Quantum Covariance Scalar Products and Efficient Estimation of Max-Ent Projections

Abstract

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 (KMB) 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.

Figures (9)

• Figure 1: Max-Ent dynamics in the Bloch's sphere. Left: Max-Ent construction. The Max-Ent manifold ${\cal M}_{\textnormal{Max-Ent}}$ spans all the states with defined $\langle{\bf Q}_{x,y}\rangle = \langle {\bf S}_{x,y}\rangle$ mean values and maximum entropy. The state $\sigma$ is the state with maximum entropy that shares these averages with $\rho$. Right: the ideas of exact and Max-Ent dynamics are contrasted, where the latter is an approximation of the former.
• Figure 2: (Color online) Different evolution schemes. a) The solid curve (blue online) represents the trajectory of ${\bf K}(t)$ following the free Schrödinger evolution. The dot-dashed line (green online) curve and the dashed line (red online) represent the Max-Ent projection of the free evolution $\Pi_B {\bf K}(t)$ and its linearization $\pi_B {\bf K}(t)$, respectively. The dotted line (orange online) represents the restricted evolution $\tilde{\bf K}_B(t)$\ref{['eq:restricted']}. b) Relation among the distances $\Delta(t)$ and $\tilde{\Delta}(t)$\ref{['eq:deftildeDelta', 'eq:defDelta']} between the instantaneous ${\bf K}(t)$, $\Pi_B{\bf K}(t)$ and $\tilde{\bf K}_B(t)$ and its different approximations. In the scheme, intrinsic KMB geometry around $\pi_{B,\rho(t)} {\bf K}(t)$ is identified with the Euclidean one. Note that the different states do not lie in the same trajectory.
• Figure 3: Evolution of the KMB-induced norms between the exact logarithm of the states and their KMB/correlation projections, in logarithmic scale, obtained from a $t=10/J$ simulation with 200 steps. We show these results for two inverse temperatures, $\beta = J/10$ (top) and $\beta = J$ (bottom). For the short-term evolution, both the KMB and covar projected states exhibit remarkable similarities amongst themselves and with the exact state.
• Figure 4: Evolution of the expectation values for the occupation operator ${\bf n}$ at inverse temperatures $\beta = 0.1$ (top) and $\beta = 1$ (bottom), regarding the exact state and their linearized Max-Ent projections (see \ref{['eq:piasbesselfourier']}) concerning the basis $B=B_4$ given by \ref{['eq:iteraterB']}. In the short-term regime ($t J \lessapprox 2$), the three dynamics yield highly similar outcomes. The subsequent lack of conservation is a consequence of the departure of the exact state trajectory from the corresponding Max-Ent manifold ${\cal M}_B$ and the limitations of the linear approximation.
• Figure 5: Evolution of the expectation values for the Hamiltonian operator ${\bf H}$ at inverse temperatures $\beta = 0.1$ (top) and $\beta = 1$ (bottom), regarding the exact state and their linearized Max-Ent projections (see \ref{['eq:piasbesselfourier']}) concerning the basis $B=B_4$ given by \ref{['eq:iteraterB']}. In the short-term regime ($t J \lessapprox 2$), the three dynamics yield highly similar outcomes. The subsequent lack of conservation is a consequence of the departure of the exact state trajectory from the corresponding Max-Ent manifold ${\cal M}_B$ and the limitations of the linear approximation.

Theorems & Definitions (48)

• Proposition 1.1
• Proposition 1.2
• Proposition 1.3
• Lemma 1.4
• Proposition 2.1
• proof
• Lemma B.1
• proof
• Lemma B.2
• proof