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arXiv:1312.1119 · Physical Review A

Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure

Wanderson M. Pimenta, Breno Marques, Thiago O. Maciel, Reinaldo O. Vianna, A. Delgado, C. Saavedra, Sebastião Pádua

Published Jul 15, 2013• 48 citations
arXivHTMLPDF

Abstract

An experimental demonstration of two-qutrit state tomography via one-qutrit symmetric informationally complete positive operator-valued measure (SIC-POVM) is presented. A two-qutrit state is encoded in the transversal paths of a spontaneous parametric down-converted photon pair. A spatial light modulator allows us to implement the necessary 81 POVM elements to reconstruct the state. The quality of the reconstruction is evaluated by comparing independent measurements with the predicted results calculated with the reconstructed state. Entanglement in the system is calculated via negativity and generalized robustness from the two-qutrit reconstructed density operator.

DOI
10.1103/physreva.88.012112

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An experimental demonstration of two-qutrit state tomography via one-qutrit symmetric informationally complete positive operator-valued measure (SIC-POVM) is presented. A two-qutrit state is encoded in the transversal paths of a spontaneous parametric down-converted photon pair. A spatial light modulator allows us to implement the necessary 81 POVM elements to reconstruct the state. The quality of the reconstruction is evaluated by comparing independent measurements with the predicted results calculated with the reconstructed state. Entanglement in the system is calculated via negativity and generalized robustness from the two-qutrit reconstructed density operator.
It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose is symmetric informationally complete positive operator-valued measurements (SIC-POVMs). SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliary system and by performing a projective measurement on the joint system. In practice, finding the appropriate coupling is rather nontrivial. Here we propose an experimental design for directly implementing SIC-POVMs using multiport devices and path-encoded qubits and qutrits, the utility of which has recently been demonstrated by several experimental groups around the world. Furthermore, we describe how these multiports can be attained in practice with an integrated photonic system composed of nested linear optical elements.
Generalized quantum measurements [also known as positive operator-valued measures (POVMs)] are of great importance in quantum information and quantum foundations but are often difficult to perform. We present an experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context. One of the most interesting POVMs, the symmetric informationally complete (SIC) POVM, is the most compact set of measurements that can be used to fully describe a quantum state. We use our technique to carry out the first experimental characterization of the state of a qutrit using SIC POVMs. Because of the highly symmetric nature of this measurement, such a representation has the unique property that it permits all other measurement outcomes to be predicted by a simple extension of the classical Bayesian sum rule, making no use of complex amplitudes or Hilbert-space operators. We demonstrate this approach on several qutrit states encoded in single photons.
Generalized quantum measurements (also known as positive operator-valued measures or POVMs) are of great importance in quantum information and quantum foundations, but often difficult to perform. We present an experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context. One of the most interesting POVMs, the symmetric, informationally complete-POVM (or SIC-POVM), is the most compact set of measurements that can be used to fully describe a quantum state. We use our technique to carry out the first experimental characterization of the state of a qutrit using SIC-POVMs. Because of the highly symmetric nature of this measurement, such a representation has the unique property that it permits all other measurement outcomes to be predicted by a simple extension of the classical Bayesian sum rule, making no use of complex amplitudes or Hilbert-space operators. We demonstrate this approach on several qutrit states encoded in single photons.
It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose are symmetric, informationally complete POVMs. SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliary system and performing a projective measurement on the joint system. In practice, finding the appropriate coupling is rather non-trivial. Here we propose an experimental design for directly implementing SIC-POVMs using multiport devices and path-encoded qubits and qutrits, the utility of which has recently been emphasized by the Zeilinger group in Vienna. Furthermore, we describe how these multiports can be attained in practice with an integrated photonic system composed of nested linear optical elements.
Quantum measurements play a fundamental role in quantum mechanics. Especially, generalized quantum measurements provide a powerful and versatile tool to extract information from quantum systems. However, how to realize them on an arbitrary higher-dimensional quantum system remains a challenging task. Here we propose a simple recipe for the implementation of a general positive-operator valued measurement (POVM) on a higher-dimensional quantum system via a one-dimensional discrete-time quantum walk with a two-dimensional coin. Furthermore, using single photons and linear optics, we realize experimentally a symmetric, informationally complete (SIC) POVM on a three-dimensional system with high fidelity. As an application, we realize a qutrit state tomography with SIC-POVM and confirm that the infidelity scaling achieved by the qutrit SIC-POVM is as good as that based on mutually unbiased bases. Our study paves the way to explore physics and information in higher-dimensional quantum systems and finds applications in various quantum information processing tasks that rely on generalized quantum measurements.
We present an experimental technique for a complete characterization of entanglement in a two-qutrit state generated using transverse spatial correlations of two parametric down-converted photons. We verify entanglement for a particular case via entanglement witness operators which are decomposed into a sum of local observables of single path and superposition projection operators. Experimentally, these operators are accomplished by using a spatial light modulator and a polarizing beam splitter which allow to modulate the amplitude of individually chosen path states. The quantification of entanglement is computed by the negativity obtained from the expectation values of the entanglement witnesses implemented.