Quantum Bio-Informatics V Proceedings of Quantum Bio-Informatics 2011 QP–PQ: Quantum Probability and White Noise Analysis* Managing Editor: W Freudenberg Advisory Board Members: L Accardi, T Hida, R Hudson and K R Parthasarathy QP–PQ: Quantum Probability and White Noise Analysis Vol 30: Quantum Bio-Informatics V eds L Accardi, W Freudenberg and M Ohya Vol 29: Quantum Probability and Related Topics eds L Accardi and F Fagnola Vol 28: Quantum Bio-Informatics IV From Quantum Information to Bio-Informatics eds L Accardi, W Freudenberg and M Ohya Vol 27: Quantum Probability and Related Topics eds R Rebolledo and M Orszag Vol 26: Quantum Bio-Informatics III From Quantum Information to Bio-Informatics eds L Accardi, W Freudenberg and M Ohya Vol 25: Quantum Probability and Infinite Dimensional Analysis Proceedings of the 29th Conference eds H Ouerdiane and A Barhoumi Vol 24: Quantum Bio-Informatics II From Quantum Information to Bio-informatics eds L Accardi, W Freudenberg and M Ohya Vol 23: Quantum Probability and Related Topics eds J C García, R Quezada and S B Sontz Vol 22: Infinite Dimensional Stochastic Analysis eds A N Sengupta and P Sundar Vol 21: Quantum Bio-Informatics From Quantum Information to Bio-Informatics eds L Accardi, W Freudenberg and M Ohya Vol 20: Quantum Probability and Infinite Dimensional Analysis eds L Accardi, W Freudenberg and M Schürmann Vol 19: Quantum Information and Computing eds L Accardi, M Ohya and N Watanabe Vol 18: Quantum Probability and Infinite-Dimensional Analysis From Foundations to Applications eds M Schürmann and U Franz *For the complete list of the published titles in this series, please visit: www.worldscientific.com/series/qp-pq QP–PQ Quantum Probability and White Noise Analysis Volume XXX Quantum Bio-Informatics V Proceedings of Quantum Bio-Informatics 2011 Tokyo University of Science, Japan – 12 March 2011 Editors Luigi Accardi Università di Roma “Tor Vergata”, Italy Wolfgang Freudenberg Brandenburgische Technische Universität Cottbus, Germany Masanori Ohya Tokyo University of Science, Japan World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library QP–PQ: Quantum Probability and White Noise Analysis — Vol 30 QUANTUM BIO-INFORMATICS V Proceedings of the Quantum Bio-Informatics 2011 Copyright © 2013 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN 978-981-4460-01-9 Printed in Singapore by Mainland Press Pte Ltd v PREFACE This volume is based on the fifth international conference of quantum bio-informatics held at the QBI Center of Tokyo University of Sciences The purpose of the conference is towards new stage making interdisciplinary bridges in mathematics, physics, information and life sciences, in particular, research for new paradigm for information science and life science on the basis of quantum theory More than 100 researchers in various fields such as mathematics, physics, information and biology come from all over the world The conference was held for nearly one week, and we had a lot of fruitful discussion In this fifth conference, particular attention is come up on quantum entanglement, simulation of bio-systems, brain function, quantum like dynamics and adaptive systems Most of speakers gave care to the relation between their own topics and the mystery of life The papers submitted in this volume are all refereed, whose contents are related to one of the following subjects: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Mathematics of Cryptography and its related topics Quantum algorithm and computation Quantum entanglement Quantum entropy and information dynamics Quantum dynamics and time operator Stochastic dynamics and white noise analysis Brain activity Quantum like models and PD game Quantum physics and superconductivity Quantum tomography and sufficiency Adaptation in Plants Alignment of sequences Luigi Accardi Wolfgang Freudenberg Masanori Ohya This page intentionally left blank vii Five years of QBIC Masanori Ohya Department of Information Sciences, Tokyo University of Science, Japan Aims of QBIC The quantum bio-informatics center (QBIC) was founded in 2006 towards new stage making interdisciplinary bridges in philosophy, mathematics, physics, information and life sciences Our research center (QBIC) tries to nd a new paradigm for information science and life science on the basis of quantum mathematics Several researchers more than 100 on mathematics, physics, information theory and biology who are interested in mathematical study worked together in QBIC during this ve years from 2006 to 2010 To solve the mystery of life is one of the most interesting problems in 21 century After discovery of DNA, people believes that one key to read the riddle will be hidden in the process how information of life is stored and its change and transmission are made Concerning the information transmission (communication), quantum information opened a new door and is expected to understand new aspects of existence in-itself More concretely, the immensely long DNA, sequence of four bases in the genome, contains information on life, and decoding or changing this sequence is involved in the expression and control of life In quantum information, meanwhile, we produce various “information” by sequences of two quantum states, and think of ways of processing, communicating and controlling them It is thought that the problems we can process in time “T” using a conventional computer can be processed in time nearly “log T” using a quantum computer However, the transmission and processing of information in the living body might be much faster than those of quantum computer and communication Seen from this very basic viewpoint, developing the mathematical principles that have been found in quantum information should be useful in constructing mathematical principles for life sciences, which have not been established yet The mechanism of processing information in life is also expected to be useful for the further growth of quantum information viii Figure QBIC Research Project The way of our research is (1) to return to the starting point of bio-informatics and quantum information, elds that and to solve these fundamental problems, and (2) to seriously attempt mutual interaction between the two, with a view to enumerating and solving the many fundamental problems they entail In our view, there is no similar research center in the world to return to the basics of bio-information and quantum information and to focus on the correlation between the two with a view to new development of each Our way with targets and goal are described in the gure below: We had more than 200 papers published in this ve years Most of them have rst published in the ve proceedings of International Conference held in Tokyo University of Science ix I will here review basic results of some achievements in this ve years of QBIC Solving the mystery of life Solving the mystery of life requires several stages (1) Metaphysical, (2) Biological & Physical, (3) Mathematical The works of the stages (1) and (2) have been done for a long time even in the "new" life science, that is, many philosophical considerations and various experiments have been done, and several (tentative) theories have been made However it is also true we had not a basic mathematical rule (theory) in the life science so that many researchers could accept it as quantum mechanics In order to make such a theory, we have to try to develop fundamentals in various elds (mathematics, physics, information theory) with intention to the goal, i.e., nding the rst principle understanding the life itself Biological systems are open systems Biological systems are multi— component and context dependent Biological subsystems in a biological system are locally interacting each other Therefore the state of the biological systems depends on its surrounding and the eve of itself These observations entail that biological systems are adaptive We have to nd a mathematical rule to describe all of those However, in order to make our dream realize, we have to develop each eld such as mathematics, quantum physics, information, structural biology and bio-informatics so that we can use the fructication to achieve new paradigm as discussed above Some works appeared in conference of QBIC I itemize some works appeared in the conferences of QBIC during this ve years It is beyond the introduction and my ability to review all works appeared in the ve years conferences, so that I only mention some mathematical trials somehow related to life sciences The fruitful results of various works can been seen in the series of the ve proceedings of QBIC conferences 3.1 Examples of researches in QBIC 3.1.1 Concerning of the Figure above • White noise, stochastic analysis and some applications to DDS (Hida, Streit, SiSi, Accardi, Volovich, Smolyanov, Fichtner, In- 178 (n,m) (n,m)  (1,1) 100% S V QBER Perror s 0.50 25% 5.67% (2,2) 75% 2.51 0.84 7.9% s (2,3) 68.75% 2 0.91 4.5% 0.82% (3,3) 0.14% 62.5% 3.17 0.96 2.1% 0.17% also for these pulses that Alice and Bob are the most likely to not end up with the same bit, thus increasing the measured QBER Unfortunately for Alice and Bob, as soon as n > or m > 1, it is also for these pulses that the probability of detection is the smallest for a xed n and m Since this probability decreases for higher-order multiple-photon absorption, it (n,m) also means that, rather counter-intuitively, the probability Perror that Eve makes a mistake guessing a bit shared by Alice and Bob decreases with higher violation of Bell inequality and with lower QBER (see Table 1) Discussion and countermeasures An unwanted feature that could betray Eve’s attack is that the sum of coincidences depends on the measurement settings A and B However, Eve can remove this unwanted eect entirely by driving dierent detection patterns for Alice and Bob, in a similar way to what was done by Larsson 10 and Gisin 11 in their hidden-variable models The simplest method would be to alternatively drive a single photon absorption on one side, and a multiple-photon absorption on the other side The sampling is then always fair on the side driven to a single photon absorption, and the total number of coincidences becomes independent of the measurement angles Monitoring the single counts as such would not betray Eve’s attack, because all the channels are treated on equal footing by the attack Similarly, monitoring the rates of coincidences is also unlikely to betray Eve’s attack because they are all rotationally invariant: they depend only on the angle dierence |B  B |, as is the case for a genuine singlet state The correlation function does dier slightly from the V cos 2(A  B ) expected for a singlet state in a lossy channel, but this dierence is small and would require more measurement settings than in the standard Ekert and BBM92 protocols 179 However, a multiple-photon absorption is a second order phenomenon, and is as such very ine!cient Eve would thus need to send more pairs per second to compensate for this additional ine!ciency The trouble for Eve would be that, assuming independent errors, the singles scale linearly with the e!ciency whereas the coincidences scale with the square of the e!ciency Alice and Bob would thus be able to spot the attack by monitoring the absolute e!ciency, calculated as the ratio of the coincidences over the singles 16 Using frequency lters so that only a known range of restricted frequencies can reach the detectors is another obvious way to limit the possibility of multiple-photon absorption in the ideal case, but Alice and Bob still need to guarantee that the dominant way to get a click in their necessarily imperfect detector is through a single-photon absorption, even at those frequencies allowed by the lters It should be emphasized that the principle of the attack is not bound to multiple-photon absorption The essential feature is really that Eve can prevent single photons to be detected by Alice and Bob, regardless of the technique used for this purpose This issue is quite critical considering the existing faked-state attacks that have already been successfully implemented against QKD protocols 17,18 , by forcing the detectors to exit the single-photon sensitive Geiger mode 18 It is conceivable that this type of blinding attack could be tailored to allow only a n-photon absorption in the detectors used by Alice and Bob Such a combination of blinding detector attack together with our source designed for a multiple-photon absorption attack could then constitute a robust attack even against Ekert protocol, something that the blinding attack alone could not 17 An active method for Alice and Bob to detect the attack would be to implement a fair sampling test 19 adapted to quantum key distribution with entangled states 20 This fair sampling test does not introduce any loss and can be performed locally and unilaterally on either side during the production of the key The setup used for the Fair Sampling test would also make the other attacks against QKD protocols more complicated (if not impossible) to implement 21 Conclusion To our knowledge the multiple-photon absorption attack is in fact the rst explicit attack against Ekert protocol with light sources The multiple-photon absorption attack does not require Eve to be phys- 180 ically located between Alice and Bob and to intercept what was intended for either of them, as in a typical intercept-and-resend attack 17,18 The only requirement is that Eve has managed at some point in the past to replace the source of entangled state by her own mixture of separable states She just needs to know the state  of each pulse and that is something that Eve could have set deterministically in the source Once this is done, the security of the QKD protocol is compromised by anyone who happens to know the polarization of the pulses as a function of time (t) By mimicking the statistics of an entangled state well enough to pass the security checks normally undertaken by Alice and Bob, the attack can in principle work against any entanglement-based quantum key distribution protocol, independently of which protocol is chosen by Alice and Bob References 10 11 12 13 14 15 16 17 18 19 20 21 V Scarani, et al Rev Mod Phys 81 1301 (2009) A.K Ekert, Phys Rev Lett 67 661 (1991) C.H Bennett, G Brassard and N.D Mermin, Phys Rev Lett 68 557 (1992) G Adenier, I Basieva, A Khrennikov and N Watanabe, arXiv:1011.4740 R Braunstein, Phys Rev 125 475 (1961) J F Clauser, M A Horne, A Shimony and R A Holt Phys Rev Lett 23 880 (1969) T Scheidl et al, New J Phys 11 085002 (2009) C Branciard, N Gisin, B Kraus and V Scarani, Phys Rev A 72 032301 (2005) P.M Pearle, Phys Rev D 1418 (1970) J.-Å Larsson, Phys Rev A 57 3304 (1998).’ N Gisin and B Gisin, Phys Lett A 260 323 (1999) G Adenier, Am J Phys 76 147 (2008) G Adenier, AIP Conf Proc 1101 (2009) X Ma, Chi-HangFred Fung, H.K Lo, Phys Rev A 76 012307 (2007) C Erven, C Couteau, R La amme, and G Weihs, Opt Express 16 16840 (2008) D N Klyshko, Sov J Quantum Electron 10, 1112-1116 (1981) V Makarov and J Skaar, Quant Inf Comp 0622 (2008) L Lydersen, C Wiechers, C Wittmann, D Elser, J Skaar and V Makarov, Optics Express 18 27938 (2010) G Adenier, J Russ Laser Res 29 409 (2008) G Adenier, N Watanabe and A Yu Khrennikov, arXiv:1004.1242 G Adenier, I Basieva, A Yu Khrennikov, M Ohya, N Watanabe, arXiv:1102.3366 Quantum Bio-Informatics V c 2013 World Scientific Publishing Co Pte Ltd pp 181–185 PROTEIN SEQUENCE ALIGNMENT TAKING THE STRUCTURE OF PEPTIDE BOND TOSHIHIDE HARA∗ , KEIKO SATO and MASANORI OHYA Department of Information Sciences, Tokyo University of Science, 2641 Yamazaki, Noda City, Chiba, Japan ∗ E-mail: hara@is.noda.tus.ac.jp In a previous paper1 we proposed a new method for performing pairwise alignment of protein sequences The method, called MTRAP, achieves the highest performance compared with other alignment methods such as ClustalW22,3 on two benchmarks for alignment accuracy In this paper, we introduce a new measure between two amino acids based on the formation of peptide bonds The measure is implemented into MTRAP software to further improve alignment accuracy Our alignment software is available at “http://www.rs.noda.tus.ac.jp/%7Eohya-m/” Keywords: Sequence analysis; Sequence alignment; Dihedral angle; Ramachandran plot; Peptide bond Introduction A protein is a polypeptide chain made up of amino acid residues Each amino acid residue in a protein has the phi and psi dihedral angles.4 Dihedral angles of amino acid residues are of importance in understanding protein structure5,6 because they specify the backbone conformation of a protein In this paper, we introduce a new measure of difference between a pair of amino acid sequences based on dihedral angles of amino acid residues Recently, we developed a high accurate alignment method called MTRAP.7 The measure is implemented into MTRAP software to further improve alignment accuracy The measure of difference between a pair of sequences used in MTRAP We use the following notations Let Ω be the set of all 20 amino acids, Ω∗ n be the union of Ω and {∗} (where ∗ is indel (gap)), and [Ω∗ ] be the set of 181 182 sequences of length n consisting of elements of Ω∗ We call an element of Ω a residue and an element of Ω∗ a symbol In addition, let Γ ≡ Ω × Ω be the direct product of two Ωs and Γ∗ ≡ Ω∗ × Ω∗ MTRAP uses two types of functions as quantitative measures of difference between two sequences A = (a1 , a2 , · · · , an ) and B = (b1 , b2 , · · · , bn ) ∈ [Ω∗ ]n : dsub (A, B) and dtrans (A, B) The function dsub (A, B) is defined as follows: n dsub (A, B) = s˜ (ai , bi ) , (1) i=1 where s˜ (a, b) is a normalized difference of symbols a and b expressed as a value between and This normalize difference is derived by converting a substitution matrix score such as PAM8 and BLOSUM.9 Gaps require special treatment (see ref for details) Furthermore, we consider the measure of difference between a pair of sequences based on two consecutive pairs of sites For the two sequences A and B, the function dtrans (A, B) is defined as follows: n−1 t˜(ui , ui+1 ) , dtrans (A, B) = (2) i=1 where ui = (ai , bi ) ∈ Γ∗ and t˜(u, v) is a normalized difference of sites u and v This difference takes value in the range of [0, 1] with indicating that the correlation between u and v is strong (see ref for details) The difference measure used in the MTRAP is defined by a weighted mixture of dsub (A, B) and dtrans (A, B): dMTRAP (A, B) = (1 − ε) dsub (A, B) + εdtrans (A, B) (3) Here ε is a degree of mixture set to 0.775 by default A new measure based on dihedral angles of amino acids Let us introduce a new measure of difference mentioned in Sec We use Ramachandran plot stored in DASSD.10 Each dot on the Ramachandran plot ((ϕ, ψ) = (−180◦ , 180◦ ) × (−180◦ , 180◦ )) shows the phi (x-axis) and psi (y-axis) value for an amino acid We first divide the each axe ϕ ψ ϕ ϕ ψ ψ N into N parts equally: ϕ = ∪N i=1 i , ψ = ∪i=1 i , i ∩ j = φ, i ∩ j = φ, then we count the number of dots in each region Let na (i, j) and nb (i, j) ψ be the number of dots in each ϕ i × j region for amino acid a and b (where 183 i, j = 1, 2, · · · , N ) Then we define the following difference of dihedral angle between amino acid a and b: r˜ (a, b) = N i=1 N na (i,j) j=1 Na − nb (i,j) Nb , (4) N N N N where Na = i=1 j=1 na (i, j), Nb = i=1 j=1 nb (i, j), and ≤ r˜ (a, b) ≤ Using r˜ (a, b), a difference between the two sequence A and B is defined as follows: n dang (A, B) = r˜ (ai , bi ) (5) i=1 Finally, we incorporate the function dang (A, B) into equation (3), the difference measure defined in this paper is a weighted mixture of dsub , dtrans and dtrans : dNEW (A, B) = (1 − ε) {(1 − γ) dsub (A, B) + γdang (A, B)}+εdtrans (A, B) , (6) where γ is a degree of mixture We calculate the above r˜ with N = 72 using the data-set obtained from DASSD10 which contains the values of dihedral angles ϕ, ψ and the “secondary structure details” for short fragments of amino acids having 1, and lengths In this paper, we use the all data of length and identify γ = 0.1 as the value minimizing cross-validation error Results and discussion We compare the accuracy of new method with ClustalW2,3 MAFFT,11 MUSCLE,11 TCoffee12 and MTRAP7 by using the PREFAB benchmark test.13 We use the all 1682 pairwise alignments on PREFAB as reference alignments The accuracy of generated alignment is measured by Q score.13 The Q score is defined as the ratio of the number