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arXiv:0910.5943 · Physical Review A

Quantum tomography via equidistant states

C. Paiva-Sánchez, Enrique B. Inostroza, O. Jiménez, A. Delgado

Published Sep 20, 2010• 29 citations
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Abstract

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 $N$ and involves the measurement of ${N}^{2}$ 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 $3{N}^{2}/2$.

DOI
10.1103/physreva.82.032115

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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.
Symmetric informationally complete positive operator-valued measures provide efficient quantum state tomography in any finite dimension.In this work, we implement state tomography using symmetric informationally complete positive operator-valued measures for both pure and mixed photonic qudit states in Hilbert spaces of orbital angular momentum, including spaces whose dimension is not power of a prime.Fidelities of reconstruction within the range of 0.81-0.96are obtained for both pure and mixed states.These results are relevant to high-dimensional quantum information and computation experiments, especially to those where a complete set of mutually unbiased bases is unknown.
We study the determination of an unknown mixed state of a ddd-dimensional quantum system by means of unambiguous state discrimination. We show that optimal and nonoptimal unambiguous state discrimination can be used to reconstruct unknown states of a qubit. This result is extended to the case of a qudit by a sequence of reconstructions in two-dimensional subspaces. The total number of projections scales approximately as 2d22{d}^{2}2d2 for ddd large; this is twice as much as in the case of tomography based on mutually unbiased bases or symmetric informationally complete positive-operator-valued measures, and less than the d3{d}^{3}d3 projections required by standard tomography.
PACS. 03.65.Ta – Foundations of quantum mechanics; measurement theory. PACS. 03.65.Wj – State reconstruction, quantum tomography. Abstract. – We propose a method for the tomographic reconstruction of qubit states for a general class of solid-state systems in which the Hamiltonians are represented by spin operators, e.g., with Heisenberg-, XXZ-, or XY-type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data. Quantum information processing requires the effective measurement of quantum states. However, a single quantum measurement can only obtain partial information of a quantum state. The reconstruction of a quantum state requires measuring a complete set of observables on an ensemble of identically prepared copies of the system. This method called quantum state tomography [1], is very important because any unknown state can be ascertained by tomographic measurements. Moreover, the full description of qubit states can increase the accuracy of quantum operations. Tomographic measurements have been experimentally implemented for, e.g., the nuclear spin state of an NMR system [2], the electromagnetic field and photon
High-dimensional quantum information processing has become a mature field of research with several different approaches being adopted for the encoding of DDD-dimensional quantum systems. Such progress has fueled the search of reliable quantum tomographic methods aiming for the characterization of these systems, most of these methods being specifically designed for a given scenario. Here, we report on a tomographic method based on multiply symmetric states and on experimental investigations to study its performance in higher dimensions. Unlike other methods, it is guaranteed to exist in any dimension and provides a significant reduction in the number of measurement outcomes when compared to standard quantum tomography. Furthermore, in the case of odd dimensions, the method requires the least possible number of measurement outcomes. In our experiment we adopt the technique where high-dimensional quantum states are encoded using the linear transverse momentum of single photons and are controlled by spatial light modulators. Our results show that fidelities of $0.984\ifmmode\pm\else\textpm\fi{}0.009$ with ensemble sizes of only $1.5\ifmmode\times\else\texttimes\fi{}{10}^{5}$ photons in dimension D=15D=15D=15 can be obtained in typical laboratory conditions, thus showing its practicability in higher dimensions.
Abstract In this paper, we establish a dynamical quantum state tomography framework. Under this framework, it is feasible to obtain complete knowledge of any unknown state of a d-level system via only one operator of a special type of positive operator-valued measure (POVM) in dimension d. We define a new channel, referred to as the time-dependent average channel. Utilizing this channel, we show that we can acquire a collection of projective operators that is informationally complete (IC) and thus obtain the corresponding informationally complete POVMs (IC-POVMs). Zauner conjectured that for any dimension d there exists a fiducial vector, such that all remaining d 2 − 1 elements of the desired symmetric informationally complete POVM (SIC-POVM) can be obtained by acting on said vector with unitary matrices representing elements of the Weyl-Heisenberg group. We show that under certain condition, it is possible to obtain infinite families of projective operators that are IC, and obtain infinite families of corresponding IC-POVMs; otherwise, Zauner’s conjecture is incorrect. We also show how to simulate a SIC-POVM on any unknown quantum state by using the time-dependent average channel.