On the spin projection operator and the probabilistic meaning of the bipartite correlation function

TL;DR

This paper explores the probabilistic interpretation of the bipartite spin correlation function, revealing that Bell inequalities may not be applicable due to the underlying probability space partitioning dependent on measurement directions.

Contribution

It provides a detailed analysis of the spin projection operator correlation function and challenges the conventional application of Bell inequalities to the singlet state.

Findings

Correlation function admits an unequivocal probabilistic interpretation

Partitioning of probability space depends on measurement directions

Questions the applicability of Bell inequalities to the singlet state

Abstract

Spin is a fundamental and distinctive property of the electron, having far-reaching consequences in wide areas of physics. Yet, further to its association with an angular momentum, the physics underpinning its formal treatment remains obscure. In this work we propose to advance in disclosing the meaning behind the formalism, by first recalling some basic facts about the one-particle spin operator. Consistently informed by and in line with the quantum formalism, we then proceed to analyse in detail the spin projection operator correlation function $C_Q(\boldsymbol{a}, \boldsymbol{b}) = \langle(\hat{\sigma}\cdot\boldsymbol{a})(\hat{\sigma}\cdot\boldsymbol{b})\rangle$ for the bipartite singlet state, and show it to be amenable to an unequivocal probabilistic reading. In particular, the calculation of $C_Q(\boldsymbol{a}, \boldsymbol{b})$ entails a partitioning of the probability space, which is dependent on the directions $(\boldsymbol{a}, \boldsymbol{b})$. The derivation of the CHSH- or other Bell-type inequalities, on the other hand, does not consider such partitioning. This observation puts into question the applicability of Bell-type inequalities to the bipartite singlet spin state.

Equations

$$|{+}_r⟩=\cos \frac{\theta }{2}|{+}_z⟩+e^{i\phi }\sin \frac{\theta }{2}|{-}_z⟩,$$

$$|{-}_r⟩=-e^{-i\phi }\sin \frac{\theta }{2}|{+}_z⟩+\cos \frac{\theta }{2}|{-}_z⟩,$$

$$\mathbf{r}=\mathbf{i}\sin \theta \cos \phi +\mathbf{j}\sin \theta \sin \phi +\mathbf{k}\cos \theta$$

$$\hat{\mathbf{\sigma }}\cdot \mathbf{k}|{\pm }_z⟩=\pm |{\pm }_z⟩,$$

$$U(\theta,\varphi)=\left(\begin{array}[]{cc}\cos\frac{\theta}{2}&-e^{-i\varphi}\sin\frac{\theta}{2}\\ e^{i\varphi}\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{array}\right)$$

$$\left|+_{z}\right\rangle=\left(\begin{array}[]{cc}1\\ 0\end{array}\right),\;\left|-_{z}\right\rangle=\left(\begin{array}[]{cc}0\\ 1\end{array}\right),$$

$$\hat{\mathbf{\sigma }}\cdot \mathbf{r}|{\pm }_r⟩=\pm |{\pm }_r⟩.$$

$$⟨{+}_r|\hat{\mathbf{\sigma }}|{+}_r⟩=\mathbf{r},⟨{-}_r|\hat{\mathbf{\sigma }}|{-}_r⟩=-\mathbf{r}$$

$$\hat{\bm{\sigma}}\cdotp\bm{a}=\left(\begin{array}[]{cc}a_{z}&a_{x}-ia_{y}\\ a_{x}+ia_{y}&-a_{z}\end{array}\right),$$

$$⟨{\pm }_r|\hat{\mathbf{\sigma }}\cdot \mathbf{a}|{\pm }_r⟩=\pm \mathbf{r}\cdot \mathbf{a}=\pm \cos {\theta }_{ra},$$

$$⟨{-}_r|\hat{\mathbf{\sigma }}\cdot \mathbf{a}|{+}_r⟩$$

$$\mathbf{\theta }$$

$$\mathbf{\phi }$$

$$\left|⟨{-}_r|\hat{\mathbf{\sigma }}\cdot \mathbf{a}|{+}_r⟩\right|=\mid \mathbf{r}\times \mathbf{a}\mid =\sin {\theta }_{ra},$$

