In E.T Jaynes' book "Probability theory: the logic of science" the maximum entropy principle is discussed as the way to choose out of all possible hypotheses agreeing with constraints, the ones that are maximally non-committal with respect to missing information.

Chapter 11 of the book describes the standard maximum entropy principle (constraints are scalar functions $f_k(x_i)$) that results in scalar probabilities

$$p_i = \frac{\exp\left[{-\lambda_0 - \sum_{l=1}^m \lambda_l f_l(x_i) }\right]}{Z}$$

where the $\lambda$ are the Lagrangian multipliers associated with the constraints $f_k$, $\lambda_0$ is associated to the probability normalization and $Z$ is the partition function.

Chapter 30 instead tackles the same problem in its matricial formulation, where the constraints are matrices and instead of probabilities $p_i$ one has to consider a density matrix $\rho = e^{-H}/\textrm{Tr}[e^{-H}]$.

I understand the passage from the scalar MAXENT described in Chapter 11 to the matricial case in Chapter 30 as the passage from classical to quantum mechanics, where observables are matrices. Is this interpretation correct?

In the case one is constraining on a single matrix, what is the meaning of its associated Lagrangian multiplier? In the classical statistical mechanics case the Lagrangian multiplier $\beta$ is an inverse temperature that scales the energy of the system.

I have hard times finding some real-world usage of the matricial formulation of maximum entropy framework, although I believe it could be a very powerful formalism, for example in deriving maximally random ensembles of graphs obeying some symmetry patterns in graph-theory. If the matrix has some internal symmetries, how tuning the Lagrangian multiplier affects them?



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