Choosing E91/BB84 basis states and angles

Question

What is the logic for choosing the angles of E91 for the specific basis states? I have noticed a variety of different angles being used as the basis states for Alice and Bob. Would adding more basis states or changing the angles of these basis states affect the way that Alice and Bob check for intruders with the CHSH inequality, or would the fundamental idea of S <= 2 remain the same?

Thank you!

Answer 1

The fundamental idea of $S\leq 2$ remains the same. However, when you design the scheme, you ideally want two things:

• When you're measuring the generate key, the two parties should be using the same bases.
• When you're measuring to test CHSH, you want to choose bases that will give maximal violation so that, if there's an eavesdropper/errors, you've still got the maximum opportunity to get $S>2$. The maximum comes (as can be derived by following the arguments of the Tsirelson bound) when $\text{Tr}(A_iB_j)=\pm\sqrt{2}$, where $A_i$, $B_j$ are the observables of Alice and Bob respectively, which also satisfy $A_i^2=B_j^2=I$.