Classical and quantum geometric information flows and entanglement of relativistic mechanical systems
aa r X i v : . [ phy s i c s . g e n - ph ] J a n Classical and quantum geometric information flowsand entanglement of relativistic mechanical systems
Sergiu I. Vacaru ∗ Physics Department, California State University at Fresno, Fresno, CA 93740, USA; andDep. Theoretical Physics and Computer Modelling, 101 Storozhynetska street, Chernivtsi, 58029, Ukraine
Laurenţiu Bubuianu † SRTV - Studioul TVR Iaşi, 28 Alexandru Lapuşneanu street, Iaşi, 700057, Romania; and
University Apollonia, 2 Muzicii street, Iaşi, 700399, Romania
January 30, 2020
Abstract
This article elaborates on entanglement entropy and quantum information theory of geometric flows of(relativistic) Lagrange–Hamilton mechanical systems. A set of basic geometric and quantum mechanicsand probability concepts together with methods of computation are developed in general covariant formfor curved phase spaces modelled as cotangent Lorentz bundles. The constructions are based on ideasrelating the Grigory Perelman’s entropy for geometric flows and associated statistical thermodynamicsystems to the quantum von Neumann entropy, classical and quantum relative and conditional entropy,mutual information etc. We formulate the concept of the entanglement entropy of quantum geometricinformation flows and study properties and inequalities for quantum, thermodynamic and geometricentropies characterising such systems.
Keywords:
Perelman W-entropy; quantum geometric information flows.PACS2010: 02.40.-k, 02.90.+p, 03.65.Ud, 04.50.-h, 04.90.+e, 05.90.+m 05.90.+mMSC2010: 53C44, 53C50, 53C80, 81P45, 82D99, 83C15, 83C55, 83C99, 83D99, 35Q75, 37J60, 37D35
Contents ∗ emails: [email protected] and [email protected] ; Address for post correspondence in 2019-2020 as a visitor senior researcher at YF CNU Ukraine:
37 Yu. Gagarin street, ap.3, Chernivtsi, Ukraine, 58008 † email: [email protected] G. Perelman & von Neumann entropies for geometric information flows 7
A generic feature of quantum physics which is absent in classical physics is that of entanglement. Therewere introduced several entanglement measures of how much quantum a given system is. Because of com-putational accessibility, the entanglement entropy plays a particulary important role together with Rényientropies, mutual information etc. For recent reviews of most important ideas and results related to quan-tum information theory, we cite [1, 2, 3, 4, 5, 6, 7, 8, 9] and references therein. Here we note that theconcept of entanglement entropy originated from quantum information theory [10]. At present, it is con-nected to a wide range of applications in condensed matter physics, gravity theories and particle physics etc.The progress in such directions included the holographic formula for entanglement entropy [11], a new typeof order parameter for quantum-phase transitions [12, 13, 14], ideas of formulating quantum gravity fromquantum entanglement and so-called ER=EPR [15, 16].There are many motivations to study quantum entanglement which depends on respective directions ofresearch. For instance, we elaborated [17, 18] on the idea that an intriguing connection exists between thePoincaré-Thurston conjecture (it became again a conjecture for relativistic Ricci flows even a proof exists forRiemannian metrics [19]) and the emergent entropic gravity and/or other type modifications. G. Perelmanintroduced and applied in his famous preprints [19, 20, 21] the F- and W-functionals from which the R.Hamilton’s Ricci flow equations for Ricci flows [22, 23, 24] can be derived. Here we note that in physicssuch equations were considered earlier by D. Friedan [25, 26, 27]. In a general context, such works andfurther developments provide strong motivations for elaborating a new direction (based on geometric flowsand associated thermodynamical models) in classical and quantum information theory. For such models,the quantum entanglement can be exploited for computational tasks which are impossible if only classicalmethods are used but for performing on new type theories unifying quantum and geometric flow evolutionscenarios. 2his article is the 5th partner in a series of works [28, 17, 18, 29] devoted to applications of G. Perelman’sentropic functionals [19] and nonholonommic geometric flow methods in classical and quantum informationtheory, geometric mechanics and thermodynamics, and modified (entropic and other types) gravity. Fora review of mathematical results on Ricci flows of Riemannian and Kähler metrics, rigorous proofs andtopological and geometric analysis methods, we cite [30, 31, 32]. In our approach, we consider nonholonomicdeformations of the G. Perelman’s functionals and elaborated on new geometric methods and applicationsin (modified) gravity, geometric mechanics; locally anisotropic kinetics, diffusion and thermodynamics; andinformation theory. Here we note that in this work we follow the notations on the so-called quantum geometricinformation flow, QGIF, theory (in brief, it is used GIF for classical models) introduced in [29]. Readers maystudy our previous works [33, 34, 35, 36, 37] and references therein, on nonholonomic (non) commutative /supersymmetric geometric flows and related kinetic and statistical thermodynamic models.The aim of this paper is to specifically address the geometric flow evolution and dynamics of the en-tanglement in quantized Lagrange–Hamilton relativistic mechanical systems. We develop our approach onelaborating new principles and methods for formulating classical and quantum information theories encodinggeometric flows and their analogous geometric thermodynamic models. The key ideas for developing suchnew directions in (quantum) information theory and applications is to extend the standard constructionsinvolving the von Neumann, and related conditional and relative entropies. We introduce into considerationgeneralizations of the concepts of W-entropy and analogous thermodynamic entropy elaborated in originalvariants by G. Perelman for Ricci flows of Riemannian metrics.We try to make this work self-contained and multi-disciplinary pedagogic enough but for advanced re-searchers working on geometry and physics, nonholonomic geometric mechanics and thermodynamics, quan-tum mechanics and quantum field theory and information theory. In our case, some typical Alice and Bobcommunicating using methods of quantum information theory should also have certain knowledge on ge-ometric flows; systems of nonlinear partial differential equations, PDEs, and their applications in modernclassical and quantum physics. It is assumed that readers have a background on modified gravity theoriesand modern astrophysics and cosmology because all such theories provide strong motivations and examplesof applications of the formalism elaborated in the cited monographs, reviews and series of works on geo-metric flows and information theory. In this article, we study entanglement for quantum geometric flows ofmechanical systems and do not concern issues on gravity, quantum field theory or condensed matter physics.On emergent gravity theories, modified Ricci flow theories and gravity, exact solutions and related classicaland quantum mechanical entropic functionals from which generalized Einstein equations can be derived, seeour recent results [33, 34, 35, 36, 37] and references therein.This article is organized as follows: In section 2, we start with reviewing the fundamentals of the theoryof geometric flows of relativistic Hamilton phase spaces. After defining the fundamental geometric objectssuch as the nonlinear connection, N-connection, and distinguished metric, d-metric, structures, we show howthe curvatures can be computed for general and preferred linear connections. Then we introduce the G.Perelman F- and W-functionals (entropic type) for W. Hamilton mechanical systems and their formulationin general N-adapted variables.Section 3 begins with a quick introduction into the statistical thermodynamic theory of geometric infor-mation flows, GIFs, when the G. Perelman approach is generalized for nonholonomic N-adapted variables.The approach is generalized for quantum geometric information flows, QGIFs, using the statistical densitymatrix and its analogous quantum density matrix. The von Neumann entropy for QGIFs and quantumgeneralizations of the W- and thermodynamic entropy are considered.In section 4, we explore the entanglement and QGIFs as quantum mechanical systems. There are definedQGIF analogs of two spin systems, thermofield double GIF states and Bell like geometric flow states. Weoutline the main properties and inequalities of the entanglement entropy for such systems with mixed geo-metric and quantum flow evolution. The entanglement and Rényi entropy and QGIFs at finite temperatureare studied. Conclusions are provided in section 5. 3
