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Choice of Measurement Sets in Qubit Tomography

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Choice of Measurement Sets in Qubit Tomography PHYSICAL REVIEW A 78, 052122 ͑2008͒ Choice of measurement sets in qubit tomography Mark D. de Burgh,1 Nathan K. Langford,1 Andrew C. Doherty,1 and Alexei Gilchrist2 1School of Physical Sciences, University of Queensland, St Lucia, Queensland 4072, Australia 2Physics Department, Macquarie University, Sydney NSW 2109, Australia ͑Received 5 July 2007; revised manuscript received 20 October 2008; published 26 November 2008͒ Optimal generalized measurements for state estimation are well understood. However, practical quantum state tomography is typically performed using a fixed set of projective measurements, and the question of how to choose these measurements has been largely unexplored in the literature. In this work, we develop theoret- ical asymptotic bounds for the average fidelity of pure qubit tomography using measurement sets whose axes correspond to faces of Platonic solids. We also present comprehensive simulations of maximum likelihood tomography for mixed qubit states using the Platonic solid measurements. We show that overcomplete mea- surement sets can be used to improve the accuracy of tomographic reconstructions. DOI: 10.1103/PhysRevA.78.052122 PACS number͑s͒: 03.65.Ta, 03.67.Ϫa I. INTRODUCTION AND BACKGROUND In this paper, we investigate how the choice of measure- ments affects the quality of tomographic reconstruction for Quantum tomography ͓1͔, the practical estimation of qubit systems. We follow Jones ͓17͔ and investigate a class quantum states through the measurement of large numbers of of measurement sets based on Platonic solids. This class copies, is of fundamental importance in the study of quantum gives close-to-optimal performance for tomography when mechanics. With the emergence of quantum-information sci- using independent measurements of fixed projectors by ence, the tomographic reconstruction of finite-dimensional spreading the projectors uniformly over the surface of the systems ͓2͔ has also become an essential technology for Bloch sphere. The different Platonic solids allow us to inves- characterizing the experimental performance of practical tigate how the number of measurement settings affects the quantum gates and state preparation. Examples include to- tomographic reconstruction quality, and we show that the mography of the polarization states of light ͓3,4͔ and elec- more overcomplete sets produce the best results. We begin tronic states of trapped ions ͓5,6͔. It is important for these by deriving analytical bounds for the performance of Pla- experiments that a systematic study is carried out to deter- tonic solid measurements in the tomography of pure states. mine efficient practical measurement and state estimation We then extend this analysis to systems of one- and two- strategies. qubit mixed states by performing comprehensive numerical There has already been much theoretical work in this area, simulations of the tomographic procedure for the two impor- and optimal bounds on state estimation as well as construc- tant cases of photonic and atomic experiments. We present tions for measurements that achieve these bounds are known our results as a function of the total number of copies of the ͓7–14͔. However, these bounds require all copies of the state quantum state used in the experiment. This work provides a to be collected and a combined measurement performed detailed study of how information is acquired during a to- across all the copies. While these collective measurements mography experiment. To support these results, it would also are known to be more powerful than performing independent be very interesting to perform such a study in some common measurements on each copy ͓8͔, they are totally infeasible experimental systems, because it would determine whether for any existing experimental implementation. the quality of the reconstructions was limited by the expected Experimental tomography in discrete quantum systems is statistics or by other experimental imperfections, such as currently performed using independent measurements in drifting sources. To the best of our knowledge, such a study fixed sets of projectors ͓15,16͔, but it remains an important has still not been performed. open question as to which fixed sets are best to use ͑although Platonic solid measurements were first proposed for to- there are several relevant theoretical discussions ͓13,17–25͔͒. mography in ͓17͔, where analytical asymptotic bounds on the For mainly historical reasons, measurement sets such as reconstruction performance were calculated for the special those described in ͓15͔ have become popular, particularly in case of pure states using mutual information as the figure of optical experiments ͑except, e.g., ͓4,26͔͒. These measure- merit. It is difficult to compare these results with more recent ment sets are attractive since they have the minimal number work, however, where the most commonly used figure of of settings, and in these experiments a significant amount of merit is the average fidelity. We therefore