Partial trace operators

Before deriving equations that determine the RDMCs, we ought to clarify precisely which are the RDMCs of interest. It is clear, from Eqs. (25a) and (25b), that Ai and A2 contain the same information as D2 and can therefore be used to calculate expectation values (IT), where W is any symmetric two-electron operator of the form given in Eq. (1). Whereas the 2-RDM contains all of the information available from the 1-RDM, and affords the value of (IT) with no additional information, the 2-RDMC in general does not determine the 1-RDM [43, 65], so both Ai and A2 must be determined independently in order to calculate (IT). More generally, Ai,...,A are all independent quantities, whereas the RDMs Dj,..., D are related by the partial trace operation. The u-RDM determines all of the lower-order RDMs and lower-order RDMCs, but... 

When dealing with composite systems, the density operator of the subsystems can be obtained through the partial trace operation over the density operator of the whole system. The partial trace operation is a sum over all the possible states of one subsystem. For instance, if is the density operator of a composite system ab), the density operator of each subsystem is given by ... 

Using the partial trace operation, which sums over all the state of the others qubits of the system, one may And the respective density matrices for each qubit in the system-... 

The next step is to introduce temperature by averaging out the bath operators appearing in the time dependent terms of Eq.(51) [137] over an adequate ensemble. To this end, the partial trace (or sum of the diagonal elements) over the surrounding subsystem has to be taken. For the system in interaction, the effective Hamiltonian of the solvent Hmeff must be defined in such a way that the sum of HSeff+Hmeff leads to the... 

Here, P is the Dyson time-ordering operator [57], Q (t)IP is the coordinate in the interaction picture with respect to the thermal bath and to the diagonal part of the Hamiltonian of the H-bond bridge, and the notation (( )e)siow has the meaning of a partial trace on the thermal bath and on the H-bond bridge coordinates. 

Of course, the reduced-density operator of the driven damped quantum harmonic oscillator at time t is the partial trace over the thermal bath of the full density operator ... 

Next, the reduced density operator at time t may be obtained by performing the partial trace tre over the thermal bath ... 

As a matter of fact, it must be described by a density matrix p which is the partial trace over the environmental degrees of freedom of the total density matrix ptot of the closed compound system quantum computer + environment p = Tren ) [ptot] p generally evolves non-unitarily according to the operator-sum representation, the matrix [/ obtained after the interaction of the computer with its environment can be written under the form... 

We use (10) to define the time evolution in the subsystem 1 by defining the linear evolution operator using the partial trace... 

In the spin-representation the two, three and four electron functions of the basis are simple products of fermion operators. Therefore, the upper limit of their occupation number is one. This upper bound value has also been adopted for the elements in a spin-adapted basis of representation. We know also that the diagonal elements must be positive. Finally, we know the value not only of the trace but also of the partial traces of the spin-adapted matrices (18, 19, 20). 

The subscript S signifies that this is a matrix element in the system space only, that is, we perform a partial trace over the system degrees of freedom, and GvX(t) is still a full Liouville space operator in the bath degrees of freedom. The angular brackets < > denote averaging over the bath degrees of freedom, that is,... 

The density operator pit) has been formulated for the entire quantum-mechanical system. For magnetic resonance applications, it is usually sufficient to calculate expectation values of a restricted set of operators which act exclusively on nuclear variables. The remaining degrees of freedom are referred to as lattice . The reduced spin density operator is defined by ait) = Tri p(f), where Tri denotes a partial trace over the lattice variables. The reduced density operator can be represented as a vector in a Liouville space of dimension... 

As the foundation of quantum statistical mechanics, the theory of open quantum systems has remained an active topic of research since about the middle of the last century [1-40]. Its development has involved scientists working in fields as diversified as nuclear magnetic resonance, quantum optics and nonlinear spectroscopy, solid-state physics, material science, chemical physics, biophysics, and quantum information. The key quantity in quantum dissipation theory (QDT) is the reduced system density operator, defined formally as the partial trace of the total composite density operator over the stochastic surroundings (bath) degrees of freedom. 

The two T squares in the right-hand side of Eq. 26 both yield an A, which corresponds to the presence of two A-quantities in Eq. 23, but the symmetrized square of / can yield only one totally symmetric Ai, which will correspond to the trace of the sextuplet. The other A-quantity thus must be an A2. This operator will be equal to the difference of the trace-operators for Ti and T2, which is precisely the operator that splits I into Ti and T2. The absence of such a mode in the distortion space leads us to the conclusion that, in this 20-graph, there is no distortion that can produce a neat separation of the two conjugate triplets. The only distortion that will partially distinguish both triplets is a distortion to D5 symmetry, using a component of the H representation, with the following subduction relation ... 

In this field, the density operator plays an important and uncontested role. It allows for more than just the above-given series expansion it can be used for consistent approximations by integrating out (read take the partial trace) of a series of states we are not particularly interested in, leading to the so-called reduced density matrix. It can also be used to find representations in spaces, for instance, the Wigner representation [35], that give more insight into the quantum distribution functions, and provide in some cases distribution functions that are more close to the classical. 

N2, and traces of PH, CO2, E, and S large furnaces generate off-gas at a rate of about 120—180 m /min. In most installations the off-gas is passed through a series of Cottrell electrostatic precipitators which remove 80—95% of the dust particles. The precipitators ate operated at temperatures above the 180°C dew point of the phosphoms. The collected dust is either handled as a water slurry or treated dry. Einal disposal is to a landfill or the dust is partially recycled back to the process. The phosphoms is typically condensed in closed spray towers that maintain spray water temperatures between 20 and 60°C. The condensed product along with the accompanying spray water is processed in sumps where the water is separated and recycled to the spray condenser, and the phosphoms and impurities ate settled for subsequent purification. 

In the last equality here we have introduced the partial Wigner transforms of the density matrix and operator. The prime on the trace indicates a trace over the subsystem degrees of freedom. All information on the quantum initial distribution is contained in pw R,P, 0). In the evaluation of this expression we assume that the time evolution of Bw R, P,t) is given by Eq. (4). This... 

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