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Quantum morphogenesis : A variation on Thom’s catastrophe theory Aerts, Diederik, 1953-; Czachor, Marek; Gabora, Liane; Kuna, Maciej; Posiewnik, Andrzej; Pykacz, Jarosław; Syty, Monika May 27, 2001

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tinaiennesyayodericosy’ aPHYSICAL REVIEW E 67, 051926 ~2003!The systems we shall consider are probabilistic. The mor-phogenesis will be described in terms of probabilities or un-certainties associated with given sets of propositions. Thecontextual nature of the propositions will require a represen-tation of probabilities different from the Kolmogorovianframework @8# of sets and commuting projectors ~character-istic functions!. Propositions will be represented by projec-tors on subspaces of a Hilbert space.Another element we regard as crucial is a feedback. Feed-back means that the system under consideration interactswith some environment. The environment is influenced bythe system and the system reacts to the changes in the envi-ronment. Even the simplest models of such interactions leadeffectively to nonlinear evolution equations @9#. Therefore,instead of modeling the interaction we will say that the feed-back is present if the dynamics of the system is nonlinear,with some restrictions on the form of nonlinearity.we mean two things: ~1! The Hamiltonian functions shouldbe typical of a very large class of dynamical systems, and ~2!the results should not crucially depend on the form of a feed-back, but more on the very fact that the feedback is present.The most universal Hamiltonian functions seem to corre-spond to Hamiltonians with equally spaced spectra or, moreprecisely, whose spectra contain equally spaced subsets. Theclass of Hamiltonians includes harmonic oscillators, quan-tum fields, spin systems, ensembles of identical objects, andmany others. Quite recently, the role of Hamiltonians of theharmonic oscillator type was shown to be relevant to thedynamics of a stock market @11#.A linear Hamiltonian dynamics of r is given by the vonNeumann equationir˙ 5vˆ r , ~1!Quantum morphogenesis: A variaDiederik Aerts,1 Marek Czachor,2 Liane Gabora,1 Maciej Ku1Centrum Leo Apostel (CLEA) and Foundations of the Exact Sc2Wydział Fizyki Technicznej i Matematyki Stosowa3Wydział Fizyki i Matematyki, Uniwer~Received 19 November 2002; published 27 MNoncommutative propositions are characteristic of band psychological! situations. In a Hilbert space mopossible propositions, are represented by density matenvironment, their dynamics is nonlinear. Nonlinear evmorphogenesis that may occur in noncommutativepresented, including ‘‘birth and death of an organism’DOI: 10.1103/PhysRevE.67.051926I. INTRODUCTIONThom’s catastrophe theory is an attempt at finding a uni-versal mathematical treatment of morphogenesis @1#, under-stood as a temporally stable change of form of a system. Thetheory works at a meta level and does not crucially dependon details of interactions that form a concrete ecosystem,organism, or society. In order to achieve this goal, the analy-sis must deal with qualitative classes of objects and has topossess certain universality properties.The purpose of the present work is similar. We define asystem by an abstract space of states. The set of propositionsthat define properties of the system is, in general, non-Boolean. In particular, propositions corresponding to thesame property may not be simultaneously measurable if con-sidered at different times. Also, at the same time there mayexist sets of mutually inconsistent propositions.Although formal logical systems of this type are wellknown from quantum mechanics @2#, it is also known that thescope of applications of non-Boolean logic is much wider@3–7#. Practically, any situation involving contexts belongsto this category. Formally, a context means that a logicalvalue associated with a given proposition depends on thehistory of the system. In particular, the order in which ques-tions are asked is not irrelevant.1063-651X/2003/67~5!/051926~13!/$20.00 67 0519on on Thom’s catastrophe theory,2 Andrzej Posiewnik,3 Jarosław Pykacz,3 and Monika Syty2ces (FUND), Brussels Free University, 1050 Brussels, Belgiumj, Politechnika Gdan´ska, 80-952 Gdan´sk, Polandtet Gdan´ski, 80-952 Gdan´sk, Poland2003; publisher error corrected 3 June 2003!th quantum and nonquantum ~sociological, biological,l, states, understood as correlations between all thees. If systems in question interact via feedback withlutions of density matrices lead to the phenomenon ofstems. Several explicit exactly solvable models arend ‘‘development of complementary properties.’’PACS number~s!: 87.23.Cc, 03.67.Mn, 05.45.YvA system interacting with the environment is statisticallycharacterized by nontrivial conditional probabilities. In thelanguage of non-Kolmogorovian probability calculus, thisimplies that states are not given by simple tensor products ofstates. On the other hand, a simple tensor describes a stateinvolving no correlations and hence neither interactions norfeedback.As a consequence, the nonlinearity representing feedbackshould disappear if the system in question and the environ-ment are in a product state. The latter property may be usedto reduce the class of admissible nonlinear evolutions. In theHilbert-space language, the state of a subsystem is repre-sented by a statistical operator r that is not a projector ~i.e.,r2Þr) whenever the state of the composite system(subsystem1environment) is not a product state. Therefore,the condition r25r characterizes states of subsystems whichdo not interact with the environment. This leads to the fol-lowing restriction: The dynamics of r is linear if r25r .The latter condition is still not restrictive enough since itcan be satisfied by both dissipative and nondissipative evo-lutions @10#. We shall restrict the dynamics to Hamiltoniansystems. In the present paper the Hamiltonian functions willbe time independent, which roughly means that the form ofthe feedback does not change in time.Finally, we want to make the discussion universal. By this©2003 The American Physical Society26-1MAREK CZACHOR et al. PHYSICAL REVIEW E 67, 051926 ~2003!with vˆ r5@H ,r# . Equation ~1! may also be regarded as anabstract representation of a harmonic oscillator. An oscillatoroccurring in many applications in biological sciences is,however, the nonlinear oscillator @12#, whose abstract ver-sion readsir˙ 5(j vˆ j f j~r!. ~2!The ‘‘generic’’ equation, which is the basis of our analysis, istherefore the von Neumann–type equationir˙ 5(j @H j , f j~r!# . ~3!The index j is responsible for the possibility of having dif-ferent parts of the system, which differently interact via thefeedback. For the sake of simplicity, in this paper we restrictthe analysis to only one H and a single f:ir˙ 5@H , f ~r!# . ~4!The only assumptions we make about f are that this is astandard operator function in the sense accepted in spectraltheory of self-adjoint operators, and that it should be linearwhenever there is no feedback. A nontrivial example satisfy-ing all the above requirements is an arbitrary polynomialf ~r!5a01a1r1•••1anrn. ~5!II. RELATION TO REACTION-DIFFUSION MODELSThe typical reaction-diffusion models are of the form@13,14#iX˙ 5vˆ X1vˆ 1 f ~X !, ~6!where vˆ 5A„2, and A and vˆ 1 are, in general complex, ma-trices and X, f (X) are vectors. Particular cases of Eq. ~6! arethe Swift-Hohenberg, l2v , and Ginzburg-Landau models@15–18#.To illustrate what kind of models we arrive at, considerthe quadratic nonlinearity f (r)5r2 and the harmonic oscil-lator Hamiltonian H5(n50‘ nun&^nu. In the simplest case ofa one-dimensional harmonic oscillator, our nonlinear vonNeumann equation ir˙ 5@H ,r2# reads in position space,ir˙ ~x ,y !5~2]x21]y21x22y2!E dzr~x ,z !r~z ,y !. ~7!So even the simplest cases lead to rather complicatedintegro-partial-differential nonlinear equations. Theno-feedback condition implies r(x ,y)5c(x)c¯ (y),*dzc¯ (z)c(z)51, and the equation can be separated yield-ing the Schro¨dinger equationic˙ ~x !5~2]x21x21const!c~x !. ~8!05192The Schro¨dinger equation may be regarded as a diffusionequation in complex time. Similarly, the nonlinear von Neu-mann equations can be mapped into diffusion type equationsby replacing t by it , or by admitting non-Hermitian H j . TheDarboux techniques we are using are not restricted to Her-mitian operators.The self-switching solutions discussed below correspondto certain r(x ,y)Þc(x)c¯ (y). The ‘‘patterns’’ we find in ex-plicit examples are illustrated by the probability densitiespt ,x5^xur tux&5r t~x ,x !. ~9!III. ENTITIES IN ENVIRONMENTSConsider two Hilbert spaces: HE describing an ‘‘environ-ment’’ and spanned by vectors uE& , and He describing an‘‘entity’’ and spanned by vectors ue&. The composite system‘‘environment1entity’’ is represented by either a state vectoruC&5(E ,eCEeuE ,e&5(E ,eCEeuE& ^ ue& ~10!or by a density matrixr5 (EE8ee8rEE8ee8uE ,e&^E8,e8u. ~11!Assuming that all expectation values of random variables arerepresented in terms of quantum averages, we can write^A&C5^CuAuC& ~12!or^A&r5Tr rA . ~13!Of particular interest are averages representing certain statis-tical quantities associated only with the entities, i.e., of theform^I ^ Ae&C5^CuI ^ AeuC&5Tr ereAe ~14!or^I ^ Ae&r5Tr r~I ^ Ae!5Tr e reAe . ~15!The reduced density matrices re are defined, respectively, byre5Tr EuC&^Cu5 (Eee8CEe* CEe8ue&^e8u ~16!orre5Tr Er5 (Eee8rEEee8ue&^e8u. ~17!In particular, for product states, i.e., those of the form6-2QUANTUM MORPHOGENESIS: A VARIATION ON . . . PHYSICAL REVIEW E 67, 051926 ~2003!uC&5(E ,ecEfeuE ,e&5uc& ^ uf& ~18!orr5 (EE8ee8%EE8see8uE ,e&^E8,e8u5% ^ s , ~19!the reduced density matrices are, respectively,re5Tr EuC&^Cu5uf&^fu ~20!andre5Tr Er5s . ~21!In such a case we say that the entity is uncorrelated with theenvironment, i.e., probabilities of events associated with theentity are independent of all the events associated with theenvironment.States of composite systems are of a product form if andonly if entities are uncorrelated with environments. Interac-tions of entities with environments destroy the product formsand introduce correlations.Reduced density matrices corresponding to nontrivial cor-relations satisfy the conditionre2Þre . ~22!Any density matrix r is Hermitian and positive. From thespectral theorem it follows that there exists a basis such thatr is diagonal. For example, any density matrix of an entitycan be written in some basis asre5(epeue&^eu. ~23!Now consider a vectoruC&5(eApeuCe& ^ ue&, ~24!where uCe&PHE are any orthonormal vectors belonging tothe Hilbert space of the environment and ue& are the eigen-vectors of re . ThenTr EuC&^Cu5(epeue&^eu. ~25!In other words, for any density matrix re one can find a stateof the composite system guaranteeing that its reduced den-sity matrix is identical to re . In what follows we shall there-fore assume that a given initial re is a result of correlationsof the entity with the environment. If re2Þre then the corre-lations are nontrivial.05192IV. FEEDBACK WITH THE ENVIRONMENTTypical systems discussed in the biophysics literature in-volve nonlinearities given by nonpolynomial functions f.One often encounters Hill and other functions which are con-tinuous approximations to step functions. A simple one-dimensional reaction-diffusion model describing experimentson regeneration and transplantation in hydra involves nonlin-earities with positive and negative powers @19#. The environ-ment is here modeled by two densities describing concentra-tion of activator and inhibitor producing cells. Essential tothe model is the symmetry breaking of the two densities, afact accounting for the nonsymmetric development of hydra.More refined models @20# do not need externally imposedinhomogeneities but involve environments acting as activechemicals. The aim of complicated feedback behaviors is toaccount for the observed symmetry breaking of the develop-ment of hydra without a need of putting the nonsymmetricelements by hand.A close quantum analog of biophysical dynamical sys-tems is a ‘‘general’’ nonlinear von Neumann equation ~4!@21,22#. If f is to represent a feedback, the nonlinear effectshould disappear if the entity is uncorrelated with the envi-ronment. Assuming that the whole system is represented by astate vector uC& , the lack of correlations implies that uC&5uc& ^ uf& and re5uf&^fu. Such a reduced density matrixsatisfies re25re . The condition ‘‘no correlations, no feed-back’’ is formally translated into@H , f ~r!#5@H ,r# if r25r . ~26!Let us note that the above restriction means that re5uf&^fu satisfies an equation equivalent toiuf˙ &5Huf&. ~27!The latter is a general linear Schro¨dinger equation. In theabsence of feedback the entity evolves according to the rulesof quantum mechanics, an assumption that is rather generaland weak.This property has also another interpretation that is en-tirely ‘‘classical.’’ Consider a system consisting of N classi-cal harmonic oscillators with frequencies v1 , . . . ,vN . De-note by H the diagonal matrix diag(v1 , . . . ,vN) and by uf&a column vector with entries fk5qk1ipk . Then Eq. ~27! isequivalent to the system of classical equations q˙ k5vkpk ,p˙ k52vkqk . As a consequence, the description we proposemay be extended even to fully classical systems modeled byensembles of oscillators evolving linearly and independentlyin the absence of a feedback.Now, what are the restrictions imposed on f by Eq. ~26!?As we have said before, a general density matrix has a formre5(epeue&^eu, where pe are probabilities. The spectraltheorem implies thatf ~re!5(ef ~pe!ue&^eu. ~28!6-3MAREK CZACHOR et al. PHYSICAL REVIEW E 67, 051926 ~2003!The condition re25re implies that pe25pe whose solutionsare 0 and 1. Therefore Eq. ~26! is satisfied by any f thatfulfills f (0)50 and f (1)51. In practical computations, onecan relax Eq. ~26! by requiring only@H , f ~r!#;@H ,r# if r25r , ~29!since having Eq. ~29! one can always reparametrize the timevariable t so that Eq. ~26! is satisfied. The polynomial men-tioned in the Introduction belongs to this category.Equation ~4! possesses a number of interesting generalproperties. For example, the quantitiesh5Tr H f ~r!, ~30!cn5Tr ~rn!, ~31!for all natural n, are time independent. h is the Hamiltonianfunction for the dynamics and, hence, plays the role of theaverage energy of the entity ~the feedback energy included!.An analogous situation occurs in nonextensive statisticswhere h has an interpretation of internal energy @22,36#. Asystem with conserved h is closed.Conservation of cn implies that eigenvalues of r are con-served. The latter property means that there are certain fea-tures of the system that occur with time independent prob-abilities. However, and this is very important, the featuresthemselves change in time in a way that is rather unusual inphysical systems and has many analogies in evolution ofbiological systems.V. SOLITON MORPHOGENESISThere exists a class of solutions of Eq. ~4! which exhibitsa kind of a three-regime switching effect @23–25#: For times2‘,t!t1 the dynamics looks as if there was not feedback,then in the switching regime t1,t,t2 a ‘‘sudden’’ transitionoccurs, driving the system into a new state that for times t2!t,‘ evolves again as if there was no feedback. Of course,the feedback is present for all times, but is ‘‘visible’’ onlyduring the switching period. Formally, the effect is verysimilar to scattering between two asymptotically linear evo-lutions ~‘‘self-scattering’’!. One can additionally complicatethe dynamics by introducing an external element that makesthe form of the feedback time dependent. We shall illustratethe effect on explicit examples.The general equation ~4! belongs to the family of equa-tions integrable by means of soliton methods. One beginswith its Lax representationzl^cu5^cu~r2lH !, ~32!2i^c˙ u51l^cu f ~r!. ~33!The construction requires two additional Lax pairszn^xu5^xu~r2nH !, ~34!051922i^x˙ u51n^xu f ~r!, ~35!zmuw&5~r2mH !uw&, ~36!iuw˙ &51mf ~r!uw& . ~37!The method of solving Eq. ~4! is based on the followingtheorem establishing the Darboux covariance of the Lax pair~32!, ~33! @25#.Theorem. Assume that ^cu, ^xu, and uw& are solutions ofEqs. ~32!, ~33!, and ~34!–~37! and ^c1u, r1 are defined by^c1u5^cuS 11n2mm2l P D , ~38!r15S 11 m2nn P D rS 11 n2mm P D , ~39!P5uw&^xu^xuw&. ~40!Thenzl^c1u5^c1u~r12lH !, ~41!2i^c˙ 1u51l^c1u f ~r1!, ~42!ir˙ 15@H , f ~r1!# . ~43!Let us note that the theorem is valid even for non-HermitianH, i.e., for open systems. However, in the present paper werestrict the analysis to closed ~conservative! systems charac-terized by self-adjoint H. Systems whose average populationdoes not change belong to this class.One of the strategies of finding the ‘‘switching solutions’’is the following. One begins with a seed solution r such thatthe operatorDa“ f ~r!2ar , ~44!where @a ,H#5@a ,r#50, satisfies @Da ,H#50 and Da is nota multiple of the identity. Now we can writeir˙ 5@H , f ~r!#5a@H ,r# ~45!andr~ t !5e2iaHtr~0 !eiaHt. ~46!Taking the Lax pairs with m5n¯ and repeating the construc-tion from Refs. @23,24#, we get6-4QUANTUM MORPHOGENESIS: A VARIATION ON . . . PHYSICAL REVIEW E 67, 051926 ~2003!r1~ t !5e2iaHt$r~0 !1~n¯2n!Fa~ t !21e2iDat/n¯3@ ux~0 !&^x~0 !u,H#eiDat/n%eiaHt, ~47!whereFa~ t !5^x~0 !uexpS i n¯2nunu2Dat D ux~0 !&and ^x(0)u is an initial condition for the solution of the Laxpair.VI. ‘‘SUDDEN’’ MUTATION OF POPULATIONIn our first example we consider the quadratic nonlinear-ity f (r)5(12h)r1hr2. The parameter h controls thestrength of the feedback. However, for any h and any densitymatrix satisfying r25r we find f (r)5r and the feedbackvanishes. This is consistent with our assumption that r25rcharacterizes systems not interacting with an environment.We take the Hamiltonian H5(n50‘ nun&^nu which may rep-resent a system whose energy is proportional to the numberof its elements. Solutions of the von Neumann equation are,in general, infinite dimensional, but in order to illustrate themorphogenesis we restrict the analysis to a finite dimension.The lowest dimension where the effect occurs is 3. There-fore, we select a subspace spanned by three subsequent vec-tors uk& , uk11&, and uk12&. We will discuss a family, pa-rametrized by aPR, of self-switching solutions r t5(m ,n502 rmnuk1m&^k1nu of Eq. ~4!. The solution is com-pletely characterized by the matrix of time-dependent coef-ficients rmn . Here we only give the final result and postponea detailed derivation to Sec. VIII where we analyze a gener-alization involving a greater number of ‘‘different species.’’The reader may check by a straightforward substitution thatthe matrixS r00 r01 r02r10 r11 r12r20 r21 r22D 5 1151A5 S 5 j~ t ! z~ t !j¯~ t ! 51A5 j~ t !z¯~ t ! j¯~ t ! 5D~48!withj~ t !5~213i2A5i !A31A5aA3~egt1a2e2gt!eiv0t,z~ t !529e2gt1~114A5i !a23~e2gt1a2!e2iv0tis indeed a solution of the von Neumann equation.The parameters are v0512(51A5)h/(151A5), g52h/(151A5).There exists a critical value h05(151A5)/(51A5) cor-responding to v050. Using the explicit position dependenceof the eigenstates of the harmonic oscillator Hamiltonian we05192can make a plot illustrating the time dependence of the prob-ability density pt ,x in position space as a function of time andh.The dynamics we encounter in this example is particularlysuggestive for h5h0 ~Fig. 1! and resembles a mutation ofthe statistical ensemble described by r . The correspondingprobability appears static for, roughly, 2‘,t,240 andthen also for 40,t,‘ . Switching is ‘‘suddenly’’ triggered ina neighborhood of t50. Figure 2 shows the evolution of theprobability density at the origin pt ,0 as a function of time forFIG. 1. Probability density pt ,x5^xur tux& for the critical valueh05(15151/2)/(5151/2) as a function of time and x for 240,t,40 ~in arbitrary units!. The three regimes are clearly visible. Theprobability interpolates between asymptotic probabilities that areconstant in time. The visible switching ~morphogenesis! beginsaround t5230 and takes approximately 30 units of time. For latertimes the probability density becomes indistinguishable from thenew asymptotic state.FIG. 2. Probability density pt ,0 at x50 as a function of time andh for h0<h<2.45. For h.h0 the switching around t50 takesplace between two different asymptotic oscillating probability den-sities. The switching is absent only for h50 ~not shown! where thedynamics is linear.6-5MAREK CZACHOR et al. PHYSICAL REVIEW E 67, 051926 ~2003!tem with equally spaced spectrum, say, a three-dimensionalharmonic oscillator! we would have obtained a differentshape of the probability density. Although different choicesof H imply different differential equations, their commonfeature is the effect of ‘‘mutation.’’VII. COMPOSITE ENTITIES: BIRTH AND DEATHOF AN ORGANISMIn this example we consider an organism, that is, a com-posite entity undergoing the feedback process as a whole. Asimple model consists of a two-qubit system described by theHamiltonianH5H1 ^ 111^ H2 . ~49!The Hamiltonian does not contain an interaction term. How-ever, the two subentities forming the ‘‘organism’’ do notevolve independently. They are coupled to each otherthrough the feedback with the environment, i.e., through thenonlinearity. As we shall see, they become asymptoticallyuncoulped at t→6‘ . In a ‘‘distant past’’ the system consistsof uncorrelated subentities that, after a period of certain jointactivity, become again uncorrelated in the future. An analogywith ‘‘birth’’ and ‘‘death’’ is striking, and justifies the name‘‘organism.’’To make the example concrete, assume thatr int~ t !512 152A7 tanh 2t 00 51A7 tanh 2t13i23A72A151iA1058 cosh 2t215i1A72A151iA1058 cosh 2t7i13A723A152iA1058 cosh 2tA71A152 cosh 2t05192H5S 0 0 21 20 0 2 21D . ~52!The density matrixr~ t !5exp@25iHt#r~0 !exp@5iHt# ~53!is a solution of Eq. ~4! with f (r)5r2. Such a r(t) describessimultaneously the dynamics of two noninteracting systemssatisfying the linear von Neumann equationir˙ 55@2sx ^ 111^ sz ,r# . ~54!To understand why this happens, it is sufficient to note thatthe solution satisfies@H ,r2#5@H ,5r#5@5H ,r# . ~55!The environment does not trigger in this solution any switch-ing, but only makes its evolution five times faster than in theabsence of the feedback. The Darboux transformation, whenapplied to r(t), produces ~for more details, cf. Ref. @23#! thesolutionr1~ t !5exp@25iHt#r int~ t !exp@5iHt# , ~56!where213i23A72A152iA1058 cosh 2t27i13A723A151iA1058 cosh 2t15i1A72A152iA1058 cosh 2tA71A152 cosh 2t51A15 tanh 2t 00 52A15 tanh 2t2 .~57!different values of h. For hÞh0 the probability density is anoscillating function of time, but in the neighborhood of t50 one observes the ‘‘mutation’’ that occurs for any hÞ0,the longer the transition period, the smaller h. Duration ofthe switching process is of the order 1/h . For h50 the dy-namics is linear ~no feedback! and there is no switching. Theexample shows that there occurs a kind of uncertainty rela-tion between the strength of the feedback and duration of theswitching: The smaller the feedback, the longer the switch-ing period.Let us note that the probability density shown in Fig. 1has this particular shape since we have used the position-space wave functions characteristic of a quantum one-dimensional harmonic oscillator ~a Gaussian times Hermitepolynomials!. Had we chosen any other system that is isos-pectral to a one-dimensional harmonic oscillator ~or any sys-H52sx ^ 111^ sz . ~50!We will start with the non-normalized density matrixr~0 !512 S 51A7 0 0 00 52A7 0 00 0 51A15 00 0 0 52A15D ,~51!which is written in such a basis that1 2 0 02 1 0 06-6QUANTUM MORPHOGENESIS: A VARIATION ON . . . PHYSICAL REVIEW E 67, 051926 ~2003!Now the switching between the two asymptotic evolutions istriggered in the neighborhood of t50.If we look at the subentities forming the organism, wenotice that they do not evolve independently. The easiest wayof seeing this is to compute the reduced density matrices ofthe two subentities. Here we write explicitly the eigenvaluesof the reduced density matrices. Both subsystems are twodimensional, so there are two eigenvalues for each reduceddensity matrix. They readp6~1 !512 6A152A720 tanh 2t ~particle 1 !, ~58!p6~2 !5126A2612A10540 cosh 2t~particle 2 !. ~59!The asymptotics arer int~2‘!512 S 52A7 0 0 00 51A7 0 00 0 52A15 00 0 0 51A15D ,~60!r int~1‘!512 S 51A7 0 0 00 52A7 0 00 0 51A15 00 0 0 52A15D5r~0 !, ~61!and therefore the dynamics represents asymptotically twononinteracting subentities. It is also interesting that the 1‘asymptotics is r1(t)’r(t). At large times an ‘‘organism’’that ‘‘dies’’ becomes practically indistinguishable from theone that never ‘‘lived.’’The ‘‘life’’ of the organism is the period of time when thetwo subentities exhibit certain joint activity. Computing thevon Neumann entropies of reduced density matrices of thetwo subentities, we can introduce a quantitative measure ofthis activity. The entropies of the two particles are shown inFig. 3. The organism lives several units of time. Similar arethe scales of time when the off-diagonal matrix elements ofr int(t) become non-negligible. It should be stressed that theentropy characterizing the entire organism is time indepen-dent @since eigenvalues of solutions of Eq. ~4! are constantsof motion for all f ].Although it is clear that the ‘‘organism’’ behaves duringthe evolution as an indivisible entity, one should not confusethis indivisibility with the so-called nonseparability dis-cussed in quantum information theory. The organism we con-sider in the example is a two-qubit system and therefore onecan check the separability of r1(t) by means of the Peres-Horodecki partial transposition criterion @26,27#: A two qubitdensity matrix r is separable if and only if its partial trans-position is positive. It turns out that partial transposition of05192r1(t) is positive for any t and, hence, r1(t) is in this senseseparable ~has ‘‘zero entanglement’’!. It is well known, how-ever, that ‘‘zero entanglement’’ does not mean ‘‘no quantumcorrelations’’ in the system. The so called three-particleGreenberger-Horne-Zeilinger state @28# is fully entangled atthe three-particle level in spite of the fact that all its two-particle subsystems are described by separable density ma-trices.VIII. MODEL WITH SEVERAL SPECIESThe models we have considered so far corresponded to aHilbert space with basis vectors un& . The only characteriza-tion of a state was in terms of the quantum number n thatcould be regarded as the number of elements of a givenpopulation. Now we want to extend the description to thesituation where we have a population consisting of severalspecies characterized by numbers n1 , . . . ,nN . The basisvectors areun&5un1 , . . . ,nN&5un1& ^ •••^ unN& ~62!and the HamiltonianH5(n j~n11•••1nN!un1 , . . . ,nN&^n1 , . . . ,nNu ~63!5(nEnun&^nu. ~64!The Hamiltonian has an equally spaced spectrum and is for-mally very similar to those we have encountered in the pre-vious sections. The difference is that now the energy eigen-states are highly degenerated, a property that is very usefulfrom the perspective of constructing multiparameter andhigher-dimensional self-switching solutions.For simplicity consider two species (N52), the quadraticnonlinearityFIG. 3. Life and death of the two-qubit organism: the von Neu-mann entropies of particles 1 ~solid curve! and 2 ~dashed curve!.The times where the particles are practically independent corre-spond to the flat parts of the plots.6-7j(1MAREK CZACHOR et al. PHYSICAL REVIEW E 67, 051926 ~2003!wherer j~0 !5a2 ~ u0 j&^0 ju1u2 j&^2 ju!1a1Aa214~b2m2!2 u1 j&^1 ju2Aa214b23~ u2 j&^0 ju1u0 j&^2 ju!. ~69!Positivity of r j(0) restricts the parameters as follows: 0,4m2,a214b,a2. The operatorDa5r~0 !22ar~0 !5bI˜2m2(j50lu1 j&^1 ju ~70!commutes with H. We denote by I˜ and H˜ the restrictions ofthe identity I and H to the 3(l11)-dimensional subspacespanned by vectors ~65!–~67!. We will write H˜ 5( j50l H j ,whereH j5 (n502~k1nm !un j&^n ju. ~71!Consider the eigenvalue problemr1~ t !5r~0 !12i~ uau21e2m2t/(12a)ubu2!21 (j , j850l@~a juwThe vectorsuF j~ t !&5a juw j(1)&1b jem2t/(12a)uw j(2)&5uf j~ t !& ^ u j&,~81!where05192@r~0 !2iH#uw&5zuw&, ~76!with the same z for anyuw&5(j50l~a juw j(1)&1b juw j(2)&). ~77!The self-switching solution can thus be constructed bymeans of uw& and readsr1~ t !5e2i[11h(a21)]Ht$r~0 !12iFa~ t !21e2hDat3@ uw&^wu,H#e2hDat%ei[11h(a21)]Ht, ~78!withFa~ t !5^wuexp~22hDat !uw&5e22hbt(j50l~ ua ju21e2hm2tub ju2!5e22hbt~ uau21e2hm2tubu2!.~79!Probabilities analogous to Fig. 1 are found if a and h aretuned in a way that eliminates the oscillating parte2i[11h(a21)]Ht, i.e., for h51/(12a). In this case)&1b jem2t/12auw j(2)&)~a¯ j8^w j8(1)u1b¯ j8em2t/(12a)^w j8(2)u!,H# .~80!uf j~ t !&5a jA2 S 2 2im1Aa214~b2m2!Aa214b uk2 j&1uk12m2 j& D 1b jem2t/(12a)uk1m2 j&~82!u2 j&5uk12m2 j , j& , ~67!j50,1, . . . ,l<k . We start with the unnormalized densitymatrixr~0 !5(j50lr j~0 !, ~68!uw j(2)&5u1 j& ~74!correspond to the same j-independent eigenvaluez5a1Aa214~b2m2!2 1~k1m !i , ~75!and thereforef ~r!5~12h !r1hr2,and take some three energy eigenvalues Ek , Ek1m , Ek12m .For each energy take l11 vectors, which will be denoted byu0 j&5uk2 j , j&, ~65!u1 j&5uk1m2 j , j& , ~66!@r j~0 !2iH j#uw j&5zuw j&. ~72!We find that the two solutionsuw j(1)&522im1Aa214~b2m2!A2Aa214bu0 j&11A2u2 j&, ~73!6-8QUANTUM MORPHOGENESIS: A VARIATION ON . . . PHYSICAL REVIEW E 67, 051926 ~2003!are orthogonal for different j. Denoting H5H1 ^ I1I ^ H2,one can easily compute the reduced density matrix of thefirst species,r1I ~ t !5Tr 2r1~ t !5r I~0 !12i~ uau21e2m2t/(12a)ubu2!213F (j50luf j~ t !&^f j~ t !u,H1G . ~83!The entire information about the dynamics of the first speciesis encoded in r1I (t). Changes of properties of the species aregiven by the matrix elements^nur1I ~ t !un8&5^nur I~0 !un8&12i~n82n !3(j50l^nuf j~ t !&^f j~ t !un8&uau21e2m2t/(12a)ubu2. ~84!An immediate conclusion from the above formula is that forn5n8 the expression is time independent. It follows that thenumber of elements of the ensemble does not change duringthe evolution. What changes are certain properties of theensemble.A. Example: k˜m˜1, j˜0,1, a˜5, b˜À4The two-species states in the subspace in question areu00&5u1,0& , ~85!u10&5u2,0& , ~86!u20&5u3,0& , ~87!u01&5u0,1& , ~88!u11&5u1,1& , ~89!u21&5u2,1& . ~90!The two-species initial seed density matrix is given byr0~0 !552 ~ u1,0&^1,0u1u3,0&^3,0u!151A52 u2,0&^2,0u232 ~ u3,0&^1,0u1u1,0&^3,0u!, ~91!r1~0 !552 ~ u0,1&^0,1u1u2,1&^2,1u!151A52 u1,1&^1,1u232 ~ u2,1&^0,1u1u0,1&^2,1u!, ~92!r~0 !5r0~0 !1r1~0 !. ~93!Assume that a05a151/A2, b05et0/4, b15et1/4. Then05192uf0~ t !&512 S 2 2i1A53 u1&1u3& D 1e (t02t)/4u2& , ~94!uf1~ t !&512 S 2 2i1A53 u0&1u2& D 1e (t12t)/4u1& . ~95!Writing the restriction of H to the six-dimensional subspaceasH˜ 5S 1 0 0 0 0 00 1 0 0 0 00 0 2 0 0 00 0 0 2 0 00 0 0 0 3 00 0 0 0 0 3D ~96!we can represent the two-species density matrix r1 in theformr15S 52 1 22iA53 j 2 32 11 22iA53 z21iA53 jT 51A53 1 ijT232 1121iA53 z2ij52 1D ,~97!where 1 is the 232 unit matrix, T denotes transposition, andj~ t !5et/4et/21et0/21et1/2S et1/4 et0/4et1/4 et0/4D , ~98!z~ t !5et/2et/21et0/21et1/2S 1 11 1 D . ~99!One can verify by a straightforward calculation that(h52 14 )ir˙ 1554 @H ,r1#214 @H ,r12# . ~100!To illustrate the time variation of statistical quantities asso-ciated with the two-species system it is sufficient to visualizethe behavior of matrix elements of r1. There are only threetypes of functions occurring in r1:F~ t !5et/2et/21et0/21et1/2, ~101!F0~ t !5e (t1t0)/4et/21et0/21et1/2, ~102!6-9FFMAREK CZACHOR et al. PHYSICAL REVIEW E 67, 051926 ~2003!proposition is a projector, i.e., an operator with eigenvalues 1and 0 ~logical ‘‘true’’ and ‘‘false’’!. Propositions that can beasked simultaneously are represented by commuting projec-tors. Propositions P1 , P2, which do not commute, are re-lated by an uncertainty relation: The more is known aboutP1, the less is known about P2, and vice versa.It is obvious that the above structures do not have to beassociated with quantum systems. Just to give an example,many psychological tests are based on questionnaires thatinvolve the same question asked many times in different con-texts. The questions commute if the answer to a given ques-tion is always the same. However, in typical situations the051926P512 S 1 0 1 00 0 0 01 0 1 00 0 0 0D ~108!corresponding to the first species as follows:p~ t !5Tr Pr1I ~ t !Tr r1I ~ t !51491A518F~ t !