nLab Bell's inequality

Idea

In quantum physics/quantum information theory, What came to be called Bell’s inequality (Bell 1964) is an inequality satisfied by the three pairwise correlation functions between three random variables defined on one and the same classical probability space. As such, it is an elementary statement about classical probability theory which as been argued (Pitowsky 1989a) to have been known already to Boole (1854).

The point of the argument by Bell 1964 was to highlight that when taking these three random variables to be the results of quantum measurements of the spin of an electron along three pairwise non-orthogonal axes (as in the Stern-Gerlach experiment) then quantum theory predicts that this inequality is violated – implying that there is no single classical probability space (called a hidden variable in the context of interpretations of quantum mechanics) on which these three quantum measurement-results are jointly random variables.

A number of experiments have sought to check Bell’s inequalities in quantum physics (“Bell tests”) and all claim to have verified that it is indeed violated in nature (see Aspect 2015), as predicted by quantum theory.

Bell’s inequality has been and is receiving an enormous amount of attention, first in discussions of interpretations of quantum mechanics, but more recently and more concretely also in the context of quantum information theory.

Statement

A transparent and compact way to derive the actual inequality of Bell 1964 (adjusting the original argument only slightly for mathematical elegance) is reviewed in Khrennikov 2008, §10.1, which we broadly follow:

Proposition

Given

1. a probability space $(\Lambda, d\rho)$ with

2. three random variables taking values in $\{\pm 1\}$ (regarded inside the real numbers):

   (1)$$S_i \;\colon\; X \longrightarrow \{\pm 1\} \hookrightarrow \mathbb{R} \,, \;\;\;\; i\,\in\, \{1,2,3\}$$

then the correlation functions

(2)$$\langle S_{i} \, S_j\rangle \;\coloneqq\; \int_{\Lambda} \; S_i(\lambda) \, S_j(\lambda) \; d\rho(\lambda)$$

satisfy this inequality:

(3)$$\big\vert \langle S_1 S_2\rangle - \langle S_3 S_2\rangle \big\vert \;\leq\; 1 - \langle S_1 S_3\rangle \,.$$

(where $\left\vert-\right\vert$ denotes the absolute value)

Proof

Recall that the expectation value of a random variable $P \,\colon\, \Lambda \longrightarrow \mathbb{R}$ is given by its Lebesgue integral against the probability measure:

$$\langle P \rangle \;\coloneqq\; \int_\Lambda P(\lambda) \, d\rho(\lambda) \,,$$

and that $d\rho$ being a probability measure implies the normalization

(4)$$\langle 1 \rangle \;\equiv\; \int_\Lambda 1 \, d\rho(\lambda) \;=\; 1 \,.$$

Moreover, the assumption (1) that the random variables $S_i$ take values in $\{\pm 1\}$ immediately implies for all $i,j \,in\, \{1,2,3\}$ that

(5)$$\big( S_i \cdot S_i \big) \,=\, 1 \,, \;\;\;\; \text{i.e.} \;\;\; \underset{\lambda \,\in\, \Lambda}{\forall} S_i(\lambda) \, S_i(\lambda) \,=\, (\pm 1)^2 \,=\, 1 \,.$$

Together this implies – by repeatedly using the Cauchy-Schwarz inequality – the bounds:

$$\big\vert \langle S_i \rangle \big\vert \;\leq\; 1 \,, \;\;\;\;\;\;\; \big\vert \langle S_i S_j\rangle \big\vert \;\leq\; 1$$

and thus, in particular:

(6)$$\big\vert \langle P \, S_i \, S_j \rangle \big\vert \;\leq\; \big\vert \langle P \rangle \big\vert \,,$$

for any random variable $P \,\colon\, \Lambda \to \mathbb{R}$.

