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arXiv:quant-ph/9911068 · Physical Review A

Reconstruction of the spin state

Z. Hradil, Johann Summhammer, G. Badurek, H. Rauch

Published Jun 9, 2000• 43 citations
arXivHTMLPDF

Abstract

An ensemble of spin-$\frac{1}{2}$ particles is observed repeatedly using Stern-Gerlach devices with varying orientations. Synthesis of such noncommuting observables is analyzed using the maximum likelihood estimation as an example of quantum-state reconstruction. Repeated incompatible observations represents a new generalized measurement. This idealized scheme will serve for analysis of future experiments in neutron and quantum optics.

DOI
10.1103/physreva.62.014101

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Abstract. The theory of quantum state reconstruction is illustrated here on sev-eral examples taken from modern experimental praxis. Maximum-likelihood estima-tion is applied to experiments on physical systems of increasing complexity, starting with a simple one-dimensional problem of quantum phase estimation, continuing with the absorption and phase neutron tomographies, further discussing quantum tomography of higher-dimensional discrete quantum systems, and closing with the homodyne tomography of an infinite dimensional system- a mode of light. All these experiments nicely demonstrate the utility of present state-of-art techniques for ma-nipulating states of a neutron and internal as well as external states of a photon. Over the past hundred years, quantum theory proved to be an extremely useful tool for describing and understanding nature on a microscopic level. Its usual applications consist in predicting the outcomes of future experimental results given a quantum state describing the physical system.

•Nov 14, 2015•0 citations
An ensemble of spin-12\frac{1}{2}21​ particles is observed repeatedly using Stern-Gerlach devices with varying orientations. Synthesis of such noncommuting observables is analyzed using the maximum likelihood estimation as an example of quantum-state reconstruction. Repeated incompatible observations represents a new generalized measurement. This idealized scheme will serve for analysis of future experiments in neutron and quantum optics.
Maximum-likelihood estimation is applied to the determination of an unknown quantum measurement. The calibrated measuring apparatus carries out measurements on many different quantum states and the positive operator-valued measure governing the measurement statistics is then inferred from the collected data via the maximum-likelihood principle. In contrast to the procedures based on linear inversion, our approach always provides a physically sensible result. We illustrate the method on the case of the Stern-Gerlach apparatus.
The question of what information about a quantum state may be inferred from a sequence of measurements made on it, is addressed. The main result is that maximum-likelihood estimation gives an arguably natural optimum approach in quantum theory. It singles out a particular choice of basis, in which the state is expressed. Various incompatible and incomplete observations made on the ensemble are treated as a complete generalized measurement. Consequently, the maximum likelihood (relative entropy) is the best measure for relating experimental data with theoretical predictions of quantum theory.
Quantum theory allows the sequential measurement of incompatible observables in the course of repeated measurements. To get information about the observed system, all observations must be synthesized. This is the main idea of quantum tomographical methods. As shown in this paper, the maximum likelihood principle provides the best measure for relating the experimental data with predictions of quantum theory. Synthesis of incompatible observations appears to be a new quantum measurement described by a positive operator-valued measure. Besides this, the procedure finds the optimal state of the system, which best fits such a measurement.
The measurement of spin-1/2 observables using a Stern-Gerlach type device is studied. A magnetic field with a dominant dipole component leads to a measurement of spin in one direction. The measurement is shown to be feasible (in principle) for electrons as well as for neutral particles. A quadrupole field leads to a joint measurement of two incompatible spin observables. Again, both the electron and the neutral case are presented. The accuracy of the joint measurement is compared to the limit imposed by the uncertainty principle.
Abstract. The theory of quantum state reconstruction is illustrated here on sev-eral examples taken from modern experimental praxis. Maximum-likelihood estima-tion is applied to experiments on physical systems of increasing complexity, starting with a simple one-dimensional problem of quantum phase estimation, continuing with the absorption and phase neutron tomographies, further discussing quantum tomography of higher-dimensional discrete quantum systems, and closing with the homodyne tomography of an infinite dimensional system- a mode of light. All these experiments nicely demonstrate the utility of present state-of-art techniques for ma-nipulating states of a neutron and internal as well as external states of a photon. Over the past hundred years, quantum theory proved to be an extremely useful tool for describing and understanding nature on a microscopic level. Its usual applications consist in predicting the outcomes of future experimental results given a quantum state describing the physical system.