Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Chapman & Hall/CRC eBooks

Observability and State Determination

Domenico D’Alessandro

Published Jul 13, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9781003051268-5

Related papers

Observability and State Determination

Domenico D'Alessandro

As is known from measurement theory (cf. section 1.2), the measurement of an observable gives a certain result with a probability depending on the state. This suggests that measurements of appropriate observables on several copies of the same system can be used to determine the state of the system with some level of confidence. 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.

Chapman & Hall/CRC eBooks•Aug 3, 2007•0 citations

On quantum state observability and measurement

Domenico D Alessandro

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.

arXiv:quant-ph/0307127•Journal of Physics A Mathematical and General•Sep 2, 2003•26 citations

State determination in continuous measurement

Andrew C. Doherty, S. M. Tan, A. S. Parkins +1 more

The possibility of determining the state of a quantum system after a continuous measurement of position is discussed in the framework of quantum trajectory theory. The initial lack of knowledge of the system and external noises are accounted for by considering the evolution of conditioned density matrices under a stochastic master equation. It is shown that after a finite time the state of the system is a pure state, and can be inferred from the measurement record alone. The relation to emerging possibilities for the continuous experimental observation of single quanta, as for example in cavity quantum electrodynamics, is discussed.

arXiv:quant-ph/9903030•Physical Review A•Sep 1, 1999•59 citations

Quantum Tomography

S. Rajasekar, R. Velusamy

This chapter provides the relevant theory and techniques of quantum tomography. Quantum tomography aims to determine the state of the system by determining its density matrix by means of a series of measurements done on fairly a large number of identically prepared copies of the same system. Wolfgang Ernst Pauli problem asks the question whether measurements of probability densities of position and momentum of a particle determine its quantum state wave function. The chapter outlines the optical homodyne tomography technique of reconstruction of Wigner function. Polarized light wave is described by the commonly used Stokes parameters introduced by George Gabriel Stokes in 1852. Quantum process tomography (QPT) aims to give full information on the dynamics of a quantum state. QPT can be used to improve the design and control of quantum hardware. Direct characterization of quantum dynamics differs in many ways from standard quantum process tomography and ancilla-assisted process tomography.

CRC Press eBooks•Oct 5, 2022•0 citations

On the observability and state determination of quantum mechanical systems

Deanna M. D’Alessandro

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

IEEE Conference on Decision and Control (CDC)•Jan 1, 2004•2 citations

Tasks and premises in quantum state determination

Claudio Carmeli, Teiko Heinosaari, Jussi Schultz +1 more

The purpose of quantum tomography is to determine an unknown quantum state from measurement outcome statistics. There are two obvious ways to generalize this setting. First, our task need not be the determination of any possible input state but only some input states, for instance pure states. Second, we may have some prior information, or premise, which guarantees that the input state belongs to some subset of states, for instance the set of states with rank less than half of the dimension of the Hilbert space. We investigate state determination under these two supplemental features, concentrating on the cases where the task and the premise are statements about the rank of the unknown state. We characterize the structure of quantum observables (positive operator valued measures) that are capable of fulfilling these type of determination tasks. After the general treatment we focus on the class of covariant phase space observables, thus providing physically relevant examples of observables both capable and incapable of performing these tasks. In this context, the effect of noise is discussed.

arXiv:1308.5502•Journal of Physics A Mathematical and Theoretical•Feb 3, 2014•24 citations
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.
As is known from measurement theory (cf. section 1.2), the measurement of an observable gives a certain result with a probability depending on the state. This suggests that measurements of appropriate observables on several copies of the same system can be used to determine the state of the system with some level of confidence. 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.
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.
The possibility of determining the state of a quantum system after a continuous measurement of position is discussed in the framework of quantum trajectory theory. The initial lack of knowledge of the system and external noises are accounted for by considering the evolution of conditioned density matrices under a stochastic master equation. It is shown that after a finite time the state of the system is a pure state, and can be inferred from the measurement record alone. The relation to emerging possibilities for the continuous experimental observation of single quanta, as for example in cavity quantum electrodynamics, is discussed.
This chapter provides the relevant theory and techniques of quantum tomography. Quantum tomography aims to determine the state of the system by determining its density matrix by means of a series of measurements done on fairly a large number of identically prepared copies of the same system. Wolfgang Ernst Pauli problem asks the question whether measurements of probability densities of position and momentum of a particle determine its quantum state wave function. The chapter outlines the optical homodyne tomography technique of reconstruction of Wigner function. Polarized light wave is described by the commonly used Stokes parameters introduced by George Gabriel Stokes in 1852. Quantum process tomography (QPT) aims to give full information on the dynamics of a quantum state. QPT can be used to improve the design and control of quantum hardware. Direct characterization of quantum dynamics differs in many ways from standard quantum process tomography and ancilla-assisted process tomography.
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
The purpose of quantum tomography is to determine an unknown quantum state from measurement outcome statistics. There are two obvious ways to generalize this setting. First, our task need not be the determination of any possible input state but only some input states, for instance pure states. Second, we may have some prior information, or premise, which guarantees that the input state belongs to some subset of states, for instance the set of states with rank less than half of the dimension of the Hilbert space. We investigate state determination under these two supplemental features, concentrating on the cases where the task and the premise are statements about the rank of the unknown state. We characterize the structure of quantum observables (positive operator valued measures) that are capable of fulfilling these type of determination tasks. After the general treatment we focus on the class of covariant phase space observables, thus providing physically relevant examples of observables both capable and incapable of performing these tasks. In this context, the effect of noise is discussed.