Reduced Density Matrix Elements Research Articles

Beyond a Richardson-Gaudin Mean-Field: Slater-Condon Rules and Perturbation Theory.

Richardson-Gaudin states provide a basis of the Hilbert space for strongly correlated electrons. In this study, optimal expressions for the transition density matrix elements between Richardson-Gaudin states are obtained with a cost comparable with the corresponding reduced density matrix elements. Analogues of the Slater-Condon rules are identified based on the number of near-zero singular values of the RG state overlap matrix. Finally, a perturbative approach is shown to be close in quality to a configuration interaction of Richardson-Gaudin states while being feasible to compute.

Entanglement entropy of Compton scattering with a witness

Unitarity and the optical theorem are used to derive the reduced density matrices of Compton scattering in the presence of a witness particle. Two photons are initially entangled wherein one photon participates in Compton scattering, while the other is a witness, i.e., does not interact with the electron. Unitarity is shown to require that the entanglement entropy of the witness photon does not change after its entangled partner undergoes scattering. The final mutual information of the electron's and witness particle’s polarizations is shown to be nonzero for low-energy Compton scattering. This indicates that the two particles became correlated in spite of no direct interaction. Assuming an initial maximally entangled state, the change in entanglement entropy of the scattered photon’s polarization is calculated in terms of Stokes parameters. A common ratio of areas occurs in the final reduced density matrix elements, von Neumann entropies, Stokes parameter, and mutual information. This common ratio consists of the Thomson scattering cross-section and an accessible regularized scattering area.

Tensor-Train Thermo-Field Memory Kernels for Generalized Quantum Master Equations.

The generalized quantum master equation (GQME) approach provides a rigorous framework for deriving the exact equation of motion for any subset of electronic reduced density matrix elements (e.g., the diagonal elements). In the context of electronic dynamics, the memory kernel and inhomogeneous term of the GQME introduce the implicit coupling to nuclear motion and dynamics of electronic density matrix elements that are projected out (e.g., the off-diagonal elements), allowing for efficient quantum dynamics simulations. Here, we focus on benchmark quantum simulations of electronic dynamics in a spin-boson model system described by various types of GQMEs. Exact memory kernels and inhomogeneous terms are obtained from short-time quantum-mechanically exact tensor-train thermo-field dynamics (TT-TFD) simulations and are compared with those obtained from an approximate linearized semiclassical method, allowing for assessment of the accuracy of these approximate memory kernels and inhomogeneous terms. Moreover, we have analyzed the computational cost of the full and reduced-dimensionality GQMEs. The scaling of the computational cost is dependent on several factors, sometimes with opposite scaling trends. The TT-TFD memory kernels can provide insights on the main sources of inaccuracies of GQME approaches when combined with approximate input methods and pave the road for the development of quantum circuits that implement GQMEs on digital quantum computers.

New method for directly computing reduced density matrices

We demonstrate the power of a first principle-based and practicable method that allows for the perturbative computation of reduced density matrix elements of an open quantum system without making use of any master equations. The approach is based on techniques from non-equilibrium quantum field theory like thermo field dynamics, the Schwinger-Keldsyh formalism, and the Feynman-Vernon influence functional. It does not require the Markov approximation and is essentially a Lehmann-Szymanzik-Zimmermann-like reduction. In order to illustrate this method, we consider a real scalar field as an open quantum system interacting with an environment comprising another real scalar field. We give a general formula that allows for the perturbative computation of density matrix elements for any number of particles in a momentum basis. Finally, we consider a simple toy model and use this formula to obtain expressions for some of the system's reduced density matrix elements.

Influence of the N-representability conditions on the variational determination of two-electron reduced density matrices for ground and excited N-electron states in the doubly occupied configuration interaction space.

In this work, we analyze the effectiveness of different sets of well-known necessary N-representability conditions that are imposed on the variational determination of two-electron reduced density matrices corresponding to the ground and excited states of N-electron Hamiltonians within doubly occupied configuration interaction subspaces. The two-electron reduced density matrix elements are determined so that they are suitably related to those of higher-order reduced density matrices and other associated matrices, up to a given order, constituting positive semidefinite matrices. These conditions up to two-, three-, and four-order have been sequentially imposed on the variational calculation of the expectation values of the dispersion operator corresponding to several Hamiltonian models, obtaining the electronic spectra. These results have been contrasted with those arising from the full configuration interaction method, illuminating the sufficiency/insufficiency of the variational N-representability conditions applied.

Simulating the dynamics of electronic observables via reduced-dimensionality generalized quantum master equations.

