# The Clauser-Horne-Shimony-Holt inequality and Tsirelson's inequality. #

We establish a version of the Clauser-Horne-Shimony-Holt (CHSH) inequality (which is a generalization of Bell's inequality). This is a foundational result which implies that quantum mechanics is not a local hidden variable theory.

As usually stated the CHSH inequality requires substantial language from physics and probability,
but it is possible to give a statement that is purely about ordered `*`

-algebras.
We do that here, to avoid as many practical and logical dependencies as possible.
Since the algebra of observables of any quantum system is an ordered `*`

-algebra
(in particular a von Neumann algebra) this is a strict generalization of the usual statement.

Let `R`

be a `*`

-ring.

A CHSH tuple in `R`

consists of

- four elements
`A₀ A₁ B₀ B₁ : R`

, such that - each
`Aᵢ`

and`Bⱼ`

is a self-adjoint involution, and - the
`Aᵢ`

commute with the`Bⱼ`

.

The physical interpretation is that the four elements are observables (hence self-adjoint)
that take values ±1 (hence involutions), and that the `Aᵢ`

are spacelike separated from the `Bⱼ`

(and hence commute).

The CHSH inequality says that when `R`

is an ordered `*`

-ring
(that is, a `*`

-ring which is ordered, and for every `r : R`

, `0 ≤ star r * r`

),
which is moreover *commutative*, we have
`A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2`

On the other hand, Tsirelson's inequality says that for any ordered `*`

-ring we have
`A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2√2`

(A caveat: in the commutative case we need 2⁻¹ in the ring, and in the noncommutative case we need √2 and √2⁻¹. To keep things simple we just assume our rings are ℝ-algebras.)

The proofs I've seen in the literature either
assume a significant framework for quantum mechanics,
or assume the ring is a `C^*`

-algebra.
In the `C^*`

-algebra case,
the order structure is completely determined by the `*`

-algebra structure:
`0 ≤ A`

iff there exists some `B`

so `A = star B * B`

.
There's a nice proof of both bounds in this setting at
https://en.wikipedia.org/wiki/Tsirelson%27s_bound
The proof given here is purely algebraic.

## Future work #

One can show that Tsirelson's inequality is tight.
In the `*`

-ring of n-by-n complex matrices, if `A ≤ λ I`

for some `λ : ℝ`

,
then every eigenvalue has absolute value at most `λ`

.
There is a CHSH tuple in 4-by-4 matrices such that
`A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁`

has `2√2`

as an eigenvalue.

## References #

- [Clauser, Horne, Shimony, Holt,
*Proposed experiment to test local hidden-variable theories*][zbMATH06785026] - [Bell,
*On the Einstein Podolsky Rosen Paradox*][MR3790629] - [Tsirelson,
*Quantum generalizations of Bell's inequality*][MR577178]

A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁`

such that
the `Aᵢ`

commute with the `Bⱼ`

.

The physical interpretation is that `A₀`

and `A₁`

are a pair of boolean observables which
are spacelike separated from another pair `B₀`

and `B₁`

of boolean observables.

## Instances For

Given a CHSH tuple (A₀, A₁, B₀, B₁) in a *commutative* ordered `*`

-algebra over ℝ,
`A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2`

.

(We could work over ℤ[⅟2] if we wanted to!)

We now prove some rather specialized lemmas in preparation for the Tsirelson inequality, which we hide in a namespace as they are unlikely to be useful elsewhere.

Before proving Tsirelson's bound, we prepare some easy lemmas about √2.

In a noncommutative ordered `*`

-algebra over ℝ,
Tsirelson's bound for a CHSH tuple (A₀, A₁, B₀, B₁) is
`A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2^(3/2) • 1`

.

We prove this by providing an explicit sum-of-squares decomposition of the difference.

(We could work over `ℤ[2^(1/2), 2^(-1/2)]`

if we really wanted to!)