Density Matrices Of Systems Research Articles

Exact expressions for ensemble functionals from particle number dependence

Some properties of exact ensemble density functionals can be determined by examining the particle number dependence of ground state ensemble density matrices for systems where the integer ground state energies satisfy a convexity condition. The results include the observation that the integral of the product of the functional derivative and Fukui function of functionals that can be expressed as the trace of an operator is particle number independent for particle numbers between successive integers and the integral itself is equal to the difference between functionals evaluated at successive integer particle numbers. Expressions that must be satisfied by 2nd and higher order functional derivatives are formulated and equations that must be satisfied point by point in space are derived. Using the analytic Hooke's atom model, it is shown that commonly used correlation functional approximations do not bear any resemblance to a spatially dependent expression derived from the exact second order functional derivative of the correlation functional. It is also shown that two expressions for the mutual Coulomb energy are not equal when approximate exchange and correlation functionals are used.

On graphs whose Laplacian matrix’s multipartite separability is invariant under graph isomorphism

Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are responsible for many nonintuitive physical phenomena. The entanglement property of normalized Laplacian matrices is in general not invariant under graph isomorphism. In recent papers, graphs were identified whose entanglement and separability properties are invariant under isomorphism. The purpose of this note is to completely characterize the set of graphs whose separability is invariant under graph isomorphism. In particular, we show that this set consists of K 2 , 2 and its complement, all complete graphs and no other graphs.

Reduced density matrices and entanglement entropy in free lattice models

We review the properties of reduced density matrices for free fermionic or bosonic many-particle systems in their ground state. Their basic feature is that they have a thermal form and thus lead to a quasi-thermodynamic problem with a certain free-particle Hamiltonian. We discuss the derivation of this result, the character of the Hamiltonian and its eigenstates, the single-particle spectra and the full spectra, the resulting entanglement and in particular the entanglement entropy. This is done for various one- and two-dimensional situations, including also the evolution after global or local quenches.

Excitation Energies in Time-Dependent (Current-) Density-Functional Theory: A Simple Perspective.

This paper gives a simple and pedagogical explanation, using density matrices of two-level systems, how to calculate excitation energies with time-dependent density-functional theory. The well-known single-pole approximation for excitation energies is derived here in an alternative way and extended to time-dependent current-density-functional theory.

Orbital-unrelaxed Lagrangian density matrices for periodic systems at the local MP2 level

In the present paper a method based on the Hylleraas functional is proposed in order to obtain correlated ground state density matrices for periodic systems at the level of local MP2. The general properties of these density matrices, namely size-extensivity, translational invariance, exponential decay of the off-diagonal elements, etc are discussed. As test examples we investigate the influence of the electron correlation on the density in diamond and strontium titanate (in the latter case via the Mulliken charges). The calculations reveal that in diamond the concentration of the electrons in the bond region decreases when the correlation is taken into account, but the change in the density relative to Hartree-Fock is small. In the case of SrTiO3, this change is more significant and causes a lowering of the ionicity of this crystal.

Orbit Classification of Qutrit via the Gram Matrix

We classify the orbits generated by unitary transformation on the density matrices of the three-state quantum systems (qutrits) via the Gram matrix. The Gram matrix is a real symmetric matrix formed from the Hilbert–Schmidt scalar products of the vectors lying in the tangent space to the orbits. The rank of the Gram matrix determines the dimensions of the orbits, which fall into three classes for qutrits.

Bloch vectors for qudits

We present three different matrix bases that can be used to decompose density matrices of d-dimensional quantum systems, so-called qudits: the generalized Gell–Mann matrix basis, the polarization operator basis and the Weyl operator basis. Such a decomposition can be identified with a vector—the Bloch vector, i.e. a generalization of the well-known qubit case—and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We present a new method to decompose density matrices via so-called standard matrices, consider the important case of an isotropic two-qudit state and decompose it according to each basis. In the case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin-1 measurements, which is appropriate for an experimental realization.

Bures metric over thermal state manifolds and quantum criticality

We analyze the Bures metric over the manifold of thermal density matrices for systems featuring a zero temperature quantum phase transition. We show that the quantum critical region can be characterized in terms of the temperature scaling behavior of the metric tensor itself. Furthermore, the analysis of the metric tensor when both temperature and an external field are varied, allows one to complement the understanding of the phase diagram including crossover regions which are not characterized by any singular behavior. These results provide a further extension of the scope of the metric approach to quantum criticality.

