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 ${\mathrm{H}}_4$.

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• Electronic structure
• Quantum many-body systems
• Density matrix methods
• Many-body techniques