$$1$$

$$\displaystyle=\left\langle+_{r}\right|\left(\hat{\bm{\sigma}}\cdotp\bm{a}\right)\big{[}\left|+_{r}\right\rangle\left\langle+_{r}\right|+\left|-_{r}\right\rangle\left\langle-_{r}\right|\big{]}\left(\hat{\bm{\sigma}}\cdotp\bm{a}\right)\left|+_{r}\right\rangle,$$

$$\left(\hat{\mathbf{\bm{\sigma}}}\cdotp\bm{a}\right)\left|+_{r}\right\rangle=\left(\begin{array}[]{c}\cos\theta_{a}\cos\frac{\theta}{2}+\sin\theta_{a}e^{i\left(\varphi-\varphi_{a}\right)}\sin\frac{\theta}{2}\\ \sin\theta_{a}e^{i\varphi_{a}}\cos\frac{\theta}{2}-\cos\theta_{a}e^{i\varphi}\sin\frac{\theta}{2}\end{array}\right).$$

$$\left(\hat{\mathbf{\bm{\sigma}}}\cdotp\bm{a}\right)\left|-_{r}\right\rangle=\left(\begin{array}[]{c}-\cos\theta_{a}e^{-i\varphi}\sin\frac{\theta}{2}+\sin\theta_{a}e^{-i\varphi_{a}}\cos\frac{\theta}{2}\\ -\sin\theta_{a}e^{-i\left(\varphi-\varphi_{a}\right)}\sin\frac{\theta}{2}-\cos\theta_{a}\cos\frac{\theta}{2}\end{array}\right),$$

$$⟨{+}_{ra}|\hat{\mathbf{\sigma }}|{+}_{ra}⟩={\mathbf{r}}_{\mathbf{a}},$$

$$|{+}_{ra}⟩\equiv \left(\hat{\mathbf{\sigma }}\cdot \mathbf{a}\right)|{+}_r⟩.$$

$${\mathbf{r}}_{\mathbf{a}}=2\left(\mathbf{r}\cdot \mathbf{a}\right)\mathbf{a}-\mathbf{r},$$

$$\left(\hat{\mathbf{\sigma }}\cdot \mathbf{a}\right)|{+}_{ra}⟩=|{+}_r⟩,$$

$$|{\Psi }^0⟩=\frac{1}{\sqrt{2}}\left(|{+}_r⟩|{-}_r⟩-|{-}_r⟩|{+}_r⟩\right),$$

$$⟨{\Psi }^0|\left(\hat{\mathbf{\sigma }}\cdot \mathbf{a}\right)\otimes \mathbb{I}|{\Psi }^0⟩$$

$$C_Q(\mathbf{a},\mathbf{b})=⟨{\Psi }^0|\left(\hat{\mathbf{\sigma }}\cdot \mathbf{a}\right)\otimes \left(\hat{\mathbf{\sigma }}\cdot \mathbf{b}\right)|{\Psi }^0⟩$$

$$C_Q(\mathbf{a},\mathbf{b})=$$

$$\displaystyle=-\big{[}(\bm{r}\cdot\bm{a})(\bm{r}\cdot\bm{b})+(\bm{\theta}\cdot\bm{a})(\bm{\theta}\cdot\bm{b})+(\bm{\varphi}\cdot\bm{a})(\bm{\varphi}\cdot\bm{b})\big{]}$$

$$=-\mathbf{a}\cdot \mathbf{b}.$$

$$|{\Psi }^1⟩$$

$$|{\Psi }^3⟩$$

$$C_Q(\mathbf{a},\mathbf{b})$$

$$F_1$$

$$F_2$$

Full text

I Introduction

Ever since Pauli’s dictum on the impossibility of a model for the electron spin, physicists have been taught to replace it with the abstract concept of spin as a dichotomic quantity living in its own Hilbert space hist . The introduction of spin as a postulate in nonrelativistic quantum mechanics and its successful treatment in terms of Pauli matrices and spinors seem to foreclose the need for a deeper reflection on its nature. And yet the very existence of the electron spin has a huge impact in various areas of physics, ranging from atomic structure to fermion statistics, the stability of matter, and quantum communication.

This paper is devoted to a close analysis of the spin operator and associated observables, with the intention to advance in the understanding of their meaning. The analysis carried out, fully within the conventional quantum formalism, serves to clarify certain assumptions usually made regarding the correspondence between the mathematical expressions involving spin operators and their geometrical interpretation. Such clarification bears particular relevance in the case of the (entangled) singlet state of the two-electron system, putting into question the interpretation of the Bell-type inequalities as conventionally applied to this case.

