Algebraic Geometry of Quantum Graphical Models

TL;DR

This paper develops algebraic geometric methods to analyze quantum graphical models, extending classical techniques to quantum states and establishing foundational properties and algorithms for their study.

Contribution

It introduces algebraic varieties associated with quantum graphical models, enabling the application of algebraic geometry to quantum statistics and information theory.

Findings

Classical models are recoverable via diagonal quantum states.

Algorithms for computing defining equations of quantum model varieties.

Proves a quantum analogue of Birch's Theorem.

Abstract

Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical models are families of quantum states satisfying certain locality or correlation conditions encoded by a graph. We lay out several ways to associate an algebraic variety to a quantum graphical model. The classical graphical models can be recovered from most of these varieties by restricting to quantum states represented by diagonal matrices. We study fundamental properties of these varieties and provide algorithms to compute their defining equations. Moreover, we study quantum information projections to quantum exponential families defined by graphs and prove a quantum analogue of Birch's Theorem.