Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systems
aa r X i v : . [ phy s i c s . g e n - ph ] M a y Geometric information flows and G. Perelman entropyfor relativistic classical and quantum mechanical systems
Sergiu I. Vacaru ∗ Physics Department, California State University at Fresno, CA 93740, USA and
Project IDEI, University "Al. I. Cuza" Iaşi, Romania
May 8, 2019
Abstract
This work consists an introduction to the classical and quantum information theory of geometric flowsof (relativistic) Lagrange–Hamilton mechanical systems. Basic geometric and physical properties of thecanonical nonholonomic deformations of G. Perelman entropy functionals and geometric flows evolutionequations of classical mechanical systems are described. There are studied projections of such F- andW-functionals on Lorentz spacetime manifolds and three dimensional spacelike hypersurfaces. Thesefunctionals are used for elaborating relativistic thermodynamic models for Lagrange–Hamilton geometricevolution and respective generalized R. Hamilton geometric flow and nonholonomic Ricci flow equations.The concept of nonholonomic W-entropy is developed as a complementary one for the classical Shannonentropy and the quantum von Neumann entropy. There are considered geometric flow generalizations ofthe approaches based on classical and quantum relative entropy, conditional entropy, mutual information,and related thermodynamic models. Such basic ingredients and topics of quantum geometric flow infor-mation theory are elaborated using the formalism of density matrices and measurements with quantumchannels for evolution of quantum mechanical systems.
Keywords:
Lagrange and Hamilton geometry, relativistic geometric flow evolution; Perelman F- andW-entropy; classical and quantum information theory.PACS2010: 02.40.-k, 02.40.Yy, 02.90.+p, 03.67.-a, 05.90.+m, 45.10.NaMSC2010: 53C44, 53C50, 53C60, 53C80, 53Z05, 82C99, 35Q75, 35Q99 37J60, 37D35
Contents ∗ The address for post correspondence: 140 Morehampton Rd, Donnybrook, Dublin 04, Ireland, D04 N2C0; the UAICaffiliation is for a former hosted IDEI project; emails: [email protected] ; [email protected] Geometric flow evolution of classical mechanical systems 11
One of the most remarkable success in modern mathematics is the proof of the Poincaré–Thurstonconjecture due to G. Perelman [1, 2, 3]. We cite here most important related works on W. Thurston’sclassification of three dimensional, 3-d, manifolds, [4, 5, 6]; then D. Friedman’s geometric flow evolutionequations derived for renorm-group considerations in quantum field theory and condensed matter physics,see [7, 8, 9]; and R. Hamilton [10, 11, 12] fundamental contributions to Ricci flow theory. The monographs[13, 14, 15] can be considered for rigorous proofs and reviews of results in geometric analysis and topology. Aseries of our works were elaborated in a ’geometry and physics’ style involving generalizations for relativisticsystems and applications in modern physics and cosmology. We cite [19, 20, 21, 22], for geometric flowsof Lagrange-Finsler spaces and nonholonomic manifolds and algebroids; [23], on noncommutative geometricflow evolution theories; [24, 25], for respective super-Ricci flows and thermodynamics of relativistic Ricciflows; and a series of works [26, 27, 28, 29] related to modified gravity theories, MGTs, and cosmology, seereviews [30, 31, 32, 33, 34, 35].Above mentioned directions for advanced studies in geometry and mathematical physics were developedusing G. Perelman’s concepts of F- and W-entropy Perelman. Such values were constructed as A. M.Lyapunov type functionals [36] which for geometric flows of Riemannian metrics are determined by Riccitensors and scalars. We defined their nonholonomic deformations (equivalently, anholonomic, i.e. subjectedto non-integrable constraints) for various generalized geometric and physical models. The W-entropy is like a"minus entropy" and it describes some nonholonomic entropic flows of various classical and quantum physicalsystems. The concept of W-entropy is different from the Shannon, von Neumann, or other type, entropy usedin modern thermodynamics and classical/ quantum information theory, see [37, 38] and references therein. We emphasize that the terms Hamilton mechanics and Hamilton equations for Ricci flows are related to the names oftwo different famous scientists. In the first case, it refers to William R. Hamilton who formulated in 1834 his Hamiltonianmechanics starting from Lagrangian mechanics (a previous reformulation for classical mechanics introduced by Joseph LouisLagrange in 1788). On mathematical and physical approaches and historical remarks on Lagrange and Hamilton mechanics,see [16, 17, 18]. In the second case, Richard Hamilton is known because of his achievements on the Ricci flows theory andapplications in topology and geometric analysis [10, 11, 12].
We develop an approach to geometrization of relativistic Lagrange and Hamilton mechanics on tangentand cotangent Lorentz manifolds (respectively,
T V and T ∗ V ) on a Lorentz manifold V of dimension dim V =4 and with local Euclidean signature (+ + + − ) , see [34, 35] for details and historical remarks. The concept ofLagrange space was proposed in [43] as an alternative geometrization for nonrelativistic mechanics outlinedin [16, 17, 18]. The main idea in such Hessian geometric models (with a so-called vertical, or covertical,metric determined by a Lagrange, or Hamilton, generating function) is to drop the homogeneity conditionfor generating functions and apply Finselr and almost Kähler geometry methods to classical field theories and3echanics [44, 45]. Here we note that other approaches on geometrization of classical mechanics and fields,for instance, the poly-simplectic formalism (see [18], references therein and further developments in modernliterature), do not allow an unified formulation of models for geometric flow evolution, thermodynamics andstatistics, (modified) gravity theories and classical and quantum information. In our works [19, 20, 21, 22, 23,26, 27, 28, 29, 25, 26, 27, 28, 29, 24], using constructions with generalized Finsler like Hessian geometrizationof Lagrange-Hamilton systems in mathematical relativity, cosmology and particle physics, various directionswere developed for classical and quantum (non) commutative / supersymetric field theories, in modifiedgravity, inhomogeneous cosmology and theory of nonholonomic geometric flows. Geometrization of classical nonrelativistic and relativistic mechanical systems can be performed on aRiemannian or Lorentz manifold V and it tangent T V and cotangent T ∗ V bundles enabled with (pseudo)Riemannian metrics with local (pseudo) Euclidean signature. We call
T V and/or T ∗ V as phase spaces or phase spacetimes depending on signatures of metrics they areenabled. In a typical case, there are considered corresponding quadratic line elements determined by totalphase space metrics with signature (+ + + − ; + + + − ) , ds = g αβ ( x k ) du α du β = g ij ( x k ) dx i dx j + η ab dy a dy b , for y a ∼ dx a /dτ ; and/ or (1) d p s = p g αβ ( x k ) d p u α d p u β = g ij ( x k ) dx i dx j + η ab dp a dp b , for p a ∼ dx a /dτ. (2)In these formulas, the local frame and dual frame (co-frame) coordinates are labeled respectively. We write u α = ( x i , y a ) , (or in brief, u = ( x, y )) , on the tangent bundle T V ; and p u α = ( x i , p a ) , (or in brief, p u = ( x, p )) , on the cotangent bundle T ∗ V. The total phase space metrics g αβ ( u ) and p g αβ ( p u ) are determined, for suchexamples, by a pseudo–Riemannian spacetime metric g = { g ij ( x ) } with the Levi-Civita connection, LC-connection, ∇ , which is metric compatible and with zero torsion. In diagonal form, the vertical metric η ab and its dual η ab are standard Minkowski metrics, η ab = diag [1 , , , − used for computations in typicalfibers of respective (co) tangent bundles. The mechanical models can be elaborated for general frame/coordinate transforms in total spaces when the metric structures can be parameterized equivalently by thesame h-components of g αβ ( x k ) and p g αβ ( x k ) = g αβ ( x k ) , but different (co) fiber metrics g ab ( x, y ) and g ab ( x, p ) than those considered in (1) and (2). A relativistic 4-d model of Lagrange space L , = ( T V, L ( x, y )) is determined by a fundamental function(equivalently, generating function) T V ∋ ( x, y ) → L ( x, y ) ∈ R , i.e. a real valued function (in brief, called aLagrangian or a Lagrange density) which is differentiable on g T V := T V / { } , for { } being the null sectionof T V, and continuous on the null section of π : T V → V. Such a relativistic model is regular if the Hessianmetric (equivalently, v-metric) e g ab ( x, y ) := 12 ∂ L∂y a ∂y b (3)is non-degenerate, i.e. det | e g ab | 6 = 0 , and of constant signature.In modern literature on geometric mechanics, kinetics and statistical mechanics of locally anisotropicprocesses (see a review of such results and references in [34, 35]), there are used constructions on cotangentbundles with such a concept: A 4-d relativistic model of Hamilton space H , = ( T ∗ V, H ( x, p )) is constructed There are used such conventions for indices: the "horizontal" indices, h–indices, run values i, j, k, ... = 1 , , , the verticalindices, v-vertical, run values a, b, c... = 5 , , , ; respectively, the v-indices can be identified/ contracted with h-indices , , , for lifts on total (co) tangent Lorentz bundles, when α = ( i, a ) , β = ( j, b ) , γ = ( k, c ) , ... = 1 , , , ... . We shall consider letterslabelled by an abstract left up/low symbol " p " (for instance, p u α and p g αβ ) in order to emphasize that certain geometric/physical objects are defined on T ∗ V. In similar forms, we can consider indices for lower and higher dimensions than , orother type signatures. V. One considers that T ∗ V ∋ ( x, p ) → H ( x, p ) ∈ R defines a real valuedfunction being differentiable on g T ∗ V := T ∗ V / { ∗ } , for { ∗ } being the null section of T ∗ V, and continuous onthe null section of π ∗ : T ∗ V → V. Such a relativistic mechanical model is regular if the Hessian (cv-metric) p e g ab ( x, p ) := 12 ∂ H∂p a ∂p b (4)is non-degenerate, i.e. det | p e g ab | 6 = 0 , and of constant signature.For Lagrange and Hamilton spaces, we can consider Legendre transforms L → H ( x, p ) := p a y a − L ( x, y ) and y a determining solutions of the equations p a = ∂L ( x, y ) /∂y a . In a similar manner, the inverse Legendretransforms can be introduced, H → L, when L ( x, y ) := p a y a − H ( x, p ) (5)for p a determining solutions of the equations y a = ∂H ( x, p ) /∂p a . The non-Riemannian total phase space geometries are characterized by nonlinear quadratic line elements ds L = L ( x, y ) , for models on T V ; d p s H = H ( x, p ) , for models on T ∗ V. (6)We can elaborate on geometric and physical theories with an effective phase spacetime modelled on (co) tan-gent Lorentz bundles endowed with generalized frame, metric and linear and nonlinear connection structuresdetermined by nonlinear quadratic line elements and (6). For certain special cases, such values transformcorrespondingly into quadratic line elements (1) and (2).The Hessians e g ab and p e g ab are labeled by a tilde "~" in order to emphasize that such conventional v- andcv–metrics are defined canonically by respective Lagrange and Hamilton generating functions. For simplicity,we can work with such regular metrics even, in principle, mechanical models with degenerate Hessians arealso studied in modern mechanics and field theories. Considering general frame/ coordinate transforms onphase spaces, we can express any "tilde" Hessian in a general quadratic form, respectively as a vertical metric(v-metric), g ab ( x, y ) , and/or co-vertical metric (cv-metric), p g ab ( x, p ) . Inversely, if a v-metric (cv-metric) isprescribed, we can introduce respective (co) frame /coordinate systems, when such values can transformedinto certain canonical ones, with "tilde" values. In general, a v-metric g ab is different from the inverse ofa cv-metric p g ab , i.e. from the p g ab . Nevertheless, certain relations between such values can be found viaLegendre transforms. We shall omit tildes on geometrical/ physical objects on respective phase spaces ifcertain formulas hold in general (not only canonical) forms and/or that will not result in ambiguities.For simplicity, the bulk of geometric constructions in this paper will be performed for (effective and/orgeneralized) Hamilton spaces if that will not result in ambiguities. We shall consider that via correspond-ing frame and Legendre transforms, or homogeneity conditions, we can generate necessary type Lagrange/Finsler/ Cartan configurations. A complete geometrization of mechanical models is not possible if we use only Lagrange-Hamilton func-tions and respective (non) linear quadratic elements. There are necessary additional concepts and definitionof new geometric objects like the nonlinear connection structure, the distinguished linear connection, variousdistinguished geometric objects etc., see details and motivations in [34, 35]. A relativistic 4-d model of Finsler space is an example of Lagrange space when a regular L = F is defined by a fundamental(generating) Finsler function subjected to certain additional conditions: 1) F is a real positive valued function which is differentialon g T V and continuous on the null section of the projection π : T V → V ;
2) it is satisfied the homogeneity condition F ( x, λy ) = | λ | F ( x, y ) , for a nonzero real value λ ; and 3) the Hessian (3) is defined by F in such a form that in any point ( x (0) , y (0) ) the v-metricis of signature (+ + + − ) . In a similar form, we can define relativistic Cartan spaces C , = ( V, C ( x, p )) , when H = C ( x, p ) is1-homogeneous on co-fiber coordinates p a . T V, or T ∗ V, is defined as a Whitney sum of conven-tional h and v –distributions, or h and cv –distributions, N : T T V = hT V ⊕ vT V, or p N : T T ∗ V = hT ∗ V ⊕ vT ∗ V. (7)Parameterizing locally the N-connections with respect to coordinate bases by corresponding coefficients N = { N ai } and p N = { p N ia } , we obtain by explicit constructions that decompositions/splitting (7) definerespective systems of N–linear (i.e. N-adapted) bases e α = ( e i = ∂∂x i − N ai ( x, y ) ∂∂y a , e b = ∂∂y b ) , e α = ( e i = dx i , e a = dy a + N ai ( x, y ) dx i ) , and/ or (8) p e α = ( p e i = ∂∂x i − p N ia ( x, p ) ∂∂p a , p e b = ∂∂p b ) , p e α = ( p e i = dx i , p e a = dp a + p N ia ( x, p ) dx i ) . The N–connection coefficients and necessary types of (co) frame/ coordinate transforms can be used forconstructing lifts of metric structures ( V, g ) to respective nonholonomic (co)tangent bundles, ( TV , N , g ) and ( T ∗ V , p N , p g ) . We can consider various type of metric structures on a tangent, TV , and/or cotangent, T ∗ V , Lorentzbundles. This can be used for elaborating mechanical models, thermodynamic and kinetic theories andgeneralizations of the Einstein gravity. Such metric structures can be parameterized by frame transforms inN–adapted form, i.e. as distinguished metrics (d-metrics) g = g αβ ( x, y ) e α ⊗ e β = g ij ( x ) e i ⊗ e j + g ab ( x, y ) e a ⊗ e a and/or (9) p g = p g αβ ( x, p ) p e α ⊗ p e β = g ij ( x ) e i ⊗ e j + p g ab ( x, p ) p e a ⊗ p e b . (10)In this work, such metrics on conventional 8-d manifolds are of signature (+ , + , + , − , + , + , + , − ) but forelaborating non-relativistic mechanical/ thermodynamical / statistical models other type signatures can beconsidered. For instance, a pseudo–Riemannian metric g ij ( x ) can be subjected to the condition that it definesa solution of the standard Einstein equations in GR, or a MGT, with a corresponding base Lorentz manifold V . For various mechanical and thermodynamical models, there are necessary additional geometrically andphysically motivated assumptions on how nonlinear quadratic elements of type or (6), and/or (9), or (10),encode local anisotropies, inhomogeneous structures, modified dispersion relations etc.
