  The term Bell's theorem refers to a set of closely related results which
imply that quantum mechanics is incompatible with local hidden variable
theories. Bell's inequality is the statement that if measurements are performed
independently on two space-like separated particles of an entangled pair, the
assumption that outcomes depend on hidden variables implies an upper bound on
the correlations between the outcomes. Quantum mechanics predicts correlations
which violate this upper bound. The CHSH inequality is a specific Bell
inequality in which classical correlation (i.e. if local hidden variables
exist) can take the maximum value of 2. Quantum mechanics violates this bound,
allowing for a higher bound on the correlation, which can take the maximum
value $2\sqrt{2}$, known as the Tsirelson bound. Popescu and Rohrlich showed
that the assumption of relativistic causality allows for an even higher bound
on the CHSH correlation, this value being 4. Why is the bound coming from
causality higher than the Tsirelson bound? Are there relativistic causal
dynamical theories which violate the Tsirelson bound? In the present paper we
answer this question in the affirmative. We show that the pre-quantum theory of
trace dynamics, from which quantum theory is emergent as a thermodynamic
approximation, permits the CHSH correlation to take values higher than
$2\sqrt{2}$. We interpret our findings to suggest that quantum theory is
approximate, and emergent from the more general theory of trace dynamics.
