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arXiv:0807.0991

Accuracy of minimal and optimal qubit tomography for finite-length experiments

Alexander E. Ling, Lamas-Linares, Antia, Kurtsiefer, Christian

Published Jul 7, 2008• 7 citations
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Abstract

Practical quantum state tomography is usually performed by carrying out repeated measurements on many copies of a given state. The accuracy of the reconstruction depends strongly on the dimensionality of the system and the number of copies used for the measurements. We investigate the accuracy of an experimental implementation of a minimal and optimal tomography scheme for one- and two-qubit states encoded in the polarization of photons. A suitable statistical model for the attainable accuracy is introduced.

DOI
10.48550/arxiv.0807.0991

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Practical quantum state tomography is usually performed by carrying out repeated measurements on many copies of a given state. The accuracy of the reconstruction depends strongly on the dimensionality of the system and the number of copies used for the measurements. We investigate the accuracy of an experimental implementation of a minimal and optimal tomography scheme for one- and two-qubit states encoded in the polarization of photons. A suitable statistical model for the attainable accuracy is introduced.
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We report an experimental realization of adaptive Bayesian quantum state tomography for two-qubit states. Our implementation is based on the adaptive experimental design strategy proposed in the work by Husz\'ar and Houlsby [F. Husz\'ar and N. M. T. Houlsby, Phys. Rev. A 85, 052120 (2012).] and provides an optimal measurement approach in terms of the information gain. We address the practical questions which one faces in any experimental application: the influence of technical noise and the behavior of the tomographic algorithm for an easy-to-implement class of factorized measurements. In an experiment with polarization states of entangled photon pairs, we observe a lower instrumental noise floor and superior reconstruction accuracy for nearly pure states of the adaptive protocol compared to a nonadaptive protocol. At the same time, we show that for the mixed states, the restriction to factorized measurements results in no advantage for adaptive measurements, so general measurements have to be used.
A fundamental difficulty in demonstrating quantum state tomography is that the required resources grow exponentially with the system size. For pure states and nearly pure states, the task of tomography can be more efficient. We proposed two methods for state reconstruction, by (1) minimizing entropy and (2) maximizing likelihood. The algorithm of compressed sampling is employed to solve the optimization problem. Experiments are demonstrated considering 4-qubit photonic states. The results show that (1) much fewer measurements than the standard tomography are sufficient to obtain high fidelity, and (2) the method of maximizing likelihood is more accurate and noise robust than the original reconstruction method of compressed sampling. Furthermore, the physical meaning of the methods of minimizing entropy and maximizing likelihood is clear.
An operational method has been proposed for estimating the efficiency of quantum cryptography protocols for quantum states with discrete variables. The method is based on the estimate of the quality of the protocol by means of a universal asymptotic distribution of the characteristic of the accuracy of the reconstruction of the quantum state, which is called fidelity. For a specially designed measurement matrix, the condition number, which is minimal for the optimal protocol, is introduced. The method has been tested in a numerical simulation and real experiments with polarization qubits. It has been shown that the optimal choice of a polarization transformer makes it possible to strongly improve the quality of the reconstruction of states.