otherfnsymbols\Bat§▲§

affil0affil0affiliationtext: Department of Physics, Santa Clara University,500
El Camino Real, Santa Clara, CA 95053, United States

# Einstein’s Equations and the pseudo–Entropy of pseudo–Riemannian Information
Manifolds

Hassan Alshal<EMAIL_ADDRESS>
###### Abstract

Motivated by the corrected form of the entropy–area law, and with the help of
von Neumann entropy of quantum matter, we construct an emergent spacetime by
the virtue of the geometric language of statistical information manifolds. We
discuss the link between Wald and Jacobson approaches of thermodynamic/gravity
correspondence and Fisher pseudo–Riemannian metric of information manifold. We
derive in detail Einstein’s field equations in statistical information
geometric forms. This results in finding a quantum origin of a positive
cosmological constant that is founded on Fisher metric. This cosmological
constant resembles those found in Lovelock’s theories in a de Sitter
background as a result of using the complex extension of spacetime and the
Gaussian exponential families of probability distributions, and we find a time
varying dynamical gravitational constant as a function of Fisher metric
together with the corresponding Ryu–Takayanagi formula of such system.
Consequently, we obtain a dynamical equation for the entropy in information
manifold using Liouville–von Neumann equation from the Hamiltonian of the
system. This Hamiltonian is suggested to be non–Hermitian, which corroborates
the approaches that relate non–unitary conformal field theories to information
manifolds. This provides some insights on resolving “the problem of time”.

###### Contents

  1. 1 Introduction
  2. 2 Information and Spacetime Thermodynamics
  3. 3 Entropy and Riemannian Geometry
    1. 3.1 Vector space construction
    2. 3.2 Density manifold
    3. 3.3 Manifold metric
    4. 3.4 The reduced spatial metric and the Euclidean structure of the space of observables
    5. 3.5 Fisher metric and Kullback-Leibler divergence
    6. 3.6 Hessian structure and Einstein tensor
    7. 3.7 Obtaining a pseudo-Riemannian information manifold
  4. 4 Entropy of the Information Manifold
  5. 5 Discussions and Conclusions
  6. 6 Appendix

##  1 Introduction

One of the main challenges in physics is to find a fundamental dynamics
between geometry/gravity and quantum matter. And many approaches such as
string theory, gravity/field correspondence, and loop quantum gravity try to
tackle this problem in different ways and frameworks; check [1] for detailed
review. In this paper, we approach the problem from the perspective of
information manifolds and entropy. In the last few years, information geometry
has earned a great interest in fields like machine learning [2] and deep
learning in physics [3]. The information manifold and entropy concepts,
particularly the _relative entropy_ , are extremely useful in understanding
many physical patterns that include, but not limited to, quantum computers
[4], chemistry[5], biological systems [6] and even economy [7]. The
entropy–area law, corrected and generalized by the outside von Neumann entropy
[8, 9, 10], is introduced to the information geometry in order to check that
geometry obeys the second law of thermodynamics and preserves information
[11]. Our proposal to approach the gravity/quantum problem arises intuitively
from looking at the information paradox in black hole physics from the
entanglement entropy perspective [12, 13] along with the formally established
the second law of thermodynamics for black holes Noetherian charges [14, 15,
16] and the quantum origin of spacetime and Einstein equations [17].

The outline of this work is organized as follows. Right after this passage and
within the introduction section, we summarize the reasons behind correlating
the holographic principle with relevant entropy, _coarse–grained entropy_ ,
Kullback–Leibler (KL) divergence, and Fisher information metric. Then in
section (2), we investigate the corrected entropy–area law by von Neumann
entropy in order to both satisfy the second law of thermodynamics and preserve
information. We compute a form of corrected entropy–area law in the light of
Liouville–von Neumann equation so that we get a dynamical relation between the
quantum Hamiltonian and the time variation of both entropy and area. Later in
the same section, we briefly review the gravity/thermodynamics correspondence,
developed by Jacobson [17] and by Wald [14], since we will use that later in
the following sections. We comment on the relation between the quantum origin
of spacetime and the dynamical equation of expansion rate. In section (3), we
discuss the importance of von Neumann entropy and its maximized coarse–grained
version [18], and we provide a detailed study of the geometric representation
of the density matrix related to both entropies. We cover the essential
structures of the information geometry, and we apply the
thermodynamics/geometry correspondence to derive an information geometry form
of Einstein field equation. The Fisher information metric will be introduced
to measure how far the cumulant probability density functions are away from
each others after being varied with respect to the microscopic variables. In
other words, the Fisher metric tells us about the quota of information the
microscopic variables carry in the statistical manifolds [19, 20]. Thus, we
study the correlations between variables using the cumulant probability
density functions as good candidates to count on, instead of the moments of
distributions, while developing the Fisher metric. For example, think of the
mean square error as a second moment of the error. When it is differentiated
with respect to the microscopic value, the varied cumulant probability density
measures the level of proficiency in the model–data fitting.111In the
statistical sciences jargon it is called score or informant of the likelihood
function [21]. This also appears in KL divergence that measures the difference
between probability distributions. The last concept plays a fundamental role
in relating Fisher metric to Shannon/von Neumann entropy, which is what we
show later. And that concept can be seen mathematically as a second derivative
with respect to the microscopic variables222In the statistical sciences jargon
it is called observed information, and is equal to the mean square error of
the same second derivative applied on the cumulant probability distribution.
acting on the KL divergence, which is nothing but the inverse of the
probability distribution of Shannon/von Neumann entropy. Such structure is
very similar to the mathematical definitions of curvatures in Riemannian
manifolds, which is addressed also in the same section. Thus, Fisher metric is
in fact a metric from which the curvature structure in the information
manifold is obtained [22] once proved to be endowed with other properties of
Riemannian manifolds. This will lead to reformulate Einstein field equations
in the corresponding information manifold. Additionally, a positive
cosmological term in the Einstein equations is obtained in the information
manifold. Later, we relate the components of the field equations to commulant
functions and get a more detailed informatic description to the gravitational
constant. In section (4), we introduce a dynamical equation of entropy in the
information manifold using only quantum information geometry without using any
classical components. It is a new combination of von Neumann master equation,
the von Neumann entropy, and the gravitational entropy formula, and its
generalization to statistical manifolds. Then, we use the RT formula for the
statistical manifold to describe its corresponding _pseudo–entropy_ [23, 24].
In section (5), we discuss and comment on the findings.

The so–far achieved developments in entropic information theory have led to
conceptually rich ideas like entanglement entropy [25]. This line of thoughts
can be traced back to the discovered relation between the area and the black
hole entropy in the Bekenstein–Hawking [9, 26]. The holographic principle,
developed by ’t Hooft [27] and Susskind [28], suggests finding a
correspondence between the $3D$ volume and the $2D$ area, which leads to the
known gauge/gravity correspondence, or the AdS/CFT as a quantum field theory
with local degrees of freedom [29], and the sufficiently described entropy at
the _microscopic level_ in contrary to Bekenstein–Hawking law. Thus, we are
allowed to address the entanglement entropy [30, 31], which describes the
quantum information load in the quantum states, as a stored information
encoded on the geometric features of the space [32]. But since different
surface geometries are indeed associated with different entanglement
entropies, the area in Bekenstein–Hawking law is suggested to be replaced by
another area law for the extremal surfaces, known as Ryu–Takayanagi (RT)
surfaces in holography models [12, 13], which provides a clue to rethink of
the entropy built of microscopic variables as the fundamental underpinnings of
the spacetime classical geometry upon reintroducing RT surfaces to Wald’s
formula for the entropy of black holes [33]. We can exploit RT law as the
commoner understanding of the law allows using it to describe the von Neumann
entropy of gravity–coupled quantum systems as a _pseudo–entropy_ , controlled
by the system’s microscopic variables, in dS space [23, 34, 24, 35] and in AdS
space [36, 37, 38, 39]. Pseudo–entropy is a property for holographic systems
obtained from generalizing the entanglement entropy using the transition
matrix of two different states living in a black hole/environment system. In
contrary to the usual definition of entropy, pseudo–entropy is allowed to take
negative or complex values; more on its properties can be found int Ref. [36,
40]. Aside from our approach, it is worth referring to the yet–to–be–revealed
relation between pseudo–entropy and the so–called _tensor network_ [41]. There
are numerous statistical gravitational tensor network models based on RT
entanglement approach [42, 43, 44, 45]. It is argued that semiclassical
coarse–graining of holographic states, as realized in tensor networks, result
in a flow in spacetimes approaching RT surface [46]. Moreover, random tensor
network is proposed as a statistical gravitational model of a massive scalar
field on a thermally fluctuating background using neural network algorithm,
which recovers the background geometry fluctuation from multi-region RT
entanglement entropy [47].

The complicated relations among the microscopic variables, together with the
difficulties associated with measuring those variables, are the motives behind
why it is always more convenient to express the statistical phenomena
corresponding to the microscopic variables in terms of the stochastic
variables that are ruled by more fundamental and relatively easier–to–measure
laws. Such reductionism in the description automatically will lead to select
some variables to be relevant and other microscopic variables to be irrelevant
[20]. Relevant variables are all parameterized as functions in time or any
other affine parameter. We can think of the relation between the microscopic
and macroscopic variables like the relation between the speed of gas molecules
and the temperature of the whole sample, both can be used to describe
different types of thermodynamical energies. These microscopic relevant
variables should be averaged, using the relevant density function, such that
they define the macroscopic or the stochastic variables. And the averaging
process could be done using any density matrix as we discuss in details in
subsection (3.1). Additionally, any two different densities, constructed from
those relevant variables, are favored over each other according to which of
them is capable of defining maximum entropy that corresponds to least amount
of data loss due to the unfavorable unavoidable effects coming from irrelevant
information [48]. Such density is known as the canonical coarse–grained
density, and its corresponding entropy is known as the relevant entropy; it is
the candidate entropy to reformulate the RT law of statistical manifolds as
discussed in section (3).

Due to the mathematical difficulties a person might encounter while trying to
find the exact states and their corresponding densities, it is suggested to
replace the von Neumann general relevant entropy with other more specified
entropies: the _coarse–grained relative entropy_ , where this entropy is
always larger or equal to the fine–grained von Neumann entropy, and we comment
on that in section (3.1). Yet we emphasize that the coarse–grained entropy is
“lossy but true” entropy [49] as it depends on the macroscopic variables of
the system, see Ref. [50] for more on that. Also, more on the relation between
the fine–grained entropy and von Neumann entropy can be found in Ref. [18]. It
is argued that one can coarse–grain any type of quantum entropy, such as
entanglement entropy, using observational entropy [51]. The technique of
coarse–graining obtained with help of the projectors acting on Hilbert space
has been previously used in information geometry333Balian et al. [20] did not
restrict the physical observables to be the Hilbert space projectors but
Safránek _et al._ did [51]. [20]. It is worth noting that there are some
generalized versions of quantum entropies, such as Rényi and Tsallis
entropies, that do not manifestly take the form of the von Neumann entropy.
Yet they can be treated as von Neumann entropy upon applying the proper limits
[52]. Notice also that observational entropy is a quantum analogue of the
classical Boltzmann entropy. More importantly, observational entropy shows the
measurement limitations when one tries to get more precise information even if
the density state is more precisely known. Thus, the process of
coarse–graining is inevitable because even pure states span over more than one
macrostate in the phase space due to superposition. Yet, observational entropy
is bounded from below by von Neumann entropy, and equals to the later if the
former satisfies the coarse–graining conditions in Ref. [51], i.e. after
several consecutive coarse–graining processes, one can end up having a
fine–grained entropy. Generally for finite–dimensional systems, the
observational entropy can be expressed as a relative entropy. Statistically
speaking, observational entropy can take the form of KL divergence, which is a
type of relative entropy, from which we obtain Fisher information metric in
information manifolds.

##  2 Information and Spacetime Thermodynamics

In a stationary black hole with perfect fluid environment, the black hole
entropy follows Bekenstein-Hawking formula, while the rest of the entropy is
allocated to the perfect fluid [8, 9, 10, 53]. However, the entropy can be
generalized to include the interactions, especially the scattering processes,
of the external entangled quanta and Hawking radiation [30, 18, 11]. In order
to obtain an entropy–area law that respects the second law of thermodynamics
and preserves information, we need the full entropy of a black hole to contain
both the entropy of what is inside the horizon and the entropy of the quantum
matter outside the horizon [11, 54, 55]. This means the generalized entropy
law [8, 9, 10] would take the form

$S_{\text{gen}}=\frac{A_{\text{H}}}{4G\hbar}+S_{\text{matter}},$ (1)

where $S_{\text{gen}}$ is the generalized entropy of the black hole,
$A_{\text{H}}$ is the area of black hole horizon that could be coarse–grained,
and $S_{\text{matter}}$ is the von Neumann entanglement entropy of the matter
outside the black hole. The constants are Planck constant $\hbar$ and Newton’s
gravitational constant $G$.
On one side, von Neumann entropy for a quantum-mechanical system described by
a density matrix $\rho$ is given by

$S_{\text{matter}}=-\operatorname{Tr}(\rho\ln\rho).$ (2)

On the other side, the time-evolution equation of the density matrix $\rho$ is
given by Liouville–von Neumann equation [56],

$\frac{d\rho}{dt}=\frac{1}{i\hbar}[H,\rho],$ (3)

where $H$ is the Hamiltonian of the considered quantum system. Since the trace
operator commutes with the differential time operator, we can in principle
write a time evolution equation for the von Neumann entropy. For that purpose,
we use the straightforward mathematical trick

$\frac{d\rho}{dt}=\frac{d}{dt}(\rho\ln\rho)-\frac{d\rho}{dt}\ln\rho.$ (4)

Using Eq. (4) in Eq. (3), we get the quantum time evolution equation as
follows

$i\hbar\frac{d}{dt}(\rho\ln\rho)-i\hbar\frac{d\rho}{dt}\ln\rho=[H,\rho].$ (5)

We take the trace of both sides and use the fact that trace operator commutes
with the differential time operator. Then, we substitute Eq. (2) in Eq. (5) to
get

$-i\hbar~{}\frac{d}{dt}S_{\text{matter}}=\operatorname{Tr}\bigg{[}i\hbar\frac{d\rho}{dt}\ln\rho+[H,\rho]\bigg{]}.$
(6)

Thus, an equation for the time evolution of von Neumann entropy is obtained.
For a black hole system, the von Neumann entropy satisfies the second law of
thermodynamics only through introducing the fine–grained entropy as we
mentioned in the introduction. The coarse–grained entropy is obtained by
tracing every possible density matrix using the environment observables [18].
Then, the fined–grained entropy is maximized to become coarse–grained one.
Thus, the semiclassical approximation guarantees that the von Neumann entropy
is that of the quanta outside the black hole in curved spacetime.
Consequently, we use Eq. (1) to rewrite the time evolution of entropy as
follows

$-i\hbar\frac{dS_{\text{gen}}}{dt}+i\frac{1}{4G}\frac{dA_{\text{H}}}{dt}=\operatorname{Tr}\bigg{[}i\hbar\frac{d\rho}{dt}\ln\rho+[H,\rho]\bigg{]}.$
(7)

The last equation introduces a a relation between black hole full entropy,
black hole horizon area, and density matrix of quantum states. As we observe
here, we did not make any additional assumptions to get to Eq. (7). It appears
that Eq. (7) is a quantum/semi–classical form of entropy-area law for the
black hole. It introduces a relation between geometry (Area) and quantum
matter (Density Matrix). We will reconsider this relation after we connect
entropy as a macroscopic quantum quantity to the area as a geometric quantity.
But before that, it is worth noting that the two concepts are related in
general within the non–dissipative systems, i.e. systems with
$dS_{\text{gen}}/dt=0$. This assumption is valid as $S_{\text{gen}}$ is the
total corrected entropy of the black hole, and when $A_{\text{H}}$ and the RT
extremal surface $A_{\text{ext.RT}}$ coincide or an “island” is realized [13,
57]. Such extremal surface realization is related to the existence of flat
plateau in the corresponding Page curve [58, 59] after minimizing the
generalized entropy of the extremal surfaces.444Generalized extremal entropy:
$S_{\text{gen.RT}}=min\left[A_{\text{ext.RT}}/4\hbar G+S_{matter}\right]$,
which looks similar to Eq. (1) [18]. Minimizing the entropy recovers the Page
curve as required and guarantees that the fine grained entropy will not exceed
the coarse grained one as emphasized in the next section. Therefore, Eq. (7)
becomes

$i\frac{1}{4G}\frac{dA_{\text{H}}}{dt}=\operatorname{Tr}\bigg{[}i\hbar\frac{d\rho}{dt}\ln\rho+[H,\rho]\bigg{]}.$
(8)

When $A_{\text{H}}$ and $A_{\text{ext.RT}}$ coincide, the von Neumann term
becomes negligible, and the area term dominates the entropy, hence the
importance of Eq. (8). Upon solving the previous equation, we have two
possible explanations. Either the area would have the fine–grained entropic
definition, which is expected as the horizon contains the information of the
entangled particles that fall inside it [60], or the fine–grained quantum part
of the black hole entropy would have a coarse–grained geometric meaning based
on the microscopic variables of the density, the entropy, and the manifold.
Before we discuss the second meaning, which is what we do in section (4), we
emphasize that all previous equations are derived assuming that Liouvillian
mechanics stems from $d\rho/dt=0$. Also, the Hamilton–Jacobi equation shows
that $H=-~{}\partial\mathcal{A}/\partial t$, where $\mathcal{A}$ is the
action. As we will see later, this Hamiltonian, and consequently the action,
depends on a cumulant function that will be treated as a scalar field to
obtain Einstein equations in the information manifold. Wald [14] noticed that
all what one needs to do is to express the entropy as a function of the
density of the state then applies Liouvillian mechanics to solve the
Hamilton–Jacobi equation in order to generate the action that will be
extremized to get the conserved quantities together with Euler–Lagrange
equations. This is why we swiftly review the gravity/thermodynamics
correspondence developed by Jacobson [61, 17] based on Wald approach [14].

