= Solution
For the <spin singlet state>, measurements along the same axis are perfectly anticorrelated. By measuring either spin component of particle $B$, one can therefore predict with certainty the corresponding result for distant particle $A$. The <Einstein–Podolsky–Rosen criterion of reality> then says that every such spin component of $A$ is an element of physical reality, because the choice at $B$ can be made without disturbing $A$. Since the quantum state assigns no simultaneous sharp values to noncommuting spin components, the EPR argument concludes that the wavefunction is not a complete description, provided this locality premise is accepted.
Let $\lambda$ denote a proposed complete hidden state, and let $a,b$ be the chosen axes.
* <Outcome determinism> says that, given $\lambda$ and the settings, each outcome is fixed rather than merely probabilistic: $A,B\in\{-1,1\}$ are definite response functions.
* <Parameter independence> says that the conditional distribution of one local outcome is independent of the distant setting. Together with outcome determinism, it gives $A=A(a,\lambda)$ and $B=B(b,\lambda)$.
* <Measurement independence> says that the preparation variable is statistically independent of the later settings: $\rho(\lambda|a,b)=\rho(\lambda)$.
For two axes on each side, every $\lambda$ obeys
$$
\begin{aligned}
S(\lambda)
&=A(a_0,\lambda)[B(b_0,\lambda)+B(b_1,\lambda)]\\
&\quad+A(a_1,\lambda)[B(b_0,\lambda)-B(b_1,\lambda)]
\in\{-2,2\}.
\end{aligned}
$$
<Measurement independence> permits averaging the same distribution of $\lambda$ for all four setting pairs, producing the <CHSH inequality>
$$
|E_{00}+E_{01}+E_{10}-E_{11}|\leq2.
$$
The singlet prediction is
$$
E(a,b)=-\mathbf a\mathbin\cdot\mathbf b.
$$
Choose coplanar unit vectors
$$
\mathbf a_0=\mathbf z,
\qquad
\mathbf a_1=\mathbf x,
\qquad
\mathbf b_0=\frac{\mathbf z+\mathbf x}{\sqrt2},
\qquad
\mathbf b_1=\frac{\mathbf z-\mathbf x}{\sqrt2}.
$$
Then
$$
E_{00}=E_{01}=E_{10}=-\frac1{\sqrt2},
\qquad
E_{11}=\frac1{\sqrt2},
$$
and therefore
$$
\boxed{|E_{00}+E_{01}+E_{10}-E_{11}|=2\sqrt2>2}.
$$
Thus the predictions of quantum theory violate the conjunction of outcome determinism, parameter independence, and measurement independence. This is <Bell theorem>; the calculation alone does not select which premise a deeper theory must abandon.
Back to article page