Using these (evident) ingredients, we directly compute as follows

$$\begin{array}{ll} \big\vert \langle S_1 S_2\rangle - \langle S_3 S_2\rangle \big\vert & \\ \;=\; \Big\vert \int_{\Lambda} S_1(\lambda) \, S_2(\lambda) \, d\rho(\lambda) - \int_{\Lambda} S_3(\lambda) \, S_2(\lambda) \, d\rho(\lambda) \Big\vert & \text{by}\;\text{(2)} \\ \;=\; \Big\vert \int_{\Lambda} \big( S_1(\lambda) - S_3(\lambda) \big) \, S_2(\lambda) \, d\rho(\lambda) \Big\vert & \text{by linearity of the integral} \\ \;=\; \Big\vert \int_{\Lambda} \big( 1 - S_1(\lambda) \, S_3(\lambda) \big) S_1(\lambda) \, S_2(\lambda) \, d\rho(\lambda) \Big\vert & \text{by}\;\text{(5)} \\ \;\leq\; \Big\vert \int_{\Lambda} \big( 1 - S_1(\lambda) \, S_3(\lambda) \big) \, d\rho(\lambda) \Big\vert & \text{by}\;\text{(6)} \\ \;=\; 1 - \langle S_1 S_3\rangle & \text{by}\;\text{(4)}\;\text{and}\; \text{(2)} \end{array}$$

This is the inequality (3).

Related concepts

• Einstein-Podolsky-Rosen paradox

• Grothendieck inequality

• quantum probability

• interpretation of quantum mechanics

• quantum information theory

References

General

The original article:

• John Bell, On the Einstein Podolsky Rosen paradox, Physics 1 195 (1964) [doi:10.1103/PhysicsPhysiqueFizika.1.195, pdf]

Review:

• Greg Kuperberg, section 1.6.2 of A concise introduction to quantum probability, quantum mechanics, and quantum computation, 2005 (pdf)

Experiments:

• Alain Aspect, Closing the Door on Einstein and Bohr’s Quantum Debate, Physics 8 123 (2015)

Relation to the Kochen-Specker theorem:

• Leandro Aolita, Rodrigo Gallego, Antonio Acín, Andrea Chiuri, Giuseppe Vallone, Paolo Mataloni, Adán Cabello, Fully nonlocal quantum correlations, Phys. Rev. A 85 032107 (2012) [arXiv:1105.3598, doi:10.1103/PhysRevA.85.032107]

See also:

• Wikipedia, Bell’s theorem

• Wikipedia, Bell test

• Wikipedia, Leggett-Garg inequality

In relation to the Grothendieck inequality:

• Boris S. Tsirelson, Quantum analogues of the Bell inequalities. The case of two spatially separated domains, Journal of Soviet Mathematics 36 (1987) 557–570 [doi:10.1007/BF01663472]

• Boris S. Tsirelson, Some results and problems on quantum Bell-type inequalities Hadronic Journal Supplement 8 4 (1993) 329-345 [pdf, pdf web]

  (but see the erratum here)

• Wikipedia, Tsirelson’s bound

In quantum field theory

In the generality of quantum field theory:

On Bell inequalities in particle physics and possible relation to the weak gravity conjecture:

• Aninda Sinha, Ahmadullah Zahed, Bell inequalities in 2-2 scattering [arXiv:2212.10213]

On BRST invariant Bell inequality in gauge field theory:

• David Dudal, Philipe De Fabritiis, Marcelo S. Guimaraes, Giovani Peruzzo, Silvio P. Sorella: BRST invariant formulation of the Bell-CHSH inequality in gauge field theories [arXiv:2304.01028]

Probabilistic opposition

Identification of Bell’s inequalities with much older inequalities in classical probability theory, due to George Boole‘s The Laws of Thought, was pointed out by (among others, called the “probabilistic opposition” in Khrennikov 2007, p. 3) by:

• Itamar Pitowsky, From George Boole To John Bell — The Origins of Bell’s Inequality, in: Bell’s Theorem, Quantum Theory and Conceptions of the Universe, Fundamental Theories of Physics 37 Springer (1989) [doi:10.1007/978-94-017-0849-4_6]

• Itamar Pitowsky, Quantum Probability – Quantum Logic, Lecture Notes in Physics 321, Springer (1989) [doi:10.1007/BFb0021186]

• Luigi Accardi, The Probabilistic Roots of the Quantum Mechanical Paradoxes, in: The Wave-Particle Dualism, Fundamental Theories of Physics 3 Springer (1984) [doi:10.1007/978-94-009-6286-6_16]

reviewed in:

• Elemer E Rosinger, George Boole and the Bell inequalities [arXiv:quant-ph/0406004]

• Andrei Khrennikov, Bell’s inequality: Physics meets Probability [arXiv:0709.3909]

• Andrei Khrennikov, Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?, Entropy 10 2 (2008) 19-32 [doi:10.3390/entropy-e10020019]