  We develop connections between generalised notions of entanglement and
quantum computational devices where the measurements available are restricted,
either because they are noisy and/or because by design they are only along
Pauli directions. By considering restricted measurements one can (by
considering the dual positive operators) construct single particle state spaces
that are different to the usual quantum state space. This leads to a modified
notion of entanglement that can be very different to the quantum version (for
example, Bell states can become separable). We use this approach to develop
alternative methods of classical simulation that have strong connections to the
study of non-local correlations: we construct noisy quantum computers that
admit operations outside the Clifford set and can generate some forms of
multiparty quantum entanglement, but are otherwise classical in that they can
be efficiently simulated classically and cannot generate non-local statistics.
Although the approach provides new regimes of noisy quantum evolution that can
be efficiently simulated classically, it does not appear to lead to significant
reductions of existing upper bounds to fault tolerance thresholds for common
noise models.