We describe a general-purpose framework for formulating the dynamics of any subset of electronic reduced density matrix elements in terms of a formally exact generalized quantum master equation (GQME). Within this framework, the effect of coupling to the nuclear degrees of freedom, as well as to any projected-out electronic reduced density matrix elements, is captured by a memory kernel and an inhomogeneous term, whose dimensionalities are dictated by the number of electronic reduced density matrix elements included in the subset of interest. We show that the memory kernel and inhomogeneous term within such GQMEs can be calculated from projection-free inputs of the same dimensionality, which can be cast in terms of the corresponding subsets of overall system two-time correlation functions. The applicability and feasibility of such reduced-dimensionality GQMEs is demonstrated on the two-state spin-boson benchmark model. To this end, we compare and contrast the following four types of GQMEs: (1) a full density matrix GQME, (2) a single-population scalar GQME, (3) a populations-only GQME, and (4) a subset GQME for any combination of populations and coherences. Using a method based on the mapping Hamiltonian approach and linearized semiclassical approximation to calculate the projection-free inputs, we find that while single-population GQMEs and subset GQMEs containing only one population are less accurate, they can still produce reasonable results and that the accuracy of the results obtained via the populations-only GQME and a subset GQME containing both populations is comparable to that obtained via the full density matrix GQMEs.

Exciton transfer in organic photovoltaic cells: A role of local and nonlocal electron-phonon interactions in a donor domain.

We theoretically investigate an exciton transfer process in a donor domain of organic photovoltaic cells focusing on the roles of local and nonlocal electron-phonon interactions. Our model consists of a three-level system described by the Holstein-Peierls Hamiltonian coupled to multiple heat baths for local and nonlocal molecular modes characterized by Brownian spectral distribution functions. We chose tetracene as a reference donor molecule, where the spectral distribution functions of the local and nonlocal modes are available. We then employ the reduced hierarchical equations of motion approach to simulate the dynamics of the system under the influence of the environment as a function of the electron-phonon coupling strength and temperature. We rigorously calculate the reduced density matrix elements to explain the time scale of dynamics under the influence of the dissipative local and nonlocal modes. The results indicate that the strong nonlocal electron-phonon interaction under high temperature conditions favors the exciton transfer process and enhances the efficiency of organic photovoltaic materials, while the lifetime of the exciton becomes shorter due to a low-frequency local mode.

Incorporating dynamic correlation into the variational determination method of the second‐order reduced density matrix in the doubly occupied configuration interaction space

AbstractThe variational determination of the second‐order reduced density matrices arising from N‐electron doubly occupied configuration interaction wave functions has recently been proposed as a method to describe systems possessing static correlation in their ground states. In this work, we propose to combine this approach with the on‐top pair density functional theory in order to incorporate dynamic correlations and improve the description of systems possessing both types of electron correlation. The procedure requires ensuring that the second‐order reduced density matrix elements satisfy determined N‐representability variational conditions, as well as other constraints arising from the doubly‐occupied configuration interaction wave function features. An analysis of the results obtained through this method, in studies of strongly correlated many‐electron systems, shows the usefulness of our proposals.

Linking phase transitions and quantum entanglement at arbitrary temperature

In this work, we establish a general theory of phase transitions and quantum entanglement in the equilibrium state at arbitrary temperatures. First, we derived a set of universal functional relations between the matrix elements of two-body reduced density matrix of the canonical density matrix and the Helmholtz free energy of the equilibrium state, which implies that the Helmholtz free energy and its derivatives are directly related to entanglement measures because any entanglement measures are defined as a function of the reduced density matrix. Then we show that the first order phase transitions are signaled by the matrix elements of reduced density matrix while the second order phase transitions are witnessed by the first derivatives of the reduced density matrix elements. Near second order phase transition point, we show that the first derivative of the reduced density matrix elements present universal scaling behaviors. Finally we establish a theorem which connects the phase transitions and entanglement at arbitrary temperatures. Our general results are demonstrated in an experimentally relevant many-body spin model.

Buildup of Fano resonances in the time domain in a double quantum dot Aharonov-Bohm interferometer

In this paper, we investigate the transient quantum transport in a nanoscale Aharonov-Bohm interferometer consisting of laterally coupled double quantum dots (DQDs) coupled to the source and drain electrodes. The tran- sient linear conductance is derived at arbitrary temperature of the leads. The resulting transient conductance can be divided into three terms contributed from different transport channels. From the transient linear conductance and the reduced density matrix elements of the DQD system, we show the buildup of Fano resonances in the time domain.

