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

Physical Review A

Quantum state tomography for strongly coupled nuclear spin systems

Fatemeh Anvari Vind, Alexandre M. Souza, Roberto S. Sarthour, Ivan S. Oliveira

Published Dec 29, 2014• 2 citations
Publisher

Abstract

We perform quantum state tomography (QST) in a three-qubit system consisting of strongly coupled nuclear spins, known in the NMR literature as $ABX$ systems. We find that the number of experiments necessary to perform QST in such systems can be reduced with respect to those containing three qubits weakly coupled, which reduces the experimental effort required for the complete density-matrix reconstruction. To test the procedure we implement the full protocol for quantum teleportation. The tomographic results demonstrate that the density matrix can be reconstructed with fewer operations and high fidelity.

DOI
10.1103/physreva.90.062339

Related papers

A tailor-made quantum state tomography approach

Daniele Binosi, Giovanni Garberoglio, Diego Maragnano +2 more

Quantum state tomography (QST) aims at reconstructing the state of a quantum system. However, in conventional QST, the number of measurements scales exponentially with the number of qubits. Here, we propose a QST protocol, in which the introduction of a threshold allows one to drastically reduce the number of measurements required for the reconstruction of the state density matrix without compromising the result accuracy. In addition, one can also use the same approach to reconstruct an approximated density matrix tailoring the number of measurements on the available resources. We experimentally demonstrate this protocol by performing the tomography of states up to 7 qubits. We show that our approach can lead to results in agreement with those obtained by QST even when the number of measurements is reduced by more than two orders of magnitude.

arXiv:2401.12864•APL Quantum•Jul 26, 2024•10 citations

State interrogation in nuclear magnetic resonance quantum-information processing

Garett M. Leskowitz, Leonard J. Mueller

Reconstruction of a reduced density operator for weakly coupled systems of spins $\frac{1}{2}$ from fits to nuclear magnetic resonance spectra is described in detail. Particular emphasis is placed on data treatment procedures that specify fewer than the ${3}^{n}$ complete spectra that are implicitly prescribed in published references to state tomography on $n$-spin systems. It is shown that if the density operator is expanded in the so-called product-operator basis, it is always possible to estimate a desired coefficient in the expansion by measuring a single spectral multiplet. This simple observation can substantially reduce the experimental effort required for either complete density-matrix reconstruction or estimation of subsets of the coefficients in the product-operator expansion. A simple iterative algorithm can be used to produce reduced measurement procedures for experiments involving small numbers of qubits.

Physical Review A•May 6, 2004•92 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

Computational framework for quantum state tomography of spin ensembles

Jiwoo Seo, Vesna F. Mitrović

We present Tomography-NMR, an open-source Python package that reconstructs quantum density matrices from spectroscopic measurement data. The package implements a complete analysis pipeline for two-qubit quantum state tomography based on the product operator formalism: raw time-domain signals are Fourier-transformed into frequency-domain spectra, spectral peak intensities are mapped to expansion coefficients of the density matrix, and the full quantum state is reconstructed. Three integration methods are provided for different use cases: direct peak height measurement and fixed-parameter numerical integration require no theoretical reference and are suited to unknown states, achieving fidelities of approximately 98\% on a benchmark set of known states, while a systematic parameter optimization against a known target state achieves reconstruction fidelities exceeding 99\% for the same benchmark states. While the detailed theoretical framework for quantum state tomography is well established, the practical procedures for extracting density matrices from experimental spectra are often inadequately documented in the literature and obscured within proprietary software. This package addresses that gap by providing a fully transparent, reproducible implementation of every analysis step, from spectral preprocessing to density matrix visualization. The software has been validated on experimentally prepared two-qubit states measured via nuclear magnetic resonance (NMR) spectroscopy of coupled $^{31}$P nuclei. Average reconstruction fidelities range from 0.975 to 0.995 across a benchmark set of 20 two-qubit states, including the computational basis states, Bell states, and the outputs of three fundamental quantum gates (CNOT, H, and T). Although developed for NMR, the modular architecture facilitates adaptation to other spectroscopic platforms and alternative measurement protocols.

quant-pharXiv:2609.11868•Sep 11, 2026•0 citations

Quantum state tomography for quadrupolar nuclei using global rotations of the spin system

J. Teles, Eduardo Ribeiro DeAzevedo, R. Auccaise +3 more

In this paper, we describe a quantum state tomography method based on global rotations of the spin system which, together with a coherence selection scheme, enables the complete density matrix reconstruction. The main advantage of this technique, in respect to previous proposals, is the use of much shorter rf pulses, which decreases significantly the time necessary for algorithm quantum state tomography. In this case, under adequate experimental conditions, the rf pulses correspond to simple spatial rotations of the spin states, and its analytical description is conveniently given in the irreducible tensor formalism. Simulated results show the feasibility of the method for a single spin 72 nucleus. As an experimental result, we exemplify the application of this method by tomographing the steps during the implementation of the Deutsch algorithm. The algorithm was implemented in a (23)Na quadrupole nucleus using the strongly modulated pulses technique. We also extended the tomography method for a 3-coupled homonuclear spin 12 system, where an additional evolution under the internal Hamiltonian is necessary for zero order coherences evaluation.

