Abstract
Pattern recognition is a central topic in learning theory, with numerous applications such as voice and text recognition, image analysis and computer diagnosis. The statistical setup in classification is the following: we are given an i.i.d. training set (X1, Y1), ... , (Xn, Yn), where Xi represents a feature and Yi∊{0, 1} is a label attached to that feature. The underlying joint distribution of (X, Y) is unknown, but we can learn about it from the training set, and we aim at devising low error classifiers f: X→Y used to predict the label of new incoming features. In this paper, we solve a quantum analogue of this problem, namely the classification of two arbitrary unknown mixed qubit states. Given a number of 'training' copies from each of the states, we would like to 'learn' about them by performing a measurement on the training set. The outcome is then used to design measurements for the classification of future systems with unknown labels. We found the asymptotically optimal classification strategy and show that typically it performs strictly better than a plug-in strategy, which consists of estimating the states separately and then discriminating between them using the Helstrom measurement. The figure of merit is given by the excess risk equal to the difference between the probability of error and the probability of error of the optimal measurement for known states. We show that the excess risk scales as n−1 and compute the exact constant of the rate.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. Statistical learning theory is a broad research field stretching over statistics and computer science, whose general goal is to devise algorithms which have the ability to learn from data. One of the central learning problems is how to recognise patterns, with practical applications in speech and text recognition, image analysis, computer-aided diagnosis and data mining. In this paper we put forward a new type of quantum statistical problem inspired by learning theory, namely quantum state classification. In a nutshell, the problem is the following: we are given a set of quantum systems prepared identically in an unknown state ρ and another set of systems prepared in a different unknown state σ, and we would like to design a measurement which discriminates between ρ and σ as well as possible.
Main results. The naive solution to the classification problem is to first estimate the states and then use the optimal (Helstrom) measurement of the estimated states as classifier. One of our results is that this 'plug-in' strategy is typically not optimal, and is outperformed by a direct learning method which aims at estimating the Helstrom measurement without passing through state estimation. We show that for qubits the additional error probability due to uncertainty about the states typically decreases as n-1 where n is the size of the training set, and we compute the exact constant factor for both the optimal and the plug-in methods.
Wider implications. Our results show that well established paradigms of statistics and learning theory have natural extensions to the quantum domain, which need to be further explored and applied to statistical problems in quantum engineering.