Diagonal N-representability as a method for solving the representability problem

The latest results of density matrix theory are summarized in terms of analysis of off-diagonal components. For a given reduced density matrix (RDM) of arbitrary order, the full density matrix is explicitly constructed, from which the RDM is derived by means of the corresponding reduction procedure. Both the matrices satisfy all the necessary conditions, such antisymmetry, normalization, hermiticity, except that the non-negative definiteness of the full DM is not guaranteed. If an RDM satisfies the necessary inequalities with the same number of indices as it has itself, then, as proved here, there is an additive correction to the full DM matrix that turns it into a non-negative definite matrix leading to the same RDM and satisfying the other requirements. This establishes a fundamental fact that these inequalities are sufficient as well, providing the N-representability of the given RDM. The construction is based on the classification, introduced in the article, of the multi-index elements of the RDM and full DM according to their degree of off-diagonality. RDM elements with a predetermined degree of off-diagonality are expressed through full DM elements with the same off-diagonality. This shows that the conventional variational approach to calculating the energy of the system by convex programming methods under the conditions of N-representability (before and even after reaching the minimum energy) generally does not lead to constructing a mixed quantum state composed of solutions to the Schrodinger equation. Therefore, except for the energy, it does not guarantee a correct description of the other properties so long as the conditions of correctness for each particular property are not expressed in terms of the RDM and not incorporated into the specific conditions of N-representability (regardless of the requirement of energy minimization).

Nonlinear selective reflection spectroscopy of V‐type atomic system at the gas‐solid interface

We theoretically present the nonlinear selective reflection spectroscopy of V‐type atomic system at gas‐solid interface in a pump‐probe scheme. The saturation and coherence effects are distinguished by solving Liouville equation in the absence and presence of reduced density matrix element between the two excited levels. When the coherence effect exists, two peaks appear in reflection spectroscopy with asymmetry lineshape. We investigate the dependence of reflection spectroscopy on pump field intensity, frequency detuning and coherent decay rate induced by collision between atoms. The lineshape can be explained based on reflection spectroscopy contributed from atoms with negative (before collision) and positive (after collision) velocities, single‐photon and two‐photon processes. This study is helpful for investigating quantum coherence and dynamic processes of atoms at gas‐solid interface.

Coffman-Kundu-Wootters inequality for fermions

We derive an inequality for three fermions with six single particle states which reduces to the sum of the famous Coffman-Kundu-Wootters inequalities when an embedded three qubit system is considered. We identify the quantities which are playing the role of the concurrence, the three-tangle and the invariant $\det \rho_A+\det \rho_B+\det \rho_C$ for this tripartite system. We show that this latter one is almost interchangeable with the von Neumann entropy and conjecture that it measures the entanglement of one fermion with the rest of the system. We prove that the vanishing of the fermionic "concurrence" implies that the two particle reduced density matrix is a mixture of separable states. Also the vanishing of this quantitiy is only possible in the GHZ class, where some genuie tripartite entanglement is present and in the separable class. Based on this we conjecture that this "concurrence" measures the amount of entanglement between pairs of fermions. We identify the well-known "spin-flipped" density matrix in the fermionic context as the reduced density matrix of a special particle-hole dual state. We show that in general this dual state is always canonically defined by the Hermitian inner product of the fermionic Fock space and that it can be used to calculate SLOCC covariants. We show that Fierz identities known from the theory of spinors relate SLOCC covariants with reduced density matrix elements of the state and its "spin-flipped" dual.

Density-matrix quantum Monte Carlo method

We present a quantum Monte Carlo method capable of sampling the full density matrix of a many-particle system at finite temperature. This allows arbitrary reduced density matrix elements and expectation values of complicated non-local observables to be evaluated easily. The method resembles full configuration interaction quantum Monte Carlo but works in the space of many-particle operators instead of the space of many-particle wave functions. One simulation provides the density matrix at all temperatures simultaneously, from $T=\infty$ to $T=0$, allowing the temperature dependence of expectation values to be studied. The direct sampling of the density matrix also allows the calculation of some previously inaccessible entanglement measures. We explain the theory underlying the method, describe the algorithm, and introduce an importance-sampling procedure to improve the stochastic efficiency. To demonstrate the potential of our approach, the energy and staggered magnetization of the isotropic antiferromagnetic Heisenberg model on small lattices, the concurrence of one-dimensional spin rings, and the Renyi $S_2$ entanglement entropy of various sublattices of the $6 \times 6$ Heisenberg model are calculated. The nature of the sign problem in the method is also investigated.

Ac Josephson transport through interacting quantum dots

We investigate the AC Josephson current through a quantum dot with strong Coulomb interaction attached to two superconducting and one normal lead. To this end, we perform a perturbation expansion in the tunneling couplings within a diagrammatic real-time technique. The AC Josephson current is connected to the reduced density matrix elements that describe superconducting correlations induced on the quantum dot via proximity effect. We analyze the dependence of the AC signal on the level position of the quantum dot, the charging energy, and the applied bias voltages.