derive new time can be spent changing the measurement settings. How- asymptotic bounds for the pure-state case based on the aver- ever, such decisions are generally made without a quantita- age fidelity. Much of the original interest in fidelity arises tive understanding of how the resulting tomographic perfor- from the fact that it bounds the distinguishability of quantum mance is affected. For example, in Fig. 1 we show that both states; see ͓27͔. Recently, a true quantum Chernoff bound on the average and worst-case fidelities of the tomographic re- the distinguishability of states has been introduced ͓28͔, and construction are greatly improved by replacing the popular this quantity is a tighter bound than the fidelity. We therefore minimal measurement set of ͓15͔ with an overcomplete set also consider this quantity as a figure of merit and show that of six measurements corresponding to the six faces of a cube. the results do not depend greatly on this choice. 1050-2947/2008/78͑5͒/052122͑15͒ 052122-1 ©2008 The American Physical Society DE BURGH et al. PHYSICAL REVIEW A 78, 052122 ͑2008͒ FIG. 1. ͑Color online͒ Performance of tomographically reconstructed states on the Bloch sphere for two measurement sets. The top set is a popular set of measurements ͑James4͒ used in the literature ͓15͔, the bottom set is a more isotropic set composed of measurements along the three spatial axes ͑a cube measurement set͒; see text for details. For each target state, the average fidelity with 400 reconstructed states is plotted by color. The ensemble of reconstructed states was generated by adding Possonian noise to a simulated experiment with an average total of 4000 counts, and then performing maximum-likelihood tomography. Very recently, Roy and Scott ͓21͔ were able to derive includes a comparison with recent analytical results based on results about mixed-state reconstruction performance by us- the mean-squared Hilbert-Schmidt distance in Sec. VID. ing another figure of merit, the mean-squared Hilbert- Schmidt distance, choosing this measure specifically because II. TOMOGRAPHIC RECONSTRUCTION AND it allowed tractable analytical solutions. In this way, they RECONSTRUCTION QUALITY identified a class of measurements, including all the Platonic solid measurements, that provide optimal reconstruction per- A d-dimensional quantum state ␳ is represented by a formance for qubits. In contrast, by using the physically dϫd positive semidefinite density matrix, with trace 1. well-motivated average fidelity and quantum Chernoff bound There are d2 −1 real parameters to be estimated. In the case figures of merit for numerical simulations, we show that of optical experiments such as ͓3,4͔, the flux must also be there is a range of performance within the class of Platonic estimated giving d2 real parameters to be estimated. solids, with the higher-order solids performing better. Per- Any setting l of an experimental apparatus designed to haps surprisingly, this indicates that the choice of figure of measure the quantum state may be described by a positive merit can have a qualitative effect on the comparison be- operator value measure ͑POVM͒. Each of the k outcomes of tween tomographic procedures. We will briefly discuss the a measurement setting is represented by a positive semidefi- ͚ ramifications and possible causes of this result. nite operator Olk. The operators satisfy kOlk=I. The prob- The structure of the paper is as follows. In Sec. II, we ability of observing outcome k is given by the Born rule ͑␳ ͒ introduce the tomography problem, discussing how to assess plk=tr Olk . In state tomography, each of the l measure- the performance of tomographic reconstructions. We then de- ments is performed on a large number of copies and esti- fine various relevant one- and two-qubit measurement sets in mates obtained for each of the plk. We denote these estimates Sec. III, including Platonic solid measurements and some pˆ lk. When the number of linearly independent Olk equals the others that have been studied in the literature for comparison number of parameters to be estimated, our measurements are purposes. In Sec. IV, we present some mostly analytical re- known as informationally complete ͓29͔. If these probability sults that motivate our more complete numerical investiga- estimates were perfect, so that pˆ lk=plk, it would be possible tions and illustrate some interesting effects that should be to reconstruct the state exactly. In this case, there is a set of studied experimentally. In particular, Sec. IV B provides operators Rlk, known as a dual basis or dual frame, such that ␳ ͚ ͑ some important intuition about the complications that arise = lkplkRlk. When the number of measurements Olk in an when reconstructing mixed states. We then describe in detail informationally complete POVM exceeds the number of pa- our numerical mixed-state simulations for atomic and photo- rameters to be estimated, the dual frame is not unique.͒ Since nic qubit systems in Sec. V and present the results for both our probability estimates are not perfect due to the finite the fidelity and the quantum Chernoff bound in Sec. VI.
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