Let us consider that a spacetime Lorentzian (or a space Riemannian) manifold V is endowed with ametric hg = { g ij ( x ) } of signature (3 , (or of Euclidean signature). Using frame/generalized coordinatetransforms on base and total spaces, metrics can be deformed to off-diagonal metrics depending on velocity/momentum coordinates, including horizontal components of Hessian type.Considering a regular curve c ( τ ) defined c : τ ∈ [0 , → x i ( τ ) ⊂ U ⊂ V, for a real parameter τ, we can construct a lifted to π − ( U ) ⊂ g T V defining a curve in the total space, when e c ( τ ) : τ ∈ [0 , → (cid:0) x i ( τ ) , y i ( τ ) = dx i /dτ (cid:1) with a non-vanishing v-vector field dx i /dτ. Using a canonical symplectic structure θ := dp i ∧ dx i on T ∗ V and a unique vector filed e X H := ∂ e H∂p i ∂∂x i − ∂ e H∂x i ∂∂p i defined by e H, we construct anequation i e X H θ = − d e H. We write ∧ for the antisymmetric product where i e X H denotes the interior producedefined by e X H . This allows us to formulate and prove using an explicit calculus for any functions f ( x, p ) and f ( x, p ) on T ∗ V and a canonical Poisson structure { f, f } := θ ( e X f , e X f ) . The canonical Hamilton-Jacobi equations are defined using above canonical Poisson structure, dx i dτ = { e H, x i } and dp a dτ = { e H, p a } . Boldface symbols are used in order to emphasize that certain geometric/physical objects are considered in N–adapted formfor certain phase spaces and/or spacetime enabled with N–connection structure and when the coefficients of tensors, spinors,and fundamental geometric objects can be computed with respect to N-elongated bases of type (8). L -dual effective phase spaces e H , and e L , is described equiva-lently by the Hamilton equations dx i dτ = ∂ e H∂p i and dp i dτ = − ∂ e H∂x i , or as Euler-Lagrange equations, ddτ ∂ e L∂y i − ∂ e L∂x i = 0 . In their turn, these equations are equivalent to the nonlinear geodesic (semi-spray) equations d x i dτ + 2 e G i ( x, y ) = 0 , (11)for e G i = e g ij ( ∂ e L∂y i y k − ∂ e L∂x i ) , with e g ij being inverse to e g ij (3).The equations (11) show that point like probing particles move not along usual geodesics as on Lorentzmanifolds but follow some nonlinear geodesic equations determined by generating Lagrange functions andtheir Hessians.Using the constructions from above subsection, we prove there are canonical N–connections determinedby generating Lagrange/ Hamilton functions following formulas p e N = ( p e N ij := 12 " { p e g ij , e H } − ∂ e H∂p k ∂x i p e g jk − ∂ e H∂p k ∂x j p e g ik and e N = ( e N ai := ∂ e G∂y i ) , (12)where p e g ij is inverse to p e g ab (4). Introducing these canonical N–connection coefficients into formulas (8),we prove that there are canonical N–adapted (co) frames e e α = ( e e i = ∂∂x i − e N ai ( x, y ) ∂∂y a , e b = ∂∂y b ); e e α = ( e e i = dx i , e e a = dy a + e N ai ( x, y ) dx i ); and (13) p e e α = ( p e e i = ∂∂x i − p e N ia ( x, p ) ∂∂p a , p e b = ∂∂p b ); p e e α = ( p e i = dx i , p e a = dp a + p e N ia ( x, p ) dx i ) . Such a canonical N-splitting e N : T T V = hT V ⊕ vT V and p e N : T T ∗ V = hT ∗ V ⊕ vT ∗ V is statedby respective generating Lagrange and/or Hamilton functions on any tangent and/or cotangent Lorentzbundle. The nonholonomic structure of phase spaces can be described in equivalent forms using canonicaldata ( e L, e N ; e e α , e e α ) , with effective Largange density e L (correspondingly, ( e H , p e N ; p e e α , p e e α ) , with effectiveHamilton density e H ). We can consider a general N-splitting without effective Lagrangians (Hamiltonians),i.e. in terms of arbitrary geometric data ( N ; e α , e α ) (correspondingly ( p N ; p e α , p e α ) ). Using tensor productsof N-adapted (co) frames on phase space, we can parameterize in N-adapted forms (canonical or general ones)arbitrary tensors fields (d-tensors), connections and d-connections and other types of geometric objects, d-objects.
There are canonical data ( e L, e N ; e e α , e e α ; e g jk , e g jk ) and/or ( e H, p e N ; p e e α , p e e α ; p e g ab , p e g ab ) when the d-metricsare parameterized in the Hessian form both for the h- and (c)v-components, e g = e g αβ ( x, y ) e e α ⊗ e e β = e g ij ( x, y ) e i ⊗ e j + e g ab ( x, y ) e e a ⊗ e e a and/or (14) p e g = p e g αβ ( x, p ) p e e α ⊗ p e e β = p e g ij ( x, p ) e i ⊗ e j + p e g ab ( x, p ) p e e a ⊗ p e e b . (15) On nonholonomic (co) tangent bundles, we can consider d–vectors if they are written in a form adapted to a prescribedN–connection structure, for instance, X = e X α e e α = e X i e e i + X b e b = X α e α = X i e i + X b e b ∈ T TV , p X = p e X α e e α = p e X i p e e i + p X b p e b = p X α p e α = p X i p e i + p X b p e b ∈ T T ∗ V . Such formulas can be written equivalently for decompositions with respect to canonical, or arbitrary, N-adapted bases. In brief,the h-v and/or h-cv decompositions can be written X α = e X α = ( e X i , X b ) = ( X i , X b ) , p X α = p e X α = ( p e X i , p X b ) = ( p X i , p X b ) . Considering X and p X as 1-forms, we have X = e X α e α = X i e i + e X a e e a = e X α e α = X i e i + X a e a ∈ T ∗ TV p X = p e X α p e α = p X i p e i + p e X a p e e a = p e X α p e α = p X i p e i + p X a p e a ∈ T ∗ T ∗ V , or, in brief, X α = e X α = ( X i , e X a ) = ( X i , X a ) , p X α = p e X α = ( p X i , p e X a ) = ( p X i , p X a )
7y frame transforms, the canonical d-metric structures (14) and (15) [with tildes] can be written, respec-tively, in general d-metric forms (9) and (10) [without tildes]. In explicit form, the general vierbein transformsare written e α = e αα ( u ) ∂/∂u α and e β = e ββ ( u ) du β . We underline the local coordinate indices in order todistinguish them from arbitrary abstract ones. In such formulas, the matrix e ββ is inverse to e αα for orthonor-malized bases. For Hamilton like configurations on cotangent bundles, we consider p e α = p e αα ( p u ) ∂/∂ p u α and p e β = p e ββ ( p u ) d p u β . There are not used boldface symbols for such transforms because they can be notadapted to a N–connection structure.Using (13), respectively, for (9) and (10) and regrouping with respect to local coordinate bases, we provethat with respect to local coordinate frames, any d–metric structures on TV and/or T ∗ V , g = g αβ ( x, y ) e α ⊗ e β = g αβ ( x, y ) du α ⊗ du β and/or p g = p g αβ ( x, p ) p e α ⊗ p e β = p g αβ ( x, p ) d p u α ⊗ d p u β . These formulas can be subjected to frame transforms, g αβ = e αα e ββ g αβ and p g αβ = p e αα p e ββ p g αβ , and writtenin equivalent off-diagonal forms: g αβ = (cid:20) g ij ( x ) + g ab ( x, y ) N ai ( x, y ) N bj ( x, y ) g ae ( x, y ) N ej ( x, y ) g be ( x, y ) N ei ( x, y ) g ab ( x, y ) (cid:21) and/or p g αβ = (cid:20) p g ij ( x ) + p g ab ( x, p ) p N ia ( x, p ) p N jb ( x, p ) p g ae p N je ( x, p ) p g be p N ie ( x, p ) p g ab ( x, p ) (cid:21) . (16)Parameterizations of type (16) for metrics are considered, for instance, in Kaluza–Klein theories onassociated vector bundles. In our cases, the constructions are on (co) tangent bundles for geometric mechanicsmodels. We conclude that if we fix a metric structure of type p e g (15), we can elaborate equivalent modelswith p g (10) determined by certain classes of nonholonomic frame transforms. Inversely, prescribing a d-metric p g , we can define nonholonomic variables when this metric structure can be represented as a p e g , i.e.in mechanical like variables, when p g = p e g . In a more general context, we can elaborate on bi-metric (andeven multi-metric theories of gravity, geometric mechanics and thermodynamics) if we consider that p e g and p g are related via certain generalized nonholonomic transforms, see details an references in [34, 35].The canonical N–connections e N and p e N define respectively certain canonical almost complex structures e J , on TV , and p e J , on T ∗ V . This follows, for instance, from such a construction on T ∗ V . Let us consider alinear operator p e J acting on p e α = ( p e i , p e b ) using formulas p e J ( p e i ) = − p e n + i and p e J ( p e n + i ) = p e i . This p e J defines globally an almost complex structure ( p e J ◦ p e J = − I , where I is the unity matrix) on T ∗ V . Suchan operator is completely determined for Hamilton spaces by a e H ( x, p ) . We note that e J and p e J are standard almost complex structures only for the Euclidean signatures,respectively, on TV and T ∗ V . Contrary, we call them as pseudo almost complex structure. It is possible toomit tildes and write J and p J for arbitrary frame/ coordinate transforms.The canonical Neijenhuis tensor fields determined by Lagrange and Hamilton generating functions, forrespective canonical almost complex structures e J on TV and/or p e J on T ∗ V , are introduced as curvaturesof respective N–connections e Ω ( e X , e Y ) := − [ e X , e Y ] + [ e J e X , e J e Y ] − e J [ e J e X , e Y ] − e J [ e X , e J e Y ] and/or p e Ω ( p e X , p e Y ) := − [ p e X , p e Y ] + [ p e J p e X , p e J p e Y ] − p e J [ p e J p e X , p e Y ] − p e J [ p e X , p e J p e Y ] , (17)for any d–vectors X , Y and p X , p Y . Such formulas can be written in general form without tilde values if thereare considered arbitrary frame transforms. In local form, a N–connection on TV , or T ∗ V , is characterizedby such coefficients of (17) (i.e. the N–connection curvature): Ω aij = ∂N ai ∂x j − ∂N aj ∂x i + N bi ∂N aj ∂y b − N bj ∂N ai ∂y b , or p Ω ija = ∂ p N ia ∂x j − ∂ p N ja ∂x i + p N ib ∂ p N ja ∂p b − p N jb ∂ p N ia ∂p b . (18)8lmost complex structures J and p J transform into standard complex structures for Euclidean signaturesif Ω = 0 and/or p Ω = 0 . For almost complex canonical structures, we can consider canonical forms with"tilde" values determined by e N = { e N bj } and p e N = { p e N ia } . Applying a straightforward N-adapted calculus using formulas e e α = ( e e i , e b ) and p e e α = ( p e e i , p e b ) , see (13)and (18), we prove that the canonical nonholonomic frame structures on TV and/or T ∗ V are characterizedby corresponding anholonomy relations [ e e α , e e β ] = e e α e e β − e e β e e α = f W γαβ e e γ and [ p e e α , p e e β ] = p e e α p e e β − p e e β p e e α = p f W γαβ p e e γ (19)with anholonomy coefficients f W bia = ∂ a e N bi , f W aji = e Ω aij , and p f W aib = ∂ p e N ib /∂p a and p f W jia = p e Ω ija . We candefine holonomic (integrable) frame configurations if the respective anholonomy coefficients in (19) are zero.In geometric mechanics, the canonical d-metric structures e g (14) and p e g (15) are described by genericoff–diagonal metrics (16) if respective anholonomy coefficients (19) are not trivial. Elaborating on different type Lagrange-Hamilton models, we are not able to perform the constructions inN-adapted anholonomic form if we work only with generalized (Finsler like) metrics determined by nonlinearquadratic forms L ( x, y ) and/or H ( x, p ) (6). The goal of this subsection is to analyze which classes oflinear connections and respective covariant derivative operators can be generated canonically by fundamentalgenerating functions. We can define a linear connection D on TV when a L –duality between the tangent and correspondingcotangent bundles which can be defined by pull–back and push–forward maps. We omit geometric detailson constructing such maps from/to base space to total space, considered, for instance, in [34, 35]. A linearconnection p D on T ∗ V is defined as follows: p D p X p Y := ( D X Y ) ∗ = p ( D X Y ) , for any vector fields p X and p Y on T ∗ V . Inversely, we can consider a linear connection p D on T ∗ V and then construct a linearconnection ◦ D on TV , following the rule ◦ D X Y := ( p D p X p Y ) ◦ , for any vector fields X and Y on TV .A distinguished connection (d–connection) is a linear connection D on TV (or p D on T ∗ V ) which iscompatible with the N–connection splitting (7).The coefficients of d–connections can be defined and computed in corresponding N-adapted forms, D e β e γ := Γ αβγ e α and p D p e β p e γ := p Γ αβγ p e α . For a h-v splitting, D e k e j := L ijk e i , D e k e b := ´ L abk e a , D e c e j := ´ C ijc e i , D e c e b := C abc e a and a h-cv splitting, p D p e k p e j := p L ijk p e i , p D e k p e b := − p ´ L ba k p e a , p D p e c p e j := p ´ C i cj p e i , p D p e c p e b := − p C bca p e a . In result, theN-adapted coefficients of d-connections on (co) tangent Lorentz bundles can be parameterized (respectively) Γ αβγ = { L ijk , ´ L abk , ´ C ijc , C abc } and p Γ αβγ = { p L ijk , p ´ L ba k , p ´ C i cj , p C bca } . These coefficients can be used forexplicit computations of h– and/or v–splitting, cv-splitting, of covariant derivatives D = ( h D , v D ) and p D = (cid:0) p h D , p v D (cid:1) , where h D = { L ijk , ´ L abk } , v D = { ´ C ijc , C abc } and p h D = { p L ijk , p ´ L ba k } , p v D = { p ´ C i cj , p C bca } . We can consider a linear connection D (which is not obligatory a d-connection) and a d-connection D bothdefined on TV . Such geometric objects are respectively denoted p D and p D on T ∗ V . For (co)vector bundles,there are nonholonomic relations with respective distortion d-tensors Z := D − D and p Z := p D − p D. Using similar definitions and theorems both for linear connections and d-connections, we prove that d–connection D , or p D , is characterized by respective curvature ( R , or p R ) , torsion ( T , or p T ) , nonmetricity, ( Q , or p Q ) , d-tensors, R ( X , Y ) := D X D Y − D Y D X − D [ X , Y ] , T ( X , Y ) := D X Y − D Y X − [ X , Y ] , Q ( X ) := D X g , or (20) p R ( p X , p Y ) := p D p X p D p Y − p D p Y p D p X − p D [ p X , p Y ] , p T ( p X , p Y ) := p D p X p Y − p D p Y p X − [ p X , p Y ] , p Q ( p X ) := p D p X p g . The N–adapted coefficients for the curvature, torsion and nonmetricity d-tensorsare provided in Appendices to [34, 35], see also references therein. The geometric d-tensors (20) are written,for instance, using tilde on symbols if such d-objects are defined and computed for Lagrange (or Hamilton)generating functions, see below. Respectively, the Ricci d–tensors are defined and computed as
Ric = { R αβ := R ταβτ } , for a d-connection D , and p Ric = { p R αβ := p R ταβτ } , for a d-connection p D , see formulas (20). In N-adapted form, we provethat the N-adapted coefficients of the Ricci d–tensors of a d-connection D (or p D ) in respective phase spacesare parameterized in h - and/or v -, or cv -form, by formulas R αβ = { R hj := R ihji , R ja := − P ijia , R bk := P abka , R bc = S abca } , or (21) p R αβ = { p R hj := p R ihji , p R aj := − p P i aji , p R bk := p P b aa k , p R bc = p S bcaa } . (22)If a phase space is enabled both with a d-connection, D (or p D ) , and d-metric, g (9) (or p g (10)) [inparticular, we can consider canonical "tilde" values with d-metrics e g (14) and/or p e g (15), and their frametransforms], we can define and compute nonholonomic Ricci scalars. In result, we obtain that the scalarcurvature of a d-connection D , or p D , can be defined and computed for the inverse d-metric g αβ , or p g αβ , s R := g αβ R αβ = g ij R ij + g ab R ab = R + S, or p s R := p g αβ p R αβ = p g ij p R ij + p g ab p R ab = p R + p S, with respective h– and v–components R = g ij R ij , S = g ab S ab , or p R = p g ij p R ij , p S = p g ab p S ab . By constructions, the Einstein d-tensors on TV and/or T ∗ V are defined: En = { E αβ := R αβ − g αβ s R } and/or p En = { p E αβ := p R αβ − p g αβ p s R } . Such values can be used in MGTs and encoding geometric and physical models in quantum computingtheories.
The Lagrange and/or Hamilton phase spaces (with a possible L –duality) can be endowed and charac-terized respectively by different type geometric and physically important linear connections and canonical/almost symplectic connections, which are equivalent if respective distorting relations are postulated. In ourapproaches to geometric mechanics and classical and quantum field/ thermodynamic and gravity theories,we use such linear connection structures: [ g , N ] ≃ [ e g , e N ] ≃ [ e θ := e g ( e J · , · ) , e P , e J , e J ] (23) = ⇒ ∇ : ∇ g = 0; T [ ∇ ] = 0 , Lagrange LC–connection ; b D : b D g = 0; h b T = 0 , v b T = 0 . canonical Lagrange d-connection ; e D : e D e θ = 0 , e D e θ = 0 almost sympl. Lagrange d-connection ;[ p g , p N ] ≃ [ p e g , p e N ] ≃ [ p e θ := p e g ( p e J · , · ) , p e P , p e J , p e J ] (24) = ⇒ p ∇ : p ∇ p g = 0; p T [ p ∇ ] = 0 , Hamilton LC-connection ; p b D : p b D g = 0; h p b T = 0 , cv p b T = 0 . canonical Hamilton d-connection ; p e D : p e D p e θ = 0 , p e D p e θ = 0 almost sympl. Hamilton d-connection.We can consider distortion relations b D = ∇ + b Z , e D = ∇ + e Z , and b D = e D + Z , determined by ( g , N ); (25) p b D = p ∇ + p b Z , p e D = p ∇ + p e Z , and p b D = p e D + p Z , determined by ( p g , p N ); b Z , e Z , and Z , on T TV , and p b Z , p e Z , and p Z , on T T ∗ V . Geometric mechanic models are characterized by respective canonical and/or almost symplectic distortiond-tensors b Z [ e g , e N ] , e Z [ e g , e N ] , and Z [ e g , e N ] , for (almost symplectic) Lagrange models, and p b Z [ p e g , p e N ] , p e Z [ p e g , p e N ] , and p Z [ p e g , p e N ] , for (almost symplectic) Hamilton models. Respective phase space geometries can be describedin equivalent forms by such data on TV : ( g , N , b D ) ⇆ ( L : e g , e N , e D ) ↔ ( e θ, e P , e J , e J , e D ) ↔ [( g [ N ] , ∇ )]; l possible L -duality & l not N-adaptedon T ∗ V : ( p g , p N , p b D ) ⇆ ( H : p e g , p e N , p e D ) ↔ ( p e θ, p e P , p e J , p e J , p e D ) ↔ [( p g [ p N ] , p ∇ )] . We can work with canonical d-connection structures on (co) tangent bundles, b D and/or p b D which allows usto decouple and integrate in most general exact and parametric forms certain effective geometric flow and/ormodified gravitational field equations. Here we note that Lagrange–Finsler variables can be introduced on4-d, and higher dimension, (pseudo) Riemannian spaces and in GR if nonholonomic fibered structures areconsidered on spacetime manifolds, see discussions and examples in Refs. [34, 35, 25, 26, 27, 28, 29, 24].An important example is that when imposing certain (in general, nonholonomic) constraints of type b Z = 0 , we obtain b D | b Z =0 ≃ ∇ even b D = ∇ . If such conditions are satisfied, we can extract (pseudo)Riemannian or effective geometric mechanical (with tilde values) LC-configurations from more (general)nonholonmic metric-affine structures. For instance, we can obtain LC-configurations for geometric modelswith b D and/or p b D for respective zero distortions, b Z and/or p b Z . Equivalently, one can be considered thezero torsion conditions for b T = { b T γαβ } = 0 and/or p b T = { p b T γαβ } = 0 . Using distortions of linear connections, we can prove in abstract and N-adapted forms that there arecanonical distortion relations encoding generating functions for respective Lagrange-Hamilton and equivalentnonholonomic variables: For the curvature d-tensors, we compute b R [ g , b D = ∇ + b Z ] = R [ g , ∇ ] + b Z [ g , b Z ] , p b R [ p g , p b D = p ∇ + p b Z ] = p R [ p g , p ∇ ] + p b Z [ p g , p b Z ] , with respective distortion d-tensors b Z , on TV , and p b Z , on T ∗ V . Similarly, we obtain for the Ricci d-tensors, b Ric [ g , b D = ∇ + b Z ] = Ric [ g , ∇ ] + b Zic [ g , b Z ] , p b Ric [ p g , p b D = p ∇ + p b Z ] = p Ric [ p g , p ∇ ] + p b Zic [ p g , p b Z ] , with respective distortion d-tensors b Zic, on TV , and p b Zic, on T ∗ V . Finally, for the scalar curvature ofcanonical d-connection b D , or p b D , p s b R [ g , b D = ∇ + b Z ] = R [ g , ∇ ] + s b Z [ g , b Z ] , p s b R [ p g , p b D = p ∇ + p b Z ] = p s R [ p g , p ∇ ] + p s b Z [ p g , p b Z ] , with respective distortion scalar functionals s b Z, on TV , and p s b Z, on T ∗ V .Above formulas can be reformulated for distortions of the almost symplectic Lagrange, or Finsler, d-connections, for instance, considering e R [ e g ≃ e θ, e D = ∇ + e Z ] = R [ e g ≃ e θ, ∇ ] + e Z [ e g ≃ e θ, e Z ] , p e R [ p e g ≃ p e θ, p e D = p ∇ + p e Z ] = p R [ p e g ≃ p e θ, p ∇ ] + p e Z [ p g ≃ p e θ, p e Z ] , and any similar geometric objects with "tilde" symbols. The goal of this section is to formulate in canonical Hamilton variables the theory of nonholonomicgeometric flows of relativistic mechanical systems. This is important for further developments in classical andquantum information theories when the Hamilton variables are used in explicit form. We shall present alsothe main results in canonical Lagrange variables because such formulas are very important for investigatingvarious connections between quantum field theory, QFT, quantum gravity, QG, and quantum informationtheory. Such a research is related to author’s project on geometric flows and applications in physics which was elaborated in2005 for a sabbatical professor fellowship at CSIC, Madrid, in Spain, and further developments supported by a project IDEI, in .1 Relativistic geometric flows and Perelman’s thermodynamics for phase spacetimes Let us consider families of nonholonomic 8–d tangent and cotangent Lorentz bundles, T V ( τ ) and T ∗ V ( τ ) parameterized by a positive parameter τ, ≤ τ ≤ τ . Such phase spacetimes are enabled with correspondingsets of canonical d-metrics of type (14) and (15), e g ( τ ) = e g ( τ, u ) and p e g ( τ ) = p e g ( τ, p u ) and canonical N–connections of type (12), p e N ( τ ) = p e N ( τ, p u ) . Any relativistic nonholonomic phase spacetime T V ⊂ T V ( τ ) and/or T ∗ V ⊂ T ∗ V ( τ ) can be enabled with necessary types of double nonholonomic (2+2)+(2+2) and(3+1)+(3+1) splitting, see details for such geometric constructions in [24, 25, 26, 27, 28]. For instance, anonholonomic (3+1)+(3+1) splitting on a T V can be chosen in such a form that any open region on a baseLorentz manifold, U ⊂ V , is covered by a family of 3-d spacelike hypersurfaces b Ξ t , or e Ξ t , parameterized bya time like parameter t. The parameterizations of hypersurfaces can be labeled in certain forms which areadapted to the type of canonical d-connection we use for our geometric constructions. In this work, we preferto use "tilde" labels/ values related to geometric mechanics. On a typical cofiber of T ∗ V , we can consider a3-d cofiber hypersurface p e Ξ E , for instance, of signature (+ + +) with a label E for parameterizations by anenergy type parameter. We can write correspondingly e Ξ = ( e Ξ t , e Ξ E ) and p e Ξ = ( e Ξ t , p e Ξ E ) for nonholonomicdistributions of base and fiber hypersurfaces with conventional splitting 3+3 of signature (+++;+++) ontotal phase space. For additional shell decompositions of type (2+2)+(2+2), we can use also a s -label, p s b Ξ = ( s b Ξ t , p s b Ξ E ) ⊂ s T ∗ V , if we shall be interested in constructing certain classes of exact or parametricsolutions of geometric flow equations. In general, we can elaborate on two generic different types of geometricphase flow theories: The fist type is with a conventional parameter τ ( χ ) admitting re-parameterizations ofa temperature like parameter used for labeling 4-d Lorentz spacetimes and their phase space configurations.The second type of models is with τ ( t ) as a time like parameter when (3+3)-d spacelike phase configurationsevolve relativistically on a "redefined" time like coordinate. In this work, we elaborate on theories of type 1. In [20], we studied geometric flows of Finsler-Lagrange theories using canonical data ( g ( τ ) , e D ( τ )) whenvarious generalizations and applications in MGTs were elaborated for the data ( g ( τ ) , b D ( τ )) , [19, 23, 24, 25,26, 27, 28, 29]. Those constructions were based on nonholonomic generalizations of Perelman’s functionals[1] and distortion relations form the Levi-Civita configurations ( g ( τ ) , ∇ ( τ )) . Let us consider how Perelman’sfunctionals can be generalized in relativistic form for geometric flow evolution of Lagrange–Hamilton spaces.
