Information Geometry, Inference Methods and Chaotic Energy Levels Statistics
aa r X i v : . [ m a t h - ph ] O c t Information Geometry, Inference Methods and Chaotic Energy Levels Statistics
Carlo Cafaro ∗ Department of Physics, State University of New York at Albany-SUNY,1400 Washington Avenue, Albany, NY 12222, USA
In this Letter, we propose a novel information-geometric characterization of chaotic (integrable)energy level statistics of a quantum antiferromagnetic Ising spin chain in a tilted (transverse) externalmagnetic field. Finally, we conjecture our results might find some potential physical applications inquantum energy level statistics.
PACS numbers: 02.50.Tt- Inference methods; 02.40.Ky- Riemannian geometry; 02.50.Cw- Probability theory;05.45.-a- Nonlinear dynamics and chaos
I. INTRODUCTION
Research on complexity [1] has created a new set of ideas on how very simple systems may give rise to very complexbehaviors. Moreover, in many cases, the ”laws of complexity” have been found to hold universally, caring not at allfor the details of the system’s constituents. Chaotic behavior is a particular case of complex behavior and it will bethe object of the present work.In this Letter we make use of the so-called Entropic Dynamics (ED) [2]. ED is a theoretical framework that arisesfrom the combination of inductive inference (Maximum Entropy Methods (ME), [3]) and Information Geometry(IG) [4]. The most intriguing question being pursued in ED stems from the possibility of deriving dynamics frompurely entropic arguments. This is clearly valuable in circumstances where microscopic dynamics may be too farremoved from the phenomena of interest, such as in complex biological or ecological systems, or where it may justbe unknown or perhaps even nonexistent, as in economics. It has already been shown that entropic arguments doaccount for a substantial part of the formalism of quantum mechanics, a theory that is presumably fundamental [5].Perhaps the fundamental theories of physics are not so fundamental; they may just be consistent, objective waysto manipulate information. Following this line of thought, we extend the applicability of ED to temporally-complex(chaotic) dynamical systems on curved statistical manifolds and identify relevant measures of chaoticity of such aninformation geometrodynamical approach to chaos (IGAC).The layout of this Letter is as follows. In section II, we present an introduction to the main features of our IGAC.In section III, using our information geometrodynamical approach to chaos and following the results provided byProsen [6], we propose a novel information-geometric characterization of chaotic (integrable) energy level statistics ofa quantum antiferromagnetic Ising chain in a tilted (transverse) external magnetic field. We emphasize that we haveomitted technical details that will appear elsewhere. However, some applications of our IGAC to low dimensionalchaotic systems can be found in our previous articles [7, 8, 9, 10]. Finally, in section IV we present our final remarks.
II. THEORETICAL STRUCTURE OF THE IGAC
IGAC arises as a theoretical framework to study chaos in informational geodesic flows describing, physical, biologicalor chemical systems. It is the information-geometric analogue of conventional geometrodynamical approaches [11, 12]where the classical configuration space Γ E is being replaced by a statistical manifold M S with the additional possibilityof considering chaotic dynamics arising from non conformally flat metrics (the Jacobi metric is always conformallyflat, instead). It is an information-geometric extension of the Jacobi geometrodynamics (the geometrization of aHamiltonian system by transforming it to a geodesic flow [13]). The reformulation of dynamics in terms of a geodesicproblem allows the application of a wide range of well-known geometrical techniques in the investigation of the solutionspace and properties of the equation of motion. The power of the Jacobi reformulation is that all of the dynamicalinformation is collected into a single geometric object in which all the available manifest symmetries are retained- themanifold on which geodesic flow is induced. For example, integrability of the system is connected with existence ofKilling vectors and tensors on this manifold. The sensitive dependence of trajectories on initial conditions, which is ∗ Electronic address: [email protected] a key ingredient of chaos, can be investigated from the equation of geodesic deviation. In the Riemannian [11] andFinslerian [12] (a Finsler metric is obtained from a Riemannian metric by relaxing the requirement that the metric bequadratic on each tangent space) geometrodynamical approach to chaos in classical Hamiltonian systems, an activefield of research concerns the possibility of finding a rigorous relation among the sectional curvature, the Lyapunovexponents, and the Kolmogorov-Sinai dynamical entropy (i. e. the sum of positive Lyapunov exponents) [14]. Usinginformation-geometric methods, we have investigated in some detail the above still open research problem [8]. Oneof the goals of this Letter is that of representing an additional step forward in that research direction.
