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Optics Communications

State reconstruction of a qutrit by a minimal set of discrete measurements

Peter A. Ivanov, Nikolay V. Vitanov

Published Jun 16, 2006• 7 citations
Publisher
DOI
10.1016/j.optcom.2005.12.080

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Spatial qudit states can be realized by using multislits to discretize the transverse momentum of a photon. The merit of this kind of spatial qudit states is that the implementation of higher-dimensional qudits is relatively easy. As we have recently shown, the quantum states of these spatial qudits can be analyzed by scanning a single interference pattern. This method of single scan tomography can also be applied at higher dimensions, but the reconstruction becomes more sensitive to smaller details of the scanned patterns as the dimensions increase. In this paper, we investigate the effect of finite measurement resolution on the single scan tomography of spatial qutrits. Realistic measurement operators describing the spatial resolution of the measurement are introduced and the corresponding pattern functions for quantum state reconstruction are derived. We use the pattern functions to analyze experimental results for entangled pairs of spatial qutrits generated by spontaneous parametric downconversion. It is shown that a reliable reconstruction of the quantum state can be achieved with finite measurement resolution if this limitation of the measurement is included in the pattern functions of single scan tomography.
Based on mutually unbiased measurements, an optimal tomographic scheme for the multiqutrit states is presented explicitly. Because the reconstruction process of states based on mutually unbiased states is free of information waste, we refer to our scheme as the optimal scheme. By optimal we mean that the number of the required conditional operations reaches the minimum in this tomographic scheme for the states of qutrit systems. Special attention will be paid to how those different mutually unbiased measurements are realized; that is, how to decompose each transformation that connects each mutually unbiased basis with the standard computational basis. It is found that all those transformations can be decomposed into several basic implementable single- and two-qutrit unitary operations. For the three-qutrit system, there exist five different mutually unbiased-bases structures with different entanglement properties, so we introduce the concept of physical complexity to minimize the number of nonlocal operations needed over the five different structures. This scheme is helpful for experimental scientists to realize the most economical reconstruction of quantum states in qutrit systems.
Tomography of polarization qutrits in a measurement scheme in a natural polarization basis with a minimum number of tunable elements has been investigated. The information completeness of protocols for states of various ranks has been analyzed theoretically using the analysis of the information matrix and compressive tomography methods was carried out; the results have been confirmed experimentally. The existence of a protocol with five dimensions that provides a complete reconstruction of arbitrary (including mixed) states has been shown, and it has been found that schemes with a smaller number of dimensions provide the complete reconstruction of pure states. The obtained protocols demonstrate high accuracy of reconstructing test states under the conditions of a limited set of rotations of the measuring basis.