Pure-state density matrices

Let the eigenvalue w be fixed and assume that fit is nondegenerate and unit-normalized. The restriction to nondegenerate eigenstates will be relaxed in Section V, but for now we consider only pure-state density matrices. The A -electron density matrix for the pure state fit is... 

Lowdin, who contributed in no small measure to the development of formal many-electron theory through his seminal work on electron correlation, reduced density matrices, perturbation theory, etc. many times expressed his concerns about the theoretical aspects of density functional approaches. This short review of the interconnected features of formal many-electron theory in terms of propagators, reduced pure state density matrices, and density functionals is dedicated to the memory of Per-Olov Lowdin. 

To describe these systems, the physical extension of the ground energy level where AC is the number of particles with AC e R, as weU as their states as a function of a continuous number of particles is needed. The most general description of the state of a quantum system is the density matrix D [26, 33]. It describes the state of an isolated system as a non-coherent convex sum of the complete set of all accessible Af-electron pure state density matrices [26, 33, 34]... 

An operational statement of the difference between a pure state and a statistical mixture can be made with respect to their diagonal representations. In such representation the pure state density matrix will have only one nonzero element on its diagonal, that will obviously take the value 1. A diagonal density matrix representing a statistical mixture must have at least two elements (whose sum is 1) on its diagonal. 

The first relation follows from the fact that if the density n is a pure state v-representable density then the minimizing density matrix for FL is a pure state density matrix. The second relation also easily follows. We take n to be an E-V-density which is not a PS-V-density. There is a ground state ensemble density matrix D[n for which we have... 

Conventional NMR deals with a large ensemble of spins. It means that the state of the system is in a statistical mixture, which is obviously inadequate for QIP. However, the NMR ability for manipulating spins states worked out by Cory et al. [24] and Chuang et al. [23] resulted in elegant methods for creating the so called effectively pure or pseudo-pure states. Behind the idea of the pseudo-pure states is the fact that NMR experiments are only sensitive to the traceless deviation density matrix. Thus, we might search for transformations that, applied to the thermal equilibrium density matrix, produce a deviation density matrix with the same form as a pure state density matrix. Once such state is created, all remaining unitary transformations will act only on such a deviation density matrix, which will transform as a true pure state. 

As an example of a reconstructed number state density matrix, we show in Fig. S our result for a coho-oit superpoation of n=0) and n=2) number states. This state is ideally suited to demonstrate the sensitivity of the reconstruction to coherences. Our result indicates that the prepared motional states in our system are very close to pure states. 

Tr(p ). For an initially thennal state the radius < 1, while for a pure state = 1. The object of cooling is to manipulate the density matrix onto spheres of increasingly larger radius. 

The density matrix p describes the pure state, as seen from the equality p = p, while p does not. The transition from (2.35a) to (2.35b) describes a strong collision , which fully localizes the particle, but in general the off-diagonal elements may not completely vanish. This however does not affect the qualitative picture. 

We may now construct the density matrix for the polarization of a one-photon state. If we choose for our basic states the states of right and left circular polarization then for an arbitrary pure state... 

Since r3 is diagonal, r3 = 1 corresponds to pure right and left polarization, respectively, and r3 = 0 to plane polarization. If we do not wish to consider just pure states, but also wish to include in our discussion partially polarized states, the density matrix for the polarization for such a mixture is still given by... 

The quantum-mechanical equivalent of phase density is known as the density matrix or density operator. It is best understood in the case of a mixed ensemble whose systems are not all in the same quantum state, as for a pure ensemble. 

Hamiltonians involving more than two electron interactions. I shall use this to illustrate the general case of arbitrary p. The second-order reduced density matrix (2-RDM) of a pure state ij/, a function of four particles, is defined as follows ... 

These definitions are easily generalized from a pure state, described by to ensemble states, described by a system density matrix V, for which an expectation value is... 

This is formulated here for a pure state, but the generalization to an ensemble state is straightforward.) Also the one-hole density matrix must be nonnegative [21] ... 

The n-particle density matrix of an w-particle state is pure-state n-representable if—for unit trace—it is idempotent. Since we normalize y as... 

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... 

Given a density matrix p of a pair of quantum systems A and B and all possible pure-state decompositions of p... 

In this section we review the known theorems that relate entanglement to the ranks of density matrices [52]. The rank of a matrix p, denoted as rank(p), is the maximal number of linearly independent row vectors (also column vectors) in the matrix p. Based on the ranks of reduced density matrices, one can derive necessary conditions for the separability of multiparticle arbitrary-dimensional mixed states, which are equivalent to sufficient conditions for entanglement [53]. For convenience, let us introduce the following definitions [54—56]. A pure state p of N particles Ai, A2,..., is called entangled when it cannot be written... 

Lemma 1 A state is pure if and only if the rank of its density matrix p is equal to 1, that is, rank(p) = 1. 

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). 

Exercise. Show that the criterion for a density matrix to be a pure state is... 

The procedure was repeated for the density matrix matrix of unpolarized excited states. In Fig. 3 we display = O p (f) / ,0 = 0) 2 for j = 134, derived from the data of Fig. 1, at different times. As in the pure case of Figs. 2, our procedure is able to prefectly reconstruct the true density-matrix at all times considered. 

It would be very useful if there were an equation of motion for the electron density or for the reduced density matrices corresponding to a pure state of a many-electron system from which these quantities could directly be determined. Unfortunately this is not the case. The quantity closest to the one-matrix that has an equation of motion from which it can be determined by well-defined approximations is the one-electron Green s function or electron propagator. We explore its connection to the electron density in the next section. 

At T = 0 the hindered rotation is a coherent tunneling process like that studied in Section 2.3 for the double well. If, for instance, the system is initially prepared in pure state localized in one of the wells, then the density matrix in the coordinate representation is given by... 

An isolated microscopic system is fully determined, in the quantum mechanical sense, when its state function y/ is known. In the Dirac formalism of quantum mechanics, the state function can be identified with a vector of state, 11//). (11) The system in the 1y/ state may equivalently be described by a Hermitian operator, the so-called density matrix p of a pure state,... 

F.6), there appears the possibility to consider the latter to be the reduction of a many-body fermionic pure state to an N-representable two-matrix. Since the density matrix above, if adapted appropriately, consequently is essentially N-representable through its relation to Coleman s extreme case [107], one might, via appropriate projections, completely recover the proper information, cf. corresponding partitioning procedures depicted in Appendix A. The structure described here is also of fundamental importance in connection with the phenomena of superconductivity and superfluidity through its intimate connections with Yang s concept of ODLRO [106], see more under Section 3.2. 

Thus, we see that in order to obtain the mean field equations of motion, the density matrix of the entire system is assumed to factor into a product of subsystem and environmental contributions with neglect of correlations. The quantum dynamics then evolves as a pure state wave function depending on the coordinates evolving in the mean field generated by the quantum density. As we have seen in the previous sections, these approximations are not valid and no simple representation of the quantum-classical dynamics is possible in terms of single effective trajectories. Consequently, in contrast to claims made in the literature [54], quantum-classical Liouville dynamics is not equivalent to mean field dynamics. 

Show that for a pure state the off-diagonal density matrix elements in... 

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