Correlations of mixed systems in confining backgrounds

$\text{Mahdis Ghodrati}^{a}$,

aAsia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea

e-mails<EMAIL_ADDRESS>
We show that the entanglement of purification and the critical distance
between the two mixed systems is a powerful measure in probing the phase
structures of QCD and confining backgrounds, as it can distinguish the scale
of chiral symmetry breaking versus the scale of confinement/deconfinement
phase transitions. For two symmetric strips with equal and finite width and
infinite length, and in the background of several confining geometries, we
numerically calculate the critical distance between them where the mutual
information vanishes and show that this quantity can probe the very rich phase
structures of these backgrounds. The geometries that we study here are AdS-
soliton, Witten Sakai Sugimoto and deformed Sakai-Sugimoto, Witten-QCD,
Klebanov-Strassler, Klebanov-Tseytlin, Klebanov-Witten, Maldacena-Nunez,
Nunez-Legramandi metric, and Domain-Wall QCD model. For each background we
also present the relation for the entanglement of purification. Finally, we
show that the Crofton forms of these geometries also behave in a universal
form where a “ well” is being observed around the IR wall, and therefore for
all confining backgrounds, the Crofton form would also be capable of
distinguishing the confining versus conformal backgrounds as it is also a tool
in the reconstruction of various bulk geometries.

###### Contents

  1. 1 Introduction
  2. 2 Mixed correlation measures in confining backgrounds
  3. 3 Critical distances in the confining backgrounds
    1. 3.1 AdS soliton background
    2. 3.2 Witten-Sakai-Sugimoto model
      1. 3.2.1 Deformed Sakai-Sugimoto
    3. 3.3 Witten-QCD
    4. 3.4 Klebanov-Strassler
    5. 3.5 Klebanov-Tseytlin
    6. 3.6 Klebanov-Witten
    7. 3.7 Maldacena-Nunez
    8. 3.8 Domain Wall AdS/QCD
    9. 3.9 Nunez-Legramandi metric
    10. 3.10 Backgrounds with smeared branes
  4. 4 Crofton form in confining backgrounds
    1. 4.1 AdS-soliton
    2. 4.2 Sakai-Sugimoto and deformed Sakai-Sugimoto
    3. 4.3 Klebanov-Strassler
    4. 4.4 Witten-QCD
    5. 4.5 Maldacena-Nunez
    6. 4.6 Klebanov-Tseytlin
    7. 4.7 Domain Wall QCD
    8. 4.8 Nunez-Legramandi metric
    9. 4.9 Other measures and applications
  5. 5 Conclusion

##  1 Introduction

From the advent of Ryu-Takayanagi (RT) prescription [1] which for calculating
the entanglement entropy of field theories uses calculations of the area of
extremal surface which is homologous to the boundary, the deep connections
between quantum information and geometry specially in the context of AdS/CFT
has been ground in. The other new dualities in this context such as
computational complexity/volume of the subregions [2, 3, 4, 5, 6, 7], and
mixed correlation measures/minimal wedge cross section [8, 9, 10] further
strengthened this connection.

However, it worths to note that for the dual of the entanglement wedge cross
section (EWCS) in the bulk, recently, several mixed correlation measures have
been proposed in the literature. These are measures of varied combinations of
quantum and classical correlations. They include entanglement of purification
(EoP)[8], logarithmic negativity [11], odd entropy [12], entanglement
distillation [13], and reflected entropy [14]. It is not still clear which of
these measures exactly would be dual to the minimal wedge cross section, or if
in the large $N$ limit, all or subset of them could work as the dual quantity,
however, here we only calculate the critical distance $D_{c}$ for two strips,
the mutual information and the minimal wedge cross section in the confining
geometries without considering any specific dual quantity for EWCS.

As the calculations of the properties of geometrical objects in the bulk are
much easier than the calculations of quantum measures in the boundary field
theories and some of these geometrical objects can deeply probe the bulk and
gather non-local information of it which would be dual to interesting quantum
measures of the boundary, one would expect that these geometrical objects can
also detect the rich phase structures of the exotic forms of matters such as
quark gluon plasmas, strange metals, high temperature superconductors,
superfluids, etc. The motivation of this work is to use minimal wedge cross
section or mutual information between two mixed systems in the background of
confining geometries to gain the properties of the phase structures of QCD.

We show in this work that the minimal wedge cross section of two symmetric
strips which as proposed in [8] could be dual to the mixed correlation
measures such as entanglement of purification between the two mixed state, can
capture a lot of information about the phase structures of various confining
models. Specially this would be true for QCD and confining systems. Using the
correlation measures of two symmetric mixed strips which would be easy to
calculate, one could gain a lot about the phase structures of QCD and
confining backgrounds, which is the motivation of this work. We found that in
addition to the direct calculation of mutual information or minimal wedge
cross section, in order to probe the phase structures of confining
backgrounds, it would actually be easier and more preferable to calculate the
critical distance between the two symmetric strips where the mutual
information suddenly drops to zero. In addition, it could even detect the
scale of chiral symmetry breaking and the confining-deconfining phase
transitions which is the main result of this work.

For doing so, we use several top-down confining backgrounds here. These top-
down, and ten-dimensional backgrounds use the fundamental models of string
theory with different Dp-brane structures. For example, the Witten model of
low-energy QCD is constructed through the type-IIA supergravity, using D4-D8
intersected branes. The boundary dual of this theory is a supersymmetric $4+1$
dimensional gauge theory which by higher temperatures dimensional reduction
would lead to a non-supersymmetric QCD in $3+1$ dimension. This theory has a
large number of colors and a large ’t Hooft coupling. Extending this model,
for instance, by adding flavor D8-branes, Sakai and Sugimoto could also
explain the chiral symmetry breaking and its restoration at high temperatures
by an interesting geometrical realization. The only free parameters in their
model is the Kaluza-Klein mass parameter $M_{KK}$ or $u_{KK}$ and the ’t Hooft
coupling $\lambda$ at that scale. Another example is Klebanov-Strassler which
contains $N$ regular and $M$ fractional D3 branes. Here for each metric we
discuss briefly about the D-brane structures of each background.

Here, we study the behavior of several quantum information quantities such as
entanglement entropy and mixed correlation measures such as EWCS/entanglement
of purification (EoP) versus these free parameters, such as $u_{KK}$, to probe
the phase structures of QCD in these backgrounds. We hope it could also help
to find the corrections needed to approach real QCD models at finite $N_{c}$
and $\lambda$ that could be gained from type-IIA string theory. Note that
specifically since the Sakai-Sugimoto model, both qualitatively and
quantitatively is very close to real QCD, our results in principle should be
close to the results of lattice QCD.

Additionally, in any holographic QCD model, one would have an IR wall where
the geometry would be terminated in the holographic direction, or there would
be some sorts of singularities there. In this work we also would like to check
the effects of these soft or hard walls and cut-offs and the terminated radius
$u_{KK}$ on the quantum information measures such as mutual information,
entanglement of purification and negativity, and the phase diagrams of various
confining backgrounds. Therefore, another aim of this work is to check how the
topology of different confining geometries would affect QCD properties and
if/how the new holographic quantum measures such as entanglement of
purification or quantum negativity, or the critical distance between two mixed
systems, (and later complexity or complexity of purification), could probe
these topological properties or dually the characteristics of QCD. This would
actually be in continuation of our previous work [5], which we used complexity
to probe the phase structures of Gubser’s models of QCD [15], dubbed $V_{1}$,
$V_{2}$, $V_{\text{QCD}}$ and $V_{\text{IHQCD}}$. In the current work however,
to construct the phase structure, we use new mixed correlations measures and
new confining backgrounds, including AdS-soliton, Witten-Sakai-Sugimoto and
deformed Sakai-Sugimoto, Witten-QCD, Klebanov-Strassler, Klebanov-Tseytlin,
Klebanov-Witten, Maldacena-Nunez, domain-wall QCD and Nunez-Legramandi metric.
So, first in section 2, we show how to generalize the new quantum measures in
confining backgrounds and then in section 3, we present the results for each
of our confining models.

Then, in section 4, we present the relations and plots for the behavior of
Crofton-from (which is a quantity in studying the Kinematic space and bulk
reconstruction [16]) for each of these confining geometries and show they all
have a universal behavior where a well could be detected around the
singularity while it would become constant at the bigger holographic
distances.

We finish with conclusion in section 5.

##  2 Mixed correlation measures in confining backgrounds

The generalization of entanglement entropy (EE) from conformal to non-
conformal cases would be from

$\displaystyle
S_{A}=\frac{1}{4G_{N}^{(d+2)}}\int_{\gamma}d^{d}\sigma\sqrt{G^{(d)}_{\text{ind}}},$
(2.1)

to

$\displaystyle S_{A}=\frac{1}{4G_{N}^{(10)}}\int d^{8}\sigma
e^{-2\Phi}\sqrt{G^{(8)}_{\text{ind}}},$ (2.2)

where $G_{N}^{(d+2)}$ is the $d+2$ dimensional Newton constant and
$G_{\text{ind}}^{(d)}$ is the induced string frame metric on $\gamma$. This
generalization would be due to the fact that, in general, in the non-conformal
theories, the volume of the $8-d$ compact dimensions and also the dilaton
would be non-constant.

Note that, because of the symmetry, or isotropy, in out setup, where we have
two equal strips, we expect that this formula works for the confining
backgrounds since the only difference with the conformal case is the wall at
the end of the geometry in the deep IR, in the large $u$, which does not break
the symmetry in the $x$ direction. Therefore, in the integral of equation 2.2,
we only need to consider the variation of this dilaton field along the radial
direction $u$, and therefore simply the term $e^{\phi}$ is multiplied by the
induced metric along the minimal wedge cross section.

Figure 1: The minimal wedge cross section for the two connected mixed systems
of $A$ and $B$, i.e, $\Sigma_{AB}$ is shown in yellow while the whole
entanglement wedge is shown in green.

We put the setup of figure 1 in the background of confining geometries (as in
figure 9 which is for the AdS soliton case) and probe various exotic features
of these top-down, supergravity and confining backgrounds and show that mixed
quantum entanglement measures and correspondingly the geometrical objects such
as minimal wedge cross section can detect the effects of singularities, walls,
throats and exotic topologies of such backgrounds.

First, note that if we write the confining gravitational backgrounds in the
string frame in the form of

$\displaystyle
ds^{2}=\alpha(\rho)[\beta(\rho)d\rho^{2}+dx^{\mu}dx_{\mu}]+g_{ij}d\theta^{i}d\theta^{j},$
(2.3)

where $\rho$ is the holographic radial coordinate in the interval
$\rho_{\Lambda}<\rho<\infty$, $x^{\mu},\ \ (\mu=0,1,...,d)$ parameterize
$\mathbb{R}^{d+1}$, and $\theta^{i},\ \ (i=d+2,...,9)$ are the $8-d$ internal
directions.

The bulk minimal wedge cross section constructed from two mixed boundary
regions, has been proposed in [8] to be dual to the entanglement of
purification between these two mixed systems of $A$ and $B$, as in figure 1.
For the simple Schwarzchild AdS black brane geometry and for two regions with
width of $l$ and distance $D$, in [9], this quantity has been found as

$\displaystyle\Gamma=\int_{z_{D}}^{z_{2l+D}}\frac{dz}{z^{d-1}\sqrt{1-\frac{z^{d}}{z_{h}^{d}}}},$
(2.4)

where $z_{D}$ and $z_{2l+D}$ are the two turning points.

This then should be generalized for the non-conformal and specifically for
confining geometry similar to the EE case. In fact, for several Dp-brane
geometries, in [17], several similar quantum information measures have been
calculated.

Here we could generalize it as

$\displaystyle\Gamma=\int_{u_{D}}^{u_{2l+D}}due^{-2\Phi}\sqrt{-\gamma_{ab}},$
(2.5)

where $\gamma_{ab}$ is the determinant of the eight dimensional induced metric
on $\Gamma$ that results from the ten dimensions by setting the two
coordinates of time and radial to zero, i.e, $dt=dr=0$.

Note that they could be some constant factors behind the integral which we
will set to one in our studied here. Also, $e^{-2\Phi}$ should be brought
here, which is due to the fact that in non-conformal geometries, the dilaton
field $\Phi$ and the volume of the compact dimensions are not constant and so
similar to the entanglement entropy case, are needed to be considered for the
formulation of the entanglement of purification as well. The dilaton field
would be denoted by $\Phi$ and the RR and NS fluxes would complete the
background which for each geometry is different.

One can define another quantity

$\displaystyle H(\rho)=e^{-4\Phi}V^{2}_{\text{int}}\alpha^{2},$ (2.6)

where $V_{\text{int}}=\int d\vec{\theta}\sqrt{\text{det}[g_{ij}]}$ is the
volume of the internal manifold described by the coordinates $\vec{\theta}$.

Then, at any confining background, the entanglement entropy (EE) could be
written as [18]

$\displaystyle
S_{C}(\rho_{0})=\frac{V_{d-1}}{2G_{N}^{(10)}}\int_{\rho_{0}}^{\infty}d\rho\sqrt{\frac{\beta(\rho)H(\rho)}{1-\frac{H(\rho_{0})}{H(\rho)}}},$
(2.7)

where $\rho_{0}$ is the minimum value of $\rho_{0}$ and the width of one
interval as a function of $\rho_{0}$ would be

$\displaystyle
L(\rho_{0})=2\int_{\rho_{0}}^{\infty}d\rho\sqrt{\frac{\beta(\rho)}{\frac{H(\rho)}{H(\rho_{0})}-1}},$
(2.8)

and the disconnected solution would be written as

$\displaystyle
S_{D}(\rho_{0})=\frac{V_{d-1}}{2G_{N}^{(10)}}\int_{\rho_{\Lambda}}^{\infty}d\rho\sqrt{\beta(\rho)H(\rho)}.$
(2.9)

The difference between the connected and disconnected solution would be
finite.

These quantities would be useful for pure states. When we have mixed system,
the main quantity that should be examined is the mutual information,
$I(A,B)=S_{A}+S_{B}-S_{AB}$, which needs these relations 2.7, 2.8 and 2.9.
From the width of one interval $l$ or the distance between intervals $D$, the
turning point for each region would be calculated and then the entanglement
entropy for each region could be calculated. In addition, from the distance
between the turning points, the minimal wedge cross section $\Gamma$ could be
found.

##  3 Critical distances in the confining backgrounds

When the distance between two mixed systems becomes larger, at a specific
critical distance $D_{c}$, the correlations, mutual information and
entanglement of purification between them vanish or rather drop to one order
of magnitude lower.

For two equal strips with the width $l$ and distance $D$, the mutual
information would be

$\displaystyle I(D,l)=S_{A}+S_{B}-S_{AB}=2S(l)-S(D)-S(2l+D),$ (3.1)

and the critical distance can be found by where it drops to zero, i.e,
$I(D_{c},l)=0$.

