Unitary evolution operator

Focusing on strictly local (i.e. nearest neighbor) interactions. Grossing and Zeilinger [gross88a] consider the following unitary evolution operator U, approximated to first order by ... 

The problem now is to find the corresponding Hamiltonian, t Hooft shows that the most obvious construction, obtained by rewriting U(t+l,t) as a product of cyclic elements, unfortunately does not work because at the end of the calculation there is no way to uniquely define the vacuum state. Given a cellular automaton with a local unitary evolution operator U = WgUg and the commutator [Ug, Ug ] 0 if [ af — af j> d for some d > 0, the real problem is therefore to find a Hamiltonian... 

The control is a set of real parameters/ , which have been combined to a vector /. These can be timings, amplitudes, and/or phases of a given number of discrete pulses, or describe a time-continuous modulation of the system. Here, we focus on time-dependent control, where the/(r) parameterize the system Hamiltonian as Hg = Hglfit)], or the unitary evolution operator 6/(t) = =... 

We assume that the Hamiltonian H has no explicit time-dependence, and, therefore, the unitary evolution operator over At may be written as... 

It is possible to verify our considerations performing appropriate numerical calculations. As, we have excluded here all damping processes, the dynamics of our system can be described by the unitary evolution. Therefore, we define the unitary evolution operator... 

Then (9.1) gives uniquely rise to the unitary evolution operator U for the composite system, that is, to the unitary dynamics of the composite system ... 

Consider a deteriiiinistic local reversible CA i.o. start with an infinite array of sites, T, arranged in some regular fashion, and a.ssume each site can be any of N states labeled by 0 < cr x) < N. If the number of sites is Af, the Hilbert space spanned by the states <7-(x is N- dimensional. The state at time t + 1, cTf+i(a ) depends only on the values cri x ) that are in the immediate neighborhood of X. Because the cellular automata is reversible, the mapping ai x) crt+i x ) is assumed to have a unique inveuse and the evolution operator U t,t + 1) in this Hilbert space is unitary,... 

U m T(R) a unitary operator time-evolution operator (Section 13.1) matrix representative of the symmetry operator R sometimes just R, for brevity... 

A fundamental limitation to coherent population control is that it is impossible to transfer 100% of the population in a mixed state. That is, the maximum value of the population transferred cannot exceed the maximum of the initial population distribution of a system without any dissipative process such as spontaneous emission. This result can be simply verified using the unitary property of the density operator, pit) = U(t, to)pito)U t, to), where p(to) is the diagonalized density operator at t = to, Uit, to) is the time-evolution operator given by... 

Now, according to Eq. (J.7), and since the time-evolution operators are unitary, the product of the two time-evolution operators involved in the ACF (J.l) is... 

T.2.A.3. H-Bond Bridge Unitary Time Evolution Operators U(t)... 

Hilbert space EL, yielding C (/). Only if the set j belongs to an extraspecial algebra restricting the erroneous evolution operators to a subgroup Q ( T m j C Qu EL) of the full unitary group in EL, a non-trivial code space... 

The unitary evolution of the combined system is interrupted repeatedly with a measurement performed on the ancilla. Formally, the effect of a projective measurement is described with an operator V that acts only in the Hilbert space of the ancilla, g — V g, and the effect of V is taken to be instantaneous. Therefore, two immediately subsequent measurements yield the same outcome and V2 = V is a projector. 

This equation of motion for the quantum state V(t)) is formally integrated by means of the (unitary) time-evolution operator U... 

In order to preserve the normalization of the wave function, the temporal evolution operator has to be unitary. From the TDSE, for an infinitesimal time step A ... 

Using the time-evolution operator (17.11), by means of the unitary transformation (18.11), we can determine the wave function in a moment t2 after the collision in terms of the wave function in the moment t. before the collision. 

Having discussed the quantum evolution in terms of unitary and statistical operators we can make the further step towards describing quantum transitions. The starting point stays, as already custom with, on the unitary evolution action on given initial state ... 

Going to analyze Ihe momentum operatorial behavior, it influences ihe translation of systems through the unitary translational operator, in the same way as introduced the time evolution operator ... 

With the unitary time-evolution operator of the unperturbed system, Ud t) = we obtain the Schrodinger equation in the interaction... 

The unitary operator U = e is called the evolution operator, or the propagator... 

It follows from a basic theorem on linear operators that, since H is selfadjoint, the evolution operator U t) is unitary (Jordan [93]). The evolution operator maps the state vector for time zero (that is, ip)) onto the corresponding vector for time t. 

The temporal dynamics is governed by a set of [Q [-dimensional transition matrices T s) = U P s),s . 4, whose components are elements of the complex unit disk and where each is a product of a unitary matrix U and a projection operator P s). U is a. [Q[-dimensional miitary evolution operator that governs the evolution of the state vector ( [. P = P s) s A is a set of projection operators—[Q[-dimensional Hermitian matrices—that determines how the state vector is measured. The operators are mutually orthogonal and span the Hilbert space P s) = 1. 

A system which consists of k non-interacting subsystems has a Hamiltonian which is the sum of k locally acting Hamiltonians. Each of the Hamiltonians can be derived from the unitary matrix that describes one subsystem. If the subsystems interact, this is not that simple. One gains insight by looking at it in the following way It is possible to write down the desired time evolution operators Uj for the k subsystems. The product of these matrices should yield the desired unitary matrix Uideai = Hi which describes the global evolution of the computer. Since the... 

Schrodinger equation. The time evolution of a state vector is (in the non-relativistic case) governed by the Schrodinger equation which gives rise to a rotation of the state vector within the Hibert space H. The time evolution of a state vector can be described by a linear, unitary time evolution operator U t). That is, a normalized state l (O)) at a time t = 0 evolves into a normalized state l (T)) = U(T)l (0)) after a time T. Quantum mechanics is linear, therefore a superposition evolves according to... 

In the Heisenberg picture the operators themselves depend explicitly on the time and the time evolution of the system is determined by a differential equation for the operators. The time-dependent Heisenberg operator AH(t) is obtained from the corresponding Schrodinger operator As by the unitary transformation... 

The column vector is indicated by square brackets, a row vector by round brackets. The quantum numbers may be determined by the complete set of her-mitian operators commuting with the generator of time evolution. Invariance of the quantum state to frame rotation, origin displacement, parity and other symmetry operations determine quantum numbers for the corresponding irreducible representations. Frame related symmetry operations translate into unitary operator acting on Hilbert space (rigged), e.g. Ta. 

The incorporation of the time evolution of the electronic wavefunction is taken care of in the parameterization of the unitary operators in orbital and configuration space given in Equation (4.93). 

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