quantum algorithms:
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.
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:
Given
a probability space $(\Lambda, d\rho)$ with
three random variables taking values in $\{\pm 1\}$ (regarded inside the real numbers):
then the correlation functions
satisfy this inequality:
(where $\left\vert-\right\vert$ denotes the absolute value)
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:
and that $d\rho$ being a probability measure implies the normalization
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
Together this implies – by repeatedly using the Cauchy-Schwarz inequality – the bounds:
and thus, in particular:
for any random variable $P \,\colon\, \Lambda \to \mathbb{R}$.
Using these (evident) ingredients, we directly compute as follows
This is the inequality (3).
The original article:
Review:
Experiments:
See also:
Wikipedia, Bell’s theorem
Wikipedia, Bell test
Wikipedia, Leggett-Garg inequality
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]
Last revised on September 8, 2022 at 11:10:12. See the history of this page for a list of all contributions to it.