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In Cavalcanti and Skrzypczyk (2017), section IIA, pag. 3, while introducing the idea of quantum steering, the authors introduce the idea of Local Hidden State model as following (emphasis mine):

A LHS model refers to the situation where a source sends a classical message $\lambda$ to one of the parties, say Alice, and a corresponding quantum state $\rho_\lambda$ to the other party, Bob. Given that Alice decides to apply measurement $x$, the variable $\lambda$ instructs Alice’s measurement device to output the result a with probability $p(a|x, \lambda)$. Additionally it is also considered that the classical message $\lambda$ can be chosen according to a distribution $\mu(\lambda)$. Bob does not have access to the classical variable $\lambda$, so the final assemblage he observes is composed by the elements $$\sigma_{a|x}=\int d\lambda \mu(\lambda)p(a|x,\lambda)\rho_\lambda.$$

I'm struggling to grasp the general picture here. Isn't this equivalent to saying that Alice and Bob share a state of the form $$\sum_\lambda |\lambda\rangle\!\langle\lambda|\otimes \rho_\lambda,$$ for some choice of a computational basis $|\lambda\rangle$ for Alice? In particular, it's the wording that "the source sends a classical message to Alice" that confuses me. What is a "classical message" here? Do we just mean that Alice is not allowed to perform measurements or operations that probe the coherence between different states $|\lambda\rangle$, for some choice of computational basis that was chosen beforehand?

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