Reduced density matrix in two-particle interferometry

Michael Horne, Abner Shimony, Anton Zeilinger

Tohru Morita

The diagram technique, which was developed to solve a hierarchy of equations of motion for the reduced density matrices, is applied in order to solve a hierarchy of the equations of motion for the two-time Green's functions. The diagrammatical representations of the two-time Green's functions are obtained either in terms of the equilibrium one- and many-particle reduced density matrices or in terms only of the equilibrium one-particle reduced density matrix. The latter representation for the one-particle two-time Green's function and the formula by which the reduced density matrix is calculated from the Green's function give us a set of coupled equations which determines both the reduced density matrix and the two-time Green's function. The calculation with the aid of the present technique is illustrated for an example of Hubbard's calculation for the “Hubbard Hamiltonian”.

Fabian Lackner, Iva Březinová, Takeshi Sato +2 more

Describing time-dependent many-body systems where correlation effects play an important role remains a major theoretical challenge. In this paper we develop a time-dependent many-body theory that is based on the two-particle reduced density matrix (2-RDM). We present a closed equation of motion for the 2-RDM by developing a reconstruction functional for the three-particle reduced density matrix (3-RDM) that preserves norm, energy, and spin symmetries during time propagation. We show that approximately enforcing $N$-representability during time evolution is essential for achieving stable solutions. As a prototypical test case which features long-range Coulomb interactions we employ the one-dimensional model for lithium hydride (LiH) in strong infrared laser fields. We probe both one-particle observables such as the time-dependent dipole moment and two-particle observables such as the pair density and mean electron-electron interaction energy. Our results are in very good agreement with numerically exact solutions for the $N$-electron wave function obtained from the multiconfigurational time-dependent Hartree-Fock method.

Michael Horne, Abner Shimony, Anton Zeilinger

Everett G. Larson

A model many-fermion Hamiltonian is presented for which the ground state is asymptotically an Antisymmetrized Geminal Powers (AGP) wave function with largest possible greatest eigenvalue for its two-particle reduced density matrix. Closed analytical expressions and plane-wave expansions are presented for the generating geminal of the AGP ground state and for its one-particle reduced density matrix. The natural orbitals for this generating geminal are plane waves. The generating geminal shows intensely local character in its intracule and corresponds to the formation of a quasi-boson from two fermions. One may appropriately modify this generating geminal to introduce zero occupation numbers of its one-particle reduced density matrix and to make all the nonzero occupation numbers of its one-particle reduced density matrix equal, thus making this geminal a generator of an extreme AGP wave function, with an extreme large eigenvalue for its two-particle reduced density matrix. Closed analytical expressions are also given for this modified geminal and for its one-particle reduced density matrix. The modified generating geminal develops anomolously long “tails” in the intracule function. Nevertheless, both generating geminals introduced here reduce to a renormalized Dirac δ-function of the intracule coordinate, in the appropriate limit. In this limit, both of these generating geminals produce an absolute extreme AGP wave function with the absolute extreme value, N/2, for the Löwdin-normalized occupation number of the generating geminal, and vanishing occupation numbers for all other natural geminals, in the two-particle reduced density matrix of this wavefunction. A demonstration is given showing the formation of quasi-bosons in this ground state and showing their relationship to the generating geminal and to the other natural geminals of the two-particle reduced density matrix of the extreme AGP wave function. The similarities and differences of the features of this model and the accepted models of the superconducting ground state of electrons in metals, and the superfluid ground state of liquid He4 are mentioned.

Gustavo E. Massaccesi, Ofelia B. Oña, Luis Lain +4 more

Reduced density matrices are central to describing observables in many-body quantum systems. In electronic structure theory, the two-particle reduced density matrix (2-RDM) suffices to determine the energy and other key properties. Recent work has used matrix completion, leveraging the low-rank structure of RDMs and approximate theoretical models, to reconstruct the 2-RDM from partial data and thus reduce computational cost. However, matrix completion is, in general, an under-determined problem. Revisiting Rosina's theorem [M. Rosina, Queen's Papers on Pure and Applied Mathematics No. 11, 369 (1968)], we here show that the matrix completion is unique under certain conditions, identifying the subset of 2-RDM elements that enables its exact reconstruction from incomplete information. Building on this, we introduce a hybrid quantum-stochastic algorithm that achieves exact matrix completion, demonstrated through applications to the Fermi-Hubbard model.