Density matrix transformations

Note that the density matrix transforms contragrediently to operator matrices. 

In the previous chapter we saw how the matrix elements of a NMR density matrix relate to spectral lines. In the context of NMR QIP, an algorithm is nothing but a radiofrequency pulse sequence which encodes quantum logic gates. Each radiofrequency pulse implements an unitary transformation, which is used to prepare the initial state, and process the information and the computation. Under a sequence of unitary operators U (ti), U(t2), U(t ), the initial equilibrium density matrix transforms according to ... 

Once the density matrix has been transformed to the NAO basis, bonds between atoms may be identified from the off-diagonal blocks. The procedure involves the following steps. 

We now regard Eq. (8-233) as analogous to Schrodinger s equation, and proceed to carry out the transformation to the interaction representation described in Chapter 7, Section 7.7. We define the transformed density matrix R and the transformed potential U by... 

Here it is taken into account that density matrix p, being a scalar, commutates with any rotation operator, and diq defined in Eq. (7.51) is used. After an analogous transformation, in master equation (7.51) there remains the Hamiltonian, which does not depend on e ... 

A transformation of the number density matrix N under a space group operation means that both variables are transformed ... 

We notice that neither the momentum distribution nor the reciprocal form factor seems to carry any information about the translational part of the space group. The non diagonal elements of the number density matrix in momentum space, on the other hand, transform under the elements of the space group in a way which brings in the translational parts explicitly. 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

In order to fulfill compatibility condition (a), the local coordinate system of each parent molecule Mk can always be reoriented, resulting in a simple similarity transformation of the original fragment density matrix P (qS(Kk)) into a compatible fragment density matrix P (cp (K)),... 

The macromolecular density matrix built from such displaced local fragment density matrices does not necessarily fulfill the idempotency condition that is one condition involved in charge conservation. It is possible, however, to ensure idempotency for a macromolecular density matrix subject to small deformations of the nuclear arrangements by a relatively simple algorithm, based on the Lowdin transform-inverse Lowdin transform technique. 

Integration of the phase density over classical phase space corresponds to finding the trace of the density matrix in quantum mechanics. Transition to a new basis is achieved by unitary transformation... 

In the previous sections we have considered the transformation of observables. In this section we consider the transformation of the density matrix. The transformed Liouville equation for p(t) = Ap(t) is... 

For the weak coupling case with Eq. (32), our master equation reduces to the well-known quantum master equation, obtained through the approximation, widely used in quantum optics. This equation describes, among other things, quantum decoherence due to Brownian motion. Hence, we have derived an exact quantum master equation for the transformed density operator p that describes exact decoherence. Furthermore, our master equation cannot keep the purity of the transformed density matrix. Indeed, one can show that if p(t) is factorized into a product of transformed wave functions at t = 0, it will not be factorized into their product for t > 0. This is consistent the nondistributivity of the nonunitary transformation (18). 

M. D. Benayoun and A. Y. Lu, Invariance of the cumulant expansion under l-particle unitary transformations in reduced density matrix theory. Chem. Phys. Lett. 387, 485 (2004). 

T. Yanai and G. K. L. Chan, Canonical transformation theory for dynamic correlations in multireference problems, in Reduced-Density-Matrix Mechanics With Application to Many-Electron Atoms and Molecules, A Special Volume of Advances in Chemical Physics, Volume 134 (D.A. Mazziotti, ed.), Wiley, Hoboken, NJ, 2007. 

Because polynomials of orbital occupation number operators do not depend on the off-diagonal elements of the reduced density matrix, it is important to generalize the Q, R) constraints to off-diagonal elements. This can be done in two ways. First, because the Q, R) conditions must hold in any orbital basis, unitary transformations of the orbital basis set can be used to constrain the off-diagonal elements of the density matrix. (See Eq. (56) and the surrounding discussion.) Second, one can replace the number operators by creation and annihilation operators on different orbitals according to the rule... 

Then, instead of performing the six-dimensional integral in Eq. (5.19) all at once, we perform successive three-dimensional integrals over s and R. The first step takes us to W R,P), the Wigner representation [130,131] of the density matrix, and the second step to the p-space density matrix, n(P — p/2 P + p/2). The reverse transformation of Eq. (5.20) can also be performed stepwise over P and p to obtain A( , p), the Moyal mixed representation [132], and then the r-space representation V R— s/2 R + s/2). These steps are shown schematically in Figure 5.2. 

Figure 5.2. The relationships among the r-space density matrix F, the p-space density matrix n, the Wigner representation W, and the Moyal representation A. Two-headed arrows with a T beside them signify reversible, three-dimensional, Fourier transformations.

To this end, we resort to a novel general approach to the control of arbitrary multidimensional quantum operations in open systems described by the reduced density matrix p(t) if the desired operation is disturbed by linear couplings to a bath, via operators S B (where S is the traceless system operator and B is the bath operator), one can choose controls to maximize the operation fidelity according to the following recipe, which holds to second order in the system-bath coupling (i) The control (modulation) transforms the system-bath coupling operators to the time-dependent form S t) (S) B(t) in the interaction picture, via the rotation matrix e,(t) a set of time-dependent coefficients in the operator basis, (Pauli matrices in the case of a qubit), such that ... 

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