A. General Formalism of the IGAC
The IGAC is an application of ED to complex systems of arbitrary nature. ED is a form of information-constraineddynamics built on curved statistical manifolds M S where elements of the manifold are probability distributions { P ( X | Θ) } that are in a one-to-one relation with a suitable set of macroscopic statistical variables { Θ } that providea convenient parametrization of points on M S . The set { Θ } is called the parameter space D Θ of the system.In what follows, we schematically outline the main points underlying the construction of an arbitrary form of entropicdynamics. First, the microstates of the system under investigation must be defined. For the sake of simplicity, weassume the system is characterized by an l -dimensional microspace with microstates { X } . The main goal of anED model is that of inferring ”macroscopic predictions” in the absence of detailed knowledge of the microscopicnature of arbitrary complex systems. More explicitly, with ”macroscopic prediction” we mean the knowledge ofthe probability function statistical parameters (expectation values, variances, etc.). Once the microstates have beendefined, we have to select the relevant information about such microstates. In other words, we have to select the macrospace of the system. It is worthwhile mentioning that the coexistence of macroscopic dynamics with microscopicdynamics for a given physical (biological, chemical) system from a dynamical and statistical point of view has alwaysbeen a very important object of investigation [15]. For example, in certain fluid systems showing Rayleigh-Benardconvection [16], the macroscopic chaotic behavior (macroscopic chaos) is a manifestation of the underlying molecularinteraction of a very large collection of molecules (microscopic or molecular chaos). Moreover, macroscopic chaosarising from an underlying microscopic chaotic molecular behavior has been observed in chemical reactions. In orderto study underlying dynamics in rate equations of chemical reactions, a mesoscopic description has been adopted,which is given by a set of transition probabilities among chemicals. In such a description, the underlying dynamics ofmacroscopic motion is that one of stochastic processes and the evolution of probability distribution of each chemicalis investigated [17].For the sake of the argument, we assume that the degrees of freedom { x k } of the microstates { X } are Gaussian-distributed. They are defined by 2 l -information constraints, for example their expectation values µ k and variances σ k . h x k i ≡ µ k and (cid:16)D ( x k − h x k i ) E(cid:17) ≡ σ k . (1)In addition to information constraints, each 1-dimensional Gaussian distribution p k ( x k | µ k , σ k ) of each degree offreedom x k must satisfy the usual normalization conditions, + ∞ Z −∞ dx k p k ( x k | µ k , σ k ) = 1 (2)where p k ( x k | µ k , σ k ) = (cid:0) πσ k (cid:1) − exp − ( x k − µ k ) σ k ! . (3)(More correctly, we should label the Gaussian probability distribution in (3) as P ( x k | µ k , σ k ), where P indicatesa specific Gaussian distribution selected from a generic parametric family of distributions). Once the microstateshave been defined and the relevant (linear or nonlinear) information constraints selected, we are left with a set of l -dimensional vector probability distributions p ( X | Θ) = l Y k =1 p k ( x k | µ k , σ k ) encoding the relevant available informationabout the system where X is the l -dimensional microscopic vector with components ( x ,..., x l ) and Θ is the 2 l -dimensional macroscopic vector with coordinates ( µ ,..., µ l ; σ ,..., σ l ). The set { Θ } defines the 2 l -dimensional spaceof macrostates of the system, the statistical manifold M S . A measure of distinguishability among macrostates isobtained by assigning a probability distribution P ( X | Θ) ∋ M S to each macrostate Θ . Assignment of a probabilitydistribution to each state endows M S with a metric structure. Specifically, the Fisher-Rao information metric g µν (Θ)[4], g µν (Θ) = Z dXp ( X | Θ) ∂ µ log p ( X | Θ) ∂ ν log p ( X | Θ) , (4)with µ , ν = 1,..., 2 l and ∂ µ = ∂∂ Θ µ , defines a measure of distinguishability among macrostates on M S . The statisticalmanifold M S , M S = ( p ( X | Θ) = l Y k =1 p k ( x k | µ k , σ k ) ) , (5)is defined as the set of probabilities { p ( X | Θ) } described above where X ∈ R N , Θ ∈ D Θ = [ I µ × I σ ] N . Theparameter space D Θ (homeomorphic to M S ) is the direct product of the parameter subspaces I µ and I σ , where(unless specified otherwise) I µ = ( −∞ , + ∞ ) µ and I σ = (0, + ∞ ) σ . It is worthwhile pointing out that the possiblechaotic behavior of the set of macrostates { Θ } is strictly related to the selected relevant information about the setof microstates { X } of the system. In other words, the assumed Gaussian characterization of the degrees of freedom { x k } of each microstate of the system has deep consequences on the macroscopic behavior of the system itself.Once M S and D Θ are defined, the ED formalism provides the tools to explore dynamics driven on