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IEEE Conference on Decision and Control (CDC)

Observability, measurement and parameter identification of quantum mechanical systems; applications to spin networks

Francesca Albertini, Domenico D’Alessandro

Published Jul 8, 2004• 4 citations
Publisher

Abstract

In this paper, we study the observability properties of quantum systems and the related problem of parameter identification with particular emphasys on networks of spin 1/2 particles. These networks model magnetic molecules. We give criteria for observability of general quantum systems and apply these to characterize the equivalence classes of spin networks that have the same input output behavior.

DOI
10.1109/cdc.2003.1272602

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In this paper, we study the observability properties of quantum systems and the related problem of parameter identification with particular emphasys on networks of spin 1/2 particles. These networks model magnetic molecules. We give criteria for observability of general quantum systems and apply these to characterize the equivalence classes of spin networks that have the same input output behavior.
The system identification problem is to estimate dynamical parameters from the output data, obtained by performing measurements on the output fields. We investigate system identification for quantum linear systems. Our main objectives are to address the following general problems: (1) Which parameters can be identified by measuring the output? (2) How can we construct a system realisation from sufficient input-output data? (3) How well can we estimate the parameters governing the dynamics? We investigate these problems in two contrasting approaches; using time-dependent inputs or time-stationary (quantum noise) inputs.
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 this paper, we consider the problem of model equivalence for quantum systems. Two models are said to be (input-output) equivalent if they give the same output for every admissible input. In the case of quantum systems, we take as output the expectation value of a given observable or, more generally, a probability distribution for the result of a quantum measurement. We link the input-output equivalence of two models to the existence of a homomorphism of the underlying Lie algebra. In several cases, a Cartan decomposition of the Lie algebra su(N)su(N)su(N) is useful to find such a homomorphism and to determine the classes of equivalent models. We consider in detail the important cases of two level systems with a Cartan structure and of spin networks. In the latter case, complete results are given generalizing previous results to networks of spin particles with arbitrary values of the spins. In treating this problem, we give independent proofs of some instrumental results on the subalgebras of su(N)su(N)su(N).
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 this paper, we consider the problem of model equivalence for quantum systems. Two models are said to be (input-output) equivalent if they give the same output for every admissible input. In the case of quantum systems, the output is the expectation value of a given observable or, more in general, a probability distribution for the result of a quantum measurement. We link the input-output equivalence of two models to the existence of a homomorphism of the underlying Lie algebra. In several cases, a Cartan decomposition of the Lie algebra su(n) is useful to find such a homomorphism and to determine the classes of equivalent models. We consider in detail the important cases of two level systems with a Cartan structure and of spin networks. In the latter case, complete results are given generalizing previous results to the case of networks of spin particles with any value of the spin. In treating this problem, we prove some instrumental results on the subalgebras of su(n) which are of independent interest.
The aim of quantum system identification is to estimate the ingredients inside a black box, in which some quantum-mechanical unitary process takes place, by just looking at its input-output behavior. Here we establish a basic and general framework for quantum system identification, that allows us to classify how much knowledge about the quantum system is attainable, in principle, from a given experimental setup. We show that controllable closed quantum systems can be estimated up to unitary conjugation. Prior knowledge on some elements of the black box helps the system identification. We present an example in which a Bell measurement is more efficient to identify the system. When the topology of the system is known, the framework enables us to establish a general criterion for the estimability of the coupling constants in its Hamiltonian.