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

arXiv:1007.3635 · Journal of Computational and Theoretical Nanoscience

An Overview on Single Apparatus Quantum Measurements

Bahar Mehmani, Theo M. Nieuwenhuizen

Published Mar 23, 2011• 4 citations
arXivHTMLPDF

Abstract

Given the state of a quantum system, one can calculate the expectation value of any observable of the system. However, the inverse problem of determining the state by performing different measurements is not a trivial task. In various experimental setups it is reasonably straightforward to reconstruct the state of a quantum system employing linear tomographic technique. In this way the elements of the density matrix can be linearly related to a set of measured quantities. But since different observables of a quantum system are not commuting with each other, one often has to perform series of successive measurements of observables which cannot be done simultaneously. Simultaneous measurement of observables cost less time and energy and is more beneficial. In this paper we review the strategy of quantum state tomography with simultaneous measurement of commuting observables. This can be done by introducing an assistant system of which the state is known. We show that the interaction between the assistant and the system of interest within different frame works allows the reconstruction of the state of the system. Specifically, we consider a two-level system and reconstruct its initial state by introducing an assistant which can be either another two-level system or a single cavity mode of the electromagnetic field.

DOI
10.1166/jctn.2011.1773

Related papers

Quantum-state tomography using a single apparatus

Bahar Mehmani, ARMEN E. ALLAHVERDYAN, Theo M. Nieuwenhuizen

The density matrix of a two-level system (spin, atom) is usually determined by measuring the three noncommuting components of the Pauli vector. This density matrix can also be obtained via the measurement data of two commuting variables, using a single apparatus. This is done by coupling the two-level system to a mode of radiation field, where the atom-field interaction is described with the Jaynes-Cummings model. The mode starts its evolution from a known coherent state. The unknown initial state of the atom is found by measuring two commuting observables: the population difference of the atom and the photon number of the field. We discuss the advantages of this setup and its possible applications.

arXiv:0708.1918•Physical Review A•Mar 31, 2008•8 citations

Determining an unknown state of a high-dimensional quantum system with a single, factorized observable

Ya‐Fei Yu, Hua Wen, Hua Li +1 more

One can uniquely identify an unknown state of a quantum system $S$ by measuring a ``quorum'' consisting of a complete set of noncommuting observables. It is also possible to determine the quantum state by repeated measurements of a single and factorized observable, when the system $S$ is coupled to an assistant system $A$ whose initial state is known. This is because that the redistribution of the information about the unknown quantum state into the composite system $A+S$ results in a one-to-one mapping between the unknown density matrix elements and the probabilities of the occurrence of the eigenvalues of a single, factorized observable of the composite system. Here we focus on quantum state tomography of high-dimensional quantum systems (e.g., a spin greater than 1/2) via a single observable. We determine the condition for the best determination and the upper bound to achieve the most robust measurements. From the experimental view we require a suitable interaction Hamiltonian to maximize the measure efficiency. For this we numerically investigate a three-level system. Moreover, the error analysis for the different-dimensional quantum states shows that the present measurement method is still very effective in determining an unknown state of a high-dimensional quantum system.

Physical Review A•Mar 25, 2011•5 citations

Determining the state of a single cavity mode from photon statistics

Kurt A. Jacobs, P. L. Knight, Vlatko Vedral

We present various schemes for measuring the quantum state of a single mode of the electromagnetic field. These involve measuring the photon statistics for the mode before and after an interaction with either one or two two-level atoms. The photon statistics conditioned on the final state of the atoms, for two choices of the initial set of atomic states, along with the initial photon statistics, may be used to calculate the complete quantum state in a simple manner. Alternatively, when one atom is used, two unconditioned sets of photon statistics, each after interaction with a single atom in different initial states, along with the initial photon statistics may be used to calculate the initial state in a simple manner. When the cavity is allowed to interact with just one atom, only pure cavity states which do not contain zeros in the photon distribution may be reconstructed. When two atoms are used we may reconstruct pure states which do not contain adjacent zeros in the photon distribution. Coherent states and number states are among those that may be measured with one-atom interaction, and squeezed states and ‚Schrödinger cats‘ are among those that may be measured with a two-atom interaction.

Journal of Modern Optics•Nov 1, 1997•6 citations

Qudit quantum-state tomography

Rob Thew, Kae Nemoto, Andrew G. White +1 more

Recently quantum tomography has been proposed as a fundamental tool for prototyping a few qubit quantum device. It allows the complete reconstruction of the state produced from a given input into the device. From this reconstructed density matrix, relevant quantum information quantities such as the degree of entanglement and entropy can be calculated. Generally, orthogonal measurements have been discussed for this tomographic reconstruction. In this paper, we extend the tomographic reconstruction technique to two new regimes. First, we show how nonorthogonal measurements allow the reconstruction of the state of the system provided the measurements span the Hilbert space. We then detail how quantum-state tomography can be performed for multiqudits with a specific example illustrating how to achieve this in one- and two-qutrit systems.

arXiv:quant-ph/0201052•Physical Review A•Jul 16, 2002•347 citations

Homodyne Tomography and the Reconstruction of Quantum States of Light

Giacomo Mauro D’Ariano, Lorenzo Maccone, Massimiliano Federico Sacchi

Quantum tomography is a procedure to determine the quantum state of a physical system, or equivalently, to estimate the expectation value of any operator. It consists in appropriately averaging the outcomes of the measurement results of different observables, obtained on identical copies of the same system. Alternatively, it consists in maximizing an appropriate likelihood function defined on the same data. The procedure can be also used to completely characterize an unknown apparatus. Here we focus on the electromagnetic field, where the tomographic observables are obtained from homodyne detection.