The paper is organized as follows. In section II, basic elements of the quantum description of the single spin $\nicefrac{{1}}{{2}}$ are reviewed. Section III contains a brief introduction to the entangled bipartite system, in preparation for the discussion on the physical meaning of the spin projection operator correlation function $C_{Q}(\bm{a},\bm{b})$ in section IV, where an appropiate disaggregation of $C_{Q}(\bm{a},\bm{b})$ in terms of the individual spin projection eigenfunctions is carried out, leading to expressions with a clear probabilistic meaning. In Section V these expressions are translated to the hidden-variable language, in order to make contact with the Bell-type inequalities. The paper concludes with a discussion on the applicability of such inequalities to the bipartite entangled spin state, and relates this discussion to relevant literature on the subject.

II Single-spin state and spin projections

Let us start by recalling that the most general one-particle (pure) spin $\nicefrac{{1}}{{2}}-$state can be expressed in terms of the spinor (except for an irrelevant overall factor $e^{i\eta}$)

[TABLE]

and the spinor orthogonal to it,

[TABLE]

with $0\leq\theta\leq\pi$ and $0\leq\varphi\leq 2\pi$, $\theta$ and $\varphi$ being the zenithal and azimuthal angles that define the unit vector

[TABLE]

in 3D space, identified in the literature as the Bloch vector (see, e. g., Ref. JA ).

In terms of the Pauli matrices $\hat{\sigma}_{i}$ ($i=x,y,z)$, the spin operator is given by $\mathbf{\hat{s}}=(\hbar/2)\mathbf{\hat{\bm{\sigma}}}$; however, we shall refer throughout to $\mathbf{\hat{\bm{\sigma}}}$ as the spin operator, for simplicity. The states $\left|\pm_{z}\right\rangle$ are such that

[TABLE]

i. e., they are eigenstates of the spin projection operator along $\mathbf{k}$. The vectors $\left|\pm_{r}\right\rangle$, in their turn, which can be obtained by applying the unitary rotation matrix

[TABLE]

to the spinors

[TABLE]

satisfy the eigenvalue equation

[TABLE]

Therefore, $\left|+_{r}\right\rangle$ and $\left|-_{r}\right\rangle$ correspond to states representing, respectively, parallel and antiparallel spin projections along the arbitrary direction $\bm{r}$. The expectation value of $\bm{\hat{\sigma}}$ in the states (1) and (2) is

[TABLE]

whence one may associate indeed the direction $\bm{r}$ with the spin state $\left|+_{r}\right\rangle$, and similarly the direction $-\bm{r}$ with the spin state $\left|-_{r}\right\rangle$.

Consider now a second arbitrary direction, determined by the unitary vector $\bm{a}=(a_{x},a_{y},a_{z})$. The projection of the spin operator along this new direction takes on the matrix form

[TABLE]

and its expectation values read

[TABLE]

with $\theta_{ra}$ the angle formed by $\bm{r}$ and $\bm{a}$. In their turn, the off-diagonal elements of $\bm{\hat{\sigma}}\cdot\bm{a}$ in the basis $\{\left|+_{r}\right\rangle,\left|-_{r}\right\rangle\}$ are given by

[TABLE]

where $\bm{\theta}$ and $\bm{\varphi}$ are the unit vectors that, together with $\bm{r}$ given by (3), define the orthogonal triad in spherical coordinates,

[TABLE]

Further, the modulus of (11) is given by

[TABLE]

which should come as no surprise, since

[TABLE]

whence, using (10), one obtains $\left|\left\langle-_{r}\right|\hat{\bm{\sigma}}\cdotp\bm{a}\left|+_{r}\right\rangle\right|{}^{2}=1-\cos\theta_{ra}^{2}=\sin^{2}\theta_{ra}$.