The Journal of Chemical Physics•Apr 17, 2007•48 citations

Spin-1 teleportation-based quantum state tomography

Gustavo Rigolin

We show that the teleportation-based quantum state tomography (QST) protocol, originally built to reconstruct qubits (spin-1/2 systems), can be extended to deal with qutrits (spin-1 systems) as well. Similarly to the original proposal, only two resources are needed to implement the spin-1 teleportation-based QST protocol: (1) Alice should be able to implement the analog of Bell measurements for spin-1 systems; and (2) she should be able to prepare a few different single qutrit states that will be teleported to Bob.

quant-pharXiv:2607.05621•Physics Letters A•Jul 8, 2026•1 citation
We perform quantum state tomography (QST) in a three-qubit system consisting of strongly coupled nuclear spins, known in the NMR literature as ABXABXABX systems. We find that the number of experiments necessary to perform QST in such systems can be reduced with respect to those containing three qubits weakly coupled, which reduces the experimental effort required for the complete density-matrix reconstruction. To test the procedure we implement the full protocol for quantum teleportation. The tomographic results demonstrate that the density matrix can be reconstructed with fewer operations and high fidelity.
Quantum state tomography (QST) aims at reconstructing the state of a quantum system. However, in conventional QST, the number of measurements scales exponentially with the number of qubits. Here, we propose a QST protocol, in which the introduction of a threshold allows one to drastically reduce the number of measurements required for the reconstruction of the state density matrix without compromising the result accuracy. In addition, one can also use the same approach to reconstruct an approximated density matrix tailoring the number of measurements on the available resources. We experimentally demonstrate this protocol by performing the tomography of states up to 7 qubits. We show that our approach can lead to results in agreement with those obtained by QST even when the number of measurements is reduced by more than two orders of magnitude.
Reconstruction of a reduced density operator for weakly coupled systems of spins 12\frac{1}{2}21​ from fits to nuclear magnetic resonance spectra is described in detail. Particular emphasis is placed on data treatment procedures that specify fewer than the 3n{3}^{n}3n complete spectra that are implicitly prescribed in published references to state tomography on nnn-spin systems. It is shown that if the density operator is expanded in the so-called product-operator basis, it is always possible to estimate a desired coefficient in the expansion by measuring a single spectral multiplet. This simple observation can substantially reduce the experimental effort required for either complete density-matrix reconstruction or estimation of subsets of the coefficients in the product-operator expansion. A simple iterative algorithm can be used to produce reduced measurement procedures for experiments involving small numbers of qubits.
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
We present Tomography-NMR, an open-source Python package that reconstructs quantum density matrices from spectroscopic measurement data. The package implements a complete analysis pipeline for two-qubit quantum state tomography based on the product operator formalism: raw time-domain signals are Fourier-transformed into frequency-domain spectra, spectral peak intensities are mapped to expansion coefficients of the density matrix, and the full quantum state is reconstructed. Three integration methods are provided for different use cases: direct peak height measurement and fixed-parameter numerical integration require no theoretical reference and are suited to unknown states, achieving fidelities of approximately 98\% on a benchmark set of known states, while a systematic parameter optimization against a known target state achieves reconstruction fidelities exceeding 99\% for the same benchmark states. While the detailed theoretical framework for quantum state tomography is well established, the practical procedures for extracting density matrices from experimental spectra are often inadequately documented in the literature and obscured within proprietary software. This package addresses that gap by providing a fully transparent, reproducible implementation of every analysis step, from spectral preprocessing to density matrix visualization. The software has been validated on experimentally prepared two-qubit states measured via nuclear magnetic resonance (NMR) spectroscopy of coupled 31^{31}31P nuclei. Average reconstruction fidelities range from 0.975 to 0.995 across a benchmark set of 20 two-qubit states, including the computational basis states, Bell states, and the outputs of three fundamental quantum gates (CNOT, H, and T). Although developed for NMR, the modular architecture facilitates adaptation to other spectroscopic platforms and alternative measurement protocols.
In this paper, we describe a quantum state tomography method based on global rotations of the spin system which, together with a coherence selection scheme, enables the complete density matrix reconstruction. The main advantage of this technique, in respect to previous proposals, is the use of much shorter rf pulses, which decreases significantly the time necessary for algorithm quantum state tomography. In this case, under adequate experimental conditions, the rf pulses correspond to simple spatial rotations of the spin states, and its analytical description is conveniently given in the irreducible tensor formalism. Simulated results show the feasibility of the method for a single spin 72 nucleus. As an experimental result, we exemplify the application of this method by tomographing the steps during the implementation of the Deutsch algorithm. The algorithm was implemented in a (23)Na quadrupole nucleus using the strongly modulated pulses technique. We also extended the tomography method for a 3-coupled homonuclear spin 12 system, where an additional evolution under the internal Hamiltonian is necessary for zero order coherences evaluation.
We show that the teleportation-based quantum state tomography (QST) protocol, originally built to reconstruct qubits (spin-1/2 systems), can be extended to deal with qutrits (spin-1 systems) as well. Similarly to the original proposal, only two resources are needed to implement the spin-1 teleportation-based QST protocol: (1) Alice should be able to implement the analog of Bell measurements for spin-1 systems; and (2) she should be able to prepare a few different single qutrit states that will be teleported to Bob.