We show that this critical distance could give interesting new information
about QCD phase structures as it in fact, would be a new scale in the theory.

The critical width for “one strip” could distinguish the
confinement/deconfinement phase transition and it has been proposed that this
critical width $L_{\text{crit}}$ is proportional to the inverse of confinement
temperature, i.e, $L_{\text{crit}}=\mathcal{O}(\Lambda_{IR}^{-1})$, where
$\Lambda_{IR}$ correspond to the singularity in each theory. However, here we
can show that the critical distance between “two strips” is a stronger
quantity and can even distinguish the scale of chiral symmetry breaking in
addition to the confinement/deconfinement scale.

Similar to the work of [19], we find that the phase structure of confining
background would be as shown in figure 2. Note that, in each of these based on
the relative sizes of $r_{s}$ or $u_{KK}$, $D_{c}$ and $l$, the entanglement
wedge cross section becomes minimum between all other solutions and therefore
that specific phase beocome dominant, and each time this happens we see a jump
in the numerical solutions.

These jumps then could be used to distinguish the scale of chiral symmetry
breaking versus confinement/deconfinement phase transitions. We present more
numerical details about how this quantity would probe phase structures of QCD
and could capture important scales in each theory.

Figure 2: The structure of phases in confining backgrounds as first presented
in [19].

###  3.1 AdS soliton background

The first background where we probe its phase structures using mixed
correlations would be AdS soliton. This geometry was first introduced in the
work of Myers and Horowitz [20] where the boundary CFT side have
supersymmetry-breaking boundary conditions and the bulk geometry have negative
energy dual to the negative Casimir energy in the boundary gauge theory side.
Therefore, the AdS-soliton would have the lowest negative boundary energy
among all solutions with the same asymptotic boundary. This negative energy
which could source and generate the AdS-soliton geometry has been calculated
as
$\frac{E}{L_{x}^{n-2}L_{y}}\Big{|}_{\text{BH}}=\frac{(n-1)r_{0}^{n}}{2\ell^{n+1}}$,
where $n$ is the dimension of the boundary, $n+1$ is the dimension of the bulk
geometry, $L_{x}$ and $L_{y}$ represent the extension of the surface on $x$
and $y$ directions and $\ell$ is the radius of AdS spacetime. In the bulk it
means that all massive and massless particles would repel each other and
therefore it means AdS-soliton would actually anti-gravitates. The effects of
this feature would then be carved in the structures of quantum states and on
the behavior of quantum information measures, and bit threads structures.

In addition, since there are more phase transitions possible between AdS-black
hole (BH) solutions and AdS-solition geometry compared with other cases like
AdS-BH and thermal AdS geometry, it would be more interesting to study the
behaviors of mixed quantum information measures like entanglement of
purification (EoP), mutual information (MI), complexity of purification (CoP),
negativity, etc and also tools such as bit threads and Wilson loops during
such phase transitions.

We choose the following background for the AdS soliton in the Poincare
coordinates [21]

$\displaystyle
ds^{2}=\frac{L^{2}_{\text{AdS}}}{z^{2}}\left(\frac{dz^{2}}{f(z)}+f(z)d\theta^{2}-dt^{2}+dr^{2}+r^{2}d\Omega_{d-3}\right),$
(3.2)

where $f(z)$ is a harmonic function and has a general form as

$\displaystyle
f(z)=\left(1-k_{1}\frac{z}{z_{0}}\right)\left(1-k_{2}\frac{z}{z_{0}}\right)(1+\sum_{n=1}c_{n}z^{n}).$
(3.3)

This metric [21] has been found by double Wick rotating the asymptotically AdS
space where the coefficients $c_{n}$ could be derived by the condition that
the $1+\sum c_{n}z^{n}$ should not contain poles and zeros at $z=z_{0}$.

Figure 3: The behavior of
$f_{p}(z)=(1-k_{1}\frac{z}{z_{0}})(1-k_{2}\frac{z}{z_{0}})(1+\sum_{n=1}c_{n}z^{n})$
which uses the perturbative sum and arbitrary values of $c_{i}$ as
$k_{1}=1,k_{2}=-2,c_{1}=1,c_{2}=2,c_{3}=3,c_{4}=4,c_{5}=5$, which is used for
the general case of soliton is shown in the left part, and the function for
the simplest AdS soliton case, $f_{s}(p)=1-(\frac{z}{z_{0}})^{8-d}$, $d=4$,
which we use here for calculations, is shown in the right part.

By changing $k_{1}$ and $k_{2}$, different IR behaviors could be yielded, but
not for any $k_{1}$ and $k_{2}$, this ansatz would become a solution of
Einstein theory. In order to get a soliton AdS geometry one of the $k_{1}$ or
$k_{2}$ should be negative, and the simplest AdS soliton would be with
$k_{1}=1$ and $k_{2}=-1$. The behavior of $f(z)$ for the general case and
simplest case are shown in figure 3. Here we take
$f(z)=1-(\frac{z}{z_{0}})^{8-d}$ as for the simplest AdS soliton.

It worths to mention that other complicated soliton solutions can also be
obtained by double Wick rotating the charged or hairy scalar AdS black hole
solutions. For example, by double Wick rotating charged $\text{AdS}_{5}$, the
function $f$ would be found as $f(z)=1-mz^{4}+q^{2}z^{6}$, which is related to
a boundary theory with non-zero magnetic flux or current density condensate.

Now, to study entanglement and other correlation measures, first, the width of
the strip versus turning point $z_{t}$, for AdS soliton, could be found as

$\displaystyle
L(z_{0})=2\int_{z_{0}}^{\infty}\frac{dz}{\left(1-\left(\frac{z}{z_{0}}\right)^{8-d}\right)\left(\left(\frac{z_{0}}{z}\right)^{2d-4}\left(1-\left(\frac{z}{z_{0}}\right)^{8-d}\right)-1\right)},$
(3.4)

and the entanglement entropy for the connected solution in this background
would be

$\displaystyle
S_{C}(z_{t})=c_{1}\int_{z_{0}}^{\infty}\frac{dz}{z^{d-2}\sqrt{1-\frac{\left(\frac{z}{z_{0}}\right)^{2d-4}}{1-\left(\frac{z}{z_{0}}\right)^{8-d}}}},$
(3.5)

and for the disconnected part one gets

$\displaystyle S_{D}=-c_{2}\int_{z_{\Lambda}}^{\infty}\frac{dz}{z},$ (3.6)

where $c_{1}$ and $c_{2}$ are just two constants that are the combination of
the parameters $G_{N}^{(10)}$, $d$, $L$, $\pi$ and $V_{d-3}$, and are not
important in our numerical studies here.

The behavior of these quantities are shown in figure 4. The noises observed in
these phase diagrams could be due to various phenomena in the system such as
presence of bound-entangled states, creation and annihilation of EPR pairs,
etc.

Figure 4: The relationship between width of a strip versus turning point
$\tau_{0}$ (in the left), and entanglement entropy versus $L$ (in the right)
in the AdS-soliton background.

The behavior of $D_{c}$ versus $u_{KK}$ in AdS-soliton background is shown in
figure 5.

Figure 5: The behavior of $D_{c}$ versus $u_{KK}$ in AdS-soliton background
with the metric 3.2 .

For the AdS-soliton metric in the form of [22]

$\displaystyle
ds^{2}=\frac{r^{2}}{\ell^{2}}\left[-dt^{2}+\left(1-\frac{r_{+}^{d}}{r^{d}}\right)d\chi^{2}+d\vec{x}^{2}\right]+\left(1-\frac{r_{+}^{d}}{r^{d}}\right)^{-1}\frac{\ell^{2}}{r^{2}}dr^{2},$
(3.7)

where there are $d-2$ components of $\vec{x}$, the function for $L(r_{0})$ and
$S(r_{0})$ would be

$\displaystyle
L(r_{0})=2\int_{r_{0}}^{\infty}dr\sqrt{\frac{\frac{\ell^{4}}{r^{4}}\left(1-\frac{r_{+}^{d}}{r^{d}}\right)^{-1}}{\left(\frac{r}{r_{0}}\right)^{2d-6}\frac{1-\left(\frac{r_{+}}{r^{d}}\right)^{d}}{1-\left(\frac{r_{+}}{r_{0}}\right)^{d}}}},$
$\displaystyle
S_{C}(r_{0})=\frac{V_{d}}{2G_{N}}\int_{r_{0}}^{\infty}dr\frac{\left(\frac{r}{\ell}\right)^{d-5}}{\sqrt{1-\left(\frac{r_{0}}{r}\right)^{2d-6}\frac{1-\left(\frac{r_{+}}{r_{0}}\right)^{d}}{1-\left(\frac{r_{+}}{r}\right)^{d}}}},$
(3.8)

For the case of $d=4$, $d=5$ and $d=6$, the behavior of $D_{c}$ versus $r_{+}$
is shown in figures 6, where the four phases are clear. The phases become less
distinct when the dimension increases, for $d\geq 4$. The case of $d=2$ also
is shown in figure 7.

Figure 6: The behavior of $D_{c}$ versus $r_{+}$ in the AdS-soliton background
with the metric 3.7, for the case of $d=4$ in the left, $d=5$ in the center,
and $d=6$ in the right part. Figure 7: $D_{c}$ versus $r_{+}$ in the AdS-
soliton background with the metric 3.7, for the case of $d=2$.

Note that the previous discussions are for one strip and for pure states. Now
we move to the case for two strips and mixed states, where the mutual
information and critical distance, $D_{c}$ between the two strips are being
used to probe the phase space. This critical distance $D_{c}$ can be
calculated by setting the mutual information 3.1 to zero, while we use the
relation 2.8, for calculating the turning points from the widths of each
region and then we use the relation 2.7 for calculating the entanglement
entropies.

Then, the position of the singularity could be changed, and the behavior of
critical distance $D_{c}$ versus the width $l$ and therefore the phase
structures of this theory could be found where the results are shown in figure
8. By changing $z_{0}$ at least four different phases could be detected here
where in each phase the behavior of $D_{c}$ is somehow distinct.

Figure 8: The four phases detected by the relationship between critical
distance between two strips, $D_{c}$ and width of the strip in AdS soliton
background, for the range of parameter $0.35<z_{0}<0.4$ in the up-left for the
1st phase and $2<z_{0}<3.5$ in the up-right for the 2nd phase, $0.2<z_{0}<0.3$
down-left in the 3rd phase and $0.4<z_{0}<0.6$ down-right in the 4th phase
which actually till $z_{0}=2$ would still be in the fourth phase.

From plotting $D_{c}$ for various positions of the end of the world or
singularity, i.e, $z_{0}$ here or $z_{\Lambda}$ in other geometries, it could
be noted that unlike the conformal case which has “two phases”, confining
geometries have “four phases”. Also, the behavior of this critical distance in
probing phases is similar to the behavior of logarithmic negativity in the
setup of the end of the world brane as recently discussed in [23] by Dong,
McBride, Weng. So, this shows that in confining cases, the measures of quantum
mixed states such as mutual information and negativity show compatibility with
each other.

One important feature of this theory is the capping or the gap where its
effect could be detected in the correlation measures such as EWCS and MI. As
the entangling surface would wrap around the compact dimension, all the UV
scaling structures specifically for the entanglement entropy and also
entanglement of purification would become different. For instance EWCS would
be proportional to $\frac{L_{\theta}}{R}S_{UV}^{(d)}$ where $L_{\theta}$ is
the fixed proper size of the compact circle, $R$ is the linear size of the
entangling surface and $S_{UV}^{(d)}$ is the UV structure of entanglement
entropy of $\text{AdS}_{d+1}$. These measures in specific regimes and phases
show monotonic RG flows as in c-theorem.

Note that AdS-soliton case could be presented in the cylinder or disc
topologies. As mentioned in figure 1 of [21], the cylinder topology
corresponds to the trivial product state after RG flow in MERA, and the disc
topology corresponds to the entangled states protected by the symmetry or the
topological orders. The critical distance in each case could capture these
quantum distinctions as well.

Our results would also be very useful in studying the dynamics of black holes
and also chaotic systems. In principle, the structures of islands and replica
worm holes could also be investigated for the black holes in such confining
backgrounds using this new quantity. Then, the interplay between the UV and IR
correlations could also be probed this way.

Note that the AdS soliton can also be written in other formats as well. One
can write the general background which is asymptotically AdS as

$\displaystyle
ds^{2}=\frac{r^{2}}{\ell^{2}}\left[-\left(1-\frac{r_{0}^{p+1}}{r^{p+1}}\right)dt^{2}+(dx^{i})^{2}\right]+\left(1-\frac{r_{0}^{p+1}}{r^{p+1}}\right)^{-1}\frac{\ell^{2}}{r^{2}}dr^{2},$
(3.9)

where $i=1,...,p$, which for certain values of $p$ would arise in the near
horizon geometry of p-branes. The energy here is $E_{p}=\frac{pV_{p}}{16\pi
G_{p+2}\ell^{p+2}}r_{0}^{p+1}$.

The AdS soliton metric could then be derived by the analytical continuation
$t\to i\tau$ and $x^{p}\to it$ and be written in the form of [20]

$\displaystyle
ds^{2}=\frac{r^{2}}{\ell^{2}}\left[\left(1-\frac{r_{0}^{p+1}}{r^{p+1}}\right)d\tau^{2}+(dx^{i})^{2}-dt^{2}\right]+\left(1-\frac{r_{0}^{p+1}}{r^{p+1}}\right)^{-1}\frac{\ell^{2}}{r^{2}}dr^{2},$
(3.10)

where there are $p-1$ $x^{i}$’s. The energy then would be
$E=-\frac{r_{0}^{p+1}\beta V_{p-1}}{16\pi G_{p+2}\ell^{p+2}}$, where
$\beta=4\pi l^{2}/(p+1))r_{0}$ is the period.

In order to understand this negative energy feature of the AdS soliton, in
[24], the geodesic motions of massive and massless particles have been studied
in this background. In the Poincare coordinate, the metrics have been written
as

$\displaystyle\text{AdS-BH}\ :\ \
ds^{2}=\frac{r^{2}}{\ell^{2}}(-hd\eta^{2}+d\vec{x}.d\vec{x}+dy^{2})+\frac{\ell^{2}dr^{2}}{r^{2}h},$
$\displaystyle\text{AdS-soliton}\ :\ \
ds^{2}=\frac{r^{2}}{\ell^{2}}(+hdy^{2}+d\vec{x}.d\vec{x}-d\eta^{2})+\frac{\ell^{2}dr^{2}}{r^{2}h},$
(3.11)

where $h=1-\frac{r_{0}^{n}}{r^{n}}$. In that work the topologies of AdS-BH and
AdS-soliton has been compared, as in figure 9, where the horizon surface of
AdS-BH has the topology of $R^{n-}$, but the $r=r_{0}$ “line” in AdS-soliton
case has the topology of $R^{n-2,1}$. The two mixed systems in the form of two
parallel strips with width $l$ and infinite lengths would be located on the
boundary side.