It is interesting to note that the operator $\bm{\hat{\sigma}}\cdot\bm{a}$, rather than simply projecting the spin vector onto $\bm{a}$, changes in a nontrivial way the direction associated with it. In terms of the angles ($\theta_{a},\varphi_{a}$) that define the orientation of $\bm{a}$, we have

[TABLE]

This vector, along with

[TABLE]

form an orthonormal basis, with a different spin orientation in 3D space (except of course when $\bm{a}$ and $\bm{r}$ are colinear, as shown in (7)). The corresponding Bloch vector, which we call $\bm{r_{a}}$, is obtained by taking the expectation value

[TABLE]

with

[TABLE]

The calculation of (17) is most easily done by using once more $\left|+_{r}\right\rangle\left\langle+_{r}\right|+\left|-_{r}\right\rangle\left\langle-_{r}\right|=\mathbb{I}$, together with Eqs. (10) and (11); the result,

[TABLE]

clearly depends on both vectors $\bm{r}$ and $\bm{a}$, not just on the angle formed by them. Interestingly, a second operation $\left(\hat{\mathbf{\bm{\sigma}}}\cdotp\bm{a}\right)$ performed on $\left|+_{r}\right\rangle$ (or on any spin state vector, for that matter) brings it back to its original form, or in terms of (18),

[TABLE]

since $\left(\hat{\mathbf{\bm{\sigma}}}\cdotp\bm{a}\right)\left(\hat{\mathbf{\bm{\sigma}}}\cdotp\bm{a}\right)=1$. In other words, the effect of $\left(\hat{\mathbf{\bm{\sigma}}}\cdotp\bm{a}\right)$ on a state vector is reversible. Footnote1

III Bipartite spin state and associated observables

Let us now consider a system of two $\nicefrac{{1}}{{2}}-$spin particles in the (entangled) singlet state

[TABLE]

in terms of the simplified (standard) notation $\left|{\phi}\right\rangle\left|{\chi}\right\rangle=\left|{\phi}\right\rangle\otimes\left|{\chi}\right\rangle,$ with $\left|{\phi}\right\rangle$ a vector in the Hilbert space of subsystem 1, and $\left|{\chi}\right\rangle$ a vector in the Hilbert space of subsystem 2. It is convenient to bear in mind that the angles $(\theta,\varphi)$ that define the direction of $\bm{r}$ are not set; the singlet state is spherically symmetric, and therefore $\bm{r}$ may be chosen arbitrarily.

Now, in the composite system, the projection of the first spin operator along an arbitrary direction $\bm{a}$ is described by $(\hat{\bm{\sigma}}\cdotp\bm{a})\otimes\mathbb{I}$, and the projection of the second spin operator along a direction $\bm{b}$ is described by $\mathbb{I}\otimes(\hat{\bm{\sigma}}\cdotp\bm{b})$. The expectation value of any of these operators in the state $\left|{\Psi^{0}}\right\rangle$ vanishes,

[TABLE]

as follows from Eq. (10). Further, their quantum correlation

[TABLE]

is given according to Eqs. (10) and (11) by

[TABLE]

Though this result is well known, it is worthwhile revisiting it from a different angle, as follows.

We take as an orthonormal and complete basis of the composite Hilbert space, the set of vectors

[TABLE]

and write

[TABLE]

Calculation of the separate terms that contribute to $C_{Q}(\bm{a},\bm{b})$ gives, with the aid of Eqs. (10) and (11),

[TABLE]

The sum of the four terms gives of course the expected expression (24). The fact that this result depends only on the angle formed by $\bm{a}$ and $\bm{b}$ is due to the spherical symmetry of the singlet spin state. It is interesting however to look at the terms separately, and observe that the sum of the first two ($F_{1}+F_{2}$), which involve intermediate states ($\left|{\Psi^{1}}\right\rangle$ and $\left|{\Psi^{2}}\right\rangle$) of antiparallel spins (along the arbitrary direction $\bm{r}$), gives the product of the projections of $\bm{a}$ and $\bm{b}$ onto $\bm{r}$, whilst the sum of the last two ($F_{3}+F_{4}$), which involve intermediate states ($\left|{\Psi^{3}}\right\rangle$ and $\left|{\Psi^{4}}\right\rangle$) of parallel spins, contains their vector products. In other words, the two spin projection operators $\hat{\bm{\sigma}}\cdotp\bm{a}$, $\hat{\bm{\sigma}}\cdotp\bm{b}$ establish a correlation not just through the intermediate states representing antiparallel spins—as one might suppose for the entangled spin-zero state—but also through the intermediate states of parallel spins, $\left|{+_{r}}\right\rangle\left|{+_{r}}\right\rangle$ and $\left|{-_{r}}\right\rangle\left|{-_{r}}\right\rangle$. Footnote2

IV Probabilistic content of the spin projection correlation function

With the above elements we have prepared the ground to address the question: what exactly is the physical content of Eq. (23)? It is commonplace to say simply that it is the average of a product of spin projections, and this is what is put experimentally to test. Here we hope to contribute to provide a fairer picture of this expression by analysing it in more detail.

