Practical measures of electronic decoherence, called distilled purities, that are applicable to many-body systems are introduced. While usual measures of electronic decoherence such as the purity employ the full N-particle density matrix which is generally unavailable, the distilled purities are based on the r-body reduced density matrices (r-RDMs) which are more accessible quantities. The r-body distilled purities are derivative quantities of the previously introduced r-body reduced purities [I. Franco and H. Appel, J. Chem. Phys. 139, 094109 (2013)] that measure the non-idempotency of the r-RDMs. Specifically, the distilled purities exploit the structure of the reduced purities to extract coherences between Slater determinants with integer occupations defined by a given single-particle basis that compose an electronic state. In this way, the distilled purities offer a practical platform to quantify coherences in a given basis that can be used to analyze the quantum dynamics of many-electron systems. Exact expressions for the one-body and two-body distilled purities are presented and the utility of the approach is exemplified via an analysis of the dynamics of oligo-acetylene as described by the Su-Schrieffer-Heeger Hamiltonian. Last, the advantages and limitations of the purity, reduced purity, and distilled purity as measures of electronic coherence are discussed.
Practical measures of electronic decoherence, called distilled purities, that are applicable to many-body systems are introduced. While usual measures of electronic decoherence such as the purity employ the full N-particle density matrix which is generally unavailable, the distilled purities are based on the r-body reduced density matrices (r-RDMs) which are more accessible quantities. The r-body distilled purities are derivative quantities of the previously introduced r-body reduced purities [I. Franco and H. Appel, J. Chem. Phys. 139, 094109 (2013)] that measure the non-idempotency of the r-RDMs. Specifically, the distilled purities exploit the structure of the reduced purities to extract coherences between Slater determinants with integer occupations defined by a given single-particle basis that compose an electronic state. In this way, the distilled purities offer a practical platform to quantify coherences in a given basis that can be used to analyze the quantum dynamics of many-electron systems. Exact expressions for the one-body and two-body distilled purities are presented and the utility of the approach is exemplified via an analysis of the dynamics of oligo-acetylene as described by the Su-Schrieffer-Heeger Hamiltonian. Last, the advantages and limitations of the purity, reduced purity, and distilled purity as measures of electronic coherence are discussed.
rarchy of measures of decoherence for many-electron systems that is based on the purity and the hierarchy of reduced electronic density matrices is presented. These reduced purities can be used to characterize electronic decoherence in the common case when the many-body electronic density matrix is not known and only reduced information about the electronic subsystem is available. Being defined from reduced electronic quantities, the interpretation of the reduced purities is more intricate than the usual (many-body) purity. This is because the nonidempotency of the r-body reduced electronic density matrix that is the basis of the reduced purity measures can arise due to decoherence or due to electronic correlations. To guide the interpretation, explicit expressions are provided for the one-body and two-body reduced purities for a general electronic state. Using them, the information content and structure of the one-body and two-body reduced purities is established, and limits on the changes that decoherence can induce are elucidated. The practical use of the reduced purities to understand decoherence dynamics in many-electron systems is exemplified through an analysis of the electronic decoherence dynamics in a model molecular system.
A hierarchy of measures of decoherence for many-electron systems that is based on the purity and the hierarchy of reduced electronic density matrices is presented. These reduced purities can be used to characterize electronic decoherence in the common case when the many-body electronic density matrix is not known and only reduced information about the electronic subsystem is available. Being defined from reduced electronic quantities, the interpretation of the reduced purities is more intricate than the usual (many-body) purity. This is because the nonidempotency of the r-body reduced electronic density matrix that is the basis of the reduced purity measures can arise due to decoherence or due to electronic correlations. To guide the interpretation, explicit expressions are provided for the one-body and two-body reduced purities for a general electronic state. Using them, the information content and structure of the one-body and two-body reduced purities is established, and limits on the changes that decoherence can induce are elucidated. The practical use of the reduced purities to understand decoherence dynamics in many-electron systems is exemplified through an analysis of the electronic decoherence dynamics in a model molecular system.
The N-representability problem for reduced density matrices remains a fundamental challenge in electronic structure theory. Following our previous work that employs a unitary-evolution algorithm based on an adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to probe pure-state N-representability of reduced density matrices [J. Chem. Theory Comput. 2024, 20, 9968], in this work we propose a practical framework for determining the ensemble N-representability of a p-body matrix. This is accomplished using a purification strategy consisting of embedding an ensemble state into a pure state defined on an extended Hilbert space, such that the reduced density matrices of the purified state reproduce those of the original ensemble. By iteratively applying variational unitaries to an initial purified state, the proposed algorithm minimizes the Hilbert-Schmidt distance between its p-body reduced density matrix and a specified target p-body matrix, which serves as a measure of the N-representability of the target. This methodology facilitates both error correction of defective ensemble reduced density matrices, and quantum-state reconstruction on a quantum computer, offering a route for density-matrix refinement. We validate the algorithm with numerical simulations on systems of two, three, and four electrons in both, simple models as well as molecular systems at finite temperature, demonstrating its robustness.
