Bell's Theorem Begs the Question

1. Introduction

1.1. Heuristics for completing quantum mechanics

The purpose of any hidden variable theory [2,17] is to reproduce the statistical predictions encoded in the quantum states $|\psi \rangle \in \mathcal{H}$ of physical systems using hypothetical dispersion-free states $|\psi ,\lambda ):=\left\{\right|\psi \rangle ,\lambda \}\in \mathcal{H}\otimes \mathcal{L}$ that have no inherent statistical character, where the Hilbert space $\mathcal{H}$ is extended by the space $\mathcal{L}$ of hidden variables $\lambda$, which are hypothesized to “complete” the states of the physical systems as envisaged by Einstein [18]. If the values of $\lambda \in \mathcal{L}$ can be specified in advance, then the results of any measurements on a given physical system are uniquely determined.

To appreciate this, recall that expectation value of the square of any self-adjoint operator $\mathsf{\Omega }\in \mathcal{H}$ in a normalized quantum mechanical state $|\psi \rangle$ and the square of the expectation value of $\mathsf{\Omega }$ will not be equal to each other in general:

$$\langle \psi \left|{\mathsf{\Omega }}^2\right|\psi \rangle \ne {〈\psi \left|\mathsf{\Omega }\right|\psi 〉}^2.$$

(1)

This gives rise to inherent statistical uncertainty in the value of $\mathsf{\Omega }$, indicating that the state $|\psi \rangle$ is not dispersion-free:

$$\mathsf{\Delta }\mathsf{\Omega }=\sqrt{\langle \psi \left|\right\{\mathsf{\Omega }-{\langle \psi \left|\mathsf{\Omega }\right|\psi \rangle \}}^2|\psi \rangle }\ne 0.$$

(2)

By contrast, in a normalized dispersion-free state $|\psi ,\lambda )$ of hidden variable theories formalized by von Neumann [2], the expectation value of $\mathsf{\Omega }$, by hypothesis, is equal to one of its eigenvalues $\omega \left(\lambda \right)$, determined by the hidden variables$\lambda$,

$$\left(\psi ,\lambda \left|\mathsf{\Omega }\right|\psi ,\lambda \right)=\omega \left(\lambda \right)⟺\mathsf{\Omega }|\psi ,\lambda )=\omega \left(\lambda \right)|\psi ,\lambda ),$$

(3)

so that a measurement of $\mathsf{\Omega }$ in the state $\left|\psi ,\lambda \right)$ would yield the result $\omega \left(\lambda \right)$ with certainty. How this can be accomplished in a dynamical theory of measurement process remains an open question [17]. But accepting the hypothesis (3) implies

$$(\psi ,\lambda |{\mathsf{\Omega }}^2|\psi ,\lambda )={(\psi ,\lambda |\mathsf{\Omega }|\psi ,\lambda )}^2.$$

(4)

Consequently, unlike in a quantum sate $|\psi \rangle$, in a dispersion-free state $|\psi ,\lambda )$ observables $\mathsf{\Omega }$ have no inherent uncertainty:

$$\mathsf{\Delta }\mathsf{\Omega }=\sqrt{(\psi ,\lambda |{\{\mathsf{\Omega }-\left(\psi ,\lambda \left|\mathsf{\Omega }\right|\psi ,\lambda \right)\}}^2|\psi ,\lambda )}=0.$$

(5)

The expectation value of $\mathsf{\Omega }$ in the quantum state $|\psi \rangle$ can then be recovered by integrating over the hidden variables$\lambda$:

$$〈\psi \left|\mathsf{\Omega }\right|\psi 〉={\int }_{\mathcal{L}}\left(\psi ,\lambda \left|\mathsf{\Omega }\right|\psi ,\lambda \right)p\left(\lambda \right)d\lambda ={\int }_{\mathcal{L}}\omega \left(\lambda \right)p\left(\lambda \right)d\lambda ,$$

(6)

where $p\left(\lambda \right)$ denotes the normalized probability distribution over the space $\mathcal{L}$ of thus hypothesized hidden variables.

As it stands, this prescription amounts to assignment of unique eigenvalues $\omega \left(\lambda \right)$ to all observables $\mathsf{\Omega }$simultaneously, regardless of whether they are actually measured. In other words, according to (6) every physical quantity of a given system represented by $\mathsf{\Omega }$ would possess a unique preexisting value, irrespective of any measurements being performed. In Section 2 of [17], Bell works out an instructive example to illustrate how this works for a system of two-dimensional Hilbert space. The prescription (6) fails, however, for Hilbert spaces of dimensions greater than two. This was proved independently by Bell [17], Kochen and Specker [19], and Belinfante [20], as a corollary to Gleason’s theorem [21,22].

