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IEEE Conference on Decision and Control (CDC)

On the observability and state determination of quantum mechanical systems

Deanna M. D’Alessandro

Published Jan 1, 2004• 2 citations
Publisher

Abstract

We consider the problem of determining the state of a quantum system via a series of controlled evolutions and nonselective measurements of a given observable. Generalizing previous results to the case of possibly degenerate observables, we give a bound on the maximum number of parameters of the state that can be determined. We give a general algorithm for the determination of the maximum number of parameters. We also discuss how indirect measurement, when the system and probe are jointly controllable, allows one to determine all the parameters of the quantum state

DOI
10.1109/cdc.2004.4609055

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We consider the problem of determining the state of a quantum system via a series of controlled evolutions and nonselective measurements of a given observable. Generalizing previous results to the case of possibly degenerate observables, we give a bound on the maximum number of parameters of the state that can be determined. We give a general algorithm for the determination of the maximum number of parameters. We also discuss how indirect measurement, when the system and probe are jointly controllable, allows one to determine all the parameters of the quantum state
In this paper, we present a collection of results on the observability of quantum mechanical systems, in the case the output is the result of a discrete nonselective measurement. nt. By defining an effective observable, we extend previous results, on the Lie algebraic characterization of observable systems, to general measurements. Further results include the characterization of a `best probe' (i.e. a minimally disturbing probe) in indirect measurement and a study of the relation between disturbance and observability in this case. We also discuss how the observability properties of a quantum system relate to the problem of state reconstruction. Extensions of the formalism to the case of selective measurements are also given.
In many experiments, a target quantum mechanical system is controlled and-or measured indirectly using an auxiliary quantum system. In the case of control, this means that the control action only affects the auxiliary system while the evolution of the target system is affected indirectly through the interaction with the auxiliary system. In the case of measurement, only the state of the auxiliary system can be measured, providing information on the initial state of the target system. Indirect controllability is the property of the target system of being driven between two arbitrary states while indirect observability refers to the possibility of extracting all the information on the state of the target system by measuring the auxiliary system. This paper has three main goals. First, we summarize recent notions and results introduced by us on the study of indirect controllability. Then, we present some new technical results on indirect controllability. In particular, we present a counterexample to the converse of a criterion for indirect controllability, which shows that the Lie algebraic condition introduced in this criterion is necessary but not sufficient for indirect controllability. This gives an open problem which is how to strengthen this condition to obtain a necessary and sufficient condition for indirect controllability. Lastly, we present a parallel treatment of indirect observability, give the relevant notions and definitions and relate the properties of indirect controllability and observability. Our final result says that strong notions of indirect controllability and observability are equivalent properties to complete controllability of the system. We discuss examples of physical systems which are controlled and-or measured indirectly. The research is motivated by experimental schemes of current interest
A general scheme is presented for controlling quantum systems using evolution driven by nonselective von Neumann measurements, with or without an additional tailored electromagnetic field. As an example, a two-level quantum system controlled by nonselective quantum measurements is considered. The control goal is to find optimal system observables such that consecutive nonselective measurement of these observables transforms the system from a given initial state into a state which maximizes the expected value of a target operator (the objective). A complete analytical solution is found including explicit expressions for the optimal measured observables and for the maximal objective value given any target operator, any initial system density matrix, and any number of measurements. As an illustration, upper bounds on measurement-induced population transfer between the ground and the excited states for any number of measurements are found. The anti-Zeno effect is recovered in the limit of an infinite number of measurements. In this limit the system becomes completely controllable. The results establish the degree of control attainable by a finite number of measurements.
We consider the problem of determining the state of a quantum system given one or more readings of the expectation value of an observable. The system is assumed to be a finite-dimensional quantum control system for which we can influence the dynamics by generating all the unitary evolutions in a Lie group. We investigate to what extent, by an appropriate sequence of evolutions and measurements, we can obtain information on the initial state of the system. We present a system theoretic viewpoint of this problem in that we study the observability of the system. In this context, we characterize the equivalence classes of indistinguishable states and propose algorithms for state identification.
In systems theory, the study of the observability of a system describes to what extent it is possible to determine the initial state by a combined action of dynamics and output measurement. This chapter briefly summarizes the main ideas of quantum state tomography which is the technique used to determine a quantum state via a series of measurements. It discusses a system theoretic treatment of the problem of state determination namely to the study of the observability of quantum systems. The chapter partitions the space of density matrices into equivalence classes of states that cannot be distinguished by appropriate control and measurement.