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Journal of Modern Optics

Quantum and classical tomography with equidistant reference angles

Ulf Leonhardt

Published Nov 1, 1997• 11 citations
Publisher

Abstract

Concepts of quantum tomography are useful for classical image reconstructions as well. The overall error caused by a finite number of equidistant reference phases is quantified in terms of the density matrix in Fock representation. Moreover, an exact condition for perfect reconstruction is derived.
DOI
10.1080/09500349708231883

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We study the possibility of performing quantum state tomography via equidistant states. This class of states allows us to propose a nonsymmetric informationally complete Positive Operator Valued Measure (POVM) based tomographic scheme. The scheme is defined for odd dimensions NNN and involves the measurement of N2{N}^{2}N2 transition probabilities and an inversion, which can be analytically carried out by Fourier transform. The scheme can be modified to allow the reconstruction of states in the case of even dimensions at the expense of increasing the number of measurements to 3N2/23{N}^{2}/23N2/2.
This chapter begins with a review of classical tomography, which reconstructs hidden structures inside an object by successive transmissions of an X-ray beam at different angles and lateral displacements. By interpreting the Wigner distribution as an analogue of the density distribution in a physical object, the mathematical methods of classical tomography — the Radon transform, the Fourier slice theorem, and the inverse Radon transform — are adapted to perform a high resolution determination of a quantum state of light. In optical homodyne tomography, the classical transmission angle is replaced by the adjustable phase of a local oscillator, and the successive lateral displacements are replaced by a series of measurements of the quadrature operator defined by the local oscillator phase. Experimental data is presented showing various properties of the Wigner distribution for a coherent state.
Quantum state tomography (QST) refers to any method that allows one to reconstruct the accurate representation of a quantum system based on data obtainable from an experiment. In this paper, we concentrate on theoretical methods of quantum tomography, but some significant experimental results are also presented. Due to a considerable body of literature and intensive ongoing research activity in the field of QST, this overview is restricted to presenting selected ideas, methods, and results. First, we discuss tomography of pure states by distinguishing two aspects—complex vector reconstruction and wavefunction measurement. Then, we move on to the Wigner function reconstruction. Finally, the core section of the article is devoted to the stroboscopic tomography, which provides the optimal criteria for state recovery by including the dynamics in the scheme. Throughout the paper, we pay particular attention to photonic tomography, since multiple protocols in quantum optics require well-defined states of light.
Quantum computation depends on quantum entanglement, a correlation between subsystems that cannot occur classically. A variety of theoretical measures exist for quantifying the degree entanglement in such schemes, all of which are functions of the system density matrix. How can the entanglement be measured experimentally? Using quantum tomography techniques developed for two photon entangled states, the density matrix can be reconstructed from the appropriate experimental data. In this case the state tomography gives the complete characterization of the physical system (for the relevant degree of freedom, such as spin or polarization). It gives information on both the degree of nonclassical correlation, that is entanglement, as well as the amount of decoherence in the system. In this proceedings we discuss the general state tomography procedure required to characterize a few qubit quantum computer, for any architecture.