We present a constructive solution to the N -representability problem-a full characterization of the conditions for constraining the two-electron reduced density matrix (2-RDM) to represent an N -electron density matrix. Previously known conditions, while rigorous, were incomplete. Here we derive a hierarchy of constraints built upon (i) the bipolar theorem and (ii) tensor decompositions of model Hamiltonians. Existing conditions D, Q, G, T 1, and T 2, known classical conditions, and new conditions appear naturally. Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems. PACS numbers: 31.10.+z The wavefunction of a many-electron quantum system contains significantly more information than necessary for the calculation of energies and properties. In 1955 Mayer proposed in Physical Review computing the ground-state energy variationally as a functional of the two-electron reduced density matrix (2-RDM) which, unlike the wavefunction, scales polynomially with the number N of electrons [1] [2] [3] . However, the 2-electron density matrix must be constrained to represent a manyelectron (or N -electron) density matrix (or wavefunction); otherwise, the minimized energy is unphysically below the ground-state energy for N > 2. Coleman called these constraints N -representability conditions [4] , and the search for them became known as the Nrepresentability problem [5] [6] [7] [8] [9] [10] . In 1995 the National Research Council ranked the N -representability problem as one of the top unsolved theoretical problems in chemical physics [11] . While progress was limited for many years, recent advances in theory and optimization [12] [13] [14] [15] [16] [17] [18] [19] [20] have enabled the application of the variational 2-RDM method to studying strong correlation in quantum phase transitions [21] Despite the recent computational results with 2-RDM methods, a complete set of N -representability conditions on the 2-RDM-not dependent upon higher-order RDMs-has remained unknown. While formal solutions of the N -representability problem were developed in the 1960s [5, 26] , practically they required the N -electron density matrix [1, 2] . In this Letter we present a constructive solution of the N -representability problem that generates a complete set of N -representability conditions on the 2-RDM. The approach is applicable to generating the N -representability conditions on the p-RDM for any p ≤ N . The conditions arise naturally as a hierarchy of constraints on the 2-RDM, which we label the (2, q)-positivity conditions, where the (2, 2)-and (2, 3)-positivity conditions include the already known D, Q, G, T 1, and T 2 conditions [4, 5, 7, 15] . The second number in (2, q) corresponds to the higher q-RDM which serves as the starting point for the derivation of the condition. A key advance in extending the (2, q)-positivity conditions for q > 3 is the use of tensor decompositions in the model Hamiltonians that expose the boundary of the N -representable 2-RDM set. The decompositions allow the terms in the model Hamiltonians to have no more than two-body interactions through the cancelation of all higher 3-to-q-body terms. A second important element is the recognition that when q = r where r is the rank of the one-electron basis set the positivity conditions are complete. The hierarchy of conditions can be thought of as a collection of model Hamiltonians [9] . For example, the 'basic' (2,2)-positivity conditions are both necessary and sufficient constraints for computing the ground-state energies of pairing model Hamiltonians [2, 14] , often employed in describing long-range order and superconductivity. Consider a quantum system composed of N fermions. A matrix is a fermionic density matrix if and only if it is: (i) Hermitian, (ii) normalized (fixed trace), (iii) antisymmetric in the exchange of particles, and (iv) positive semidefinite. A matrix is positive semidefinite if and only if its eigenvalues are nonnegative. The p-particle reduced density matrix (p-RDM) can be obtained from the N -particle density matrix by integrating over all but the first p particles The set of N D is a convex set which we denote as P N while the set p D is a convex set which we denote as P p N , the set of N -representable p-particle density matrices. A set is convex if and only if the convex combination of any two members of the set is also contained in the set where 0 ≤ w ≤ 1. The integration in Eq. (1) defines a linear mapping from P N to P p N , which preserves its convexity. The energy of a quantum system in a stationary state can be computed from the Hamiltonian traced against
In 2012 Mazziotti established a hierarchy of representability conditions providing a formal solution of the representability problem for the two-body RDMs of N -particle density matrices REF .
Structure of Fermionic Density Matrices: Complete N-representability Conditions
Phys. Rev. Lett. 108, 263002 (2012)
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