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.
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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 theo...
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I am an undergraduate student in theoretical physics and mathematics. The intention of this web blog is to share what I learn in theoretical physics and mathematics.
Aayush Verma
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