Figure 9: The picture in the left is the AdS-BH solution, the one in the
middle is an AdS-soliton geometry and the one in the right shows our system of
two symmetric strips in the AdS soliton background.

Note that in AdS/CFT setup, and for the AdS soliton case, the cold and large
black holes are dual to the deconfining phase and the hot and small black
holes are dual to the confining phase.

###  3.2 Witten-Sakai-Sugimoto model

In this work we are more interested in the top-down holographic QCD models
which are engineered by intersecting D-branes, like D4-D8 brane intersections
known as Witten-Sakai-Sugimoto model shown in figure 10. This theory in fact
can nicely model the $SU(N_{f})_{L}\times SU(N_{f})_{R}$ chiral flavor
symmetry breaking.

In this setup, $x^{4}$ is the spatial coordinate which is being compactified
on $S^{1}$, which for fermions would have anti-periodic boundary conditions.
The gauge theory is coupled to $N_{f}$ left-handed quarks and $N_{f}$ right-
handed quarks which are localized at different points of the compact circle.
The D8-branes are at $x^{4}=0$ and the anti-D8 branes are located in parallel
and at $x^{4}=\pi R$, and $R$ is the radius of $S^{1}$.

Figure 10: Intersecting D4-D8 brane model of Witten-Sakai-Sugimoto. Usually
the D8-branes are treated as probe branes.

In the above picture, the D4-branes correspond to color sector and the open
strings which end on them produce the color/gluon gauge field. On the other
hand, the D8-branes are flavor branes and the strings ending on them give the
flavor/meson gauge field. The strings which are stretched between D4 and D8
branes would then give rise to the quarks. The closed strings also correspond
to the glue-balls. Using this picture, the color/flavor duality would then
mapped to open/closed string duality.

The entanglement and correlations structures among D8-D8, D4-D4 and D8-D4
branes would be different. The different correlation measures could then probe
all these structures which we for the first step, we show that it can be
captured first by the critical distance $D_{c}$.

The metric of D4-branes is

$\displaystyle
ds^{2}_{D4}=\left(\frac{u}{R_{D4}}\right)^{3/2}(-dt^{2}+\delta_{ij}dx^{i}dx^{j}+f(u)(dx^{4})^{2})+\left(\frac{R_{D4}}{u}\right)^{3/2}\left(\frac{du^{2}}{f(u)}+u^{2}d\Omega_{4}^{2}\right),$
(3.12)

where the dilaton, the Ramond-Ramond field, the function $f(u)$ and the AdS
radius are defined as

$\displaystyle e^{\Phi}=g_{s}\left(\frac{u}{R_{D4}}\right)^{3/4},\ \ \
F_{4}\equiv dC_{3}=\frac{2\pi N_{c}}{V_{4}}\epsilon_{4},\ \ \ f(u)\equiv
1-\frac{u_{KK}^{3}}{u^{3}},\ \ \ R^{3}_{D4}\equiv\pi g_{s}N_{c}l_{s}^{3}.$
(3.13)

Here, $N_{c}$ is the number of colors in the gauge group, $g_{s}$ is the
string coupling, $l_{s}$ is the string length which has the relation
$l_{s}^{2}=\alpha^{\prime}$, $V_{4}$ is the volume of the unit four sphere
$S^{4}$, and $s_{4}$ is the volume form of $S^{4}$. In addition, $u$ is the
holographic radial direction, which is in the region of $u_{KK}\leq
u\leq\infty$.

Then, assuming the position of D8 brane at $x^{4}=0$ and anti-brane at
$x^{4}=\pi R$, leading to relation $dx^{4}/du=0$, the induced metric of the
D8-brane becomes

$\displaystyle
ds_{D8}^{2}=\left(\frac{u}{R_{D4}}\right)^{3/2}(-dt^{2}+\delta_{ij}dx^{i}dx^{j})+\left(\frac{R_{D4}}{u}\right)^{3/2}\left(\frac{du^{2}}{f(u)}+u^{2}d\Omega_{4}^{2}\right).$
(3.14)

As there are two metrics, one for D4-branes and one for D8 branes, here one
might be confused that which one should be employed to calculate the
holographic mixed quantum measures. In fact, the calculations should be done
for the flavor D-branes metric which represent the quark sector, which here
for the case of Sakai-Sugimoto, would correspond to the D8-branes. In works
such as [25], for calculating the pair production rates, also the imaginary
part of D8-branes have been evaluated, which again is the main part for mixed
correlations too. Therefore, for the calculation of Sakai-Sugimoto case, we
consider the above metric 3.14.

We find $S_{C}(u_{t})$, i.e, entanglement entropy at turning point $u_{t}$ of
the strip with width $L(u_{t})$ as

$\displaystyle
S_{C}(u_{t})=\frac{V_{3}V_{4}R_{D_{4}}^{3}}{2g_{s}^{2}G_{N}^{(10)}}\int_{u_{t}}^{\infty}du\frac{u}{\sqrt{\left(1-\frac{u_{KK}^{3}}{u^{3}}\right)\left(1-\frac{u_{t}^{5}}{u^{5}}\right)}},$
(3.15)

and the width of the strip in terms of turning point as

$\displaystyle
L(u_{t})=2R_{D_{4}}^{\frac{3}{2}}\int_{u_{t}}^{\infty}du\frac{1}{\sqrt{u^{3}\left(1-\frac{u_{KK}^{3}}{u^{3}}\right)\left(\frac{u^{5}}{u_{t}^{5}}-1\right)}}.$
(3.16)

The plot of $L(u_{t})$ versus turning point $u_{t}$ and entanglement entropy
versus $L$ is shown in figure 11.

Figure 11: The plot of $L(u_{t})$ vs. turning point $u_{t}$ and $S(u_{t})$ vs.
$L(u_{t})$ for the case of Sakai-Sugimoto, for $u_{KK}=0.5$.

Next, in this confining model, we would like to examine the behavior of
critical distance between the two strips $D_{c}$ as a function of $u_{KK}$ and
detect various phases in this specific background.

Again, at least four distinct phases could be detected. In the first phase
shown in figure 12, increasing $u_{KK}$ would increase $D_{c}$. When $u_{KK}$
becomes zero the critical distance would be around $D_{c}=0.899$. The upper
bound for $u_{KK}$ in this phase is $u_{KK}=0.5$ corresponding to
$D_{c}=0.97$. After that increasing $u_{KK}$ decreases $D_{c}$ and we enter
the next phase which by increasing $u_{KK}$, $D_{c}$ would decrease.

Figure 12: The plot of $D_{c}$ vs. $l$ for different $u_{KK}$ in phase 1.

In the second phase, shown in the left side of figure 13, one could note that
increasing $u_{KK}$ would decrease critical distance $D_{c}$ unlike the
previous phase of figure 12. Also, at smaller $I$ one could see that $D_{c}$
falls and then become saturated for the larger $l$. It could see that at the
end of the second phase and the beginning of the third phase, i.e, for
$u_{KK}=0.9$, the behavior is becoming more like the third phase in the right
side.

Figure 13: The plot of $D_{c}$ vs. $l$ for different $u_{KK}$, in phase 2 in
the left side and in phase 3 in the right side.

In the third phase shown in the right side of figure 13, again one can see
that after reaching its maximum at the previous phase, increasing $u_{KK}$
decreases $D_{c}$, and the behavior is different from the second phase.

In figure 14, we show in more details the behaviour of critical distance
$D_{c}$ at various $u_{KK}$ for a constant $l$.

So both the confinement and approximate spontaneous chiral symmetry breaking
would affect the correlations between mixed systems in these confining
backgrounds where mutual information and other mixed correlation measures such
as entanglement of purification (and probably complexity of purification)
could detect their effects.

Note that in this background, the massless fields living on the D8-branes
would lead to the low-spin mesons, while the strings which fall from the
D8-branes down to the wall at $u=u_{KK}$ and then going back up again would
lead to the higher spin mesons. This is another reason that changing the
position of $u=u_{KK}$ would have such a strong effect on the entanglement and
correlations between two mixed systems in this confining background as we have
observed in figures 12 and 13. Also, this change of position of $u_{KK}$ would
change the behavior of the fluctuations of the gauge fields on the branes
which are responsible for the creation of pseudo-vectors, scalar mesons and
massless pions. These particles could also mediate information between the two
mixed systems and therefore affect the phase structures of QCD probing by
these quantum measures.

Figure 14: The plots of $D_{c}$ vs. $u_{KK}$ in Witten-Sakai-Sugimoto
background.

These phase transitions could also be examined from the point of view of
traversable wormholes [26, 27]. During these first order phase transitions,
the traversable wormholes would exchange their configurations. Also, it could
be seen as Hawking-Page-like phase transition where the quark-anti-quark
potential changes from the linear behavior in the low-temperature and finite
values of $\sqrt{g_{tt}g_{xx}}$ and the free energy of the order $N_{c}^{0}$
in the confined phase to the decaying potential in higher temperatures and
zero value of $\sqrt{g_{tt}g_{xx}}$, and free energy of the order $N_{c}^{2}$
in the deconfined phase.

In [5], we also established the connections between complexity and potential
in the QCD models. The similarities between the behavior of potential and
entanglement and complexity of purification could also be observed in these
confining models, specifically in Sakai-Sugimoto and Klebanov-Strassler case.

One should note that as mentioned in [28], in these confining backgrounds
there is another critical separation denoted by $L_{c}$ where when the quark
separation is $L>L_{c}\simeq 0.97*R$ the chiral symmetry would be restored at
the temperature $T_{d}=1/2\pi R$, and when $L<L_{c}\simeq 0.97*R$ in an
intermediate deconfined phase, the chiral symmetry is broken and again the
chiral symmetry would be restored at $T_{\chi SB}\simeq 0.154/L$. Here $R$ is
the radius of the circle where the $4+1$ supersymmetric gauge theory is
compactified on. Note also that here all the phase transitions are first
order.

The competitions between chiral symmetry breaking/restoration and the
vanishing of mutual information due to increasing the distance between the two
strips would in fact create the four phase transitions also observed in 14. In
general the scale of chiral symmetry breaking and confinement would be
independent, as the scale of chiral symmetry breaking would mostly depend on
the distance between D8-branes or the fermions on the compactified circle,
while the scale of confinement would depend on the radius where D4-brane is
wrapped around, or the radius of the circle which the $SU(N_{c})$ gauge theory
is compactified on it.

So in the Sakai-Sugimoto model, the chiral symmetry breaking/restoration which
is related to the mass scale of mesons, ($\frac{1}{L}$), is related to the
distance between D8 branes. More precisely, the low spin mesons are described
by strings attached between the two D8 branes while the high spin mesons fall
from D8 branes on the wall at $u=u_{KK}$, being completely stretched on the
wall and then go back to the D8 brane. The confinement/deconfinement phase
transition happens at the scale which is related to the mass of
glueballs,$\frac{1}{R}$, which is related to the radius where D4 brane is
wrapped around.

Also, note that in this model, the unstable modes have been observed and
studied in the literature which specifically in high-energy regimes could
significantly affect the quantum measures that we use here.

The relation for the minimal wedge cross section could be found using the
relation

$\displaystyle\Gamma_{WSS}=R_{D4}^{3}\int_{u_{D}}^{u_{2l+D}}du\frac{u^{5}}{1-\frac{u^{3}_{KK}}{u^{3}}}.$
(3.17)

Based on the AdS/CFT dictionary we have relations between the parameters
$M_{KK},g_{YM},N_{c}$ in the boundary gauge side and $R_{D_{4}},u_{KK},g_{s}$
in the gravity side as

$\displaystyle R_{D_{4}}^{3}=\frac{1}{2}\frac{\lambda l_{s}^{2}}{M_{KK}},\ \ \
u_{KK}=\frac{2}{9}\lambda M_{KK}l_{s}^{2},\ \ \
g_{s}=\frac{1}{2\pi}\frac{\lambda}{M_{KK}N_{c}l_{s}}.$ (3.18)

The ’t Hooft coupling $\lambda$ is $\lambda=g_{YM}^{2}N_{c}$ and the gauge
coupling $g_{YM}$ at the cutoff scale $M_{KK}$ can be written as
$g_{YM}^{2}=(2\pi)^{2}g_{s}l_{s}/\delta x^{4}$ and also we have the relation
$M_{KK}\equiv\frac{2\pi}{\delta x^{4}}$[25].

It could be seen that the string coupling $g_{s}$, the number of colors of
gauge group $N_{c}$, the string length $l_{s}$ and the cutoff energy $M_{KK}$
increase the entanglement of purification while the periodicity of the
boundary condition $\delta x^{4}$ would have an inverse effect and decreases
the EWCS. The increasing behavior of EWCS versus $M_{KK}$ and its decrease
versus $\delta x^{4}$ are shown in figure 15.

Figure 15: The behavior of entanglement of purification versus $M_{KK}$ and
$\delta x_{4}$

####  3.2.1 Deformed Sakai-Sugimoto

One can then consider the deformed Sakai-Sugimoto geometry [28, 25], where
$x^{4}$ instead of being constant, depends on the coordinate $u$.

It has the configuration as shown in figure 16 and its metric is as follows

$\displaystyle
ds^{2}_{D8}=\left(\frac{u}{R_{D4}}\right)^{3/2}(-dt^{2}+\delta_{ij}dx^{i}dx^{j})+\left(\frac{u}{R_{D4}}\right)^{3/2}\frac{du^{2}}{h(u)}+\left(\frac{R_{D4}}{u}\right)^{3/2}u^{2}d\Omega_{4}^{2},$
(3.19)

where

$\displaystyle
h(u)\equiv\left[f(u)\left(\frac{dx^{4}(u)}{du}\right)^{2}+\left(\frac{R_{D4}}{u}\right)^{3}\frac{1}{f(u)}\right]^{-1}.$
(3.20)

Figure 16: The geometry of deformed Sakai-Sugimoto.

The entanglement entropy of a strip in terms of the turning point is

$\displaystyle
S(u_{t})=\frac{V_{3}V_{4}R_{D_{4}}^{3}}{2g_{s}^{2}G_{N}^{(10)}}\int_{u_{t}}^{\infty}du\frac{u}{\sqrt{(1-\frac{u_{KK}^{3}}{u^{3}})(1-\frac{u_{t}^{5}}{u^{5}})}}+\frac{V_{3}V_{4}R^{\frac{3}{2}}_{D_{4}}}{2g_{s}^{2}G_{N}^{(10)}}\int_{u_{t}}^{\infty}du\left(\frac{dx^{4}(u)}{du}\right)\sqrt{\frac{u^{5}(1-\frac{u_{KK}^{3}}{u^{3}})}{1-\frac{u_{t}^{5}}{u^{5}}}}.$
(3.21)

Note that the first term is similar to the previous usual Sakai-Sugimoto case,
while in the deformed case the second term is added which increases the
entanglement entropy compared to the previous case of relation 3.15. The
gradient of the coordinate $x^{4}(u)$ versus $u$ controls the difference.

Similarly, the width of the strip versus the turning point $u_{t}$ could be
written as

$\displaystyle
L(u_{t})=2R_{D_{4}}^{\frac{3}{2}}\int_{u_{t}}^{\infty}du\frac{1}{\sqrt{u^{3}\left(1-\frac{u_{KK}^{3}}{u^{3}}\right)\left(\frac{u^{5}}{u_{t}^{5}}-1\right)}}+2\int_{u_{t}}^{\infty}du\left(\frac{dx^{4}(u)}{du}\right)\sqrt{\frac{1-\frac{u_{KK}^{3}}{u^{3}}}{\frac{u^{5}}{u_{t}^{5}}-1}},$
(3.22)

which again the second term has been added to the previous case of relation
3.16.

For this case then, the relation for the minimal wedge cross section would be

$\displaystyle\Gamma_{\text{Deformed
SS}}=R_{D4}^{3}\int_{u_{D}}^{u_{2l+D}}du\frac{u^{5}}{1-\frac{u^{3}_{KK}}{u^{3}}}+\int_{u_{D}}^{u_{2l+D}}duu^{8}\left(1-\frac{u^{3}_{KK}}{u^{3}}\right)\left(\frac{dx^{4}(u)}{du}\right)^{2}.$
(3.23)

So again in the deformed case the second term is being added. Therefore, the
minimal wedge cross section of deformed Sakai-Sugimoto is bigger than the
normal case, or rather the entanglement of purification of two mixed system is
higher and the difference is being controlled by the derivative of $x^{4}(u)$
relative to $u$.

By calculating the DBI action and then calculating its Hamiltonian which is
conserved one would specifically get the relation for $u^{\prime}=du/dx^{4}$
[28] as

$\displaystyle u^{\prime
2}=f^{2}(u)\left(\frac{u}{R_{D_{4}}}\right)^{3}\left(\frac{f(u)}{f(u_{0})}\frac{u^{8}}{u_{0}^{8}}-1\right),$
(3.24)

Then, we can insert the inverse of this relation into all the previous results
for the deformed Sakai-Sugimo and find the numerical solutions for this case
as well. However, since the second integral contains a square term, the
numerical solutions become much more complicated and noisy, and we leave its
direct calculations for the future works.

###  3.3 Witten-QCD

In the string frame, the metric of Witten-QCD and its dilaton field is written
as [29, 30, 31, 32],

$\displaystyle
ds^{2}=(\frac{u}{R})^{3/2}\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\frac{4R^{3}}{9u_{t}}f(u)d\theta^{2}\right)+\left(\frac{R}{u}\right)^{3/2}\frac{du^{2}}{f(u)}+R^{3/2}u^{1/2}d\Omega_{4}^{2},$
$\displaystyle f(u)=1-\frac{u_{t}^{3}}{u^{3}},\ \ \ \ \ \ \ \ R=(\pi
Ng_{s})^{\frac{1}{3}}{\alpha^{\prime}}^{\frac{1}{2}},\ \ \ \ \ \
e^{\Phi}=g_{s}\frac{u^{3/4}}{R^{3/4}}.$ (3.25)

Figure 17: The relationship between width of a strip versus turning point
$\tau_{0}$ (in the left), and entanglement entropy versus $L$ (in the right)
in the background on Witten-QCD. Figure 18: The behavior of $D_{c}$ vs $l$ for
various $u_{t}$ is shown where the region for non-zero mutual information for
each parameter is the region covered below each curve. Figure 19: The behavior
of $D_{c}$ versus $u_{KK}$ in Witten-QCD background.

The turning point of the RT surface depends on the width of the strip
$L(u_{0})$ and has been found in [18, 33] as

$\displaystyle
L(u_{0})=2\int_{u_{0}}^{\infty}du\sqrt{\frac{(\frac{R}{u})^{3}\frac{1}{f(u)}}{\frac{f(u)u^{5}}{f(u_{0})u_{0}^{5}}-1}},$
(3.26)

and the entropy of the connected solution $S_{C}$ would be [33]

$\displaystyle
S_{C}(u_{0})=\frac{V_{2}}{G_{N}^{(10)}}\frac{8\pi^{2}R^{\frac{9}{2}}}{9g_{s}^{2}\sqrt{u_{t}}}\int_{u_{0}}^{\infty}du\sqrt{\frac{u^{2}}{1-\frac{u_{0}^{5}}{u^{5}}.\frac{f(u_{0})}{f(u)}}}.$
(3.27)

Using the above relations, the phase structures can be numerically probed
where the results are shown in figures 18 and 19.

The behavior of $D_{c}$ versus $u_{KK}$ in Witten-QCD background is shown in
figure 19. Note that specifically in figure 19, the first phase is when
$D_{c}$ versus $u_{KK}$ is constant, then, around $u_{KK}=1$ there is a drop
and then a singularity or rather a jump which is the second phase, then
another one around $u_{KK}=1.3$ and the next around $u_{KK}=1.7$, and then
$D_{c}$ decreases by increasing $u_{KK}$ which is the last phase, a similar
behavior to AdS-soliton case.

Then, if we use relation 2.5, for this case, and then assuming that the
coordinate $\theta(u)$ is the only coordinate as the function of $u$, then the
minimal wedge cross section $\Gamma$ would be found as

$\displaystyle\Gamma_{\text{WQCD}}=\frac{2R^{\frac{9}{2}}}{3\sqrt{u_{t}}g_{s}^{2}}\int_{u_{D}}^{u_{2l+D}}duu^{\frac{7}{4}}\sqrt{1+\frac{4}{9u_{t}}\left(\frac{d\theta}{du}\right)^{2}f^{2}(u)},$
(3.28)

which its relation with various parameters of the theory could be seen.

###  3.4 Klebanov-Strassler

Another interesting geometry worths to study and check the behavior of quantum
information measures and $D_{c}$ in it, would be the Klebanov-Strassler (KS)
case [34]. This type-IIB supergravity solution has various fluxes and warp
factors and are dual to the confining $\mathcal{N}=1$ SYM theories which
describe the baryonic branch of the dual gauge theory.

The KS metric is actually being constructed by the collection of N regular and
M fractional D3-branes in the geometry of the deformed conifold. The
fractional D3-branes could be thought as D5-branes that wrap the two-cycle of
$T^{1,1}$ which collapse at the apex $\tau=0$ in a way that the world-volume
would effectively becomes that of a 3-brane.

There are two free parameters in the KS model which the dual interpretations
of them are baryonic VEV and the gauge coupling constant. The Maldacena-Nunez
solution which we study later, could also be considered as the end point of a
flow from KS by changing these two parameters.

The gauge group on the stack of N D3-branes and M fractional D3-branes is
$SU(N+M)\times SU(N)$ [35]. The limit where $M<<N$ corresponds to the
Klebanov-Tseytlin solution which we also check later.

The fluxes follow the relations

$\displaystyle\int_{T^{1,1}}F_{5}\propto N,\ \ \ \int_{S^{3}}F_{3}\propto M,$
(3.29)

and the NSNS-flux would be

$\displaystyle H_{3}\propto\frac{g_{s}M}{\tau}d\tau\wedge\omega_{2},\ \ \ \ \
\text{therefore:}\ \ \int_{S^{2}}B_{2}=g_{s}M\log(\tau/\tau_{0}),$ (3.30)

where $\omega_{2}$ is the volume form of the two-sphere of $T^{1,1}$ and
$\tau_{0}$ is just an integration constant.

The form of the metric is

$\displaystyle
ds_{\text{def}}^{2}=\frac{1}{2}\epsilon^{\frac{4}{3}}K\Big{[}\frac{1}{3K^{3}}(d\tau^{2}+g_{5}^{2})+\sinh^{2}(\frac{\tau}{2})(g_{1}^{2}+g_{2}^{2})+\cosh^{2}(\frac{\tau}{2})(g_{3}^{2}+g_{4}^{2})\Big{]},$
(3.31)

and $K$ is a decreasing function of the radial coordinate $\tau$, as

$\displaystyle K(\tau)=\frac{(\sinh
2\tau-2\tau)^{\frac{1}{3}}}{2^{\frac{1}{3}}\sinh\tau}.$ (3.32)

For these confining geometries which has an exotic throat as shown in figure
20, the tools of bulk reconstruction such as Kinematic space, Crofton form,
tensor networks or even the form of bit threads or “quantum bit threads” [36,
37], entanglement contours and Page curves could give further information
about the interplay between geometry and topology in the bulk, and the
information in the boundary. The complicated relation for the Crofton form of
KS metric is presented in equation 4.3.

Figure 20: Klebanov-Strassler is the part of the warped throat in this
geometry.

The Klebanov-Strassler (KS) metric which is known also as warped deformed
conifold is obtained by a collection of $N$ regular and $M$ fractional
D3-branes [34].

The metric is

$\displaystyle
ds_{10}^{2}=h^{-\frac{1}{2}}(\tau)dx_{\mu}dx^{\mu}+h^{\frac{1}{2}}(\tau){ds_{6}}^{2},$
(3.33)

where again the $ds_{6}^{2}$ is the metric of the deformed conifold which is

$\displaystyle{ds_{6}}^{2}=\frac{1}{2}\epsilon^{\frac{4}{3}}K(\tau)\Big{[}\frac{1}{3K^{3}(\tau)}(d\tau^{2}+(g^{5})^{2})+\cosh^{2}(\frac{\tau}{2})[(g^{3})^{2}+(g^{4})^{2}]+\sinh^{2}(\frac{\tau}{2})[(g^{1})^{2}+(g^{2})^{2}]\Big{]}.$
(3.34)

The parameters of the metric are

$\displaystyle
K(\tau)=\frac{(\sinh(2\tau)-2\tau)^{\frac{1}{3}}}{2^{\frac{1}{3}}\sinh\tau},\
\ \ \ \ \ \ \ \
h(\tau)=(g_{s}M\alpha^{\prime})^{2}2^{2/3}\epsilon^{-8/3}I(\tau),$
$\displaystyle I(\tau)=\int_{\tau}^{\infty}dx\frac{x\coth
x-1}{\sinh^{2}x}(\sinh(2x)-2x)^{\frac{1}{3}},$ (3.35)

and

$\displaystyle
g^{1}=\frac{1}{\sqrt{2}}[-\sin\theta_{1}d\phi_{1}-\cos\psi\sin\theta_{2}d\phi_{2}+\sin\psi
d\theta_{2}],\ \ \ \ \
g^{2}=\frac{1}{\sqrt{2}}[d\theta_{1}-\sin\psi\sin\theta_{2}d\phi_{2}-\cos\psi
d\theta_{2}],$ $\displaystyle
g^{3}=\frac{1}{\sqrt{2}}[-\sin\theta_{1}d\phi_{1}+\cos\psi\sin\theta_{2}d\phi_{2}-\sin\psi
d\theta_{2}],\ \ \ \ \
g^{4}=\frac{1}{\sqrt{2}}[d\theta_{1}+\sin\psi\sin\theta_{2}d\phi_{2}+\cos\psi
d\theta_{2}],$ $\displaystyle
g^{5}=d\psi+\cos\theta_{1}d\phi_{1}+\cos\theta_{2}d\phi_{2}.$ (3.36)

Note that $h(\tau)$ is the warp factor here. Also, as mentioned, $ds_{6}^{2}$
is the deformed conifold which is a cone over $S$ pace that is topologically
$S^{3}\times S^{2}$. The $(g_{1},g_{2})$ would build the $S^{2}$ part which
shrinks to zero size at the apex of the deformed conifold while
$(g_{3},g_{4},g_{5})$ build the $S^{3}$ part which have a finite size.

In [38], it has been mentioned that the IR behavior of the function $I$ acts
as $I(\tau\to 0)\to a_{0}-a_{2}\tau^{2}+\mathcal{O}(\tau^{4})$, and since
there is no linear term in $\tau$, the expansion is really around the end of
space and the Wilson loop would find it more favorable to arrange themselves
there. This point then should show its effects on the behaviours of other
quantum information measures such as entanglement of purification, mutual
information and negativity for the two strips in the mixed setup as we examine
here.

The length of the strip versus turning point of the RT surface can be written
as [9]

$\displaystyle
L(\tau_{0})=\frac{2^{\frac{5}{6}}\epsilon^{\frac{2}{3}}}{\sqrt{3}}\int_{\tau_{0}}^{\infty}d\tau\frac{\sinh(\tau)}{(\sinh(2\tau)-2\tau)^{\frac{1}{3}}}\sqrt{\frac{h(\tau)}{\frac{\sinh^{2}(\tau)}{\sinh^{2}(\tau_{0})}\left(\frac{\sinh(2\tau)-2\tau}{\sinh(2\tau_{0})-2\tau_{0}}\right)^{\frac{2}{3}}-1}},$
(3.37)

and the entanglement entropy for the connected solution is

$\displaystyle
S_{C}(\tau_{0})=\frac{V_{2}\pi^{3}\epsilon^{4}}{3G_{N}^{(10)}}\int_{\tau_{0}}^{\infty}d\tau\frac{h(\tau)\sinh^{2}(\tau)}{\sqrt{1-\left(\frac{\sinh\tau_{0}}{\sinh\tau}\right)^{2}\left(\frac{\sinh(2\tau_{0})-2\tau_{0}}{\sinh(2\tau)-2\tau}\right)^{\frac{2}{3}}}}.$
(3.38)

Figure 21: The relationship between width of a strip versus turning point
$\tau_{0}$ (in the left), and entanglement entropy versus $L$ (in the right)
in Klebanov-Strassler background.

The determinant of the eight-dimensional matrix which is needed would be found
where only one of the coordinate $\theta_{1}$ would be considered as a
function of the radial coordinate $\tau$.