Assuming Rindler frame of references, Jacobson found that the Einstein
equations can be obtained from the entropy–horizon area relation together with
the laws of thermodynamics [17]. According to Unruh’s radiation [62], the
radiation temperature detected by a Rindler observer is directly proportional
to the uniform acceleration $``a"$ of that observer. Applying the equivalence
principle to Rindler transformations [63, 64] guarantees that the vacuum state
in the generic spacetime is locally more or less the same as the Minkowski
vacuum, i.e. the time involved in defining the quantum operators in vacuum,
measured by the accelerated observer, should be close to that measured by an
observed in a flat spacetime [65]. Thus, each point in the spacetime has its
own local Rindler horizons, both past and future, with Killing fields in the
null directions of the horizons. The relations between heat flux and the black
hole hairs, i.e. the mass and the angular momentum,555Electromagnetic energy
is better represented separately inside the density matrix $\rho$. are
discussed in [53, 66, 14, 67] through the Hamiltonian formulation of the first
law of thermodynamics. In brief, the heat flow of such system could be defined
to obey the averaged null energy condition along a time–like geodesic

$\delta Q=-a\int T_{\mu\nu}k^{\mu}k^{\nu}tdt\delta A,$ (9)

where $T_{\mu\nu}$ is the energy momentum tensor, $k^{\mu}$ is the null
vector, $\delta A=\sqrt{\gamma}dA$ is the differential element in the
congruence cross sectional area of the horizon, and $\sqrt{\gamma}$ is the
determinant of the induced metric of the spatial area element of the horizon
element $dA$. The last equation stems originally from Wald’s formula [61, 68].
Also, the last equation is well founded in flat spacetime and independent of
the Hamiltonian formulation of the first law of thermodynamics [69]. But the
Hamiltonian formulation can also be used to obtain Eq. (9) with careful
considerations upon applying it on the Rindler wedge, e.g., in the holographic
approach the considered Hamiltonian should be _modular_ [70, 71]. Now for the
null geodesic of $k^{\mu}$ that generates the horizon,666Higher order
contributions obtained from the shear terms should not be ignored in
nonequilibrium thermodynamical systems [65]. the Raychaudhuri equation gives

$\theta=-tR_{\mu\nu}k^{\mu}k^{\nu}.$ (10)

where the affine parameter $\theta$ is the rate of change of $\sqrt{\gamma}$
[72], i.e.

$\theta=\frac{1}{\sqrt{\gamma}}\frac{d}{dt}\left(\sqrt{\gamma}\right),$ (11)

which also describes the expansion of $\delta A$. The last two equations can
be combined to give

$\theta A\frac{d\theta}{dA}=-R_{\mu\nu}k^{\mu}k^{\nu}.$ (12)

As in [17], Eq. (9) and Eq. (10) yield the relation

$\frac{\delta Q}{\delta A}=\frac{a\hbar}{8\pi\ell^{2}_{P}},$ (13)

which demands that the Einstein equations become

$G_{\mu\nu}=\frac{8\pi\ell^{2}_{P}}{\hbar}T_{\mu\nu}~{},$ (14)

where the $G_{\mu\nu}$ is the Einstein tensor and $\ell_{P}$ is the Planck
length.

It is worth mentioning that another approach of understanding gravity as an
entropic force is suggested by Verlinde in [73, 74] but we will not review it
here. Rather, one might say that nothing new in this section. But Eq. (12) and
Eq. (13) entice us to consider the existence of a new relation between
$dS=\delta Q/T$ and $\delta\theta$ at the microscopic level of $\rho$
considering the von Neumann definition of entropy. Such $(S,\theta)$ relation
is indeed a dynamical relation, and we will study that relation in the context
of information geometry in section (4). But first we need to study the meaning
of the geometric quantities $G_{\mu\nu}$ and $\theta$ in the classical
spacetime using Fisher Riemannian metric of information manifold. In the next
section, we adopt the Fisher metric as it guarantees extending the geometric
quantities to any quantum theory. Moreover, the observational entropy is
realized as the KL divergence, which is another information quantity from
which we can obtain the Fisher metric.

##  3 Entropy and Riemannian Geometry

With help of Lie group thermodynamics in the framework of the covariant
formalization of geometrized thermodynamics [75, 76], we can expand the
analysis that resulted in Eq. (8) in the speculative direction of the
geometric interpretation of the fine–grained quantum part of the black hole
entropy. Balian _et al._ gauge theory of thermodynamics [20], as an extension
of Fisher Riemannian metric of information manifold, guarantees extending the
geometric interpretation to any quantum theory. Later, we introduce Balian _et
al._ Riemann metric of the density matrix space as the Hessian of the
fine–grained entropy. This metric stems directly from the endeavors of
Ruppeiner information metric [77]. The metric is situated as a canonical
function in between the space of states and the space of observables. This
involves Legendre transforms just like those in Liouvillian mechanics. We also
comment on the similarities between the Hessian structure in our approach and
that of gravity in the framework of superstrings [78].

###  3.1 Vector space construction

According to Balian _et al._ [20], it is useful to focus only on the space of
the density states as the density state is more suitable for information
gathering in comparison with observables themselves. But when it comes to the
density state space metric, we would rather use the fine–grained entropy to
define such metric. This choice is made as the density state could concede an
incomplete description for the information relative to the observables, i.e.
finding the average value of any observable $\hat{\mathcal{O}}$ using the
usual relation

$\langle\hat{\mathcal{O}}\rangle=\text{Tr}\left[\hat{\mathcal{O}}~{}\hat{\rho}\right]$
(15)

does not define a unique $\rho$ as there are different $\rho$’s that
sufficiently define this value. Notice that the fine–grained entropy described
in Eq. (2) has a global maximum as we infer from Fig. (1). Then, all densities
satisfying Eq. (15) give equivalent entropies, and hence, the loss in the
information of the observables due to the different $\rho$’s is irrelevant.
But the density $\rho_{0}$ corresponding to maximum entropy is more favorable
to describe the density space metric as, by construction, it has the minimum
information to calculate different types of observables. Even for the
irrelevant information, $\rho_{0}$ contains the least of them, and thus it
corresponds to the entropy that “ _quantifies our ignorance about the precise
quantum state of the system_ ” [18]. This is why the relevant von Neumann
entropy $S[\rho_{0}]$ is that of the macroscopic thermal phenomena such as the
thermodynamically gravitational quantities in the dissipative systems. To
elaborate the above discussion, we focus in Eq. (15) on the
$\hat{\rho}\hat{\mathcal{O}}$ components. The density
$\hat{\rho}_{\hat{\mathcal{O}}}$ is chosen to be the density proportional to
the eigenprojectors of $\hat{\mathcal{O}}$ in the corresponding sub–Hilbert
space. Therefore, all the off–diagonal elements in the general $\hat{\rho}$
will be disregarded, and the left information is stored in the
$\hat{\rho}_{\hat{\mathcal{O}}}$. Since $\hat{\rho_{0}}$ is the density that
has all the relevant information of all observables similar to
$\hat{\mathcal{O}}$, it is clear that $S[\rho_{0}]>S[\rho]$. Additionally,
$S[\rho]_{coarse}\geqslant S[\rho_{0}]$, where $S[\rho]_{coarse}$ is the
coarse–grained entropy that provides a measure to $S_{\text{gen}}$ and could
match fine–grained later when the previously mentioned inequality becomes a
direct equal relation between the two types of entropy [18].

Now we construct a geometric interpretation of the previously discussed
thoughts such that the “ _variation of the relevant entropy represents a
transfer of information between the relevant and the irrelevant variables_ ”
and “ _dissipation appears as a leakage of the relevant information_ ” [20].
With the help of the algebraic structure endowed in the observable space, the
geometric construction in the language of manifolds777In that sense, the
density matrix $\rho$ should be seen as a _pre–probability_ , see eq. (15.46)
and section (15.4) of Ref. [79]. would help us in explaining Eq. (7–8) in a
purely quantum information way. The observable $\hat{\mathcal{O}}$ has
eigenvectors $|\alpha^{k}\rangle$ in the space states, and therefore, it can
be written as

$\langle\hat{\mathcal{O}}\rangle=\sum\limits_{k}\langle\alpha^{k}|\rho|\alpha^{k}\rangle\text{Tr}\Big{[}|\alpha^{k}\rangle\langle\alpha^{k}|\mathcal{O}\Big{]}.$
(16)

To define the vector space of the observable $\hat{\mathcal{O}}$, we set the
components and the bases as

$\displaystyle\mathfrak{o}_{\mu}$
$\displaystyle:=\langle\alpha^{k}|\rho|\alpha^{k}\rangle,$ (17a)
$\displaystyle f^{\mu}$
$\displaystyle:=|\alpha^{k}\rangle\langle\alpha^{k}|\mathcal{O},$ (17b)

respectively as the components and the bases of the Liouville vector
$\vec{\mathcal{O}}=\mathfrak{o}_{\mu}f^{\mu}$.

As the set of all observables $\hat{\mathcal{O}}$ is characterized by the
$\rho$, and $\rho$ itself is an operator that can be written in terms of any
set of orthonormal bases $\\{|i\rangle\\}$. Then, we can rewrite Eq. (15–17)
such that we define the density operator as a vector $\vec{\rho}$ with
components

$\rho^{\mu}:=\langle f^{\mu}\rangle=\text{Tr}\left[f^{\mu}\rho\right].$ (18)

Then, when $\mathcal{O}$ becomes $\rho$, Eq. (15) becomes

$\overrightarrow{\langle\hat{\rho}\rangle}:=\vec{\rho}=e_{\mu}\rho^{\mu},$
(19)

where $\rho^{\mu}$ act as the components of the averaged observable
$\langle\hat{\rho}\rangle$ in its Liouville vector representation
$\overrightarrow{\langle\hat{\rho}\rangle}$, and the basis components for such
vector are defined as $e_{\mu}:=|i\rangle\langle i|$ that are dual to
$f^{\mu}$, i.e. $e_{\mu}\cdot f^{\nu}=\delta_{\mu}^{~{}\nu}$. In such
representation, Eq. (15) can be seen as a bilinear relation

$||\vec{\mathcal{O}}||=\langle\vec{\mathcal{O}},\vec{\rho}\rangle:=\mathfrak{o}_{\mu}\rho^{\mu}.$
(20)

To make the above construction more elaborate, let’s say that
$\hat{\mathcal{O}}$ commutes with the spatial observable $\hat{x}$ in a space
with Euclidean signature, i.e. the $\vec{\mathcal{O}}$ is a function in the
components of $\hat{x}$. Then, the density $\rho^{\mu}$ would correspond to
the averaged components $\langle x^{\mu}\rangle$. As the density space is made
of all the $\langle x^{\mu}\rangle$ including the irrelevant ones, this means
it is valid to embed the function $\vec{\mathcal{O}}$ as a hypersurface in the
space spanned by $\rho^{\mu}$ after being expressed in terms of the dual basis
of $\vec{\rho}$. This surface is called the surface of reduced states, and it
is extremized at $\rho_{0}$. At the same time, it is much easier to describe
the density in terms of the microscopic variables $\xi^{\mu}$ and the
stochastic variables $\mathbb{X}^{\mu}$, both as functions parameterized by
the spacetime observables $x^{\nu}$ or their averages, and we will do that
later in subsection (3.5). Speaking of irrelevant averaged observables,
relevant bases in are those of the density space needed for the calculation of
mean values of specified observable, e.g. the metric, and can be isolated from
the irrelevant bases. One can separate them using a special operator called
the _projection operator_ , we will encounter this operator in subsection
(3.4). Conceptually, the relevant components of the
$\hat{\rho}_{\hat{\mathcal{O}}}$ are those of $\rho_{0}$. But the time rate
change of the relevant parts is indeed a function in some relevant and some
irrelevant parts [80]. This is why we will upgrade Eq. (7–8) to become _master
equations_ as we will see in section (4). But for now, we just finished
setting the stage up for the debut of the density space. Next, we show this
space could be promoted to a density manifold.

Figure 1: Density manifold and the evolution of entropy surfaces away from
reduced state $\rho_{0}$.

###  3.2 Density manifold

In the density space, $\rho$ is guaranteed to be a function in time such that
Eq. (3) is satisfied. Geometrically this means that $\rho(t)$, as a point in
such space, evolves from the point $P(\rho_{0})$ on the surface $\Sigma_{P}$,
where $S[\rho]=S[\rho_{0}]$, to another point on the same $\Sigma_{P}$ with
$S[\rho]$ along some trajectory $\lambda$ [81]. This is defined as an
exponential function just like how we relate in Riemannian geometry a vector
to the points belonging to the trajectory that the vector is tangent to. Thus

$\exp_{P}:t\cdot\rho^{\mu}\to\lambda_{\rho^{\mu}}(t).$ (21)

As $t\in[0,1]$, then $\exp(1.\rho^{\mu})=\lambda_{\rho^{\mu}}(1)$ is the final
point $P(\rho)(t)$ at the trajectory. Meanwhile
$\exp(0.\rho^{\mu})=\lambda_{\rho^{\mu}}(0)$ is the initial point
$P(\rho_{0})$. The $\rho^{\mu}_{0}$ can take any direction. Therefore, its
$i^{th}$ component $\rho^{i}$ along the tangent of the trajectory is given by

$\rho^{i}_{0}\equiv\frac{d\lambda(t)}{dt}\bigg{|}_{t=0}~{}.$ (22)

Moreover, the $i^{th}$ component of $\rho^{i}_{0}$ is allowed to be in the
direction of any irrelevant basis $\rho^{\mu}$ as long as it is tangent to the
trajectory $\lambda$. This motivates considering the exponential function (21)
as a diffeomorphism between the neighborhood of point $P$, which belongs to
the state space, and the vector $\rho^{\mu}_{0}$, which belongs to the tangent
space $\text{T}_{P}\Sigma$ at point $P$. If we normalize $\rho^{i}_{0},\forall
i$, then the orthonormal basis $\\{\rho^{\mu}_{0}\\}$ of $\text{T}_{P}\Sigma$
provides the isomorphism

$E:\mathbb{R}^{n}\xrightarrow{\sim}\text{T}_{P}\Sigma,$ (23)

such that $E(\xi^{1},\cdots,\xi^{n})|_{P}=c_{i}\rho^{i}_{0}$.
The previous manifold–related definitions are valid regardless whether the
surface is extremized on not. Therefore, there exists charts $\psi$ on
manifold $\mathcal{M}$ containing all $\lambda$’s and $\text{T}_{Q}\Sigma,$
for all $Q\in\mathcal{M}$, such that

$\psi:=(\exp\cdot E)^{-1}:\mathcal{M}\to\mathbb{R}^{n}.$ (24)

Also, the exponential map of the density manifold acts similarly to how the
known the exponential map $\exp(tV)=\lambda_{V}(t)$ works between any Lie
group and its Lie algebra, where $V\in$ the algebra, $\lambda_{V}(t)\in$ the
group, and $t\in\mathbb{R}$. Then, there exists an analytic diffeomorphism in
a neighborhood $U$ of $V=0$ such that, for the coordinates
$\xi^{\mu}\in\mathbb{R}^{n}$ defined by the isomorphism $E$ in Eq. (23), we
find that $\exp(\xi_{i}V^{i})\in\exp(U)$. This would define the necessary
canonical chart.

Moreover, the entropy $S[\rho]$ never loses information as long as the initial
point $P(\rho_{0})$ evolves in a Hamiltonian trajectory. However, when we
disregard the irrelevant information, we do something similar to the shift and
lapse such that the entropy $S[\rho]=S[\rho_{0}]$ evolves in time through the
extemized and non-extremized surfaces, i.e. we can practice push-forward,
$f_{*}:\Sigma_{Q}\to\Sigma_{P}$, (or pull-back) transformations between
$\Sigma_{P}$ and $\Sigma_{Q}$, together with applying the rules of Lie
derivatives in order to relate the surfaces to each others. Now we are ready
to introduce a metric on this information manifold.

###  3.3 Manifold metric

We notice that the exponent function in Eq. (21) transfers $\rho^{\mu}$ to a
tangent space with same properties but at different point. Since the
logarithmic function is the opposite to the exponent, then, in light of Eq.
(20), we can safely say that $(\ln\rho)$ does the same job of $\rho$ except
that the domain here becomes the dual tangent space. This implies that
$S[\rho]$ acts as a bilinear map between the density $\rho$ and the
_information content_ $(-\ln\rho)$. Information content is a unit of
information regarding some random variable and follows Shannon’s definition of
self–information. If the logarithm is to the base 2, the information content
is called _bit_. And for the natural logarithm, the Shannon information
entropy coincides with the Gibbs entropy, which can be quantized to be the von
Neumann entropy, and information content is called _nit_. Information content
plays a fundamental role in constructing the Fisher metric of information
manifolds as we will see in subsection (3.5). As both $\rho$ and $(-\ln\rho)$
are unique geometric vectors, then $S[\rho]$ provides us with a linear
isomorphic relationship between the tangents and the dual tangent spaces, i.e.
the duality

$\rho\equiv\widetilde{(\ln\rho)}$ (25)

is legitimate. Therefore, there exists a map $\mathpzc{G}$ on the $\rho$ space
such that

$\displaystyle\mathpzc{G}$
$\displaystyle:(\ln\rho)_{\mu}\mapsto\mathpzc{G}_{\mu\nu}\rho^{\nu},$ (26a)
$\displaystyle\mathpzc{G}$
$\displaystyle:(\rho)^{\mu}\mapsto\mathpzc{G}^{\mu\nu}(\ln\rho)_{\nu}~{}.$
(26b)

The map $\mathpzc{G}$ is shown to be symmetric, real, and has positive
eigenvalues in the Liouville vector representation [20]. It is the best
function to play the role of metic in the $\rho$ space. Then, we can use Eq.
(26) to demonstrate the entropy as the bilinear function

$S\langle\rho,\ln\rho\rangle:=-\mathpzc{G}_{\mu\nu}(\rho(\xi))\rho^{\mu}\rho^{\nu}.$
(27)

The $\rho$ space is dense enough–satisfying the topological features of
manifolds–such that we can introduce the infinitesimal change $d\rho$.
Therefore, Eq. (27) can be redefined infinitesimally such that the second
differential in the entropy becomes the metric itself.888Such metric metric
can be considered a Ruppeiner information metric [77]. And the $\rho$ space
becomes eligible for a promotion to be a Riemannian manifold endowed with the
invariant distance

$-ds^{2}:=d^{2}S\langle\rho,\ln\rho\rangle=-\mathpzc{G}_{\mu\nu}(\rho(\xi))d\rho^{\mu}d\rho^{\nu},$
(28)

where the map (26) is explicitly defined as

$\mathpzc{G}(e_{\mu},e_{\nu}):=\mathpzc{G}_{\mu\nu}(\rho(\xi))=-\frac{d^{2}S}{d\rho^{\mu}d\rho^{\nu}}.$
(29)

We find Eq. (29) leads us to define the dual vector $(-\ln\rho)$ as

$\displaystyle-(\ln\rho)$ $\displaystyle:=\varrho_{\mu}f^{\mu},$ (30a)
$\displaystyle\varrho_{\mu}$
$\displaystyle:=(-\ln\rho)_{\mu}\equiv\frac{\partial S}{\partial\rho^{\mu}}.$
(30b)

Notice that if the density is defined as usual $\rho=\exp(-\beta H)/Z$, it
renders $\phi-\varrho=-\beta H$ as in Eq. (A17), where $Z$ is the partition
function, the energy $H$ is related to the entropy $S$ [15, 82] and is a
function in the cumulant partition function $\phi$ that we use later to define
the metric of the manifold. This notice encourages us to comment on the
Hamiltonian operator in such system and study its relation to Eq. (7–8). So we
introduce a _hermitian entanglement Hamiltonian_ to Eq. (7-8) as a
superoperator 999The prefix super has no Grassmann rings, i.e. has nothing to
do with Supersymmetry or graded algebra in general. $\mathcal{H}$ such that

$\displaystyle\mathcal{H}:=\mathcal{H}^{~{}\mu}_{\nu}e_{\mu}\otimes f^{\nu},$
(31a)
$\displaystyle\mathcal{H}^{~{}\nu}_{\mu}=\langle\mathcal{H}e_{\mu},f^{\nu}\rangle.$
(31b)

In order to double check that the Hamiltonian in the information manifold is
that of $\phi$, we follow Ref. [83]. Eq. (2–6) define the hermitian
entanglement Hamiltonian similar to those in Ref. [60]. Additionally, the full
Hilbert space of the black hole and the environment makes defines the state,
and consequently the density, as that of the black hole and environment
together [84]. Thus, we define the Hamiltonian as
$\mathcal{H}=\mathcal{H}_{o}+\mathcal{H}_{i}$, where $\mathcal{H}_{o}$ is the
non-interaction Hamiltonian–mainly that of the black hole and that of the
environment separately–and $\mathcal{H}_{i}$ is the interaction Hamiltonian
that _connects the black hole state to the environment throughout information
exchange_ [84]. Now, Eq. (3) does not restrict the Hamiltonian to be that of
the quantum states alone or their interactions. This means that the
entanglement entropy $S_{\text{matter}}$ has a piece that depends on
$\mathcal{H}_{i}$. And the entropy corresponding to $\mathcal{H}_{i}$, that is
responsible for the mutual information exchange between the black hole and the
environment, is another von Neumann entropy, i.e. it depends on $\rho$, as a
function of $\phi$, and other operators acting independently on the
environment and the black hole [85]. Consequently, The Hamiltonian in Eq.
(7–8) is expected to be $\mathcal{H}_{i}$ as a function of $\phi$. This result
will help us when we reconsider Eq. (7–8) in section (4). For non–hermitain
Hamiltonian, see Ref. [83].
If we set $\mathcal{O}=\mathcal{H}$, then Eq. (3), or the commutator in Eq.
(15), becomes the Liouville superoperator

$\frac{d\rho}{dt}=\mathscr{L}\rho=-i[\mathcal{H},\rho].$ (32)

Eq. (31) helps defining the components of $\mathscr{L}$ as

$\mathscr{L}_{\mu}^{~{}\nu}=-i\text{Tr}f^{\nu}[\mathcal{H},e_{\mu}]=-i\text{Tr}[f^{\nu},\mathcal{H}]e_{\mu}.$
(33)

Reintroducing the Liouville operator as a superoperator excavates its “super
power” such that it manifestly plays the role of the Lie derivatives on the
Riemannian $\rho$ manifold. The Jacobian of transformations between
$\Sigma_{P}$ and $\Sigma_{Q}$ is given by

$J:=\text{det}\left[\frac{\partial\rho(t+\delta
t)}{\partial\rho(t)}\right]=1-i\mathscr{L}\delta t$ (34)

or

$i\mathscr{L}_{\nu}^{~{}\mu}\rho^{\nu}:=\lim\limits_{\delta t\to
0}\frac{\rho^{\mu}(Q)(t+\delta t)-\rho^{\mu}_{0}(P)(t)}{\delta t}.$ (35)

It is obvious that, with the above geometric interpretation, we also can
introduce the evolution superoperator

$\mathscr{U}:=\exp[-i\mathscr{L}(\delta t)],$ (36)

which plays a role similar to that of the Lie groups, or their corresponding
Killing fields, over the usual Riemannian manifolds.
We now introduce the Legendre transformation

$\displaystyle\mathfrak{S}(\varrho_{\mu})$ $\displaystyle=$ $\displaystyle
S-\langle\ln(\rho)\rangle$ (37) $\displaystyle=$ $\displaystyle
S+\varrho_{\mu}\rho^{\mu}.$

This transform reintroduces the $\rho$ components to be

$\rho^{\mu}:=\frac{\partial\mathfrak{S}}{\partial\varrho_{\mu}}.$ (38)

Therefore, the metric in Eq.(29) can be contravariantized as

$\displaystyle\mathpzc{G}^{\mu\nu}(\rho(\xi))$ $\displaystyle:=$
$\displaystyle\frac{\partial^{2}\mathfrak{S}}{\partial\varrho_{\mu}\varrho_{\nu}}$
(39) $\displaystyle=$
$\displaystyle\frac{\partial\rho^{\kappa}}{\partial\varrho_{\mu}}\frac{\partial\rho^{\lambda}}{\partial\varrho_{\nu}}\frac{\partial^{2}\mathfrak{S}}{\partial\rho^{\kappa}\partial\rho^{\lambda}}.$

Consequently, Eq. (28) is transformed into

$d^{2}\mathfrak{S}=d^{2}S-\left[d^{2}\varrho_{\mu}\rho_{\mu}+2d\varrho_{\mu}d\rho_{\mu}+\varrho_{\mu}d^{2}\rho_{\mu}\right].$
(40)

The metricity $\nabla\mathpzc{G}=0$, or the parallel transport along geodesics
$\nabla_{\rho}\rho=0$, implies that there exists a set of connection
coefficients $\\{\\}$ on the $\rho$ manifold similar to the Levi–Civita
connections on the Einstein manifold. This means that both metric and
connections are related through

$\displaystyle\mathpzc{G}(\nabla_{e^{\lambda}}e_{\mu},e_{\nu})$
$\displaystyle:=\big{\\{}^{\;\kappa}_{\lambda\mu}\big{\\}}\mathpzc{G}_{\kappa\nu}(\rho(\xi)),$
(41a)
$\displaystyle\big{\\{}^{\;\kappa}_{\lambda\mu}\big{\\}}=\frac{1}{2}\mathpzc{G}^{\kappa\iota}(\rho(\xi))\bigg{[}\partial_{\mu}\mathpzc{G}_{\iota\lambda}(\rho$
$\displaystyle(\xi))+\partial_{\lambda}\mathpzc{G}_{\iota\mu}(\rho(\xi))-\partial_{\iota}\mathpzc{G}_{\mu\lambda}(\rho(\xi))\bigg{]},$
(41b)

which is enough to introduce geodesic equations, Riemann curvature tensor and
related other tensors. Additionally, Eq. (29) and Eq. (39) reveal a Hessian
structure on the density manifold such that Eq. (41a) can be rearranged to get
the corresponding connection coefficients of the first kind

$\displaystyle\big{\\{}{\scriptstyle\lambda\mu\nu}\big{\\}}$
$\displaystyle=-\frac{1}{2}\frac{\partial^{3}S}{\partial\rho^{\lambda}\partial\rho^{\mu}\partial\rho^{\nu}}$
$\displaystyle=\frac{1}{2}\frac{\partial~{}\mathpzc{G}_{\lambda\mu}(\rho(\xi))}{\partial\rho^{\nu}}.$
(42)

In light of Balian _et al._ metric, Eq. (32) and Eq. (30a) yield

$\mathscr{L}^{~{}\nu}_{\mu}\varrho_{\nu}=\mathpzc{G}_{\mu\lambda}\mathscr{L}^{~{}\lambda}_{\nu}\rho^{\nu}.$
(43)

In light of Eq. (26), the symmetric property of Balian _et al._ metric, and
the covariant form of the Liouville superoperator
$\mathscr{L}_{\mu\nu}=\mathpzc{G}_{\mu\lambda}(\rho(\xi))\mathscr{L}^{\lambda}_{\nu}$,
we differentiate Eq. (43) such that

$0=\frac{1}{2}\frac{\partial~{}\mathpzc{G}_{\mu\lambda}(\rho(\xi))}{\partial\rho^{\kappa}}\mathscr{L}^{~{}\lambda}_{\nu}\rho^{\nu}+\frac{1}{2}\left(\mathscr{L}_{\mu\kappa}+\mathscr{L}_{\kappa\mu}\right).$
(44)

Then, we substitute Eq. (3.3) and Eq. (32) in Eq. (44) such that

$-i\frac{d\mathpzc{G}_{\mu\nu}(\rho(\xi))}{dt}=\mathscr{L}_{\mu\nu}+\mathscr{L}_{\nu\mu}~{},$
(45)

which is the reason why we said before that $\mathscr{U}$, as defined in Eq.
(36), plays a role similar to that of Lie groups, or their corresponding
Killing fields, over the usual Riemannian manifolds.

###  3.4 The reduced spatial metric and the Euclidean structure of the space
of observables

Since we deal with thermodynamical gauge degrees of freedom in information
manifolds [75], it would be more convenient if we consider implementing
Einstein’s vierbein101010Vierbeins in 4D manifolds become vielbeins in
arbitrary dimensional manifolds. (frame) field theory as a gauge field theory
of gravity [86]. We summarize this approach by defining an observable
$\vec{\mathbb{W}}\equiv\mathbb{w}^{\mu}\mathpzc{e}_{\mu}$, where
$\\{\mathpzc{e}_{\mu}\\}$ is the coordinate non–orthnormal bases set.
Obviously $\mathpzc{G}_{\mu\nu}\equiv\mathpzc{e}_{\mu}\mathpzc{e}_{\nu}$. And
for the non–coordinate orthnormal bases set $\\{E_{a}\\}$, as defined in the
beginning of subsection (3.2), we have
$E_{a}=e_{a}^{~{}\mu}\mathpzc{e}_{\mu}$, where $e_{a}^{~{}\mu}$ is the
vielbein structure in such manifold. As expected
$\eta_{ab}=\mathpzc{G}_{\mu\nu}~{}e_{a}^{~{}\mu}e_{b}^{~{}\nu}$. Therefore, we
can connect this Euclidean space to the Riemannian density manifold through
the spatial parts of vielbein structure $e^{~{}\mu}_{i}$ such that the
Euclidean flat metric corresponding to this structure is defined as
$\delta_{ij}=\mathpzc{G}_{\mu\nu}~{}e_{i}^{~{}\mu}e_{j}^{~{}\nu}$, and
$e_{i}^{~{}\mu}$ is indeed a function in the observable component
$\mathbb{w}^{i}$. Moreover, the Euclidean metric $\delta_{ij}$ is the
($n-$1)–reduced dimensional metric of the $n$–dimensional Lorentzian metric
$\eta_{ab}$. This strongly demands extracting the curved reduced spatial
metric $\mathfrak{g}_{ij}$, that of a spatial slice, out of the metric
$\mathpzc{G}_{\mu\nu}$. So for a vector $\vec{\mathbb{W}}$ that is tangent to
the whole manifold and to a curve that lies in a slice $\Sigma_{(E_{a})}$ at
constant orthnormal coordinate $E_{a}$, we define a _normal_ cotangent vector
$n\equiv n_{\mu}E^{\mu}$, with components $n_{\mu}$, that is normal to
$\Sigma_{(E_{a})}$ such that $n(\mathbb{W})=\mathbb{w}^{\nu}n_{\mu}=0$. And
for constant temporal moment $t$ with $E_{0}$, the slices $\Sigma$’s become
spatial, and the normal covector is defined as

$\displaystyle n_{\mu}(t)$
$\displaystyle=\frac{E_{0}~{}e^{~{}0}_{\mu}}{\sqrt{\mathpzc{G}^{00}}}$ (46a)
$\displaystyle\sigma\equiv\mathpzc{G}^{\mu\nu}n_{\mu}n_{\nu}$
$\displaystyle=n_{\mu}n^{\mu}=\pm 1$ (46b)

Now we define the projection operator to obtain the spatial metric
$\mathfrak{g}_{ij}$. The projected part of $\mathbb{W}$ is defined as

$\forall~{}\mathbb{W}\in\mathpzc{T_{\text{\emph{P}}}(M)},~{}P\in\mathpzc{M},~{}\exists~{}\mathbb{W}_{{\perp}}\in
T_{P}(\Sigma)\subset\mathpzc{T_{\text{\emph{P}}}(M)}~{}\text{such
that}~{}\mathbb{W}_{{\perp}}\equiv\mathscr{P}(\mathbb{W}).$

This means $\\{\mathpzc{e}_{\mu}\\}$ and $\\{E_{a}\\}$ can be projected to
$\\{(\mathpzc{e}_{\perp})_{\mu}\\}$ and $\\{(E_{\perp})_{a}\\}$ respectively
as follows

$\displaystyle\mathbb{W}_{{\perp}}=[\mathscr{P}(\mathbb{W})]^{\mu}\mathpzc{e}_{\mu}=[\mathscr{P}^{~{}\mu}_{\nu}\mathpzc{e}_{\mu}]\mathbb{w}^{\nu}=\mathbb{w}^{\nu}(\mathpzc{e}_{{\perp}})_{\nu},$
(47a)
$\displaystyle\mathbb{W}_{\perp}=[\mathscr{P}(\mathbb{W})^{a}]E_{a}=[\mathscr{P}^{~{}a}_{b}E_{a}]\mathbb{w}^{b}=\mathbb{w}^{b}(E_{{\perp}})_{b}~{}.$
(47b)

Moreover, we can define another sets of bases $\\{(\mathpzc{e}_{||})_{\mu}\\}$
and $\\{(E_{||})_{a}\\}$ that are tangents to the slices $\Sigma_{t}$ as
follows

$\displaystyle(\mathpzc{e}_{||})_{\mu}=\mathpzc{e}_{\mu}-(\mathpzc{e}_{\perp})_{\mu}$
(48a) $\displaystyle(E_{||})_{a}=E_{a}-(E_{\perp})_{a}$ (48b)
$\displaystyle\mathpzc{G}_{\mu\nu}=(\mathpzc{e}_{||})_{\mu}(\mathpzc{e}_{||})_{\nu}+(\mathpzc{e}_{\perp})_{\mu}(\mathpzc{e}_{\perp})_{\nu}$
(48c)

The density manifold has an Euclidean substructure as in the example mentioned
at the end of subsection (3.1). This can help determining the relevant and the
irrelevant part of the density [87, 80]. Also,this means for any observable
$\vec{\mathbb{W}}$, including the probability density,111111Remember in the
interaction picture the probability density can be treated as an observable
operator. every spatial relevant component $\mathbb{w}^{i}$ has orthogonal
irrelevant projections $\mathscr{P}\vec{\mathbb{W}}$ such that

$\langle\vec{\mathbb{W}}-\mathscr{P}\vec{\mathbb{W}},\vec{\mathbb{W}}\rangle=\Big{\langle}\left(\vec{\mathbb{W}}-\mathfrak{o}\mathbb{w}\right)^{j},\mathbb{w}_{i}\Big{\rangle}=\delta^{~{}j}_{i}~{},$
(49)

where the bilinear form is defined according to the map (26), i.e. we can
infer that $\mathscr{P}$ is a superoperator acting on both Euclidean and
Riemannian manifolds. In order to obtain the components of this operator, we
combine Eq. (46) and Eq. (48) such that we get

$\mathscr{P}_{\mu}^{~{}\nu}=\delta_{\mu}^{~{}\nu}-\sigma n_{\mu}n^{\nu}.$ (50)

If we covariantize the projection operator, we get

$\mathscr{P}_{\mu\nu}=\mathpzc{G}_{\mu\nu}-\sigma n_{\mu}n_{\nu}.$ (51)

This helps defining the metric in form of the _covariant_ projection operator
and the normal vectors as

$\mathpzc{G}_{\mu\nu}=\mathscr{P}_{\mu\nu}+\sigma n_{\mu}n_{\nu}.$ (52)

This means that the covariant projection operator defines the reduced spatial
metric $\mathfrak{g}_{ij}$ if we realize that $\mathscr{P}$ is made only of
spatial components $\mathscr{P}_{ij}$. So we apply $n^{i}n^{j}$ on both sides
of the last equation, with the help of the normal vector definition in Eq.
(46), to get

$\displaystyle\mathfrak{g}_{ij}\equiv e^{~{}\mu}_{0}e^{~{}0}_{i}$
$\displaystyle~{}\mathpzc{G}_{\mu\nu}~{}e^{~{}\nu}_{0}e^{~{}0}_{j}=\mathpzc{G}_{\mu\nu}\delta^{~{}\mu}_{i}\delta^{~{}\nu}_{j},$
(53a) $\displaystyle\mathfrak{g}^{jk}\mathfrak{g}_{ki}=\delta^{~{}j}_{i}.$
(53b)

In an information manifold this metric plays the same role the spatial
spacetime metric $\gamma_{ij}$ does in defining the expansion parameter in Eq.
(11). It is worth noting that the vielbein acts on the density vector as a
projector operator to yield the components of the density vector in the
Liouville space, which is another way to define the axes in Fig. (1), i.e. we
could start from the projector operator and the vielbein structure backward
until we reach the Liouville vector representation of the density operator;
both approaches therefore are equivalent. So if we return back to Fig. (1) and
choose a point $P^{\prime}(t_{0})\in\Sigma_{Q}$ along the curve $t_{0}$, then

$\displaystyle\mathscr{P}\rho(t_{0})$ $\displaystyle=\rho_{0}(t_{0}),$ (54a)
$\displaystyle\rho^{i}:=\langle\mathbb{w}^{i}\rangle$
$\displaystyle=\langle\mathbb{w}^{i},\rho\rangle=\langle\mathbb{w}^{i},\rho_{0}\rangle,~{}~{}\forall
i.$ (54b)

Now it is safe to infer that the distance on the surfaces $\Sigma$ in Fig. (1)
are given by

$\displaystyle ds^{2}_{\Sigma}$ $\displaystyle=$
$\displaystyle\mathfrak{g}_{ij}(\rho(\xi))d\rho^{i}d\rho^{j}$ (55)
$\displaystyle=$
$\displaystyle\mathfrak{g}^{ij}(\rho(\xi))d\rho_{i}d\rho_{j}~{},$

where $\mathfrak{g}_{ij}(\rho(\xi))$ could be not equal to
$\mathfrak{g}^{ij}(\rho(\xi))$ in general, i.e.
$d\rho_{i}\neq\mathfrak{g}_{ij}(\rho(\xi))d\rho^{j}$ necessarily as
$d\rho_{i}$ is more like the component $\mathfrak{o}_{i}$ of
$\vec{\mathcal{O}}$ as defined previously. Another important manifold
structure can be obtained by combining Eq. (22) and Eq.(54) such that

$\langle\mathbb{w}^{i},\mathcalligra{r}\rangle=0~{},$ (56)

where the vector $\mathcalligra{r}=(\rho-\rho^{i}_{0})$ is the tangent along
the curve $t_{0}$, see Fig. (1). Thus, we have a vector bundle structure,
where the base is the surface $\Sigma_{P}$ and the fibres are the curves
$t_{0}+n\delta t,n\in\mathbb{N}$. See Ref. [88] for more about the relation
with the _blurred space_.

###  3.5 Fisher metric and Kullback-Leibler divergence

The previously constructed density manifold has an Euclidean signature. Then,
we target constructing an information manifold with a Lorentzian signature
diag$(-1,+1,+1,\cdots)$, and we achieve this goal in subsection (3.7). For
now, we focus on relating Balian _et al._ metric to Fisher metric. As the
vacuum spacetime is in a “continuous” experience of quantum fluctuations, it
can be optimized stochastically such that the expectation values of the
operators over spacetime, i.e. the stochastic variables,121212For more on the
properties of the stochastic variables, see Ref. [89]. are invariant under
coordinate transformations between different frames of references [90]. This
means we can define the density vectors $\rho^{\mu}$, which is a function in
the classical variables $x^{\mu}$ characterizing the spacetime itself, as
function in the stochastic variables
$\mathbb{X}^{\mu}\equiv\mathbb{X}^{\mu}(\langle
x^{\mu}\rangle,\sigma_{x^{\mu}})$ that are functions in a spatial space of
averages $\mathbb{X}^{\mu}(\langle x^{\mu}\rangle)$ and standard deviations
$\sigma_{x^{\mu}}$.131313See Ref. [91] for more on the $2D$ manifold of
averages and standard deviations in the exponential families of normal
distribution. Then, we may guess that Balian _et al._ metric
$\mathpzc{G}_{\mu\nu}\left(\rho(\xi^{\mu})\right)$ in Eq. (28) could be
expressed as an explicit function of $\xi^{\mu}$ and $\mathbb{X}^{\mu}$
variables, i.e.
$\mathpzc{G}_{\mu\nu}\equiv\mathpzc{G}_{\mu\nu}(\xi^{\mu};\mathbb{X}^{\mu})$.
In order to check the validity of this guess, we need to check that this
Ruppeiner metric maintains the same form when coordinate transformations are
considered. The best candidate to test this requirement is the
Kullback–Leibler divergence

$D_{\text{KL}}=\sum\limits_{\xi^{i};\mathbb{X}^{i}}\rho(\xi^{\mu};\mathbb{X}^{\mu})\ln\frac{\rho(\xi^{\mu};\mathbb{X}^{\mu})}{\rho(\xi^{\mu}+d\xi^{\mu};\mathbb{X}^{\mu})}\Delta\mathbb{X}^{i}.$
(57)

As the spacetime variables are infinitesimally changing and $\rho\to d\rho$,
then Eq. (57) becomes

$\displaystyle D_{\text{KL}}=-\int\limits_{\mathcal{M}}$
$\displaystyle\rho(\xi;\mathbb{X})\Big{[}\frac{1}{\rho(\xi;\mathbb{X})}\frac{\partial\rho(\xi;\mathbb{X})}{d\xi^{\mu}}d\xi^{\mu}-\frac{1}{2}\frac{1}{\rho^{2}(\xi;\mathbb{X})}\frac{\partial\rho(\xi;\mathbb{X})}{d\xi^{\mu}}\frac{\partial\rho(\xi;\mathbb{X})}{d\xi^{\nu}}d\xi^{\mu}d\xi^{\nu}\Big{]}d\mathbb{X},$
(58)

where $\rho(\xi;\mathbb{X})\equiv\rho(\xi^{\mu};\mathbb{X}^{\mu})$. The second
term in Eq. (58) is nothing but the Fisher metric

$ds^{2}:=\sum\limits_{i}\frac{(d\rho_{i})^{2}}{\rho_{i}}~{},$ (59)

which is the same metric in Eq. (28) but as an explicit function in
$\xi^{\mu}$ and $\mathbb{X}^{\mu}$ [20]. Applying Fisher metric, chain rule,
and the completeness relation of the density to Eq. (57) yields

$D_{\text{KL}}=\int\limits_{\mathcal{M}}\frac{1}{2}\mathpzc{G}_{\mu\nu}(\xi;\mathbb{X})d\xi^{\mu}d\xi^{\nu}d\mathbb{X}.$
(60)

Eq. (57) can be seen as

$\displaystyle D_{\text{KL}}\sim-d^{2}S=d\bigg{[}\frac{\partial
S}{\partial\rho_{\mu}}d\rho^{\mu}\bigg{]}=\frac{\partial^{2}S}{\partial\rho^{\mu}\partial\rho^{\nu}}\frac{\partial\rho^{\mu}}{\partial\xi^{\kappa}}\frac{\partial\rho^{\nu}}{\partial\xi^{\lambda}}d\xi^{\kappa}d\xi^{\lambda}.$
(61)

By comparing Eq. (39), Eq. (60), and Eq. (61), and as the KL divergence is
nothing but a modified Shannon entropy, the covariant form of Eq. (39) says
that

$\mathpzc{G}_{\mu\nu}(\varrho(\xi;\mathbb{X}))=\frac{\partial\varrho}{\partial\xi^{\mu}}\frac{\partial\varrho}{\partial\xi^{\nu}}.$
(62)

As we notice, the density is no longer a vector, it is just a function, and
the vectors of the new manifold are
$\partial_{\mu}\equiv\partial/\partial\xi^{\mu}$. If we strict expressing all
densities as as functions in stochastic variables $\mathbb{X}^{\mu}$ rather
than the canonical ones $\xi^{\mu}$, i.e. $\rho\equiv\rho(\mathbb{X}^{\mu})$,
then, with help of Eq. (15–20), we suppress Fisher metric density
$\xi^{\mu}$–dependence, i.e. from now on the Balian _et al._ metric141414We
move from the information manifold of Balian _et al._ [20] to the information
manifold of Amari [92]. $\mathpzc{G}_{\mu\nu}$ is not the metric we use. And
Eq. (62) should be improved as

$\langle\mathpzc{G}_{\mu\nu}(\xi;\mathbb{X})\rangle=\langle\frac{\partial\varrho}{\partial\xi^{\mu}}\frac{\partial\varrho}{\partial\xi^{\nu}}\rangle.$
(63)

In order to understand the last result, we need to get back to the Fisher
metric. Without loss of generality, Eq. (58) describes the relation between
the entropy at the reduced state $\rho_{0}$ and any other state $\rho$, see
the Points $P$ and $Q$ in Fig. (1). Then, in information manifold the
probabilistic average of the stochastic Fisher metric $\mathbb{g}_{\mu\nu}$
plays the same role the spacetime metric does in the Riemannian manifolds,
i.e. $g_{\mu\nu}\simeq\mathbb{g}_{\mu\nu}$. Thus,

$\mathbb{g}_{\mu\nu}:=\langle\mathpzc{G}_{\mu\nu}(\xi,\mathbb{X})\rangle~{}.$
(64)

So we confirm that the metric of the information manifold maintains its form
when coordinate transformations are considered. This is crucial for defining
Einstein tensor and the analogue of the gravitational constant in the
information manifold as we will see by the end of the next subsection (3.6).

###  3.6 Hessian structure and Einstein tensor

For an information manifold endowed with exponential family of distribution,
there exists a potential function $\phi$ such that its Hessian defines the
metric of that manifold [93]. Non–exponential families do the same but in a
less straightforward way,151515This is mainly because the Bayesian hypothesis
[94] related to the Hessian in non–exponential families needs variational
Bayesian approximations, like Laplace approximation and parameter separation
parameterization, to obtain the posterior distribution [95]. for more on that
see Ref. [96, 97] and references there. For simplicity we discuss the Hessian
structure of exponential families, but our discussion is applicable to
non–exponential ones too. We share with Ref. [98] constructing Einstein tensor
from the Fisher metric in information manifold. Also, we find Einstein tensor
to be endowed with relevant information to construct the energy–momentum
tensor from the varying cumulant partition function defined as a scalar field.
This is guaranteed naturally since the entropy data of the system underpin the
field strength of the system. Thus, we can consider Einstein equations in
information manifold as the equations of coarse–grained states for the
original microscopic system of quantum field theories behind the classical
ones. The major difference between this work and the endeavor of Ref. [98] is
that, without assuming any family of exponentials, we find a _positively
definite cosmological–like background term_ in the coarse–grained Einstein
equations, particularly in the terms containing the derivatives of the
Christoffel connections. This suggests redefining Einstein tensor in
information manifold to become a Lovelock tensor. Consequently,
coarse–graining the states could reveal extra disguised higher ordered
curvature terms in the theory. Additionally, in contrary to Ref. [98] choice
of real probability distributions that suit AdS/CFT, we impose the family of
exponentials to be defined as complex probability distributions such that we
construct Einstein tensor in a information manifold with Lorentzian signature.
This does not change the result that the gravitational constant is indeed
dynamical. However, _it changes the sign of the extremal area and the
entropy_. This result is consistent with the fact that complex probabilities
are associated with non–Hermitian Hamiltonians and their non–unitary
transformations. Ref. [98] admits the problem of controlling the energy scale
of the quantum field theory in the information geometric approach. In order to
resolve that, we suggest repeating the process coarse–graining until
fine–graining is achieved [51]. Consequently, the information approach would
have a renormalization process, and the von Neumann equation stays
fine–grained as suggested in Ref. [11]. Thus, our modifications may introduce
quantum information geometry approach to dS/CFT [99, 100] with the help of the
obtained pseudo–entropy [23, 24].