Geometric flows of relativistic Hamilton phase spaces
We present a short review of the geometry of relativistic Hamilton phase spaces modelled on cotangentbundle T ∗ V of a nonholonomic Lorentz manifold V , see an axiomatic approach and details in [38, 39]. Thereare provided formulas for respective generalizations of G. Perelman’s F- and W-entropy functionals for whichwe follow the conventions from [17, 18, 29], see proofs and references therein. We consider a cotangent Lorentz bundle T ∗ V, dim V = 4 , enabled with local coordinates p u α = ( x i , p a ) , (inbrief, p u = ( x, p )) , where x i are base manifold coordinates and p a are momentum like typical fiber coordinates.Such a model of relativistic phase spacetime is enabled in any point with a total metric structure (phasespace metric) of signature (+ + + − ; + + + − ) , which for corresponding frames/coordinates transforms canbe represented in the form d p s = p g αβ d p u α d p u β = η ij dx i dx j + η ab dp a dp b , for p a ∼ dx a /dτ. (1)In these formulas, η ij = diag [1 , , , − and η ab = diag [1 , , , − . In a more general context, we can elab-orate on physical models on curved phase spaces when the metric structure (1) is determined by coefficientsof type p g αβ ( p u ) = [ g ij ( x ) , p g ab ( x, p )] . A relativistic Hamilton space H , = ( T ∗ V, H ( x, p )) is determined by a fundamental function H ( x, p ) (itcan be used a generating Hamilton function, Hamiltonian or Hamilton density). For classical models, it isconsidered that a map T ∗ V ∋ ( x, p ) → H ( x, p ) ∈ R defines a real valued function being differentiable on g T ∗ V := T ∗ V / { ∗ } , for { ∗ } being the null section of T ∗ V, and continuous on the null section of π ∗ : T ∗ V → V. In a more general context, a H ( x, p ) can be quantized following prescriptions for a respective quantum model(quantum mechanics, QM, or quantum field theory, QFT, with corresponding quasi-classical relativistic andnon-relativistic limits). In this work, we elaborate on relativistic mechanical models which are regular if theHessian (cv-metric) p e g ab ( x, p ) := 12 ∂ H∂p a ∂p b (2)for a H = e H is non-degenerate, i.e. det | p e g ab | 6 = 0 , and of constant signature.For Lagrange and Hamilton spaces, we can perform 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 ) for p a determining solutions of theequations y a = ∂H ( x, p ) /∂p a . In this work, we consider Hamilton structures which allow canonical Hamiltonformulations of some QM models and respective quasi-classical limits.Any e H defines a canonical nonlinear connection (N-connection) structure p e N : T T ∗ V = hT ∗ V ⊕ vT ∗ V (3)and a N-adapted canonical distinguished metric (d-metric) structure parameterized with conventional hori-zontal, h, and covertical, cv, components, 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 , (4) We follow such conventions: the "horizontal" indices, h–indices, run values i, j, k, ... = 1 , , , the vertical indices, v-vertical, run values a, b, c... = 5 , , , ; respectively, the v-indices can be identified/ contracted with h-indices , , , for liftson total (co) bundles, when α = ( i, a ) , β = ( j, b ) , γ = ( k, c ) , ... = 1 , , , ... . There are used letters labelled by an abstract leftup/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. Similar formulas can be derived on
T V for geometric objects labeled without " p ". Boldface symbols are used for geometricobjects on spaces endowed with nonlinear connection structure (see below formula (3)). p e e α = ( e i , p e e a ) are canonically determined by data ( e H, p e g ab ) . Considering general frame (vierbein) transforms, e α = e αα ( u ) ∂/∂u α and e β = e ββ ( u ) du β , any N-connectionand d-metric structure on a cotangent Lorentz bundle T ∗ V can be written in general form (without "tilde"on symbols), p N = { p N ij ( x, p ) } , with arbitrary coefficients ; p g = p g αβ ( x, p ) p e α ⊗ p e β = p g ij ( x, p ) e i ⊗ e j + p g ab ( x, p ) p e a ⊗ p e b . So, any classical regular Hamilton mechanics can be geometrized in general form on a phase spacetime T ∗ V by some nonholonomic data ( p N , p g ) . Inversely, using respective frame transforms on a nonholo-nomic cotangent bundle, we can always consider a relativistic Hamilton space model defined by some data ( e H, p e N ; p e e α , p e e α ; p e g ab , p e g ab ) . A physically realistic geometrization of physical models on T ∗ V is possible if such a phase space isenabled with a linear (affine) connection structure. Using p g , we can define in standard form the Levi-Civitaconnection p ∇ (as a unique metric compatible and with zero torsion) but such a geometric object is notadapted to the N-connection structure. To elaborate N-adapted geometric models we have to consider theconcept of distinguished connection (d–connection) which is a linear connection p D on T ∗ V preserving underparallel transports a N–connection splitting p N . With respect to a N-adapted basis the coefficients of a d-connection p D are labelled p Γ αβγ = { p L ijk , p ´ L ba k , p ´ C i cj , p C bca } . This involves an explicit h– and cv–splitting,of covariant derivatives p D = (cid:0) p h D , p cv D (cid:1) , where p h D = { p L ijk , p ´ L ba k } and p cv D = { p ´ C i cj , p C bca } . Prescribing a d-connection structure p D , we can work alternatively with an arbitrary linear connectiona linear connection D (which is not obligatory a d-connection) p D on T ∗ V . For such covector bundles,there are nonholonomic deformation relations with a respective distortion distinguished tensor, d-tensor, p Z := p D − p D. For any linear connection and/or d-connection structure, p D and/or p D , we can define in standard formrespective curvature, p R and/or p R , torsion, p T and/or p T , nonmetricity, p Q and/or p Q , d-tensors, 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 ] , (5) 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 , and/or 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 and/or coordinate formulas for coefficients of such geometric objects can be computed inexplicit form, see appendices to [38, 39] and references therein.Using (5), we can define and compute respective Ricci tensors/ d-tensors, scalar curvatures etc. Forinstance, the Ricci d–tensor of a p D is defined and computed p Ric = { p R αβ := p R ταβτ } . The N-adapted The coefficients of the canonical N-connection are computed following formulas p e N = n p e N ij := h { 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 io , where p e g ij is inverse to p e g ab (2). The canonical N–adapted (co)frames are 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 ) , being characterized by corresponding anholonomy relations [ 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 , with anholonomycoefficients 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 . Such a frame is holonomic (integrable) if therespective anholonomy coefficients are zero. p D in respective phase spaces are parameterized in h -and/or cv -form by formulas 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 } . (6)Such formulas for p D can be written in a similar "underlined" form. Hereafter, for simplicity, we shallprovide the formulas only for a general d-connection p D if that will not result in ambiguities.If a phase space is enabled both with a d-connection, p D , and a d-metric, p g , structures, we can defineand compute nonholonomic Ricci scalars, 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, (7)with respective h– and v–components, p R = p g ij p R ij and p S = p g ab p S ab . The geometric objects (5), (6) and (7) can be defined for any special classes of linear connection structures.In next subsection, we consider three important classes of linear and/or d-connections determined by a d-metric structure p e g or p g . Any relativistic phase space T ∗ V can be described as a Hamilton space using the canonical data ( p e N , p e g ) and/or in general nonholonomic (pseudo) Riemannian form for some ( p N , p g ) . Respective canonical N–connections p e N and/or p N define correspondingly certain canonical almost complex structures p e J and/or p J . For instance, we can consider a linear operator p J acting on p e α = ( p e i , p e b ) using formulas p J ( p e i ) = − p e n + i and p J ( p e n + i ) = p e i . Such a p J defines globally an almost complex structure ( p J ◦ p J = − I , where I is the unity matrix) on T ∗ V . Using p e J and p J , we can define respective (canononical) almost symplecticstructures, p e θ := p e g ( p e J · , · ) and p θ := p g ( p J · , · ) . In result, we can construct such preferred linear/distinguishedconnections: 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. (8)The geometric objects in (8) are related via corresponding distortion relations 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 ); with distortion d-tensors p b Z , p e Z , and p Z , on T T ∗ V . In principle, we can work with any such linear connectionstructure even they have different geometric and physical meaning. The corresponding curvatures and Ricci d-tensors and scalar curvatures can be computed by introducing such distortion relations in respective formulas(5), (6) and (7). The goal of this subsection is to generalize G. Perelman’s functionals and formulate and approach tothe theory of nonholonomic geometric flows of relativistic mechanical systems. We shall consider canonicalHamilton variables and nonholonomic deformations to a general d-connection structure. This is importantfor further developments in classical and quantum information theories when the Hamilton variables areused in explicit form for analyzing certain analogous mechanical and thermodynamic models and, latter, theresults are reformulated in general covariant forms.We consider a family of nonholonomic cotangent Lorentz bundles T ∗ V ( τ ) enabled with correspondingsets of canonical N–connections p e N ( τ ) = p e N ( τ, p u ) and d-metrics p e g ( τ ) = p e g ( τ, p u ) all parameterized by6 positive parameter τ, ≤ τ ≤ τ . In general frame form, such sets of geometric objects are respectivelydenoted p N ( τ ) = p N ( τ, p u ) and p g ( τ ) = p g ( τ, p u ) . Let us write correspondingly e Ξ = ( e Ξ t , e Ξ E ) and p e Ξ = ( e Ξ t , p e Ξ E ) for nonholonomic distributions of base and fiber hypersurfaces with conventional splitting3+3 of signature (+++;+++) on total phase space T ∗ V . On a typical cofiber of such a phase space, we canconsider a 3-d cofiber hypersurface p e Ξ E , for instance, of signature (+ + +) with a label p = E for an energytype parameter. Using N–adapted (3+1)+(3+1) frame and coordinate transforms of metrics with additionaldependence on a flow parameter, we can parameterise the d-metric in the form 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 , (9)where p e α s (for a = 5 , are N-adapted bases. This ansatz is written as an extension of a couple of 3–dmetrics, 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 . We consider ˘ N as the lapse function on the baseand p ˇ N as the lapse function in the co-fiber. In this work, we elaborate on geometric phase flow theories ona conventional temperature like parameter τ. The theory of geometric flows of relativistic Hamilton mechanical systems was formulated [18] in explicitform using canonical data ( p e g ( τ ) , p e D ( τ )) , in terms of geometric objects with "tilde" values. Consideringnonholonomic frame transforms and deformations of d-connections, and redefining the normalizing functions,we can postulate such generalizations of the G. Perelman functionals: p F = p Z e − p f q | p g αβ | d p u ( p s R + | p D p f | ) and (10) p W = p Z p µ q | p g αβ | d p u [ τ ( p s R + | p h D p f | + | p v D p f | ) + p f − . (11)In these formulas, we use a brief notation for the integrals on phase space variables and the normalizingfunction p f ( τ, p u ) is subjected to the conditions p Z p µ q | p g αβ | d p u = Z t t Z Ξ t Z E E Z p Ξ E p µ q | p g αβ | d p u = 1 for a classical integration measure p µ = (4 πτ ) − e − p f and the Ricci scalar p s R is taken for the Ricci d-tensor p R αβ of a d-connection p D . Similar F- and W–functionals can be postulated for nonholonomic geometric flows on T ∗ V using data ( p g ( τ ) , p b D ( τ )) , or ( p e g ( τ ) , p e D ( τ )) , and other type ones related via distorting relations, with correspondinglyredefined integration measures and normalizing functions, and respective hypersurfaces. LC-configurationscan be extracted for certain conditions when p D | p T =0 = p ∇ . Geometric flows of Riemannian metrics are characterized by a statistical thermodynamic model whichcan be elaborated in a self-consistent form using a W-functional of type (11) defined for Riemannian metrics For simplicity, we shall write, for instance, p e N ( τ ) instead of p e N ( τ, p u ) if that will not result in ambiguities. Relativisticnonholonomic phase spacetimes can be enabled with necessary types double nonholonomic (2+2)+(2+2) and (3+1)+(3+1)splitting [35, 36, 37, 28, 17]. Local (3+1)+(3+1) coordinates are labeled in the form 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 , . The insices ` ı, ` j, ` k, ... = 1 , , , respectively, ` a, ` b, ` c, ... = 5 , , can be used for corresponding spacelike hyper surfaces on a base manifold andtypical cofiber. For relativistic geometric flows of mechanical systems described by Hamiltonians [18, 29], the thermo-dynamic generating function can be written in the form p e Z [ p e g ( τ )] = p eR e − p e f p | p e g αβ | d p u ( − p e f + 16) , on T ∗ V , where the integral p eR is considered for canonical mechanical variables and the corresponding func-tional dependence is determined by p e g ( τ ) . With respect to general frames (or with necessary (3+1)+(3+1)decomposition and a d-metric of type (9)), the integration measure can be re-defined in a form which allowsus to consider p Z [ p g ( τ )] = p Z e − p f q | p g αβ | d p u ( − p f + 16) , for T ∗ V . (12)A variational N-adapted calculus for p Z and geometric data ( p N , p g , p D ) allows us to compute such relativisticthermodynamic values: p E = − τ p Z e − p f q | q q q ˘ N p q p q p q p ˇ N | δ p u ( p s R + | p D p f | − τ ) , (13) p S = − p Z e − p f q | q q q ˘ N p q p q p q p ˇ N | δ p u (cid:2) τ (cid:0) p s R + | p D p f | (cid:1) + p f − (cid:3) , p η = − p Z e − p f q | q q q ˘ N p q p q p q p ˇ N | δ p u [ | p R αβ + p D α p D β p f − τ p g αβ | ] . Such values can be written in Hamilton mechanical variables with tilde as in (12) or re-defining the normal-izing functions for the canonical d-connection p b D , see (8) and respective distorting relations.Using the first two formulas in (13) for two d-metrics g and g , we can define the respective free energyand relative entropy, p F ( g ) = E ( g ) − τ p S ( g ) and p S ( g q g ) = β [ p F ( g ) − p F ( g )] , where E ( g ) = − τ Z e − f q | q q q ˘ N q q q ˇ N | δ u [ s R ( g ) + | D ( g ) f ( τ, u ) | − τ ] , S ( g ) = − Z e − f q | q q q ˘ N q q q ˇ N | δ u (cid:2) τ (cid:0) s R ( g ) + | D ( g ) f ( τ, u ) | (cid:1) + f ( τ, u ) − (cid:3) are computed using the phase spacetime measures, the Ricci scalar and canonical d–connection are deter-mined respectively by g and g , Hereafter, we shall not write such dependencies in explicit form if that will not result in ambiguities.
8n this work, we study the geometric flow evolution of thermodynamics systems that preserves the thermalequilibrium at temperature β for maps g → g . A realistic physical interpretation for such systems existsif S ( g q g ) ≥ S ( g q g ) , i.e. F ( g ) ≥ F ( g ) . (14)These aspects connect general frame and mechanical variables flow models to the second low of thermody-namics. Values of type (13) are in relativistic thermodynamic relation if the second thermodynamic law(14) is satisfied. Such conditions impose additional constraints on the class of normalizing and generatingfunctions. In this subsection, we develop the density matrix formalism for applications in the theory of classicaland quantum geometric information flows (respectively, GIFs and QGIFs), see sections 4 and 5 in [29] fora formulation in Hamilton mechanical variables. Nonholonomic deformations of G. Perelman entropy likefunctionals will be used for relativistic formulations of the von Neumann entropy and QGIFs in arbitraryframes.
The thermodynamic generating function p Z [ p g ( τ )] (12) with free energy p E can be used for defining thestate density p σ ( β, p E , p g ) = p Z − e − β p E , (15)with β = 1 /T, τ = T. This value is the a classical analog of the density matrix in QM. We shall use it forelaborating models of QGIFs.We can consider that a density state p σ [ p g ] is associated to p g αβ , when but the geometric evolution mayinvolve another density p ρ [ p e g ] , where the left label 1 is used for distinguishing two d-metrics p g and p g . In result, the concept of relative entropy between any state density p ρ ( β, p E , p g ) and p e σ ( β, p E , p g ) canbe introduced. It can be computed for a prescribed measure ω ( E ) on a cotangent Lorentz bundle with E considered as a thermodynamical energy parameter associated to p E . The conditional entropy for GIFs is introduced p S ( p ρ q p σ ) = β [ p F ( p ρ ) − p F ( p σ )] , (16)where the free energy corresponding to p ρ is defined by formula p F ( p ρ ) := p E ( p ρ ) − T p S ( p ρ ) with the average energy p E ( p ρ ) = R p ρEdω ( E ) . The thermodynamic entropy in (16) is computed followingformula p S ( p ρ ) := β p E ( p ρ ) + log p Z ( p ρ ) . The condition p S ( p σ q p σ ) = 0 is satisfied if log p Z is independent on p ρ. Using canonical mechanical variables ( e H, p e g ab ) , we can study special QM systems described by purestates. In a more general context, QM involves probabilities considered not for a quantum state but fordensities matrices. In this subsection, we elaborate on how GIFs of classical mechanical systems can begeneralized to QGIFs using basic concepts of quantum mechanics, QM, and information theory. We shall9laborate on quantum models of GIFs described in terms of density matrices defined as quantum analogs ofstate densities of type p σ (15).For any point p u ∈ T ∗ V of a typical relativistic phase space used for modeling a classical GIF system A = [ p E , p S , p η ] (13), we associate a typical Hilbert space H A , which is denoted e H A for canonical Hamiltonmechanical variables. A state vector e ψ A ∈ e H A can be defined as an infinite dimensional complex vectorfunction. For applications in quantum information theory, there are considered approximations with finitedimensions. Such a e ψ A is a solution of the Schrödinger equation with a Hamiltonian b H constructed as awell-defined quantum version of a canonical Hamiltonian e H. In a a more general context, we can work withgeneral covariant variables (or certain versions with (3+1)+(3+1) splitting), when "non-tilde" d-metrics p g (see (9)) are used for definition of certain quantum measures. Considering unitary transforms of type e ψ A → U ψ A , we can describe the system A by an abstract Hilbert space H A , or to associate a complexvector space of dimension N with Hermitian product, see details in [1, 7].The complex geometric arena for QGIFs models consists from complex bundles H A ( T ∗ V ) = ∪ p u H p u A assoctiated to T ∗ V and constructed as unities of Hilbert spaces H p u A for p u ∈ T ∗ V , or a points of asubspace of such a phase space. We consider that there are nonholonomic variables when ψ A → e ψ A ( p u ) and the integration measure is determined by a p e g , or its frame transforms to a p g . It is assumed that suchconstructions are possible at least for perturbanions nearly a flat "double" Minkowski metric (1) nearly apoint p u. This way a perturbative QGIF model with quasi-classical limits can be always elaborated. GIFsdescribe flow evolution of mechanical systems in causal relativistic classical forms.The combined Hilbert space is defined as a tensor product, H A ⊗ H B , with an associate Hilbert space H A considered for a complementary system A . Here we note that symbols A , B , C etc. are used as labels forcertain systems under geometric evolutions described by respective thermodynamical modles of type (13).The state vectors for a combined QGIF system are written ψ AB = ψ A ⊗ ψ B ∈ H AB = H A ⊗ H B for ψ A = 1 A taken as the unity state vector. Quantum systems subjected only to quantum evolution and not to geometricflows are denoted A, B, C, ...
Entangled states:
In QM and QGIF theories, a pure state ψ AB ∈ H AB may be not only a tensor productvector but also entangled and represented by a matrix of dimension N × M if dim H A = N and dim H B = M .We underline such symbols in order to avoid ambiguities with the N-connection symbol N . A Schmidtdecomposition can be considered for any pure state, ψ AB = X i √ p i ψ i A ⊗ ψ i B , (17)for any index i = 1 , , .... (up to a finite value). The state vectors ψ i A can be taken to be orthonormal, < ψ i A , ψ j A > = < ψ i B , ψ j B > = δ ij , where δ ij is the Kronecker symbol. If p i > and P i p i = 1 , we can treat p i as probabilities. In general, such ψ i A and/or ψ i B do not define bases of H A and/or H B because we can takesome vectors when, in principle, it is not enough for such bases. We can consider aht such values split theGIFs into certain probable evolution scenarios. The quantum density matrix for a QGIF-associated system A is defined ρ A := X a p a | ψ a A >< ⊗ ψ a A | (18)10s a Hermitian and positive semi-definite operator with trace T r H A ρ A = 1 . Using such a ρ A , we can computethe expectation value of any operator O A characterizing additionally such a system, < O > AB = < ψ AB |O A ⊗ B | ψ AB > = X i p i < ψ i A |O A | ψ i A >< ψ i B | B | ψ i B > = < O > A = X i p i < ψ i A |O A | ψ i A > = T r H A ρ A O A . (19)Such values encode both quantum information and geometric flow evolution of bipartite systems of type A , B , and AB with both quantum and geometric entanglement defined by density matrices. Joint probabilities for bipartite quantum systems and measurements:
Bipartite QGIFs systemsare described in general form by quantum density matrices of type ρ AB or (in canonical mechanical variables) ρ e A e B . In the classical probability theory, we describe a bipartite system XY 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 details in [1, 7] and, for GIFs, [29].Considering AB as a bipartite quantum system with Hilbert space H A ⊗ H B , we can define and param-eterize a QGIF density matrix ρ AB in standard QM form: ρ AB = X a,a ′ ,b,b ′ ρ aa ′ bb ′ | a > A ⊗| b > B A < a ′ | ⊗ B < b ′ | . In this formula, | a > A , a = 1 , , ...,n is an orthonormal basis of H A and | b > B , b = 1 , , ...,m as an orthonormalbasis of H B . A measurement of the system A is characterized by a reduced density matrix obtained by respectivecontracting of indices, ρ A = T r H B ρ AB = X a,a ′ ,b,b ρ aa ′ bb | a > A A < a ′ | , for | b > B B < b | = 1 . In a similar form, we can define and compute ρ B = T r H A ρ AB . For cotangent bundle constructions, we candistinguish the geometric and physical objects putting left labels " p ", p ρ B = T r H A p ρ AB . Using such formulas,we can elaborate on QGIFs models and quantum information theory formulated in conventional mechanicalvariables or in a general covariant form.
The quantum density matrix p σ AB for a state density p σ (15) can be defined and computed using formulas(19), p σ AB = < p σ > AB = < ψ AB | p σ ⊗ B | ψ AB > = X i p i < ψ i A | p σ | ψ i A >< ψ i B | B | ψ i B > = p σ A = < p σ > A = X i p i < ψ i A | p σ | ψ i A > = T r H A p ρ A p σ, (20)where the density matrix p ρ A is taken for computing the QGIF density matrix p σ A . This matrix is determinedby a state density of the thermodynamical model for GIFs of a classical system p σ which can be parameterizedin nonholonomic variables of a mechanical Hamiltonian system p e σ. For quantum systems, we can work with quantum density matrices p σ AB and p σ A and respective partialtraces p σ A = T r H B p σ AB and p σ B = T r H A p σ AB . Such formulas can be written in coefficient forms p σ AB = X a,a ′ ,b,b ′ p σ aa ′ bb ′ | a > A ⊗| b > B A < a ′ | ⊗ B < b ′ | and p σ A = X a,a ′ ,b,b p σ aa ′ bb | a > A A < a ′ | . QGIFs can be described in standard QM form for the von Neumann entropy determined by p σ A (20) asa probability distribution, p q S ( p σ A ) := T r p σ A log p σ A . (21)Hereafter we shall write the trace in a simplified form without a label for the corresponding Hilbert space ifthat will not result in ambiguities. We use also a left label q to state the quantum character of such values.It should be also emphasized that such an entropy is a quantum analog of a p e S used in the thermodynamicmodel for geometric flow evolution of Hamilton mechanical systems. Tilde can be omitted for general frametransforms when p S encode a different frame structure. Such a QGIF entropy satisfies two conditions: p q S ( p σ A ) ≥ and it is manifestly invariant under a unitary transformation p σ A → U p σ A U − . The von Neuman entropy for QGIFs, p q S ( p σ A ) , has a purifying property which does not have a classicalanalog. Considering a bipartite system ψ AB = P i √ p i ψ i A ⊗ ψ i B and ρ A := P i p i | ψ i A > ⊗ < ψ i A | , we compute p σ A := X a,a ′ ,b,b p X k σ aa ′ bb p k A < a ′ || ψ k A >< ⊗ ψ k A || a > A , p σ B := X a,a ′ ,b,b p X k σ aa ′ bb p k B < a ′ || ψ k B >< ⊗ ψ k B || b > B . (22)In these formulas, we have the same probabilities p k for two formulas with different matrices and bases. Thisproves that p q S ( p σ A ) = p q S ( p σ B ) when a system A and a purifying system B have the same von Neumannentropy. QGIFs can be characterized not only by a von Neumann entropy of type (21) but also by quantum analogsof entropy values used for classical geometric flows. We can consider both an associated thermodynamicsentropy and a W-entropy in classical variants and then quantize such systems using a respective Hamiltonianwhich allows a self-consistent QM formulation. Such values can be introduced and computed in explicit formusing respective formulas (20), (22) for classical conditional (16) and mutual entropy considered for GIFsand in information theory [1, 7, 29]. We define respectively p q W AB = T r H AB [( p σ AB )( p AB W )] and p q W A = T r H A [( p σ A )( p A W )] , p q W B = T r H B [( p σ B )( p B W )]; p q S AB = T r H AB [( p σ AB )( p AB S )] and p q S A = T r H A [( p σ A )( p A S )] , p q S B = T r H B [( p σ B )( p B S )] . Such values describe corresponding entropic properties of quantum systems with rich geometric structureunder geometric flow evolution.The quantum probabilistic characteristics are described by the von Neumann entropy p q S ( p σ A ) (21) andcorresponding generalizations for AB and B systems p q S ( p σ AB ) := T r p σ AB log p σ AB and p q S ( p σ A ) := T r p σ A log p σ A , p q S ( p σ B ) := T r p σ B log p σ B . Such values also encode thermodynamic, geometric flow and probabilistic properties of QGIFs and can be usedfor elaborating a standard approach to quantum information theory for systems with geometric mechanicalHamilton flows and their covariant frame transforms.12
Entanglement and QGIFs of quantum mechanical systems
Originally, the notion of bipartite entanglement was introduced for pure states and density matrix gen-eralizations in description of finite-dimensional QM systems, see review of results in [1, 7, 3, 6]. In thissection, we analyze how the concept of entanglement can be generalized for QGIFs when, for instance, thereare considered two relativistic mechanical systems under geometric flow evolution. Such systems and theirthermodynamic and QM analogs are characterized by a set of entropies like G. Perelman’s W-entropy andgeometric thermodynamic entropy and the nontrivial entanglement entropy in the von Neumann sense. Eachof such entropic values characterise classical and quantum correlations determined by geometric flow evo-lution and quantifies the amount of quantum entanglement. A set of inequalities involving Pereleman andentanglement entropies play a crucial role in definition and description of such systems. We provide suchformulas without rigorous proofs following two reasons: The W-entropy p W (11), thermodynamic entropy p S (13) and related von Neumann p q S (21) realizations are well-defined classical and quantum entropic typevalues. For physicists, such formulas have a natural and intuitive motivation and interpretation in termsof thermodynamical generation functions and density matrices for GIFs. Rigorous mathematical proofs onhundreds of papers use methods of geometric analysis [19, 30, 31, 32]. On main ideas and key steps forchecking such results and selecting causal and realistic physical scenarios, we discuss in footnote 10 of ourpartner work [29]. The goal of this subsection is to study how the concept of quantum entanglement can be developed forQGIF systems characterized by an associated statistical thermodynamic model with respective generatingfunction which transforms into a respective density matrix in a related quantum theory.
For any (relativistic) mechanic model, continuous or a lattice model of quantum field theory, thermo-fieldtheory, QGIF model etc., we can associate a QM mechanical model with a pure ground state | Ψ > for atotal Hilbert space t H when the density matrix is p t ρ = | Ψ >< Ψ | (23)can be normalized following the conditions < Ψ | Ψ > = 1 so that the total trace t tr ( p t ρ ) = 1 . Such aconventional total quantum system is divided into a two subsystems A and B . In this section, we consider that A = [ p E , p S , p η ] (13) is a typical GIF system (in mechanical, or general covariant variables) for with a QGIFmodel is elaborated. A similar model (in principle, for a different associated relativistic Hamiltonian andd-metric g ) is considered for B = (cid:2) p E , p S , p η (cid:3) . Such subsystems A and B = A are complimentary to eachother if in a n -dimensional cotangent bundle space there is a common boundary ∂ A = ∂ B of codimension 2,where the non-singular geometric flow evolution A transforms into a necessary analytic class of flows on A . In principle, we can consider two completely different and classically separated GIF systems A and B whichare correlated as quantum systems. We can consider that for bipartite QGIFs t H = H AB = H A ⊗ H B aswe considered in subsection 3.2. Such an approximation is less suitable, for instance, if there are consideredtheories with gauge symmetries, see discussion and references in footnote 3 of [6] (we omit such constructionsin this work).The measure of entanglement of a QGIF subsystem A is just the von Neumann entropy p q S (21) butdefined for the reduced density matrix p ρ A = T r H B ( p t ρ ) , when the entanglement entropy of A is p q S ( p ρ A ) := T r ( p ρ A log p ρ A ) . (24)13uch a p ρ A is associated to a state density p ρ ( β, p E , p g ) of type (15). We note that the total entropy p t S = 0 for a pure grand state (23). Considering {| a > A ; a = 1 , , ...k a } ∈ H A and {| b > B ; b = 1 , , ...k b } ∈ H B as orthonormal bases, we canparameterize a pure total ground state in the form | Ψ > = X ab c ab | a > A ⊗| b > B , (25)where c ab is a complex matrix of dimension dim H A × dim H B . When such coefficients factorize, c ab = c a c b , we obtain a separable ground state (equivalently, pure product state), when | Ψ > = | Ψ A > ⊗| Ψ B >, for | Ψ A > = X a c a | a > A and | Ψ B > = X b c b | b > B . The entanglement entropy p q S ( p ρ A ) = 0 if and only if the pure ground state is separable. For QGIFs,such definitions are motivated because corresponding sub-systems are described by corresponding effectiverelativistic Hamilton functions, e H A and e H B , and/or effective thermodynamics energies, p A E and p B E . A ground state | Ψ > (25) is entangled (inseparable ) if c ab = c a c b . For such a state, the entanglemententropy is positive, p q S ( p ρ A ) > . Using quantum Schmidt decompositions (17) and (18), we prove that p q S = − min( a,b ) X a p a log p a and p q S | max = log min( a, b ) for