F- and W-functionals in canonical J. Lagrange variables:
Considering canonical Lagrange data ( e g ( τ ) , e D ( τ )) on tangent Lorentz bundles in order to postulate the functionals: e F = eZ e − e f q | e g αβ | d u ( s e R + | e D e f | ) and (26) f W = eZ e µ q | e g αβ | d u [ τ ( s e R + | h e D e f | + | v e D e f | ) + e f − , (27) Romania; and related visiting projects at CERN (Switzerland); M. Planck Institute, Munich, and A. Einstein Institute, Postdam,(Germany) etc. Those projects were on applications of nonholonomic geometric methods in classical and quantum mechanicsand physics, with various generalizations to deformation quantization, noncommutative geometry etc. A sub-direction of formerresearch was devoted to studies on flow evolution of Lagrange-Hamilton systems geometrized on (co) tangent bundles, whichresulted in a series of works on the nonholonomic geometric evolution of Finsler-Lagrange-Hamilton space spaces, see historicalremarks and a comprehensive bibliography in Appendix B.4.17 to Ref. [34]. Here we note that nonholonomic generalizations ofG. Perelman functionals and R. Hamilton geometric evolution equations were considered for Finsler–Lagrange systems in Refs.[19, 20], see further generalizations for almost Kähler –Lagrange-Hamilton models on Lie algebroids , relativistic Lagrange-Hamilton mechanics etc. [21, 22, 50]. In principle, Finsler-Lagrange-Hamilton variables can be introduced on any (non)commutative / (super) manifold, which allows to re-write in effective (super/noncommutative) mechanic forms all results ongeometric flows of physical theories elaborated in [23, 24, 25, 26, 27, 28, 29], see also references therein. Additionally to coordinate and index conventions from footnote 2, we label the local (3+1)+(3+1) coordinates in theform p u = { p u α = p u α s = ( x i , y a ; p a , p a ) = ( x ` ı , u = y = t ; p ` a , p = E ) } for i , j , k , ... = 1 , a , b , c , ... = 3 , a , b , c , ... = 5 , a , b , c , ... = 7 , and ` ı, ` j, ` k, ... = 1 , , , respectively, ` a, ` b, ` c, ... = 5 , , can be used for correspondingspacelike hyper surfaces on a base Lorentz manifold and typical cofiber. e f ( τ, u ) satisfies the conditions eZ e µ q | e g αβ | d u := Z t t Z e Ξ t Z y y Z p e Ξ E e µ q | e g αβ | d u = 1 for e µ = (4 πτ ) − e − e f , when the coefficients
16 = 2 × is for 8-d spaces. For 3-d hypersurface LC-configurationswith ∇ , such values transform into the standard G. Perelman functionals. The Ricci scalar s e R is taken for theRicci d-tensor e R αβ (21) constructed for the canonical Lagrange data ( e g , e D ) . Re-defining the normalizationfunctions and using corresponding nonholonomic frame transforms and d-connection distortions, we can re-write the functionals (26) and (27) in "hat" variables, b F and c W , see similar constructions in [19, 24, 25, 26,27, 28]. F- and W-functionals in canonical W. Hamilton variables:
We use canonical data ( p e g ( τ ) , p e D ( τ )) on cotangent Lorentz bundles and postulate the functionals: p e F = p eZ e − p e f q | p e g αβ | d p u ( p s e R + | p e D p e f | ) and (28) p f W = p eZ p e µ q | p e g αβ | d p u [ τ ( p s e R + | p h e D p e f | + | p v e D p e f | ) + p e f − , (29)where the normalizing function p e f ( τ, p u ) satisfies p eZ p e µ q | p e g αβ | d p u := Z t t Z e Ξ t Z E E Z p e Ξ E p e µ q | p e g αβ | d p u = 1 for p e µ = (4 πτ ) − e − p e f , when the coefficient
16 = 2 × is taken for 8-d spaces. The Ricci scalar p s e R is takenfor the Ricci d-tensor p e R αβ (22) constructed using the canonical Hamilton data ( p e g , p e D ) . Similar functionals can be postulated for nonholonomic geometric flows on T ∗ V using data ( p g ( τ ) , p b D ( τ )) and redefined integration measures and normalizing functions on respective hypersurfaces. Considering LC-configurations with p e D | p e T =0 = p ∇ and/or p b D | p b T =0 = p ∇ , the values (28) and (29) transform respectivelyinto 8-d phase space versions of the so called Perelman’s F-entropy and W-entropy. It should be noted that p f W and/or p c W do not have a character of entropy for pseudo–Riemannian metrics but can be treated as avalue characterizing relativistic geometric hydrodynamic phase space flows. Nonholonomic lapse and shift variables:
Using N–adapted diadic shell and/or double (2+2)+(2+2)and (3+1)+(3+1) frame and coordinate transforms of metrics with additional dependence on a flow pa-rameter, we can introduce various parameterizations of geometric objects on phase spacetimes. To definethermodynamic like variables for geometric flow evolution of stationary configurations on T ∗ V , we take p g = p g α ′ β ′ ( τ, p u ) d p e α ′ ⊗ d p e β ′ = q i ( τ, x k ) dx i ⊗ dx i + q ( τ, x k , y ) e ⊗ e − [ ˘ N ( τ, x k , y )] e ⊗ e + p q a ( τ, x k , y , p b ) p e a ⊗ p e a + p q ( τ, x k , y , p b , p b ) p e ⊗ p e − [ p ˇ N ( τ, x k , y , p b , p b )] p e ⊗ p e , where, for instance, p e α s are N-adapted bases on total space of respective cotangent Lorentz bundles. Thisansatz for metrics is a general N-adapted one for a 8–d phase space metric which can be written as anextension of a couple of 3–d metrics, q ij = diag ( q ` ı ) = ( q i , q ) on a hypersurface e Ξ t , and p q ` a ` b = diag ( p q ` a ) =( p q a , p q ) on a hypersurface p e Ξ E , if q = g , ˘ N = − g and p q = p g , p ˇ N = − p g , (30)where ˘ N is the lapse function on the base and p ˇ N is the lapse function in the co-fiber (here we note that"the inverse hat" labels are a bit different for the 4-th and 8-th coordinate).13n T V , the nonholonomic lapse and shift variables are introduced in a similar way, which results ind–matric parameterizations g = g α ′ β ′ ( τ, u ) d e α ′ ⊗ d e β ′ = q i ( τ, x k ) dx i ⊗ dx i + q ( τ, x k , y ) e ⊗ e − [ ˘ N ( τ, x k , y )] e ⊗ e + q a ( τ, x k , y , y b ) e a ⊗ e a + q ( τ, x k , y , y b , y b ) e ⊗ e − [ ˇ N ( τ, x k , y , y b , y b )] e ⊗ e . (31) We consider respective hypersurface formulas, q ij = diag ( q ` ı ) = ( q i , q ) on a hypersurface e Ξ t , and q ` a ` b = diag ( q ` a ) = ( q a , q ) on a hypersurface e Ξ E , if q = g , ˘ N = − g and q = g , ˇ N = − g , where ˘ N is thelapse function on the base and p ˇ N is the lapse function in the fiber G. Perelman’s very original idea was that the geometric flows of Riemannian metrics can be characterizedby an analogous thermodynamic model [1]. In this work, we consider relativistic mechanical generalizationsrelated to geometric flow approaches to classical mechanics [19, 20].
Some basic concepts from statistical thermodynamics:
To elaborate analogous thermodynamicalmodels we can consider a partition function Z = R exp( − βE ) dω ( E ) for the canonical ensemble at temper-ature β − = T (one should not confuse this T for thermodynamics with standard tensor notations with T containing indices for respective for energy-momentum tensors and/or torsion in MGTs) being defined bythe measure taken to be the density of states ω ( E ) . The thermodynamical values are computed in standardform for the average energy, E = h E i := − ∂ log Z/∂β, the entropy S := β h E i + log Z and the fluctuation η := D ( E − h E i ) E = ∂ log Z/∂β . Using Z, we can define a conventional state density (generalized as adensity matrix, it is important for elaborations in geometric flow thermodynamics and information theory,see next sections) σ ( β, E ) = Z − e − βE . Considering log σ = − β E− log Z, we define the relative entropy between any state density ρ and σ, S ( ρ q σ ) := −S ( ρ ) + Z ( β E + log Z ) ρdω ( E ) = β [ E ( ρ ) − T S ( ρ )] + log Z, where the average energy computed in the density matrix ρ is E ( ρ ) = R E ρdω ( E ) . The free energy corre-sponding to ρ is F ( ρ ) := E ( ρ ) − T S ( ρ ) . (32)We note that if log Z is independent on ρ (as we consider in above formulas) we have S ( σ q σ ) = 0 . Thisallows us to write S ( ρ q σ ) = β [ F ( ρ ) − F ( σ )] . (33)In this work, we study the geometric flow evolution of thermodynamics systems that preserves the thermalequilibrium at temperature β but maps ρ → ρ ′ (such density states are different ones) keeping the samedensity state σ. We can provide a realistic physical interpretation for such systems if S ( ρ q σ ) ≥ S ( ρ ′ q σ ) , i.e. F ( ρ ) ≥ F ( ρ ′ ) . (34)So, we should elaborate on thermodynamic geometric flows that preserve the thermal equilibrium and canonly reduce the free energy. These aspects connect mechanical flow models to the second low of thermody-namics. It should be noted here that G. Perelman treated τ = β − as a temperature parameter and that he introduced the conceptof W–entropy following an analogy to formulas for the entropy in statistical mechanics. We reproduce here the Remark 5.3 andnext paragraph, just before section 6 in [1]: "An entropy formula for the Ricci flow in dimension two was found by Chow [C]; thereseems to be no relation between his formula and ours. .... The interplay of statistical physics and (pseudo)-riemannian geometry hermodynamic values for relativistic Lagrange-Hamilton flows: For relativistic geometric flowsof mechanical systems, we introduce respective thermodynamic generating functions e Z [ e g ( τ )] = eZ e − e f q | e g αβ | d u ( − e f + 16) , for T V ; (35) p e Z [ p e g ( τ )] = p eZ e − p e f q | p e g αβ | d p u ( − p e f + 16) , for T ∗ V , where the respective functional dependence is given by [ e g ( τ )] and [ p e g ( τ )] (we shall not write such dependen-cies if that will not result in ambiguities). For a thermodynamic analogous interpretation we can considerthat a density state σ is associated to e g αβ , we can write in functional form σ [ e g ] , but the geometric evolutionmay involve densities ρ [ e g ] and ρ ′ [ e g ] , where the left label 1 is used in order to distinguish two d-metrics e g and e g . On cotangent bundles, such values are written respectively p σ [ p e g ] , p ρ [ p e g ] and ρ ′ [ p e g ] . Generalizing for nonholonomic deformations of metrics and d-connections respective formulas related torespective entropy like functionals (26), (27) and (28), (29), we can define and compute such relativisiticthermodynamic values for geometric evolution flows of Lagrange mechanical systems,average flow energy: e E = − τ eZ e − e f q | q q q ˘ N q q q ˇ N | δ u ( s e R + | e D e f | − τ ) , (36)flow entropy: e S = − eZ e − e f q | q q q ˘ N q q q ˇ N | δ u h τ (cid:16) s e R + | e D e f | (cid:17) + ˜ f − i , flow fluctuation: e η = − eZ e − e f q | q q q ˘ N q q q ˇ N | δ u [ | e R αβ + e D α e D β ˜ f − τ g αβ | ] , where δ u contains N-elongated differentials of type (8) (when we compute such integrals in N-adapted form).Using such values, we can compute the respective free energy (32) and relative entropy (33), e F ( e g ) = e E ( e g ) − β − e S ( e g ) and e S ( e g q σ ) = β [ e F ( e g ) − e F ( e g )] , where e E ( e g ) = − τ eZ e − e f q | q q q ˘ N q q q ˇ N | δ u [ s e R ( e g ) + | e D ( e g ) e f ( τ, u ) | − τ ] , e S ( e g ) = − eZ e − e f q | q q q ˘ N q q q ˇ N | δ u h τ (cid:16) s e R ( e g ) + | e D ( e g ) e f ( τ, u ) | (cid:17) + ˜ f ( τ, u ) − i . Such values are in relativistic thermodynamic relation if the second thermodynamic law (34) is satisfied.This impose certain constraints on the class of normalizing and generating functions we consider for thetermodynamic description of such relativistic Lagrange systems.For geometric evolution flows of Hamilton mechanical systems, the relativistic thermodynamic values are p e E = − τ p eZ e − p e f q | q q q ˘ N p q p q p q p ˇ N | δ p u ( p s e R + | p e D p e f | − τ ) , (37) p e S = − p eZ e − p e f q | q q q ˘ N p q p q p q p ˇ N | δ p u h τ (cid:16) p s e R + | p e D p e f | (cid:17) + p ˜ f − i , p e η = − p eZ e − p e f q | q q q ˘ N p q p q p q p ˇ N | δ p u [ | p e R αβ + p e D α p e D β p ˜ f − τ p g αβ | ] . occurs in the subject of Black Hole Thermodynamics, developed by Hawking et al. Unfortunately, this subject is beyond myunderstanding at the moment." It should be also emphasized that G. Perelman had not specified what type of underlyingmicrostates and their energy should be taken in order to explain the geometric flows corresponding to certain thermodynamicaland gravity models. In this work, we are interested in geometric mechanics and the classical and quantum information theorydeveloping our approaches elaborated in [19, 23, 24, 25, 26, 27, 28, 29]. b D and p b D if we consider nonholonomic deformations to certain systems of nonlinear partialdifferential equations with general decoupling. The geometric flow evolution of 4-d (pseudo) Riemannian configurations is described by nonholonomicallymodified Perelman’s functionals integrated on (co) fiber variables (26), (27) and/or (28), (29). A subclassof such relativistic flows are generated for parameterizations with d-metrics (1) and (2)). Re-defining thenormalizing functions, e f → b f ( x , x , y , y ) and/or e f → p b f , for general frame transforms on a base Lorentzmanifold, we obtain such functionals: b F = Z t t Z b Ξ t e − b f q | q q q ˘ N | δ u ( s b R + | b D b f | ) and (38) c W = Z t t Z b Ξ t (4 πτ ) − e − f q | q q q ˘ N | δ u [ τ ( s b R + | h b D b f | + | v b D b f | ) + b f − . In these formulas, geometric fllows of s b R are for respective b D = ( h b D , v b D ) on a family of bases V ( τ ) , wherethe normalizing function b f ( τ, u ) satisfies the conditions R t t R e Ξ t b µ q | q q q ˘ N | δ u = 1 for b µ = (4 πτ ) − e − b f , when the coefficient × is taken for 4-d manifolds.Using formulas for distortions of connections (25) re-defined for 4-d nonholonomic manifolds, the func-tionals (38) can re-written using geometric data ( e g , e D ) and/or ( g , ∇ ) . Such F- and W–functionals definenonholonomic geometric evolution flows of vacuum gravitational fields in MGTs and GR, see details in Refs.[25, 26, 27, 28]. We can consider that, in principle, (modified) gravitational interactions are induced ascertain emergent fields from geometric evolution flows of mechanical Lagrange/ Hamilton systems.The thermodynamic generating function corresponding to (38) can be defined in the form b Z = Z t t Z b Ξ t e − b f q | q q q ˘ N | δ u ( − b f + 8) , for V . In result, we can characterize emergent (pseudo) Riemannian geometries by such relativistic thermodynamicvalues, b E = − τ Z t t Z b Ξ t e − b f q | q q q ˘ N | δ u ( s b R + | b D b f | − τ ) , (39) b S = − Z t t Z b Ξ t e − b f q | q q q ˘ N | δ u h τ (cid:16) s b R + | b D b f | (cid:17) + b f − i , b η = − Z t t Z b Ξ t e − b f q | q q q ˘ N | δ u [ | b R αβ + b D α b D β ˜ f − τ g αβ | ] , where all geometric objects and indices are for 4-d base manifolds. Up to nonholonomic frame transforms anddeformations of connections, such vaules encode explicit information (integrated on fiber variables and/orprojected on base spacetime manifolds) on certain total space Lagrange/ Hamilton generating functions.There are different approaches for elaborating models of 3–d Ricci flow evolution of mechanical systemsand (emergent of prescribe) 4–d spacetimes with pseudo–Euclidean signature. In principle, there are twogeneral possibilities. In the first case, is to approach the problem as in the theories of stochastic / diffusion andkinetic processes with local anisotropy, fractional geometric evolution etc. For such models, one elaborates onthermofield models of Ricci flow evolution on imaginary time τ = − it (0 ≤ τ ≤ /κT, where κ is Boltzmann’sconstant. In corresponding formulas, the symbol T is used for the temperature (such a letter with respective16ndices for torsion and energy-momentum tensors is also used in gravity theories). In result, the pseudo–Riemannian spacetime is transformed into a Riemannian configuration space as one elaborates in thermaland/or finite temperature quantum field theory. The second class consists from theories modelled on 3-dhypersurfaces and evolving relativistically, for instance, on a 4-d Ricci soliton configuration. In such cases,the evolution parameter τ ∼ t is a time like coordinate. In this work, we study evolution of relativisticmechanics systems on a temperature like parameter τ ∼ T. Lagrange and Hamilton mechanical systems on Lorentz manifolds can be also characterized by 3-d spacelike hypersurface functionals. Such values can be defined respectively for (38) and (39) for any 3+1 splittingwith 3-d closed hypersurface fibrations b Ξ t . We denote by ◦ b D = b D | b Ξ t the canonical d–connection b D defined on a 3-d hypersurface b Ξ t . In a similarform, there are defined hypersurface "tilde" variables with ◦ e D = e D | e Ξ t determined as a projection of 8-dcanonical Lagrange-Hamilton d–connection defined on a 3-d hypersurface e Ξ t . For geometric flow evolution,all such values depend on a temperature like parameter τ ( τ ′ ) with possible scale re-definitions for anotherparameter τ ′ etc. We define also s ◦ b R := s b R | b Ξ t and s ◦ e R := s e R | e Ξ t . Using q ` ı ( τ ) = [ q i ( τ ) , q ( τ )] in a family ofd-metrics (see, for instance, (31)), we define 3-d F- and W-functionals parameterized in N–adapted form forthe canonical d-connection, ◦ b F = Z b Ξ t e − ◦ b f p | q q q | δ ` x h ( s ◦ b R + | ◦ b D ◦ b f | ) i , and (40) ◦ c W = Z b Ξ t ◦ b µ p | q q q | δ ` x (cid:20) τ (cid:16) s ◦ b R + | h ◦ b D ◦ b f | + | v ◦ b D ◦ b f | (cid:17) + ◦ b f − (cid:21) . (41)These functionals are for a redefined normalization function ◦ b f . We can always chose a necessary type scalingfunction ◦ b f which satisfies normalization conditions R b Ξ t ◦ b µ p | q q q | δ ` x = 1 for ◦ b µ = (4 πτ ) − e − ◦ b f . Fortopological considerations, the type of normalization is not important. Such conditions can be imposed as viaframe/coordinate transforms and deformations of linear connections which allows to solve derived geometricflow evolution equations in explicit form. For certain applications, we can consider ◦ b f as an undeterminedscalar function which can be related to certain conformal transforms or re-parameterizations.Using ◦ b F (40) and the thermodynamic generating function ◦ b Z = exp[ R b Ξ t ◦ b µ p | q q q | δ ` x ( − ◦ b f + 6)] ,we can define and compute such 3-d hypersurface thermodynamic values: ◦ b E = − τ Z b Ξ t ◦ b µ p | q q q | δ ` x (cid:18) s ◦ b R + | ◦ b D ◦ b f | − τ (cid:19) , (42) ◦ b S = − Z b Ξ t ◦ b µ p | q q q | δ ` x h τ (cid:16) s ◦ b R + | ◦ b D ◦ b f | (cid:17) + ˜ f − i , ◦ b η = 2 τ Z b Ξ t ◦ b µ p | q q q | δ ` x [ | ◦ b R ` ı ` j + ◦ b D ` ı ◦ b D ` j ˜ f − τ q ` ı ` j | ] . These formulas can be considered for 4–d configurations (39) taking the lapse function ˘ N = 1 for N-adaptedGaussian coordinates. We can also write such formulas in equivalent form using geometric data ( e q , ◦ e D ) and/or ( q , ◦ ∇ ) for respectively re-defined normalizing functions. For LC-configurations, the 3-d hypersurfaceformulas (40), (41) and (42) transform into the standard ones from G. Perelman’s preprint [1]. The maindifference is that in our approach such Riemannian hypersufrace flow evolution scenarios are determined byLagrange-Hamilton mechanical systems. In this section we show that Lagrange and/or Hamilton mechanical systems are characterized not onlyby dynamical equations (which is well-known from classical mechanics [16, 17, 18]) but also by certain17lasses of geometric flow evolution equations [19, 20]. Relativistic variants of such systems of nonlinearPDEs can be proven by applying a variational N-adapted calculus for respective F- and W-functionals as in[19, 23, 24, 25, 26, 27, 28, 29]. For holonomic Riemannian manifolds, such proofs can be found in [1, 13, 14, 15].
Applying a N–adapted variational procedure on a 3-d hypersurface to a functional (40) or (41) definedby data ( e g ` ı ` j , e ∇ ) , we obtain such equations in the form ∂ e g ` ı ` j ∂τ = − e R ` ı ` j , (43)where τ is an evolution real parameter. There are used local coordinates u ` ı with indices ` ı, ` j = 1 , , andRicci tensor e R ` ı ` j for a 3-d Riemannian manifold (in this work constructed as an emergent from geometricmechanics curve space). These equations are equivalent to the (non-relativistic) Ricci flow evolution equationspostulated heuristically by R. Hamilton [10, 11, 12]. G. Perelman proved such equations using his F- andW-functionals.The equations (43) describe a nonlinear diffusion process for geometric flow evolution of relativisticmechanical systems encoded up to frame transforms into 3-d Riemannian metrics (we can omit tilde andwrite g ` ı ` j in certain general covariant form). For models with small deformations of a 3–d Euclidean metric g ` ı ` j ≈ δ ` ı ` j + h ` ı ` j , with δ ` ı ` j = diag [1 , , and h ` ı ` j | ≪ , the Ricci tensor approximates the 3-d Laplace operator ∆ = ∂ ( ∂u ) + ∂ ( ∂u ) + ∂ ( ∂u ) . On 3-d hypersurfaces and "slow" evolution, the geometric flows of mechanicalsystems are described by a linear diffusion equation with R ` ı ` j ∼ ∆ h ` ı ` j . For relativistic models, we haveto elaborate on hydrodynamic anisotropic like transports of entropic fields and derived geometric objects[24, 25, 26].
Applying a N-adapted variational procedure with a corresponding re-definition of normalizing function for e F (26) determined by geometric data ( e g = { e g µν = [ e g ij , e g ab ] } , e N = { e N ai } , e D ) , we obtain a system of nonlinearPDEs generalizing the R. Hamilton equations for geometric flow evolution of relativistic Lagrange systems, ∂ τ e g ij = − e R ij ; ∂ τ e g ab = − e R ab ; (44) e R ia = e R ai = 0; e R ij = e R ji ; e R ab = e R ba ; ∂ τ e f = − e (cid:3) e f + (cid:12)(cid:12)(cid:12) e D e f (cid:12)(cid:12)(cid:12) − s e R ) . In these formulas, e (cid:3) ( τ ) = e D α ( τ ) e D α ( τ ) and the conditions e R ia = 0 and e R ai = 0 for the Ricci tensor Ric [ e D ] = { e R αβ = [ e R ij , e R ia , e R ai , e R ab ] } are imposed in order to keep a symmetric metric evolution.For the geometric flow evolution of relativisitic Hamilton mechanical systems, the analogs of (44) can bewritten (in principle, such equations can be proven in abstract form dualizing geometric objects from thetangent Lorentz bundles to respective cotangent bundles and functional p e F (28)) for the geometric data ( p e g = { p e g µν = [ p e g ij , p e g ab ] } , p e N = { p e N ai } , p e D ) ,∂ τ p e g ij = − p e R ij ; ∂ τ p e g ab = − p e R ab ; (45) p e R ia = p e R ai = 0; p e R ij = p e R ji ; p e R ab = p e R ba ; ∂ τ p e f = − p e (cid:3) e f + (cid:12)(cid:12)(cid:12) p e D p e f (cid:12)(cid:12)(cid:12) − p s e R ) , where p e (cid:3) ( τ ) = p e D α ( τ ) p e D α ( τ ) . Using nonholonomic deformations of d-connections (25), respective frame transforms and re-definitionof normalizing functions, the geometric flow evolution equations can be written in "hat" variables or for18C-configurations. Imposing corresponding classes of nonholonomic constraints, we may drive the flows ofgeometric objects in a "pure" mechanical form, or mix the frames and indices and generate new classes ofnonholonomic phase spacetimes.
For self-similar configurations in a fixed point τ = τ , the geometric flows (43) are described by nonholo-nomic Ricci soliton equations e R ` ı ` j − λ e g ` ı ` j = e ∇ ` ı v ` j + e ∇ ` j v ` ı , (46)for λ = ± , and a vector field v ` j . In these formulas, λ is taken for a corresponding normalization function,which defines a 3-d hypersurface version of the Einstein equations with cosmological constant. We keep tildeon symbols in order to emphasize that the geometric objects are determined by certain Lagrange or Hamiltongenerating function on a 8-d (co) tangent bundle.In a similar form, we can consider self-similar point τ = τ configurations for the systems of nonlinearPDEs (44) and/or (45), when ∂ τ e g µν = 0 and/or ∂ τ e g µν = 0 , with a corresponding choice of the normalizinggeometric flow functions (for simplicity, we can take a zero vector field v α = 0) , the equations (44) transforminto relativistic nonholonomic Ricci soliton equations e R ij = λ e g ` ı ` j , e R ab = λ e g ab , e R ia = e R ai = 0 , on T V ; (47) p e R ij = λ p e g ` ı ` j , p e R ab = λ p e g ab , p e R ia = p e R ai = 0 , on T ∗ V . Such equation can be written in hat and/or LC-variables using nonholonomic deformations of d-connections(25) and frame transforms. Projecting (47) on a base 4-d Lorentz manifold V , we obtain nonholonomicallydeformed vacuum Einstein equations with cosmological constant λ. In this work, we do not study gravitational and matter field geometric field interactions. Nevertheless,we note that in our nonholonomic geometric flow approach to investigating the evolution of Lagrange-Hamilton systems, the gravitational field equations emerge from geometric flows of mechanical systemsbeing characterized by a W-entropy (38) and respective thermodynamical values (39). The gravitationalconstant can be introduced for identifications with respective spherical symmetric solutions with an additionalassumption that at long distances the standard Newton gravitational potential is generated. In certain sense,for such theories, a W-entropy acts as an entropic force for the E. Verlinde model [51, 52], see proofs in[27, 28, 29].
This section is a short introduction to basic aspects of classical and quantum geometric informationflow (respectively, GIF and QGIF) models and related subjects from the theory of geometric evolutionof relativistic mechanical systems (elaborated in previous sections). Using modified G. Perelman entropyfunctionals and the nonholonomically adapted von Neumann entropy for quantum density matrices, thereare defined quantum conditional entropy, relative entropy, and mutual information values as basic ingredientsof the QGIF theory.
Classical information theory is based on fundamental concepts of Shannon, conditional and relativeentropies [37, 38, 47, 48, 49]. To elaborate on classical aspects of geometric information flow, GIF, modelswe have to define analogous values determined by (modified) Perelman entropy functionals and associatedthermodynamical models. 19 .1.1 Shannon entropy and geometric flow entropy in information theories
Let us remember the general definition of the Shannon entropy S B of a probability distribution for arandom variable B taking certain values b , b , ..., b k (for instance, to send a long message N ≫ with k letters) with respective probabilities to observe such values p , p , ..., p k . By definition, S B := − k X j =1 p j log p j ≥ with k X j =1 p j = 1 . This is for the probability theory with random variables. In classical information models,
N S B is the numberof bits of information which can be extracted from a message with N symbols which are randomly generated.For engineering applications, N S B is the number of bits to which a message with N letters can be compressed.Typically, such messages are not randomly generated but contain certain information. To encode certainreal messages with correlations between letters (for instance, words for grammar and syntax) and loose lessmodifications is a more complex random process. In the ideal gaze limit (ignoring correlations), we canconsider that the entropy of a long message is just N S , when S is the entropy of a message consisting of onlyone letter. We can formalize the constructions as models of statistical mechanics if we introduce a classicalHamiltonian H determining the probability of a i -th symbol b i in a statistical thermodynamical model viaformula p i = 2 − H ( b i ) . The theory of geometric flows is different from the standard theory of random processes, classical infor-mation models and "simple" engineering applications. The flow evolution is characterized by the W-entropyand (which is important for our further developments) additional assumptions on associated statistical ther-modynamic values like mean energy, entropy and fluctuation. For classical mechanical systems, such valuesare canonically determined by generating functions e L and e H, see formulas f W (27) and p f W (29), and, respec-tively, for flow evolution of Hessian metrics, by h e E , e S , e η i (36) and h p e E , p e S , p e η i (37). On a discrete networkwith random variables, we can introduce probabilities, for instance, e p n = 2 − e H ( b n ) and p e p n = 2 − p e H ( b n ) , or,for statistical ansambles, e p n = 2 − e E ( b n ) and p e p n = 2 − p e E ( b n ) . In result, it is possible to elaborate classicalinformation theories determined by effective Hamiltonians e H, or energy functionals e E and p e E . This is forcertain discrete versions with probability models and correlations encoding information on geometric flowsof mechanical systems.In this subsection, we elaborate on continuous information flow models encoding geometric evolution ofmechanical systems using the thermodynamic entropies e S [ e g ( τ )] and p e S [ p e g ( τ )] without involving in the con-structions probability distributions which appear for random variables. Geometric flows can be described by e S [ e g ( τ )] and p e S [ p e g ( τ )] . We can elaborate equivalent constructions for W-entropies f W [ e g ( τ )] and p f W [ p e g ( τ )]) .Systems under geometric flows are denoted as e B = e B [ e g ( τ )] and p e B = p e B [ p e g ( τ )] determined by correspondingcanonical d-metrics on phase spacetimes. In information theory, there are studied various conventional models with communicating humans called,for instance, Alice and Bob, see [37, 38]. Let us suppose that Alice sends a message via a noisy telephoneconnection with many letters (any letter is a random variable X taking possible values x , ..., x k ). Bobreceives instead of X a random variable Y consisting from possible letters y , ..., y r . In classical informationtheory, one computes how many bits of information does Bob receives form Alice’s message with N letters? In this section, we should not confuse symbols for probabilities p i with similar notations for cofiber coordinates; and anumber N is different from the symbol N used for the N-connections. Here we note that it is almost impossible and notoptimal to elaborate an unified system of notations with completely different symbols in an article involving different directionsin differential geometry, geometric mechanics, probability and diffusion, classical and quantum information theory. We try tokeep traditional notations for different directions in mathematics or physics but (if necessary) underly symbols and providerespective remarks allowing to avoid notation ambiguities. X, Y, Z etc. For one variable, the probability to observe X = x i is denoted P X ( x i ) subjected to the condition that P i P X ( x i ) = 1 . The communication between Aliceand Bob is a random process of two variables defined by a joint distribution P X,Y ( x i , y j ) as the probabilitythat Alice sends X = x i and Bob hears Y = y j . It is considered that the value P Y ( y j ) = P i P X,Y ( x i , y j ) is the probability that Bob hears Y = y j (summation is over all choices of what Alice could send). The conditional probability P X | Y ( x i | y j ) := P X,Y ( x i , y j ) P Y ( y j ) is by definition a value characterizing that if Bob hear Y = y j , he can estimate the probability that Alicesent x i . We can write for Alice’s messages P X ( x i ) = P j P X,Y ( x i , y j ) , or consider P X ( x i ) as an independentprobability density. Using these formulas, one defines such important values: S X | Y = y j := − X i P X | Y ( x i | y j ) log P X | Y ( x i | y j ) , the Shannon entropy of the conditional probability ; S XY := − X i,j P X,Y ( x i , y j ) log P X,Y ( x i , y j ) , the entropy of joint distribution ; S Y := − X i,j P X,Y ( x i , y j ) log P Y ( y j ) , the total information content received by Bob ; S X := − X i,j P X,Y ( x i , y j ) log P X ( x i ) , the total information content in Alice’s message ; (48) S X | Y := S ( X | Y ) = X j P Y ( y j ) S X | Y = y j , the conditional entropy . Using such formulas, one prove that (this can be violated by quantum systems) S ( X | Y ) = S XY − S Y ≥ (49)and the mutual information between X and Y (a measure of how much we learn about X observing Y ) I ( X ; Y ) := S X − S XY + S Y ≥ . (50)Now, let us analyse another type of communications between Alice and Bob. We suppose that they areresearch scientists and know advanced differential geometry, classical mechanics, information theory, andtheory of geometric flows. Alice sends to Bob not only simple messages consisting from letters and densityprobabilities but messages encoding that (in her world ) she study geometric flow evolution processes of amechanical system of type e A = e A [ e g ( τ )] , or p e A = p e A [ p e g ( τ )] , determined by flows of Hessian metrics. Bobwill receive Alice’s message (it may be a short letter) and knows that Alice plays a game with geometricflow modeling. We denote Bob’s geometric evolution systems as e B = e B [ e g ( τ )] , or p e B = p e B [ p e g ( τ )] . Inelaborating such GIF models, Alice and Bob could work or not with probability densities. In principle, thethermodynamic generating functions e Z [ e g ( τ )] and/or p e Z [ p e g ( τ )] from (35) can be considered as geometricflow analogs of probability densities but they may use directly the W-entropy f W (27), or p f W (29), and,respectively, for ansambles of Hessian metrics, by h e E , e S , e η i (36), or h p e E , p e S , p e η i (37). For simplicity, weanalyze here how they may GIF-communicate using instead of messages with random letters certain geometricflow transfers of information encoding concepts of mechanical dual phase spacetimes for Lorentz cotangentbundles. In such a case, we have to use the geometric flow thermodynamic entropy p e S [ p e g ( τ )] associated toW-entropy p f W [ p e g ( τ )] and formulas considered in subsection 3.1.2. We shall use also geometric flow modelson T ∗ V ⊗ T ∗ V with one cotangent bundle for Alice and another one for Bob. The local coordinates on suchproducts of cotangent bundles are labeled ( p u, p u ) and the normalizing functions are of type p AB e f ( p u, p u ) . The canonical d-metric structure on such tensor products of phase spacetimes is of type p AB e g = { p e g = [ q , q , q , ˘ N , p q , p q , p q , p ˇ N ] , p e g = [ q , q , q , ˘ N , p q , p q , p q , p ˇ N ] } . p AB e D = p e D + p B e D and respective scalar curvature p sAB e R = p s e R + p s e R. We work with p e S [ e A ] and p e S [ e B ] defined by respective formulas for p e g ( τ ) and p e g ( τ ) as in (37). Theyare analogs of S X and S Y in above formulas. As an analog of S XY for GIF, we consider the thermodynamicgenerating function (as a generalization of (35)) p AB e Z [ p e g ( τ ) , p e g ( τ )] = p fZ p eZ e − p AB e f q | p e g αβ | q | p e g αβ | d p u d p u ( − p AB e f + 32) , for T ∗ V ⊗ T ∗ V , and resulting entropy function p AB e S = p e S [ e A, e B ] = − p eZ p eZ e − p AB e f q | q q q ˘ N p q p q p q p ˇ N | q | q q q ˘ N p q p q p q p ˇ N | δ p u d p u h τ (cid:16) p s e R + p s e R + | p e D p AB e f + p e D p AB e f | (cid:17) + p AB ˜ f − i . Using such formulas, we claim that for GIFs the formulas for the conditional entropy (37) and mutualinformation (37) are respectively generalized p e S [ e A | e B ] := p AB e S − p e S [ e B ] ≥ and (51) p e J [ e A ; e B ] := p e S [ e A ] − p AB e S + p e S [ e B ] ≥ . (52)Similar claims can be formulated if we use the W-entropy p f W (29): p f W [ e A | e B ] := p AB f W − p f W [ e B ] ≥ and p e J [ e A ; e B ] := p f W [ e A ] − p AB f W + p f W [ e B ] ≥ , with respective formulas computed for the W–entropy instead of the S-entropy in the standard probabilitytheory. For relativistic information flows, such formulas can be applied without additional assumptions onformulating associated statistical thermodynamic models. Finally, we note that above formulas can be defined and proven respectively, and in similar forms, on T V ,T V ⊗ TV , and other tensor products and lower dimension projections involving Lagrange generatingfunctions. For instance, e S [ e A | e B ] := AB e S − e S [ e B ] ≥ and e J [ e A ; e B ] := e S [ e A ] − AB e S + e S [ e B ] ≥ f W [ e A | e B ] := AB f W − f W [ e B ] ≥ and e J f W [ e A ; e B ] := f W [ e A ] − AB f W + f W [ e B ] ≥ . Such values can satisfy certain Legendre conditions and duality conditions to respective formulas (51) and(51) and W-analogs. The models for cotangent bundles are important for elaborating quantum mechanicaltheories of GIFs with Hamilton generating functions. In their turn, the GIF models on tangent bundles areimportant for encoding quantum field theories formulated using the Lagrange formalism. Let us explain why we use the word "claim" for these formulas. In principle, the conditions of non–negativity of respectivevalues can be violated if Alice sends to Bob GIFs as solutions, for instance, of generalized R. Hamilton geometric flow equations(45). For such variants, we use the claims (51) and (52) as criteria for selecting physically realistic and viable solutions forthe information theory of geometric flow evolution of W. Hamilton mechanical systems. Nevertheless, working on cotangentLorentz bundles, such claims can be transformed into theorems and proven if we consider a causal axiomatic approach toFinsler-Lagrange-Hamilton theories elaborated in [34, 35]. Here we sketch the idea and key steps for proving such formulas.For physicists, such formulas seem to be natural ones; rigorous mathematical proofs require hundreds of pages and applicationof a corresponding interference of methods outlined in [1, 13, 14, 15] together with [37, 38, 47, 48, 49] and, for nonholonomicconfigurations, in our works [19, 23, 24, 25, 26, 27, 28, 29]. The W-entropy and respective thermodynamic values can be definedon a 3-d hypersurface as in (40), (41) and (42), and then extended for evolution on a time like curve to formulas (38) and (39).Then the formulas are dualized to momentum type local coordinates on some open regions on T ∗ V ⊗ T ∗ V . Such causal curvescan be defined to cover a subspace on respective phase spacetimes, their tensor products, and projections on lower dimensions.Here we note that in any point of a causal curve in T ∗ V and related tensor products/ projection spaces and subspaces we candefine entopies of type (48). This way, the geometric flow information values can be completed with certain random variables.Alice’s letters to Bob will encode not only GIFs but also random bit information processes. We can associate entropies of type p f W and/or p e S to probabilistic entropies. .1.3 Relative GIF entropy and monotonicity In the standard probability theory, the concept of relative entropy is introduced if (for a random variable X ) there are considered two probability distributions P X and Q X , where for X = x i , labeled by i = { , , ...s } , one obtains p i = P X ( x i ) and q i = Q X ( x i ) , let say, for some long messages with N letters. The key issue is todecide which distribution describe a random process more realistically. The relative entropy per observation(or Kullback–Liebler divergence) is defined S ( P X || Q X ) := P i p i (log p i − log q i ) ≥ under assumption that N S ( P X || Q X ) ≫ . This is an asymmetric value on P X and Q X and measure the difference between thesetwo probability distributions when we consider that P X is a correct answer and Q X is an initial hypothesis.Let us study a pair of random variables X and Y for which we consider two probability distributions.The fist one is a possible correlated joint distribution P X,Y ( x i , y j ) and P X ( x i ) := X j P X,Y ( x i , y j ) , P Y ( y j ) := X i P X,Y ( x i , y j ) . (53)A second probability distribution Q X,Y ( x i , y j ) = P X ( x i ) P Y ( y j ) can be defined to ignore correlations between X and Y. In a general context, Q X,Y ( x i , y j ) can be with correlations when Q X ( x i ) := P j Q X,Y ( x i , y j ) . Formore general constructions, we can introduce three random variables
X, Y, Z described by a joint probabilitydistribution and related values: P X,Y,Z ( x i , y j , z k ) and P X ( x i ) := X j,k P X,Y,Z ( x i , y j , z k ) , P Y,Z ( y j , z k ) := X i P X,Y,Z ( x i , y j , z k ) . If we forget the correlations between X and Y Z, we define Q X,Y,Z ( x i , y j , z k ) := P X ( x i ) P Y,Z ( y j , z k ) . Othertype values can be defined if we observe the subsystem
XY, when P X,Y ( x i , y j ) := X k P X,Y,Z ( x i , y j , z k ) , Q X,Y ( x i , y j ) := X k Q X,Y,Z ( x i , y j , z k ) = P X ( x i ) P Y ( y j ) . Now, we can calculate the relative entropy S and mutual information I between two distributions S ( P X || Q X ) := X i,j P X,Y ( x i , y j )[log P X,Y ( x i , y j ) − log( P X ( x i ) P Y ( y j ))] = S X − S XY + S Y = I ( X ; Y ); S ( P X,Y || Q X,Y ) := S X − S XY + S Y = I ( X ; Y ); S ( P X,Y,Z || Q X,Y,Z ) := S XY − S XY Z − S Y Z = I ( X ; Y Z ) . In result, one proves by explicit calculations such properties I ( X ; Y ) := S X + S Y − S XY ≥ , subadditivity of entropy ; S ( P X,Y || Q X,Y ) ≥ S ( P X || Q X ) , S ( P X,Y,Z || Q X,Y,Z ) ≥ S ( P X,Y || Q X,Y ) , monotonicity of relative entropy . There is also the condition of strong subadditivity S X − S XY Z − S Y Z ≥ S X − S XY + S Y , or S XY + S Y Z ≥ S Y + S XY Z , which is equivalent for the condition of monotonity of mutual information I ( X ; Y Z ) ≥ I ( X ; Y ) . Above formulas for S and I can be generalized for respective relative entropy and mutual information ofgeometric flows of mechanical systems (for simplicity, we consider formulas generated by certain relativisticHamilton generating functions H ( x, p ) ). For such evolution systems, there are considered p A e Z := p e Z [ p e g ( τ )] and p B e Z := p e Z [ p e g ( τ )] , see (35), as analogs of p i = P X ( x i ) and q i = Q X ( x i ) , see also formulas in the previoussubsection. In general, there are considered three evolution flow canonical mechanical systems e A, e B, e C. In23esult, we claim (and can prove following the method sketched in footnote 10) by explicit integral N-adaptedcalculations on T ∗ V ⊗ T ∗ V ⊗ T ∗ V such properties p e J [ e A ; e B ] := p e S [ e A ] − p AB e S + p e S [ e B ] ≥ , subadditivity of entropy ; p e S [ p AB e Z|| p AB e Z ] ≥ p e S [ p A e Z|| p A e Z ] , p e S [ p ABC e Z|| p ABC e Z ] ≥ p e S [ p AB e Z|| p AB e Z ] , monotonicity of relative entropy . The conditions of strong subadditivity for GIF entropies are claimed p A e S − p ABC e S − p BC e S ≥ p A e S − p AB e S + p B e S , or p AB e S + p BC e S ≥ p B e S + p ABC e S . In equivalent form, these formulas can be written as the condition of monotonicity of GIFs mutual informa-tion, p e J [ e A ; e B e C ] ≥ p e J [ e A ; e B ] . Above formulas involve, for instance, the thermodynamic generating function (as a generalization of (35)) p ABC e Z [ p e g ( τ ) , p e g ( τ ) , p e g ( τ )] = p fZ p eZ p eZ e − p ABC e f q | p e g αβ | q | p e g αβ | q | p e g αβ | d p u d p u d p u ( − p ABC e f + 48) , for T ∗ V ⊗ T ∗ V ⊗ T ∗ V , with a normalizing function p ABC e f ( p u, p u, p u ) , when the local coordinates on such such products ofcotangent bundles are labeled ( p u, p u, p u ) . The canonical d-metric structure on such tensor products ofphase spacetimes is of type p ABC e g = { p e g = [ q , q , q , ˘ N , p q , p q , p q , p ˇ N ] , p e g = [ q , q , q , ˘ N , p q , p q , p q , p ˇ N ] , p e g = [ q , q , q , ˘ N , p q , p q , p q , p ˇ N ] } . We can consider a canonical d–connection p ABC e D = p e D + p B e D + p C e D and respective scalar curvature p sABC e R = p s e R + p s e R + p s e R. The resulting entropy function p ABC e S = p e S [ e A, e B, e C ] = − p eZ p eZ p eZ e − p ABC e f q | q q q ˘ N p q p q p q p ˇ N | q | q q q ˘ N p q p q p q p ˇ N | q | q q q ˘ N p q p q p q p ˇ N | δ p u d p u d p u h τ (cid:16) p s e R + p s e R + + p s e R + | p e D p ABC e f + p e D p ABC e f + p e D p ABC e f | (cid:17) + p ABC ˜ f − i . Similar formulas can be derived for W-entropies and for Lagrange GIFs on T V ⊗ TV ⊗ TV . We conclude this introduction to the GIF theory of canonical classical mechanical systems with tworemarks: First, such constructions can be generalized for stochastic maps and nonholonomic flow evolutionand kinetic processes of Lagrange-Hamilton systems as we studied in [53, 54, 55]. Here, we shall analysea QGIF analog when the quantum relative entropy is monotonic in any quantum channel, including thoseassociated to evolution of Hamiltonian quantum mechanical systems.Second, we shown that we are able both in the probability theory and for geometric flow models todefine conditional on some observation entropies. There is not a good analog of the probability conditionaldistribution in the quantum mechanical case. Nevertheless, there is a miracle that many conclusions havequantum analogs [38]. For GIFs of mechanical Hamilton systems with a H ( τ, x, p ) , this is not a miraclebecause the flow evolution of Hessian Hamilton metrics p e g ab ( τ, x, p ) := ∂ H/∂p a ∂p b (4) and respectivecanonical d-metrics p e g ( τ ) (14) are characterized by well–defined concepts of W-entropy p f W (27) and respec-tive thermodynamical variables h p e E , p e S , p e η i (37). In result, we can introduce GIF formulas for conditionalentropy and mutual entropy and their W-analogs. For quantum developments in next subsection, we shallspeculate on strong subadditivity of quantum entropy which holds also for quantum analogs of mechanicalHamilton systems. 24 .2 Basic ingredients of the quantum geometric information flow theory The goal of this subsection is to analyze how the main concepts and formulas for GIFs of mechanicalsystems can be extended to quantum theory and formulate an approach to the theory of QGIFs. We note thata noncommutative version of geometric flow theory was elaborated in [23]. Those results can be extendedfor elaborating noncommutative models of quantum information theory. In a more simplified approach, wecan consider quantum mechanical models, and respective quantum geometric flows, by quantizing certainrelativistic mechanical Hamiltonians H ( τ, x, p ) , when in the quasi-classical limits the geometric mechanicstheory with Hessian metrics p e g ab ( τ, x, p ) emerges. In this work, the main goal is to elaborate on quantuminformation theory for geometric flows of mechanical systems characterized by geometric thermodynamicaldata h p f W ; p e E , p e S , p e η i , see (27) and (37). The thermodynamic gener-ating function p e Z [ p e g ( τ )] (35) with canonical geometric objects determined by a Hamilton function e H, seealso subsection 3.1.2, can be used for defining the state density p e σ ( β, e H , p e g ) = p e Z − e − β e H , (54)with β = 1 /T, τ = T, as a classical analog of the density matrix in quantum mechanics. The relative entropybetween any state density p e ρ ( β, e H, p e g ) and p e σ ( β, e H , p e g ) is computed for a prescribed measure ω ( e H ) , forinstance, on a cotangent Lorentz bundle with E considered as a thermodynamical energy parameter.Using formulas (33) and (32), we define for the conditional entropy for geometric flows of Hamiltonmechanical systems p e S ( p e ρ q p e σ ) = β [ p e F ( p e ρ ) − p e F ( p e σ )] , (55)where the free energy corresponding to p e ρ is p e F ( p e ρ ) := p e E ( p e ρ ) − T p e S ( p e ρ ) . In these formulas, the averageenergy is computed p e E ( p e ρ ) = R p e ρ e Hdω ( e H ) (i.e. using the density matrix p e ρ ) and the thermodynamic entropyis p e S ( p e ρ ) := β p e E ( p e ρ ) + log p e Z ( p e ρ ) . Both values p e E ( p e ρ ) and p e S ( p e ρ ) can be written equivalently to (37). Wenote that if log p e Z is independent on p e ρ (as we consider in above formulas) we have p e S ( p e σ q p e σ ) = 0 . In this subsection, we elaborate on how GIFs of classical mechanical systems can be generalized to QGIFsusing basic concepts of quantum mechanics, QM, and information theory. QM involves probabilities not asclassical probability distributions for a quantum state but, in general, as densities matrices. Certain specialQM systems can be described by pure states. Nevertheless, to study quantum models of GIFs systems isnecessary to consider density matrices as quantum analogs of state densities of type p e σ (54). Density matrix for quantum information theory and associated Hamilton mechanical systems:
In an idealized case, a Hamiltonian GIF system e A = h p e E , p e S , p e η i (37) can be described by a Hilbert space e H A . A state vector e ψ A ∈ e H A can be defined as infinite dimensional complex vector solving the Schrödingerequation with a Hamiltonian b H taken as a well-defined quantum version of a canonical Hamiltonian e H. Inthe quasi-classical limit, from a quantum mechanical model with b H, we obtain a relativistic e H and respectiveHessian p e g ab ( x, p ) (4) and canonical d-metric p e g (15) (from which "non-tilde" d-metrics p g (10) emerge forgeneral frame and coordinate transforms on a T V ) . We can consider unitary transforms of type e ψ A → U ψ A and describe the system e A in an abstract Hilbert space H e A (we put tilde on certain symbols if it is necessary toemphasize that the constructions are related to quantization of a canonical mechanical Hamiltonian system).For applications in the information theory, a Hilbert space is approximated to a complex vector space ofdimension N with Hermitian product, see details in [37, 38].We can consider a complementary system B (we write e B if it is a quantum mechanical analog of a classicalHamilton mechanics) with an associate Hilbert space H B , or H e B , with state vectors of type ψ B ∈ H B and/or25nitary transforms of type e ψ B → ψ B V ∈ H e B . The combined Hilbert space is defined as a tensor product, H A ⊗ H B and/or H e A ⊗ H e B . The state vectors for the combined system are of type ψ AB = ψ A ⊗ ψ B ∈ H AB = H A ⊗ H B , where, for instance, ψ B = 1 B is considered as the unity state vector. For such products, the predictionsabout a system e A can be made using the state vector e ψ A and forgetting about the system B . In general,a generic pure state ψ AB ∈ H AB is not a tensor product vector but is "entangled". This means that if therespective dimension dim H A = N and dim H B = M then a generic state ψ AB is described by an N × M matrix. In quantum information theory, it is considered that any pure state can be written as a Schmidtdecomposition ψ AB = X i √ p i ψ iA ⊗ ψ iB or e ψ AB = X i √ p i e ψ iA ⊗ e ψ iB . (56)In such formulas, the state vectors are orthonormal: for instance, < ψ iA , ψ jA > = < ψ iB , ψ jB > = δ ij , where δ ij is the Kronecker symbol. If p i > and P i p i = 1 (this is equivalent to the condition that, for instance, ψ AB is a unit vector), we can treat p i as probabilities. Here we note that ψ iA , or ψ iB , may not be bases of H A , or H B (in principle, they may be not enough for such bases).The quantum density matrix for a system A, or e A, is defined ρ A := X i p i | ψ iA >< ⊗ ψ iA | or ρ e A := X i p i | ψ i e A >< ⊗ ψ i e A | . This operator is Hermitian and positive semi-definite, with trace
T r H A ρ A = T r H e A ρ e A = 1 . Using ρ A , or ρ e A , we can compute the expectation value of any operator O A , or O e A , following, for instance, the rules < O > AB = < ψ AB |O A ⊗ B | ψ AB > = X i p i < ψ iA |O A | ψ iA >< ψ iB | B | ψ iB > = < O > A = X i p i < ψ iA |O A | ψ iA > = T r H A ρ A O A . (57)In above formulas, we considered a bipartite system AB, or e A e B. Such systems are described in generalform by quantum denstity matrices of type ρ AB , or ρ e A e B . Here we note that in the classical probabilitytheory a bipartite system XY is described by a joint probability distribution P X,Y ( x i , y j ) , where P X ( x i ) := P j P X,Y ( x i , y j ) , see (53). For AB as a bipartite quantum system with Hilbert space H A ⊗ H B , the densitymatrix ρ AB is defined in standard quantum mechanical form: Let us consider | i > A , i = 1 , , ..., n as anorthonormal basis of H A and | b > B , b = 1 , , ..., m as an orthonormal basis of H B . We write ρ AB = X i,i ′ ,b,b ′ ρ ii ′ bb ′ | i > A ⊗| b > B A < i ′ | ⊗ B < b ′ | . For measurements of the system A, it is considered the reduced density matrix obtained by respectivecontracting of indices, ρ A = T r H B ρ AB = X i,i ′ ,b,b ρ ii ′ bb | i > A A < i ′ | , for | b > B B < b | = 1 . In a similar form, it is defined ρ B = T r H A ρ AB . Using such formulas, we can elaborate on quantum informationtheory (see reviews [37, 38]) and develop the approach for QGIFs.26 uantum density matrix for GIFs of mechanical Hamilton systems:
Using formulas (57), we cancompute expectation values of a state density p e σ (54) and define a respective quantum density p e σ AB = < p e σ > AB = < ψ AB | p e σ ⊗ B | ψ AB > = X i p i < ψ iA | p e σ | ψ iA >< ψ iB | B | ψ iB > = p e σ A = < p e σ > A = X i p i < ψ iA | p e σ | ψ iA > = T r H A ρ A p e σ. (58)Here the density matrix ρ A is taken for computing the QGIF density matrix p e σ A determined by a statedensity of the thermodynamical model for GIFs of a classical mechanical Hamiltonian system p e σ. For suchsystems, we can work directly with quantum density matrices p e σ AB and p e σ A and respective partial traces p e σ A = T r H B p e σ AB and p e σ B = T r H A p e σ AB . (59)In coefficient form, we obtain such formulas p e σ AB = X i,i ′ ,b,b ′ p e σ ii ′ bb ′ | i > A ⊗| b > B A < i ′ | ⊗ B < b ′ | and p e σ A = X i,i ′ ,b,b p e σ ii ′ bb | i > A A < i ′ | . Let us discuss a concrete example with density matrices. Consider an isolated classical mechanical Hami-tonian systems for which a QM model can be constructed. To describe thermodynamically the geometricflow evolution of both classical and quantum models we need respective state density and quantum densitymatrix. In a pure state formalism, the mathematical machinery gets bigger and bigger involving differentialgeometric concepts, quantum mechanics and probability theories. This can be organized as quantum infor-mation flow evolution model. Using a density matrix encoding the data for Hamilton mechanical system, wecan compute respective thermodynamical values.
Using p e σ A , wecan describe QGIF in a standard QM form when the respective von Neumann entropy is used instead of theShannon entropy for a probability distribution, p q e S ( p e σ A ) := T r p e σ A log p e σ A , (60)where the trace is written in a simplified form without a label for the corresponding Hilbert space. We usea left label q as "quantum" and emphasize that such an entropy is a quantum analog of p e S used in thethermodynamic model for geometric flow evolution of Hamilton mechanical systems. The QGIF entropy p q e S ( p e σ A ) ≥ and is manifestly invariant under a unitary transformation p e σ A → U p e σ A U − . The quantum value p q e S ( p e σ A ) has a purifying property which is typical for quantum information theoryand does not have a classical analog. For a bipartite system e ψ AB = P i √ p i e ψ iA ⊗ e ψ iB (56) and ρ A := P i p i | ψ iA > ⊗ < ψ iA | , we write p e σ A := X i,i ′ ,b,b p X k e σ ii ′ bb p k A < i ′ || ψ kA >< ⊗ ψ kA || i > A , (61) p e σ B := X j,j ′ ,b,b p X k e σ jj ′ bb p k B < j ′ || ψ kB >< ⊗ ψ kB || j > B . In both these formulas, we have the sam probabilities p k even the matrices and bases are different. So, it isclear that p q e S ( p e σ A ) = p q e S ( p e σ B ) , which proves that a system A and a purifying system B always have thesame QGIF von Neumann entropy. This holds true if e A is taken for GIFs of a mechanical Hamilton system.27ecause p q e S ( p e σ ) is a typical von Neumann entropy, it has another very important concavity property.Let explain this for QGIFs because there are involved certain important features induced by geometric flowevolution. This mean that for any two density mechanical matrices p e σ and p e σ we can introduce p e σ ( λ ) = λ p e σ + (1 − λ ) p e σ , for ≤ λ ≤ , and prove that d p q e S ( p e σ ) /dλ ≤ . In result, one obtains p q e S ( p e σ D ) ≥ p q e S ( p e σ ) , here D is from diagonal, which means that dropping the off-diagonal part of density matrix (thisholds in any basis) results in entropy increasing. This has important implications, for instance, in gravitymodels emerging from (quantum) mechanical evolution theories. Pure diagonal configurations have higherentropy than the generic off-diagonal ones. Quantum generalizations of W- and thermodynamic entropy of mechanical systems:
QGIFscan characterized not only by a von Neumann entropy of type (60) but also by quantum analogs of entropyvalues used for classical geometric flows (associated thermodynamics entropy and W-entropy). Such valuescan be introduced and computed in explcity form using respective formulas (58), (59), (61) for classicalconditional and mutual entropy used in formulas (51) and (52). The quantum formulas introduced in thissubsection can be considered for geometric flows of arbitrary systems and not only for mechanical ones. So,we write
A, B, ... instead of e A, e B, ... and define p q e S AB = T r H AB [( p e σ AB )( p AB e S )] and p q e S A = T r H A [( p e σ A )( p A e S )] , p q e S B = T r H B [( p e σ B )( p B e S )] . Similar formulas can be provided for the quantum version of W-entropy, p q f W AB = T r H AB [( p e σ AB )( p AB f W )] and p q f W A = T r H A [( p e σ A )( p A f W )] , p q f W B = T r H B [( p e σ B )( p B f W )] . Such values describe QGIFs of Hamiltonian (quantum) mechanical systems.The quantum probabilistic characteristics are described by the von Neumann entropy p q e S ( p e σ A ) (60) andcorresponding generalizations for AB and B systems p q e S ( p e σ AB ) := T r p e σ AB log p e σ AB and p q e S ( p e σ A ) := T r p e σ A log p e σ A , p q e S ( p e σ B ) := T r p e σ B log p e σ B . Finally, we note that the entropies p q e S A , p q f W A , and p q e S ( p e σ A ) characterize respectively different thermody-namic, geometric flow and probabilistic properties of QGIFs of geometric mechanical Hamilton flows. Ina similar form, we can omit the label " p " and derive respective formulas for quantum flows of Lagrangesystems. Such a formalism is more sophisticate mathematically because the Lagrange generating functionscan not be used directly for constructing base vectors for respective Hilbert spaces. Conditional and relative quantum entropy for QGIFs of mechanical systems:
For QGIFs, we canimitate formally many classical definitions for GIFs. As it is stated in section 3.4 of [38], the quantum versionsare potentially misleading or not good or usual notions. This is not surprising in the case of geometric flowsbecause they are characterized not only by certain probabilistic quantum entropies but also by G. PerelmanW-entropy and geometric thermodynamic entropy. Let us outline the main equations for respective vonNeumann and conditional and relative entropy of quantum mechanical geometric flows.Using quantum matrix computations with formulas of type (58), (59), (61), we prove such quantumproperties of entropies for QGIFs: p q e S [ A | B ] = p q e S AB − p q e S B and p q e J [ A ; B ] = p q e S A + p q e S AB + p q e S B ≥ . (62)Similar claims can be formulated (from small quantum perturbations, we can prove respective theorems) forthe Neumann (60) and quantum W-entropy (29), p q e S ( p e σ A | B ) := p q e S ( p e σ AB ) − p q e S ( p e σ B ) and p q e J ( p e σ A ; B ) := p q e S ( p e σ A ) − p q e S ( p e σ AB ) + p q e S ( p e σ B ); p q f W [ A | B ] = p q f W AB − p q f W B and p q e J f W [ A ; B ] = p q f W A + p q f W AB + p q f W B ≥ .