We focus again on the quantum correlation for the bipartite singlet spin state as expressed in Eq. (23), and make an alternative calculation, this time resorting to the eigenvalue equations

[TABLE]

to construct a new orthonormal basis for the bipartite system:

[TABLE]

[TABLE]

Instead of (26) we then write

[TABLE]

The basis $\left\{\left|\phi^{k}\right\rangle\right\}$ is a very convenient one, given that, in view of (31) and (32),

[TABLE]

with

[TABLE]

where $\alpha_{k},\beta_{k}$ are the individual eigenvalues corresponding to the bipartite state $\left|\phi^{k}\right\rangle$. Thus from Eq. (34) we get

[TABLE]

with

[TABLE]

From Eqs. (31-35) it follows that

[TABLE]

Note that the coefficients in Eq. (36) have an unambiguous meaning: $A_{k}$ is the eigenvalue of the operator $\left(\hat{\bm{\sigma}}\cdotp\bm{a}\otimes\hat{\bm{\sigma}}\cdotp\bm{b}\right)$ corresponding to the bipartite state $\left|\phi^{k}\right\rangle$. In its turn, $C_{k}$ is the relative weight of the eigenvalue $A_{k}$. Further, the $C_{k}$ are nonnegative and add to give

[TABLE]

Consequently, $C_{k}$ can be identified with the joint probability associated with the corresponding $A_{k}$,

[TABLE]

or in explicit terms, using (32) (the superindex $k$ is now redundant),

[TABLE]

[TABLE]

The marginal probability $P_{ab}(\alpha)$ is obtained from these expressions by considering the two possible outcomes $\beta=\pm 1$ for a given $\alpha$, so for instance (using (37))

[TABLE]

where \textrm{Tr}_{2}\big{(}|\Psi^{0}\rangle\langle\Psi^{0}|\big{)} denotes the partial trace of $|\Psi^{0}\rangle\langle\Psi^{0}|$ over the degrees of freedom of subsystem $2$, and hence represents the (reduced) density matrix of the first subsystem, $\rho_{1}$. Since $\left|{\Psi^{0}}\right\rangle$ is a maximally entangled state, $\rho_{1}=(1/2)\mathbb{I}$, which leads directly to the result (43). The same applies of course to all four marginal probabilities.

The joint probability (40) can also be written as the product of the marginal probability of occurrence of a given $\beta$ and the conditional probability $P_{ab}(\alpha\mid\beta)$. Thus for instance,

[TABLE]

with $P_{ab}(+\mid-)$ the probability of occurrence of $\alpha=+$ under the condition that $\beta=-1$. A further interesting result in support of the probabilistic meaning just described, is obtained by integrating $P_{ab}(\alpha,\beta)$ over all possible orientations of $\bm{b}$ to get the probability of $\alpha$ having a given value (say, $\alpha=1)$, with $\beta=1$ in any direction ($\Omega_{b}$ is the solid angle),

[TABLE]

Since the basis $\left\{\left|\phi^{k}\right\rangle\right\}$ was constructed in terms of the individual eigenvectors $\left|{\pm_{a}}\right\rangle$ and $\left|{\pm_{b}}\right\rangle$, it is essential to use this basis consistently in the calculations leading to $C_{Q}(\bm{a},\bm{b})$; in other words, both the relative weights (or joint probabilities) $C_{k}$ and the eigenvalues $A_{k}$ are anchored to this basis. To stress this point, in Eq. (36) the dependence on $\bm{a}$ and $\bm{b}$ has been introduced explicitly in the notation. If a different direction $\bm{b}^{\prime}$ is chosen for the calculation of $C_{Q}(\bm{a},\bm{b}^{\prime})$, a new vector basis $\{\left|\phi^{\prime k}\right\rangle\}$ will have to be used, with elements involving the eigenstates of the operator $\hat{\bm{\sigma}}\cdotp\bm{b}^{\prime}$. Clearly this will lead in general to different values for the individual coefficients $C_{k}(\bm{a},\bm{b}^{\prime})$.

