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Table of Contents
                            Bell’s Theorem and Quantum Realism
1 Introduction
	1.1 Quantum Realism and Bell's Theorem
		1.1.1 Opening Remarks
		1.1.2 David Bohm's Theory of Hidden Variables
	1.2 Topics to be Covered
	1.3 Review of the Formalism of Quantum Mechanics
		1.3.1 The State and its Evolution
		1.3.2 Rules of Measurement
	1.4 Von Neumann's Theorem and Hidden Variables
		1.4.1 Introduction
		1.4.2 Von Neumann's Theorem
		1.4.3 Von Neumann's Impossibility Proof
		1.4.4 Refutation of the Impossibility Proof
		1.4.5 Summary and Further Remarks
		1.4.6 Schrödinger's Derivation of Von Neumann's Proof
2 Contextuality
	2.1 Gleason's Theorem
	2.2 Kochen and Specker's Theorem
	2.3 Contextuality and Gleason's, and Kochen  and Specker's Impossibility Proofs
		2.3.1 Procedure to Measure the Kochen  and Specker Observables
	2.4 Contextuality Theorems and Spectral Incompatibility
	2.5 Albert's Example and Contextuality
		2.5.1 Bohmian Mechanics and Albert's Example
3 The Einstein--Podolsky--Rosen Paradox,  Bell's Theorem and Nonlocality
	3.1 Introductory Comments
	3.2 Review of the Einstein--Podolsky--Rosen Paradox
		3.2.1 Rotational Invariance of the Spin Singlet State  and Perfect Correlations
		3.2.2 The EPR Incompleteness Argument  and Objective Realism
		3.2.3 The Einstein--Podolsky--Rosen Theorem
	3.3 Bell's Theorem
		3.3.1 Proof of Bell's Theorem
	3.4 Einstein--Podolsky--Rosen, Bell's Theorem, and Nonlocality
		3.4.1 Complex Arguments
		3.4.2 Conjunction of EPR Analysis with Bell's Theorem
		3.4.3 The `No-Superdeterminism' Assumption
	3.5 Summary
4 Schrödinger's Paradox and Nonlocality
	4.1 Introduction
	4.2 Schrödinger's Generalization of EPR
		4.2.1 The Einstein--Podolsky--Rosen Quantum State
		4.2.2 Schrödinger's Generalization and Maximal  Perfect Correlations
	4.3 Schrödinger's Paradox, and Incompleteness
		4.3.1 Perfect Correlations and Procedure of Measurement
		4.3.2 Schrödinger's Theorem
	4.4 EPR Quantum State and Other Maximally  Entangled States
		4.4.1 Generalized Form of the EPR State
		4.4.2 The General Form of a Maximally Entangled State
		4.4.3 Maximally Entangled State with Two Spin-1 Particles
	4.5 Schrödinger Nonlocality
		4.5.1 Schrödinger Paradox and Spectral  Incompatibility Theorems
		4.5.2 What is Proven by the Conway--Kochen  Free Will Theorem
	4.6 Schrödinger's Paradox and Von Neumann's Theorem
	4.7 Summary and Conclusions
Document Text Contents
Page 1

SpringerBriefs in Physics

For further volumes:

Editorial Board

Egor Babaev, University of Massachusetts, USA
Malcolm Bremer, University of Bristol, UK
Xavier Calmet, University of Sussex, UK
Francesca Di Lodovico, Queen Mary University of London, UK
Maarten Hoogerland, University of Auckland, New Zealand
Eric Le Ru, Victoria University of Wellington, New Zealand
James Overduin, Towson University, USA
Vesselin Petkov, Concordia University, Canada
Charles H.-T. Wang, The University of Aberdeen, UK
Andrew Whitaker, Queen’s University Belfast, UK

Page 54


42 3 The Einstein–Podolsky–Rosen Paradox, Bell’s Theorem and Nonlocality

Rosen paradox, what follows is a stronger conclusion. Not only do we have a conflict
of quantum mechanics with local realism, but Bell’s Theorem implies the incompat-
ibility of locality itself with quantum theory. I.e., one is left to conclude that quantum
theory is irreducibly nonlocal.

