# Worst Bell inequality violation with non-maximally entangled state?

I'm familiar with CHSH game and the strategy that allows Alice and Bob to succeed with a probability of $$\frac{1+\tfrac{1}{\sqrt 2}}{2}\approx 85\%$$ if they share a maximally entangled state such as $$\lvert\Phi_+\rangle$$ and use the measurements described by the observables $$\sigma_x,\sigma_y$$ on their respective qubits to determine their choice of move, depending on the bit they receive from Charlie. I've also seen the proof of the Tsirelson bound which effectively shows that $$\big\lVert A_1 \otimes B_1 + A_1\otimes B_2 + A_2\otimes B_1 - A_2\otimes B_2\big\rVert_2 \leq 2\sqrt{2}$$ if $$A_1,A_2$$ are Hermitian operators on a space $$\mathcal H_A$$ with spectrum $$\pm 1$$ and $$B_1, B_2$$ are Hermitian operators on $$\mathcal H_B$$ with spectrum $$\pm 1$$, and it happens that this maximum is attained when $$A_1=B_1=\sigma_x$$ and $$A_2=B_2=\sigma_y$$ with $$\mathcal H_A=\mathcal H_B = \mathbb C^2$$. (And the singular vector that gives the matrix norm its value happens to be maximally entangled.)

My question: Suppose we are only allowing Alice and Bob to start with some given quantum state that is entangled but not maximally entangled. For instance, say we give them the state $$\lvert\Psi\rangle = \tfrac{3}{5}\lvert 00\rangle + \tfrac{4}{5}\lvert 11\rangle$$ They can still outperform the classical maximum winning probability of $$75\%$$, and in fact by performing a $$e^{\pi i/4}$$ phase shift on the second component of their shared state and using the Pauli matrices just as is done in the optimal solution to the CHSH game, they can attain a winning probability of $$\frac{1+\tfrac{1}{\sqrt 2}\cdot\tfrac{24}{25}}{2}\approx 84\%$$ but I'd like to know if this solution is optimal for the given entangled state, or if some other slightly different choice of observables could give a better performance. More generally, does being given a different pure entangled state change their optimal strategy for the CHSH game, and how can we prove the answer?

• Why the downvote? Can I improve this question somehow? Mar 18 at 8:35

What you're after is Gisin's theorem. He proved that the maximal violation of the CHSH inequality with a fixed state $$|\psi\rangle = c_0 |00\rangle + c_1|11\rangle$$ is given by $$2\sqrt{1+4 |c_0 c_1|^2}$$. He also gives the optimal strategy.
• Am I being silly, or does this paper have a small error? I'm trying to follow his algebra, but I'm finding that the expression $|\cos\beta-\cos\beta'| + 2c_1c_2[\sin\beta + \sin\beta']$ is maximized at $x = (1+4|c_1c_2|^2)^{-1/2}$, as you write in your question, not $x = (1+4|c_1c_2|)^{-1/2}$. Maybe there is some strange notational issue I'm missing in the paper? Mar 18 at 17:20