V Discussion

A corollary of the anaysis carried out above is that the partition of the probability space $\Lambda$ into subspaces appropriate for the construction of $C_{Q}(\bm{a},\bm{b})$, cannot be the same as that used to construct $C_{Q}(\bm{a},\bm{b}^{\prime})$. This important restriction, which here emerges as a direct consequence of the operator algebra, is nevertheless often overlooked or not well understood.

To clarify this point, let us go back to Eq. (36) and note that by using the basis constructed with the eigenvectors of the individual (commuting) spin projection operators $\left(\hat{\bm{\sigma}}\cdotp\bm{a}\right)$, $\left(\hat{\bm{\sigma}}\cdotp\bm{b}\right)$, pertaining to particles 1 and 2, respectively, we have been able to write $C_{Q}(\bm{a},\bm{b})$ as a sum of (product) eigenvalues $A_{k}$ with their corresponding statistical weights $C_{k}$. This means that the ensemble of systems represented by the entangled state vector $\left|{\Psi^{0}}\right\rangle$, has been partitioned into four subensembles that are mutually exclusive and complementary: the subensembles that produce the outcomes $(+,-)$, $(-,+)$, $(+,+)$ and $(-,-)$ for $(\alpha,\beta)$, given a certain pair of directions $(\bm{a},\bm{b})$. These subensembles are represented by the basis vectors $\left|\phi^{k}\right\rangle$, $k=1,2,3,4$. Every (bipartite) element of the full ensemble belongs to one and only one of such subensembles. For a different pair $(\bm{a},\bm{b}^{\prime})$, the partitioning of the (same) ensemble of systems represented by $\left|{\Psi^{0}}\right\rangle$ will be into four (mutually exclusive and complementary) subensembles different from the previous ones: those that produce the outcomes $(+,-^{\prime})$, $(-,+^{\prime})$, $(+,+^{\prime})$ and $(-,-^{\prime})$ for a pair of directions $(\bm{a},\bm{b}^{\prime})$, and are represented by the basis vectors $\left|\phi^{\prime k}\right\rangle$ .

It is clear from this discussion that an expression that combines eigenvalues $A_{k}$, $A_{k}^{\prime}$ pertaining to different pairs $(\bm{a},\bm{b})$, $(\bm{a},\bm{b}^{\prime})$ is physically meaningless, as it would entail a mixture of elements pertaining to different subdivisions of the ensemble represented by $\left|{\Psi^{0}}\right\rangle$; in other words, it would imply the simultaneous use of two partitionings of the probability space which are incommensurable. Yet the procedure of combining under one formula the eigenvalues that correspond to different pairs of directions is central in the derivation of Bell-type inequalities for the bipartite singlet spin state CHSH . For clarity in the argument, let us translate it to the hidden-variable language as follows.

The partitioning of the probability space $\Lambda$ corresponding to Eq. (36) can be expressed as

[TABLE]

where

[TABLE]

is the probability space spanned by the subensemble represented by $\left|{\phi^{k}}\right\rangle$;

[TABLE]

is the corresponding statistical weight, with

[TABLE]

in agreement with (39), and

[TABLE]

Equation (45) expresses the fact that the partitioning depends on the directions $\bm{a}$, $\bm{b}$ and the eigenvalues $\alpha_{k}$, $\beta_{k}$. The individual eigenvalues $\alpha_{k}$, $\beta_{k}$ depend of course on $\bm{a}$ and $\bm{b}$, respectively, as indicated in Eq. (48), whilst the hidden variables $\lambda$ themselves do not; it is only the domain $\Lambda_{k}$ which is determined by the choice of $\bm{a}$ and $\bm{b}$, for the reasons given above.

