Quantum covariance scalar products and efficient estimation of maximum-entropy projections

The maximum-entropy principle (max-ent) is a valuable and extensively used tool in statistical mechanics and quantum information theory. It provides a method for inferring the state of a system by utilizing a reduced set of parameters associated with measurable quantities. However, the computational cost of employing max-ent projections in simulations of quantum many-body systems is a significant drawback, primarily due to the computational cost of evaluating these projections. In this work, a different approach for estimating max-ent projections is proposed. The approach involves replacing the expensive max-ent induced local geometry, represented by the Kubo-Mori-Bogoliubov scalar product, with a less computationally demanding geometry. Specifically, a new local geometry is defined in terms of the quantum analog of the covariance scalar product for classical random variables. Relations between induced distances and projections for both products are explored. Connections with standard variational and dynamical mean-field approaches are discussed. The effectiveness of the approach is calibrated and illustrated by its application to the dynamic of excitations in a $XX$ Heisenberg spin-$\frac{1}{2}$ chain model.