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2014 | OriginalPaper | Buchkapitel

1. Quantum-Source Independent Component Analysis and Related Statistical Blind Qubit Uncoupling Methods

verfasst von : Yannick Deville, Alain Deville

Erschienen in: Blind Source Separation

Verlag: Springer Berlin Heidelberg

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Abstract

Quantum Information Processing (QIP) is an emerging field which yields new capabilities beyond classical, i.e., non-quantum, information processing. QIP methods manipulate quantum bit (qubit) states instead of classical bit values. Undesired coupling between these individual quantum states is expected, in the same way as classical systems involve undesired signal coupling. Methods for recovering individual quantum states from their coupled version are therefore required. To solve this problem, we recently introduced the field of Quantum Source Separation (QSS). We showed how to convert qubit states with cylindrical-symmetry Heisenberg coupling into classical-form data, mixed according to a specific nonlinear model, which was not previously studied in the literature. We therefore started to develop methods for unmixing such data. While we restricted ourselves to nonblind QSS methods and a basic blind approach in those previous works, we here proceed much further for the more difficult, i.e., blind, case: we introduce the concept of Quantum-Source Independent Component Analysis (QSICA), and we develop related QSS methods using various statistical signal processing tools, namely mutual information, likelihood and moments. The performance of the proposed approaches is validated by means of numerical tests. This especially shows the attractiveness of our method focused on second-order moments.

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Vorheriges Kapitel Erratum to: Performance Study for Complex Independent Component Analysis
Nächstes Kapitel Blind Source Separation Based on Dictionary Learning: A Singularity-Aware Approach
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Fußnoten
1
In the field of (B)SS, the term “source” sometimes refers to a physical object which provides (e.g., emits) a signal, but it is more often used as an abbreviation for “source signal”, since (B)SS is more concerned with the processing of these signals than with the objects which provide them. This appears in the name of SS itself: performing “SS” of course does not mean that one physically extracts source objects one from another or from their overall set, but that one extracts the signals associated with such objects (by using the measured mixtures of these signals): to be precise, the field of SS is not “SS” but “source signal separation”. For the sake of simplicity, we also often use the term “source” as an abbreviation for “source signal” in the field of QSS. For instance, in the sentence containing this footnote, “retriev[ing] the quantum sources” means “retrieving the quantum source signals”, i.e. “retrieving the signals associated with quantum sources”, where these quantum sources (i.e., these objects) consist of physical implementations of qubits. Similarly, the “source vector” considered further in this chapter is the vector composed of the values of source signals.
 
2
These vectors \( | + \rangle \) and \( | - \rangle \) are often respectively denoted as \(|0 \rangle \) and \(|1 \rangle \) (see e.g., [38]). We had to use the notations \( | + \rangle \) and \( | - \rangle \) in [21], to avoid confusion, and we keep them here.
 
3
It should be noted that the observed signals involved in this QSS problem have a specific nature, as compared to standard nonquantum BSS problems. In the latter problems, each value of an observed signal is usually the value of a measured physical quantity, such as the value of a voltage measured at a given time. On the contrary, as shown by (1.13), each value of an observed signal is here the value of a probability (which is estimated in practice). The overall signal composed of all successive values of a given observation (e.g., all values of \(x_{1}\)) therefore consists of a set of values of probabilities (e.g., all values of \(p_1\)), which depend on the values of the states used for initializing the qubits.
 
4
Our configuration is also an extension of ANC in the sense that (i) it involves signals which initially have a quantum form and (ii) reference signals are not directly available as observations here, but only after the adequate fixed processing (1.16)–(1.17) of some observations, which yields the signals defined by (1.19)–(1.20). A reference-based model is thus obtained for the global model, not directly for the mixing model, unlike in ANC.
 
5
For example only one sign indeterminacy for the Heisenberg global model, as detailed hereafter.
 
6
The index \(i\) of these coefficients associated with \( w_ {i}(.) \) is omitted for readability, i.e., \( j \) is used as the overall single index of all coefficients of \( w_ {i}(.) \).
 
7
One may also choose to define the concept of ICA for BSS in a broader sense, i.e., as the estimation of statistically independent source signals from their mixtures, using any suitable approach. The ML-based approach then completely belongs to ICA.
 
8
We here aim at avoiding any ambiguity between the actual “fixed data” of the considered problem and the corresponding variables introduced in the ML approach. We therefore use different notations for these corresponding quantities, e.g., \(v\) for the fixed (unknown) mixing parameter and \(\tilde{v}\) for the corresponding variable of the ML approach. In the framework of BSS, this type of notations was especially introduced in [7].
 
9
See function \(\phi \) defined on p. 13.
 
10
The expression “negentropy” is often used by the signal processing community for the quantity \( \mathcal{J}( U) \) defined in (1.67). We call this quantity “shifted negentropy” because, whereas negentropy literally means “negative of entropy” [5], the shifted negentropy \( \mathcal{J}( U) \) of \(U\) has the property of never being negative [33]. The expression “shifted negentropy” is quite compatible with two other uses of the word negentropy. The first one occurs in the context of living organisms, since Schrödinger first spoke of “negative entropy” in [45], in order to describe the ability of living organisms to fight against the tendency to disorder. The second appeared in the field of information theory, when Brillouin explicitly introduced the word negentropy, in [5], when establishing a link between information processing and the behavior of the physical systems making this processing.
 
11
In practice, they are estimated from a sequence of i.d. ( therefore possibly i.i.d ) source values.
 
12
This performance assessment procedure can only be used when developing and testing the considered BQSS methods, with actual source values \(s\) which are known (but which are not used in the BQSS methods themselves). On the contrary, in the actual setup which is to be eventually used, the actual sources are unknown, and one precisely aims at estimating them ! They cannot therefore be compared to their estimated values.
 
13
The above conditions for each elementary test are the same as in [21].
 
14
We therefore here perform more exhaustive tests than in [21], where only one elementary test was performed for each set of conditions (and we avoided the complex-valued outputs mentioned below).
 
15
We here use the standard definition of the RMSE, which was detailed in [21] for an elementary test, and which is straightforwardly extended to the set of elementary tests which yield real-valued separating system outputs.
 
16
Low values of \(K_{m}\) are not considered here, because higher numbers of measurements are more easily accepted in the single initial chararacterization of the system (mixture estimation stage) than in its subsequent permanent use (source estimation stage), and because these higher numbers of measurements are preferred, in order to better estimate the mixing parameter and thus to achieve better performance.
 
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Metadaten
Titel
Quantum-Source Independent Component Analysis and Related Statistical Blind Qubit Uncoupling Methods
verfasst von
Yannick Deville
Alain Deville
Copyright-Jahr
2014
Verlag
Springer Berlin Heidelberg
Buch
Blind Source Separation

Print ISBN: 978-3-642-55015-7

Electronic ISBN: 978-3-642-55016-4

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-642-55016-4_1

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