Do all entangled states violate some Bell inequality? Suppose I have a multipartite pure entangled two-level system. Is it always possible to form a bell-inequality based on this state such that the observed correlations cannot be reproduced classically? Is there any straightforward way to form such inequalities?
 A: It is true if you restrict to pure states, but not for general states. There are (bipartite) entangled states which admit local hidden variable models.
The standard references on this are Werner's 1989 PRA and Wiseman et al. 2007.
Werner's way of showing this is to explicitly construct a local hidden variable model for a state that is not separable.
They consider what are now called Werner states, which are bipartite states $W\in\mathrm D(\mathbb C^d\otimes\mathbb C^d)$ --- where $\mathrm D(\mathcal H)$ denotes the set of density matrices on the Hilbert space $\mathcal H$ --- such that $(U\otimes U)W(U^\dagger\otimes U^\dagger)=W$ for any unitary $U:\mathbb C^d\to \mathbb C^d$.
These can also be given a more explicit form, as
$$W_{p,n}=p\frac{\Pi_+}{\binom{n+1}{2}} + (1-p)\frac{\Pi_-}{\binom{n}{2}},$$
where $\Pi_\pm$ are the projections onto the $\pm1$ eigenspaces of the swap operator: $\mathrm{SWAP}|i,j\rangle=|j,i\rangle$:
$$\Pi_\pm\equiv\frac12(I \pm \mathrm{SWAP}).$$
The binomial coefficients are due to the dimensions of the symmetric and asymmetric subspaces on which these projectors operate. See Watrous' book for more details on this.
One can see that Werner states are entangled for $p\in[0,1/2)$, e.g. via PPT criterion, and also that they are separable for $p\ge1/2$ with another approach (again, discussed in Watrous' book). However, as Werner shows, local hidden variable models can be found also for $p\ge1/2$. I asked a question concerning the explicit construction on qc.SE here.
Another related post on qc.SE is What is an example of an entangled state whose correlations are describable with a local hidden variable model?.
