A Rigorous Analysis of the Clauser–Horne–Shimony–Holt Inequality Experiment When Trials Need Not be Independent

Abstract

The Clauser–Horne–Shimony–Holt (CHSH) inequality is a constraint that local hidden variable theories must obey. Quantum Mechanics predicts a violation of this inequality in certain experimental settings. Treatments of this subject frequently make simplifying assumptions about the probability spaces available to a local hidden variable theory, such as assuming the state of the system is a discrete or absolutely continuous random variable, or assuming that repeated experimental trials are independent and identically distributed. In this paper, we do two things: first, show that the CHSH inequality holds even for completely general state variables in the measure-theoretic setting, and second, demonstrate how to drop the assumption of independence of subsequent trials while still being able to perform a hypothesis test that will distinguish Quantum Mechanics from local theories. The statistical strength of such a test is computed.

Appendix

In this paper, we worked in the most general measure-theoretic setting. In addition to requiring more work, the general setting can make it harder to gain an intuitive grasp of the probabilistic assumptions in the model. In contrast, the system of Brandenburger and Yanofsky [15] involves a simplifying assumption that the state of the system, $\lambda$, is a discrete random variable with finitely many outputs. With this assumption, notions such as “locality” and “$\lambda$-independence” are easier to formulate and easier to understand. Though the finite-$\lambda$ assumption restricts the type of theories one can model, this is not as egregious as it might seem: in some hidden-variable situations, any possible correlation scenario can be modeled by a finite-output $\lambda$, as discussed in [6].

In this appendix, we investigate what happens to the system of Sect. 2 if we make the additional assumption that $\lambda$ is a discrete random variable with finitely many outputs. This will allow us to directly compare our system to the system of [15], as well as to illustrate and clarify the nature of our particular choices of assumptions.

Before restricting ourselves to the finite-$\lambda$ setting, we can show that, working in the system of Sect. 2, we can derive the following alternate version of (4):

Experimental Assumption 3*:

$$\begin{aligned} (A, B) \perp \!\!\!\perp \lambda . \end{aligned}$$

(45)

The above alternative version of (4) is more similar to the “$\lambda$-independence” assumption as formulated in [15]. (45) is also a stronger assumption than (4): one can verify that (45) directly implies (4), but the converse does not hold. However, if we also assume (2) and (5), we can derive (45) from (4).

Proposition 5

The condition (4), in conjunction with (2) and (5), implies (45).

Proof

We show that $P\big ( (A, B) = (a, b) | \lambda \big ) = P\big ( (A, B) = (a, b)\big )$; the proof for the three other cases $(a, b')$, $(a', b)$, and $(a', b')$ is the same. We have

$$\begin{aligned} P\big ( (A, B) \!=\! (a, b) | \lambda \big ) \!=\! \sum _{i=\pm 1}\sum _{j=\pm 1}P\big (\{A\!=\!a\}\cap \{B=b\} \cap (\{D_1 = i\} \cap \{D_2 \!=\! j\})\big |\lambda \big ), \end{aligned}$$

by Lemma 2. Applying (5) to the above expression yields

$$\begin{aligned} \sum _{i=\pm 1}\sum _{j=\pm 1} P\big (\{D_1=i\}\cap \{A=a\}\big |\lambda \big )\cdot P\big (\{D_2=j\}\cap \{B=b\}\big |\lambda \big ). \end{aligned}$$

Factoring the expression and applying Lemma 2, we obtain

$$\begin{aligned}&P\big (+_2\cap \{B=b\}\big |\lambda \big )\bigg [ P\big (+_1\cap \{A=a\}\big |\lambda \big ) + P\big (-_1\cap \{A=a\}\big |\lambda \big )\bigg ] \\&\quad + P\big (-_2\cap \{B=b\}\big |\lambda \big )\bigg [ P\big (+_1\cap \{A=a\}\big |\lambda \big ) + P\big (-_1\cap \{A=a\} \big |\lambda \big )\bigg ] \\&\qquad = P\big (+_2\cap \{B=b\}\big |\lambda \big )\cdot P(A=a|\lambda ) +P\big (-_2\cap \{B=b\}\big |\lambda \big )\cdot P(A=a|\lambda ) \\&\qquad =P(A = a|\lambda )P(B = b|\lambda ). \end{aligned}$$

Now, by applying (4) and then (2), we have

$$\begin{aligned} P(A = a|\lambda )P(B = b|\lambda ) = P(A = a)P(B = b) = P\big ( (A, B) = (a, b) \big ). \end{aligned}$$

$\square$

Proposition 5 rules out the possibility of $\lambda$ having some dependence on the joint distribution of $A$ and $B$.

