Representability for Quantum Theory beyond Particle-Number Conservation

TL;DR

This paper develops a new framework for the representability problem of two-particle reduced density matrices in quantum systems that do not conserve particle number, providing explicit conditions and a unified approach.

Contribution

It introduces a systematic hierarchy of representability conditions for nonconserving quantum systems using geometric and linear algebraic methods.

Findings

Derived explicit linear equations for the polar cone of 2-RDMs.

Established a hierarchy of conditions independent of higher RDMs or wave functions.

Unified treatment of particle-number conserving and nonconserving systems with variance augmentation.

Abstract

Representability determines when a two-particle reduced density matrix (2-RDM) corresponds to a physical quantum state, enabling many-particle quantum calculations with 2-RDMs rather than the wave function. In this Letter, we present a solution of the representability problem for quantum systems without particle-number conservation. The physically allowed set of 2-RDMs can be characterized from a geometrically `orthogonal' set, the polar cone. We derive explicit linear equations for the two-body operators in the polar cone -- the intersection of the $p$-positive cone with the two-body operator space -- to obtain a systematic hierarchy of representability conditions that do not depend on higher RDMs or the wave function. Moreover, by augmenting these conditions with the particle-number variance, we obtain a unified framework for treating both particle-number-conserving and nonconserving systems. We illustrate with a spin system and molecular H$_4$.