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Alice and Bob are each in possession of one half of a maximally entangled pair of particles. Alice can make either of two observations, $A_1$ or $A_2$. Bob can make either of two observations, $B_1$ or $B_2$. (Observations have values $1$ or $-1$). Write $E(x,y)$ for the expected value of the product $xy$. Then Tsirelson's Bound says that $|E(A1,B2)-E(A1,B1)-E(A2,B1)-E(A2,B2)|$ is bounded above by $2 \sqrt{2}$.

Question: Is the converse true? That is, suppose I have four numbers $x,y,z,w$. Suppose they are all bounded by one in absolute value and that they satisfy $|x-y-z-w| < 2 \sqrt{2}$. Does it follow that there are observables $A_1, A_2, B_1, B_2$ such that $x=E(A_1,B_2)$, $y=E(A_1,B_1)$, etc.?

If not, what other conditions do I need on $x, y, z, w$ in order for such observables to exist?

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No. The necessary and sufficient condition is well-known; see for instance Tsirelson 1993, or L. Landau.

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