These proofs – known as the Kochen-Specker theorem – do not exclude contextual hidden variable theories in which the complete state $|\psi ,\lambda )$ of a system assigns unique values to physical quantities only relative to experimental contexts [22]. If we denote the observables as $\mathsf{\Omega }\left(c\right)$ with c being the environmental contexts of their measurements, then the non-contextual prescription (6) can be easily modified to accommodate contextual hidden variable theories as follows:

$$〈\psi \left|\mathsf{\Omega }\right(c\left)\right|\psi 〉={\int }_{\mathcal{L}}\left(\psi ,\lambda \left|\mathsf{\Omega }\right(c\left)\right|\psi ,\lambda \right)p\left(\lambda \right)d\lambda ={\int }_{\mathcal{L}}\omega (c,\lambda )p\left(\lambda \right)d\lambda .$$

(7)

Each observable $\mathsf{\Omega }\left(c\right)$ is still assigned a unique eigenvalue $\omega (c,\lambda )$, but now determined cooperatively by the complete state $|\psi ,\lambda )$ of the system and the state c of its environmental context. Consequently, even though some of its features are no longer intrinsic to the system, contextual hidden variable theories do not have the inherent statistical character of quantum mechanics, because outcome of an experiment is a cooperative effect, just as it is in classical physics [22]. Therefore, such theories interpret quantum entanglement at the level of the complete state $|\psi ,\lambda )$ only epistemically.

For our purposes here, it is also important to recall that in the Hilbert space formulation of quantum mechanics [2] the correspondence between observables and Hermitian operators is one-to-one. Moreover, a sum $\tilde{\mathsf{\Omega }}\left(\tilde{c}\right)={\sum }_{i=1}^n{\mathsf{\Omega }}_i\left(c_i\right)$ of several observables such as ${\mathsf{\Omega }}_1\left(c_1\right),{\mathsf{\Omega }}_2\left(c_2\right),{\mathsf{\Omega }}_3\left(c_3\right),\cdots ,{\mathsf{\Omega }}_n\left(c_n\right)$ is also an observable representing a physical quantity, and consequently the sum of the expectation values of ${\mathsf{\Omega }}_i\left(c_i\right)$ is the expectation value of the summed operator $\tilde{\mathsf{\Omega }}\left(\tilde{c}\right)$,

$$\sum _{i=1}^n〈\psi |{\mathsf{\Omega }}_i\left(c_i\right)|\psi 〉=\langle \psi |\left[\sum _{i=1}^n{\mathsf{\Omega }}_i\left(c_i\right)\right]|\psi \rangle ,$$

(8)

regardless of whether the observables are simultaneously measurable or mutually commutative [17]. The question then is, since within any contextual hidden variable theory characterised by (7) all of the observables ${\mathsf{\Omega }}_i\left(c_i\right)$ and their sum $\tilde{\mathsf{\Omega }}\left(\tilde{c}\right)$ are assigned unique eigenvalues ${\omega }_i(c_i,\lambda )$ and $\tilde{\omega }(\tilde{c},\lambda )$, respectively, would these eigenvalues satisfy the equality

$$\sum _{i=1}^n\left[{\int }_{\mathcal{L}}{\omega }_i(c_i,\lambda )p\left(\lambda \right)d\lambda \right]\stackrel{?}{=}{\int }_{\mathcal{L}}\left[\sum _{i=1}^n{\omega }_i(c_i,\lambda )\right]p\left(\lambda \right)d\lambda$$

(9)

in dispersion-free states $|\psi ,\lambda )$ of physical systems in analogy with the linear quantum mechanical relation (8)? The answer is: Not in general, because the eigenvalue of the summed operator $\tilde{\omega }(\tilde{c},\lambda )\ne {\sum }_{i=1}^n{\omega }_i(c_i,\lambda )$ for specific hidden variables $\lambda$, unless all of the constituent observables ${\mathsf{\Omega }}_i\left(c_i\right)$ are mutually commutative or simultaneously measurable. As Bell points out in Section 3 of [17], the linear relation (8) is an unusual property of quantum mechanical states $|\psi \rangle$. There is no reason to demand it individually of the dispersion-free states $|\psi ,\lambda )$, whose function is to reproduce the measurable features of quantum systems only when averaged over, as in (7). I will come back to this point in Section 1.5.

Figure 1. In an EPR-Bohm-type experiment, a spin-less fermion, such as a neutral pion, is assumed to decay from a source into an electron-positron pair, as depicted. Then, measurements of the spin components of each separated fermion are performed at space-like separated observation stations $\mathbf{1}$ and $\mathbf{2}$, obtaining binary results $\mathcal{A}=\pm 1$ and $\mathcal{B}=\pm 1$ along directions $\mathbf{a}$ and $\mathbf{b}$. The conservation of spin momentum dictates that the total spin of the system remains zero during its free evolution. After Ref. [4].

Figure 1. In an EPR-Bohm-type experiment, a spin-less fermion, such as a neutral pion, is assumed to decay from a source into an electron-positron pair, as depicted. Then, measurements of the spin components of each separated fermion are performed at space-like separated observation stations $\mathbf{1}$ and $\mathbf{2}$, obtaining binary results $\mathcal{A}=\pm 1$ and $\mathcal{B}=\pm 1$ along directions $\mathbf{a}$ and $\mathbf{b}$. The conservation of spin momentum dictates that the total spin of the system remains zero during its free evolution. After Ref. [4].

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