So we get

$\displaystyle\Gamma_{\text{KS}}$
$\displaystyle=\int_{\tau_{D}}^{\tau_{2l+D}}d\tau\frac{h\epsilon^{4}\sin\left(\theta_{1}\right)}{192\sqrt{6}K^{3/2}}\left(3K^{3}\left(\frac{\text{d$\theta$}_{1}}{\text{d$\tau$}}\right)^{2}\cosh(\tau)+2\right)^{\frac{1}{2}}\times$
$\displaystyle\left(\cosh(\tau)\left(4\cos\left(2\theta_{2}\right)\sinh^{2}(\tau)-1\right)+6K^{3}\sin^{2}\left(\theta_{2}\right)\sin^{2}(\psi)(\cosh(2\tau)+\cos(2\psi))+\cosh(3\tau)\right)^{\frac{1}{2}}$

The parameter $M$, $\alpha^{\prime}$ or $g_{s}$ could be changed which
correspondingly change $h(\tau)$, and therefore the behavior of the critical
$D$ in this background can be studied which probes the phase structures.

Generally, the behavior of $D_{c}$ versus the width of the two symmetric
strips $l$ in the Klebanov-Strassler background would be as shown in figure
22.

Figure 22: The behavior of critical distance, $D_{c}$, between the two strips
in mixed systems, in the Klebanov-Strassler background.

It could be seen that for small width of the strips the critical distance
would increase by increasing $l$, but after reaching to a maximum it would
decrease at then becomes constant.

###  3.5 Klebanov-Tseytlin

The Klebanov-Tseytin (KT) metric is a singular solution which is dual to the
chirally symmetric phase of the Klebanov-Strassler model which has D3-brane
charges that dissolve in the flux [39].

The metric is

$\displaystyle
ds_{10}^{2}=h(r)^{-1/2}\left[-dt^{2}+d\vec{x}^{2}\right]+h(r)^{1/2}\left[dr^{2}+r^{2}ds_{T^{1,1}}^{2}\right].$
(3.39)

Here $ds_{T^{1,1}}^{2}$ is a base of a cone with the definition of

$\displaystyle
ds_{T^{1,1}}^{2}=\frac{1}{9}(g^{5})^{2}+\frac{1}{6}\sum_{i=1}^{4}(g^{i})^{2}.$
(3.40)

It is actually the metric on the coset space $T^{1,1}=(SU(2)\times
SU(2))/U(1)$.

Also, $g^{i}$ are some functions of the angles
$\theta_{1},\theta_{2},\phi_{1},\phi_{2},\psi$ as

$\displaystyle
g^{1}=(-\sin\theta_{1}d\phi_{1}-\cos\psi\sin\theta_{2}d\phi_{2}+\sin\psi
d\theta_{2})/\sqrt{2},\ \ \ \
g^{2}=(d\theta_{1}-\sin\psi\sin\theta_{2}d\phi_{2}-\cos\psi
d\theta_{2})/\sqrt{2},$ $\displaystyle
g^{3}=(-\sin\theta_{1}d\phi_{1}+\cos\psi\sin\theta_{2}d\phi_{2}-\sin\psi
d\theta_{2})\sqrt{2},\ \ \ \ \
g^{4}=(d\theta_{1}+\sin\psi\sin\theta_{2}d\phi_{2}+\cos\psi
d\theta_{2})/\sqrt{2},$ $\displaystyle
g^{5}=d\psi+\cos\theta_{1}d\phi_{1}+\cos\theta_{2}d\phi_{2},$ (3.41)

and also we have the relations

$\displaystyle h(r)=\frac{L^{4}}{r^{4}}\ln{\frac{r}{r_{s}}},\ \ \ \ \ \ \
L^{4}=\frac{81}{2}g_{s}M^{2}\epsilon^{4}.$ (3.42)

Figure 23: The relationship between width of a strip $L(r_{0})$ versus turning
point $r_{0}$ (in the left), and entanglement entropy $S(r_{0})$ versus $L$
(in the right) in the Klebanov-Tseytlin background. Figure 24: The
relationship between the critical distance $D_{c}$ between the two strips
versus $l$ in the Klebanov-Tseytlin geometry. The regions below each curve for
each $r_{s}$ correspond to non-zero mutual information and $r_{s}$ corresponds
to the position of the singularity. Figure 25: The relationship between the
critical distance $D_{c}$ between the two strips versus $u_{KK}$ in the
Klebanov-Tseytlin geometry.

In this frame, the asymptotic flat region has been eliminated. Also, $r=r_{s}$
is where the naked singularity is located.

The width of one strip versus turning point of RT surface in KT background can
be written as [33, 18]

$\displaystyle
L(r_{0})=9\sqrt{2}M\sqrt{g_{s}}\epsilon^{2}\int_{r_{0}}^{\infty}dr\frac{\sqrt{\ln\frac{r}{r_{s}}}}{r^{2}\sqrt{\frac{r^{6}}{r_{0}^{6}}\frac{\ln\frac{r}{r_{s}}}{\ln\frac{r_{0}}{r_{s}}}-1}},$
(3.43)

and the entanglement entropy of the connected solution is

$\displaystyle
S_{C}(r_{0})=\frac{12V_{2}\pi^{3}M^{2}g_{s}\epsilon^{4}}{G_{N}^{(10)}}\int_{r_{0}}^{\infty}dr\frac{r\ln\frac{r}{r_{s}}}{\sqrt{1-\frac{r_{0}^{6}}{r^{6}}\frac{\ln\frac{r_{0}}{r_{s}}}{\ln\frac{r}{r_{s}}}}},$
(3.44)

where their behavior is shown in figure 23.

The behavior of $D_{c}$ versus $L$ is shown in figure 24 and the behavior of
$D_{c}$ versus $u_{KK}$ is shown in figure 25.

Then, the determinant of the induced metric for calculating $\Gamma$ could be
found and then we get

$\displaystyle\Gamma_{\text{KT}}=\frac{1}{648}h^{3/4}r^{4}\sin\left(\theta_{1}\right)\left(r^{2}\left(\frac{\text{d$\theta$}_{1}}{\text{dr}}\right){}^{2}+6\right)^{\frac{1}{2}}\left(\left(2\sqrt{h}r^{2}+1\right)\cos\left(2\theta_{2}\right)+2\sqrt{h}r^{2}-5\right)^{\frac{1}{2}}.$
(3.45)

The phase structure coming from changing $r_{s}$ is presented in figure 24.
From figure 24, it could be seen that the relation of critical distance
between the two strip versus $l$ is not a monotonic function in all the phase
space, and again, by studying the behavior of $D_{c}$ and therefore EWCS, at
least four different phases could be detected in these top-down QCD models,
and again this is consistent with the result of [19].

Note that again the numerics are very sensitive to the parameter $r_{s}$ which
is the position of naked singularity here, specially for smaller value of
$r_{s}$.

###  3.6 Klebanov-Witten

The Klebanov-Witten solution is similar to the KT throat geometry but with no
logarithmic warping [40]. Unlike the other four mentioned metrics, it is just
a conformal geometry. We study this background to compare our previous results
in confining cases with the conformal one.

The metric is

$\displaystyle
ds^{2}=h^{-\frac{1}{2}}g_{\mu\nu}dx^{\mu}dx^{\nu}+h^{\frac{1}{2}}(dr^{2}+r^{2}ds_{T^{1,1}}^{2}),$
(3.46)

where

$\displaystyle h=\frac{L^{4}}{r^{4}},\ \ \ \ \ \ \text{and}\ \ \ \
L^{4}=\frac{27\pi}{4}g_{s}N(\alpha^{\prime})^{2}.$ (3.47)

Since the Klebanov-Witten solution is very similar to Klebanov-Tseytlin, the
functional for calculating $\Gamma$ would be similar, and only the relation
for $h$ is different, as it has no Logaritmic term.

$\displaystyle\Gamma_{\text{KW}}=\frac{1}{108\sqrt{6}}\int_{r_{D}}^{r_{2l+D}}drh^{2}(r)r^{5}\int_{0}^{2\pi}\int_{0}^{2\pi}d\theta_{1}d\theta_{2}\sqrt{-3+\cos
2\theta_{1}+2\cos 2\theta_{2}}\sin\theta_{1}.$ (3.48)

The width of the strip in terms of the turning point and the entanglement
entropy could be written as

$\displaystyle
L(r_{0})=2L^{2}\int_{r_{0}}^{\infty}dr\frac{1}{r^{2}\sqrt{\frac{r^{6}}{r_{0}^{6}}-1}},$
(3.49)

and

$\displaystyle
S(r_{0})=\frac{8V_{d-1}\pi^{4}L}{27G_{N}^{(10)}}\int_{r_{0}}^{\infty}dr\frac{r}{\sqrt{1-\frac{r_{0}^{6}}{r^{6}}}}.$
(3.50)

where their behavior is shown in figure 26.

Figure 26: The relationship between width of a strip versus turning point
$r_{0}$ (in the left), and entanglement entropy versus $L$ (in the right) in
the Klebanov-Witten (KW) background which shows similar behavior to AdS case.

The behavior of $D_{c}$ versus $l$ is shown in figure 27. Since this geometry
is conformal and there is no position of the wall here, as in $r_{s}$, no rich
phase structure could be detected, but the behavior of $D_{c}$ is similar to
the first phase of Klebanov-Tseytlin shown in figure 24, as one expects.

Figure 27: The behavior of $D_{c}$ versus $l$ in KW background in the right
part and non-zero region of mutual information in the left.

###  3.7 Maldacena-Nunez

The Maldacena-Nunez (MN) metric is obtained by a large number of D5-branes
wrapping on $S^{2}$ [41]. In the string frame the metric and the fields are
written as [38]

$\displaystyle
ds^{2}_{10}=e^{\Phi}[-dt^{2}+{dx_{1}}^{2}+{dx_{2}}^{2}+{dx_{3}}^{2}+e^{2h(r)}({d\theta_{1}}^{2}+\sin^{2}\theta_{1}{d\phi_{1}}^{2})+dr^{2}+\frac{1}{4}(w^{i}-A^{i})^{2}],$
(3.51)

where

$\displaystyle A^{1}=-a(r)d\theta_{1},\ \ \ \ \ \ \ \
A^{2}=a(r)\sin\theta_{1}d\phi_{1},\ \ \ \ \ \ \ \
A^{3}=-\cos\theta_{1}d\phi_{1},$ (3.52)

and the $\omega^{i}$s parametrize the 3-sphere compactification, which are

$\displaystyle w^{1}=\cos\psi d\theta_{2}+\sin\psi\sin\theta_{2}d\phi_{2},\ \
\ \ \ \ \ \ \ w^{2}=-\sin\psi d\theta_{2}+\cos\psi\sin\theta_{2}d\phi_{2},$
$\displaystyle w^{3}=d\psi+\cos\theta_{2}d\phi_{2},$ (3.53)

and also the other parameters of the metric would be

$\displaystyle a(r)=\frac{2r}{\sinh 2r},\ \ \ \
e^{2h}=r\coth{2r}-\frac{r^{2}}{{\sinh 2r}^{2}}-\frac{1}{4},\ \ \ \
e^{-2\Phi}=e^{-2\Phi_{0}}\frac{2e^{h}}{\sinh 2r}.$ (3.54)

For this geometry we get

$\displaystyle\alpha(r)=e^{\Phi},\ \ \ \ \ \beta(r)=1,\ \ \ \ \
H(r)=64e^{4h+4\Phi}=64\left(r\coth 2r-\frac{r^{2}}{\sinh
2r^{2}}-\frac{1}{4}\right)^{2}.$ (3.55)

So using the relation 2.8 the width of the strip versus turning point can be
written as

$\displaystyle L(r_{0})=2\int_{r_{0}}^{\infty}dr\frac{1}{\sqrt{\frac{r\coth
2r-\frac{r^{2}}{\sinh 2r^{2}}-\frac{1}{4}}{r_{0}\coth
2r_{0}-\frac{r_{0}^{2}}{\sinh 2r_{0}^{2}}-\frac{1}{4}}-1}},$ (3.56)

and the entanglement entropy of the connected solution of the strip from the
relation 2.7 would be found as

$\displaystyle
S_{C}(r_{0})=\frac{4V_{2}}{G_{N}^{(10)}}\int_{r_{0}}^{\infty}\frac{r\coth
2r-\frac{r^{2}}{\sinh 2r^{2}}-\frac{1}{4}}{\sqrt{1-\frac{r_{0}\coth
2r_{0}-\frac{r_{0}^{2}}{\sinh 2r_{0}^{2}}-\frac{1}{4}}{r\coth
2r-\frac{r^{2}}{\sinh 2r^{2}}-\frac{1}{4}}}}.$ (3.57)

The behavior of $L(r_{0})$ and $S$ are shown in figure 28.

Figure 28: The relationship between width of a strip versus turning point
$r_{0}$ (in the left), and entanglement entropy versus $L$ (in the right) in
the Maldacena-Nunez background.

The behavior of the critical distance versus $L$ (one strip) is then shown in
figure 29. Note that unlike other confining geometries, we don’t have many
free parameters in the MN case and therefore similar to the conformal case of
KW, no phase structure could also be detected. The only difference here would
be the early oscillations and noises in the behavior of $D_{c}$ for small
values of width, $l$, (for the case of two strips) which we have explained
their origins in the confining backgrounds before.

Figure 29: The relationship between width of a strip $L(r_{0})$ versus turning
point $r_{0}$ (in the left), and the entanglement entropy $S(L)$ versus $L$
(in the right) in the Maldacena-Nunez background.

Then, for this geometry, the determinant of the eight-dimensional induced
metric $\Gamma$ could be calculated and then after simplifications we get the
following result for the minimal wedge cross section as

$\displaystyle\Gamma_{\text{MN}}$
$\displaystyle=\int_{r_{D}}^{r_{2l+D}}dr\frac{e^{4\Phi}}{64}\left(\left(a^{2}+4e^{2h}\right)\left(\frac{d\theta_{1}}{dr}\right)^{2}+4\right)^{\frac{1}{2}}$
$\displaystyle\Bigg{(}-2e^{2h}(-a^{2}\sin^{2}(\theta_{1})(\cos(2\theta_{2})(6\cos(2\psi)-4)+3\cos(2\psi)+1)$
$\displaystyle-8a\sin(\theta_{1})\sin(\theta_{2})\cos(\theta_{2})\cos(\theta_{1})\cos(\psi)+2\cos(2\theta_{2})\cos^{2}(\theta_{1}))-a^{2}\sin^{2}(\psi)(a(a\sin^{2}(\theta_{1})(9\cos^{2}(\theta_{2})$
$\displaystyle-\sin^{2}(\theta_{2})(\cos(2\psi)-2))+\sin(2\theta_{1})\sin(2\theta_{2})\cos(\psi))+\cos^{2}(\theta_{1}))+16e^{4h}\sin^{2}(\theta_{1})(2\cos(2\theta_{2})+1)\Bigg{)}^{\frac{1}{2}},$

where again the only free parameters are $\Phi_{0}$ or $r_{0}$ and no rich
phase structure could be found here.

###  3.8 Domain Wall AdS/QCD

In the recent work [42, 43], using the domain wall structure, a new
holographic model for QCD has been constructed which consists of probe D7
branes in a D5 brane geometry. When one dimension of the D5 geometry is being
compactified, the confinement will come into play and affects the gauge
degrees of freedom in the theory.