Closed-form expressions for correlated density matrices: Application to dispersive interactions and example of He2

Empirically correlated density matrices of N-electron systems are investigated. Closed-form expressions are derived for the one- and two-electron reduced density matrices from a pairwise correlated wave function. Approximate expressions are then proposed which reflect dispersive interactions between closed-shell centrosymmetric subsystems. Said expressions clearly illustrate the consequences of second-order correlation effects on the reduced density matrices. Application is made to a simple example: the He(2) system. Reduced density matrices are explicitly calculated, correct to second order in correlation, and compared with approximations of independent electrons and independent electron pairs. The models proposed allow for variational calculations of interaction energies and equilibrium distance as well as a clear interpretation of dispersive effects on electron distributions. Both exchange and second order correlation effects are shown to play a critical role on the quality of the results.

Solving gapped Hamiltonians locally

We show that any short-range Hamiltonian with a gap between the ground and excited states can be written as a sum of local operators, such that the ground state is an approximate eigenvector of each operator separately. We then show that the ground state of any such Hamiltonian is close to a generalized matrix product state. The range of the given operators needed to obtain a good approximation to the ground state is proportional to the square of the logarithm of the system size times a characteristic ``factorization length.'' Applications to many-body quantum simulation are discussed. We also consider density matrices of systems at non zero temperature.

On the asymptotics of some large Hankel determinants generated by Fisher–Hartwig symbols defined on the real line

We investigate the asymptotics of Hankel determinants of the form detj,k=0N−1[∫ΩdxωN(x)∏i=1m∣μi−x∣2qixj+k] as N→∞ with q and μ fixed, where Ω is an infinite subinterval of R and ωN(x) is a positive weight on Ω. Such objects are natural analogs of Toeplitz determinants generated by Fisher–Hartwig symbols, and arise in random matrix theory in the investigation of certain expectations involving random characteristic polynomials. The reduced density matrices of certain one-dimensional systems of trapped impenetrable bosons can also be expressed in terms of Hankel determinants of this form. We focus on the specific cases of scaled Hermite and Laguerre weights. We compute the asymptotics by using a duality formula expressing the N×N Hankel determinant as a 2(q1+⋯+qm)-fold integral, which is valid when each qi is natural. We thus verify, for such q, a recent conjecture of Forrester and Frankel derived using a log-gas argument.

Finite-level systems, Hermitian operators, isometries and a novel parametrization of Stiefel and Grassmann manifolds

In this paper we obtain a description of the Hermitian operators acting on the Hilbert space $\C^n$, description which gives a complete solution to the over parameterization problem. More precisely we provide an explicit parameterization of arbitrary $n$-dimensional operators, operators that may be considered either as Hamiltonians, or density matrices for finite-level quantum systems. It is shown that the spectral multiplicities are encoded in a flag unitary matrix obtained as an ordered product of special unitary matrices, each one generated by a complex $n-k$-dimensional unit vector, $k=0,1,...,n-2$. As a byproduct, an alternative and simple parameterization of Stiefel and Grassmann manifolds is obtained.

Quantum statistical mechanics with Gaussians: Equilibrium properties of van der Waals clusters

The variational Gaussian wave-packet method for computation of equilibrium density matrices of quantum many-body systems is further developed. The density matrix is expressed in terms of Gaussian resolution, in which each Gaussian is propagated independently in imaginary time beta=(k(B)T)(-1) starting at the classical limit beta=0. For an N-particle system a Gaussian exp[(r-q)(T)G(r-q)+gamma] is represented by its center qinR(3N), the width matrix GinR(3Nx3N), and the scale gammainR, all treated as dynamical variables. Evaluation of observables is done by Monte Carlo sampling of the initial Gaussian positions. As demonstrated previously at not-very-low temperatures the method is surprisingly accurate for a range of model systems including the case of double-well potential. Ideally, a single Gaussian propagation requires numerical effort comparable to the propagation of a single classical trajectory for a system with 9(N(2)+N)/2 degrees of freedom. Furthermore, an approximation based on a direct product of single-particle Gaussians, rather than a fully coupled Gaussian, reduces the number of dynamical variables to 9N. The success of the methodology depends on whether various Gaussian integrals needed for calculation of, e.g., the potential matrix elements or pair correlation functions could be evaluated efficiently. We present techniques to accomplish these goals and apply the method to compute the heat capacity and radial pair correlation function of Ne(13) Lennard-Jones cluster. Our results agree very well with the available path-integral Monte Carlo calculations.

Existence of short-time approximations of any polynomial order for the computation of density matrices by path integral methods.