Moving forward, we now make the assumption that the random variable $\lambda$, introduced in Sect. 2, is of the form

$$\begin{aligned} \lambda : \Omega \rightarrow \Lambda = \{l_1, \ldots , l_n\}, \end{aligned}$$

(46)

so $\Lambda$ is now taken to be a finite set containing $n$ elements. (The nature of its constituent elements $l_i$ is not characterized, or important.) We also assume that for all $i$, $P(\lambda = l_i)> 0$; any zero-probability event has no observable effect on the behavior of the model, so we remove such events from consideration. Henceforth we will use $l_i$ to refer to the event $\{\lambda = l_i\} = \lambda ^{-1}(l_i)\subseteq \Omega$, unless doing so could create ambiguity.

With the assumption that $\lambda$ is finite, the expression (4) is now equivalent to

$$\begin{aligned}&\forall i\in \{1, \ldots , n\}, \mathbf{a} \in \{a, a'\}, \,\, \mathrm{and } \,\, \mathbf{b} \in \{b, b'\},\\&P(A=\mathbf{a}\cap l_i) = P(A=\mathbf{a})P(l_i) \quad \mathrm{and }\\&P(B=\mathbf{b}\cap l_i) = P(B=\mathbf{b})P(l_i). \end{aligned}$$

The interpretation of (5) is simplified as well. This is because for any event $E$, $P(E|\lambda )$ will now just be a simple function that is equal to $P(E|l_i)$ on each set $l_i$. Now (5) is equivalent to the condition

$$\begin{aligned}&\forall i\in \{1,\ldots ,n\}, \mathbf{a} \in \{a, a'\}, \mathbf{b} \in \{b, b'\}, j_a \in \{+_1, -_1\}, \,\, \mathrm{and} \,\, k_b\in \{+_2, -_2\},\nonumber \\&P(j_a, \mathbf{a}, k_b, \mathbf{b}|l_i) = P(j_a, \mathbf{a}|l_i)P(k_b, \mathbf{b}|l_i). \end{aligned}$$

(47)

Note that the conditionals in (47) are events, not random variables, resulting in a simpler construction when compared to expressions like (6). So now, all of the assumptions (2)–(5) can be expressed in terms of elementary probabilistic statements concerning a finite collection of events.

We now show that the axiomatization of Sect. 2 is essentially equivalent to the axiomatization of [15], when applied to the relevant experimental setup (i.e., an experiment with two detectors, two detector settings, and two outcomes). To do this, note that if we assume (2), (3), and replace (4) with the stronger (45), then the following statement,

$$\begin{aligned} \forall i, \mathbf{a}, \mathbf{b}, j_a, k_b,\quad {P(j_a, \mathbf{a}, k_b, \mathbf{b},l_i)\over P(l_i)} = {P(j_a, \mathbf{a},l_i)P(k_b, \mathbf{b},l_i)\over P(l_i)^2} \end{aligned}$$

(48)

is equivalent to

$$\begin{aligned} \forall i, \mathbf{a}, \mathbf{b}, j_a, k_b, \quad {P(j_a, \mathbf{a}, k_b, \mathbf{b},l_i)\over P(\mathbf{a}, \mathbf{b}, l_i)} = {P(j_a, \mathbf{a},l_i)P(k_b, \mathbf{b},l_i)\over P(\mathbf{a}, l_i)P(\mathbf{b}, l_i)}. \end{aligned}$$

(49)

Demonstrating the above biconditional is a straightforward exercise. Note that (48) is equivalent to (47). Now, recall that by Proposition 5, we have the following logical relationship between assumptions,

$$\begin{aligned} (2), (3), (4), (5) \quad \Leftrightarrow \quad (2), (3), (45), (5), \end{aligned}$$

and in the finite-$\lambda$ setting, we have (5) $\Leftrightarrow$ (47), so we can say that

$$\begin{aligned} (2), (3), (45), (5) \quad \Leftrightarrow \quad (2), (3), (45), (49). \end{aligned}$$

The collection of assumptions on the right side of the above equivalence is closely related to the framework of [15] as it would apply to the 2-detector, 2-setting, 2-outcome scenario. (45) is equivalent to Definition 2.4 (“$\lambda$-independence”) in [15], and (49) is equivalent to Definition 2.10 (“locality”). So in a finite-$\lambda$ setting, our framework—i.e., the set of conditions (2)–(5)—is equivalent to the Brandenburger/Yanofsky framework applied to a 2-dector, 2-setting, 2-outcome scenario where measurement choices are independent from each other (the condition (2)) and none of the measurement settings have trivial probabilities (the condition (3)).