/3151A5. ~109!r1II5S 2 321iA53 F12iF0151A52D . ~107!Of course, all the density matrices are not normalized so thataverages must be computed according to ^A&5Tr Ar1 /Tr r1, etc. ~note that Tr r1 is time independent!.IX. MORPHOGENESIS OF COMPLEMENTARITYAccording to the sufficiency of subsystem correlations~SSC! @30,31# a density matrix r is uniquely determined bycorrelations between all the possible propositions associatedwith a given system. Each matrix element of a r can begiven an interpretation in terms of probabilities associatedwith some proposition. In the Hilbert space language ations asked in all the possible orders. In the Hilbert spaceformalism, where the questions are represented by projec-tors, an ideal questionnaire encodes all the possible correla-tions and thus, via the SSC theorem, is equivalent to a den-sity matrix.It is also known that there exist simple examples of sys-tems whose logic is non-Boolean, but which do not allow aHilbert space formulation @29#. The density matrix languagewill probably not suffice here and one has to admit a possi-bility of other state spaces and other nonlinear evolutions.The richness of available structures is immense.Let us finally give examples of propositions whose aver-ages ~i.e., probabilities! change in time according to selectedmatrix elements of the self-switching solutions. The functionF(t) shown in Fig. 4 is associated with the propositionF1~ t !5e (t1t1)/4et/21et0/21et1/2. ~103!The two parameters, t0 and t1, control two types of three-regime behaviors of r1. The function F is responsible forasymptotic properties of r1 ~via z). Functions F0 and F1determine properties of the switching regime ~via j). Fort0!t1 one finds F0(t)’0 for all t and the switching is con-trolled by F(t) and F1(t); the ‘‘moment’’ of switching isshifted proportionally to t1. For t0@t1 one finds F1(t)’0for all t and the switching is controlled by F(t) and F0(t);r1I 515222iA53 F121iA53 F1 51A52232 121iA53 F21iA53 F02i0 232 121iA53151A5 22iA5F11iF0the ‘‘moment’’ of switching does not depend on t1 and isdetermined by t0. Therefore, the two types of switches arecharacterized by vanishing of those matrix elements of r1which contain either F0 or F1.The asymptotic behavior of the system is given byF0~6‘!5F1~6‘!5F~2‘!50, ~104!F~1‘!51. ~105!The reduced density matrices of single species are232 122iA53 F022iA53 F01iF1 232 122iA53 F1 51A52iF02iF0522 , ~106!same question has different answers within a single question-naire. An ideal questionnaire involves all the possible ques--10QUANTUM MORPHOGENESIS: A VARIATION ON . . . PHYSICAL REVIEW E 67, 051926 ~2003!Here p(t) is the probability of the answer ‘‘true’’ associatedwith P. Analogously, F1(t) shown at Fig. 5 is associatedwith the propositionP1512 S 1 1 0 01 1 0 00 0 0 00 0 0 0D ~110!by means ofp1~ t !5Tr P1r1I ~ t !Tr r1I ~ t !514151A518F1~ t !/3151A5. ~111!The evolution of the probabilities resembles the well knownevolutions typically modeled by Hill functions @12# in the socalled sigmoidal response models @32–35,37#. Square devia-tions associated with the two propositions satisfy the uncer-tainty relationFIG. 4. F(t) as a function of t and t1 for t05150.FIG. 5. F1(t) as a function of t and t1 for t05150.051926DPDP1>12 UTr @P ,P1#r1I ~ t !Tr r1I ~ t ! U5UA5~et/21e (t1t0)/4!2~31A5 !e (t1t1)/412~151A5 !~et/21et0/21et1/2! U .~112!For any fixed t0 , t1, the right-hand side of the inequalityvanishes for t→2‘ and approaches A5/@12(151A5)# fort→1‘ . Figure 6 shows this function for t150. The twopropositions that were not complementary in the past evolveinto propositions satisfying an uncertainty relation.In application to psychology, a density matrix may repre-sent an ideal questionnaire and, hence, a state of personalityof a given individual. The morphogenesis we have discussedis a simple model of development of two complementaryconcepts. The model is simplified and perhaps too farfetched. However, philosophically this is not very far fromthe approaches of Thom @1# and particularly of Zeeman @38#in their catastrophe theory models of the brain. More inter-esting in this context may be infinite dimensional caseswhose preliminary analysis in terms of Darboux transforma-tions for arbitrary f (r) can be found in Ref. @25#.X. DISCUSSIONThe model we have described satisfies the assumptionsimposed by Thom on a system of forms in evolution ~Chap.1.2.A of Ref. @1#!. The model is continuous and the morpho-genesis is a result of soliton dynamics. In this respect, theconstruction is analogous to nonlinear sigmoidal responsemodels used in biochemistry @37#. What makes our construc-tion essentially different from the models one finds in theliterature is the role of noncommutativity of the system ofpropositions.FIG. 6. Morphogenesis of complementarity. The right-hand sideof the uncertainty relation for standard deviations DP and DP1 as afunction of time t and the parameter t0 (t150). Propositions P andP1 are the more complementary, the greater the value of this func-tion.-11The class of solutions which has an interpretation in terms The work of M.C. and M.K. is a part of the KBN ProjectMAREK CZACHOR et al. PHYSICAL REVIEW E 67, 051926 ~2003!of morphogenesis has features that do not crucially dependon the form of the nonlinearity, but more on the very pres-ence of a feedback. The exact time development ~say, dura-@1# R. Thom, Structural Stability and Morphogenesis: An Outlineof a General Theory of Models ~Benjamin, Reading, MA,1975!.@2# E.G. Beltrametti and G. Cassinelli, The Logic of Quantum Me-chanics ~Addison-Wesley, Reading, MA, 1981!.@3# D. Aerts and S. Aerts, Found. Sc. 1, 85 ~1994!.@4# L. Gabora, Doctoral thesis, CLEA, Brussels Free University,2001.@5# D. Aerts, B. Coecke, and B. D’Hooghe, Helv. Phys. Acta 70,793 ~1997!.@6# D. Aerts, S. Aerts, J. Broekaert, and L. Gabora, Found. Phys.30, 1387 ~2000!.@7# E.W. Piotrowski and J. Sl”adkowski, Acta Phys. Pol. B 32, 3873~2001!.@8# D. Aerts, J. Math. Phys. 24, 2441 ~1983!.