28t should be noted that different entropies and related mutual information values characterize differentproperties of the QGIFs of mechanical Hamilton systems. The von Neumann type values p q e S ( p e σ A | B ) and p q e J ( p e σ A ; B ) can be used for proofs of entanglement and purifcation properties of such systems following stan-dard methods of quantum information theory. Unlike the classical case, the quantum conditional entropy isnot conditional on certain classical or quantum processes. But for QGIFs, the systems are with nonholonomicstructure encoding classical and/or quantum mechanical systems. The conditional properties of such systemsare encoded in p q e S A and p q e J [ A ; B ] , for thermodynamical models of QGIFs, and p q f W A and p q e J f W [ A ; B ] , forquantum geometric evolution flows. Monotonicity and monogamy of entanglement of relative entropy for QGIFs:
The relative en-tropies for QGIFs are positive just as for the classical GIFs. Using p q e S ( p e σ A | B ) , we can prove that such aquantum entropy is also monotonic (for proofs, we can use the same methods as in [56, 38], and posses alsoa strong subadditivity property as in [57]). The intuition behind the classical theory of probability is notapplicable in a direct way for geometric flows and/or quantum systems. In this sense, the monotonicity ofquantum relative entropies is a miracle.Let us consider a very basic property of QGIFs described by the von Neumann entropy p q e S ( p e σ A ) . For abipartite system AB with two density matrices p e ρ AB and p e σ AB , we can define the corresponding reduceddensity matrices on A, p e ρ A = T r B ( p e ρ AB ) and p e σ A = T r B ( p e σ AB ) . The partial trace can only reduce therelative quantum entropy, p q e S ( p e ρ AB q p e σ AB ) ≥ p q e S ( p e ρ A q p e σ A ) . (63)see also (34) and (55).For a tripartite system ABC with QGIF density matrix p e ρ ABC and above montonicity property, we canproved a strong subadditivity property for geometric flows of quantum mechanical Hamilton systems. Thereare used reduced density matrices and corresponding second density matrices p e ρ A = T BC p e ρ ABC , p e ρ BC = T A p e ρ ABC , p e ρ AB = T C p e ρ ABC and p e σ ABC = p e ρ A ⊗ p e ρ BC , p e σ AB = T C p e σ ABC = p e ρ A ⊗ p e ρ B . (64)Using above monotonicity property, we can write p q e S ( p e ρ ABC q p e σ ABC ) ≥ p q e S ( p e ρ AB q p e σ AB ) and/or as the monotonicity of mutual information p q ˘ J ( A ; BC ) ≥ p q ˘ J ( A ; B ) , (65)which is equivalent to the condition of strong subadditivity p q e S AB + p q e S BC ≥ p q e S B + p q e S ABC . (66)These formulas follow from (64); notations of type p q e S ( p e σ A ) = p q ˘ S A , p q e S ( p e σ AB ) = p q ˘ S AB , p q e S ( p e σ ABC ) = p q ˘ S ABC ; and definitions p q e S ( p e ρ ABC q p e σ ABC ) = p q e S ( p e ρ ABC q p e ρ A ⊗ p e ρ BC ) = p q ˘ J ( A ; BC ) := p q ˘ S A + p q ˘ S BC − p q ˘ S ABC ; p q e S ( p e ρ AB q p e σ AB ) = p q e S ( p e ρ AB q p e ρ A ⊗ p e ρ B ) = p q ˘ J ( A ; B ) := p q ˘ S A + p q ˘ S B − p q ˘ S AB . The von Neumann entropy for QGIFs allows us to deduce an important property related to the monogamyof entanglement when a given qubit in a QGIF system e C can be entangled with e D (reducing p q ˘ S CD ) or with e B (reducing p q ˘ S BC ) , but not with both systems for set of 4 QGIF systems e A e B e C e D (for mechanical systems,we can use tilde on symbols, which can be omitted for general GIFs). This follows from the possibility ofpurification of this type of entropy, which allows to find various equivalent systems. If we consider ABCD in29 pure state, then p q ˘ S AB = p q ˘ S CD , p q e S ABC = p q ˘ S D . The inequality (66) becomes p q ˘ S CD + p q ˘ S BC ≥ p q ˘ S B + p q ˘ S D . We can consider, for instance, that p q ˘ S ( C | D ) = p q ˘ S CD − p q ˘ S D < , or p q ˘ S ( C | B ) = p q ˘ S BC − p q ˘ S B < , whenthe monogamy of entanglement follows from the non negative condition p q ˘ S ( C | D ) + p q ˘ S ( C | B ) ≥ . (67)Above important conditions (65), (66) and (67) for the von Neumann entropy for QGIFs can be provenin a standard form for quantum information theory [56, 38, 57]. It is not clear if similar results can be provenfor the thermodynamic entropy p q e S A or W-entropy p q f W A . In principle, such values characterize certaincomplementary properties of QGIFs and relativistic quantum mechanical systems.
In QM, measurements involve projection onto orthogonal subspaces of a Hilbert space H A . The sameformalism can be applied to QGIFs if we work with a density matrix p e σ of type (58) or (59). Generalized measurements for QGIFs of mechanical systems:
Let us introduce a system of s =1 , ..., k orthogonal Hermitian projection operators π s subjected to the conditions P ks =1 π s = 1; ( π s ) = π s ; and π s π s ′ = 0 for s = s ′ . Applying such a π s to a pure quantum system | ψ > ∈ H , we obtain an outcome s with probability p s = < ψ | π s | ψ >, when the properties of π s result in P ks =1 p s = 1 . If a system e A encodes aQGIF of a mechanical system characterized by a density matrix p e σ, the outcome s is p e p s = T r H π s p e σ. Weendow such a probability p e p s with a typical label for a canonical Hamilton quantum system and respectivegeometric flows. A measurement with an outcome s changes the QGIFs and results in a new density matrix p e σ s = π s p e σπ s / p e p s (68)encoding quantum information both from the geometric flows and the mechanical Hamilton structure.In a more general context, measurements can be performed using an auxiliary system C. Such a systemis not obligatory a mechanical one, of type e C (it can be an electric device etc.). A procedure with auxiliary C is called a "positive operator-valued measurement" or POVM) with Hilbert space C . Conventionally, sucha e C is k -dimensional with a basis consisting from k vectors/states | s > ∈ C , for s = 1 , , ...,k . We can initializesuch a C -system in the state | >, then consider a combined system C ⊗ H and a corresponding unitarytransform U which, for instance, adjusts a time- and flow parameter - dependent Hamiltonian H (if we agoing to study quantum geometric flows of mechanical systems). The operator U can be chosen that for any ψ ∈ H , the result of such a transform is parameterized using arbitrary linear operators E s ,U ( | > ⊗ ψ ) = k X s =1 | s > ⊗ E s ψ when k X s =1 E † s E s = 1 (69)follows from the condition of unitarity (the symbol † is used for the Hermitian conjugation). We canlabel such values with "tilde" if they are considered for geometric mechanical flows, for instance, using e U and e E s . In princile, one can be used arbitrary operators, U and E s , even the quantum density matrices,see below, will be taken for QGIFs. In general, projective measurements of the system C ⊗ H can beperformed using the commuting projection operators π s = | s >< s | ⊗ when the probability of outcome s is p s = | E s | ψ > | = < ψE † s E s | ψ > . The described above POVM procedure can be applied for measurements of a QGIF system defined by adensity matrix p e σ, when the probability of outcome s is p e p s = T r H E † s E s p e σ. We can treat the numbers p e p s asprobabilities for any p e σ because E † s E s ≥ for any s and (together with (69)) this results in P ks =1 p e p s = 1 . Itshould be noted that E † s E s are nonnegative Hermitian operators that add to 1 but not orthogonal projectionones. After a measurement with an outcome s, the combined system C ⊗ H can be described by the density30atrix for a "pure" quantum system. It can be parameterized in the form ( p s ) − | s >< s | ⊗ E s | ψ >< ψE † s , see (68), and, taking the partial trace over C , we obtain a conventional density matrix ( p s ) − E s | ψ >< ψE † s for the orginal system H . QGIFs of such quantum mechanical systems can be described by mixed statedwith density matrix p e σ, when ( e p s ) − E s p e σE † s results in an outcome s. Finally, we note that above POVM constructions can be generalized for any Hilbert space of type
C ⊗ ( H ⊕ H ′ ) with linear transforms E s : H → H ′ , which is useful for elaborating on generalized quantum modelsand information theory. Quantum channels for QGIFs:
For modeling quantum information flow theories, a corresponding den-sity matrix evolves both in a QM form and as a geometric flow evolution process. The usual Hamiltonianevolution of a state | ψ > → U | ψ > can be described by a unitary operator U a Hamiltonian b H correspondingto a canonical relativistic Hamiltonian e H (and respective Hessian p e g ab ( x, p ) (4) and canonical d-metric p e g (15)) or by a thermodynamic GIF system e A = h p e E , p e S , p e η i (37). In all cases, we can introduce the vonNeumann entropy p q e S ( p e σ A ) (60), and conditional entropy p q e S [ e A | e B ] (62), which are invariant under unitarytransforms p e σ A ∈ U p e σ A U − . Such QGIFs are also characterized by W-entropy p q f W A (29) and or p q e S A (37).Let us analyze how the notion of quantum channels can be elaborated for QGIFs of mechanical Hamiltonsystems. We consider again an extended system C ⊗ H enabled with a density matrix p ˘ σ = | >< | p e σ, where p e σ is a density matrix on H . Unitary maps p ˘ σ → p ˘ σ ′ , and with a trace induced matrix p e σ ′ on H ,can be parameterized in the form (69), p ˘ σ ′ = U p ˘ σU − = k X s,s ′ =1 | s >< s ′ | ⊗ E s p e σE † s and p σ ′ = T r C p ˘ σ ′ = k X s =1 E s p e σE † s . In result, we can define certain "quantum channels" for evolution of QGIF density matrices for mechanicalsystems as operations p e σ → P ks =1 E s p e σE † s , where the so-called Kraus operators E s are subjected to thecondition P ks =1 E s E † s = 1 . If we consider only one Kraus operator, we obtain as a special case the unitaryevolution of a QGIF system.We can consider quantum channels for the relative entropy and respective inequality conditions (63)which are written in the form p q e S ( p e ρ q p e σ ) ≥ p q e S ( p e ρ q p e σ ) for p e ρ → P ks =1 E s p e ρE † s and p e σ → P ks =1 E s p e σE † s , when the fist step of initialization consists in replacing p e ρ and p e σ, respectively, by | >< ⊗ p e ρ and | >< ⊗ p e σ. This is a very general statement on monotonicity ofrelative entropy and the von Neumann entropy for QGIFs of mechanical systems. The properties of Krausoperators for quantum channels are similar to those outlined in paragraphs (1)-(6) in section 3.7 of [38], seealso references therein. There are two differences: the first one is that we consider geometric flow evolution ofdensity matrices and that such rich quantum and geometric flow evolutions are characterized by additionalinequalities for the quantum versions of thermodynamic entropy and W-entropy.
Thermodynamics of QGIFs and quantum channels:
Let us consider a thermal quantum densitymatrix as in QM, p q e σ = p q e Z − e − β e H , with β = 1 /T, τ = T. We define for the conditional quantum entropyfor geometric flows of Hamilton mechanical systems p q e S ( p q e ρ q p q e σ ) = β [ p q e F ( p q e ρ ) − p q e F ( p q e σ )] , where the free energy corresponding to a second density matrix p q e ρ is p q e F ( p q e ρ ) := p q e E ( p q e ρ ) − T p q e S ( p q e ρ ) . Theenergy operator is defined and computed as p q e E ( p q e ρ ) = T r [( p q e ρ ) e H ] and the thermodynamic entropy is p q e S ( p q e ρ ) := β p q e E ( p q e ρ ) + log p q e Z ( p q e ρ ) . log p q e Z is independent on p q e ρ, we obtain p a e S ( p q e σ q p q e σ ) = 0 . For any quantum channel preserving thethermal equilibrium at temperature T, there is a map p q e σ to itself and transforms p q e ρ to a general densitymatrix p q e ρ ′ . In such a quantum channel the entropy decreases following formulas p q e S ( p q e ρ q p q e σ ) ≥ p q e S ( p q e ρ ′ q p q e σ ) and p q e F ( p q e ρ ) ≥ p q e F ( p q e ρ ′ ) . For quasi-classical approximations, we consider that such formulas transform into similar ones, see (55),for the state densities of type p e σ (54). In this paper, we put emphasis on the roles of entropic values derived from Perelman-Lyapunov typefunctionals [1, 36] in elaborating relativistic models of geometric flow evolution of Lagrange-Hamilton me-chanical systems and possible applications in classical and quantum information theory. Our aim was to seekanswer to wether the incorporation of fundamental geometric objects in relativistic mechanics into canoni-cal noholonomic structures on (co) tangent Lorentz bundles allow a new (J. Kern type) geometrization interms of certain generalized (pseudo) Riemannian and Finsler-Lagrange-Hamilton spaces [43, 45, 34, 35].Due to Grigory Perelman, such geometric constructions can be characterized by W-entropy functionals andrespective statistical/ geometric thermodynamic functionals like average flow energy, flow entropy and flowfluctuation, see further developments and applications in physics [53, 19, 20, 21, 22, 25, 26, 50].Here it should be emphasized that such concepts of "non-area, non-holographic, non-conformal ... "entropy are more general that those based on the Bekenstein-Hawking thermodynamics [39, 40, 41, 42]. Inour approach, the fundamental geometric and physical objects are defined by analogous metrics, nonlinearand linear connections, and their curvatures, canonically determined by Hessians of respective Lagrangeand/or Hamilton generating functions. Corresponding entropic and thermodynamic type values can becomputed for various classes of exact and parametric solutions (not only black hole type ones) in geometricflow evolution and (modified) gravity theories.The work presented here indicates that G. Perelman’s ideas and geometric methods with W-entropyand associated thermodynamic models for Ricci flows presented not only an important tool for proving thePoincaré-Thurston hypothesis. The constructions can be generalized for various types of relativistic and/ornon-Riemannian geometries which allow to elaborate on further developments for noncommutative, super-symmetric, stochastic and quantum geometries [23, 54, 55, 24, 27]. Although in this paper we investigatedonly flows of geometric mechanical Lagrange-Hamilton models elaborated on (co) tangent Lorentz bundles,and did the hole analysis based on classical and quantum mechanical Hamilton structures, our study shedslight on the importance of such constructions in elaborating new directions in quantum information theory[37, 38, 47, 48, 49] . We note that the conjecture that gravity can be thought of as an entropic force [51, 52]can be proven for certain classes of nonholonomic deformations of G. Perelman’s functionals [27, 28, 29].Using the results of this and partner works [27, 28, 29], we conclude that such proofs can be performed forthe emergent gravity from classical and quantum mechanical Lagrange-Hamilton theories.The results of section 4 support also the conclusion that using advanced geometric methods we canelaborate on basic ingredients of the geometric flow information, QGIF, theory. We close with the remarkthat in our future works there will be considered some more special topics of QGIFs such as teleportation andconditional geometric flow entropy; relative entropy and hypothesis geometric flow testing; how to encodeclassical geometric flow information in quantum states; geometric classical and quantum flow entanglementand emergent gravity theories.
Acknowledgments:
This research develops author’s former programs partially supported by IDEI, PN-II-ID-PCE-2011-3-0256, CERN and DAAD and contains certain results for new grant proposals. A secondUAIC affiliation refers to the Project IDEI hosted by that University during 2012-2015, when the bulk ofmain ideas and results of this and partner works were elaborated. Author is grateful to D. Singleton and P.Stavrinos for supporting his research on geometric methods in physics.32 eferences [1] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/0211159[2] G. Perelman, Ricci flow with surgery on three–manifolds, arXiv: math.DG/0303109[3] G. Perelman, Finite extintion time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math.DG/0307245[4] W. Thurston, Three-dimensional geometry and topology, Vol. 1. Edited by S. Levy, Princeton Mathe-matical Series, 35 (Princeton University Press, Princeton, NJ, 1997)[5] W. Thurston, The Geometry and Topology of Three-Manifolds, Princeton lectures notes on geometricstructures on 3-manifolds (1980), see updated electronic version following a MSRI link:http://library.msri.org/books/gt3m/[6] W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, AmericanMathematical Society. Bulletin New Series 6 (3), 357-381[7] D. Friedan, Nonlinear models in ε dimensions, PhD Thesis (Berkely) LBL-11517, UMI-81-13038,Aug 1980. 212pp[8] D. Friedan, Nonlinear models in ε dimensions, Phys. Rev. Lett. 45 (1980) 1057-1060[9] D. Friedan, Nonlinear models in εε