Quantum theory imposes a strict limit on the strength of nonlocal correlations. It only allows for a violation of the Clauser, Horne, Shimony, and Holt (CHSH) inequality up to the value 22 , known as Tsirelson’s bound. In this paper, we consider generalized CHSH inequalities based on many measurement settings with two possible measurement outcomes each. We demonstrate how to prove Tsirelson bounds for any such generalized CHSH inequality using semidefinite programming. As an example, we show that for any shared entangled state and observables X1,…,Xn and Y1,…,Yn with eigenvalues ±1 we have ∣⟨X1Y1⟩+⟨X2Y1⟩+⟨X2Y2⟩+⟨X3Y2⟩+⋯+⟨XnYn⟩-⟨X1Yn⟩∣⩽2ncos[π/(2n)] . It is well known that there exist observables such that equality can be achieved. However, we show that these are indeed optimal. Our approach can easily be generalized to other inequalities for such observables.
Physical Review A
- Pub Date:
- February 2006
- Entanglement and quantum nonlocality;
- Quantum Physics
- 9 pages, LateX, V2: Updated reference . To appear in Physical Review A