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Conference on Lasers and Electro-Optics

Optimal Experiment Design for Minimal Tomography

J. Nunn, Brian J. Smith, Graciana Puentes, Jeff S. Lundeen, Ian A. Walmsley

Published Jan 1, 2010• 0 citations
Publisher

Abstract

Given an experimental set-up and a fixed number of measurements, how should one take data in order to optimally reconstruct the state of a quantum system? We show how to calculate the optimal design explicitly.
DOI
10.1364/qels.2010.qff2

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Given an experimental setup and a fixed number of measurements, how should one take data to optimally reconstruct the state of a quantum system? The problem of optimal experiment design (OED) for quantum state tomography was first broached by Kosut et al. [R. Kosut, I. Walmsley, and H. Rabitz, e-print arXiv:quant-ph/0411093 (2004)]. Here we provide efficient numerical algorithms for finding the optimal design, and analytic results for the case of `minimal tomography'. We also introduce the average OED, which is independent of the state to be reconstructed, and the optimal design for tomography (ODT), which minimizes tomographic bias. Monte Carlo simulations confirm the utility of our results for qubits. Finally, we adapt our approach to deal with constrained techniques such as maximum-likelihood estimation. We find that these are less amenable to optimization than cruder reconstruction methods, such as linear inversion.
Quantum state tomography is an indispensable but costly part of many quantum experiments. Typically, it requires measurements to be carried out in a number of different settings on a fixed experimental setup. The collected data are often informationally overcomplete, with the amount of information redundancy depending on the particular set of measurement settings chosen. This raises a question about how one should optimally take data so that the number of measurement settings necessary can be reduced. Here, we cast this problem in terms of integer programming. For a given experimental setup, standard integer-programming algorithms allow us to find the minimum set of readout operations that can realize a target tomographic task. We apply the method to certain basic and practical state-tomographic problems in nuclear-magnetic-resonance experimental systems. The results show that considerably fewer readout operations can be found using our technique than by using the previous greedy search strategy. Therefore, our method could be helpful for simplifying measurement schemes to minimize the experimental effort.
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.
In the absence of experimental constraints, optimal measurement schemes for quantum state tomography are well understood. We consider the scenario where the experimenter doesn't have arbitrary freedom to construct their measurement set, and may therefore not be able to implement a known optimal scheme. We introduce a simple procedure for minimizing the uncertainty in the reconstructed quantum state for an arbitrary tomographic scheme. We do this by defining a figure of merit based on the equally weighted variance of the measurement statistics. This figure of merit is straightforwardly based on the singular value decomposition of the measurement matrix, making it well suited for optimization.