 1932. Then for a third of a century everyone cited von Neumann’s work carelessly, till Bell in 1966 indicated that there was an unreasonable assumption in von Neumann’s proof and urged for further examination of the problem (see [6] and [1] for more historical details). Bell also constructed a hidden variables model for one qubit that could assign the outcome of any observable in a way consistent with quantum mechanics. He also proved two theorems that forbid non-contextual hidden variables theories using counterexamples with larger quantum systems. Yet surprisingly he himself thought that hidden variables do exists and considered these theorems only as characterizations of the hidden variables theory we should look for. In this essay we will discuss several versions of one of the two Bell’s theorems. The theorem we will consider was also independently proven by Kochen and Specker in 1967 (see [5]), hence it is called Bell-Kochen-Specker theorem. For shortness and in order not to mix this theorem with another Bell’s theorem from now on we will refer to it as Kochen-Specker theorem. We will be interested in proving several versions of this theorem and the following lemma will turn out to be of great use: