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2014 | OriginalPaper | Chapter

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

Authors : Yannick Deville, Alain Deville

Published in: Blind Source Separation

Publisher: 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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previous chapter Erratum to: Performance Study for Complex Independent Component Analysis

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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.

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.

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.

For example only one sign indeterminacy for the Heisenberg global model, as detailed hereafter.

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 \( w_ {i}(.) \).

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.

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].

See function \(\phi \) defined on p. 13.

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.

In practice, they are estimated from a sequence of i.d. ( therefore possibly i.i.d ) source values.

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.

The above conditions for each elementary test are the same as in [21].

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).

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.

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.

Almeida, L.B.: MISEP-linear and nonlinear ICA based on mutual information. J. Mach. Learn. Res. 4, 1297–1318 (2003)

Belouchrani, A., Abed-Meraim, K., Cardoso, J.-F., Moulines, E.: A blind source separation technique using second-order statistics. IEEE Trans. Signal Process. 45(2), 434–444 (1997)CrossRef

Bennett, C.H., Shor, P.W.: Quantum information theory. IEEE Trans. Inf. Theory 44(6), 2724–2742 (1998)MATHMathSciNetCrossRef

Bienvenu, G., Kopp, L.: Optimality of high resolution array processing using the eigensystem approach. IEEE Trans. Acoust. Speech Signal Process. ASSP-31( 5), 1235–1247 (1983)

Brillouin, L.: Science and Information Theory. Academic Press, New York (1956)MATH

Cardoso, J.-F., Souloumiac, A.: Blind beamforming for non Gaussian signals. IEE Proc. F 140(6), 362–370 (1993)

Cardoso, J.-F.: Infomax and maximum likelihood for blind source separation. IEEE Signal Process. Lett. 4(4), 112–114 (1997)CrossRef

Cardoso, J.-F.: Blind signal separation: statistical principles. Proc. IEEE 86(10), 2009–2025 (1998)CrossRef

Chaouchi, C., Deville, Y., Hosseini, S.: Nonlinear source separation: a maximum likelihood approach for quadratic mixtures. In: Proceedings of the 30th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt 2010), Chamonix, France, 4–9 July 2010

10.

Comon, P.: Independent component analysis, a new concept ? Signal Process. 36(3), 287–314 (1994)MATHCrossRef

11.

Comon, P., Jutten, C. (eds.): Handbook of Blind Source Separation. Independent Component Analysis and Applications. Academic Press, Oxford (2010)

12.

Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1991)MATHCrossRef

13.

Darbellay, G.A., Vajda, I.: Estimation of the information by an adaptive partitioning of the observation space. IEEE Trans. Inf. Theory 45(4), 1315–1321 (1999)MATHMathSciNetCrossRef

14.

De Lathauwer, L., De Moor, B., Vanderwalle, J.: Fetal electrocardiogram extraction by blind source subspace separation. IEEE Trans. Biomed. Eng. 47(5), 567–572 (2000)CrossRef

15.

Delfosse, N., Loubaton, P.: Adaptive blind separation of independent sources: a deflation approach. Signal Process. 45(1), 59–84 (1995)MATHCrossRef

16.

Deville, Y., Damour, J., Charkani, N.: Multi-tag radio-frequency identification systems based on new blind source separation neural networks. Neurocomputing 49, 369–388 (2002)CrossRef

17.

Deville, Y., Deville, A.: Blind separation of quantum states: estimating two qubits from an isotropic Heisenberg spin coupling model. In: Proceedings of the 7th International Conference on Independent Component Analysis and Signal Separation (ICA 2007), vol. LNCS 4666, pp. 706–713. Springer, London, 9–12 Sept 2007. Erratum: replace two terms \( E \{ r_i \} E \{ q_i \} \) in (33) of [17] by \( E \{ r_i q_i \} \), since \( q_i \) depends on \( r_i \): see (5) in the current paper

18.

Deville, Y.: Traitement du signal : signaux temporels et spatiotemporels—Analyse des signaux, théorie de l’information, traitement d’antenne, séparation aveugle de sources. Ellipses Editions Marketing, Paris (2011)

19.

Deville, Y., Hosseini, S., Deville, A.: Effect of indirect dependencies on maximum likelihood and information theoretic blind source separation for nonlinear mixtures. Signal Process. 91(4), 793–800 (2011)MATHCrossRef

20.

Deville, Y.: ICA-based and second-order separability of nonlinear models involving reference signals: general properties and application to quantum bits. Signal Process. 92(8), 1785–1795 (2012)CrossRef

21.

Deville, Y., Deville, A.: Classical-processing and quantum-processing signal separation methods for qubit uncoupling. Quantum Inf. Process. 11(6), 1311–1347 (2012)MATHMathSciNetCrossRef

22.

Deville, Y., Deville, A.: A quantum/classical-processing signal separation method for two qubits with cylindrical-symmetry Heisenberg coupling. In: Deloumeaux, P., Gorzalka, J.D. (eds.) Information Theory: New Research, pp. 145–170. Nova Science Publishers, Hauppauge, NY (2012). ISBN: 978-1-62100-325-0 (Chapter 5)

23.

DiVincenzo, D.P.: Quantum computation. Science 270, 255–261 (1995)MATHMathSciNetCrossRef

24.

Duarte, L.T., Jutten, C.: A mutual information minimization approach for a class of nonlinear recurrent separating systems. In: IEEE International Workshop on Machine Learning for Signal Processing, pp. 122–127. Thessaloniki, Greece, 27–29 Aug 2007

25.

Ehlers, F., Schuster, H.G.: Blind separation of convolutive mixtures and an application in automatic speech recognition in a noisy environment. IEEE Trans. Signal Process. 45(10), 2608–2612 (1997)CrossRef

26.

Fety, L.: Méthodes de traitement d’antenne adaptées aux radiocommunications. Ph.D, ENST, Paris, France, June 3 (1988)

27.

Gaeta, M., Lacoume, J.-L.: Sources separation without a priori knowledge: the maximum likelihood solution. In: Fifth European Signal Processing Conference (EUSIPCO-90), pp. 621–624. Barcelona, Spain, 18–21 Sept 1990

28.

Guidara, R., Hosseini, S., Deville, Y.: Blind separation of nonstationary Markovian sources using an equivariant Newton-Raphson algorithm. IEEE Signal Process. Lett. 16(5), 426–429 (2009)CrossRef

29.

Guidara, R., Hosseini, S., Deville, Y.: Maximum likelihood blind image separation using non-symmetrical half-plane Markov random fields. IEEE Trans. Image process. 18(11), 2435–2450 (2009)MathSciNetCrossRef

30.

Hosseini, S., Jutten, C., Pham, D.T.: Markovian source separation. IEEE Trans. Signal Process. 51(12), 3009–3019 (2003)MathSciNetCrossRef

31.

Hosseini, S., Deville, Y.: Blind maximum likelihood separation of a linear-quadratic mixture. In: Proceedings of the Fifth International Conference on Independent Component Analysis and Blind Signal Separation (ICA 2004), vol. LNCS 3195, pp. 694–701. Springer, Granada, Spain, 22–24 Sept 2004. Erratum: see also “Correction to “Blind maximum likelihood separation of a linear-quadratic mixture“”, available on-line at http://arxiv.org/abs/1001.0863

32.

Hyvarinen, A., Oja, E.: A fast fixed-point algorithm for independent component analysis. Neural Comput. 9, 1483–1492 (1997)CrossRef

33.

Hyvarinen, A., Karhunen, J., Oja, E.: Independent Component Analysis. Wiley, New York (2001)CrossRef

34.

Jutten, C., Hérault, J.: Blind separation of sources, Part I: an adaptive algorithm based on neuromimetic architecture. Signal Process. 24(1), 1–10 (1991)MATHCrossRef

35.

Kendall, M., Stuart, A.: The Advanced Theory of Statistics, vol. 1. Charles Griffin, London, High Wycombe (1977)MATH

36.

Mokhtari, F., Babaie-Zadeh, M., Jutten, C.: Blind separation of bilinear mixtures using mutual information minimization. In: Proceedings of IEEE MLSP, France, Grenoble, 2–4 Sept 2009

37.

