Is it possible to propose a local hidden-variable framework that reproduces the predictions of quantum mechanics?

Is it possible to propose a local hidden-variable framework that reproduces the predictions of quantum mechanics? In Sakurai’s book Modern Quantum Mechanics, section on Einstein’s Locality Principle and Bell’s Inequality (p.229), consider a trivial hidden-variable framework in which λis uniformly distributed: Prˆa`,ˆ b`s ď Prˆa`,ˆc`s ` Prˆc`,ˆ b`s.(1) In this case, every possible outcome has an identical chance of occurring: Prˆa`,ˆ b`s “ Prˆa`,ˆc`s “ Prˆc`,ˆ b`s “ 2 8. We then can write (1) as: 1 2sin2θď1 2sin2θ`1 2sin2θ, (2) with θ“45.0˝. Notice that (2) can be written as: psin2θq2ďsin2θ, (3) with θ“45.0˝. The inequalities psin2θq2ďsin2θand sin2θď |sin θ|are equivalent @θPR˝. Consequently, from (3): sin245.0˝ď |sin 45.0˝|(4) Then one can rewrite (4) as: 1´sin245.0˝ďsin 45.0˝,(5) or even as: 1´sin 45.0˝ďsin245.0˝.(6) And making use of the half-angle identity: 1´sin 45.0˝“2 sin222.5˝,(7) we obtain from (6): 2 sin222.5˝ďsin245.0˝.(8) Rearranging (8): 1 2sin222.5˝`1 2sin222.5˝ď1 2sin245.0˝,(9) which are the quantum-mechanical predictions shown on p.230. 1