Bell's Theorem

Bell’s theorem supplies quantum predictions outside the class of measurement-independent, Bell-local models. In a finite measurement scenario, this becomes a geometric statement: the predicted probability vector lies outside a compact polytope, and a linear functional separates it from that polytope. This page develops the separation and its error margin; the linked article explains the physical premises and explicit quantum violation.

Required background. Local Hidden Variables defines factorization and its deterministic refinement; Bell’s Theorem establishes the existence of a quantum behavior outside the local set.

The finite local polytope

Take finite setting sets $X,Y$ and finite outcome sets $A,B$. Arrange all conditional probabilities $p(a,b\mid x,y)$ in a vector in $\mathbb R^N$. Normalization and marginal consistency constrain this vector, so the relevant affine subspace usually has dimension smaller than $N$.

For each pair of deterministic response tables $f:X\to A$ and $g:Y\to B$, write

$$D_{f,g}(a,b\mid x,y) =\mathbf1_{a=f(x)}\mathbf1_{b=g(y)}.$$

The deterministic-refinement result gives

$$\mathcal L=\operatorname{conv}\{D_{f,g}\}.$$

There are $|A|^{|X|}|B|^{|Y|}$ such tables. Therefore $\mathcal L$ is convex and compact: it is the continuous image of a finite probability simplex. This step depends on a finite scenario. Infinite settings or continuous outcomes require a specified topology and separate closure arguments; one cannot silently import finite polytope language.

Constructing a separating Bell functional

Let $p_*\notin\mathcal L$. Compactness provides a nearest point $q_*$ in Euclidean norm. Set $c=p_*-q_*$, which is nonzero. For every $q\in\mathcal L$, convexity puts $q_*+t(q-q_*)$ in $\mathcal L$ for $0\le t\le1$. The squared distance from $p_*$ has a minimum at $t=0$, so its right derivative gives

$$-2c\cdot(q-q_*)\ge0.$$

Consequently, with $\beta=c\cdot q_*$,

$$\begin{aligned} c\cdot q&\le\beta &&(q\in\mathcal L),\\ c\cdot p_*-\beta&=\|c\|_2^2>0. \end{aligned}$$

Thus every exterior behavior violates some linear Bell inequality. Conversely, violation of any valid local inequality proves that the behavior is exterior. This is a geometric existence proof; finding useful coefficients efficiently is a separate optimization problem. The separating inequality need not define a facet, and different coefficient arrays can represent the same functional on the normalized affine subspace.

The CHSH example has the local bound $2$ and an attainable quantum value $2\sqrt2$. Its explicit operator calculation is in CHSH Inequality. Separation explains why linear witnesses suffice in every finite scenario, not merely that example.

A witness gives a quantitative margin

Suppose $c\cdot p_*-\beta=\gamma>0$. For another normalized behavior $p$, Hölder’s inequality gives

$$|c\cdot(p-p_*)| \le\|c\|_\infty\|p-p_*\|_1.$$

Hence $p$ still violates the witness whenever $\|p-p_*\|_1<\gamma/\|c\|_\infty$. Here the norm sums over all setting-and-outcome entries; it is not the total-variation distance of a single joint experiment unless a setting distribution has also been specified. Scaling both $c$ and $\beta$ leaves the criterion unchanged.

This is a deterministic perturbation bound on probability vectors. Turning finite counts into a confidence statement requires a statistical model for the trials, losses, setting choices, and dependence across trials. A witness margin alone is not a significance level.

Infinite-dimensional separability still implies locality

A normal separable state is in the trace-norm closed convex hull of product density operators. It need not admit a countable product-state ensemble. Fix a finite family of local POVMs and approximate such a state by finite product mixtures $\rho_n$ in trace norm. Each corresponding behavior $p_n$ is local by the canonical product-state argument.

Every effect $M_{a|x}\otimes N_{b|y}$ is bounded by the identity. Trace duality gives

$$|p_n(a,b\mid x,y)-p(a,b\mid x,y)| \le\|\rho_n-\rho\|_1.$$

There are finitely many coordinates, so $p_n\to p$ in $\mathbb R^N$. Closedness of $\mathcal L$ yields $p\in\mathcal L$. This proves locality in every fixed finite scenario without assuming a countable separable decomposition. See Separable and Entangled Mixed States for the definition. It does not prove the converse: entangled states can have local statistics for substantial measurement classes.

Exercise: bound the distance from the local set

For a valid witness with gap $\gamma>0$ at $p$, show that every $q\in\mathcal L$ obeys $\|p-q\|_1\ge\gamma/\|c\|_\infty$. Explain why the bound may be nonoptimal.

Solution

Since $c\cdot q\le\beta$, $\gamma=c\cdot p-\beta\le c\cdot(p-q) \le\|c\|_\infty\|p-q\|_1$. Taking the infimum over local $q$ proves the distance bound. A chosen witness need not be the dual functional that optimizes this norm distance; Hölder’s inequality can also be strict. The result is a certified lower bound, not automatically the exact distance.

References

• Bell, John S., “On the Einstein Podolsky Rosen paradox,” Physics Physique Fizika 1, 195–200 (1964), doi:10.1103/PhysicsPhysiqueFizika.1.195 — the locality obstruction.
• Brunner, Nicolas, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419–478 (2014), doi:10.1103/RevModPhys.86.419 — finite correlation sets and Bell functionals.
• Holevo, Alexander S., Maksim E. Shirokov, and Reinhard F. Werner, “Separability and entanglement-breaking in infinite dimensions,” Russian Mathematical Surveys 60, 359–360 (2005), arXiv:quant-ph/0504204 — trace-norm separability and ensemble caveats.