of residue pairs correctly aligned in the test alignment (i.e., the alignment generated by a specified method such as MTRAP) to the total number of residue pairs aligned in the reference alignment When all pairs are correctly aligned, the score have a maximum value 1, and when no-pairs are correctly aligned, the score have a minimum value The performance is evaluated on the three different range of sequence identity of the references: 0-15%, 15-30% and 0-100% The number of alignments in each range is 423, 911 and 1682, respectively In Table we report the accuracy of the new method, the original MTRAP,7 ClustalW23 ver 2.0.12, MAFFT11 ver 6.864b, MUSCLE13 ver 3.7 and TCoffee12 ver 8.47 184 The new method achieves the best results in all ranges However, the alignment accuracy is similar to the original MTRAP The new method shows a slight improvement in alignment accuracy Table Method New Method MTRAP ClustalW2 MAFFT MUSCLE TCoffee Median and average of Q scores on PREFAB database Median / Average of Q scores for each %ID range (PREFAB) 0-15%(423) 15-30%(911) 0-100%(1682) 0.190 / 0.250 0.750 / 0.678 0.729 / 0.616 0.188 / 0.250 0.750 / 0.676 0.728 / 0.615 0.123 / 0.199 0.727 / 0.647 0.701 / 0.585 0.102 / 0.170 0.686 / 0.627 0.650 / 0.568 0.138 / 0.205 0.701 / 0.636 0.675 / 0.581 0.133 / 0.198 0.706 / 0.645 0.680 / 0.585 Note: The highest values in each %ID range are shown in bold style The number in parentheses denotes the number of alignments in each sequence identity range ClustalW2, MUSCLE and TCoffee run with their default parameters MAFFT runs with “linsi” option Conclusion Although the new method shows the best performance in comparison with the other method, it gives slightly better accuracy than the original MTRAP contrary to our expectation From this result, we come to the conclusion that to consider correlations between two consecutive pairs of sites might covers to consider dihedral angles of amino acid residues References T Hara, K Sato and M Ohya, QP-PQ: Quantum Probability and White Noise Analysis (Quantum Bio-Informatics IV) 28, 129 (2011) J D Thompson, D G Higgins and T J Gibson, Nucleic Acids Res 22, 4673(Nov 1994) M A Larkin, G Blackshields, N P Brown, R Chenna, P A McGettigan, H McWilliam, F Valentin, I M Wallace, A Wilm, R Lopez, J D Thompson, T J Gibson and D G Higgins, Bioinformatics 23, 2947(Nov 2007) G Ramachandran, C Ramakrishnan and V Sasisekharan, Journal of molecular biology 7, p 95 (1963) R Laskowski, M MacArthur, D Moss and J Thornton, Journal of applied crystallography 26, 283 (1993) R Hooft, C Sander and G Vriend, Computer applications in the biosciences: CABIOS 13, 425 (1997) T Hara, K Sato and M Ohya, BMC Bioinformatics 11, p 235 (2010) 185 M O Dayhoff, R M Schwartz and B C Orcutt, Atlas of Protein Sequence and Structure 5(3), 345 (1978) S Henikoff and J G Henikoff, Proc Natl Acad Sci U.S.A 89, 10915(Nov 1992) 10 S Dayalan, N Gooneratne, S Bevinakoppa and H Schroder, Bioinformation 1, p 78 (2006) 11 K Katoh, K Misawa, K Kuma and T Miyata, Nucleic Acids Res 30, 3059(Jul 2002) 12 C Notredame, D G Higgins and J Heringa, J Mol Biol 302, 205(Sep 2000) 13 R C Edgar, Nucleic Acids Res 32, 1792 (2004) This page intentionally left blank Quantum Bio-Informatics V c 2013 World Scientific Publishing Co Pte Ltd pp 187–191 SPACE - TIME - NOISE (RAUM - ZEIT - RAUSCHEN) TAKEYUKI HIDA Professor Emeritus, Nagoya University, Japan E-mail: takeyuki@math.nagoya-u.ac.jp Introduction In the mathematical study of a random complex phenomenon, there is an ideal road map, by which we start with Reduction This means that it would be ne if we can nd a system of independent random variables that contains the same information as the given random phenomena, and it is expressed as a (non-random) function of those random variables The random variables may be ideal random functions which are assumed to be i.i.d (independent identically distributed) and atomic Such a system of idealized elemental random variables will be called a noise b The well-known good examples are the (Gaussian) white noise B+t, and the Poisson noise Pb +t, They