X a p a = 1 and p a = 1 / min( a, b ) , ∀ a. (26)In summary, an entangled state of QGIFs is a superposition of several quantum states associated toGIFs. An observer having access only to a subsystem A will find him/ herself in a mixed state when thetotal ground state | Ψ > is entangled following such conditions: | Ψ > : separable ←→ p ρ A : pure state , | Ψ > : entangled ←→ p ρ A : mixed state . The von Neumann entanglement entropy p q S encodes two types of information: 1) how geometric evolutionis quantum flow correlated and 2) how much a given QGIF state differs from a separable QM state. Amaximum value of quantum correlations is reached when a given QGIF state is a superposition of all possiblequantum states with an equal weight. Additional GIF properties are characterized by W-entropy p W (11)and thermodynamic entropy p S (13) which can be computed in certain quasi-classical QM limits, for a 3+1splitting, for instance, along a time like curve. The most simple example of an entangled system [1, 7, 3, 6] is that of two particles A and B with spin / . In the information theory, such quantum spin systems can be used to encode binary information as bitsand, with further generalizations, to elaborate on quantum bits, qubits. Respective theoretical descriptionsuse density matrices and the von Neumann entropy.To study similar entanglement properties of geometric flows in classical and quantum information theorywe can consider two thermodynamical models of general covariant mechanical systems A = [ g , p E , p S , p η ] and B = (cid:2) g , p E , p S , p η (cid:3) , see formulas (13). A respective QGIF model with entanglement is elaborated fordifferent associated relativistic Hamiltonians and respective d-metrics g and g . For simplicity, we consider14hat the conventional Hilbert spaces are spanned by two orthonormal basic states in the form {| a > A ; a =1 , } ∈ H A and {| b > B ; b = 1 , } ∈ H B , when A , B < a | b > A , B = δ ab . The total Hilbert space H AB = H A ⊗ H B has a 4-dim orthonormal basis H AB = {| >, | >, | >, | > } , where | ab > = | a > A ⊗| b > B are tensorproduct states.As a general state, we can consider | Ψ > = cos θ | > − sin θ | >, (27)where ≤ θ ≤ π/ . The corresponding entanglement entropy (24) is computed p q S ( p ρ A ) = − cos θ log(cos θ ) − sin θ log(sin θ ) . Above formulas show that for θ = 0 , π/ we obtain pure product states with zero entanglement entropy.For a system | Ψ > = √ ( | > −| > ) , when the density matrix p ρ A = 12 ( | > A A < | + | > A A < | ) = 12 diag (1 , results in p q S ( p ρ A ) = − tr A ( p ρ A log p ρ A ) = log 2 . So, the maximal entanglement is for θ = π/ . If the GIF structure is "ignored" for such a quantum system(or (27)), we can treat it as conventional QM system, for instance, with up-spin | > and down-spin | > . In a general context, QGIFs with nonholonomic structure determined by Hamilton mechanical systems arecharacterized additionally by respective values of W-entropy p W (11) and thermodynamic entropy p S (13). Inorthonormal quantum bases, the entanglement entropy is the measure of "pure" quantum entanglement. Theinformation flows with rich nonholonomic geometric structure are characterized additionally by geometrictype entropies. If the evolution parameter β = T − is treated as a temperature one like in the standard G. Perelman’sapproach, we can consider respective geometric flow theories as certan classical and/or quantum thermofieldmodels. Such a nontrivial example with entanglement and a thermofield double GIFs state is defined by aground state (25) parameterized in the form | Ψ > = Z − / X k e − βE k / | k > A ⊗| k > B , (28)where the normalization of the states is take for the partition function Z = P k e − βE k / . Such values areassociated to the thermodynamic generating function p Z [ p g ( τ )] (12) and state density matrix p σ ( β, p E , p g ) (15) the energy p E A = { E k } is considered quantized with a discrete spectrum for a QGIF system A =[ g , p E , p S , p η ] . The density matrix for this subsystem determining a Gibbse state is computed p ρ A = Z − X k e − βE k / | k > A ⊗ A < k | = Z − e − β p E A . In above formulas, we consider p E A as a (modular) Hamiltonian p E A such that p E A | k > A = E k | k > A .In principle, the thermofield double states for QGIFs consist certain entanglement purifications of thermalstates with Boltzman weight p k = Z − P k e − βE k , see discussions related to formulas (22). Coping the state15ectros {| k > B } from H A to H B , we can purify the QGIF thermal system A in the extended Hilbert space H A ⊗ H B . In result, every expectation value of local operators in A can be represented using the thermofielddouble state | Ψ > (28) of the total system A ∪ B . For such models, the entanglement entropy is a measureof the thermal entropy of the subsystem A when p S ( p ρ A ) = − tr A [ p ρ A ( − β p E A − log Z )] = β ( < p E A > − p F A ) , where the thermal free energy is computed p F A = − log Z. Here we note that for the thermofield values itis omitted the label "q" considered, for instance, for p q S (24), see also formulas (16).Thermofield GIF configurations are also characterized by the respective W-entropy p W (11) which canbe defined even thermodynamic models are not elaborated. For nonholonomic kinetic, diffusion and ther-modynamic structures including relativistic Ricci flows, such models were studied in detail in [33, 36, 44],see references therein. We also cite some important works on geometric thermodynamics and thermofieldtheories, see [42, 43, 45] and references. The thermofield double states were considered in black hole ther-modynamics and QFT, see reviews of results in [1, 2, 7, 3, 6]. In a two QGIF system, a state (27) is maximally entangled for θ = π/ . Analogs of Bell state (orEinstein-Podosly-Rosen pairs) in quantum geometric flow theory are defined | Ψ B > = 1 √ | > + | > ) , | Ψ B > = 1 √ | > −| > ) , (29) | Ψ B > = 1 √ | > + | > ) , | Ψ B > = 1 √ | > −| > ) . In QM models, these states violate the Bell’s inequalities. Such inequalities hold in a hidden variable theoryfor the probabilistic features of QM with a hidden variable and a probability density. In this work, the states(29) encode also information of geometric flows characterized by W-entropy.
EPR pairs and multi-qubits for QGIFs:
The constructions can be extended for systems of k quabits.The first example generalizes the concept of Greenberger-Horne-Zelinger, GHZ, states [46, 47, 6], | Ψ GHZ B > = 1 √ | > ⊗ k + | > ⊗ k ) . In quantum information theory, thre are used another type of entangled states (called W states; do notconfuse with W-entropy) [49], | Ψ W B > = 1 √ | ... > + | ... > + ... + | ... > ) . We emphasize that | Ψ GHZ B > is fully separable but not | Ψ W B > which we shall prove in the example below. Tripartite QGIFs:
For k = 3 with subsystems A , B and C , we write | Ψ GHZ B > = 1 √ | > + | > ) and | Ψ W B > = 1 √ | > + | > + | > ) . Considering
T r C , we define the reduced density matrices for the system A ∪ B , p ρ GHZ
A∪B = 12 ( | >< | + | >< | ) and p ρ W A∪B = 23 | Ψ B >< Ψ B | + 13 | >< | . p ρ GHZ
A∪B = P k =1 p k p ρ k A ⊗ p ρ k B , where p k = 1 / and p ρ A , B = | >< | and p ρ A , B = | >< | . Because ofthe Bell state | Ψ B > (29), the p ρ W A∪B can not be written in a separable form. So, the state | Ψ W B > is stillentangled even we have taken T r C . This establishes a quantum correlation between QGIFs. Additionally,such values are characterized by W-entropies of type p W (11) computed for A , B , C and A ∪ B . We summarize several useful properties of the entanglement entropy (24) for QGIFs formulated in termsof the density matrix of type p ρ A = T r H B ( p t ρ ) . We omit explicit cumbersome and techniqual proofs becausethey are similar to derivations in [10]. For any p ρ A associated to a state density p ρ ( β, p E , p g ) of type (15), wecan compute the respective W-entropy and geometric thermodynamic entropy taking measures determinedby p g and/or respective Hamilton mechanical variables. Rigorous mathematical proofs involve a geometricanalysis technique summarized in [19, 30, 31, 32]. For applications in modern gravity and particle physicstheories, we can elaborate on alternative approaches using the anholonomic frame method of constructingoff-diagonal solutions in relativistic geometric flow theories and generalizations [36, 37, 28]. Using explicitclasses of solutions and re-defining normalizing functions, we can always compute Perelman’s like entropyfunctionals at least in the quasi-classical limit with respective measures and related to p q S (24) for a QGIFor a thermofield GIF model. We present four important properties of QGIFs which result in the strong subadditivity property ofentanglement and Perelman’s entropies.
Entanglement entropy for complementary subsystems: If B = A , the entanglement entropies arethe same p q S A = p q S A which follows from formulas (26) for a pure ground state wave function. Similar equalities for the W-entropy p W (11) and/or thermodynamic entropy p S (13) can be proven only for the same d-metrics p g and respectivenormalizations on A and A . Here we note that p q S A = p q S B if A ∪ B is a mixed state, for instance, at a finitetemperature. So, in general, p q S A = p q S B and p q W A = p q W B . We have to consider a subclass of nonholonomic deformations when conditions transform into equalities forrespective relativistic flow evolution scenarios and associated thermodynamic and QM systems.
Subadditivity:
For disjoint subsystems A and B , there are satisfied the conditions of subadditivity p q S A∪B ≤ p q S A + p q S B and | p q S A − p q S B | ≤ p q S A∪B . (30)The second equation transforms into the triangle inequality [50]. In the quasi-classical limit, we obtainsimilar inequalities for the thermodynamic entropy p S (13). We claim that similar conditions hold for theW-entropy p W (11). They can be computed as quantum perturbations in a QFT associated to a bipartiteQGIF model p q W A∪B ≤ p q W A + p q W B and | p q W A − p q W B | ≤ p q W A∪B . Such flow evolution and QM scenarios are elaborated for mixed geometric and quantum probabilistic infor-mation flows. 17 trong subadditivity:
Considering three disjointed QGIF subsystems A , B and C and certain conditionsof convexity of a function built from respective density matrix and unitarity of systems [51, 52, 7, 6], onehold the following inequalities of strong subbadditivity : p q S A∪B∪C + p q S B ≤ p q S A∪B + p q S B ∪C and p q S A + p q S C ≤ p q S A∪B + p q S B ∪C . From these conditions, the conditions of subadditivity (30) can be derived as particular cases. Along causalcurves on respective cotangent Lorentz manifolds, we can prove similar formulas for the W-entropy and smallquantum perturbations p q W A∪B∪C + p q W B ≤ p q W A∪B + p q W B ∪C and p q W A + p q W C ≤ p q W A∪B + p q W B ∪C . We claim such properties for respective QGIFs. They play vital roles in the entropic proofs of the so-called c - F -theorems for renormalization group flows in QFT, see review of results in section VIII of [6]. In ourapproach, we elaborate on a different geometric formalism with nonholonomic flow evolution and respectiveapplications in quantum information theory. There are several measures of quantum entanglement which are determined by geometric and thermody-namic values for QGIFs. We begin with the concept of relative entropy in geometric information theories. p S ( p ρ A q p σ A ) = T r H B [ p ρ A (log p ρ A − log p σ A )] , (31)where p S ( p ρ A q p ρ A ) = 0 . This value is a measure of "distance" between two QGIFs with a norm || p ρ A || = tr ( q ( p ρ A )( p ρ †A )) . For thermodynamical GIF systems, it transforms into the conditional entropy (16). Itwas introduced and studied for standard densiti matrices in QM and information theory, respectively, in[53] and [54, 55], see reviews [1, 7, 6]. In straightforward form, we can check that there are satisfied certainimportant properties and inequalities.
Two QGIF systems are characterized by formulas and conditions:1. for tensor products of density matrices, p S ( p ρ A ⊗ p ρ A q p σ A ⊗ p σ A ) = p S ( p ρ A q p σ A ) + p S ( p ρ A q p σ A );
2. positivity: p S ( p ρ A q p σ A ) ≥ || p ρ A − p σ A || , i.e. p S ( p ρ A q p σ A ) ≥
3. monotonicity: p S ( p ρ A q p σ A ) ≥ p S ( tr s p ρ A | tr s p σ A ) , where tr s is the trase for a subsystem of A . Using above positivity formula and the Schwarz inequality || X || ≥ tr ( XY ) / || X || , we obtain that p S ( p ρ A q p σ A ) ≥
12 ( hOi ρ − hOi σ ) ||O|| for any expectation value hOi ρ of an operator O computed with the density matrix p ρ A , see formulas (19).The relative entropy p S ( p ρ A q p σ A ) (31) can be related to the entaglement entropy p q S ( p ρ A ) (24) usingformula p S ( p ρ A q A /k A ) = log k A − p q S ( p ρ A ) , (32)where A is the k A × k A unit matrix for a k A -dimensional Hilbert space associated to the region A . Aboveproperties can be re-defined by the entanglement entropy p q S , see similar formulas for QGIFs in Hamiltonmechancal variables in [29]. 18 hree QGIF systems: Let us denote by p ρ A∪B∪C the density matrix of three QGIFs subsystems
A ∪ B ∪ C and, for instance, p ρ A∪B for its restriction on
A ∪ B and p ρ B for its restriction on B . Usingthe formula for computing traces of reduced density matrices, tr A∪B∪C [ p ρ A∪B∪C ( O A∪B ⊗ C /k C )] = tr A∪B ( p ρ A∪B O A∪B ) we prove such identities p S ( p ρ A∪B∪C q A∪B∪C /k A∪B∪C ) = p S ( p ρ A∪B q A∪B /k A∪B ) + p S ( p ρ A∪B∪C q p ρ A∪B ⊗ C /k C ) , p S ( p ρ B∪C q B∪C /k B∪C ) = p S ( p ρ B q B /k B ) + p S ( p ρ B∪C q p ρ B ⊗ C /k C ); and inequalities p S ( p ρ A∪B∪C q p ρ A∪B ⊗ C /k C ) ≥ p S ( p ρ B∪C q p ρ B ⊗ C /k C ) , p S ( p ρ A∪B∪C q A∪B∪C /k A∪B∪C ) + p S ( p ρ B q B /k B ) ≥ p S ( p ρ A∪B q A∪B /k A∪B ) + p S ( p ρ B∪C q B∪C /k B∪C ) . These formulas can be re-written (after corresponding applications of the rule (32)) for the entanglemententropies p q S and Hamilton mechanical variables with "tilde" [29]. The correlation between two QGIF systems A and B (it can be involved also a third system C ) ischaracterized by the mutual information p J ( A , B ) and respective inequalities which follow from aboveformulas for relative entropy, p J ( A , B ) := p S A + p S B − p S A∪B ≥ and p J ( A , B ∪ C ) ≤ p J ( A , B ) . The mutual information is related to the relative entropy following formula p J ( A , B ) = p S ( p ρ A∪B q p ρ A ⊗ p ρ B ) , (33)which allows to consider similar concepts and inequalities for the entanglement of QGIF systems: p q J ( A , B ) := p q S A + p q S B − p q S A∪B ≥ , p q J ( A , B ∪ C ) ≤ p q J ( A , B ) , for p q J ( A , B ) = p q S ( p ρ A∪B q p ρ A ⊗ p ρ B ) . In the classical variant of GIFs, one hold similar formulas for GIFs and associated thermodynamic modelswith statistical density p ρ ( β, p E , p g ) (15). For relativistic geometric flows, we claim that similar propertieshold for the constructions using the W-entropy. In particular, this can be proven for causal configurationsin nonholonomic Hamilton variables [29].The mutual information between two QGIFs shows how much for an union A ∪ B the density matrix p ρ A∪B differs from a separable state p ρ A ⊗ p ρ B . Quantum correlations entangle even spacetime disconnectedregions of the phase spacetime under geometric flow evolution. For bounded operators O A and O B undergeometric evolution in respective regions, one holds true (the proof is similar to that in [56]) the inequality p J ( A , B ) ≥
12 ( hO A O B i − hO A ihO B i ) ||O A || ||O B || . Such formulas can be proven for associated thermodynamic systems to classical GIFs using the statisticaldensity if, for instance, A and B are certain subsystems of phase spaces and respective geometric flows.19 .2.4 The Rényi entropy for QGIFs We can introduce another type of parametric entropy which provides us more information about theeigenvalues of reduced entropy matrices thant the entanglement entropy. This is the Rényi entropy [57]which is important for computing the entanglement entropy of QFTs using the replica method, see sectionIV of [6]. Such constructions are possible in QGIF theory because the thermodynamic generating function p Z [ p g ( τ )] (12) and related statistical density p ρ ( β, p E , p g ) (15) can be used for defining p σ A (20) as aprobability distribution. Replica method and G. Perelman’s thermodynamica model:
Let us consider an integer r called asthe replica parameter and introduce the Rényi entropy p r S ( A ) := 11 − r log[ tr A ( p ρ A ) r ] (34)for a QGIF system determined by a density matrix p ρ A . We use the symbol r for the replica parameter(and not n as in the typical works in information theory) because the symbol n is used in our works for thedimension of base manifolds. To elaborate a computational formalism one considers an analytic continuationof r to a real number which allows us to define the limit p q S ( p ρ A ) = lim r → p r S ( A ) , with the normalization tr A ( p ρ A ) for r → , when the Rényi entropy (34) reduces to the entanglement entropy (24).There are satisfied certain important inequalities for derivatives on replica parameter, ∂ r , of the Rényientropy p r S (proofs are similar to [58]): ∂ r ( p r S ) ≤ , (35) ∂ r (cid:18) r − r p r S (cid:19) ≥ , ∂ r [( r − p r S ] ≥ , ∂ rr [( r − p r S ) ≤ . These formulas have usual thermodynamical interpretations for a system with a modular Hamiltonian H A and effective statistical density p ρ A := e − πH A . Considering β r = 2 πr as the inverse temperature, weintroduce the effective "thermal" statistical generation (partition) function, p r Z ( β r ) := tr A ( p ρ A ) r = tr A ( e − β r H A ) . similarly to p Z [ p g ( τ )] (12). In analogy to the thermodynamical model for geometric flows (13), we computeby canonical relations such statistical mechanics values p r E ( β r ) := − ∂ β r log[ p r Z ( β r )] ≥ , for. the modular energy ; p r ˘ S ( β r ) := (1 − β r ∂ β r ) log[ p r Z ( β r )] ≥ , for. the modular entropy ; p r C ( β r ) := β r ∂ β r log[ p r Z ( β r )] ≥ , for. the modular capacity . These inequalities are equivalent to the second line in (35) and characterize the stability if GIFs as a thermalsystem with replica parameter regarded as the inverse temperature for a respective modular Hamiltonian.Such replica criteria of stability were not considered in the original works on Ricci flows [19, 30, 31, 32]. Theydefine a new direction for the theory of geometric flows and applications in modern physics with respectivegenrealizations for nonholonomic structures. [40, 41, 33, 34, 35, 36, 37, 17, 18, 29, 38, 39].We note that the constructions with the modular entropy can be transformed into models derived withthe Rényi entropy and inversely. Such transforms can be performed using formulas p r ˘ S := r ∂ r (cid:18) r − r p r S (cid:19) and, inversely, p r S = rr − Z r dr ′ p r ′ ˘ S ( r ′ ) . We use the symbol r for the replica parameter (and not n as in the typical works in information theory) because the symbol n is used in our works for the dimension of base manifolds. Relative Rényi entropy for QGIFs:
The concept of relative entropy p S ( p ρ A q p σ A ) (31) can be extendedto that of relative Rényi entropy [59, 60] (for a review, see section II.E.3b in [6]). For a system QGIFs withtwo density matrices p ρ A and p σ A , we introduce p r S ( p ρ A q p σ A ) = 1 r − h tr (cid:16) ( p σ A ) (1 − r ) / r p ρ A ( p σ A ) (1 − r ) / r (cid:17) r i , for r ∈ (0 , ∪ (1 , ∞ ); (36)or p S ( p ρ A q p σ A ) = p S ( p ρ A q p σ A ) and p ∞ S ( p ρ A q p σ A ) = log || ( p σ A ) − / p ρ A ( p σ A ) − / || ∞ . Such definitions allow us to prove certain monotonic properties, p r S ( p ρ A q p σ A ) ≥ p r S ( tr s p ρ A | tr s p σ A ) and ∂ r [ p r S ( p ρ A q p σ A )] ≥ , and to reduce the relative Rényi entropy to the Rényi entropy using a formula similar to (32), p r S ( p ρ A q A /k A ) = log k A − p r S ( A ) . Nevertheless, the values (36) do not allow a naive generalization of the concept of mutual informationand interpretation as an entanglement measure of quantum information because of possible negative valuesof relative Rényi entropy for r = 1 [61]. This problem is solved by the r -Rényi mutual information [62], p r J ( A , B ) := min p σ B p r S ( p ρ A∪B q p ρ A ⊗ p σ B ) ≥ , when the minimum is taken over all p σ B . This formula reduced to the mutual information (33) for r = 1 . Inresult, we can elaborate a self-consistent geometric-information thermodynamic theory for QGIFs. This ispossible if the statistical density p ρ ( β, p E , p g ) (15) is used for defining p σ A (20) as a probability distributionand respective von Neumann density matrix formulation of the quantum models. It is not clear at presentif a version of relative Rényi entropy can be elaborated for the W-entropy. The geometric flows of Riemannian metrics can be characterized by G. Perelman’s W-entropy and associ-ated statistical thermodynamic model with respective mean energy, mean entropy and fluctuation parameter[19]. Such constructions can be generalized for nonholonomic geometric flows (subjected to certain non-integrable, i.e. anholonomic, equivalently, nonholonomic conditions) with generalized entropy type function-als and related locally anisotropc diffusion, kinetic and thermodynamic theories [33, 44, 36]. In result, we canelaborate on advanced geometric methods for modeling relativistic geometric flows of classical and quantummechanical systems, and modified commutative and noncommutative/ supersymmetric gravity theories etc.[36, 37, 35].A series of our recent works, see [17, 29] and refereces therein, is devoted to formulation and applicationson the theory of geometric information flows, GIFs, and quantum information flows, QGIFs. In such ap-proaches, the geometric thermodynamic models involve G. Perelman like entropic constructions [18] whichare more general than those elaborated using the Bekenstein-Hawking surface-area entropy and respectiveholographic, dual CFT-gauge theory generalizations etc. [63, 64, 66, 67, 68, 69]. New classes of genericoff-diagonal solutions (various locally anisotropic cosmological ones, generalized black hole metrics) withthe coefficients of metrics and generalized connections depending, in principle, on all spacetime and possi-ble phase space coordinates can be constructed [39, 28] in general relativity and modified gravity theories.21uch new classes of exact and parametric solutions, and related quantized models, are characterized by G.Perelman entropies and do not have Bekenstein-Hawking analogs.In this article, we have focused on developing the notion of entanglement for quantum mechanical, QM,and geometric thermodynamic models derived for QGIFs. This specific problem is of utmost importancewithin vast domains of studies of properties of entanglement entropy of general relativistic quantum systemsand, for instance, new types of QGIF teleportation, geometric flow testing, and encoding classical mechancalflow information in quantum states. In addition to the results of [40, 41] formulated for nonholonomicLagrange and Hamilton variables, we elaborated such constructions for covariant classical and quantummechanical systems and explicit applications in quantum information theory.Finally, we note that important questions connected to entanglement of QGIF and modified gravitytheories still remain as open challenges and promising research directions in modern geometric classical andquantum mechanics, thermodynamics, and modified gravity, see [28, 17, 18, 29].
Acknowledgments:
This research develops former programs partially supported by IDEI, PN-II-ID-PCE-2011-3-0256, CERN 2012-2014, DAAD-2015, QGR 2016-2017. S. V. is grateful to D. Singleton, S.Rajpoot and P. Stavrinos for collaboration and supporting his research on geometric methods in physics.
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