The D5 geometry is

$\displaystyle
ds^{2}=\frac{u}{R}\big{(}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dx_{4}^{2}+dx_{5}^{2}f(u;u_{\Lambda})\big{)}+\frac{R}{u}\frac{du^{2}}{f(u;u_{\Lambda})}+R\
u\ d\Omega_{3}^{2},$ (3.58)

where

$\displaystyle f(u;u_{\Lambda})=\left(1-\frac{u_{\Lambda}^{2}}{u^{2}}\right),$
(3.59)

and the non trivial dilaton field and 3-form flux are

$\displaystyle e^{\Phi}=g_{s}\frac{u}{R},\ \ \ \ \
F_{3}=\frac{2R^{2}}{g_{s}}\Omega_{3},$ (3.60)

where $\Omega_{3}$ is the volume form of the unit 3-sphere and the parameter
$R$ is related to the string parameters as $R^{2}=g_{s}N_{c}\alpha^{\prime}$.

The entanglement entropy as the function of turning point $u_{t}$ could be
found as

$\displaystyle
S_{C}(u_{t})=\frac{(2\pi)^{2}V_{2}V_{3}}{2g_{s}^{2}G_{N}^{(10)}}\int_{u_{t}}^{\infty}du\frac{u}{\sqrt{1-\frac{u_{t}^{4}}{u^{4}}\left(\frac{1-\frac{u_{\Lambda}^{2}}{u_{t}^{2}}}{1-\frac{u_{\Lambda}^{2}}{u^{2}}}\right)}},$
(3.61)

and the width of the strip versus the turning point is

$\displaystyle
L(u_{t})=2R\int_{u_{t}}^{\infty}\frac{du}{u}\frac{1}{\sqrt{\left(1-\frac{u_{\Lambda}^{2}}{u^{2}}\right)\left(\frac{u^{4}}{u_{t}^{4}}\frac{1-\frac{u_{\Lambda}^{2}}{u^{2}}}{1-\frac{u_{\Lambda}^{2}}{u_{t}^{2}}}-1\right)}}.$
(3.62)

Their behavior is shown in figure 30, where again the typical and universal
behavior of the QCD models, as the butterfly geometry for $S$, can be
detected.

Figure 30: The behavior of $L(u_{t})$ versus $u_{t}$ and $S(u_{t})$ versus
$L(u_{t})$ in the domain wall QCD.

Then, the critical distance $D_{c}$ between the two strips versus their width
$l$ could be studied, where again different values of $u_{\Lambda}$ can
capture many interesting phase structures in this background, where the
results are shown in figures 31, 32, and 33.

Figure 31: Phase one in D5 background.

Figure 32: Phase two in D5 background.

Increasing $u_{\Lambda}$ after 0.49 would make $D_{c}$ negative which is not
an acceptable range of parameter for this quantity, but still the behavior is
shown in figure 33.

Figure 33:  Phase three or the range of $u_{\Lambda}$ which $D_{c}$ is
negative is shown in the left part, and the phase four is shown in the right
part.

More phases can be detected by increasing $u_{\Lambda}$, but it would become
more noisy.

For this case, the results are less noisy but again some sharp jumps can be
observed. The behavior of the quasi-normal modes would have significant
effects on the quantum information measures in this case.

The behavior of $D_{c}$ versus $u_{KK}$ (or $u_{\Lambda}$) in Domain-Wall
metric is shown in figure 34.

Figure 34: The behavior of $D_{c}$ versus $u_{KK}$ (or $u_{\Lambda}$) in the
Domain-Wall metric.

It would then be interesting to compare the behavior of quasi-normal modes
(QNM) of D-branes, and the geodesic motions in each of these models as in [44]
and check the connections between the quantum Sieberg-Witten (SW) curves and
the gravitational perturbations dubbed SW-QNM correspondence. Specially, the
behavior of QNMs around the phase transitions and the connections with the
critical distance, mutual information, EoP and negativity could be studied.

###  3.9 Nunez-Legramandi metric

Another interesting confining geometry that can be studied, is the new
geometry constructed in [45], written in the form of

$\displaystyle
ds_{10,st}^{2}=f_{1}\Big{[}ds^{2}(\text{AdS}_{6})+f_{2}ds^{2}(S^{2})+f_{3}(d\sigma^{2}+d\eta^{2})\Big{]},$
$\displaystyle
f_{1}=\frac{2}{3}\sqrt{\sigma^{2}+\frac{3\sigma\partial_{\sigma}V}{\partial^{2}_{\eta}V}},\
\ \ f_{2}=\frac{\partial_{\sigma}V\partial^{2}_{\eta}V}{3\Lambda},\ \ \
f_{3}=\frac{\partial^{2}_{\eta}V}{3\sigma\partial_{\sigma}V},$
$\displaystyle\Lambda=3\partial_{\eta}^{2}V\partial_{\sigma}V+\sigma\left[(\partial^{2}_{\eta\sigma}V)^{2}+(\partial^{2}_{\eta}V)^{2}\right],$
$\displaystyle
f_{4}=\frac{2}{9}\left(\eta-\frac{(\sigma\partial_{\sigma}V)(\partial_{\sigma}\partial_{\eta}V)}{\Lambda}\right),\
\ \
f_{5}=4\left(V-\frac{\sigma\partial_{\sigma}V}{\Lambda}(\partial_{\eta}V(\partial_{\sigma}\partial_{\eta}V)-3(\partial^{2}_{\eta}V)(\partial_{\sigma}V))\right),$
$\displaystyle
f_{6}=18^{2}\frac{3\sigma^{2}\partial_{\sigma}V\partial^{2}_{\eta}V}{(3\partial_{\sigma}V+\sigma\partial^{2}_{\eta}V)^{2}}\Lambda,\
\ \
f_{7}=18\left(\partial_{\eta}V+\frac{(3\sigma\partial_{\sigma}V)(\partial_{\sigma}\partial_{\eta}V)}{3\partial_{\sigma}V+\sigma\partial^{2}_{\eta}V}\right),$
(3.63)

or , it could be written as

$\displaystyle
ds^{2}_{\text{st}}=f_{1}\left[\frac{2\tilde{g}^{2}}{9}\left(e^{2\rho}(-d\tau^{2}+d\vec{x}_{3}^{2})+\frac{d\rho^{2}}{g(\rho)}+e^{2\rho}g(\rho)d\psi^{2}\right)+f_{2}ds^{2}(S^{2})+f_{3}(d\sigma^{2}+d\eta^{2})\right],$
$\displaystyle B_{2}=f_{4}\text{Vol}(S^{2}),\ \ \ \
C_{2}=f_{5}\text{Vol}(S^{2}),\ \ \ \ C_{0}=f_{7},\ \ \ \ \ e^{-2\Phi}=f_{6},$
(3.64)

where in order to avoid conical singularities, the function $g$ has been
chosen as $g(\rho)=1-e^{5(\rho_{*}-\rho)}$ with the constant
$e^{2\rho_{*}}=\frac{4}{125}$.

Here $V(\sigma,\eta)$ are potential function which solves the Laplace partial
differential equation

$\displaystyle\partial_{\sigma}(\sigma^{2}\partial_{\sigma}V)+\sigma^{2}\partial^{2}_{\eta}V=0,$
(3.65)

and each $V$ which solves this equation, is a solution in Type IIB which
contains a circle that asymptotes to $AdS_{6}\times
S^{2}\times\Sigma_{2}(\sigma,\eta)$ and a four dimensional Minkowski space-
time.

In this metric the flavor branes are not probe branes and actually their back-
reactions have been included in the construction of the geometry unlike the
case of Sakai-Sugimoto or domain wall, which makes it very interesting in our
discussion of using quantum information quantities to probe confinement in the
bulk geometry 111We thank Carlos Nunez for this comment..

For this geometry we find

$\displaystyle\alpha=f_{1}\frac{2\tilde{g}^{2}}{9}e^{2\rho},\ \ \ \
\beta(\rho)=\frac{1}{e^{2\rho}g(\rho)},$ $\displaystyle
H(\rho)=\frac{2^{4}}{9^{4}}\tilde{g}^{8}V_{5}^{2}g(\rho)e^{8\rho}f_{1}^{8}f_{2}^{2}f_{3}^{2}f_{6}^{2},$
(3.66)

Note that $f_{i}(\sigma,\eta)$s are not a function of the coordinate $\rho$.
So for width and entanglement entropy we get

$\displaystyle
L(\rho_{0})=2\int_{\rho_{0}}^{\infty}d\rho\frac{e^{-\rho}}{\sqrt{\frac{\left(1-\frac{2^{5}}{(5\sqrt{5})^{5}}e^{-5\rho}\right)^{2}}{1-\frac{2^{5}}{(5\sqrt{5})^{5}}e^{-5\rho_{0}}}e^{8(\rho-\rho_{0})}-1}},$
$\displaystyle S_{C}(\rho_{0})=\frac{2}{9^{2}G_{N}^{(10)}}\tilde{g}^{4}\int
dV_{\text{int}}f_{1}^{4}f_{2}f_{3}f_{6}\int_{\rho_{0}}^{\infty}d\rho\frac{e^{3\rho}}{\sqrt{1-\frac{e^{3\rho_{0}}}{e^{3\rho}}\frac{e^{5\rho_{0}}-\frac{2^{5}}{(5\sqrt{5})^{5}}}{e^{5\rho}-\frac{2^{5}}{(5\sqrt{5})^{5}}}}},$
(3.67)

which their behaviors are shown in 35.

Figure 35: $S$ and $L$ in Nunez-Legramandi metric.

Again, using the relation for the entanglement entropy in this background, the
mutual information for two strips can be calculated and then the distance $D$
where the mutual information drops to zero is the critical distance $D_{c}$
which its behavior versus width of the strip $l$ has been shown in figure 36.
This behavior is expected since this geometry is confining and therefore
$D_{c}$ first goes up and then comes down as $l$ being increased and then
becomes constant.

Figure 36: $D_{c}$ in Nunez-Legramandi metric.

###  3.10 Backgrounds with smeared branes

The previous calculations were for the cases with fully-localized D3/D7
intersection, where the branes are located at a fix location in the transverse
direction. The results can also be compared with other backgrounds and also
with the geometries where these branes are smeared.

The metric for those backgrounds could be written as

$\displaystyle
ds^{2}_{10}=h^{-1/2}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+h^{1/2}\Big{(}d\rho^{2}+\rho^{2}d\Omega_{3}^{2}+e^{-\phi}(d\omega^{2}+\omega^{2}d\theta^{2})\Big{)},$
(3.68)

where here the (near-core) wrap factor $h=h(\rho,\omega)$ is

$\displaystyle
h(\rho,\omega)=1+\frac{R^{4}}{(\rho^{2}+e^{-\phi}\omega^{2})^{2}}\ ,$ (3.69)

where $R^{4}=4\pi g_{s}N\alpha^{\prime 2}$ and the dilaton field $\phi$ and
axion field $\chi$ can be found from the relations
$e^{-\phi(\omega)}=\beta_{0}\log\frac{\omega^{2}_{\Lambda}}{\omega}$ and
$\chi(\theta)=\frac{N_{f}}{2\pi}\theta$, where the integration constants are
$\beta_{0}=\frac{N_{f}}{4\pi}$ and
$\omega^{2}_{\Lambda}=\omega_{0}^{2}e^{1/(g_{s}\beta_{0})}$.

The main difference about this geometry is that the factor $h(\rho,\omega)$ is
a factor of both of the coordinates $\rho$ and $\omega$ which makes the
calculations more difficult and we leave it for the future works. Note that,
for the small values of $\omega$, the near-core region of D3/D7 intersection
corresponds to the IR region of the dual field theory. This background
contains RR four-form potential of the D3-brane solution which then its
effects on the correlations between the two mixed strips could be studied.

Other anisotropic holographic models for heavy quarks which consist of
Einstein-dilaton-three-Maxwell action have been constructed in [46] and [47],
where the effects of magnetic field have been investigated. In these models
also the mixed quantum information measures such as negativity or entanglement
of purification, or Crofton form and kinematic space could be studied.

Note also that for constructing these results we just used the connected part
of the entanglement entropy, i.e, $S_{C}$. We could also use the regularized
entropy, $S_{C}-S_{D}$, where $S_{D}$ is the solution for the “disconnected
part” which is just a constant, and then in the numerical codes we could take
the minimum of $S_{C}$ and $S_{D}$. For example for the Witten-QCD case, we
could get somehow a different phase diagram as shown in figure 37. But the
general results would be similar and no more phases cab be detected using this
quantity instead of $S_{C}$.

Figure 37: The phase diagram for calculating $D_{c}$, coming from
$S_{C}-S_{D}$.

##  4 Crofton form in confining backgrounds

For checking the interconnections between geometry and topology, and the
quantum information measures, it would be interesting to calculate the Crofton
form for these confining models as well and compare its behaviors for each
case.

In [16], using ideas from integral geometry, the authors found the connections
between the length of a curve and the number of geodesics, or “random” lines
it would be intersected and therefore its connection to the Crofton formula.
This formula can give further intuitions about the structures of these various
confining backgrounds from the perspective of the bulk reconstruction and can
shed more lights on their various specific properties such as singularities,
throats, bulges, walls, etc and their effects on the correlations between the
two strips in the mixed setup in such confining models. Therefore, the
behavior of the Crofton form for each space will be presented here.

###  4.1 AdS-soliton

The Crofton form for the AdS-soliton geometry with the metric 3.2 would be

$\displaystyle\omega_{\text{AdS-
soliton}}=\frac{\left(2z_{0}-\frac{(d-6)z^{8}\left(\frac{z}{z_{0}}\right){}^{-d}}{z_{0}^{7}}\right)\left(1-\left(\frac{z_{t}}{z_{0}}\right){}^{8-d}\right)}{2z_{0}z_{t}^{2}\left(1-\left(\frac{z}{z_{0}}\right){}^{8-d}\right)^{2}\left(1-\frac{z^{2}\left(1-\left(\frac{z_{t}}{z_{0}}\right){}^{8-d}\right)}{\left(1-\left(\frac{z}{z_{0}}\right){}^{8-d}\right)z_{t}^{2}}\right)^{\frac{3}{2}}}-\frac{1}{z^{2}\sqrt{1-\frac{z^{2}\left(1-\left(\frac{z_{t}}{z_{0}}\right){}^{8-d}\right)}{\left(1-\left(\frac{z}{z_{0}}\right){}^{8-d}\right){}^{2}z_{t}}}},$
(4.1)

which its behavior is shown in figure 38.

Figure 38: The plot of Crofton form $\omega$ vs. $z$ in AdS-soliton
background.

