On quantum statistics in data analysis

Originally, quantum probability theory was developed to analyze statistical phenomena in quantum systems, where classical probability theory does not apply, because the lattice of measurable sets is not necessarily distributive. On the other hand, it is well known that the lattices of concepts, that arise in data analysis, are in general also non-distributive, albeit for completely different reasons. In his recent book, van Rijsbergen argues that many of the logical tools developed for quantum systems are also suitable for applications in information retrieval. I explore the mathematical support for this idea on an abstract vector space model, covering several forms of data analysis (information retrieval, data mining, collaborative filtering, formal concept analysis...), and roughly based on an idea from categorical quantum mechanics. It turns out that quantum (i.e., noncommutative) probability distributions arise already in this rudimentary mathematical framework. We show that a Bell-type inequality must be satisfied by the standard similarity measures, if they are used for preference predictions. The fact that already a very general, abstract version of the vector space model yields simple counterexamples for such inequalities seems to be an indicator of a genuine need for quantum statistics in data analysis.Comment: 7 pages, Quantum Interaction 2008 (Oxford, April 2008) v3: added two diagrams, changed some wording

Experimentally exploring compressed sensing quantum tomography

In the light of the progress in quantum technologies, the task of verifying the correct functioning of processes and obtaining accurate tomographic information about quantum states becomes increasingly important. Compressed sensing, a machinery derived from the theory of signal processing, has emerged as a feasible tool to perform robust and significantly more resource-economical quantum state tomography for intermediate-sized quantum systems. In this work, we provide a comprehensive analysis of compressed sensing tomography in the regime in which tomographically complete data is available with reliable statistics from experimental observations of a multi-mode photonic architecture. Due to the fact that the data is known with high statistical significance, we are in a position to systematically explore the quality of reconstruction depending on the number of employed measurement settings, randomly selected from the complete set of data, and on different model assumptions. We present and test a complete prescription to perform efficient compressed sensing and are able to reliably use notions of model selection and cross-validation to account for experimental imperfections and finite counting statistics. Thus, we establish compressed sensing as an effective tool for quantum state tomography, specifically suited for photonic systems.Comment: 12 pages, 5 figure

A blind hierarchical coherent search for gravitational-wave signals from coalescing compact binaries in a network of interferometric detectors

We describe a hierarchical data analysis pipeline for coherently searching for gravitational wave (GW) signals from non-spinning compact binary coalescences (CBCs) in the data of multiple earth-based detectors. It assumes no prior information on the sky position of the source or the time of occurrence of its transient signals and, hence, is termed "blind". The pipeline computes the coherent network search statistic that is optimal in stationary, Gaussian noise, and allows for the computation of a suite of alternative statistics and signal-based discriminators that can improve its performance in real data. Unlike the coincident multi-detector search statistics employed so far, the coherent statistics are different in the sense that they check for the consistency of the signal amplitudes and phases in the different detectors with their different orientations and with the signal arrival times in them. The first stage of the hierarchical pipeline constructs coincidences of triggers from the multiple interferometers, by requiring their proximity in time and component masses. The second stage follows up on these coincident triggers by computing the coherent statistics. The performance of the hierarchical coherent pipeline on Gaussian data is shown to be better than the pipeline with just the first (coincidence) stage.Comment: 12 pages, 3 figures, accepted for publication in Classical and Quantum Gravit

Strong coupling constant at NNLO from DIS data

We discuss the results of our recent analysis [1] of deep inelastic scattering data on F2 structure function in the non-singlet approximation with next-to-next-to-leading-order accuracy. The study of high statistics deep inelastic scattering data provided by BCDMS, SLAC, NMC and BFP collaborations was performed with a special emphasis placed on the higher twist contributions. For the coupling constant the following value alfa_s(MZ2) = 0.1167 +- 0.0022 (total exp. error) was found.Comment: 6 pages, 2 figures, PoS(QFTHEP2010)050, presented on the XIXth International Workshop on High Energy Physics and Quantum Field Theory, 8-15 September 2010, Golitsyno, Moscow, Russi

Learning quantum models from quantum or classical data

In this paper, we address the problem how to represent a classical data distribution in a quantum system. The proposed method is to learn quantum Hamiltonian that is such that its ground state approximates the given classical distribution. We review previous work on the quantum Boltzmann machine (QBM) and how it can be used to infer quantum Hamiltonians from quantum statistics. We then show how the proposed quantum learning formalism can also be applied to a purely classical data analysis. Representing the data as a rank one density matrix introduces quantum statistics for classical data in addition to the classical statistics. We show that quantum learning yields results that can be significantly more accurate than the classical maximum likelihood approach, both for unsupervised learning and for classification. The data density matrix and the QBM solution show entanglement, quantified by the quantum mutual information $I$. The classical mutual information in the data $I_c\le I/2=C$, with $C$ maximal classical correlations obtained by choosing a suitable orthogonal measurement basis. We suggest that the remaining mutual information $Q=I/2$ is obtained by non orthogonal measurements that may violate the Bell inequality. The excess mutual information $I-I_c$ may potentially be used to improve the performance of quantum implementations of machine learning or other statistical methods.Comment: 28 pages, 7 figure

Lifetime statistics of quantum chaos studied by a multiscale analysis

In a series of pump and probe experiments, we study the lifetime statistics of a quantum chaotic resonator when the number of open channels is greater than one. Our design embeds a stadium billiard into a two dimensional photonic crystal realized on a Silicon-on-insulator substrate. We calculate resonances through a multiscale procedure that combines graph theory, energy landscape analysis and wavelet transforms. Experimental data is found to follow the universal predictions arising from random matrix theory with an excellent level of agreement.Comment: 4 pages, 6 figure