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

0 Orthogonality relations in Quantum Tomography

Giacomo Mauro D’Ariano, Lorenzo Maccone, Matteo G. A. Paris

Published Aug 16, 2016• 18 citations
OpenAlex

Abstract

Quantum estimation of the operators of a system is investigated by analyzing its Liouville space of operators. In this way it is possible to easily derive some general characterization for the sets of observables (i.e. the possible quorums) that are measured for the quantum estimation. In particular we analyze the reconstruction of operators of spin systems.

Related papers

Quorum of observables for universal quantum estimation

G Mauro D'Ariano, Lorenzo Maccone, Matteo G A Paris

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.

arXiv:quant-ph/0006006•Journal of Physics A Mathematical and General•Dec 19, 2000•53 citations

Quorum of observables for universal quantum estimation

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

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.

•Jun 1, 2000•44 citations

Quantum Tomography: General Theory and New Experiments

Giacomo Mauro D’Ariano

Quantum tomography is a general method for estimating the ensemble average of all operators of an arbitrary quantum system from a set of measurements of a quorum of observables. A procedure for deconvolving instrumental noise is available, which makes the tomographic method feasible in different physical contexts. Recent developments are presented and new experiments are proposed, which are now made possible by the tomographic technique.

Fortschritte der Physik•May 1, 2000•6 citations

Quantum tomography of input-output processes

Alfredo Luis

We demonstrate the possibility of practical tomographic determination of arbitrary input-output transformations. It is shown that the Liouville-space formalism provides a generalization of standard tomography encompassing the reconstruction of processes as well as of quantum states.

Physical Review A•Oct 11, 2000•8 citations

Quantum tomography for solid state qubits

Yu-xi Liu, L. F. Wei, Franco Nori

PACS. 03.65.Ta – Foundations of quantum mechanics; measurement theory. PACS. 03.65.Wj – State reconstruction, quantum tomography. Abstract. – We propose a method for the tomographic reconstruction of qubit states for a general class of solid-state systems in which the Hamiltonians are represented by spin operators, e.g., with Heisenberg-, XXZ-, or XY-type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data. Quantum information processing requires the effective measurement of quantum states. However, a single quantum measurement can only obtain partial information of a quantum state. The reconstruction of a quantum state requires measuring a complete set of observables on an ensemble of identically prepared copies of the system. This method called quantum state tomography [1], is very important because any unknown state can be ascertained by tomographic measurements. Moreover, the full description of qubit states can increase the accuracy of quantum operations. Tomographic measurements have been experimentally implemented for, e.g., the nuclear spin state of an NMR system [2], the electromagnetic field and photon

•Jan 1, 2004•19 citations

Quantum tomography and Schwinger’s picture of quantum mechanics*

Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort +1 more

Abstract In this paper the problem of tomographic reconstruction of states is investigated within the so-called Schwinger’s picture of quantum mechanics in which a groupoid is associated with every quantum system. The attention is focussed on spin tomography: in this context the groupoid of interest is the groupoid of pairs over a finite set. In a nutshell, this groupoid is made up of transitions between all possible pairs of outcomes belonging to a finite set. In addition, these transitions possess a partial composition rule, generalizing the notion of groups. The main goal of the paper consists in providing a reconstruction formula for states on the groupoid-algebra associated with the observables of the system. Using the group of bisections of this groupoid, which are special subsets in one-to-one correspondence with the outcomes, a frame is defined and it is used to prove the validity of the tomographic reconstruction. The special case of the set of outcomes being the set of integers modulo n, with n odd prime, is considered in detail. In this case the subgroup of discrete affine linear transformations, whose graphs are linear subspaces of the groupoid, provides a quorum in close analogy with the continuous case.

arXiv:2205.00170•Journal of Physics A Mathematical and Theoretical•Jun 2, 2022•1 citation
Quantum estimation of the operators of a system is investigated by analyzing its Liouville space of operators. In this way it is possible to easily derive some general characterization for the sets of observables (i.e. the possible quorums) that are measured for the quantum estimation. In particular we analyze the reconstruction of operators of spin systems.
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.
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.
PACS. 03.65.Ta – Foundations of quantum mechanics; measurement theory. PACS. 03.65.Wj – State reconstruction, quantum tomography. Abstract. – We propose a method for the tomographic reconstruction of qubit states for a general class of solid-state systems in which the Hamiltonians are represented by spin operators, e.g., with Heisenberg-, XXZ-, or XY-type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data. Quantum information processing requires the effective measurement of quantum states. However, a single quantum measurement can only obtain partial information of a quantum state. The reconstruction of a quantum state requires measuring a complete set of observables on an ensemble of identically prepared copies of the system. This method called quantum state tomography [1], is very important because any unknown state can be ascertained by tomographic measurements. Moreover, the full description of qubit states can increase the accuracy of quantum operations. Tomographic measurements have been experimentally implemented for, e.g., the nuclear spin state of an NMR system [2], the electromagnetic field and photon
Abstract In this paper the problem of tomographic reconstruction of states is investigated within the so-called Schwinger’s picture of quantum mechanics in which a groupoid is associated with every quantum system. The attention is focussed on spin tomography: in this context the groupoid of interest is the groupoid of pairs over a finite set. In a nutshell, this groupoid is made up of transitions between all possible pairs of outcomes belonging to a finite set. In addition, these transitions possess a partial composition rule, generalizing the notion of groups. The main goal of the paper consists in providing a reconstruction formula for states on the groupoid-algebra associated with the observables of the system. Using the group of bisections of this groupoid, which are special subsets in one-to-one correspondence with the outcomes, a frame is defined and it is used to prove the validity of the tomographic reconstruction. The special case of the set of outcomes being the set of integers modulo n, with n odd prime, is considered in detail. In this case the subgroup of discrete affine linear transformations, whose graphs are linear subspaces of the groupoid, provides a quorum in close analogy with the continuous case.