We determine the necessary and sufficient conditions which ensure that an $N=2m$-particle fermionic or bosonic state has the form $|\mathrm{\Psi }\rangle \propto {\left(A^{†}\right)}^m|0\rangle$, where $A^{†}=\frac{1}{2}{\sum }_{i,j}{A}_{ij}{c}_i^{†}{c}_j^{†}$ is a general pair creation operator. These conditions can be cast as an eigenvalue equation for a modified two-body density matrix and enable an exact reconstruction of the operator $A^{†}$, providing as well a measure of the proximity of a given state to an exact pair condensate. Through a covariance-based formalism, it is also shown that such states are fully characterized by a set of $L$ “conserved” one-body operators which have $|\mathrm{\Psi }\rangle$ as exact eigenstate, with $L$ determined just by the single-particle space dimension involved. The whole set of two-body Hamiltonians having $|\mathrm{\Psi }\rangle$ as exact eigenstate is in this way determined, while a general subset having $|\mathrm{\Psi }\rangle$ as nondegenerate ground state is also identified. Extension to states $\propto f\left(A^{†}\right)|0\rangle$ with $f$ an arbitrary function is also discussed.

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• Bosons
• Fermions
• Pairing mechanisms
• Quantum entanglement
• Quantum many-body systems
• Approximation methods for many-body systems
• Many-body techniques
• Trial wave functions