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Quorum of observables for universal quantum estimation

Giacomo Mauro D’Ariano, Lorenzo Maccone, Paris, M G A

Published Jun 1, 2000• 44 citations
OpenAlex

Abstract

Any method for estimating the ensemble average of arbitrary operator (observables or not, including the density matrix) relates the quantity of interest to a complete set of observables, i.e. a quorum}. This corresponds to an expansion on an irreducible set of operators in the Liouville space. We give two general characterizations of these sets. All the known unbiased reconstruction techniques, i.e. ``quantum tomographies'', can be described in this framework. New operatorial resolutions are given that can be used to implement novel reconstruction schemes.

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Quorum of observables for universal quantum estimation

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We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e.g., estimating the ensemble average of an observable by measuring other observables. A main ingredient in this approach is that of informationally complete (infocomplete) measurements, which allow us to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of $AB$-informationally complete measurements, by which one can estimate jointly all the moments of two incompatible observables $A$ and $B$. After analyzing all general properties of such measurements, we address the problem of their optimality, and we completely solve the case of qubits, showing that a ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{y}$-infocomplete measurement is less noisy than any infocomplete one. We will also discuss the relation between the concept of $AB$ infocompleteness and the notion of joint measurement of observables $A$ and $B$.

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We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e. g. estimate the ensemble average of an observable by measuring other observable. A main ingredient in this approach is that of informationally complete (infocomplete in short) measurements, which allow to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of AB-informationally complete measurements, by which one can estimate jointly all the moments of two incompatible observables A and B. After analyzing all general properties of such measurements, we address the problem of their optimality, and we completely solve the case of qubits, showing that a sigma_x sigma_y-infocomplete measurement is less noisy than any infocomplete one. We will also discuss the relation between the concept of AB-infocompleteness and the notion of joint measurement of observables A and B.

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Quantum state determination: Quorum for a particle in one dimension

William Band, James L Park

In this paper we wish to illuminate the axiom that the quantal state describes a statistical ensemble of similar systems identically prepared, and is not to be identified with any single system. The mathematical representative of the general quantum state is the density matrix or statistical operator in Hilbert space. We demonstrate how this operator may be determined empirically by calculations involving only the measured mean values of a set of observables we call a ’’quorum.’’ As an example of this approach a state determination procedure is described for a spinless particle moving in one dimension; the corresponding quorum turns out to involve only position data associated with various instants subsequent to the act of preparation.

American Journal of Physics•Feb 1, 1979•42 citations
Any method for estimating the ensemble average of arbitrary operator (observables or not, including the density matrix) relates the quantity of interest to a complete set of observables, i.e. a quorum}. This corresponds to an expansion on an irreducible set of operators in the Liouville space. We give two general characterizations of these sets. All the known unbiased reconstruction techniques, i.e. ``quantum tomographies'', can be described in this framework. New operatorial resolutions are given that can be used to implement novel reconstruction schemes.
Quantum tomography is the process of reconstructing the ensemble average of an arbitrary operator (observable or not, including the density matrix), which may not be directly accessible by feasible detection schemes, starting from the measurement of a complete set of observables i.e. a quorum . The measurement of a quorum thus represents a complete characterization of the quantum state. The operator expression in terms of a quorum corresponds to an expansion on an irreducible set of operators in the Liouville space. We give two general characterizations of these sets, and show that all the known quantum tomographies can be described in this framework. New operatorial resolutions are also given that may be used in novel reconstruction schemes.
We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e.g., estimating the ensemble average of an observable by measuring other observables. A main ingredient in this approach is that of informationally complete (infocomplete) measurements, which allow us to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of ABABAB-informationally complete measurements, by which one can estimate jointly all the moments of two incompatible observables AAA and BBB. After analyzing all general properties of such measurements, we address the problem of their optimality, and we completely solve the case of qubits, showing that a ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{y}$-infocomplete measurement is less noisy than any infocomplete one. We will also discuss the relation between the concept of ABABAB infocompleteness and the notion of joint measurement of observables AAA and BBB.
Quantum indirect estimation theory and ABABAB-infocompleteness
We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e. g. estimate the ensemble average of an observable by measuring other observable. A main ingredient in this approach is that of informationally complete (infocomplete in short) measurements, which allow to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of AB-informationally complete measurements, by which one can estimate jointly all the moments of two incompatible observables A and B. After analyzing all general properties of such measurements, we address the problem of their optimality, and we completely solve the case of qubits, showing that a sigma_x sigma_y-infocomplete measurement is less noisy than any infocomplete one. We will also discuss the relation between the concept of AB-infocompleteness and the notion of joint measurement of observables A and B.
In this paper we wish to illuminate the axiom that the quantal state describes a statistical ensemble of similar systems identically prepared, and is not to be identified with any single system. The mathematical representative of the general quantum state is the density matrix or statistical operator in Hilbert space. We demonstrate how this operator may be determined empirically by calculations involving only the measured mean values of a set of observables we call a ’’quorum.’’ As an example of this approach a state determination procedure is described for a spinless particle moving in one dimension; the corresponding quorum turns out to involve only position data associated with various instants subsequent to the act of preparation.