Local Hidden Variables

The EPR argument concluded that an entangled pair must carry predetermined values that quantum mechanics omits. To test that idea we have to turn it into mathematics. The framework that does so is the local hidden-variable (LHV) model, and the assumption it encodes is called local realism.

The setup

Alice and Bob share many copies of a two-particle system. On each run Alice freely picks a measurement setting $a$ (say, a spin axis) and records an outcome $A \in \{+1, -1\}$ ; Bob picks a setting $b$ and records $B \in \{+1, -1\}$ . Repeating, they tabulate the joint statistics $P(A, B \mid a, b)$ .

A local hidden-variable model tries to explain those statistics with predetermined properties.

The two pillars of local realism

A hidden variable $\lambda$ summarizes everything the pair "carries" away from the source. It is distributed according to some probability density $\rho(\lambda)$ with $\int \rho(\lambda)\,d\lambda = 1$ , fixed at the source and independent of what Alice and Bob will later choose.

Realism. Each particle's outcome is a definite function of its own setting and of $\lambda$ . We write Alice's response as $A(a, \lambda) \in \{+1, -1\}$ and Bob's as $B(b, \lambda) \in \{+1, -1\}$ . The value exists before and independent of whether the measurement is performed — measurement merely reveals it.

Locality. Alice's outcome does not depend on Bob's setting, and vice versa. This is built into the notation: $A$ is a function of $a$ and $\lambda$ only — never of $b$ . Equivalently, conditioned on $\lambda$ the two outcomes are independent,

P(A, B \mid a, b, \lambda) = P(A \mid a, \lambda)\, P(B \mid b, \lambda).

Together, realism plus locality is local realism, and a theory respecting it is a local hidden-variable theory.

The correlation function in an LHV model

The quantity we will test is the correlation of the two outcomes for settings $a$ and $b$ , $E(a,b) = \langle AB \rangle$ , the average of the product over many runs. In any LHV model it is forced to take the form

E(a, b) = \int \rho(\lambda)\, A(a, \lambda)\, B(b, \lambda)\, d\lambda,

with $A, B \in \{+1, -1\}$ . This single expression is the entire content of local realism for our purposes. Everything Bell derives flows from the boxed structure: outcomes are $\pm 1$ valued, each depends only on its own local setting and the shared $\lambda$ , and we average over a fixed distribution $\rho(\lambda)$ .

A concrete (deterministic) model reproduces perfect anticorrelation

It is easy to build an LHV model matching the EPR singlet along a single shared axis. Let $\lambda$ be a unit vector $\hat\lambda$ , uniformly distributed, with $A(\hat n, \hat\lambda) = \text{sign}(\hat n \cdot \hat\lambda)$ and $B = -A$ . Whenever Alice and Bob use the same axis they get opposite signs — perfect anticorrelation, exactly as quantum mechanics predicts for the singlet. So far, local realism looks perfectly capable.

Why this matters

Notice what we have not assumed: nothing about specific particles, forces, or even quantum mechanics. The LHV form $E(a,b) = \int \rho\, A B\, d\lambda$ is a constraint on any local-realistic theory of the world. If experiment violates a bound derived from it, then no local-realistic description — quantum or otherwise — can be correct. That is what makes Bell's theorem, derived next, so powerful: it is an experimental test of a worldview, not of a particular equation.