Now, the usual starting point in the derivation of Bell-type inequalities is the correlation written in the form CHSH

[TABLE]

with the probability space $\Lambda$ spanned by the ensemble represented by the entangled state vector $\left|{\Psi^{0}}\right\rangle$. All Bell-type derivations involve products with at least three different orientations, say $\bm{a}$, $\bm{b}$ and $\bm{b}^{\prime}$. In the case of the CHSH inequality, which is the more widely used Bell-type inequality, four different orientations $\bm{a},\bm{a}^{\prime},\bm{b},\bm{b}^{\prime}$ are introduced, to write, with $\alpha=\alpha(\bm{a},\lambda)$, a.s.o.,

[TABLE]

The inequality in the last row is obtained as an algebraic exercise when $\alpha$, $\alpha^{\prime}$, $\beta$, $\beta^{\prime}$ take the values $\pm 1$ only. Notice that here the same probability space $\Lambda$ is used to construct the four correlations, without considering the need to subdivide the ensemble in different ways depending on the pair of orientations $(\bm{a},\bm{b})$.

According to Eqs. (44-48), instead, one should write, with $A_{k}=\alpha_{k}\beta{}_{k}=\alpha(\bm{a},\lambda)\beta{}_{k}(\bm{b},\lambda)$, a.s.o., and $\rho=\rho(\lambda)$,

[TABLE]

where $\left\{\Lambda_{k}\right\}$, $\left\{\Lambda_{l}\right\}$, $\left\{\Lambda_{m}\right\}$, $\left\{\Lambda_{n}\right\}$ represent the four different partitionings of $\Lambda$ corresponding to the four different pairs of vectors. Consequently, one cannot group the four terms under the same integral sign, as is done in passing from the first to the second line of Eq. (50). Ergo, there is no reason why the quantum correlation $C_{Q}(\bm{a},\bm{b})$ should obey the inequality (50).

Translated to the experimental domain, this is equivalent to saying that the spin projections $(\alpha,\beta)$, $(\alpha,\beta^{\prime})$, a.s.o., belong to different series of experiments. Of course the experimentalist may choose to reset the orientation of the apparatus from $\bm{b}$ to $\bm{b}^{\prime}$ after the first event, and then back to $\bm{b}$ after the second one… But eventually, after a large number of measurements, the experimental correlation $C_{E}(\bm{a},\bm{b})$ will be given by the average value of the projection products $\left(\alpha\beta\right)_{\bm{ab}}$, and $C_{E}(\bm{a},\bm{b}^{\prime})$ by the average value of the products $\left(\alpha\beta^{\prime}\right)_{\bm{ab}^{\prime}}$; the experimentalist does not mix the data from the two series of measurements for the calculation of the average values. If different series of measurements are made, for different pairs of directions $(\bm{a},\bm{b})$, one should expect the experiment to eventually confirm the functional dependence predicted by quantum mechanics; i. e., $C_{E}(\bm{a},\bm{b})=-\bm{a}\cdot\bm{b}$.

The outcome of our present analysis leaves no room for interpretations. As stated in Ref. Sven in connection with the weak values, it must be the theory that decides what meaning to ascribe to them. The same statement applies to operators and their eigenfunctions, as seen here in the particular case of the bipartite singlet spin state. The Hilbert-space formalism is a powerful and elegant way of dealing with an ensemble characterized by a common feature or physical parameter (in our case, the total spin zero represented by $\left|\Psi^{0}\right\rangle$), and of subdividing this ensemble according to some additional (set of) physical parameter(s) (in our case, the pair of spin projections onto $\bm{a}$ and $\bm{b}$). The choice of a different physical parameter (say a spin projection along a direction $\bm{b}^{\prime}$) implies a different partition of the ensemble. This feature needs to be taken into account in any probabilistic analysis of the quantum correlations.

Our conclusions, carried out entirely within the quantum formalism, finds a counterpart in the literature in the form of the measurement-dependence or contextuality argument. The assumption of noncontextuality (or so-called contextuality loophole) associated with the Bell- and CHSH theorems has been pointed out in different ways; for early works see Refs. PeCeBr ; Kup ; TAB . More recently, it is raised anew by an increasing number of authors (see e. g. Ade ; Khrenn ; Vervoort ; TMNiew ; Kup2 ), stressing that (1) probabilities belong to experiments and not to objects or events per se, and (2) any probability depends at least in principle on the context, including all detector settings of the experiment Khrenn ; Vervoort . In other words, a hidden-variable model suffers from a contextuality loophole if it pretends to describe different sets of incompatible experiments using a unique probability space and a unique joint probability distribution Khrenn ; TMNiew .

Acknowledgements.

The authors acknowledge support from DGAPA-UNAM through project PAPIIT IA101918.