If you are interested in the Quantum Mechanics version of Fréchet bounds, then I have just edited the Fréchet inequalities page in wikipedia to show that similar bounds can also be obtained in quantum mechanics for separable density matrices. These bounds were derived in our paper:

Quantum mechanics: The Bayesian theory generalized to the space of Hermitian matrices. In: Physics Review A, 94 , pp. 042106, 2016.

It is worth to point out that entangled states violate these Fréchet bounds. Entangled states exhibit a form of *stochastic dependence* stronger than the *strongest classical dependence* and in fact they violate Fréchet like bounds. Another example of violation of probabilistic bounds is provided by the famous Bell’s inequality.