Generalizations of McWeeny's purification formula are developed within the formalism of the particle-hole duality from the theory of reduced density matrices. Each of the generalized purification formulas is expressed as a sum of the one-particle reduced density matrix (1-RDM) and a finite series in the product of the one-particle and the one-hole RDMs, a product which vanishes in the limit that the 1-RDM is idempotent. Two categories of purification formulas are explored: (i) formulas which treat the "occupied" and the "virtual" occupation numbers equivalently and (ii) formulas which treat these occupation numbers differently. The latter category includes and extends the purification formulas derived in the context of the 1,2-contracted Schrödinger equation [D. A. Mazziotti, J. Chem. Phys. 115, 8305 (2001)]. While the McWeeny purification minimizes the absolute error in the occupation numbers quadratically, the generalized purification formulas exhibit faster than quadratic convergence of the 1-RDM towards idempotency. Application of these purification formulas in existing algorithms for linear scaling will be explored and discussed including illustrative calculations on sodium wires of length 10, 20, 30, and 40 atoms.
It is well known that determining the energy of molecules and other quantum many-body systems reduces in the standard approximation to optimizing a simple linear functional of a 12-variable object, the two-electron reduced density matrix (2-RDM). The difficulty is that the variation ensemble for that functional has never been satisfactorily determined. This is known as the N-representability problem of quantum chemistry (which to a large extent is a problem of quantum information theory). The situation has given rise to competing research programs, typically trading more complicated functionals for simpler representability conditions. Chief among them is density functional theory, based on a three-variable object for which the N-representability is trivial, whereas the exact functional is very strange indeed, and probably forever unknowable. An intermediate position is occupied by 1-RDM functional theory. Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers, accordingly allowing no more than one electron in each quantum state. This is a necessary and sufficient condition for a 1-RDM to be the contraction of an ensemble N-body density matrix. The fermionic one-body quantum marginal problem asks whether given natural occupation numbers can arise from an antisymmetric (ensemble or pure) N-particle state. The configuration interaction method affords optimal descriptions of quantum states of atoms and molecules by expanding the wave function in terms of orbital-based configurations of Slater determinants. For these systems, the dimension of the Hilbert space grows binomially with m, the number of spin-orbitals in the basis of the one-particle Hilbert space, and N, the number of electrons of the system. It has been observed that for the rank-six approximation of a pure-state N=3 system, the occupation numbers satisfy some additional constraints, stronger than the Pauli exclusion principle. The recent analysis by Alexander Klyachko and coworkers of the one-body marginal problem of the pure N-fermion state establishes a systematic approach to this type of constraints. In fact, for a pure quantum system of N electrons arranged in m spin-orbitals the occupation numbers satisfy a set of linear inequalities, known as generalized Pauli constraints (GPC). These inequalities define a convex polytope of allowed states in R^m. They are conditions for a 1-RDM to be the contraction of a N-body state. When one of the GPC is completely saturated, the system is said to be pinned, and it lies on one of the facets of the polytope. The nature of those conditions has been explored till now only in a few systems: a model of three spinless fermions confined to a one-dimensional harmonic potential, the lithium isoelectronic series and ground and excited states of some three- and four-electron molecules for the rank being at least twice the number of electrons. For all these systems the inequalities are (quite often) nearly saturated. This is the so-called quasipinning phenomenon. In this PhD thesis we have taken up the challenge of using numerical and analytical methods to examine pinned and quasipinned states, for atoms and molecules, starting from scratch with configuration-interaction and multiconfiguration self-consistent methods. This procedure serves to study the occurrence of quasipinning in realistic systems. A second goal is to show how the subsets of pinned states defined by GPC give rise to the most efficient approach from a computational viewpoint, yielding the leading order of the electron-electron correlations. As a consequence, we underline in this thesis the theoretical and practical importance of Klyachko's approach to the quantum marginal problem and its impact on the competing research programs to determine feasible electronic densities, 1-RDM, 2-RDM, intracular distributions or Wigner density quasiprobabilities. In relation with the above, our research provides a new variational optimization method for few-fermion ground states. We quantitatively confirm its high accuracy for quasipinned systems and derive an upper bound on the error of the correlation energy given by the ratio of the numerical value of the Klyachko inequality and the distance to the Hartree-Fock point. Depending on the details of the algorithm, we are able to reach 98%-99% of the correlation energy for such systems.
Electronic decoherence processes in molecules and materials are usually thought and modeled via schemes for the system-bath evolution in which the bath is treated either implicitly or approximately. Here we present computations of the electronic decoherence dynamics of a model many-body molecular system described by the Su-Schrieffer-Heeger Hamiltonian with Hubbard electron-electron interactions using an exact method in which both electronic and nuclear degrees of freedom are taken into account explicitly and fully quantum mechanically. To represent the electron-nuclear Hamiltonian in matrix form and propagate the dynamics, the computations employ the Jordan-Wigner transformation for the fermionic creation/annihilation operators and the discrete variable representation for the nuclear operators. The simulations offer a standard for electronic decoherence that can be used to test approximations. They also provide a useful platform to answer fundamental questions about electronic decoherence that cannot be addressed through approximate or implicit schemes. Specifically, through simulations, we isolate basic mechanisms for electronic coherence loss and demonstrate that electronic decoherence is possible even for one-dimensional nuclear bath. Furthermore, we show that (i) decreasing the mass of the bath generally leads to faster electronic decoherence; (ii) electron-electron interactions strongly affect the electronic decoherence when the electron-nuclear dynamics is not pure-dephasing; (iii) classical bath models with initial conditions sampled from the Wigner distribution accurately capture the short-time electronic decoherence dynamics; (iv) model separable initial superpositions often used to understand decoherence after photoexcitation are only relevant in experiments that employ delta-like laser pulses to initiate the dynamics. These insights can be employed to interpret and properly model coherence phenomena in molecules.