Molgedey, L., Schuster, H.G.: Separation of a mixture of independent signals using time delayed correlation. Phys. Rev. Lett. 72(23), 3634–3637 (1994)CrossRef

38.

Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATH

39.

Pham, D.T.: Blind separation of instantaneous mixture of sources via an independent component analysis. IEEE Trans. Signal Process. 44(11), 2768–2779 (1996)CrossRef

40.

Pham, D.T., Garat, P.: Blind separation of mixture of independent sources through a quasi-maximum likelihood approach. IEEE Trans. Signal Process. 45(7), 1712–1725 (1997)MATHCrossRef

41.

Pham, D.-T., Cardoso, J.-F.: Blind separation of instantaneous mixtures of nonstationary sources. IEEE Trans. Signal Process. 49(9), 1837–1848 (2001)MathSciNetCrossRef

42.

Pham, D.T.: Mutual information approach to blind separation of stationary sources. IEEE Trans. Inf. Theory 48(7), 1935–1946 (2002)MATHMathSciNetCrossRef

43.

Schmidt, R.: Multiple emitter location and signal parameter estimation. In: Proceedings of the RADC Spectrum Estimation Workshop, pp. 243–258. Rome,NY (1979)

44.

Schmidt, R.: Multiple emitter location and signal parameter estimation. In: IEEE Transactions on Antennas and Propagation, vol. AP-34, no. 3, pp. 276–280, Mar 1986

45.

Schrödinger, E.: What Is Life ?. Cambridge University Press, Cambridge (1944)

46.

Shannon, C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Urbana and Chicago (1949)MATH

47.

Shor, P.W.: Progress in quantum algorithms. Quantum Inf. Process. 3(1–5), pp. 5–13 (2004)

48.

Taleb, A., Jutten, C.: Source separation in post-nonlinear mixtures. IEEE Trans. Signal Process. 47(10), 2807–2820 (1999)CrossRef

49.

Taleb, A.: A generic framework for blind source separation in structured nonlinear models. IEEE Trans. Signal Process. 50(8), 1819–1830 (2002)MathSciNetCrossRef

50.

P. Tichavsky’s home page: http://si.utia.cas.cz/Tichavsky.html

51.

Tong, L., Liu, R., Soon, V.C., Huang, Y.-F.: Indeterminacy and identifiability of blind identification. IEEE Trans. Circuits Syst. 38(5), 499–509 (1991)MATHCrossRef

52.

Vandersypen, L.M.K., Chuang, I.L.: NMR techniques for quantum control and computation. Rev. Mod. Phys. 76, 1037–1069 (2004)CrossRef

53.

Vandersypen, L.: Dot-To-Dot Design. In: IEEE Spectrum, pp. 34–39. Elsevier publishing co., New York (2007)

54.

Van Trees, H.L.: Optimum Array Processing, Part IV of Detection, Estimation and Modulation Theory. Wiley, New York (2002)

55.

White, A.G., Gilchrist, A.: Measuring two-qubit gates. J.Opt. Soc. Am. B 24(2), 172–183 (2007)CrossRef

56.

Widrow, B., Glover, J.R., McCool, J.M., Kaunitz, J., Williams, C.S., Hearn, R.H., Zeidler, J.R., Dong, E., Goodlin, R.C.: Adaptive noise cancelling: principles and applications. Proc. IEEE 63(12), 1692–1716 (1975)CrossRef

57.

Zhang, J., Khor, L.C., Woo, W.L., Dlay, S.S.: A maximum likelihood approach to nonlinear convolutive blind source separation.In: Proceedings of the Sixth International Conference on Independent Component Analysis and Blind Signal Separation (ICA 2006), vol. LNCS 3889, pp. 926–933. Springer, Charleston, SC, USA, 5–8 Mar 2006

Title: Quantum-Source Independent Component Analysis and Related Statistical Blind Qubit Uncoupling Methods
Authors: Yannick Deville
Alain Deville
Publisher: Springer Berlin Heidelberg
Book: Blind Source Separation
Print ISBN: 978-3-642-55015-7

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

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-3-642-55016-4_1

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