@9# V.P. Belavkin and V.P. Maslov, Theor. Math. Phys. 33, 17~1977!.@10# S. Gheorghiu-Svirschevski, Phys. Rev. A 63, 022105 ~2001!.@11# J. Sl”adkowski, e-print cond-mat/0211083.@12# L. Glass and M.C. Mackey, From Clocks to Chaos: TheRhythms of Life ~Princeton University Press, Princeton, 1988!.@13# J.D. Murray, Nonlinear Differential Equation Models in Biol-ogy ~Clarendon, Oxford, 1977!.@14# P.C. Fife, Mathematical Aspects of Reacting and Diffusing Sys-051926No. 5 P03B 040 20. We would like to acknowledge the sup-port of the Flemish Fund for Scientific Research ~FWOProject No. G.0335.02!.tems, Lecture Notes in Biomathematics Vol. 28 ~Springer, NewYork, 1979!.@15# J. Swift and P.C. Hohenberg, Phys. Rev. A 15, 319 ~1977!.@16# L.N. Howard and N. Kopell, Stud. Appl. Math. 56, 95 ~1977!.@17# V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064~1950!.@18# K. Stewartson and J.T. Stuart, J. Fluid Mech. 48, 529 ~1971!.@19# A. Gierer and H. Meinhardt, Kybernetik 12, 30 ~1972!.@20# W. Kemmner, Differentiation 26, 83 ~1984!.@21# M. Czachor, Phys. Lett. A 225, 1 ~1997!.@22# M. Czachor and J. Naudts, Phys. Rev. E 59, R2497 ~1999!.@23# S.B. Leble and M. Czachor, Phys. Rev. E 58, 7091 ~1998!.@24# M. Czachor, H.D. Doebner, M. Syty, and K. Wasylka, Phys.Rev. E 61, 3325 ~2000!.@25# N.V. Ustinov, M. Czachor, M. Kuna, and S.B. Leble, Phys.Lett. A 279, 333 ~2001!.@26# A. Peres, Phys. Rev. Lett. 77, 1413 ~1996!.@27# M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A223, 1 ~1996!.@28# D.M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger,Am. J. Phys. 58, 1131 ~1990!.@29# D. Aerts, J. Math. Phys. 27, 202 ~1986!.@30# S. Bergia, F. Cannata, A. Cornia, and R. Livi, Found. Phys. 10,723 ~1980!.Neumann equations. The formalism allows to consider mor-phogenesis of a completely new type, for example, a devel-opment of complementary properties. ACKNOWLEDGMENTSNoncommutative propositions are related by uncertaintyprinciples and are typical of systems that cannot, without anessential destruction, be separated into independent parts.The examples can be taken not only from quantum physics,but also from sociology ~communities!, psychology ~person-alities!, or biology ~organisms!. In all these cases the dynam-ics of a system consists of two parts: One generated by in-ternal interactions, and the other corresponding to couplingswith environment. We have considered only the simplestcase where the internal dynamics is given a priori by aHamiltonian of a harmonic oscillator type, and different partsof an organism ~community, etc.! are coupled to each otheronly via the environment.The coupling with environment leads to a feedback and,hence, nonlinear evolution. The systems we consider areconservative, but without difficulties can be generalized toexplicitly time-dependent environments or non-HermitianHamiltonians.We model propositions by projectors on subspaces of aHilbert space. States of the systems are represented by all thepossible correlations between all the possible, even noncom-muting, propositions. The choice of the Hilbert space lan-guage leads us therefore to a density matrix representation ofstates, and the dynamics is given in terms of nonlinear vontion! of morphogenesis does depend on initial conditions orthe form of nonlinearity. However, the modifications do notinfluence the asymptotics, which is the qualitative element ofthe dynamics. For example, an organism that was ‘‘born’’ hasto ‘‘die’’ but when and how will this occur depends on manydetails that are qualitatively irrelevant.Instead of conclusions, let us quote Thom’s final remarksfrom his early work on topological models in biology @39#.‘‘Practically any morphology can be given such a dynami-cal interpretation, and the choice between possible modelsmay be done, frequently, only by qualitative appreciation andmathematical sense of elegance and economy. Here we donot deal with a scientific theory, but more precisely with amethod. And this method does not lead to specific tech-niques, but, strictly speaking, to an art of models. What maybe, in that case, the ultimate motivation to build such mod-els? They satisfy, I believe, a very fundamental epistemologi-cal need . . . . If scientific progress is to be achieved by othermeans than pure chance and lucky guess, it relies necessarilyon a qualitative understanding of the process studied. Ourdynamical schemes . . . provide us with a very powerfultool to reconstruct the dynamical origin of any morphologi-cal process. They will help us, I hope, to a better understand-ing of the structure of many phenomena of animate and in-animate nature, and also, I believe, of our own structure.’’-12@31# N.D. Mermin, Am. J. Phys. 66, 753 ~1998!.@32# C. Walter, R. Parker, and M. Ycas, J. Theor. Biol. 15, 208~1967!.@33# L. Glass and S.A. Kauffman, J. Theor. Biol. 34, 219 ~1972!.@34# L. Glass and S.A. Kauffman, J. Theor. Biol. 39, 103 ~1973!.@35# J.J. Hopfield and D.W. Tank, Science ~Washington, DC, U.S.!233, 625 ~1986!.@36# C. Tsallis, R.S. Mendes, and A.R. Plastino, Physica A 261, 534~1998!.@37# S.A. Kauffman, The Origins of Order. Self-Organization andSelection in Evolution ~Oxford University Press, Oxford,1993!.@38# E.C. Zeeman, Int. J. Neurosci. 6, 39 ~1973!.@39# R. Thom, in Towards a Theoretical Biology, edited by C.H.Waddington ~Aldine, Chicago, 1970!, Vol. 3.QUANTUM MORPHOGENESIS: A VARIATION ON . . . PHYSICAL REVIEW E 67, 051926 ~2003!051926-13

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