Von Neumann densities

Reduced density operators were first introduced by K. Husimi (Proc. Phys. Math. Soc. Japan 22 [1940], 264) to describe subsets of the IV-electron distribution (first-order for one-particle distributions, second-order for pair distributions, etc.) and are obtained from the full Mh-order (von Neumann) density operator electronic coordinates see, e.g., E. R. Davidson, Reduced Density Matrices in Quantum Chemistry (New York, Academic Press, 1976) and note 31. 

A Hermitian operator p is a von Neumann density if it is nonnegative and has unit trace. In more concrete terms, if is the finite-dimensional Fock space for a quantum model where electrons are distributed over a finite number of states, then p is a von Neumann density if (i) v,pv) > 0 for all operators v on and (ii) l)g = 1. By the formula v,pv) we mean the trace scalar product of the operators v and pv, that is, v,pv) = traceg(u pu) since (p, l)g = tracegp = 1 we have used this scalar product to express the trace condition. More generally. 

By relaxing the condition that a von Neumann density be positive semidefinite, a graded family of approximations can be constructed. Since an operator can be represented as a polynomial in the annihilation and creation operators, it can be... 

The fe-densities approximate von Neumann densities to A th order they are k-positive with unit trace. We denote the cone of all fe-positive operators by and the convex set of all fe-densities by g. The -densities satisfy the relations gjg C C C C and a S= 0-... 

Rl. The k-matrix for the von Neumann density (X/dim1 is /dim V ) I Using the properties of the monomial basis for the entries for the matrix representation are given by... 

Since the convex set of -densities contains the convex set of von Neumann densities, the following inequality holds ... 

As p varies over the set of von Neumann densities vector of matrix... 

Altematively, in the case of incoherent (e.g. statistical) initial conditions, the density matrix operator P(t) I 1>(0) (v(01 at time t can be obtained as the solution of the Liouville-von Neumann equation ... 

Note that, since the von Neumann equation for the evolution of the density matrix, 8 j8t = — ih H, / ], differs from the equation for a only by a sign, similar equations can be written out for p in the basis of the Pauli matrices, p = a Px + (tyPy -t- a p -t- il- In the incoherent regime this leads to the master equation [Zwanzig 1964 Blum 1981]. For this reason the following analysis can be easily reformulated in terms of the density matrix. 

Fig. 3.61 Some snapshots of the evolution of a 5-neighbor von Neumann neighborhood percolating voting rule V3 the initial densities are (a) p = 0.35 < pc and (b) p = 0.50 = pc-...

Table 3.7 list,s the critical density and type of process for several von Neumann and Moore neighborhood rules. In the first and fourth columns, the rules are defined by the fractions (m/n), which specify a threshold of rn cr = 1 sites out of a total of n possible votes. The table entries for are taken from published results [vich84j using the CAM-6 hardware simulator [marg87] whether some or all of these values can be determined analytically remains an open problem. 

Table 3.7 Critical densities pc and types of processes for a ft w selectefi voting rules defined on von Neumann and Moore neighboriioods.

Fig. 5.6. Some snapshots of the evolution of the 5-neighbor (von Neumann neighborhood) majority rule (p2d majority with threshold b = 2, The initial state is random with density po = 0.075.

The density operator r(<) is a Hermitian and positive function of time, and satisfies the generalized Liouville-von Neumann (LvN) equation(47, 45)... 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

The main problem is to calculate (/ (q, H-r)/(q, t- -r)) of Eq. (2). To achieve this goal, one first considers E(r,f) as a well-defined, deterministic quantity. Its effect on the system may then be determined by treating the von Neumann equation for the density matrix p(f) by perturbation theory the laser perturbation is supposed to be sufficiently small to permit a perturbation expansion. Once p(i) has been calculated, the quantity... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

By analogy the propagation of a density matrix, which corresponds to the solution of the Liouville-von Neumann equation 231... 

In 1927 Landau [I] and von Neumann [2] introduced the density matrix into quantum mechanics. The density matrix for the A-electron ground-state... 

For a pure bipartite state, it is possible to show that the von Neumann entropy of its reduced density matrix, S p ) = —Tr(pjg log2 Pred)> above... 

This result for entanglement is equivalent to the von Neumann entropy of the reduced density matrix p. For our model system of the form AB in the ground state the reduced density matrix pj = TrB(pAB) he basis set (t, 1) is given by... 

As we mentioned before, when a biparticle quantum system AB is in a pure state, there is essentially a unique measure of the entanglement between the subsystems A and B given by the von Neumann entropy S = —Tr[p log2 PaI- This approach gives exactly the same formula as the one given in Eq. (26). This is not surprising since all entanglement measures should coincide on pure bipartite states and be equal to the von Neumann entropy of the reduced density matrix (uniqueness theorem). 

Let us start the statistical theory with the von Neumann equation for the density matrix... 

The evolution of the density operator is described by the Liouville-von Neumann equation... 

It is convenient to start with an isolated system which includes the object and medium, with hamiltonian H. Its state is given by the density operator f (t) which satisfies the Liouville-von Neumann (L-vN) equation of motion... 

The powerful mathematical tools of linear algebra and superoperators in Li-ouville space can be used to proceed from the identification of molecular phenomena, to modelling and calculation of physical properties to interpret or predict experimental results. The present overview of our work shows a possible approach to the dissipative dynamics of a many-atom system undergoing localized electronic transitions. The density operator and its Liouville-von Neumann equation play a central role in its mathematical treatments. 

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