CHSH in one page

This is the shortest post in the walkthroughs, because the argument is one page long and its consequence for this programme is one line. But the line is a wall, and several seductive research programmes have crashed into it, so it is worth a page of your time.

The inequality

Two observers, far apart, each choose between two measurements. Call the observers Alice and Bob, their choices a and b, and their outcomes plus one or minus one. Write E(a, b) for the correlation of their outcomes when those settings are chosen. Local realism, the claim that outcomes are fixed by pre-existing properties plus whatever is carried from one side to the other no faster than light, makes a prediction about how correlations can combine. For any choice of settings, the Bell-Clauser-Horne-Shimony-Holt expression S equals E(a,b) plus E(a,b prime) plus E(a prime,b) minus E(a prime,b prime) is bounded: S is at most 2. The bound follows from ordinary algebra once you assume the outcomes are numbers that existed before the measurements.

Quantum mechanics predicts, and experiments confirm to exquisite precision, that S can reach 2 root 2. That number is the Tsirelson bound, and it exceeds 2 by root 2 minus nothing. Nature does this: the experiments have closed the serious loopholes, shared randomness and detection and locality, and the result stands.

The one-line consequence

A theory with pre-existing local values cannot reproduce correlations of 2 root 2. So: no local deterministic classical substrate can underlie quantum statistics. Write that in the direction this programme cares about: if you want to simulate a multiverse on a substrate that is classical, deterministic, and local, with the quantum statistics of the branches emerging as the rendered appearance, then your rendering must fail to reproduce the correlations every laboratory has seen. The assumption is not exotic. It is ruled out.

The framework’s position takes this seriously and it shapes the architecture. The substrate cannot be a big classical computer printing branch states from hidden variables. If branches are computed, the substrate must carry quantum structure natively, in exactly the way the topological-category papers describe: the substrate is a quantum theory, branches are its superselection sectors, and the quantum correlations inside a branch are as native as the branch itself.

What is and is not excluded

What is excluded: local hidden-variable renderings, computable in the ordinary sense, deterministic, with no quantum substrate. What is not excluded: a classical computer simulating quantum mechanics by representing amplitudes and evolving them, which is what we do every day on ordinary machines. The distinction is between the substrate’s own ontology and its data structures. A classical machine can carry a quantum state as data and compute its evolution, at the well-known exponential cost in memory. It cannot make the correlations emerge from classical ignorance, but it can compute them from explicit amplitudes.

That escape hatch has its own price, which is the cost of storing amplitudes, and for a multiverse the price is prohibitive: a branch space of dimension D costs memory exponential in D, which is how the fact-budget argument in the computational series shows that the top of the complexity spectrum cannot be rendered faithfully by explicit amplitude storage. The wall does not move. The framework’s answer is quantum substrate natively, sectors and all, and the honest summary of this page is: everyone proposing classical substrates for quantum worlds is in the same negotiation with 2 root 2, and the negotiation has never once gone well.