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arXiv:1608.07983

Data processing for qubit state tomography: An information geometric approach

Akio Fujiwara, Koichi Yamagata

Published Aug 29, 2016• 1 citation
arXivHTMLPDF

Abstract

A statistically feasible data post-processing method for the conventional qubit state tomography is studied from an information geometrical point of view. It is shown that the space $(-1,1)^3$ of the Stokes parameters $(ξ_1, ξ_2,ξ_3)$ that specify qubit states should be regarded as a Riemannian manifold endowed with a metric $g_{ij}:=δ_{ij}/(1-(ξ_i)^2)$, and that the data processing based on the maximum likelihood method is realized by the orthogonal projection from the empirical distribution onto the Bloch sphere with respect to the metric $g_{ij}$. An efficient algorithm for computing the maximum likelihood estimate is also proposed.

DOI
10.48550/arxiv.1608.07983

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A statistically feasible data post-processing method for the conventional qubit state tomography is studied from an information geometrical point of view. It is shown that the space (−1,1)3(-1,1)^3(−1,1)3 of the Stokes parameters (ξ1,ξ2,ξ3)(ξ_1, ξ_2,ξ_3)(ξ1​,ξ2​,ξ3​) that specify qubit states should be regarded as a Riemannian manifold endowed with a metric gij:=δij/(1−(ξi)2)g_{ij}:=δ_{ij}/(1-(ξ_i)^2)gij​:=δij​/(1−(ξi​)2), and that the data processing based on the maximum likelihood method is realized by the orthogonal projection from the empirical distribution onto the Bloch sphere with respect to the metric gijg_{ij}gij​. An efficient algorithm for computing the maximum likelihood estimate is also proposed.
A new qubit tomography protocol is introduced, based on a continuous positive operator valued measure, which is supported by the set of pure states, and equivariant under the symmetry group SO(3,R) of the qubit state space. Thus the sample data for the tomography protocol is a discrete set of points on the Bloch sphere obtained by a series of independent measurements on identical replicas of the qubit. Although the maximum likelihood estimator uniquely exists, it cannot be explicitly calculated. Instead, we present another convenient, simple estimator, show that it is unbiased and calculate its variance. Furthermore, a detailed account is given about known properties of related discrete qubit state estimation protocols (like ``minimal qubit tomography''), and the results are compared, discussed.
Quantum state tomography is the experimental procedure of determining an unknown state. It is not only essential for the verification of resources and processors of quantum information but is also important in its own right with regard to the foundation of quantum mechanics. Standard methods have been elusive for large systems because of the enormous number of observables to be measured and the exponential complexity of data post-processing. Here, we propose a new scheme of quantum state tomography that requires the measurement of only three observables (acting jointly on the system and pointer) regardless of the size of the system. The system is coupled to a "pointer" of single qubit, and the wavefunction of the system is "reaped" onto the pointer upon the measurement of the system. Subsequently, standard two-state tomography on the pointer and classical post-processing are used to reconstruct the quantum state of the system. We also developed an efficient and scalable iterative maximum likelihood algorithm to estimate states from statistically incomplete data.
Abstract As quantum tomography is becoming a key component of the quantum engineering toolbox, there is a need for a deeper understanding of the multitude of estimation methods available. Here we investigate and compare several such methods: maximum likelihood, least squares, generalised least squares, positive least squares, thresholded least squares and projected least squares. The common thread of the analysis is that each estimator projects the measurement data onto a parameter space with respect to a specific metric, thus allowing us to study the relationships between different estimators. The asymptotic behaviour of the least squares and the projected least squares estimators is studied in detail for the case of the covariant measurement and a family of states of varying ranks. This gives insight into the rank-dependent risk reduction for the projected estimator, and uncovers an interesting non-monotonic behaviour of the Bures risk. These asymptotic results complement recent non-asymptotic concentration bounds of Guta et al (2018 (arXiv:1809.11162)) which point to strong optimality properties, and high computational efficiency of the projected linear estimators. To illustrate the theoretical methods we present results of an extensive simulation study. An app running the different estimators has been made available online.
The study of Riemannian geometry offers a fundamental framework for understanding the properties of complex manifolds and their relationship to information. This paper investigates the potential of utilizing geometric information encoding as a novel method for compression and quantum information theory, aiming to address limitations in current data representation techniques. We propose a 'Geometric Information Encoding' – a process of mapping data points onto the geometry of a manifold, enabling efficient compression and retrieval. The core mechanism revolves around establishing a mapping between data points and geometric primitives, allowing for the creation of compressed representations and the exploration of novel quantum information processing paradigms. We will discuss the theoretical implications of this approach, including potential advantages over existing methods and challenges in implementation. The focus is on demonstrating the feasibility of this new technique through conceptual analysis and potential simulation.
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