Pure states, quantum

Theorem Pure-state quantum algorithms may be efficiently simulated classically, provided there is a bounded amount of global entanglement. 

For the given values of numbers of particles and parameters, and for values of energy greater than the ground-state energy Eg, the boundary EgE at S - 0 corresponds to all the pure states of the system, namely, to all states that can be described quantum mechanically by wave functions or Idempotent matrices. Thus, pure-state quantum mechanics is zero-entropy physics. 

The form of the action principle given above was first applied to quantum mechanics to describe the time evolution of pure states (i.e. 

The applications of NN to solvent extraction, reported in section 16.4.6.2., suffer from an essential limitation in that they do not apply to processes of quantum nature therefore they are not able to describe metal complexes in extraction systems on the microscopic level. In fact, the networks can describe only the pure state of simplest quantum systems, without superposition of states. Neural networks that indirectly take into account quantum effects have already been applied to chemical problems. For example, the combination of quantum mechanical molecular electrostatic potential surfaces with neural networks makes it possible to predict the bonding energy for bioactive molecules with enzyme targets. Computational NN were employed to identify the quantum mechanical features of the... 

In some applications, such mixed-state systems may be of interest, for which their interacting and equivalent non-interacting pure-state systems are described by the same sets of quantum numbers, A = /I, and they are decomposed in Eqs. (253) and (195) with the same probabilities ... 

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

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). 

The remarkable conclusion is that the microscopic quantum state, specified by the wave function ip, can be described on a macroscopic level by the probability distribution Pj. A single pure state corresponds to a macroscopic ensemble. The interference terms that are typical for quantum mechanics no longer appear. Incidentally, this resolves the paradox of Schrodinger s cat and, in general, the quantum mechanical measurement problem. )... 

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,... 

A version of the reduced density operator to be used in the mixed quantum classical description may be obtained if we replace p(t) by the pure state... 

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. 

Thus far we have dealt with the idealized case of isolated molecules that are neither -subject to external collisions nor display spontaneous emission. Further, we have V assumed that the molecule is initially in a pure state (i.e., described by a wave function) and that the externally imposed electric field is coherent, that is, that the " j field is described by a well-defined function of time [e.g., Eq. (1.35)]. Under these. circumstances the molecule is in a pure state before and after laser excitation and S remains so throughout its evolution. However, if the molecule is initially in a mixed4> state (e.g., due to prior collisional relaxation), or if the incident radiation field is notlf fully coherent (e.g., due to random fluctuations of the laser phase or of the laser amplitude), or if collisions cause the loss of quantum phase after excitation, then J phase information is degraded, interference phenomena are muted, and laser controi. is jeopardized. < f... 

This Wigner representation of the density pw q, p) proves particularly useful since it, satisfies a number of properties that are similar to the classical phase-space distribu tion pd(q, p). For example, if p = pure state, then fdppw = probability density in coor- dinate space. Similarly, integrating pw over q gives the probability density in ( momentum space. These features are shared by the classical density p p, q) in phase space. Note, however, that pw is not a probability density, as evidenced by if the fact that it can be negative, a reflection of quantum features of the dynamics, ) [165], 3... 

Product Operator. For multiple-quantum coherences, the pure zero-quantum and double-quantum states are shown with their Cartesian product operator equivalents. 

As far as linear superpositions or quantum states are concerned, the view (1) asserts that a pure state provides a complete and exhaustive description of an individual system (e.g., an electron, molecule, and molecular objects) and the view exposed in case (2) asserts that a pure state provides a description of certain statistical properties of an ensemble of similarly prepared systems, but need not provide a complete description of an individual system. These statements correspond to those given in Refs. [7-9]. For details, the reader can go back to Refs. [1,9,10]. 

The combined use of IR and NMR spectroscopy provides a complete picture of the isomeric equilibria of these compounds134 193 194. Quantum-mechanical calculations of the structure, relative energies, solvation185 186 and vibrational spectra195 of a series of model 2-acyl-2-nitroenamines have confirmed the structural assignment of the isomers either in solution or in the pure state. 

It is well known from basic quantum theory that the value of an observable quantity A, represented by the operator A, can be found for any pure state i as the expectation value, CT.)W ... 

Polythiazyl. This polymer, (SN), known since 1910, can now be obtained in a pure state. It is golden bronze in color and displays metallic type electrical conductance more remarkable still is the fact that at 0.26 K it becomes a superconductor. In the crystal the kinked, nearly planar, chains (12-IX) lie parallel and conductance takes place along the chains, in which n electrons are extensively delocalized according to molecular quantum mechanical calculations. A partially brominated substance, (SNBr04) , is an even better conductor. 

Averaging over all different possible stochastic behaviors in SMS yields the master equation used in ensemble spectroscopy, but the averaged master equation does not determine the dynamics of the (pure) states of individual molecules. Certain attempts have been made to derive a proper theory of individual behavior of single quantum systems, but a rigorous interpretation is still lacking. 

Hence, the chemical structure, dynamics, and spectroscopy of single molecules cannot be rigorously discussed in the present formalism of quantum mechanics and the problem is to construct a quantum theory for individual molecules. One possible starting point is an averaged (e.g., thermal) description and an averaged dynamics over an ensemble of molecules. The average ranges over all the pure states of the molecules in the ensemble. In technical terms, an individual quantum theory is related to a decomposition of nonpure states into pure ones (for a precise definition, see subsequent text). This decomposition is not unique. One... 

There are good reasons to believe that a strictly isolated molecule can be prepared experimentally in any (pure) state. " However, not all molecular states are equally stable under (small) external stochastic perturbations. The molecular environment is thus of decisive importance in all these discussions on molecular structure. Examples of environments that cannot be completely screened out are, e.g., quantum radiation and gravitational fields. Here an attempt is made to incorporate external environmental stochastic perturbations and to look for decompositions that are stable under these perturbations. This opens up a possible route to enable us to ... 

---