Representability for Quantum Theory beyond Particle-Number Conservation

Area of Science:

• Quantum mechanics
• Computational chemistry
• Quantum information theory

Background:

• Representability of two-particle reduced density matrices (2-RDMs) is crucial for quantum calculations, determining if a 2-RDM corresponds to a physical quantum state.
• Current methods often rely on the wave function or higher-order density matrices, posing computational challenges.

Purpose of the Study:

• To solve the representability problem for quantum systems without particle-number conservation.
• To develop a systematic hierarchy of representability conditions independent of higher RDMs or the wave function.
• To establish a unified framework for both particle-number-conserving and nonconserving systems.

Main Methods:

• Characterizing physically allowed 2-RDMs using the polar cone, an 'orthogonal' geometric set.
• Deriving explicit linear equations for two-body operators within the polar cone.
• Augmenting conditions with particle-number variance.

Main Results:

• A systematic hierarchy of representability conditions for 2-RDMs in systems without particle-number conservation.
• These conditions are derived from the polar cone and do not require higher RDMs or the wave function.
• A unified framework is presented for both particle-number-conserving and nonconserving systems.

Conclusions:

• The developed conditions provide a practical approach to assessing 2-RDM representability.
• This work simplifies quantum many-body calculations by focusing on 2-RDMs.
• The unified framework enhances the applicability of 2-RDM methods across different quantum systems.