Norm and correlations

Continuing the discussion from last time, today I want to talk about the norm of a linear operator and it's implication for the maximum correlations which can be achieved in nature: Tsirelson's bound. The very same norm definition would later on play a key role in what would unexpectedly become in the end a "geometrization" of the Standard Model coupled with (unquantized) gravity.

In a Hilbert space, the definition of the norm of a bounded linear operator is:

\(||A|| = sup (||Au||/||u||)\) for \(u\ne 0\)

The most important properties of the norm for bounded operators are the triangle inequality:

\(||A+B|| \leq ||A|| + ||B||\)

and a multiplication identity which guarantees the continuity of multiplication:

 \(||AB|| \leq ||A||  ||B||\)

(can we call this a triangle inequality for multiplication?)

On the basis of the triangle inequality, one may be tempted to explore the association of the notion of physical distance with the notion of the norm of an operator in a Hilbert space, but this is a dead end. The triangle inequality for operators is essential for quantum mechanics because it ensures the usual notions of convergence in functional analysis (most of functional analysis follows from it). The name of this blog is elliptic composability, and the "elliptic" part follows from the triangle inequality above. If one imagines a quantum mechanics where the triangle inequality is reversed, then one arrives at the unphysical hyperbolic quantum mechanics based on split-complex numbers which violates positivity which in turns prevents the usual definition of probability as a positive quantity.

There turns out however to be a deep and completely counter-intuitive relationship between the "sup" in the norm definition and the notion of physical distance, but more on this in subsequent posts-don't want to spoil the surprise, I only want to whet the (mathematical) appetite a bit.

Now back to correlations. Suppose we have four operators \(\sigma_\alpha, \sigma_\beta, \sigma_\gamma, \sigma_\delta\) such that:

\({\sigma_\alpha}^2 = {\sigma_\beta}^2 = {\sigma_\gamma}^2 = {\sigma_\delta}^2 = 1\)

and

\([\sigma_\alpha, \sigma_\beta] = [\sigma_\beta, \sigma_\gamma] = [\sigma_\gamma, \sigma_\delta] = [\sigma_\delta, \sigma_\alpha] = 0\)

If we define an operator \(C\) as follows:

\(C= \sigma_\alpha \sigma_\beta + \sigma_\beta \sigma_\gamma + \sigma_\gamma \sigma_\delta - \sigma_\delta \sigma_\alpha\)

Then it is not hard to show that:

\(C^2 = 4 + [\sigma_\alpha, \sigma_\gamma][\sigma_\beta, \sigma_\delta]\)

From the triangle inequalities we have in general that:

\(||[A, B]|| = ||AB - BA|| = ||AB + (-B)A|| \leq ||AB|| + ||-BA|| \)

\(= ||AB|| + ||BA|| \leq ||A||||B|| + ||A||||B|| = 2||A||||B||\)

and so 

\(|| [\sigma_\alpha, \sigma_\gamma]|| \leq 2 ||\sigma_\alpha|| ||\sigma_\gamma|| = 2\)

\(|| [\sigma_\beta, \sigma_\delta]|| \leq 2 ||\sigma_\beta|| ||\sigma_\delta|| = 2\)

Therefore:

\(||C^2|| \leq 4+4\)

\(||C|| \leq 2 \sqrt{2}\)

And this is Tsirelson's bound because \(C\) appears in the left hand side of the CHSH inequality.

Now one can read about textbook derivation of Tsirelson bound in many places, but the key point is that quantum correlations have their origin in the notion of operator norms in a Hilbert space. Nowhere in the derivation we have used the notion of distance or causality. Quantum correlations are a mathematical consequence of the quantum formalism which are in turn is a consequence of considerations of composition and information. Quantum mechanics is nothing but composition and information, and correlations (both quantum and classical) are nothing but considerations of composition and information as well. 

The usual way we understand classical correlations as generated by a common cause is a parochial view due to our classical intuition. Yes, a common cause can generate correlations, but correlation does not imply causation. 

So what does all of this have to do with the notion of  distance and that of space-time? To uncover the link we would need first to generalize the notion of a space in geometry. Crazy talk? Not when it is based on von Neumann algebras research which led to a Fields medal. Please stay tuned...