Tsirelson bound

In plain terms

Quantum mechanics isn't as strange as it could be. In the standard Bell test, the results of two separated measurements are combined into a single correlation score, and any theory in which measurements have answers fixed in advance keeps that score at 2 or below. Quantum mechanics reaches about 2.83, the famous violation, but sheer logic permits a score of 4, and theories that reach 4 exist on paper, cause no paradoxes, and allow no faster-than-light messages.

So relativity isn't what stops quantum mechanics at 2.83. Something else does, and nobody fully knows what. The ceiling is called the Tsirelson bound, and explaining why nature's ceiling sits exactly there rather than at the logical maximum has become a research programme in its own right.

The honest position is that we can prove where the quantum ceiling is, and we can't yet say, from first principles anyone agrees on, why that's the ceiling the world chose.

In CSD

A reconstruction has to get the ceiling right and not just the violation. Reproducing the Bell violation with the wrong maximum would mean having built a different theory that agrees with quantum mechanics only qualitatively, which would be interesting but wouldn't be quantum mechanics. So the corpus proves the bound over the whole family of two-qubit systems, a qubit being the simplest quantum system, which means every state of a pair of qubits and every choice of measurement directions and not just the singlet at the textbook angles.

Saturation is proved separately, so there really are settings at which the singlet reaches 2.83 exactly. Ceiling and attainment are two theorems rather than one observation, and keeping them apart is what lets each be checked on its own.

Mathematically

For every unit two-qubit state and all detector settings, the CHSH combination of Pauli correlations satisfies absolute value at most 2 root 2. The route is algebraic but not operator-algebraic, as the load-bearing lemma bounds the expression Re<alpha,beta> - Re<alpha,beta'> + Re<alpha',beta> + Re<alpha',beta'> for any four unit vectors in a complex inner product space, by the parallelogram law and Cauchy- Schwarz, and the quantum bound follows by the standard Tsirelson substitution alpha = (sigma.a tensor I)psi, beta = (I tensor sigma.b)psi.

Be precise about the form, because it's easy to overstate. The operator-form Loewner-order statement, the inequality about products of anticommuting involutions with no state mentioned, isn't what the corpus has. It was the original target and is blocked upstream on a missing ordered-module instance for matrices, as the module records. The vector form is general in its own right and yields the same expectation bound, and saturation by the singlet is exhibited explicitly and separately.

The name

Boris Tsirelson (1950-2020) was born in Leningrad and trained there under Ibragimov, mostly in probability, on Brownian motion, exotic noises and what he called black noise. The quantum bound was almost a side project.

It appeared in 1980 in Letters in Mathematical Physics, written from inside the Soviet Union, and it sat largely unread in the West for years because the field that would need it didn't yet exist. He emigrated to Israel in 1991 and spent the rest of his career at Tel Aviv University. In later life he became one of the most careful mathematics editors on Wikipedia, which he treated as seriously as journal work.

Related entries

• Bell and CHSHNo pre-written answer sheet for all four measurement settings can match what a singlet pair of particles does, and the corpus proves this as a theorem about every possible answer sheet.
• CGLMP inequalityBell's inequality is built for measurements with two outcomes, so once you go beyond two you need a new inequality, and the conflict with local realism grows stronger rather than weaker.
• Mermin inequalityAdd more particles and the conflict between quantum mechanics and common sense grows exponentially, so it doesn't average out but compounds.
• No-signallingNothing done to the far wing of an entangled pair changes the statistics seen at the near wing, and CSD proves this exactly, for every state, entangled ones included.
• Does CSD violate Bell's theorem?No. Bell's theorem stands, CSD proves its own versions of it, and CSD escapes its conclusion by the one route the theorem itself leaves open, measurement outcomes that depend on the measurement context, paid for openly.