M S by entropicarguments. Specifically, given a known initial macrostate Θ (initial) (probability distribution), and that the systemevolves to a final known macrostate Θ (final) , the possible trajectories of the system are examined in the ED approachusing ME methods.We emphasize ED can be derived from a standard principle of least action (of Jacobi type). The geodesic equationsfor the macrovariables of the Gaussian ED model are given by nonlinear second order coupled ordinary differentialequations, d Θ µ dτ + Γ µνρ d Θ ν dτ d Θ ρ dτ = 0. (6)The geodesic equations in (6) describe a reversible dynamics whose solution is the trajectory between an initial Θ (initial) and a final macrostate Θ (final) . The trajectory can be equally well traversed in both directions. Given the Fisher-Raoinformation metric, we can apply standard methods of Riemannian differential geometry to study the information-geometric structure of the manifold M S underlying the entropic dynamics. Connection coefficients Γ ρµν , Ricci tensor R µν , Riemannian curvature tensor R µνρσ , sectional curvatures K M S , scalar curvature R M S , Weyl anisotropy tensor W µνρσ , Killing fields ξ µ and Jacobi fields J µ can be calculated in the usual way [11, 12].To characterize the chaotic behavior of complex entropic dynamical systems, we are mainly concerned with thesigns of the scalar and sectional curvatures of M S , the asymptotic behavior of Jacobi fields J µ on M S , the existenceof Killing vectors ξ µ (or existence of a non-vanishing Weyl anisotropy tensor, the anisotropy of the manifold under-lying system dynamics plays a significant role in the mechanism of instability) and the asymptotic behavior of theinformation-geometrodynamical entropy (IGE) S M S (see (9)). It is crucial to observe that true chaos is identified bythe occurrence of two features [12]: 1) strong dependence on initial conditions and exponential divergence of the Jacobivector field intensity, i.e., stretching of dynamical trajectories; 2) compactness of the configuration space manifold,i.e., folding of dynamical trajectories. The negativity of the Ricci scalar R M S , R M S = R µνρσ g µρ g νσ = X ρ = σ K M S ( e ρ , e σ ) , (7)implies the existence of expanding directions in the configuration space manifold M s . Indeed, since R M S is the sumof all sectional curvatures of planes spanned by pairs of orthonormal basis elements (cid:8) e ρ = ∂ Θ ρ (cid:9) , the negativity of theRicci scalar is only a sufficient (not necessary) condition for local instability of geodesic flow. For this reason, thenegativity of the scalar provides a strong criterion of local instability. Scenarios may arise where negative sectionalcurvatures are present, but the positive ones could prevail in the sum so that the Ricci scalar is non-negative despitethe instability in the flow in those directions. Consequently, the signs of K M S are of primary significance for theproper characterization of chaos.A powerful mathematical tool to investigate the stability or instability of a geodesic flow is the Jacobi-Levi-Civitaequation (JLC equation) for geodesic spread [11], D J µ Dτ + R µνρσ ∂ Θ ν ∂τ J ρ ∂ Θ σ ∂τ = 0. (8)The JLC-equation covariantly describes how nearby geodesics locally scatter and relates the stability or instability of ageodesic flow with curvature properties of the ambient manifold. Finally, the asymptotic regime of diffusive evolutiondescribing the possible exponential increase of average volume elements on M s provides another useful indicator ofdynamical chaoticity. The exponential instability characteristic of chaos forces the system to rapidly explore largeareas (volumes) of the statistical manifold. It is interesting to note that this asymptotic behavior appears also in theconventional description of quantum chaos where the entropy (von Neumann) increases linearly at a rate determinedby the Lyapunov exponents. The linear increase of entropy as a quantum chaos criterion was introduced by Zurekand Paz [18]. In my information-geometric approach a relevant quantity that can be useful to study the degree ofinstability characterizing ED models is the information geometrodynamical entropy (IGE) defined as [8], S M s ( τ ) def = lim τ →∞ log V M s with V M s ( τ ) = 1 τ τ Z dτ ′ Z M s √ gd l Θ (9)and g = | det ( g µν ) | . IGE is the asymptotic limit of the natural logarithm of the statistical weight defined on M s andrepresents a measure of temporal complexity of chaotic dynamical systems whose dynamics is underlined by a curvedstatistical manifold. In conventional approaches to chaos, the notion of entropy is introduced, in both classical andquantum physics, as the missing information about the systems fine-grained state [19]. For a classical system, supposethat the phase space is partitioned into very fine-grained cells of uniform volume ∆ v , labelled by an index j . If onedoes not know which cell the system occupies, one assigns probabilities p j to the various cells; equivalently, in thelimit of infinitesimal cells, one can use a phase-space density ρ ( X j ) = p j ∆ v . Then, in a classical chaotic evolution, theasymptotic expression of the information needed to characterize a particular coarse-grained trajectory out to time τ is given by the Shannon information entropy (measured in bits) [19], S (chaotic)classical = − Z dXρ ( X ) log ( ρ ( X ) ∆ v ) = − X j p j log p j ∼ K τ . (10)where ρ ( X ) is the phase-space density and p j = v j ∆ v is the probability for the corresponding coarse-grained trajectory. S (chaotic)classical is the missing information about which fine-grained cell the system occupies. The quantity K represents thelinear rate of information increase and it is called the Kolmogorov-Sinai entropy (or metric entropy) ( K is the sum ofpositive Lyapunov exponents, K = P j λ j ). K quantifies the degree of classical chaos. III. APPLICATION OF THE IGAC TO QUANTUM ENERGY LEVEL STATISTICS
The relevant indicators of chaoticity within the IGAC are the Ricci scalar curvature R M s (or, more correctly, thesectional curvature K M S ), the Jacobi vector field intensity J M S and the IGE S M s once the line element on the curvedstatistical manifold M s underlying the entropic dynamics has been specified. In what follows, selected two specialline elements, we focus exclusively on the asymptotic temporal behavior of the IGE S M s related to the ED arisingfrom them.We apply the IGAC to study the entropic dynamics on curved statistical manifolds induced by classical probabilitydistributions of common use in the study of regular and chaotic quantum energy level statistics. In doing so, we suggestan information-geometric characterization of a special class of regular and chaotic quantum energy level statistics.As we said previously, we have omitted technical details that will appear elsewhere. However, our previous works(especially ([8])) may be very useful references in order to clarify the following application. Recall that the theoryof quantum chaos (quantum mechanics of systems whose classical dynamics are chaotic) is not primarily relatedto few-body physics. Indeed, in real physical systems such as many-electron atoms and heavy nuclei, the origin ofcomplex behavior is the very strong interaction among many particles. To deal with such systems, a famous statisticalapproach has been developed which is based upon the Random Matrix Theory (RMT). The main idea of this approachis to neglect the detailed description of the motion and to treat these systems statistically bearing in mind that theinteraction among particles is so complex and strong that generic properties are expected to emerge. Once again, thisis exactly the philosophy underlining the ED approach to complex dynamics. It is known that the asymptotic behaviorof computational costs and entanglement entropies of integrable and chaotic Ising spin chains are very different [6] .Here Prosen considered the question of time efficiency implementing an up-to-date version of the t-DMRG for a familyof Ising spin chains in arbitrary oriented magnetic field, which undergoes a transition from integrable (transverseIsing) to nonintegrable chaotic regime as the magnetic field is varied. An integrable (regular) Ising chain in a generalhomogeneous transverse magnetic field is defined through the Hamiltonian H regular (0, 2), where H ( h x , h y ) = n − X j =0 σ xj σ xj +1 + n − X j =0 (cid:0) h x σ xj + h y σ yj (cid:1) . (11)In this case, the computational cost shows a polynomial growth in time, D (regular) ε ( t ) τ →∞ ∝ τ , while the entanglemententropy is characterized by logarithmic growth, S regular (0, 2) = S (0, 2)von Neumann τ →∞ ∝ c log τ + c ′ (12)The constant c depends exclusively on the value of the fixed transverse magnetic field intensity B ⊥ , while c ′ dependson B ⊥ and on the choice of the initial local operators of finite index used to calculate the operator space entanglemententropy. Instead, a quantum chaotic Ising chain in a general homogeneous tilted magnetic field is defined throughthe Hamiltonian H chaotic (1, 1), where H is defined in (11). In this case, the computational cost shows an exponentialgrowth in time, D (chaotic) ε ( t ) τ →∞ ∝ exp ( K q τ ), while the entanglement entropy is characterized by linear growth, S chaotic (1, 1) = S (1, 1)von Neumann τ →∞ ∝ K q τ . (13)The quantity K q is a constant, asymptotically independent of the number of indexes of the initial local operatorsused to calculate the operator space entropy, that depends only on the Hamiltonian evolution and not on the detailsof the initial state observable or error measures, and can be interpreted as a kind of quantum dynamical entropy.It is well known the quantum description of chaos is characterized by a radical change in the statistics of quantumenergy levels [21]. The transition to chaos in the classical case is associated with a drastic change in the statistics ofthe nearest-neighbor spacings of quantum energy levels. In the regular regime the distribution agrees with the Poissonstatistics while in the chaotic regime the Wigner-Dyson distribution works very well. Uncorrelated energy levels arecharacteristic of quantum systems corresponding to a classically regular motion while a level repulsion (a suppression ofsmall energy level spacing) is typical for systems which are classically chaotic. A standard quantum example is providedby the study of energy level statistics of an Hydrogen atom in a strong magnetic field. It is known that level spacingdistribution (LSD) is a standard indicator of quantum chaos [22]. It displays characteristic level repulsion for stronglynonintegrable quantum systems, whereas for integrable systems there is no repulsion due to existence of conservationlaws and quantum numbers. In [6], the authors calculate the LSD of the spectra of H regular (0, 2) and H chaotic (1, 1).They find that for H regular (0, 2), the nearest neighbor LSD is described by a Poisson distribution. For H chaotic (1, 1),they find the nearest neighbor LSD is described by a Wigner-Dyson distribution. Therefore, they conclude that H regular (0, 2) and H chaotic (1, 1) indeed represent generic regular and quantum chaotic systems, respectively. We willencode the relevant information about the spin-chain in a suitable composite-probability distribution taking accountof the quantum spin chain and the configuration of the external magnetic field in which they are immersed.In the ME method [3], the selection of relevant variables is made on the basis of intuition guided by experiment; itis essentially a matter of trial and error. The variables should include those that can be controlled or experimentallyobserved, but there are cases where others must also be considered. Our objective here is to choose the relevantmicrovariables of the system and select the relevant information concerning each one of them. In the integrable case,the Hamiltonian H regular (0, 2) describes an antiferromagnetic Ising chain immersed in a transverse homogeneousmagnetic field ~B transverse = B ⊥ ˆ B ⊥ with the level spacing distribution of its spectrum given by the Poisson distribution p Poisson ( x A | µ A ) = 1 µ A exp (cid:18) − x A µ A (cid:19) , (14)where the microvariable x A is the spacing of the energy levels and the macrovariable µ A is the average spacing. Thechain is immersed in the transverse magnetic field which has just one component B ⊥ in the Hamiltonian H regular (0, 2).We translate this piece of information in our IGAC formalism, coupling the probability (14) to an exponential bath p (exponential) B ( x B | µ B ) given by p (exponential) B ( x B | µ B ) = 1 µ B exp (cid:18) − x B µ B (cid:19) , (15)where the microvariable x B is the intensity of the magnetic field and the macrovariable µ B is the average intensity.More correctly, x B should be the energy arising from the interaction of the transverse magnetic field with the spin particle magnetic moment, x B = (cid:12)(cid:12)(cid:12) − ~µ · ~B (cid:12)(cid:12)(cid:12) = |− µB cos ϕ | where ϕ is the tilt angle. For the sake of simplicity, let usset µ = 1, then in the transverse case ϕ = 0 and therefore x B = B ≡ B ⊥ . This is our best guess and we justify itby noticing that the magnetic field intensity is indeed a relevant quantity in this experiment (see equation (12)) andits components (intensity) are quantities that are varied during the transitions from integrable to chaotic regimes.In the regular regime, we say the magnetic field intensity is set to a well-defined value h x B i = µ B . Furthermore,notice that the Exponential distribution is identified by information theory as the maximum entropy distribution ifonly one piece of information (the expectation value) is known. Finally, the chosen composite probability distribution P (integrable) ( x A , x B | µ A , µ B ) encoding relevant information about the system is given by, P (integrable) ( x A , x B | µ A , µ B ) = 1 µ A µ B exp (cid:20) − (cid:18) x A µ A + x B µ B (cid:19)(cid:21) . (16)Again, we point out that our probability (16) is our best guess and, of course, must be consistent with numericalsimulations and experimental data in order to have some merit. We point out that equation (16) is not fully justifiedfrom a theoretical point of view, a situation that occurs due to the lack of a systematic way to select the relevantmicrovariables of the system (and to choose the appropriate information about such microvariables). Let us denote M integrable S the two-dimensional curved statistical manifold underlying our information geometrodynamics. The lineelement ds on M integrable S is given by, ds = ds + ds = 1 µ A dµ A + 1 µ B dµ B . (17)Applying our IGAC (see ([8])) to the line element in (17), we obtain polynomial growth in V integrable M s and logarithmicIGE growth, V (integrable) M s ( τ ) τ →∞ ∝ exp( c ′ IG ) τ c IG , S (integrable) M s ( τ ) τ →∞ ∝ c IG log τ + c ′ IG . (18)The quantity c IG is a constant proportional to the number of Exponential probability distributions in the compositedistribution used to calculate the IGE and c ′ IG is a constant that depends on the values assumed by the statisticalmacrovariables µ A and µ B . Equations in (18) may be interpreted as the information-geometric analogue of thecomputational complexity D (regular) ε ( τ ) and the entanglement entropy S regular (0, 2) defined in standard quantuminformation theory, respectively. We cannot state they are the same since we are not fully justifying, from a theoreticalstandpoint, our choice of the composite probability (16).In the chaotic case, the Hamiltonian H chaotic (1, 1) describes an antiferromagnetic Ising chain immersed in a tiltedhomogeneous magnetic field ~B tilted = B ⊥ ˆ B ⊥ + B k ˆ B k with the level spacing distribution of its spectrum given by thePoisson distribution p Wigner-Dyson ( x ′ A | µ ′ A ) p Wigner-Dyson ( x ′ A | µ ′ A ) = πx ′ A µ ′ A exp (cid:18) − πx ′ A µ ′ A (cid:19) . (19)where the microvariable x ′ A is the spacing of the energy levels and the macrovariable µ ′ A is the average spacing. Thechain is immersed in the tilted magnetic vector field which has two components B ⊥ and B k in the Hamiltonian H chaotic (1, 1). We translate this piece of information in our IGAC formalism, coupling the probability (19) to aGaussian p (Gaussian) B ( x ′ B | µ ′ B , σ ′ B ) given by, p (Gaussian) B ( x ′ B | µ ′ B , σ ′ B ) = 1 p πσ ′ B exp − ( x ′ B − µ ′ B ) σ ′ B ! . (20)where the microvariable x ′ B is the intensity of the magnetic field, the macrovariable µ ′ B is the average intensity of themagnetic energy arising from the interaction of the tilted magnetic field with the spin particle magnetic moment,and σ ′ B is its covariance: during the transition from the integrable to the chaotic regime, the magnetic field intensityis being varied (experimentally). It is being tilted and its two components ( B ⊥ and B k ) are being varied as well.Our best guess based on the experimental mechanism that drives the transitions between the two regimes is thatmagnetic field intensity ( actually the microvariable µB cos ϕ ) is Gaussian-distributed (two macrovariables) duringthis change. In the chaotic regime, we say the magnetic field intensity is set to a well-defined value h x ′ B i = µ ′ B withcovariance σ ′ B = rD ( x ′ B − h x ′ B i ) E . Furthermore, the Gaussian distribution is identified by information theory asthe maximum entropy distribution if only the expectation value and the variance are known. Therefore, the chosencomposite probability distribution P (chaotic) ( x ′ A , x ′ B | µ ′ A , µ ′ B , σ ′ B ) encoding relevant information about the system isgiven by, P (chaotic) ( x ′ A , x ′ B | µ ′ A , µ ′ B , σ ′ B ) = π (cid:0) πσ ′ B (cid:1) − µ ′ A x ′ A exp " − πx ′ A µ ′ A + ( x ′ B − µ ′ B ) σ ′ B ! . (21)Let us denote M (chaotic) S the three-dimensional curved statistical manifold underlying our information geometrody-namics. The line element ds on M (chaotic) S is given by, ds = ds + ds = 4 µ ′ A dµ ′ A + 1 σ ′ B dµ ′ B + 2 σ ′ B dσ ′ B . (22)Applying our IGAC (see ([8])) to the line element in (22), we obtain exponential growth for V chaotic M s and linear IGEgrowth V (chaotic) M s ( τ ) τ →∞ ∝ C IG exp ( K IG τ ) , S (chaotic) M s ( τ ) τ →∞ ∝ K IG τ . (23)The constant C IG encodes information about the initial conditions of the statistical macrovariables parametrizingelements of M (chaotic) S . The constant K IG , K IG τ →∞ ≈ d S M s ( τ ) dτ τ →∞ ≈ lim τ →∞ (cid:20) τ log (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) J M S ( τ ) J M S (0) (cid:13)(cid:13)(cid:13)(cid:13)(cid:19)(cid:21) def = λ J , (24)is the model parameter of the chaotic system and depends on the temporal evolution of the statistical macrovariables.It plays the role of the standard Lyapunov exponent of a trajectory and it is, in principle, an experimentally observablequantity. The quantity J M S ( τ ) is the Jacobi field intensity and λ J may be considered the information-geometricanalogue of the leading Lyapunov exponent in conventional Hamiltonian systems. Given an explicit expression of K IG in terms of the observables µ ′ A and µ ′ B and σ ′ B , a clear understanding of the relation between the IGE (or K IG ) and the entanglement entropy (or K q ) becomes the key point that deserves further study. Equations in (23)are the information-geometric analogue of the computational complexity D (chaotic) ε ( τ ) and the entanglement entropy S chaotic (1, 1) defined in standard quantum information theory, respectively. This result is remarkable, but deserves adeeper analysis in order to be fully understood.One of the major limitations of our findings is the lack of a detailed account of the comparison of the theorywith experiment. This point will be one of our primary concerns in future works. However, some considerationsmay be carried out at the present stage. The experimental observables in our theoretical models are the statisticalmacrovariables characterizing the composite probability distributions. In the integrable case, where the couplingbetween a Poisson distribution and an Exponential one is considered, µ A and µ B are the experimental observables.In the chaotic case, where the coupling between a Wigner-Dyson distribution and a Gaussian is considered, µ ′ A and µ ′ B and σ ′ B play the role of the experimental observables. We believe one way to test our theory may bethat of determining a numerical estimate of the leading Lyapunov exponent λ max or the Lyapunov spectrum forthe Hamiltonian systems under investigation directly from experimental data (measurement of a time series) andcompare it to our theoretical estimate for λ J [23]. However, we are aware that it may be extremely hard to evaluateLyapunov exponents numerically. Otherwise, knowing that the mean values of the positive Lyapunov exponents arerelated to the Kolmogorov-Sinai (KS) dynamical entropy, we suggest to measure the KS entropy K directly froma time signal associated to a suitable combination of our experimental observables and compare it to our indirecttheoretical estimate for K IG from the asymptotic behaviors of our statistical macrovariables [24]. We are awarethat the ground of our discussion is quite qualitative. However, we hope that with additional study, especially inclarifying the relation between the IGE and the entanglement entropy, our theoretical characterization presented inthis Letter will find experimental support in the future. Therefore, the statement that our findings may be relevantto experiments verifying the existence of chaoticity and related dynamical properties on a macroscopic level in energylevel statistics in chaotic and regular quantum spin chains is purely a conjecture at this stage. IV. CONCLUSIONS
In this Letter, we proposed a theoretical information-geometric framework suitable to characterize chaotic dynamicalbehavior of complex systems on curved statistical manifolds. Specifically, an information-geometric characterizationof regular and chaotic quantum energy level statistics appearing in a quantum Ising spin chain in external magneticfield was presented. It is worthwhile emphasizing the following points: the statements that spectral correlations ofclassically integrable systems are well described by Poisson statistics and that quantum spectra of classically chaoticsystems are universally correlated according to Wigner-Dyson statistics are conjectures, known as the BGS (Bohigas-Giannoni-Schmit, [25] and BTG (Berry-Tabor-Gutzwiller, [26]) conjectures, respectively. These two conjectures arevery important in the study of quantum chaos, however their validity finds some exceptions. Several other cases maybe considered. For instance, chaotic systems having a spectrum that does not obey a Wigner-Dyson distributionmay be considered. A chaotic system can also have a spectrum following a Poisson, semi-Poisson, or other types ofcritical statistics [27]. Moreover, integrable systems having a spectrum that does not obey a Poisson distribution maybe considered as well. For instance, the Harper model would represent such a situation. Moreover, it is worthwhilepointing out that not every chaotic system characterized by entropy-like quantities growing linearly in time has aspectrum described by a Wigner-Dyson distribution. Well-known examples presenting such a situation are the catmaps [28] and the famous kicked rotator [29] where its spectrum follows a Poisson distribution in cylinder represen-tation and a Wigner-Dyson in torus representation but the properties of entropy-like quantities are the same in bothrepresentations (at least classically). All these cases are not discussed in our characterization.Therefore, at present stage, because of the above considerations and because of the lack of experimental evidencein support of our theoretical construct, we can only conclude that the IGAC might find some potential applications incertain regular and chaotic dynamical systems and this remains only a conjecture. However, we hope that our workconvincingly shows that this information-geometric approach may be considered a serious effort trying to provide aunifying criterion of chaos of both classical and quantum varieties, thus deserving further research and developments.
Acknowledgments
I am grateful to Saleem Ali, Ariel Caticha and Adom Giffin for very useful discussions and for their previouscollaborations. I extend thanks to Cedric Beny, Michael Frey and Jeroen Wouters for their interest and/or usefulcomments on my research during the NIC@QS07 in Erice, Ettore Majorana Centre. Finally, I sincerely thank the twoanonymous Referees for constructive criticism and for very helpful suggestions. [1] M. Gell-Mann, ”What is Complexity”, Complexity, vol. 1, no. 1 (1995).[2] A. Caticha, ”Entropic Dynamics”, in
Bayesian Inference and Maximum Entropy Methods in Science and Engineering , ed.by R.L. Fry, AIP Conf. Proc. , 302 (2002).[3] A. Caticha and R. Preuss, ”Maximum entropy and Bayesian data analysis: Entropic prior distributions”, Phys. Rev.
E70 ,046127 (2004).[4] S. Amari and H. Nagaoka,
Methods of Information Geometry , American Mathematical Society, Oxford University Press,2000.[5] A. Caticha, ”Insufficient Reason and Entropy in Quantum Theory” Found. Phys. , 227 (2000).[6] T. Prosen and M. Znidaric, ”Is the efficiency of classical simulations of quantum dynamics related to integrability?”, Phys.Rev. E75 , 015202 (2007); T. Prosen and I. Pizorn, ”Operator space entanglement entropy in transverse Ising chain”, Phys.Rev.