As shown in Appendix 2.7 of Ref. [95], the variational Bayesian inferences
limited to exponential families, like the KL divergence, can be generalized
through the technique of _parameter separation parameterization_. The
technique is applicable to both exponential and non–exponential families of
distributions by _linearly_ relating $\rho$ to a _real valued potential
function_.161616See function $h(y)$ defined in Eq. 2.14 in Ref. [95]. The
potential function $\phi$ would help defining the metric of the information
manifold as shown in the Appendix. Without loss of generality, the equality
$\partial_{\mu}\partial_{\nu}\phi=\mathbb{g}_{\mu\nu}$ in Eq. (A20) is very
similar to the bilinear relation $\text{Hess}(\phi)=\nabla\nabla\phi$, or the
Hessian, in Riemannian geometry [92]. We are allowed to do this comparison
because of the bundle structure in the density manifold we referred to at the
end of subsection (3.4). Remember that $\varrho\in\text{T}^{*}_{P}\Sigma$,
i.e. the Hessian acts on the potential function $\phi$ to get an element in
the sections of tangent bundles
$\text{Hess}(\phi)\in\Gamma(\text{T}^{*}\Sigma\otimes\text{T}^{*}\Sigma)$.
Also, the density manifold and the canonical variable manifold, which could be
the conventional spacetime, are related through transforming $\varrho$ into
$\partial_{\mu}\varrho$. Then, we can apply a derivative on Eq. (A21) to get

$\displaystyle\partial_{\lambda}\mathbb{g}_{\mu\nu}=\partial_{\lambda}\partial_{\mu}\partial_{\nu}\phi=\langle\partial_{\lambda}(\partial_{\mu}\partial_{\nu}\varrho)\rangle=\langle\partial_{\lambda}(\partial_{\mu}\varrho\partial_{\nu}\varrho)\rangle=-\langle\partial_{\lambda}\varrho\partial_{\mu}\varrho\partial_{\nu}\varrho\rangle,$
(65)

where the tensor relation between the vectors $\partial_{\mu}\varrho$ is
suppressed, the last equality comes from
$\langle\partial_{\mu}(\partial_{\nu}\varrho)\rangle=\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\rangle$,
and the negative sign in last equality comes from the metric definition in Eq.
(27). More obviously from the behavior of the derivatives, we have

$\partial_{\lambda}\mathbb{g}_{\mu\nu}=\langle\partial_{\lambda}\partial_{\mu}\varrho\partial_{\nu}\varrho\rangle+\langle\partial_{\mu}\varrho\partial_{\lambda}\partial_{\nu}\varrho\rangle.$
(66)

Using the symmetric property of $\mathbb{g}_{\mu\nu}$, the last two equations
lead us to

$\partial_{\lambda}\mathbb{g}_{\mu\nu}=\frac{1}{2}\Big{[}\langle\partial_{\lambda}\partial_{\mu}\varrho\partial_{\nu}\varrho\rangle+\langle\partial_{\mu}\varrho\partial_{\lambda}\partial_{\nu}\varrho\rangle-\langle\partial_{\lambda}\varrho\partial_{\mu}\varrho\partial_{\nu}\varrho\rangle\Big{]}.$
(67)

In light of Eq. (41b), the last result is very enticing to define the
Christoffel connection corresponding to the canonical variable manifold as
[91, 92]

$\displaystyle\mathbb{\Gamma}^{\lambda}_{\mu\nu}$
$\displaystyle=-\frac{1}{2}\mathbb{g}^{\lambda\kappa}\partial_{\kappa}\partial_{\mu}\partial_{\nu}\phi$
$\displaystyle=\mathbb{g}^{\lambda\kappa}\Big{[}\langle\partial_{\kappa}\varrho\partial_{\mu}\partial_{\nu}\varrho\rangle-\frac{1}{2}\langle\partial_{\kappa}\varrho\partial_{\mu}\varrho\partial_{\nu}\varrho\rangle\Big{]}.$
(68)

Next, we use Eq. (3.6) to calculate the Ricci tensor and the Ricci scalar as
functions in the dual density variables and their derivatives. The Ricci
tensor is obtained as usual from the contracted Riemann tensor

$\mathbb{R}_{\mu\nu}=\mathbb{g}^{\alpha\beta}\mathbb{R}_{\alpha\mu\beta\nu}=\partial_{\alpha}\mathbb{\Gamma}^{\alpha}_{\mu\nu}-\partial_{\nu}\mathbb{\Gamma}^{\alpha}_{\mu\alpha}+\mathbb{\Gamma}^{\alpha}_{\beta\alpha}\mathbb{\Gamma}^{\beta}_{\mu\nu}-\mathbb{\Gamma}^{\alpha}_{\beta\nu}\mathbb{\Gamma}^{\beta}_{\mu\alpha}.$
(69)

Meanwhile the Ricci scalar also is obtained as usual for the contacted Ricci
tensor

$\mathbb{R}=\mathbb{g}^{\mu\nu}\mathbb{R}_{\mu\nu}.$ (70)

Then, the Einstein tensor in information manifold becomes

$\mathbb{G}_{\mu\nu}=\mathbb{R}_{\mu\nu}-\frac{1}{2}\mathbb{g}_{\mu\nu}\mathbb{R}.$
(71)

As shown in the Appendix, we deconstruct Eq. (71) into two main pieces. We
analyze these pieces using the definitions of the metric and Christoffel
connections in information manifolds such that the Hessian structure is
realized. So, we can reintroduce Eq. (71) as

$\displaystyle\mathbb{G}_{\mu\nu}=\Lambda\mathbb{g}_{\mu\nu}+(\tilde{\mathbb{R}}_{\mu\nu}$
$\displaystyle-\frac{1}{2}\mathbb{g}_{\mu\nu}\tilde{\mathbb{R}})+\frac{1}{2}\mathbb{g}_{\mu\nu}\tilde{\mathbb{R}},$
(72a)
$\displaystyle\text{or}~{}~{}\tilde{\mathbb{G}}_{\mu\nu}+\Lambda\mathbb{g}_{\mu\nu}$
$\displaystyle=\mathbb{G}_{\mu\nu}-\frac{1}{2}\mathbb{g}_{\mu\nu}\tilde{\mathbb{R}},$
(72b)

where $\tilde{\mathbb{R}}_{\mu\nu}$ is explained in Eq. 3.6, and
$\tilde{\mathbb{G}}_{\mu\nu}+\Lambda\mathbb{g}_{\mu\nu}$ plays a rule similar
to that of Lovelock tensor $A_{\mu\nu}=G_{\mu\nu}+\Lambda g_{\mu\nu}$ in
Einstein spacetime manifold [101].

The first new piece obtained from rearranging then deconstructing Eq. (71),
which is Eq. (A9-A$9^{\prime}$), gives a cosmological-like term in the
information manifold as

$\Big{(}\partial_{\alpha}\mathbb{\Gamma}^{\alpha}_{\mu\nu}-\partial_{\nu}\mathbb{\Gamma}^{\alpha}_{\mu\alpha}\Big{)}-\frac{1}{2}\mathbb{g}_{\mu\nu}\mathbb{g}^{\kappa\lambda}\Big{(}\partial_{\alpha}\mathbb{\Gamma}^{\alpha}_{\kappa\lambda}-\partial_{\kappa}\mathbb{\Gamma}^{\alpha}_{\lambda\alpha}\Big{)}=\frac{1}{2}D(\frac{D}{2}-1)\mathbb{g}_{\mu\nu}\equiv\frac{1}{2}\Lambda\mathbb{g}_{\mu\nu}~{},$
(73)

where $\Lambda=D(D/2-1)$ is defined for $D\geqslant 2$ dimensional manifold as
in Lovelock theory of gravity [102]. This helps our information manifold pass
the necessary condition, but not sufficient on its own, to develop an
additional Gauss–Bonnet term for the corresponding Riemann tensor that is
expressed in its information form. The appearance of $D(D/2-1)$ term in the
theory is very tempting to study the effects of having higher–curvature terms
in the context of holography. We leave that to be discussed hopefully in a
future study. We notice that the cosmological constant we obtained does not
depend on the manifold parameters. It only depends on the number of
dimensions, and it vanishes when we consider $D=2$. This may point to
background symmetry behind the cosmological constant, and it may have a
relation with vanishing cosmological constant in the context of some M/string
theory [103]. More importantly, this piece in the
$\tilde{\mathbb{G}}_{\mu\nu}$ tensor should exist for all kinds of exponential
or non–exponential probability distributions. It neither demands $\rho$ to be
exponential, Gaussian or complex one, nor to be in dS or AdS spaces. And this
is our major difference between this work and the endeavors in Ref. [98].

The second piece after deconstructing Eq. (71) appears written in Eq. (6) in
the Appendix. This piece of non differentiated Christoffels, obtained from
both $\mathbb{R}_{\mu\nu}$ and $\mathbb{g}_{\mu\nu}\mathbb{R}$, defines a new
tensor

$\displaystyle\mathbb{\widetilde{R}}_{\mu\nu}$
$\displaystyle=\Big{(}\mathbb{\Gamma}^{\alpha}_{\beta\alpha}\mathbb{\Gamma}^{\beta}_{\mu\nu}-\mathbb{\Gamma}^{\alpha}_{\beta\nu}\mathbb{\Gamma}^{\beta}_{\mu\alpha}\Big{)}-\frac{1}{2}\mathbb{g}_{\mu\nu}\mathbb{g}^{\kappa\lambda}\Big{(}\mathbb{\Gamma}^{\alpha}_{\beta\alpha}\mathbb{\Gamma}^{\beta}_{\kappa\lambda}-\mathbb{\Gamma}^{\alpha}_{\beta\kappa}\mathbb{\Gamma}^{\beta}_{\lambda\alpha}\Big{)}$
$\displaystyle=\frac{1}{4}\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Big{[}\langle\partial_{\alpha}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\lambda}\varrho\rangle-\langle\partial_{\nu}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\varrho\partial_{\alpha}\varrho\partial_{\lambda}\varrho\rangle\Big{]}.$
(74)

Without loss of generality,171717Generality is not lost as the crucial steps
in Eq.(A20–A21) are guaranteed by applying previously mentioned technique of
parameter separation parametrization on non–exponential families [95]. we
follow the exponential family example in Ref. [98]. We use Eq. (A15) and after
so Eq. (3.6) becomes

$\displaystyle\mathbb{\widetilde{R}}_{\mu\nu}=\frac{1}{4}\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Bigg{\\{}$
$\displaystyle\Big{[}\mathpzc{G}_{\alpha\beta}\mathpzc{G}_{\kappa\lambda}-\mathbb{g}_{\alpha\beta}\mathbb{g}_{\kappa\lambda}\Big{]}+\mathpzc{G}_{\kappa\lambda}\Big{[}\dot{\mathpzc{G}}_{\alpha\beta}\langle\mathbb{X}-\langle
x\rangle\rangle+\frac{1}{2}\ddot{\mathpzc{G}}_{\alpha\beta}\langle(\mathbb{X}-\langle
x\rangle)^{2}\rangle+\cdots\Big{]}$
$\displaystyle+\mathpzc{G}_{\alpha\beta}\Big{[}\dot{\mathpzc{G}}_{\kappa\lambda}\langle\mathbb{X}-\langle
x\rangle\rangle+\frac{1}{2}\ddot{\mathpzc{G}}_{\kappa\lambda}\langle(\mathbb{X}-\langle
x\rangle)^{2}\rangle+\cdots\Big{]}\Bigg{\\}}\times\partial_{\mu}\phi\partial_{\nu}\phi.$
(75)

Or

$\mathbb{\widetilde{R}}_{\mu\nu}=\frac{1}{2}\mathbb{G}_{D}~{}\partial_{\mu}\phi\partial_{\nu}\phi.$
(76)

where the dynamical entity $\mathbb{G}_{D}$ is defined as

$\displaystyle\mathbb{G}_{D}=\frac{1}{2}\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Bigg{\\{}$
$\displaystyle\Big{[}\mathpzc{G}_{\alpha\beta}\mathpzc{G}_{\kappa\lambda}-\mathbb{g}_{\alpha\beta}\mathbb{g}_{\kappa\lambda}\Big{]}+\mathpzc{G}_{\kappa\lambda}\Big{[}\dot{\mathpzc{G}}_{\alpha\beta}\langle\mathbb{X}-\langle
x\rangle\rangle+\frac{1}{2}\ddot{\mathpzc{G}}_{\alpha\beta}\langle(\mathbb{X}-\langle
x\rangle)^{2}\rangle+\cdots\Big{]}$
$\displaystyle+\mathpzc{G}_{\alpha\beta}\Big{[}\dot{\mathpzc{G}}_{\kappa\lambda}\langle\mathbb{X}-\langle
x\rangle\rangle+\frac{1}{2}\ddot{\mathpzc{G}}_{\kappa\lambda}\langle(\mathbb{X}-\langle
x\rangle)^{2}\rangle+\cdots\Big{]}\Bigg{\\}},$ (77)

where

$\dot{\mathpzc{G}}_{\mu\nu}(\langle
x\rangle)=\lim\limits_{\mathbb{X}\to\langle
x\rangle}\frac{\partial}{\partial\mathbb{X}}{\mathpzc{G}}_{\mu\nu}(\mathbb{X})$
(78)

and $\ddot{\mathpzc{G}}_{\mu\nu}(\langle x\rangle)$ is the usual second
derivative of the Taylor expansion expressed in Eq. (A28). The limit closes
the stochastic variable to the more probable value of the random variable.
Using the cumulant partition function $\phi$ as a scalar field in the
information manifold, we can define a Lagrangian in $D=4$ for an effective
field theory as

$\mathbb{L}=\mathbb{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi,$ (79)

and the corresponding energy–momentum tensor is

$\mathbb{T}_{\mu\nu}=\mathbb{g}_{\mu\lambda}\frac{\partial\mathbb{L}}{\partial(\partial_{\lambda}\phi)}\partial_{\nu}\phi-\mathbb{g}_{\mu\nu}\mathbb{L}=\frac{1}{\mathbb{G}_{4}}\Bigg{(}\mathbb{\widetilde{R}}_{\mu\nu}-\frac{1}{2}\mathbb{g}_{\mu\nu}\mathbb{\widetilde{R}}\Bigg{)}=\frac{1}{\mathbb{G}_{4}}\mathbb{\widetilde{G}}_{\mu\nu}~{},$
(80)

where obviously the last equation is Einstein equation with its reduced
tensors $\mathbb{\widetilde{G}}_{\mu\nu}$ and
$\mathbb{\widetilde{R}}_{\mu\nu}$ as functions in the stochastic variables
$\mathbb{X}$ in such information geometry. Also, we can add the cosmological
constant term we obtained before in Eq. (73). Comparing Eq. (80) with Eq.
(14), we get

$\mathbb{G}_{4}\simeq\frac{8\pi\ell^{2}_{P}}{\hbar},$ (81)

which again is nothing but the gravitational constant in the information form
for such $4D$ geometry. This indicates quantum information geometry induces
gravity phenomena, and the gravitational constant is no longer constant
besides its $\hbar$ dependence. This may support studies that implies inducing
gravity from quantum mechanics [104, 105, 106, 107, 108]. Also, It could be
linked to realizing space and time as approximate macroscopic concepts
stemming fundamentally from quantum field theories [109]. Additionally, a
varying gravitational constant may give a clue to Dirac’s large numbers
hypothesis [110] that also implies a varying gravitational constant based on
the simple analysis of dimensionless constants that are provided by nature.