If one uses other forms of metric for AdS soliton, for instance the form of
metric of 3.9, one could get plots with a general similar behavior as shown in
figures 39 or 40.

Figure 39: The plot of Crofton form $\omega$ vs. $z$ in AdS-soliton
background.  Figure 40: The plot of Crofton form $\omega$ vs. $z$ in AdS-
soliton background.

So one could see that the standard behavior is similar, excepts for some
constants, as a well is being formed around the singularity, or the turning
points, while in the larger AdS-radius, $\omega$ increases, with also a
decreasing gradient, or as we see later, it would become constant for
$u\to\infty$.

Note that a similar quantity to Crofton form could also be defined by
$\frac{\partial^{2}S}{\partial L^{2}}$ as well, which after $L_{\text{crit}}$
it becomes zero and before that, it is the second derivative of the green
curve of figure 41, and for large values of $u$, it becomes constant. This new
quantity could also give further information.

In addition, in works such as [48], it has been shown that not all the curves
in the entanglement wedge are reconstructable from entanglement entropies and
rather there are “entanglement shades” which for their reconstruction, in
addition to differential entropy, the slightly generalized version of
entanglement of purification, called “differential purification” would also be
needed 222We thank the referee for this comment.. So based on this idea,
another version of Crofton form can also be defined using this differential
purification, as $\frac{\partial^{2}\text{(EoP)}}{\partial L^{2}}$, which then
can be implemented in confining geometries to probe further regions of the
bulk and probably find more phases. We leave such studies for the future works
and continue with the usual Crofton form here.

Figure 41: The general behavior $S$ vs. $L$ in confining models. The second
derivate of the green curve (versus $L$ instead of $u$) could be used to
define a new quantity similar to the Crofton form which could give further
information about the interplay between geometry and information.

###  4.2 Sakai-Sugimoto and deformed Sakai-Sugimoto

The Crofton form, $\omega$, for the Sakai-Sugimoto geometry would be

$\displaystyle\omega_{\text{Sakai-
Sugimoto}}=\frac{V_{3}V_{4}R^{3}_{D_{4}}}{2g_{s}^{2}G_{N}^{(10)}}\frac{u^{4}\Big{(}10u_{\text{KK}}^{3}u_{t}^{5}-5u^{5}u_{\text{KK}}^{3}-7u^{3}u_{t}^{5}+2u^{8}\Big{)}}{2\Big{(}(u^{3}-u_{\text{KK}}^{3})(u^{5}-u_{t}^{5})\Big{)}^{\frac{3}{2}}}du\wedge
d\theta.$ (4.2)

The plot of $\omega$ versus $u$ for $u_{KK}=1$ and $u_{t}=2$ has been shown in
figure 42. One can again detect a well around $u_{t}$ and it becomes constant
in the large values of $u$.

Figure 42: The plot of Crofton form $\omega$ vs. $r$ in the Sakai-Sugimoto
background.

For the deformed case we get two additional term as

$\begin{split}\omega_{\text{deformed Sakai-
Sugimoto}}&=\frac{V_{3}V_{4}R^{3}_{D_{4}}}{2g_{s}^{2}G_{N}^{(10)}}\Bigg{[}\frac{u^{4}\Big{(}10u_{\text{KK}}^{3}u_{t}^{5}-5u^{5}u_{\text{KK}}^{3}-7u^{3}u_{t}^{5}+2u^{8}\Big{)}}{2\Big{(}(u^{3}-u_{\text{KK}}^{3})(u^{5}-u_{t}^{5})\Big{)}^{\frac{3}{2}}}\\\
&+\frac{d^{2}x^{4}(u)}{du^{2}}\sqrt{\frac{u^{5}(1-\frac{u^{3}_{KK}}{u^{3}})}{1-\frac{u_{t}^{5}}{u^{5}}}}\\\
&+\frac{dx^{4}(u)}{du}\frac{u^{5/2}\left(7u_{\text{KK}}^{3}u_{t}^{5}-2u^{5}u_{\text{KK}}^{3}-10u^{3}u_{t}^{5}+5u^{8}\right)}{2\Big{(}(u^{3}-u_{\text{KK}}^{3})(u^{5}-u_{t}^{5})\Big{)}^{\frac{3}{2}}}\Bigg{]}du\wedge
d\theta,\end{split}$

Then, using the relation 3.24, we get

$\begin{split}\omega_{\text{deformed Sakai-
Sugimoto}}&=\frac{V_{3}V_{4}R^{3}_{D_{4}}}{2g_{s}^{2}G_{N}^{(10)}}\frac{u^{4}}{2\left({\frac{u^{8}-u^{5}u_{\text{KK}}^{3}}{u_{0}^{8}-u_{0}^{5}u_{\text{KK}}^{3}}-1}\right)^{\frac{1}{2}}\left(u^{3}-u_{\text{KK}}^{3}\right){}^{\frac{3}{2}}\left(u^{5}-u_{t}\right){}^{\frac{3}{2}}}\times\\\
&\Bigg{(}\frac{R_{D_{4}}^{\frac{3}{2}}(3u_{0}^{8}u^{3}-(2u^{5}+3u_{0}^{5})u_{\text{KK}}^{6}+(13u^{8}-3u_{0}^{5}u^{3}+3u_{0}^{8})u_{\text{KK}}^{3}-11u^{11})(u^{5}-u_{t}^{5})}{u_{0}^{5}u_{\text{KK}}^{3}-u^{5}u_{\text{KK}}^{3}+u^{8}-u_{0}^{8}}\\\
&+R_{D_{4}}^{\frac{3}{2}}\left(7u_{\text{KK}}^{3}u_{t}^{5}-2u^{5}u_{\text{KK}}^{3}-10u^{3}u_{t}^{5}+5u^{8}\right)\\\
&+\Big{(}\frac{u^{8}-u^{5}u_{\text{KK}}^{3}}{u_{0}^{8}-u_{0}^{5}u_{\text{KK}}^{3}}-1\Big{)}^{\frac{1}{2}}\left(10u_{\text{KK}}^{3}u_{t}^{5}-5u^{5}u_{\text{KK}}^{3}-7u^{3}u_{t}^{5}+2u^{8}\right)\Bigg{)}du\wedge
d\theta,\end{split}$

The general behavior of $\omega_{\text{deformed Sakai-Sugimoto}}$ vs. $u$ is
shown in figure 43.

Figure 43: The plot of Crofton form $\omega$ vs. $r$ in the deformed Sakai-
Sugimoto background.

One could see that for the intermediate and bigger values of $u$ the behavior
is similar to the Sakai-Sugimoto case, as one would expect.

###  4.3 Klebanov-Strassler

The Crofton form in the KS background would be

$\begin{split}\omega_{\text{KS}}&=\frac{2^{2/3}\pi^{3}V_{2}\epsilon^{-8/3}\epsilon^{4}(\alpha_{p}{g_{s}}M)^{2}}{3G_{N}}\frac{h(\tau)\sinh(2\tau)}{\sqrt{1-\sinh^{2}\left(\tau_{0}\right)\text{csch}^{2}(\tau)\left(\frac{\tau_{0}-\sinh\left(\tau_{0}\right)\cosh\left(\tau_{0}\right)}{\tau-\sinh(\tau)\cosh(\tau)}\right){}^{\frac{2}{3}}}}-\\\
&\frac{h(\tau)\sinh^{2}\left(\tau_{0}\right)\left(2\tau_{0}-\sinh\left(2\tau_{0}\right)\right)(12\tau\coth(\tau)-5\sinh(3\tau)\text{csch}(\tau)+3)}{6(\sinh(2\tau)-2\tau)^{2}\sqrt[3]{\frac{\tau_{0}-\sinh\left(\tau_{0}\right)\cosh\left(\tau_{0}\right)}{\tau-\sinh(\tau)\cosh(\tau)}}\left(1-\sinh^{2}\left(\tau_{0}\right)\text{csch}^{2}(\tau)\left(\frac{\tau_{0}-\sinh\left(\tau_{0}\right)\cosh\left(\tau_{0}\right)}{\tau-\sinh(\tau)\cosh(\tau)}\right){}^{\frac{2}{3}}\right){}^{\frac{3}{2}}}+\\\
&\frac{\sinh^{2}(\tau)h^{\prime}(\tau)}{\sqrt{1-\sinh^{2}\left(\tau_{0}\right)\text{csch}^{2}(\tau)\left(\frac{\tau_{0}-\sinh\left(\tau_{0}\right)\cosh\left(\tau_{0}\right)}{\tau-\sinh(\tau)\cosh(\tau)}\right){}^{\frac{2}{3}}}}.\end{split}$

Due to the integral in the relation of $h(\tau)$ function, the analytic
solution could not be found and the Crofton form for the KS could only be
studied numerically. We expect its behavior would be very similar to the KT
case, as in figure 46.

###  4.4 Witten-QCD

The Crofton form for the Witten-QCD model is

$\begin{split}\omega&=\frac{-1}{2}\partial^{2}_{u}Sdu\wedge d\theta=\\\
&\frac{V_{2}}{G^{(10)}_{N}}\frac{4\pi^{2}R^{\frac{9}{2}}}{9g_{s}^{2}\sqrt{u_{t}}}\left(\frac{u\left(2\left(u^{2}-2u_{0}^{2}\right)u_{t}^{6}+\left(7u_{0}^{2}u^{3}+4u_{0}^{5}-4u^{5}\right)u_{t}^{3}+2u^{8}-7u_{0}^{5}u^{3}\right)}{2\left(u^{5}-u_{0}^{5}-u^{2}\left(u_{t}^{3}-u_{0}^{2}\right)\right){}^{\frac{3}{2}}\sqrt{u^{3}-u_{t}^{3}}}\right)du\wedge
d\theta.\end{split}$

Note that again we only use the connected solution for entanglement entropy
here.

The behavior of Crofton form for $u_{0}=2$ and $u_{t}=1$ in WQCD background
has been shown in figure 44 where one can see that around $u_{t}$ and $u_{0}$
we have an infinite well.

Figure 44: The plot of Crofton form $\omega$ vs. $u$ in WQCD background.

###  4.5 Maldacena-Nunez

The Crofton form in MN background would be

$\displaystyle\omega_{MN}=c_{MN}\frac{2\sinh(4r)\left(\sinh^{4}(2r)-2\sinh^{4}\left(2r_{0}\right)\right)}{\left(\sinh^{4}(2r)-\sinh^{4}\left(2r_{0}\right)\right)\sqrt{1-\sinh^{4}\left(2r_{0}\right)\text{csch}^{4}(2r)}},$
(4.3)

where $c_{MN}=\frac{V_{2}\pi^{3}e^{4\phi_{0}}}{G_{N}^{(10)}}$.

The plot of Crofton form for $r_{0}=2$ in MN case has been shown in figure 45.
Again a deep well could be detected.

Figure 45: The plot of Crofton form $\omega$ vs. $r$ in the MN background.

###  4.6 Klebanov-Tseytlin

The Crofton form for the KT background would be

$\displaystyle\omega_{KT}=c_{KT}\frac{2r^{9}\log^{\frac{3}{2}}(\frac{r}{r_{s}})\big{(}\log(\frac{r}{r_{s}})+1\big{)}-r_{0}^{6}r^{3}\log(\frac{r_{0}}{r_{s}})\log^{\frac{1}{2}}(\frac{r}{r_{s}})\big{(}8\log(\frac{r}{r_{s}})+3\big{)}}{2\left(r^{6}\log\big{(}\frac{r}{r_{s}}\big{)}-r_{0}^{6}\log\big{(}\frac{r_{0}}{r_{s}}\big{)}\right)^{\frac{3}{2}}},$
(4.4)

where $c_{KT}=\frac{12V_{2}\pi^{3}M^{2}g_{s}\epsilon^{4}}{G_{N}^{(10)}}$ is
just a constant.

The plot of $\omega_{KT}$ versus $r$ for singularity point of $r_{s}=2$ and
turning point of $r_{0}=1$ is shown in figure 46.

Figure 46: The plot of Crofton form $\omega$ vs. $r$ in the KT background.

Again, one can see similar to the WQCD case, for the Crofton form, there is a
well around the singularity point.

###  4.7 Domain Wall QCD

The Crofton form for the D5 black brane 3.58 which were used in the domain
wall background 3.8 is

$\displaystyle\omega_{\text{Domain-
Wall}}=\frac{u\left(u^{2}\left(3u_{t}^{2}u_{\Lambda}^{2}-3u_{t}^{4}+u_{\Lambda}^{4}\right)+2u_{t}^{2}u_{\Lambda}^{2}\left(u_{t}^{2}-u_{\Lambda}^{2}\right)+u^{6}-2u^{4}u_{\Lambda}^{2}\right)}{(u^{2}-u_{\Lambda}^{2})^{\frac{1}{2}}\left(\left(u^{2}-u_{t}^{2}\right)\left(u_{t}^{2}+u^{2}-u_{\Lambda}^{2}\right)\right)^{\frac{3}{2}}}$
(4.5)

Figure 47: The Crofron form for D5 black brane geometry for $u_{t}=2$ and
$u_{\Lambda}=1$.

Again, the hole around the $u_{\Lambda}$ and $u_{t}$ could be detected while
for the larger values of $u$ it would become constant and again it would point
to the structure of this background as the Crofron form can be used as a
holographic tool in the bulk reconstruction.

###  4.8 Nunez-Legramandi metric

The scalar part of the Crofton for this metric would be

$\displaystyle\omega_{NL}=\left(\frac{3e^{3\rho}}{\sqrt{1-\frac{\left(e^{5\text{$\rho$0}}-\frac{32}{78125\sqrt{5}}\right)e^{3\text{$\rho$0}-3\rho}}{e^{5\rho}-\frac{32}{78125\sqrt{5}}}}}-\frac{e^{3\rho}\left(\frac{3\left(e^{5\text{$\rho$0}}-\frac{32}{78125\sqrt{5}}\right)e^{3\text{$\rho$0}-3\rho}}{e^{5\rho}-\frac{32}{78125\sqrt{5}}}+\frac{5\left(e^{5\text{$\rho$0}}-\frac{32}{78125\sqrt{5}}\right)e^{2\rho+3\text{$\rho$0}}}{\left(e^{5\rho}-\frac{32}{78125\sqrt{5}}\right)^{2}}\right)}{2\left(1-\frac{\left(e^{5\text{$\rho$0}}-\frac{32}{78125\sqrt{5}}\right)e^{3\text{$\rho$0}-3\rho}}{e^{5\rho}-\frac{32}{78125\sqrt{5}}}\right)^{3/2}}\right).$
(4.6)

Its behavior is shown in figure 48.

Figure 48: The Crofron form for D5 black brane geometry for $u_{t}=2$ and
$u_{\Lambda}=1$.