A76 , 032316 (2007).[7] C. Cafaro, S. A. Ali and A. Giffin, ”An Application of Reversible Entropic Dynamics on Curved Statistical Manifolds”,in
Bayesian Inference and Maximum Entropy Methods in Science and Engineering, ed. by Ali Mohammad-Djafari, AIPConf. Proc. , 243-251 (2006).[8] C. Cafaro and S. A. Ali, ”Jacobi Fields on Statistical Manifolds of Negative Curvature”, Physica
D234 , 70-80 (2007).[9] C. Cafaro, ”Information Geometry and Chaos on Negatively Curved Statistical Manifolds”, in
Bayesian Inference andMaximum Entropy Methods in Science and Engineering, ed. by K. Knuth, et al ., AIP Conf. Proc. , 175 (2007).[10] A. Caticha and C. Cafaro, ”From Information Geometry to Newtonian Dynamics”, in Bayesian Inference and MaximumEntropy Methods in Science and Engineering, ed. by K. Knuth, et al ., AIP Conf. Proc. , 165 (2007).[11] L. Casetti, C. Clementi, and M. Pettini, ”Riemannian theory of Hamiltonian chaos and Lyapunov exponents”, Phys. Rev. E54 , 5969-5984 (1996).[12] M. Di Bari and P. Cipriani, ”Geometry and Chaos on Riemann and Finsler Manifolds”, Planet. Space Sci. , 1543 (1998).[13] C. G. J. Jacobi, ”Vorlesungen uber Dynamik”, Reimer, Berlin (1866).[14] T. Kawabe, ”Indicator of chaos based on the Riemannian geometric approach”, Phys. Rev. E71 , 017201 (2005); T. Kawabe,”Chaos based on Riemannian geometric approach to Abelian-Higgs dynamical system”, Phys. Rev.
E67 , 016201 (2003).[15] T. Shibata and K. Kaneto, ”Collective Chaos”, Phys. Rev. Lett. , 4116 (1998).[16] E. N. Lorenz, ”Deterministic nonperiodic flow”, J. Atmos. Sci. , 130 (1963).[17] R. F. Fox and J. Keizer, ”Amplification of intrinsic fluctuations by chaotic dynamics in physical systems”, Phys. Rev. A43 , 1709 (1991). [18] W. H. Zurek and J. P. Paz, ”Decoherence, Chaos, and the Second Law”, Phys. Rev. Lett. , 2508 (1994); ”QuantumChaos: a decoherent definition”, Physica D83 , 300 (1995).[19] C. M. Caves and R. Schack, ”Unpredictability, Information, and Chaos”, Complexity , 46-57 (1997); A. J. Scott, T. A.Brun, C. M. Caves, and R. Schack, ”Hypersensitivity and chaos signatures in the quantum baker’s map”, J. Phys. A39 ,13405 (2006).[20] W. H. Zurek, ”Preferred States, Predictability, Classicality and Environment-Induced Decoherence”, Prog. Theor. Phys. , 281 (1993).[21] G. Casati and B. Chirikov, Quantum Chaos, Cambridge University Press (1995); M. V. Berry, ”Chaotic Behavior inDynamical Systems”, ed. G. Casati (New York, Plenum), 1985; M. Robnik and T. Prosen, ”Comment on energy levelstatistics in the mixed regimes”, arXiv: chao-dyn/9706023, (1997).[22] F. Haake, ” Quantum Signatures of Chaos ”, Springer-Verlag, Berlin (1991) (2nd enlarged edition, 2000).[23] A. Wolf et. al . ”Determining Lyapunov Exponents form Time Series, Physica D16 , 285-317 (1985); J. Wright, ”Methodfor calculating a Lyapunov exponent”, Phys. Rev.
A29 , 2924-2927 (1984).[24] P. Grassberger and I. Procaccia, ”Estimation of the Kolmogorov entropy from a chaotic signal”, Phys. Rev.
A28 , 2591-2593(1983).[25] O. Bohigas et. al .,”Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws”, Phys. Rev.Lett. , 1 (1984).[26] M. C. Gutzwiller, ” Chaos in Classical and Quantum Mechanics ”, Springer-Verlag, New York (1990)[27] A. M. Garcia-Garcia and J. Wang, ”Universality in quantum chaos and the one parameter scaling theory”, arXiv: 0707.3964(2007); ”Anderson Localization in Quantum Chaos: Scaling and Universality”, Acta Physica Polonica
A112 , 635-653(2007).[28] Y. Gu, ”Evidences of classical and quantum chaos in the time evolution of nonequilibrium ensembles”, Phys. Lett.
A149 ,95-100 (1990); J. P. Keating, ”Asymptotic properties of the periodic orbits of the cat maps”,
Nonlinearity , 277-307(1991); J. P. Keating, ”The cat maps: quantum mechanics and classical motion”, Nonlinearity , 309-341 (1991).[29] F. M. Izrailev, ”Simple models of quantum chaos: spectrum and eigenfunctions”, Phys. Rep.196