The results obtained in Eq. (3.6–73) are discussed in details in the Appendix
of the manuscript. There, the family of exponentials is not assumed to be
either complex, which is what we discuss in the next subsection, or real as in
Ref. [98], since we keep everything in the appendix in terms of an arbitrary
$\varrho$. Additionally in Ref. [98], after the Gaussian exponential was
imposed, the “whole” Einstein tensor is equated to a “negative” cosmological
constant. This is not what we obtain. Rather, we say that the Einstein tensor
$\mathbb{G}_{\mu\nu}$ should be split into $\partial\mathbb{\Gamma}$ terms
that correspond to a “positively” cosmological constant defined in terms of
the dimension $D$ as in Eq. (73), and $\mathbb{\Gamma}\mathbb{\Gamma}$ terms
that introduce a modified Ricci tensor $\tilde{\mathbb{R}}_{\mu\nu}$ as the
kinetic term $\partial_{\mu}\phi\partial_{\nu}\phi$ in Eq. (3.6). Notice that
the Lagrangian
$\mathbb{L}\sim\mathbb{g}^{\mu\nu}\tilde{\mathbb{R}}_{\mu\nu}\sim\Box\phi$
renders the reduced Einstein tensor $\tilde{\mathbb{G}}_{\mu\nu}$ upon
applying the variational principle with respect to $\phi$, while the
$\Lambda\mathbb{g}_{\mu\nu}$ piece is produced from varying $\mathbb{L}$ with
respect to $\mathbb{g}_{\mu\nu}=\partial_{\mu}\partial_{\nu}\phi$. This
reminds us with the $Y$ piece in the modified Lagrangian of gravity in the
framework of superstrings [78], both share the same Hessian structure. In
brief, we suggest that the Einstein tensor $G_{\mu\nu}$ in Ref. [98] should
rather be treated as a Lovelock tensor $A_{\mu\nu}=G_{\mu\nu}+\Lambda
g_{\mu\nu}$. Such tensor should be split into a cosmological term
$\Lambda\mathbb{g}_{\mu\nu}$ and a modified Ricci tensor
$\tilde{\mathbb{R}}_{\mu\nu}$, and the later can be used to introduce a new
modified Einstein tensor $\tilde{\mathbb{G}}_{\mu\nu}$ that has no
cosmological terms in it. All that can be obtained without assuming the
densities to be expressed as family of exponentials or any other family.

###  3.7 Obtaining a pseudo-Riemannian information manifold

A question remains about how to construct an arbitrary $(D+1)$–dimensional
information geometry with a Lorentzian signature diag$(-1,+1,\cdots)$. And
without loss of generality, we follow the complex Gaussian ansatz in Ref.
[111], and we comment on the consequences of this ansatz in the discussion
section, to get a $(1+1)$–dimensional information manifold with a Lorentzian
signature diag$(-1,+1)$ by assuming the following distribution

$\rho(t)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(t-i\langle
t\rangle)^{2}}{2\sigma}\right]=\exp\left[-\ln(\sqrt{2\pi}\sigma)-\frac{t^{2}}{2\sigma^{2}}+\frac{it\langle
t\rangle}{\sigma^{2}}+\frac{\langle t\rangle^{2}}{2\sigma^{2}}\right],$ (82)

which requires components of the corresponding Fisher metric to be

$\displaystyle\mathbb{g}_{00}$ $\displaystyle=\int
d^{2}\xi\frac{1}{\rho(\xi)}\left(\frac{\partial\rho(\xi)}{\partial\xi^{0}}\right)^{2}=-1,$
(83a) $\displaystyle\mathbb{g}_{11}$ $\displaystyle=\int
d^{2}\xi\frac{1}{\rho(\xi)}\left(\frac{\partial\rho(\xi)}{\partial\xi^{1}}\right)^{2}=1,$
(83b) $\displaystyle\mathbb{g}_{01}$ $\displaystyle=\int
d^{2}\xi\frac{1}{\rho(\xi)}\left(\frac{\partial\rho(\xi)}{\partial\xi^{0}}\right)\left(\frac{\partial\rho(\xi)}{\partial\xi^{1}}\right)=0.$
(83c)

The Lorentzian signature of the metric requires defining a complex exponential
family as in Eq. (82) and maintains the general form

$\rho(\xi^{\mu})=\exp\left[\xi^{\mu}\mathbb{E}_{\mu}(\mathbb{X}^{\nu})-\phi(\xi^{\nu})\right],$
(84)

where the function $\mathbb{E}_{\mu}(\mathbb{X}^{\nu})$ plays the role of any
physical property related to the corresponding canonical or intensive
variables $\xi^{\nu}$, and $\phi(\xi^{\nu})$ is the cumulant partition
function of the $\mathbb{E}_{\mu}(\mathbb{X}^{\nu})$ states [92], see Eq.
(A15) and after in the Appendix. By comparing Eq. (82) to Eq. (84), it is easy
to notice that

$\displaystyle\mathbb{E}_{\mu}$
$\displaystyle:=(\mathbb{E}_{0},\mathbb{E}_{1})=(t,t^{2}),$ (85a)
$\displaystyle\xi^{\mu}$
$\displaystyle:=(\xi^{0},\xi^{1})=\left(\frac{i\langle
t\rangle}{\sigma^{2}},-\frac{1}{2\sigma^{2}}\right),$ (85b)
$\displaystyle\phi(\xi^{\nu})$
$\displaystyle:=\frac{1}{2}\ln\left(-\frac{\pi}{\xi^{1}}\right)-\frac{(\xi^{0})^{2}}{4\xi^{1}}=\ln\left(\sqrt{2\pi}\sigma\right)+\frac{\langle
t\rangle^{2}}{2\sigma^{2}},$ (85c)

which defines the cumulant distribution as

$\displaystyle\varrho:=-\ln(\rho(\xi))$
$\displaystyle\equiv\phi(\xi^{\nu})-\xi^{\mu}\mathbb{E}_{\mu}(\mathbb{X}^{\nu})$
$\displaystyle=\ln(\sqrt{2\pi}\sigma)+\frac{t^{2}}{2\sigma^{2}}-\frac{it\langle
t\rangle}{\sigma^{2}}-\frac{\langle t\rangle^{2}}{2\sigma^{2}}.$ (86)

Notice that $\mathbb{E}_{\mu}(\mathbb{X}^{\nu})$ is a function in $t$, which
is one of the components $x^{\mu}$ of the classical spacetime, i.e. the
stochastic variables $\mathbb{X}^{\nu}$ can be parameterized generally by the
classical $x^{\mu}$ components of the spacetime as we emphasized before in the
beginning of subsection (3.5) when we mentioned the coarse–graining the
properties of the manifold. Also, $\xi^{\mu}$ variables are functions in the
averages and the standard deviations of $x^{\nu}$, i.e. $\xi^{\mu}$ are
functions in statistical variables, which means we can suppress the variable
$\xi$ in the metric such that
$\mathpzc{G}_{\mu\nu}\left(\xi,\mathbb{X}\right)\equiv\mathpzc{G}_{\mu\nu}\left(\mathbb{X}\right)$.
Moreover, defining the exponential family of probability distributions to be
either complex, like how we have just done, or a real, as in Ref. [98], will
not change the definition of the cumulant probability distribution after
applying the $(-\ln)$ function on Eq. (84). Therefore, the results in the
previous subsection hold correct in both approaches, where our approach is
suitable for dS spaces while Ref. [98] fits for AdS. Constructing arbitrary
$D$–dimensional pseudo–Riemannian information manifold from a classical
spacetime is left as an exercise to the reader.181818This can be obtained in a
similar fashion of how AdS is obtained from complexifying dimensions of CFT,
see Eq. (9–12) in Ref. [112], or see the $(u,v)$ _thermal_ coordinates in Ref.
[113].

##  4 Entropy of the Information Manifold

As we have seen in section (2) for non dissipative systems, the time rate
change of the entropy is directly related to the content of information
expressed in Liouville–von Neumann equation. But in the previous section we
found that there is a correspondence between the spacetime and the statistical
information manifold, and this correspondence makes the spacetime metric
corresponds to Fisher metric even when we compare the Lorentzian signature of
both [111, 114, 115]. As the spatial entropic area $A$ is part of the
spacetime, we can construct an area $\mathbb{A}$ in the information manifold
that corresponds to the $A$, and the spatial metric of that information area
exists in light of our discussion in subsection (3.4). More about the area
$\mathbb{A}$ of such _blurred spatial space from quantum entanglement
contours_ can be found in Ref. [88, 116]. Then, it is now clear that
$\mathbb{A}$ could be obtained also after complexifying the exponential
distribution families.191919$\mathbb{A}$ could be developed for AdS as in Ref.
[117]. And if we accept the existence of such correspondence between $A$ and
$\mathbb{A}$, then we can find a equation for the spatial–like expansion rate
in the information manifold black$\theta$ that corresponds to $\theta$ as
mentioned before in section (2), i.e.

${\color[rgb]{1,1,1}\contour{black}{\theta}}=\frac{1}{\mathbb{A}}\frac{d}{dt}\left(\mathbb{A}\right),$
(87)

where, in comparison to Eq. (11) and Eq. (53a), the spatial–like expansion
rate in the information manifold black$\theta$ is defined using the
determinant of the averaged spatial components of the Fisher metric
black$\gamma$ =
det(black$\gamma$${}_{ij})=\text{det}(\langle\mathfrak{g}_{ij}\rangle)$, i.e.

${\color[rgb]{1,1,1}\contour{black}{\theta}}=\frac{1}{\sqrt{{\color[rgb]{1,1,1}\contour{black}{\gamma}}}}\frac{d}{dt}\left(\sqrt{{\color[rgb]{1,1,1}\contour{black}{\gamma}}}\right).$
(88)

Since the exponential family is assumed to be either real or complex, the
Hamiltonian is allowed to be non–Hermitian with non-unitary transformation,
and it is expected to be a stochastic Hamiltonian in comparison with regular
Hamiltonians in regular phase spaces. Thus, the Hamiltonian should be modified
to obey Lindblad master equation [118], which is the most general form of
Liouville equation, and we comment on this in the discussion. Then for a
stochastic Hamiltonian $\mathbb{H}$ as a function in $\mathbb{X}$ and its
conjugate momentum, Eq. (8) in the information manifold becomes

$i\frac{1}{4\mathbb{G}_{4}}{\color[rgb]{1,1,1}\contour{black}{\theta}}\mathbb{A}=\operatorname{Tr}\bigg{[}i\hbar\frac{d\rho}{dt}\ln\rho+[\mathbb{H},\rho]\bigg{]}.$
(89)

And for dissipative systems, Eq. (7) becomes

$-i\hbar\frac{dS_{\text{gen}}}{dt}=-i\frac{1}{4\mathbb{G}_{4}}{\color[rgb]{1,1,1}\contour{black}{\theta}}\mathbb{A}+\operatorname{Tr}\bigg{[}i\hbar\frac{d\rho}{dt}\ln\rho+[\mathbb{H},\rho]_{\text{Lb}}\bigg{]},$
(90)

which is the entropy of the black hole in the information manifold with no
classical components from the spacetime itself, just information geometry.
Based on Lindblad master equation, the $[\mathbb{H},\rho]_{\text{Lb}}$ term is
the time evolution of the density under the influence of the interaction
Hamiltonian $\mathbb{H}_{i}$ in an open dissipative system [83]

$\frac{d\rho(t)}{dt}\equiv[\mathbb{H},\rho]_{\text{Lb}}:=\frac{1}{i\hbar}[\mathbb{H}_{i}(t),\rho(0)]-\frac{1}{\hbar^{2}}\int^{t}_{0}dt^{\prime}[\mathbb{H}_{i}(t),[\mathbb{H}_{i}(t^{\prime}),\rho(t)]].$
(91)

The question now is: what is the meaning of the area $\mathbb{A}$ in the
information manifold, and how is it precisely related to the spatial extremal
area $A$ in the spacetime? The answer lies in the definition of extremal
surface in dS space, which corresponds to the so–called _pseudo–entropy_ [23,
24]. Such dS extremal surface [119, 120] could be compared with that RT
surface in the AdS space. In order to get the extremal surfaces in dS space,
we must complexify time such that average speed on the timelike surfaces,
defined by the ratio of the shortest spatial angular length $l$ of a dS space
with a radius $R_{dS}$ to the shortest time $\epsilon$, becomes the
determining factor of the size of the extermal surface, i.e.

$A_{dS}=-\pi R^{2}_{dS}\left(\frac{l}{c\epsilon}-1\right).$ (92)

The last equation shows that for some cases, $l/\epsilon>1$, the area could be
negative,202020The area could be imaginary for dS spaces with different
dimensions, see Ref. [120]. which would lead to negative or even complex
valued entropy! Before the last comment gets “frown upon”, this could be
necessary to avoid the disappearance of the spatial surfaces of dS Rindler
wedge [121]. This might not be well–appreciated as it says there could be
non–unitary states in CFT, i.e, the corresponding Hamiltonian $\mathbb{H}$
could be non–Hermitian. Very recently, it is proved that the non–Hermiticity
stems from the fact that the non–unitary CFT, dual to dS, lives on a
space–like surface and the time coordinate emerging from an Euclidean CFT [34]
related to the previously mentioned blurred space. When we shift to the
language of information manifolds, we surprisingly find some “untimely
meditations” about the necessity of complexfying the spacetime and probability
distributions so that we get a Fisher metric as an averaged metric over
spacetime fluctuations with a Lorentzian signature [111]. Consequently,
claiming the necessity of non–Hermitian Hamiltonian corresponding to the
Einstein–Hilbert formulation of GR in information manifold suggests that we
can study the dynamics of a Wheeler-deWitt Hamiltonian [122, 123] described by
the spatial metric black$\gamma$ on information manifold. An example of the
pseudo–Hermitian Wheeler–deWitt Hamiltonian is discussed in details in [124],
and detailed calculations of such Hamiltonian in information manifold could be
followed from Ref. [125, 126, 127] but they are left for a future study.

One last thing to be said about the RT formula of pseudo–entropy. By comparing
the approach212121See Ref. [120] for the analysis of RT formula in $2D$
information manifolds. followed in [128], the RT formula in $4D$ dS becomes

$S_{\text{gen.RT}}=\frac{A_{dS}}{4G_{4}}=-\frac{\pi
R^{2}_{dS}}{4G_{4}}\ln(\frac{l}{c\epsilon})\sim-\frac{\pi
R^{2}_{dS}}{4G_{4}}\left(\frac{l}{c\epsilon}-1\right),$ (93)

which is the formula of the entropy of black hole when the horizon coincides
with the RT extremal surface. And to get the expression of the entropy of the
extremal surface in information manifold, simply replace $A\to\mathbb{A}$ and
$G_{4}\to\mathbb{G}_{4}$, where $G_{4}$ is the gravitational constant in $4D$
spacetime as given in Eq. (81). For more details on the leading divergent term
in the previous equation and its relation to the holographic entanglement
entropy, see Ref. [129].

##  5 Discussions and Conclusions

As we have seen, our study suggests reducing the geometrical properties,
including spacetime itself, to an information geometry language in a way that
could evolve the insight on the deep connection between physics and
information. In this work, we studied in details the
coarse–grained/fined–grained entropy of the black hole that obeys the second
law of thermodynamics. We analyzed the entropy–area law corrected by von
Neumann entropy of the quantum matter outside its event horizon in order to
obey second law of thermodynamics and to preserve information. We constructed
the corresponding form of this corrected entropy–area law in quantum
information geometric language. Consequently, a corresponding spacetime
emerges from the quantum information. We discussed the link between Wald and
Jacobson approaches of thermodynamic/gravity correspondence and Fisher
pseudo–Riemannian metric of information manifold that guarantees extending the
geometric interpretation to any quantum theory. We formulated Einstein’s field
equations in information geometry forms, and we obtained a modified Ricci
tensor that helped constructing a Lagrangian of such theory. Also, we used the
modified Ricci tensor to introduce the reduced Einstein tensor
$\tilde{\mathbb{G}}_{\mu\nu}$, which is directly related to the
energy–momentum tensor in such manifold. The formulated Einstein’s field
equations led into two interesting outcomes stemming fundamentally from
information geometry. The first result is finding an informatic origin of a
positive cosmological constant that is founded on Fisher metric. This
cosmological constant resembles those found in Lovelock’s theories in de
Sitter background due to complexifying time and the Gaussian exponential
families of probability distributions. The second result is a time varying
gravitational constant that resembles the idea of Dirac’s large number
hypothesis and predicts varying of gravitational constant based on simple
analysis of nature constants. We extended our analysis into the information
manifold and wrote down a dynamical equation for the entropy in an information
manifold using Liouville–von Neumann equation for the quantum system
Hamiltonian. According to our results, the Hamiltonian in the information
manifold is allowed to be non–Hermitian. The resulting dynamical equation
provides a clue to a direction that could ameliorate the problem of time.

It is worth noting that relating Jacobson endeavors to information geometry
requires considering non–equilibrium thermodynamics of spacetime and its
associated dissipative gravity [130, 65]. This relation comes from the
coarse–graining process and the suggested quantum thermodynamical origin of
spacetime. To achieve fine–graining, the observational entropy will match von
Neumann entropy after several consecutive coarse–graining processes [51]. And
for finite–dimensional systems like spacetime itself, as $x^{\mu}$ are
obviously finite, the observational entropy can be expressed as relative
entropy. Moreover, a non–local heat fluxes is proven to take place in
non–equilibrium thermodynamical gravitational systems, for both Einstein’s
general relativity and the scalar–tensor modifications, due to their
dissipative characters [65]. When applied to Rindler spacetime, The thermal
character of such heat flux extends the vacuum thermal state to include all
possible Rindler wedges not just that of a single observer. So, an accelerated
observer would access information, due to entanglement entropy, on spacelike
slices. Consequently, the restricting the neighborhood of the Rindler wedge
origin, as spacelike slice around it, determines the expansion coefficient. As
the time rate change of the expansion coefficient is related to the
dissipative energy coupled to the bulk and shear viscosity, then it relates
the entropy, both internal at the irreversible level and exchanged at the
reversible level, to the Equivalence Principle in such dissipative systems.
Moreover, the time rate change of the expansion coefficient states a universal
relation between viscosity and internal viscous part of the entropy density as
found in the AdS/CFT [29].

We know that the collapse of a pure state results in a mixed state, which is a
process the unitary transformations cannot achieve in irreversible processes
due to the problem of preserving the norm of the wave functions. Moreover,
there is no such realistic quantum system that could be described by a pure
state [131, 132]. So, we are left with non–unitary transformations. As the
quantum time–reversible processes governed by Schrodinger evolution equation
are always unitary, then there is an obvious contradiction in assuming those
processes to comprise the non–reversible macro physics such as the second law
of thermodynamics. There is a long debate on the optimal epistemic and/or
ontological way to resolve this contradiction, it is usually discussed under
the umbrella of wave function jump and dissipative systems, see Ref. [133] for
more details. One of the remarkable suggestions is that the microscopic
variables indeed evolve according to non–unitary processes [134]. So, based on
the non–unitary irreversible von Neumann’s measurement transitions that render
mixed states [135], Ref. [136] provides an account on assuming that wave
function collapse takes place spontaneously and randomly in space and time. We
know that von Neumann entropy is a function in basis–independent density
operators, meanwhile the reversible Shannon entropy, related to Gibbs and
Boltzmann ones, is a function in the probability density matrices. If von
Neumann and Shannon entropies are related, then there must be some stochastic
variables that correspond to coarse–graining Liouville equation. Classically
this is associated with a loss in information, which is discussed in Balian
_et al._ endeavor [20]. At the quantum level, this is associated with a loss
in phase coherence in quantum states. This is suggested to happen in a
non–unitary transformation from a pure state to mixed one [136]. This
guarantees the validity of the second law of thermodynamics despite having
Liouville equation to be governed by phase–space Hamiltonians. However, this
does not demand making change to the Schrodinger equation; rather it says that
Liouville equation is physically broken at the quantum level. The price could
be replacing the regular Liouville equation with the more general Lindblad
master equation of open quantum systems [118], which allows the corresponding
Hamiltonian to have non–Hermitian parts [83]. These non–unitary quantum
processes are found in many high energy systems, such as CFTs with zero and
negative central charges that exhibit entanglement entropy [137, 138], in
condensed matter systems [139], and even in quantum electronics [140].

A final comment on complexifying time related to what is mentioned in Isham’s
report on the problem of time [141]. In the example, if we focus only on the
properties of an open quantum system, then the relevant state becomes that of
the reduced density matrix obtained by summing over–in another word by tracing
out–the states of the surroundings. If the states of the surroundings are
approximately orthogonal–Balian _et al._ assume that too—then the density of
quantum system exhibits decoherence. This is guaranteed to be true as the
reduced density matrix is proved to be governed in different examples of
Lindblad master equation [142] such as spatial decoherence [143] and quantum
Brownian motion [144]. Isham emphasizes that the inability to find a
satisfactory unitary Hamiltonian for Wheeler–de Witt equation should not be
considered necessarily as a disaster. Rather, this “might reflect something of
genuine physical significance”. For example, in Hartle and Hawking [145] and
Vilenkin [146] endeavors, time becomes complex due to the non–unitary
evolution.

Aknowledgement

The author would like to thank Ahmed Farag Ali for for discussions and
comments during the preparation of this work.

##  6 Appendix

This appendix is dedicated for the detailed calculations of the connection
terms and their derivatives in Einstein equation in information manifold.

If we substitute Eq. (3.6) in the derivative terms of Eq. (69) we get

$\displaystyle\medmath$
$\displaystyle\medmath{=\mathbb{g}^{\alpha\beta}\partial_{\alpha}\left[\langle\partial_{\mu}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle\right]-\mathbb{g}^{\alpha\beta}\partial_{\mu}\left[\langle\partial_{\alpha}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\alpha}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle\right]}$
$\displaystyle\medmath{+\partial_{\alpha}(\mathbb{g}^{\alpha\beta})\left[\langle\partial_{\mu}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle\right]-\partial_{\mu}(\mathbb{g}^{\alpha\beta})\left[\langle\partial_{\alpha}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\alpha}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle\right]}$

Also, if we substitute Eq. (3.6) in the multiplicative terms of Eq. (69) we
get

$\displaystyle\medmath{\mathbb{\Gamma}^{\alpha}_{\beta\alpha}\mathbb{\Gamma}^{\beta}_{\mu\nu}-\mathbb{\Gamma}^{\alpha}_{\beta\nu}\mathbb{\Gamma}^{\beta}_{\mu\alpha}=\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Bigg{\\{}}$
$\displaystyle\medmath{\left[\langle\partial_{\alpha}\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle-\frac{1}{2}\langle\partial_{\alpha}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\right]\times\left[\langle\partial_{\mu}\partial_{\nu}\varrho\partial_{\lambda}\varrho\rangle-\frac{1}{2}\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\lambda}\varrho\rangle\right]}$
$\displaystyle-$
$\displaystyle\medmath{\left[\langle\partial_{\nu}\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle-\frac{1}{2}\langle\partial_{\nu}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\right]\times\left[\langle\partial_{\mu}\partial_{\alpha}\varrho\partial_{\lambda}\varrho\rangle-\frac{1}{2}\langle\partial_{\mu}\varrho\partial_{\alpha}\varrho\partial_{\lambda}\varrho\rangle\right]\Bigg{\\}}}$
(A2)

We distribute the outer derivative applied on first line in Eq. (6), together
with exploiting the properties in Eq. (65), to get

$\displaystyle\medmath{\mathbb{g}^{\alpha\beta}\partial_{\alpha}\left[\langle\partial_{\mu}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle\right]-\mathbb{g}^{\alpha\beta}\partial_{\mu}\left[\langle\partial_{\alpha}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\alpha}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle\right]=}$
$\displaystyle\medmath{\mathbb{g}^{\alpha\beta}\Bigg{\\{}\langle\partial_{\mu}\partial_{\nu}\varrho\partial_{\alpha}\partial_{\beta}\varrho\rangle-\langle\partial_{\alpha}\varrho\partial_{\mu}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\Big{[}\langle\partial_{\mu}\partial_{\alpha}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle+\langle\partial_{\mu}\varrho\partial_{\alpha}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle+\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\alpha}\partial_{\beta}\varrho\rangle\Big{]}}$
$\displaystyle\medmath{-\langle\partial_{\alpha}\partial_{\nu}\varrho\partial_{\mu}\partial_{\beta}\varrho\rangle+\langle\partial_{\mu}\varrho\partial_{\alpha}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle+\frac{1}{2}\Big{[}\langle\partial_{\alpha}\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle+\langle\partial_{\alpha}\varrho\partial_{\mu}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle+\langle\partial_{\alpha}\varrho\partial_{\nu}\varrho\partial_{\mu}\partial_{\beta}\varrho\rangle\Big{]}\Bigg{\\}}=}$
$\displaystyle\medmath{\mathbb{g}^{\alpha\beta}\Big{[}\langle\partial_{\mu}\partial_{\nu}\varrho\partial_{\alpha}\partial_{\beta}\varrho\rangle+\frac{1}{2}\langle\partial_{\mu}\varrho\partial_{\alpha}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle+\frac{1}{2}\langle\partial_{\mu}\varrho\partial_{\beta}\partial_{\nu}\varrho\partial_{\alpha}\varrho\rangle}$
$\displaystyle\hskip
167.87108pt\medmath{-\langle\partial_{\mu}\partial_{\beta}\varrho\partial_{\nu}\partial_{\alpha}\varrho\rangle-\frac{1}{2}\langle\partial_{\alpha}\varrho\partial_{\mu}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\mu}\varrho\partial_{\alpha}\partial_{\beta}\varrho\partial_{\nu}\varrho\rangle\Big{]}}$
(A3)

We contract Eq. (6) using $\mathbb{g}_{\mu\nu}$ to get

$\displaystyle\medmath{\mathbb{g}^{\alpha\beta}\partial_{\alpha}\left[\langle\partial_{\kappa}\partial_{\kappa}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\kappa}\varrho\partial_{\kappa}\varrho\partial_{\beta}\varrho\rangle\right]-\mathbb{g}^{\alpha\beta}\partial_{\kappa}\left[\langle\partial_{\alpha}\partial_{\kappa}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\alpha}\varrho\partial_{\kappa}\varrho\partial_{\beta}\varrho\rangle\right]=}$
$\displaystyle\medmath{\mathbb{g}^{\alpha\beta}\Big{[}\langle\partial_{\kappa}\partial_{\kappa}\varrho\partial_{\alpha}\partial_{\beta}\varrho\rangle+\frac{1}{2}\langle\partial_{\kappa}\varrho\partial_{\alpha}\partial_{\kappa}\varrho\partial_{\beta}\varrho\rangle+\frac{1}{2}\langle\partial_{\kappa}\varrho\partial_{\beta}\partial_{\kappa}\varrho\partial_{\alpha}\varrho\rangle}$
$\displaystyle\hskip
167.87108pt\medmath{-\langle\partial_{\kappa}\partial_{\beta}\varrho\partial_{\kappa}\partial_{\alpha}\varrho\rangle-\frac{1}{2}\langle\partial_{\alpha}\varrho\partial_{\kappa}\partial_{\kappa}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\kappa}\varrho\partial_{\alpha}\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\Big{]}}$
(A4)

Now, we multiply Eq. (6) by $-\frac{1}{2}\mathbb{g}_{\mu\nu}$ then pick terms
from the result of such multiplication that match with the terms in Eq. (6) to
show that Eq. (6), i.e. the derivative parts in Eq. (69), corresponds to the
cosmological coupling constant term in Lovelock theory of gravity [102]. The
middle steps are obtained using the definition of the $\mathpzc{G}_{\mu\nu}$
in Eq. (A20) and the definition of $\mathbb{g}_{\mu\nu}$ in Eq. (A21). The
$\mathbb{g}^{\mu\nu}\mathbb{g}_{\mu\nu}=\mathpzc{G}^{\mu\nu}\mathpzc{G}_{\mu\nu}=D$,
where $D$ is the dimension of the manifold. Collect the fourth term in the
last two lines of Eq. (6) with $-\frac{1}{2}\mathbb{g}_{\mu\nu}\times$the
fourth term in the last two lines of Eq. (6) to get

$-\mathbb{g}^{\alpha\beta}\langle\partial_{\mu}\partial_{\alpha}\varrho\partial_{\nu}\partial_{\beta}\varrho-\frac{1}{2}\mathbb{g}_{\mu\nu}\partial_{\kappa}\partial_{\alpha}\varrho\partial_{\kappa}\partial_{\beta}\varrho\rangle=-(1-D/2)\mathbb{g}_{\mu\nu}$
(A5)

Collect the second and the third terms in the last two lines of Eq. (6) with
$-\frac{1}{2}\mathbb{g}_{\mu\nu}\times$the second and third terms in the last
two lines of Eq. (6) to get

$\frac{1}{2}\mathbb{g}^{\alpha\beta}\langle\partial_{\alpha}\partial_{\nu}\varrho\partial_{\beta}\partial_{\mu}\varrho+\partial_{\alpha}\partial_{\mu}\varrho\partial_{\beta}\partial_{\nu}\varrho-\mathbb{g}_{\mu\nu}\partial_{\kappa}\partial_{\alpha}\varrho\partial_{\kappa}\partial_{\beta}\varrho\rangle=(1-D/2)\mathbb{g}_{\mu\nu}$
(A6)

It is obvious that the last two results, Eq. (A5) and Eq. (A6), cancel each
other. Collect the last term in the last two lines of Eq. (6) with
$-\frac{1}{2}\mathbb{g}_{\mu\nu}\times$the last term in the last two lines of
Eq. (6) to get

$-\frac{1}{2}\mathbb{g}^{\alpha\beta}\langle\partial_{\alpha}\partial_{\beta}\varrho(\partial_{\mu}\partial_{\nu}\varrho-\frac{1}{2}\mathbb{g}_{\mu\nu}\partial_{\kappa}\partial_{\kappa}\varrho)\rangle=-(D/2-D^{2}/4)\mathbb{g}_{\mu\nu}$
(A7)

Collect the first term in the last two lines of Eq. (6) with
$-\frac{1}{2}\mathbb{g}_{\mu\nu}\times$the first term in the last two lines of
Eq. (6) to get

$\frac{1}{2}\mathbb{g}^{\alpha\beta}\langle\partial_{\alpha}\partial_{\beta}\varrho(\partial_{\mu}\partial_{\nu}\varrho-\frac{1}{2}\mathbb{g}_{\mu\nu}\partial_{\kappa}\partial_{\kappa}\varrho)\rangle=(D/2-D^{2}/4)\mathbb{g}_{\mu\nu}$
(A8)

It is obvious that the last two results, Eq. (A7) and Eq. (A8), cancel each
other. We are left with the fifth term in the last two lines of Eq. (6) and
the $-\frac{1}{2}\mathbb{g}_{\mu\nu}\times$fifth term in the last two lines of
Eq. (6). We combine both to get

$-\frac{1}{2}\mathbb{g}^{\alpha\beta}\langle\partial_{\alpha}\varrho\partial_{\beta}\varrho(\partial_{\mu}\partial_{\nu}\varrho-\frac{1}{2}\mathbb{g}_{\mu\nu}\partial_{\kappa}\partial_{\kappa}\varrho)\rangle=-(D/2-D^{2}/4)\mathbb{g}_{\mu\nu}$
(A9)

Therefore, Eq. (A9) is the only part that contributes to Eq. (6)

$\Big{(}\partial_{\alpha}\mathbb{\Gamma}^{\alpha}_{\mu\nu}-\partial_{\nu}\mathbb{\Gamma}^{\alpha}_{\mu\alpha}\Big{)}-\frac{1}{2}\mathbb{g}_{\mu\nu}\mathbb{g}^{\kappa\lambda}\Big{(}\partial_{\alpha}\mathbb{\Gamma}^{\alpha}_{\kappa\lambda}-\partial_{\kappa}\mathbb{\Gamma}^{\alpha}_{\lambda\alpha}\Big{)}=\frac{1}{2}D(\frac{D}{2}-1)\mathbb{g}_{\mu\nu}$
(A$9^{\prime}$)

where $\Lambda=D(D/2-1)$ is defined for $D\geqslant 2$ dimensional manifold.