arXiv:quant-ph/0507078•IMPERIAL COLLEGE PRESS eBooks•Feb 1, 2007•8 citations

Determining a Quantum State by Means of a Single Apparatus

ARMEN E. ALLAHVERDYAN, Roger Balian, Theo M. Nieuwenhuizen

The unknown state rho of a quantum system S is determined by letting it interact with an auxiliary system A, the initial state of which is known. A one-to-one mapping can thus be realized between the density matrix rho and the probabilities of the occurrence of the eigenvalues of a single and factorized observable of S+A, so that rho can be determined by repeated measurements using a single apparatus. If S and A are spins, it suffices to measure simultaneously their z components after a controlled interaction. The most robust setups are determined in this case for an initially pure or a completely disordered state of A. They involve an Ising or anisotropic Heisenberg coupling and an external field.

arXiv:cond-mat/0311488•Physical Review Letters•Mar 25, 2004•90 citations
Given the state of a quantum system, one can calculate the expectation value of any observable of the system. However, the inverse problem of determining the state by performing different measurements is not a trivial task. In various experimental setups it is reasonably straightforward to reconstruct the state of a quantum system employing linear tomographic technique. In this way the elements of the density matrix can be linearly related to a set of measured quantities. But since different observables of a quantum system are not commuting with each other, one often has to perform series of successive measurements of observables which cannot be done simultaneously. Simultaneous measurement of observables cost less time and energy and is more beneficial. In this paper we review the strategy of quantum state tomography with simultaneous measurement of commuting observables. This can be done by introducing an assistant system of which the state is known. We show that the interaction between the assistant and the system of interest within different frame works allows the reconstruction of the state of the system. Specifically, we consider a two-level system and reconstruct its initial state by introducing an assistant which can be either another two-level system or a single cavity mode of the electromagnetic field.
The density matrix of a two-level system (spin, atom) is usually determined by measuring the three noncommuting components of the Pauli vector. This density matrix can also be obtained via the measurement data of two commuting variables, using a single apparatus. This is done by coupling the two-level system to a mode of radiation field, where the atom-field interaction is described with the Jaynes-Cummings model. The mode starts its evolution from a known coherent state. The unknown initial state of the atom is found by measuring two commuting observables: the population difference of the atom and the photon number of the field. We discuss the advantages of this setup and its possible applications.
One can uniquely identify an unknown state of a quantum system SSS by measuring a ``quorum'' consisting of a complete set of noncommuting observables. It is also possible to determine the quantum state by repeated measurements of a single and factorized observable, when the system SSS is coupled to an assistant system AAA whose initial state is known. This is because that the redistribution of the information about the unknown quantum state into the composite system A+SA+SA+S results in a one-to-one mapping between the unknown density matrix elements and the probabilities of the occurrence of the eigenvalues of a single, factorized observable of the composite system. Here we focus on quantum state tomography of high-dimensional quantum systems (e.g., a spin greater than 1/2) via a single observable. We determine the condition for the best determination and the upper bound to achieve the most robust measurements. From the experimental view we require a suitable interaction Hamiltonian to maximize the measure efficiency. For this we numerically investigate a three-level system. Moreover, the error analysis for the different-dimensional quantum states shows that the present measurement method is still very effective in determining an unknown state of a high-dimensional quantum system.
We present various schemes for measuring the quantum state of a single mode of the electromagnetic field. These involve measuring the photon statistics for the mode before and after an interaction with either one or two two-level atoms. The photon statistics conditioned on the final state of the atoms, for two choices of the initial set of atomic states, along with the initial photon statistics, may be used to calculate the complete quantum state in a simple manner. Alternatively, when one atom is used, two unconditioned sets of photon statistics, each after interaction with a single atom in different initial states, along with the initial photon statistics may be used to calculate the initial state in a simple manner. When the cavity is allowed to interact with just one atom, only pure cavity states which do not contain zeros in the photon distribution may be reconstructed. When two atoms are used we may reconstruct pure states which do not contain adjacent zeros in the photon distribution. Coherent states and number states are among those that may be measured with one-atom interaction, and squeezed states and ‚Schrödinger cats‘ are among those that may be measured with a two-atom interaction.
Recently quantum tomography has been proposed as a fundamental tool for prototyping a few qubit quantum device. It allows the complete reconstruction of the state produced from a given input into the device. From this reconstructed density matrix, relevant quantum information quantities such as the degree of entanglement and entropy can be calculated. Generally, orthogonal measurements have been discussed for this tomographic reconstruction. In this paper, we extend the tomographic reconstruction technique to two new regimes. First, we show how nonorthogonal measurements allow the reconstruction of the state of the system provided the measurements span the Hilbert space. We then detail how quantum-state tomography can be performed for multiqudits with a specific example illustrating how to achieve this in one- and two-qutrit systems.
Quantum tomography is a procedure to determine the quantum state of a physical system, or equivalently, to estimate the expectation value of any operator. It consists in appropriately averaging the outcomes of the measurement results of different observables, obtained on identical copies of the same system. Alternatively, it consists in maximizing an appropriate likelihood function defined on the same data. The procedure can be also used to completely characterize an unknown apparatus. Here we focus on the electromagnetic field, where the tomographic observables are obtained from homodyne detection.
The unknown state rho of a quantum system S is determined by letting it interact with an auxiliary system A, the initial state of which is known. A one-to-one mapping can thus be realized between the density matrix rho and the probabilities of the occurrence of the eigenvalues of a single and factorized observable of S+A, so that rho can be determined by repeated measurements using a single apparatus. If S and A are spins, it suffices to measure simultaneously their z components after a controlled interaction. The most robust setups are determined in this case for an initially pure or a completely disordered state of A. They involve an Ising or anisotropic Heisenberg coupling and an external field.