Matrix representations for adjoint and anti-adjoint operators in multi-spin 1/2 systems

Abstract

This paper focuses on investigating the problems of matrix representations of adjoint and anti-adjoint operators as well as computations for these matrices in multi-spin 1/2 systems. By introducing a multi-index transformation mapping, adjoint and anti-adjoint operators on tensor space as well as their matrix representations are defined to describe dynamics of multi-spin 1/2 systems. Formulas for computing these matrices of the adjoint and anti-adjoint operators in multi-spin 1/2 systems are given in terms of matrix representations of the adjoint and anti-adjoint operators in single-spin 1/2 systems.

Optimal control of single spin‐1/2 quantum systems

“…This observation will be used in our controllability analysis of single spin-1/2 systems. The matrix representations of the adjoint operators for multi-spin-1/2 systems can also be constructed by using the adjoint matrices of single spin-1/2 systems [17].…”

“…Using the right invariance property, one has R(I , T ) 0 = R( 0 , T ) for all 0 ∈ SO(3) and all T . The following lemma summarises properties of the reachable sets for the system (17). From the above lemma, along with (17), the reachable set for the system (14) can also be characterised by applying (t) to act on the initial state of (14).…”

“…With the time T > 0, the initial state x(0) and the final state x(T ) given, suppose that the cost functional J (u) for the given control problem has the form of (16), then we shall address the problem of acquiring the optimal control law which minimises the cost functional subject to the constraints that the system evolves according to (15) and the boundary conditions are satisfied. As the previous controllability analysis subsection has shown, the controllability of the vector system (15) is linked with that of the matrix Lie group system (17), thus we will attack the problem based on the system (17). In order to deal with the problem we need to introduce a costate matrix (t), and define the system Hamiltonian H as follows…”

“…Theorem 1: Suppose thatû(t) is the optimal Lebesgue integrable control for the system (17), in the sense that, it transfers the state of system from the initial state 0 (i.e. the identity matrix) to a prescribed terminal state T , in time T and simultaneously minimises the cost functional J (u) given in (16).…”

“…The state space of the system (15) can be described by the Bloch sphere S 2 , that is, the system in itself evolves on a manifold in R 3 . The controllability analysis of right invariant systems can be facilitated with the introduction of matrix differential equations associated with (15), given bẏ (t) = (A + uB) (t), (0) = I (17) and the trajectory x(t) determined by (15) can also be obtained by allowing (t) to act on the initial state x(0) via usual matrix-vector multiplication. Thus, a new control problem on matrix Lie group can be defined which is in connection with the original problem in question.…”