Partial trace

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... 

The remarkable fact, first demonstrated by Nakatsuji [18], is that for each p >2, CSE(p) is equivalent (in a necessary and sufficient sense) to the original Hilbert-space eigenvalue equation, Eq. (2), provided that CSE(p) is solved subject to boundary conditions (A -representability conditions) appropriate for the (p + 2)-RDM. CSE(p), in other words, is a closed equation for the (p+ 2)-RDM (which determines the (p + 1)- and p-RDMs by partial trace) and has a unique A -representable solution Dp+2 for each electronic state, including excited states. Without A -representability constraints, however, this equation has many spurious solutions [48, 49]. CSE(2) is the most tractable reduced equation that is still equivalent to the original Hilbert-space equation, and ultimately it is CSE(2) that we wish to solve. Importantly, we do not wish to solve CSE(2) for... 

Partial traces of cumulants are also extensive, unlike those of the ROMs themselves. Starting from Eq. (25c), for example, one may show that... 

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... 

In Fig. 4(a) we show a typical diagram in the expansion of A3 that cannot be incorporated into any ladder-type diagram because it involves simultaneous correlation between three particles [69]. As it appears in CSE(2) and ICSE(2), however, A3 is always traced over coordinate X3, and in Fig. 4(b) we show the effect of tT3 on the diagram in Fig. 4(a). Diagram 4(b) is included in the partial trace of a third-order ladder-type diagram, namely, the one shown in Fig. 4(c). Thus the presence of tr3 in the two-particle equations allows one to incorporate three- and higher-body effects that would not otherwise be present in a ladder approximation for the three- and four-electron cumulants. 

GENERALIZED NORMAL ORDERING and partial trace relations... 

Let us now consider a generalized one-particle approximation. We no longer require that 2 = 0, but only that ks = 0. Then we can use the following partial trace relations, which hold for ks = 0 in an NSO basis [25] ... 

In the next section, we will show that the R, R) necessary conditions take an especially simple form. If T satisfies the R, R) conditions, then the Q-matrix obtained from the partial trace of Tg,... 

This relation, however, docs not strictly extend beyond the first member, although it may be partially traced in the relations of malonic and adipic acids to paralaotio and paraleucic... 

In the above expressions we have introduced the (primed) partial trace over the Hilbert space of the susbsystem, Tr pw(R,P) = Pe(R,P), and the symbols Ir and Tre refer to taking the partial trace over the subsystem and... 

Wigner-Weisskopf approximation with the above constraint, after performing the partial trace (117), that is,... 

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 ... 

Figure 1. Active site of bovine super oxide dismutase showing the coordinated amino acids and partial tracing of the peptide chain (19, 20, 21)...

The partial trace in nuclear degrees of freedom in Eq. [13] is replaced in Eq. [18] by the constraint imposed on the collective reaction coordinate X representing the energy gap between the two levels involved in the transition. This reduces the many-body problem of calculating the activation dynamics in the coordinate space q to the dynamics over just one coordinate X. As we show in the discussion of optical transition below, the same Boltzmann factor as in Eq. [18] comes into expressions for optical profiles of CT bands. The solvent component of the FCWD then becomes... 

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 3-RDM is calculated and the partial traces for the spin-adapted blocks are used in the renormalization of its diagonal. 

The cleaning of all the matrices by making equal to zero all quantities which in absolute value are smaller than 10-17 (in double precision) must be rendered more systematic. In this way we expect the accumulation of errors to diminish noticeably. The introduction of the Newton method as well as Ait.ken s extrapolation and other standard techniques to speed up convergency may be suitable in future application. However, before having recourse to these standard convergence techniques, we hope to find other more fundamental basic conditions - such as the spin equation which smoothed the oscillations away — in order to attain a complete control of the process. The most important question which remains open is the way in which the renormalizations of the p-RDM s is performed. Another possible improvement is to extend the spin-adaptation to the renormalization of the 4-RDM in order to make a thorough use of the partial traces of the different symmetries. 

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,... 

Figure 4. A partial trace including descriptive comments for the functionality "nitro from the interpretation of the spectral data of nitrotoluene.

Hjelmar, O. (1999) Determination of total or partial trace element in soil and inorganic waste material, NT Technical report 446, 44p. 

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. 

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 ... 

Like any other entropy, S p) is a measurement of the knowledge we have about the system. The base of the logarithm on Equation (3.7.3) is 2, and kj are the eigenvalues of p. Since the state of Equation (3.7.2) is pure, S p) = 0. However, if we calculate the entropy of each qubit using the partial trace concept, one finds that ... 

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