In this chapter, we review and elaborate on these arguments. We attempt to present
the issues in a very gradual manner, so that the essential nature of each step may
be easily grasped. We present first the Einstein–Podolsky–Rosen paradox and Bell’s
Theorem individually, and then consider the implications of their conjunction. It
is worth stressing here that the full ramifications of this chapter follow from this
combination which we present in the final section of the chapter.6

3.2 Review of the Einstein–Podolsky–Rosen Paradox

3.2.1 Rotational Invariance of the Spin Singlet State
and Perfect Correlations

The well-known work of Einstein, Podolsky, and Rosen, first published in 1935,
was not designed to address the possibility of nonlocality, as such. The title of the
paper was “Can Quantum Mechanical Description of Physical Reality be Considered
Complete?”, and the goal of these authors was essentially the opposite of authors
such as von Neumann: Einstein, Podolsky, and Rosen wished to demonstrate that the
addition of hidden variables to the description of state is necessary for a complete
description of a quantum system. According to these authors, the quantum mechan-
ical state description given by ψ is incomplete, i.e., it cannot account for all the
objective properties of the system. This conclusion is stated in the paper’s closing
remark: “While we have thus shown that the wave function does not provide a com-
plete description of the physical reality, we left open the question of whether such a
description exists. We believe, however, that such a theory is possible.”

Einstein, Podolsky, and Rosen arrived at this conclusion having shown that for
the system they considered, each of the particles must have position and momentum
as simultaneous “elements of reality”. Regarding the completeness of a physical
theory, the authors state: [1] (emphasis due to EPR) “Whatever the meaning assigned
to the term complete, the following requirement for a complete theory seems to be
a necessary one: every element of the physical reality must have a counterpart in

Footnote 5 (Continued)
a proof that realism is impossible in quantum physics. See Bethe [22], Gell-Mann [23], p. 172, and
Wigner [19], p. 291.
6 In particular, one cannot take the conclusion of the EPR paradox—the existence of noncontextual
hidden variables—as received and final. This conclusion is based not only on quantum mechanical
predictions, but also on the assumption of locality, which will ultimately be seen to fail. The status
of EPR becomes clearer when one recognizes that the analysis is in fact equivalent to a theorem, as
we demonstrate in Sect. 3.2.3.

Page 55


3.2 Review of the Einstein–Podolsky–Rosen Paradox 43

the physical theory”. This requirement leads them to conclude that the quantum
theory is incomplete, since it does not account for the possibility of position and
momentum as being simultaneous elements of reality. To develop this conclusion
for the position and momentum of a particle, the authors make use of the following
“sufficient condition” for a physical quantity to be considered as an element of
reality: “If without in any way disturbing a system, we can predict with certainty (i.e.
with probability equaling unity) the value of a physical quantity, then there exists an
element of physical reality corresponding to this quantity”.

What we shall present here7 is a form of the EPR paradox which was developed
in 1951, by David Bohm. Bohm’s EPR analysis involves the properties of the spin
singlet state of a pair of spin– 12 particles. Within his argument, Bohm shows that
various components of the spin of a pair of particles must be elements of reality
in the same sense as the position and momentum were for Einstein, Podolsky, and
Rosen. We begin with a discussion of the formal properties of the spin singlet state,
and then proceed with our presentation of the EPR incompleteness argument.

Because the spin and its components all commute with the observables associated
with the system’s spatial properties, one may analyze a particle with spin by separately
analyzing the spin observables and the spatial observables. The spin observables may
be analyzed in terms of a Hilbert space Hs (which is a two-dimensional space in the
case of a spin 12 particle) and the spatial observables in terms of L2(IR). The full
Hilbert space of the system is then given by tensor product Hs ⊗ L2(IR) of these
spaces. Hence, we proceed to discuss the spin observables only, without explicit
reference to the spatial observables of the system. We denote each direction in space
by its θ and φ coordinates in spherical polar coordinates, and (since we consider spin
2 particles) the symbol σ denotes the spin. To represent the eigenvectors of σθ, φ
corresponding to the eigenvalues + 12 and − 12 , we write | ↑ θ, φ〉, and | ↓ θ, φ〉,
respectively. For the eigenvectors of σz,we write simply |↑〉 and |↓〉.Often, vectors
and observables are expressed in terms of the basis formed by the eigenvectors of
σz . The vectors |↑ θ, φ〉, and |↓ θ, φ〉 when expressed in terms of these are

|↑ θ, φ〉 = cos(θ/2)| ↑〉 + sin(θ/2)eiφ | ↓〉
|↓ θ, φ〉 = sin(θ/2)e−iφ |↑〉 − cos(θ/2)|↓〉. (3.1)