For the second line in Eq. (6), we use

$\displaystyle\partial_{\mu}(\mathbb{g}^{\alpha\beta})$
$\displaystyle=-\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\partial_{\mu}\mathbb{g}_{\kappa\lambda}=\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\langle\partial_{\mu}\varrho\partial_{\kappa}\varrho\partial_{\lambda}\varrho\rangle$
(A10)

so that the second line in Eq. (6) is

$\displaystyle\medmath{\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Bigg{\\{}\langle\partial_{\alpha}\varrho\partial_{\kappa}\varrho\partial_{\lambda}\varrho\rangle\left[\langle\partial_{\mu}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle\right]-\langle\partial_{\mu}\varrho\partial_{\kappa}\varrho\partial_{\lambda}\varrho\rangle\left[\langle\partial_{\alpha}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\alpha}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle\right]\Bigg{\\}}}$
(A11)

Next, we expand Eq. (6). The terms with no $1/4$ resulting from such expansion
are

$\displaystyle\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Big{[}$
$\displaystyle\langle\partial_{\alpha}\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\partial_{\nu}\varrho\partial_{\lambda}\varrho\rangle-\frac{1}{2}\langle\partial_{\alpha}\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\lambda}\varrho\rangle-\frac{1}{2}\langle\partial_{\alpha}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\partial_{\nu}\varrho\partial_{\lambda}\varrho\rangle$
$\displaystyle-\langle\partial_{\nu}\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\partial_{\alpha}\varrho\partial_{\lambda}\varrho\rangle+\frac{1}{2}\langle\partial_{\nu}\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\varrho\partial_{\alpha}\varrho\partial_{\lambda}\varrho\rangle+\frac{1}{2}\langle\partial_{\nu}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\partial_{\alpha}\varrho\partial_{\lambda}\varrho\rangle\Big{]}$
(A12)

For the first line in Eq. (6), exchange $\beta$ and $\lambda$ in the first
term, change the sign in that term according to Eq. (65). Then, collect that
term with the third term in the same line. And for the second line in Eq. (6),
exchange $\kappa$ and $\alpha$ in the first term, change the sign in that term
according to Eq. (65). Then, collect that term with the third term in the same
line. Also for the second and fifth term in Eq.(6), change the sign in those
terms according to Eq. (65). Thus, Eq. (6) becomes

$\displaystyle\medmath{-\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Bigg{\\{}\langle\partial_{\alpha}\varrho\partial_{\kappa}\varrho\partial_{\lambda}\varrho\rangle\left[\langle\partial_{\mu}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle\right]-\langle\partial_{\mu}\varrho\partial_{\kappa}\varrho\partial_{\lambda}\varrho\rangle\left[\langle\partial_{\alpha}\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle-\frac{1}{2}\langle\partial_{\alpha}\varrho\partial_{\nu}\varrho\partial_{\beta}\varrho\rangle\right]\Bigg{\\}}}$
$\displaystyle\medmath{+\frac{1}{2}\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\bigg{[}\langle\partial_{\alpha}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\lambda}\varrho\rangle-\langle\partial_{\nu}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\varrho\partial_{\alpha}\varrho\partial_{\lambda}\varrho\rangle\bigg{]}}$
(A13)

We see that the first line in Eq. (6) cancels with Eq. (A11), which is
obtained from the second line in Eq.(6). Moreover, we add the second line in
Eq. (6) to the terms with $1/4$ in Eq. (6). Then, we contract the $\mu\nu$
indices of the result of such addition, multiply it with
$-\frac{1}{2}\mathbb{g}_{\mu\nu}$, and add that to the original terms before
the $\mu\nu$ contraction such that we introduce a new tensor

$\displaystyle\mathbb{\widetilde{R}}_{\mu\nu}$
$\displaystyle=\Big{(}\mathbb{\Gamma}^{\alpha}_{\beta\alpha}\mathbb{\Gamma}^{\beta}_{\mu\nu}-\mathbb{\Gamma}^{\alpha}_{\beta\nu}\mathbb{\Gamma}^{\beta}_{\mu\alpha}\Big{)}-\frac{1}{2}\mathbb{g}_{\mu\nu}\mathbb{g}^{\kappa\lambda}\Big{(}\mathbb{\Gamma}^{\alpha}_{\beta\alpha}\mathbb{\Gamma}^{\beta}_{\kappa\lambda}-\mathbb{\Gamma}^{\alpha}_{\beta\kappa}\mathbb{\Gamma}^{\beta}_{\lambda\alpha}\Big{)}$
$\displaystyle=\frac{1}{4}\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Big{[}\langle\partial_{\alpha}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\partial_{\lambda}\varrho\rangle-\langle\partial_{\nu}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\partial_{\mu}\varrho\partial_{\alpha}\varrho\partial_{\lambda}\varrho\rangle\Big{]}$
(A14)

Here we follow Ref. [98]. In order to relate the Fisher metric with the
entropy defined as in Eq. (29), or Eq. (39), we know the density means also
the relative share of certain energy state $E(\mathbb{X}^{\mu})$ from the
total collection of all energy states in the partition function $Z$ [91], i.e.

$\rho(\xi^{\mu};\mathbb{X}^{\mu})=\exp\left[-\beta E(\mathbb{X}^{\mu})-\ln
Z(\xi^{\mu})\right]$ (A15)

which reintroduces the probability distributions to the family of
exponentials. Then, we can define the density generally as

$\rho(\xi^{\mu})=\exp\left[\xi^{\mu}\mathbb{E}_{\mu}(\mathbb{X}^{\nu})-\phi(\xi^{\nu})\right]$
(A16)

which is Eq. (84). The corresponding dual density becomes

$\varrho=-\ln\rho=\phi(\xi^{\nu})-\xi^{\mu}\mathbb{E}_{\mu}$ (A17)

Applying the first and the second derivative with respect to $\xi^{\mu}$ on
Eq. (A17) yields

$\displaystyle\partial_{\mu}\varrho$
$\displaystyle=\partial_{\mu}\phi-\mathbb{E_{\mu}(\mathbb{X}^{\nu})}$ (A18)
$\displaystyle\partial_{\mu}\partial_{\nu}\varrho$
$\displaystyle=\partial_{\mu}\partial_{\nu}\phi$ (A19)

In light of Eq. (29), Eq. (39) and Eq. (63), the last Eq. (A18–A19) can be
rearranged to get

$\langle\mathpzc{G}_{\mu\nu}\rangle=\langle\partial_{\mu}\partial_{\nu}\varrho\rangle=\partial_{\mu}\partial_{\nu}\phi=\mathbb{g}_{\mu\nu}$
(A20)

Since $\langle\partial_{\mu}\varrho\rangle=0$, then we apply another
differentiation and use Eq. (A20) to get

$\mathbb{g}_{\mu\nu}=\langle\partial_{\mu}\partial_{\nu}\varrho\rangle=\langle\partial_{\mu}\varrho\partial_{\nu}\varrho\rangle$
(A21)

despite that
$\partial_{\mu}\partial_{\nu}\varrho\neq\partial_{\mu}\varrho\partial_{\nu}\varrho$.
Moreover, since

$\langle\mathbb{E}_{\mu}\rangle=\partial_{\mu}\phi~{},$ (A22)

then the last three equations give

$\displaystyle\mathbb{g}_{\mu\nu}$
$\displaystyle=\langle\mathbb{E}_{\mu}\mathbb{E}_{\nu}\rangle-\langle\mathbb{E}_{\mu}\rangle\langle\mathbb{E}_{\nu}\rangle$
(A23)
$\displaystyle=\langle\mathbb{E}_{\mu}\mathbb{E}_{\nu}\rangle-\partial_{\mu}\phi\langle\mathbb{E}_{\nu}\rangle=-\langle\partial_{\mu}\varrho\mathbb{E}_{\nu}\rangle$
(A24)

The last three relations will help us to construct the Christoffel symbol [91,
92], from the connections in Eq. (41b), and consequently the Riemann curvature
tensor as functions in the density vectors as we will see in a little bit.

Now, we substitute Eq. (62) and Eq. (A18) into Eq. (6), then expand, we obtain

$\displaystyle\mathbb{\widetilde{R}}_{\mu\nu}=\frac{1}{4}$
$\displaystyle\Bigg{\\{}D\partial_{\mu}\phi\partial_{\nu}\phi-\mathbb{g}^{\beta\kappa}\partial_{\mu}\phi\langle\mathbb{E}_{\nu}\mathpzc{G}_{\beta\kappa}\rangle-\mathbb{g}^{\alpha\lambda}\partial_{\nu}\phi\langle\mathbb{E}_{\mu}\mathpzc{G}_{\alpha\lambda}\rangle$
$\displaystyle+\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Big{[}\langle\mathbb{E}_{\mu}\mathpzc{G}_{\alpha\lambda}\rangle\langle\mathbb{E}_{\nu}\mathpzc{G}_{\beta\kappa}\rangle-\langle\partial_{\alpha}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\mathbb{E}_{\mu}\mathbb{E}_{\nu}\partial_{\lambda}\varrho\rangle\Big{]}$
$\displaystyle+\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\langle\partial_{\alpha}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\Big{[}\partial_{\mu}\phi\langle\mathbb{E}_{\nu}\partial_{\lambda}\varrho\rangle+\partial_{\nu}\phi\langle\mathbb{E}_{\mu}\partial_{\lambda}\varrho\rangle\Big{]}\Bigg{\\}}$
(A25)

We focus on the last line of Eq. (6). Eq. (A18) and Eq. (A20) yield
$\langle\partial_{\alpha}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle=\langle\partial_{\beta}\varrho\mathpzc{G}_{\alpha\kappa}\rangle=\partial_{\beta}\phi\mathbb{g}_{\alpha\kappa}-\langle\mathbb{E}_{\beta}\mathpzc{G}_{\alpha\kappa}\rangle$.
And the terms
$\langle\partial_{\mu}\varrho\mathbb{E}_{\nu}\rangle=-\mathbb{g}_{\mu\nu}$ as
we infer from Eq. (A24). Then, we expand Eq. (6) to get

$\mathbb{\widetilde{R}}_{\mu\nu}=\frac{1}{4}\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Big{[}\langle\mathbb{E}_{\mu}\mathpzc{G_{\alpha\beta}}\rangle\langle\mathbb{E}_{\nu}\mathpzc{G_{\kappa\lambda}}\rangle-\langle\mathbb{E}_{\mu}\rangle\langle\mathbb{E}_{\nu}\rangle\mathbb{g}_{\alpha\beta}\mathbb{g}_{\kappa\lambda}-\langle\partial_{\alpha}\varrho\partial_{\beta}\varrho\partial_{\kappa}\varrho\rangle\langle\mathbb{E}_{\mu}\mathbb{E}_{\nu}\partial_{\lambda}\varrho\rangle\Big{]}$
(A26)

The last term in Eq. (A26) is negligable as
$\langle\mathbb{E}_{\mu}\mathbb{E}_{\nu}\partial_{\lambda}\varrho\rangle\sim-\langle\partial_{\lambda}(\mathbb{E}_{\mu}\mathbb{E}_{\nu})\rangle$
as Eq. (A19) says that
$\partial_{\lambda}\mathbb{E}_{\mu}=0=\mathbb{E}_{\nu}\partial_{\lambda}\mathbb{E}_{\mu}$,
which means
$\langle\mathbb{E}_{\mu}\partial_{\lambda}\mathbb{E}_{\nu}+\mathbb{E}_{\nu}\partial_{\lambda}\mathbb{E}_{\mu}\rangle=\langle\partial_{\lambda}(\mathbb{E}_{\mu}\mathbb{E}_{\nu})\rangle=0$.
Therefore, Eq. (A26) becomes

$\mathbb{\widetilde{R}}_{\mu\nu}=\frac{1}{4}\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Big{[}\langle\mathbb{E}_{\mu}\mathpzc{G_{\alpha\beta}}\rangle\langle\mathbb{E}_{\nu}\mathpzc{G_{\kappa\lambda}}\rangle-\langle\mathbb{E}_{\mu}\rangle\langle\mathbb{E}_{\nu}\rangle\mathbb{g}_{\alpha\beta}\mathbb{g}_{\kappa\lambda}\Big{]}$
(A27)

As we defined the stochastic variables
$\mathbb{X}^{\mu}\equiv\mathbb{X}^{\mu}(\langle
x^{\nu}\rangle,\sigma_{x^{\nu}})$ in the beginning of subsection (3.5), the
same can be done for the stochastic metric
$\mathpzc{G}_{\mu\nu}(\xi,\mathbb{X})$ as we expand it around the $\langle
x^{\mu}\rangle$ while we keep $\sigma_{x^{\mu}}$ as it is. So,
$\mathbb{X}\equiv\mathbb{X}^{\mu}(\langle x\rangle)$, And the metric becomes

$\mathpzc{G}_{\mu\nu}(\mathbb{X})=\mathpzc{G}_{\mu\nu}(\langle
x\rangle)+\dot{\mathpzc{G}}_{\mu\nu}(\langle
x\rangle)\Big{(}\mathbb{X}-\langle
x\rangle\Big{)}+\frac{1}{2}\ddot{\mathpzc{G}}_{\mu\nu}(\langle
x\rangle)\Big{(}\mathbb{X}-\langle x\rangle\Big{)}^{2}+\cdots$ (A28)

where

$\dot{\mathpzc{G}}_{\mu\nu}(\langle
x\rangle)=\lim\limits_{\mathbb{X}\to\langle
x\rangle}\frac{\partial}{\partial\mathbb{X}}{\mathpzc{G}}_{\mu\nu}(\mathbb{X})$
(A29)

and $\ddot{\mathpzc{G}}_{\mu\nu}(\langle x\rangle)$ is the usual second
derivative of the above equation. Defining $\mathpzc{G}_{\mu\nu}$ as a
function in $\langle x\rangle$ allows us get
$\langle\mathpzc{G}_{\mu\nu}(\langle
x\rangle)\rangle=\mathpzc{G}_{\mu\nu}(\langle x\rangle)=\mathpzc{G}_{\mu\nu}$
as averaging the average is a redundant process. Now we substitute Eq. (A28)
in Eq. (A27), together with the help of Eq. (A18-A22) and the approximation
$\langle\mathbb{E}_{\mu}(\mathbb{X}-\langle
x\rangle)^{n}\rangle\sim\partial_{\mu}\phi\langle(\mathbb{X}-\langle
x\rangle)^{n}\rangle$, to obtain

$\displaystyle\mathbb{\widetilde{R}}_{\mu\nu}=\frac{1}{4}\mathbb{g}^{\alpha\kappa}\mathbb{g}^{\beta\lambda}\Bigg{\\{}$
$\displaystyle\Big{[}\mathpzc{G}_{\alpha\beta}\mathpzc{G}_{\kappa\lambda}-\mathbb{g}_{\alpha\beta}\mathbb{g}_{\kappa\lambda}\Big{]}+\mathpzc{G}_{\kappa\lambda}\Big{[}\dot{\mathpzc{G}}_{\alpha\beta}\langle\mathbb{X}-\langle
x\rangle\rangle+\frac{1}{2}\ddot{\mathpzc{G}}_{\alpha\beta}\langle(\mathbb{X}-\langle
x\rangle)^{2}\rangle+\cdots\Big{]}$
$\displaystyle+\mathpzc{G}_{\alpha\beta}\Big{[}\dot{\mathpzc{G}}_{\kappa\lambda}\langle\mathbb{X}-\langle
x\rangle\rangle+\frac{1}{2}\ddot{\mathpzc{G}}_{\kappa\lambda}\langle(\mathbb{X}-\langle
x\rangle)^{2}\rangle+\cdots\Big{]}\Bigg{\\}}\times\partial_{\mu}\phi\partial_{\nu}\phi$
(A30)

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