In this paper I provide significant mathematical evidence in support of the existence of direct short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials. While for Theorem 2, which is "experimental," I only provide a "physicist's" proof, I believe the present development is mathematically sound. As a verification, I explicitly construct two short-time approximations to the density matrix having convergence orders 3 and 4, respectively. Furthermore, in Appendix B, I derive the convergence constant for the trapezoidal Trotter path integral technique. The convergence orders and constants are then verified by numerical simulations. While the two short-time approximations constructed are of sure interest to physicists and chemists involved in Monte Carlo path integral simulations, the present paper is also aimed at the mathematical community, who might find the results interesting and worth exploring. I conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real-time Feynman path integral simulations.

Gaussian quantum operator representation for bosons

We introduce the Gaussian quantum operator representation, using the most general multi-mode Gaussian operator basis. The representation unifies and substantially extends existing phase-space representations of density matrices for Bose systems, and also includes generalized squeezed-state and thermal bases. It enables first-principles dynamical or equilibrium calculations in quantum many-body systems, with quantum uncertainties appearing as dynamical objects. Any quadratic Liouville equation for the density operator results in a purely deterministic time evolution. Any cubic or quartic master equation can be treated using stochastic methods.

Separability and correlations in composite states based on entropy methods

This work is an enquiry into the circumstances under which entropy methods can give an answer to the questions of both quantum separability and classical correlations of a composite state. Several entropy functionals are employed to examine the entanglement and correlation properties guided by the corresponding calculations of concurrence. It is shown that the entropy difference between that of the composite and its marginal density matrices may be of arbitrary sign except under special circumstances when conditional probability can be defined appropriately. This ambiguity is a consequence of the fact that the overlap matrix elements of the eigenstates of the composite density matrix with those of its marginal density matrices also play important roles in the definitions of probabilities and the associated entropies, along with their respective eigenvalues. The general results are illustrated using pure and mixed state density matrices of two-qubit systems. Two classes of density matrices are found for which the conditional probability can defined: (1) density matrices with commuting decompositions and (2) those which are decohered in the representation where the density matrices of the marginals are diagonal. The first class of states encompass those whose separability is currently understood as due to particular symmetries of the states. The second are a new class of states which are expected to be useful for understanding separability. Examples of entropy functionals of these decohered states including the crucial isospectral case are discussed.

Distance and entropy for density matrices

Density matrices for systems with a finite number of states are considered as elements in a vector space. A density matrix that is a convex combination of density matrices that are unitary transformations of some initial density matrix will be closer to, or possibly at the same distance from, the density matrix for the ensemble in which all states are equally occupied, compared with the initial density matrix. This distance is correlated, for few-state systems, with a function simply related to the entropy. This function increases (entropy decreases) with distance from the equal-occupancy density matrix. Other properties of the function are also established.

Separable approximations of density matrices of composite quantum systems

We investigate optimal separable approximations (decompositions) of states rho of bipartite quantum systems A and B of arbitrary dimensions MxN following the lines of Ref. [M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261 (1998)]. Such approximations allow to represent in an optimal way any density operator as a sum of a separable state and an entangled state of a certain form. For two qubit systems (M=N=2) the best separable approximation has a form of a mixture of a separable state and a projector onto a pure entangled state. We formulate a necessary condition that the pure state in the best separable approximation is not maximally entangled. We demonstrate that the weight of the entangled state in the best separable approximation in arbitrary dimensions provides a good entanglement measure. We prove in general for arbitrary M and N that the best separable approximation corresponds to a mixture of a separable and an entangled state which are both unique. We develop also a theory of optimal separable approximations for states with positive partial transpose (PPT states). Such approximations allow to decompose any density operator with positive partial transpose as a sum of a separable state and an entangled PPT state. We discuss procedures of constructing such decompositions.

On Polymer Expansions for Equilibrium Systems of Oscillators with Ternary Interaction

For Gibbs lattice systems characterized by a measurable space at sites of a d-dimensional hypercubic lattice and potential energy with pair complex potential, we formulate conditions that guarantee the convergence of polymer (cluster) expansions. We establish that the Gibbs correlation functions and reduced density matrices of classical and quantum systems of linear oscillators with ternary interaction can be expressed in terms of correlation functions of these systems.

Direct determination of second-order density matrix using density equation: Open-shell system and excited state

We formulated the density equation theory (DET) using the spin-dependent density matrix (SDM) as a basic variable and calculated the density matrices of the open-shell systems and excited states, as well as those of the closed-shell systems, without any use of the wave function. We calculated the open-shell systems, Be(3S), Be−(2S), B+(3S), B(2S), C2+(3S), C+(2S), N3+(3S), and N2+(2S), and the closed-shell systems, Be, Be2−, B+, B−, C2+, N3+, H2O, and HF. The new properties calculated are the transition energies and the spin densities at the nuclei. Generally speaking, the accuracy of the present results is slightly worse than that of the previous one using the spin-independent density matrix.