are depending on the time parameter t The time derivative of a compound Poisson process is not quite a noise, since it is decomposed into (formally speaking) atomic Poisson noises, In this note we shall show that another kind of noise, in fact, depending not on the time, but on a space parameter u, does arise naturally We shall also give a plausible probabilistic interpretation on this fact, and discuss its properties of new noise by comparing with those depending on time parameter; some properties are in parallel and some others are in a dual manner Some of other related properties will be discussed as much as space-time permits If the family of random variables are parametrized by a discrete parameter, either time or space, then we consider a sequence of i.i.d random variables However, if we come to a continuous parameter case, a continuous analogue of this is not acceptable For one thing, in this case the probability distribution associated with such family can not be an abstract Lebesgue space So, calculus does not follow smoothly 187 188 We now change our eyes slightly Instead of i.i.d we take sum of random variables with stationary independent increments As a counter part of this we can consider the continuous parameter case, for which we take an additive process with stationary increments With this understanding we proceed to nd possible noises Two kinds of noise We shall show that two dierent kind of noises can arise naturally from a limit theorem in probability theory In fact, our interpretation comes from very basic and elementary background Case I Take the unit interval I @ ^3, 4`, Let n @ {nj ,  j  5n } be the partition of I To x the idea, we assume that |nj | @ 5n To each subinterval nj we associate a random variable Xjn We assume that {Xjn } are i.i.d with mean and nite variance v Let n be larger, Then, we can appeal to the central limit theorem to have a standard Gaussian distribution by observing n X s Sn @ + Xjn ,/ 5n v If we take subinterval ^a, b` of I, then the same trick gives us a Gaussian distribution N +3, b  a, We may therefore consider Sn as an approximation of a Brownian motion and each Xjn approximates elemental random variable Because of the central limit theorem, Gaussian noise is involved Case II The interval I and its partition n are the same as in I The independent random variables Xjn are i.i.d., but all are subject to a probability distribution such that P +Xjn @ 4, @ pn , P +Xjn @ 3, @  pn Let pn be getting small as n is getting large keeping the relation 5n pn @  for some constant  > Then, by the law of small probability, the sum Sn is an approximation of a Poisson random variable P +, with intensity  We then come to a partition of I as in Case I, however the idea to form the partial sums of random variables is dierent Keep the random variables Xjn just as above and divide the sum Sn into partial sums in a 189 manner that Sn @ X Snk , k Pj(k+1) where Snk @ j(k)+1 Xjn , with < j+4, < j+5, < · · · j+m, @ 5n Then, assuming tacitly that each j+k, is large, we see that each Snk is an approximation of Poisson random variable P +k , From the method of forming the partial sum we can easily prove the following result Theorem 2.1 The Poisson random variables P +k , thus obtained are mutually independent and X P +k , P +, @ k Before we come to further observation on the above results, we should like to mention some short remarks Remark On the type of probability distributions We claim that all the Gaussian distributions are of the same type (a constant, the exceptional Gaussian variable is excluded It is also excluded in Case I.) On the other hand, Poisson type distributions with dierent intensities are not of the same type This can be proved by the formula of characteristic function By the way, Poisson type distribution means a distribution of uP +, c, c may be ignored Namely, we can compare two characteristic functions iz *1 +z, @ e1 (e *2 +z, @ e 1) 2 (eiz 1) These two functions of z can not be exchanged by any a!ne transformation of z if 1 6@ 2 - end of Remark - We can say that 1) We have a