The Quill 35: Quantum Nonlocality as an Axiom

We will present something of a different nature in this Quill. The concepts of indeterminacy and nonlocality were baffling to classical theorists. Such ideas required a definition of theories dubbed 'realist'. Einstein, Podolsky, and Rosen famously rejected the idea of nonlocality and proposed local hidden variables. Bell, also famously (all of these famously, by the way), wrote down an inequality about the correlation of measurements. Say $A, A'$ refer to the measurements done by Alice and $B, B'$ by Bob, and we denote the joint probability $P_{AB}(a,b)$ of obtaining $A = a$  and $B= b$ after measurements. A Bell inequality we are interested in is the CHSH inequality: $$E_{A B} + E_{AB'} + E_{A'B} - E_{A'B'} \leq 2$$ where the correlations are defined as $$E_{AB} = P_{AB}(1,1) + P_{AB}(-1,-1)-P_{AB}(-1,1)-P_{AB}(1,-1)$$ Any sensible theory of realism should follow this bound [also called the LHV bound], but Quantum Mechanics comically violates it, and as calculated by Tsirelson, it is $2 \sqrt{2}$. But the question is: why? Could this inequality be more violated? What limits quantum mechanics from violating it more? Is that the no-signaling principle (no superluminal transfer of information)? The algebraic maximum of the inequality is $4$; hence, should there be a theory following non-locality (or maybe no-signaling) that violates the inequality more? And would that be a better theory than quantum theory? If not, why is nature so specific about Tsirelson's bound of $2 \sqrt{2}$? These questions were raised by Popescu-Rohrlich in a paper (https://link.springer.com/article/10.1007/BF02058098). They hypothesized a theory, which we may call superquantum, in which both nonlocality and no-signaling were available, yet the bound was still violated to the algebraic maximum of $4$. In the paper, they ignore the quantum dynamics, like what Aharonov was thinking about, such as magnetic flux. However, the paper is an interesting read if one wonders why nature chooses quantum mechanics as the theory of what it is with Tsirelson's bound.


The original Popescu-Rohrlich paper uses a nonlocal correlation based on the relative angle between axes used for measurements of, say, spins. However, we will present an equivalent setup here. setting up a machine, which we call a PR box, where Alice and Bob enter inputs $x,y \in \{0,1\}$ and get outputs $a,b \in \{0,1\}$. There is a condition that Alice and Bob cannot communicate about their inputs. The output is given by $a \oplus b = xy$, where the addition is modulo $2$. So, $$ a \oplus b = \begin{cases} 0, & xy= 0 \\ 1, & xy=1 \end{cases}$$ If we write the conditional probability $P_{xy}(a,b)$, then $P_{00}(0,0) = P_{00}(1,1)= 1/2$, and the others are zero (because of the above gate operation). Other inputs have similar outputs and conditional probabilities, except $P_{11}(10)=P_{11}(01)=1/2$. Now, the correlation of measurements, or correlator, is $$E_{xy} = \sum_{a,b} (-1)^{a + b} P_{xy}(a,b)$$. But due to our definition of the output, we have $E_{xy} = (-1)^{xy}$ after simplification. We note that $E_{00} = 1, E_{10} = E_{01} = 1$ and $E_{11} = -1$. Now, from the CHSH inequality, we get that $E_{00}+E_{10}+E_{01}-E_{11} = 4$!

So, while the axioms of nonlocality and relativistic causality give this violation, this is not the quantum theory. Popescu-Rohrlich argued that this means that quantum mechanics needs more axioms than two. However, the paper opened a line of research on computation and communication. 

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