For a system consisting of two spin 12 particles,
8 the states are often classified

in terms of the total spin S = σ (1) + σ (2) of the particles, where σ (1) is the spin
of particle 1 and σ (2) is the spin of particle 2. The spin singlet state, in which
we shall be interested, is characterized by S = 0. The name given to the state
reflects that it contains just one eigenvector of the z-component Sz of the total spin:
that corresponding to the eigenvalue 0. In fact, as we shall demonstrate, the spin
singlet state is an eigenvector of all components of the total spin with an eigenvalue

7 The Bohm spin singlet version and the original version of the EPR paradox differ essentially in
the states and observables with which they are concerned. We shall consider the original EPR state
more explicitly in Sect. 4.2.1.
8 See for example, Messiah [24], and Shankar [25].

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96 4 Schrödinger’s Paradox and Nonlocality

The linkage they make to human free will is quite superfluous, since the assumption
of free will can be dropped or even reversed in the argument without affecting its
conclusion. The argument’s assumptions SPIN, TWIN and FIN65 are just as reason-
able as the authors claim, but more powerful than they appear to have realized. By
the logical implications of the Schrödinger paradox (and Schrödinger’s Theorem)
these axioms are precisely what is needed to derive the inevitable existence of non-
contextual hidden variables on the appropriate components of S2

θ, φ
. The fact that the

Kochen and Specker theorem contradicts this leads finally to the conclusion that any
theory satisfying FIN must conflict with the quantum predictions.66 This is just the
same as the Schrödinger nonlocality argument we gave in Sect. 4.5.1, in which we
showed how a complex argument may be formed based on the Schrödinger paradox
and the Kochen and Specker theorem. What Conway and Kochen have developed is
a special case of the Schrödinger nonlocality proof.

All this might bring us to inquire how Schrödinger himself regarded his results,
and what further conclusions he drew within his remarkable paper. Clearly, it would
have been possible for him to argue for quantum nonlocality had he anticipated the
results of any of a wide variety of theorems including at least Gleason’s, Kochen
and Specker’s, or any other spectral incompatibility theorem. Instead, as we saw
in Chap. 1, Schrödinger essentially reproduced the von Neumann argument against
hidden variables, in his observation of a set of observables that obey a linear relation-
ship not satisfied by the set’s eigenvalues. What may be seen from von Neumann’s
result is just what Schrödinger noted—relationships constraining the observables do
not necessarily constrain their values. Schrödinger continued this line of thought by
speculating on the case for which no relation whatsoever constrained the values of the
various observables. In light of his generalization of the EPR paradox, these notions
led Schrödinger to the idea that the quantum system in question might possess an
infinite number of degrees of freedom, which concept is actually quite similar to that
of Bohmian mechanics.

If, on the other hand, Schrödinger had made von Neumann’s error, i.e., had con-
cluded the impossibility of a map from observables to values, this mistaken line of
reasoning would have allowed him to reach the conclusion of quantum nonlocal-
ity. Thus, insofar as one might regard the von Neumann proof as “almost” leading
to the type of conclusion that follows from Gleason’s theorem, one must consider
Schrödinger as having come precisely that close to a proof of quantum nonlocality.

It is quite interesting to see that many of the issues related to quantum mechanical
incompleteness and hidden variables were addressed in the 1935 work [8–10] of
Erwin Schrödinger. One could make the case that Schrödinger’s work is the most
far-reaching of the early analyses addressed to the subject. Schrödinger presented
here not only his famous “cat paradox,” but he developed results beyond those of the
Einstein, Podolsky, Rosen paper—an extension of their incompleteness argument,

65 As we noted in Sect. 4.5.2 there is nothing gained by appealing to the strong free will theorem
in which FIN is replaced by MIN.
66 Conflict with quantum mechanics cannot be laid at the feet of TWIN nor SPIN, since these are
taken from the theory itself.

Page 109

4.7 Summary and Conclusions 97

and an analysis of this result in terms closely related to von Neumann’s theorem.
It seems clear that the field of foundations of quantum mechanics might have been
greatly advanced had these features of Schrödinger’s paper been more widely appre-
ciated at the time it was first published.


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