Partial decoherence in mesoscopic systems

The coupling of a mesoscopic system with its environment usually causes total decoherence: at long times the reduced density matrix of the system evolves in time to a limit that is independent of its initial value, losing all the quantum information stored in its initial state. Under special circumstances, a subspace of the system's Hilbert space remains coherent or ‘decoherence-free’, and the reduced density matrix approaches a non-trivial limit that contains information on its initial quantum state, despite the coupling with the environment. This situation is called ‘partial decoherence’. In this paper, we found the conditions for partial decoherence in a mesoscopic system (with N quantum states) that is coupled to its environment. When the Hamiltonian of the system commutes with the total Hamiltonian, one has ‘adiabatic decoherence’, which yields N − 1 time-independent combinations of the reduced density matrix elements. In the presence of a magnetic flux, one can measure circulating currents around loops in the system even at long times and use them to retrieve information on the initial state. For N = 2, we demonstrate that partial decoherence can happen only under adiabatic decoherence conditions. However, for N > 2 we find partial decoherence even when the Hamiltonian of the system does not commute with the total Hamiltonian, and we obtain the general conditions for such a non-adiabatic partial decoherence. For an electron moving on a ring, with N > 2 single-level quantum dots, non-adiabatic partial decoherence can arise only when the total flux through the ring vanishes (or equals an integer number of flux quanta), and therefore there is no asymptotic circulating current.

Application of resonance perturbation theory to dynamics of magnetization in spin systems interacting with local and collective bosonic reservoirs

We apply our recently developed resonance perturbation theory to describe the dynamics of magnetization in paramagnetic spin systems interacting simultaneously with local and collective bosonic environments. We derive explicit expressions for the evolution of the reduced density matrix elements. This allows us to calculate explicitly the dynamics of the macroscopic magnetization, including characteristic relaxation and dephasing timescales. We demonstrate that collective effects (i) do not influence the character of the relaxation processes but merely renormalize the relaxation times and (ii) significantly modify the dephasing times, leading in some cases to a complicated (time inhomogeneous) dynamics of the transverse magnetization, governed by an effective time-dependent magnetic field.

Evolution of entanglement

We analyze the dynamics of entanglement between two qubits which interact through collective and local environments. Our approach is based on a resonance theory which assumes a small interaction between qubits and environments and which gives rigorous perturbation theory results, valid for all times. We obtain expressions for (i) characteristic time-scales for decoherence, relaxation, disentanglement, and for (ii) the evolution of observables, valid uniformly in time $t\geq 0$. We introduce a classification of decoherence times based on clustering of the reduced density matrix elements, persisting on all time-scales. We examine characteristic dynamical properties such as creation, death and revival of entanglement. We discuss possible applications of our results for superconducting quantum computation and quantum measurement technologies.

Evolution of entanglement of two qubits interacting through local and collective environments

We analyze the dynamics of entanglement between two qubits which interact through collective and local environments. Our approach is based on a resonance theory which assumes a small interaction between qubits and environments and which gives rigorous perturbation theory results, valid for all times. We obtain expressions for (i) characteristic time-scales for decoherence, relaxation, disentanglement, and for (ii) the evolution of observables, valid uniformly in time t ≥ 0. We introduce a classification of decoherence times based on clustering of the reduced density matrix elements, persisting on all time-scales. We examine characteristic dynamical properties such as creation, death and revival of entanglement. We discuss possible applications of our results for superconducting quantum computation and quantum measurement technologies.

Multistate electron transfer dynamics in the condensed phase: Exact calculations from the reduced hierarchy equations of motion approach

Multiple displaced oscillators coupled to an Ohmic heat bath are used to describe electron transfer (ET) in a dissipative environment. By performing a canonical transformation, the model is reduced to a multilevel system coupled to a heat bath with the Brownian spectral distribution. A reduced hierarchy equations of motion approach is introduced for numerically rigorous simulation of the dynamics of the three-level system with various oscillator configurations, for different nonadiabatic coupling strengths and damping rates, and at different temperatures. The time evolution of the reduced density matrix elements illustrates the interplay of coherences between the electronic and vibrational states. The ET reaction rates, defined as a flux-flux correlation function, are calculated using the linear response of the system to an external perturbation as a function of activation energy. The results exhibit an asymmetric inverted parabolic profile in a small activation regime due to the presence of the intermediate state between the reactant and product states and a slowly decaying profile in a large activation energy regime, which arises from the quantum coherent transitions.