Domenico D’Alessandro

Small time controllability of systems on compact Lie groups and spin angular momentum

In this article, we develop some general results on the properties of the reachable sets for right invariant bilinear systems with state varying on compact Lie groups. The main results consist of a characterization of the set of states reachable in arbitrary time from the identity of the group. This, under suitable assumptions, is proved to be a Lie subgroup of the underlying Lie group. We apply these results to the analysis of the controllability of particles with spin. The results are motivated by and generalize the results in another work [D. D’Alessandro, Sys. Control Lett. 41, 213–221 (2000)], where the specific model of a spin 12 particle system in an electro-magnetic field was considered.

Decompositions of unitary evolutions and entanglement dynamics of bipartite quantum systems

We describe a decomposition of the Lie group of unitary evolutions for a bipartite quantum system of arbitrary dimensions. The decomposition is based on a recursive procedure that systematically uses the Cartan classification of the symmetric spaces of the Lie group SO(n). The resulting factorization of unitary evolutions clearly displays the local and entangling character of each factor.

Minimum time optimal synthesis for two level quantum systems

For the time optimal problem of an invariant system on SU(2), with two independent controls and a bound on the norm of the control, the extremals of the Pontryagin maximum principle are explicit functions of time. We use this fact here to perform the optimal synthesis for these systems, i.e., to find all time optimal trajectories. Although the Lie group SU(2) is three dimensional, time optimal trajectories can be described in the unit disk of the complex plane. We find that a circular trajectory separates optimal trajectories that reach the boundary of the unit disk from the others. Inside this separatrix circle, another trajectory (the critical trajectory) plays an important role in that all optimal trajectories end at an intersection with this curve. The results allow us to find the minimum time needed to achieve a given evolution of a two level quantum system.

Connection between continuous and discrete time quantum walks. From D-dimensional lattices to general graphs

I obtain the dynamics of the continuous time quantum walk on a d-dimensional lattice with periodic boundary conditions as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main result of [15] which proved this limit for the case of the quantum walk on the infinite line and the quantum cellular automaton proposed in [4]. By highlighting the main features of the limiting procedure, I then extend it to general regular graphs. For a given discrete time quantum walk on a graph, I single out the type of continuous dynamics (Hamiltonians) that can be obtained as a limit of the discrete time dynamics.

Equivalence between indirect controllability and complete controllability for quantum systems

Abstract We consider a control scheme where a quantum system S is put in contact with an auxiliary quantum system A and the control can affect A only, while S is the system of interest. The system S is then controlled indirectly through the interaction with A . Complete controllability of S + A means that every unitary state transformation for the system S + A can be achieved with this scheme. Indirect controllability means that every unitary transformation on the system S can be achieved. We prove in this paper, under appropriate conditions and definitions, that these two notions are equivalent in finite dimension. We use Lie algebraic methods to prove this result.

Controllability of quantum walks on graphs

In this paper, we consider discrete time quantum walks on graphs with coin, focusing on the decentralized model, where the coin operation is allowed to change with the vertex of the graph. When the coin operations can be modified at every time step, these systems can be looked at as control systems and techniques of geometric control theory can be applied. In particular, the set of states that one can achieve can be described by studying controllability. Extending previous results, we give a characterization of the set of reachable states in terms of an appropriate Lie algebra. Controllability is verified when any unitary operation between two states can be implemented as a result of the evolution of the quantum walk. We prove general results and criteria relating controllability to the combinatorial and topological properties of the walk. In particular, controllability is verified if and only if the underlying graph is not a bipartite graph and therefore it depends only on the graph and not on the particular quantum walk defined on it. We also provide explicit algorithms for control and quantify the number of steps needed for an arbitrary state transfer. The results of the paper are of interest in quantum information theory where quantum walks are used and analyzed in the development of quantum algorithms.

Input-Output Equivalence of Spin Networks under Multiple Measurements

Abstract.Two quantum control systems that are driven by an external field are said to be input–output equivalent if, for any control field, the measured value of a given observable is the same. Equivalent models cannot be distinguished by experiments involving state evolutions and measurements. In this paper, we characterize the equivalent models of networks of spin driven by electro-magnetic fields for which the expectation value of the total magnetization is measured. Extending previous results and definitions that only dealt with the case of a single measurement, we describe the class of equivalent models under a sequence of Von Neumann measurements. The results are motivated by the problem of parameter identification for Heisenberg spin systems modeling molecular magnets.

Control of a two-level quantum system in a coherent feedback scheme

We consider a scheme for the control of a spin , S, where an auxiliary spin , A, plays the role of the controller, and an Ising interaction is assumed between the spins. We demonstrate that even if there is no complete controllability on the system S + A we can have full control on the system S if and only if the initial state of A is pure. This provides the simplest example where the system S is indirectly controllable while the whole system S + A is not completely controllable. We also give an explicit algorithm for the indirect control of S.

Directions in the Theory of Quantum Control

We survey a number of directions in the current research on the control of finite dimensional bilinear quantum systems. We describe the model as well as the role of Lie algebra theory in determining controllability properties. We also discuss techniques for constructive controllability of these systems.

Analysis of quantum walks with time-varying coin on d-dimensional lattices

In this paper, we present a study of discrete time quantum walks whose underlying graph is a d-dimensional lattice. The dynamical behavior of these systems is of current interest because of their applications in quantum information theory as tools to design quantum algorithms. We assume that, at each step of the walk evolution, the coin transformation is allowed to change so that we can use it as a control variable to drive the evolution in a desired manner. We give an exact description of the possible evolutions and of the set of possible states that can be achieved with such a system. In particular, we show that it is possible to go from a state where there is probability 1 for the walker to be found in a vertex to a state where all the vertices have equal probability. We also prove a number of properties of the set of admissible states in terms of the number of steps needed to obtain them. We provide explicit algorithms for state transfer in low dimensional cases as well as results that allow to reduce...