freedom to choose  arbitrary Hence, we can form, by the sum of i.i.d random variables, random variables with dierent type 2) The intensity is taken to be a space parameter This also comes from our construction The above construction also shows it is additive in  3) Multiplication by a constant to Poisson type variable is possible but the constant can be a label So, take a constant u @ u+, as a label of the intensity The function u+, is therefore univalent In view of this fact, we can form an inverse function  @ +u, which is to be monotone 190 A noise with space parameter It is meaningful to consider a system of Poisson type random variables with dierent intensities From the viewpoint of information theory There is an important property that is deserved to be mentioned This is a simple rephrasement of the swell-known Raikov theorem Proposition 3.1 A Poisson type random variable is atomic, i.e it is elemental Proof Suppose a Poisson variable X were decomposed into two independent random variables X @ Y Z Then, both Y and Z are Poisson variable Except a trivial case, the sigmaeld generated by X is strictly smaller than that generated by two variables We shall now think of the sum of independent Poisson random variables with dierent intensities like X X@ u+j ,P +j ,, where j ’s are dierent and so are u+j ,’s Then, the characteristic function *X +z, of X is given by Y izu(j ) 1) *+z, @ ej (e By using the one-to-one correspondence between u and  we have Y izuj * +z, @ e(uj )+e 1, The passage to innity gives us a characteristic functional C+,, It is of the form Z ³ ¸ ´ C +, @ h{s eiu(u)   +u, du which has been discussed by Si Si in this conference 10 Concluding remarks [I] The idea of "Reduction" can be seen in the lecture notes by J -L Lions Also, we note that a concrete expression of realizing the idea of reduction can be seen through Lévy’s stochastic innitesimal equation See 191 [II] The term "noise" is more familiar in the communication theory in electrical engineering There a noise is a disturbance against smooth transmission of message On the other hand, it, e.g white noise, contains maximum information ( in terms of entropy) under the limited power References T Hida, Analysis of Brownian functionals Carleton Math Lecture Notes no 13, July, 1975 T Hida and Si Si, Lectures on white noise functionals World Scientic Pub Co 2008 T Hida, The concept of innite dimension in the analysis of white noise generalized functionals RIMS Ojima Conf Oct 2007 (in Japanes.) P Lévy, Processus stochastiques et mouvement brownien Gauthier-Villars, 1948, 2éme ed 1965 P Lévy, Problèmes concrets d’analyse fonctionnelle Gauthier-Villars, 1951 P Lévy, Univ of California Publication, 1953, J -L Lions, The earth, planet the role of mathematics and of super computers (in Spanish), Inst Espana Espas a Calpe 1990 J Mikusinski, On the square of the Dirac Delta-distribution Bull de l’Academie Polonaise des Sciences Ser Sci math astr et Phys XIV, no (1966) 511-513 Si Si, An aspect of quadratic Hida distributions in the realization of a duality between Gaussian and Poisson noises Innite Dim Analysis, Quantum Prob and Related Topics vol 11.(2008), 109-118 10 Si Si, A new noise depending on a space parameter and its application QBIC2011 Conf paper 2011 11 L Streit, Feynman integrals as generalized functions on path space: Things done and open problems 12 H Weyl, Raum Zeit Materie; English translation.by H.L.Brose Dover Pub.1922 This page intentionally left blank ... Ohya, M., Quantum Bio- Informatics IV (Quantum Probability and White Noise Analysis, Vol 28), World Scientic, 2011 Accardi, L., Freudenberg, W., Ohya, M., Quantum Bio- Informatics V (Quantum Probability... www.worldscientific.com/series/qp-pq QP–PQ Quantum Probability and White Noise Analysis Volume XXX Quantum Bio- Informatics V Proceedings of Quantum Bio- Informatics 2011 Tokyo University of Science, Japan –... Topics eds L Accardi and F Fagnola Vol 28: Quantum Bio- Informatics IV From Quantum Information to Bio- Informatics eds L Accardi, W Freudenberg and M Ohya Vol 27: Quantum Probability and Related