Again, the behavior of the Crofton form for this new confining metric for
large $\omega$ is not what we expect from a good holographic confining
geometry and therefore it probably need to be modified in order to satisfy the
expected quantum information properties. However, a well or discontinuity can
be observed around $\rho_{0}$ or the end of geometry or horizon as we expect
and in this regard, this geometry satisfy this particular requirement.

###  4.9 Other measures and applications

Other correlation measures could also employed to detect these phases in the
confining geometries. We specifically expect measures, such as logarithmic
negativity, complexity of purification or reflected entropy could also detect
these four phases. For instance, recently in [23], the entanglement negativity
has been studied in a toy model of evaporating black hole which consists of JT
gravity with an end-of-the-world (EOW) brane and similar to our confining
models, four phases have been found there which were called, disconnected,
cyclically connected, anti-cyclically connected and pairwise connected
geometries where the last of these geometries are new replica wormholes that
break the replica symmetry spontaneously. The transitions between these
geometries have been studied analytically there.

Other methods such as convex optimization theory or bit threads [49, 50, 51,
52, 37, 53, 54], could also help to depict the correlation structures in the
confining geometries, and specifically shows the effects of the wall in the
bulk on the entanglement pattern.

Other tools and measures, such as balanced partial entanglement and the
entanglement wedge cross section [55, 56], entanglement contours [56, 57],
horocycles [58], lambda lengths of Penner [59, 60], Holevo information [61],
etc, could be employed for analyzing the entanglement pattern in these
backgrounds.

As some applications of these study, it worths to mention that in works such
as [62], the “holographic” neutron stars have been studied where the boundary
states were considered as degenerate composite operators which are dual to
holographic degenerate stars in the AdS side. Their model is like a set of
$n_{F}$ “shells” which are distinguished by the number of derivatives. The
quantum correlations between these shells, then, could be probed by mixed
entanglement measures and the phase structure of neutron stars could be
constructed. As we showed here that the critical distance $D_{c}$ could
distinguish the scale of chiral breaking/restoration and
confinement/deconfinement, it could be a new tool to probe the phase structure
of neutron stars, in addition to the entanglement entropy.

##  5 Conclusion

In order to analyze the mixed correlation patterns in the confining
backgrounds and holographic top-down QCD models, in this work we used a
symmetric setup with two equal size strips with width $l$ and infinite length,
where there is a distance $D$ among them. These two subsystems are entangled
with each other and with the rest of the Hilbert space, creating a mixed
correlated system, and therefore entanglement entropy cannot be a correct
measure of their correlations. We propose here that, in addition to other
mixed correlation measures, the critical distance $D_{c}$ between the two
strips, where the mutual information $I$ between them suddenly drops to zero,
could act as a new tool for probing the deep in the bulk, and it can detect
the dual QCD phase structures. Specifically, in our models, by varying the
relative sizes of $l$, $D$ and $u_{KK}$, the critical distance could
distinguish the scale of confinement/deconfinement phase transition versus the
scale of chiral symmetry breaking/restoration.

In addition, for these confining geometries, we calculated the Crofton form,
which can further exhibit the interconnections between entanglement and
geometries. For these confining models, we showed that Crofton form would
display a universal behavior, as it diverges around the infrared point and
becomes constant in the large radiuses, which again demonstrates the effects
of the IR wall and confinement on the correlation patterns.

In future works, we plan to apply other correlation measures to derive the
phase structures of confining models and also solutions with black holes, and
by comparing the properties of such phase diagrams, derive results about the
universal properties of the pattern of mixed correlations.

## Acknowledgments

I would like to thank Matti Järvinen for supports and helpful discussions. I
would also like to thank MDPI publisher for supports in the form of 2019
Universe and 2021 Galaxies travel awards which were helpful in participating
various conferences to discuss this work. I am grateful to Prof. Sang Pyo Kim
and the organizers of the 17th Italian-Korean symposium for relativistic
astrophysics and workshop on cosmology and quantum space time (CQUeST 2021)
for their kind invitation. This work has been supported by an appointment to
the JRG Program at the APCTP through the Science and Technology Promotion Fund
and Lottery Fund of the Korean Government. It has also been supported by the
Korean Local Governments – Gyeongsangbuk do Province and Pohang City – and by
the National Research Foundation of Korea (NRF) funded by the Korean
government (MSIT) (grant number 2021R1A2C1010834).

## References

  * [1] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602, [hep-th/0603001].
  * [2] M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015), no. 12 126009, [arXiv:1509.06614].
  * [3] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016), no. 19 191301, [arXiv:1509.07876].
  * [4] M. Ghodrati, Complexity growth in massive gravity theories, the effects of chirality, and more, Phys. Rev. D 96 (2017), no. 10 106020, [arXiv:1708.07981].
  * [5] M. Ghodrati, Complexity growth rate during phase transitions, Phys. Rev. D 98 (2018), no. 10 106011, [arXiv:1808.08164].
  * [6] M. Ghodrati, Complexity and emergence of warped AdS3 space-time from chiral Liouville action, JHEP 02 (2020) 052, [arXiv:1911.03819].
  * [7] Y.-T. Zhou, M. Ghodrati, X.-M. Kuang, and J.-P. Wu, Evolutions of entanglement and complexity after a thermal quench in massive gravity theory, Phys. Rev. D 100 (2019), no. 6 066003, [arXiv:1907.08453].
  * [8] T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys. 14 (2018), no. 6 573–577, [arXiv:1708.09393].
  * [9] M. Ghodrati, X.-M. Kuang, B. Wang, C.-Y. Zhang, and Y.-T. Zhou, The connection between holographic entanglement and complexity of purification, JHEP 09 (2019) 009, [arXiv:1902.02475].
  * [10] M. Ghodrati, Entanglement wedge reconstruction and correlation measures in mixed states: Modular flows versus quantum recovery channels, Phys. Rev. D 104 (2021), no. 4 046004, [arXiv:2012.04386].
  * [11] J. Kudler-Flam and S. Ryu, Entanglement negativity and minimal entanglement wedge cross sections in holographic theories, Phys. Rev. D 99 (2019), no. 10 106014, [arXiv:1808.00446].
  * [12] K. Tamaoka, Entanglement Wedge Cross Section from the Dual Density Matrix, Phys. Rev. Lett. 122 (2019), no. 14 141601, [arXiv:1809.09109].
  * [13] C. A. Agón, J. De Boer, and J. F. Pedraza, Geometric Aspects of Holographic Bit Threads, JHEP 05 (2019) 075, [arXiv:1811.08879].
  * [14] S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, JHEP 03 (2021) 178, [arXiv:1905.00577].
  * [15] S. S. Gubser and A. Nellore, Mimicking the QCD equation of state with a dual black hole, Phys. Rev. D 78 (2008) 086007, [arXiv:0804.0434].
  * [16] B. Czech, L. Lamprou, S. McCandlish, and J. Sully, Integral Geometry and Holography, JHEP 10 (2015) 175, [arXiv:1505.05515].
  * [17] A. Lala, Entanglement measures for nonconformal D-branes, Phys. Rev. D 102 (2020), no. 12 126026, [arXiv:2008.06154].
  * [18] U. Kol, C. Nunez, D. Schofield, J. Sonnenschein, and M. Warschawski, Confinement, Phase Transitions and non-Locality in the Entanglement Entropy, JHEP 06 (2014) 005, [arXiv:1403.2721].
  * [19] P. Jain and S. Mahapatra, Mixed state entanglement measures as probe for confinement, Phys. Rev. D 102 (2020) 126022, [arXiv:2010.07702].
  * [20] G. T. Horowitz and R. C. Myers, The AdS / CFT correspondence and a new positive energy conjecture for general relativity, Phys. Rev. D 59 (1998) 026005, [hep-th/9808079].
  * [21] M. Ishihara, F.-L. Lin, and B. Ning, Refined Holographic Entanglement Entropy for the AdS Solitons and AdS black Holes, Nucl. Phys. B872 (2013) 392–426, [arXiv:1203.6153].
  * [22] A. P. Reynolds and S. F. Ross, Complexity of the AdS Soliton, Class. Quant. Grav. 35 (2018), no. 9 095006, [arXiv:1712.03732].
  * [23] X. Dong, S. McBride, and W. W. Weng, Replica Wormholes and Holographic Entanglement Negativity, arXiv:2110.11947.
  * [24] H.-q. Shi and D.-f. Zeng, Geodesic Motions in AdS Soliton Background Space-time, arXiv:1603.08624.
  * [25] K. Hashimoto, T. Oka, and A. Sonoda, Electromagnetic instability in holographic QCD, JHEP 06 (2015) 001, [arXiv:1412.4254].
  * [26] J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491.
  * [27] T. Numasawa, Four coupled SYK models and Nearly AdS2 gravities: Phase Transitions in Traversable wormholes and in Bra-ket wormholes, arXiv:2011.12962.
  * [28] O. Aharony, J. Sonnenschein, and S. Yankielowicz, A Holographic model of deconfinement and chiral symmetry restoration, Annals Phys. 322 (2007) 1420–1443, [hep-th/0604161].
  * [29] E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505–532, [hep-th/9803131].
  * [30] N. Itzhaki, J. M. Maldacena, J. Sonnenschein, and S. Yankielowicz, Supergravity and the large N limit of theories with sixteen supercharges, Phys. Rev. D 58 (1998) 046004, [hep-th/9802042].
  * [31] O. Aharony, The NonAdS / nonCFT correspondence, or three different paths to QCD, in NATO Advanced Study Institute and EC Summer School on Progress in String, Field and Particle Theory, pp. 3–24, 12, 2002.  hep-th/0212193.
  * [32] E. Witten, Branes and the dynamics of QCD, Nucl. Phys. B 507 (1997) 658–690, [hep-th/9706109].
  * [33] M. Ghodrati, Schwinger Effect and Entanglement Entropy in Confining Geometries, Phys. Rev. D 92 (2015), no. 6 065015, [arXiv:1506.08557].
  * [34] I. R. Klebanov and M. J. Strassler, Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities, JHEP 08 (2000) 052, [hep-th/0007191].
  * [35] S. S. Gubser and I. R. Klebanov, Baryons and domain walls in an N=1 superconformal gauge theory, Phys. Rev. D 58 (1998) 125025, [hep-th/9808075].
  * [36] A. Rolph, Quantum bit threads, arXiv:2105.08072.
  * [37] C. A. Agon and J. F. Pedraza, Quantum bit threads and holographic entanglement, arXiv:2105.08063.
  * [38] P. Basu, D. Das, A. Ghosh, and L. A. Pando Zayas, Chaos around Holographic Regge Trajectories, JHEP 05 (2012) 077, [arXiv:1201.5634].
  * [39] I. Bena, A. Buchel, and O. J. C. Dias, Horizons cannot save the Landscape, Phys. Rev. D 87 (2013), no. 6 063012, [arXiv:1212.5162].
  * [40] D. Kaviani and A. E. Mosaffa, Temperature in the Throat, Nucl. Phys. B 910 (2016) 724–753, [arXiv:1503.02026].
  * [41] J. M. Maldacena and C. Nunez, Towards the large N limit of pure N=1 superYang-Mills, Phys. Rev. Lett. 86 (2001) 588–591, [hep-th/0008001].
  * [42] N. Evans and J. Mitchell, Domain Wall AdS/QCD, arXiv:2108.12152.
  * [43] R. A. Janik, M. Jarvinen, and J. Sonnenschein, A simple description of holographic domain walls in confining theories — extended hydrodynamics, JHEP 09 (2021) 129, [arXiv:2106.02642].
  * [44] M. Bianchi, D. Consoli, A. Grillo, and J. F. Morales, More on the SW-QNM correspondence, arXiv:2109.09804.
  * [45] A. Legramandi and C. Nunez, Holographic description of SCFT5 compactifications, arXiv:2109.11554.
  * [46] I. Y. Aref’eva, K. Rannu, and P. Slepov, Holographic anisotropic model for light quarks with confinement-deconfinement phase transition, JHEP 06 (2021) 090, [arXiv:2009.05562].
  * [47] I. Y. Aref’eva, K. Rannu, and P. Slepov, Holographic model for heavy quarks in anisotropic hot dense QGP with external magnetic field, JHEP 07 (2021) 161, [arXiv:2011.07023].
  * [48] R. Espíndola, A. Guijosa, and J. F. Pedraza, Entanglement Wedge Reconstruction and Entanglement of Purification, Eur. Phys. J. C 78 (2018), no. 8 646, [arXiv:1804.05855].
  * [49] J. Evslin and L. Martucci, D-brane networks in flux vacua, generalized cycles and calibrations, JHEP 07 (2007) 040, [hep-th/0703129].
  * [50] D. Lust, F. Marchesano, L. Martucci, and D. Tsimpis, Generalized non-supersymmetric flux vacua, JHEP 11 (2008) 021, [arXiv:0807.4540].
  * [51] P. Koerber and L. Martucci, Deformations of calibrated D-branes in flux generalized complex manifolds, JHEP 12 (2006) 062, [hep-th/0610044].
  * [52] I. Bakhmatov, N. S. Deger, J. Gutowski, E. Ó. Colgáin, and H. Yavartanoo, Calibrated Entanglement Entropy, JHEP 07 (2017) 117, [arXiv:1705.08319].
  * [53] J. F. Pedraza, A. Russo, A. Svesko, and Z. Weller-Davies, Lorentzian threads as ’gatelines’ and holographic complexity, arXiv:2105.12735.
  * [54] J. F. Pedraza, A. Russo, A. Svesko, and Z. Weller-Davies, Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity, arXiv:2106.12585.
  * [55] Q. Wen, Balanced Partial Entanglement and the Entanglement Wedge Cross Section, JHEP 04 (2021) 301, [arXiv:2103.00415].
  * [56] M. Han and Q. Wen, First Law and Quantum Correction for Holographic Entanglement Contour, arXiv:2106.12397.
  * [57] D. S. Ageev, Shaping contours of entanglement islands in BCFT, arXiv:2107.09083.
  * [58] B. Czech, L. Lamprou, S. Mccandlish, and J. Sully, Modular Berry Connection for Entangled Subregions in AdS/CFT, Phys. Rev. Lett. 120 (2018), no. 9 091601, [arXiv:1712.07123].
  * [59] R. Penner Commun. Math. Phys.113299-339 (1987).
  * [60] B. Boldis and P. Lévay, Cluster algebraic description of entanglement patterns for the BTZ black hole, arXiv:2108.10638.
  * [61] N. Bao, J. Harper, and G. N. Remmen, Holevo Information of Black Hole Mesostates, arXiv:2103.06888.
  * [62] J. de Boer, K. Papadodimas, and E. Verlinde, Holographic Neutron Stars, JHEP 10 (2010) 020, [arXiv:0907.2695].