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AIP conference proceedings

Quantum Tomography of Light Polarization: Fundamentals and Information Aspects

V. P. Karassiov

Published Jan 1, 2007• 0 citations
Publisher

Abstract

Methods of quantum tomography of light polarization are discussed as indirect measurements of the polarization density operator R̂ describing only polarization properties of quantum light. They are shown to be adequately formulated by means of quasi — spectral expansions of R̂ in partially‐orthogonal projectors determining spectral resolutions of some tomographic observables (“measurers” of R̂). Probabilistic interpretations of results obtained are briefly discussed.

DOI
10.1063/1.2713474

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Methods of quantum tomography of light polarization are discussed as indirect measurements of the polarization density operator R̂ describing only polarization properties of quantum light. They are shown to be adequately formulated by means of quasi — spectral expansions of R̂ in partially‐orthogonal projectors determining spectral resolutions of some tomographic observables (“measurers” of R̂). Probabilistic interpretations of results obtained are briefly discussed.
Quantum-state tomography is proposed as a means to achieve a complete measurement of the quantum polarization state of a light wave. A set of measurements using dual-polarization balanced homodyne detection is shown to be tomographi-cally complete with respect to the statistics of the SU(2) Stokes operators on the Poincaré sphere. Complete reconstruction of the polarization sector of the density matrix of a partially polarized optical field can be achieved while randomizing the overall phase of the dual-polarization local oscillator.
Conceptual and operational aspects of the description of the polarization of light in classical and quantum optics are discussed. Primary emphasis is placed on the concept of P -quasispin, which ensures an adequate analysis for arbitrary quantum light fields. In terms of this concept, the polarization density matrix, which is obtained by reduction of the total field density operator and which completely describes the polarization state of quantum radiation, as well as the operational (measured in polarization experiments) characteristics and the transformations of the polarization dynamics, are determined. Applications of the formalism developed to biphoton optics are discussed.
One of the fundamental problems of experimental quantum physics is the determination of the state of a system, which provides its accurate description and contains all the available information about it. The standard solution is quantum state tomography, a technique that employs a large number of measurements of complementary observables to reconstruct a general density operator. Recently, new protocols have been proposed which allow to obtain the matrix elements one by one with fewer and less complicated observations by exploiting so-called weak measurements, that perturb the system less than ideal projections at the cost of being less precise. This thesis presents an evolution of these schemes that eschews the weakness of the measurement, thus gaining in the accuracy of the results and in practical feasibility. We realized an experiment that uses these methods to reconstruct the density operator of the polarization state of light in single photons regime.
This chapter provides the relevant theory and techniques of quantum tomography. Quantum tomography aims to determine the state of the system by determining its density matrix by means of a series of measurements done on fairly a large number of identically prepared copies of the same system. Wolfgang Ernst Pauli problem asks the question whether measurements of probability densities of position and momentum of a particle determine its quantum state wave function. The chapter outlines the optical homodyne tomography technique of reconstruction of Wigner function. Polarized light wave is described by the commonly used Stokes parameters introduced by George Gabriel Stokes in 1852. Quantum process tomography (QPT) aims to give full information on the dynamics of a quantum state. QPT can be used to improve the design and control of quantum hardware. Direct characterization of quantum dynamics differs in many ways from standard quantum process tomography and ancilla-assisted process tomography.
In standard optical tomographic methods, the off-diagonal elements of a density matrix ρ are measured indirectly. Thus, the reconstruction of ρ, even if it is based on linear inversion, typically magnifies small errors in the experimental data. Recently, an optimal tomography solution measuring all the elements of ρ one-by-one without error magnification has been theoretically proposed. We implemented this method for two-qubit polarization states. For comparison, we also experimentally implemented other well-known tomographic protocols, either based solely on local measurements (of, e.g., the Pauli operators and James-Kwiat-Munro-White projectors) or with mutually unbiased bases requiring both local and global measurements. We reconstructed seventeen separable, partially and maximally entangled two-qubit polarization states. Our experiments show that our method has the highest stability against errors in comparison to other quantum tomographies. In particular, we demonstrate that each optimally-reconstructed state is embedded in an uncertainty circle of the smallest radius, both in terms of trace distance and disturbance. We explain how to experimentally estimate uncertainty radii for all the implemented tomographies and show that, for each reconstructed state, the relevant uncertainty circles intersect indicating the approximate location of the corresponding physical density matrix.
The works devoted to studying the polarization properties of a two-photon light generated upon spontaneous parametric down-conversion in the collinear frequency-degenerate regime are briefly overviewed, with emphasis on the studies carried out by us over a period from 1999 to 2001 within the framework of the project “Polarization Optics of Biphotons” of the Russian Foundation for Basic Research. In particular, the polarization state of a two-photon light was analyzed and its pictorial mapping onto the Poincaré sphere was proposed. The experiments on polarization transformations of a two-photon light were performed; based on these transformations, a method was suggested for ternary quantum information coding. A two-photon state with the orthogonal photon polarizations was synthesized experimentally from the two beams of identically polarized correlated photons, and the spectral properties of this state were investigated. Finally, a method was suggested for measuring the polarization state of a two-photon light in the collinear frequency-degenerate case (“tomography”).