arXiv:2409.nnnn

Kaluza-Klein discreteness of the entropy:

Symmetrical bath and CFT subsystem

Harvendra Singh

Theory Division, Saha Institute of Nuclear Physics

1/AF Bidhannagar, Kolkata 700064, India

Homi Bhabha National Institute (HBNI)

Anushaktinagar, Mumbai 400094, India

Abstract

We explore the entanglement entropy of CFT systems in contact with large bath
system, such that the complete system lives on the boundary of $AdS_{d+1}$
spacetime. We are interested in finding the HEE of a bath (system-B) in
contact with a central subsystem-A. We assume that the net size of systems A
and B together remains fixed while allowing variation in individual sizes.
This assumption is simply guided by the conservation laws. It is found that
for large bath size the island entropy term are important. However other
subleading (icebergs) terms do also contribute to bath entropy. The
contributions are generally not separable from each other and all such
contributions add together to give rise a fixed quantity. Further when
accounted properly all such contributions will form part of higher entropy
branch for the bath. Nevertheless the HEE of bath system should be subjected
to minimality principle. The quantum minimality principle
$S_{quantum}[B]=\\{S[A],S_{total}+S[A]\\}_{min}$, is local in nature and gives
rise to the Page curve. It is shown that the changes in bath entropy do
capture Kaluza-Klein discreteness. The minimality principle would be
applicable in finite temperature systems as well.

##  1 Introduction

The AdS/CFT holographic duality [1] has provided a big insight to our
understanding of entanglement in strongly coupled quantum theories. Our focus
here is on simple cases of the entanglement between two similar type of
quantum systems with common interfaces. Generally one believes that sharing of
quantum information between systems is guided by unitarity and the locality.
Under this principle the understanding of the formation of gravitational black
holes like states evolving from a loosely bound pure quantum matter state and
the subsequent evaporation process (via Hawking radiation) still remains an
unsolved puzzle. Though it is believed that the whole evolution process would
be unitary and all information inside the black hole interior (behind the
horizon) will be recovered once the black hole fully evaporated. Related to
this aspect there is a proposal that the entanglement entropy curve for the
bath radiation should bend when the half Page-time is crossed [6]. This
certainly is often true when a pure quantum system is divided into two smaller
subsystems. But for mixed states of the hawking radiation, or for finite
temperature CFTs dual of the AdS-black holes, it is not that straight forward
to obtain the Page curve. However, important progress has been made recently
in certain models by coupling holographic CFT to an external radiation system
(bath), and in some other examples by involving nonperturbative techniques
such as wormholes, riplicas and islands [4, 5]. Some answers to the difficult
questions have been attempted. 111 Also see a review on information paradox
along different paradigms [7] and for list of related references therein; see
also [[8]-[11]].

Particularly the AMM proposal for generalized entanglement entropy [4]
involves the hypothesis of islandic $(I)$ contribution, and by including
gravitational entropy of respective island boundary $(\partial I)$. According
to this proposal the ‘quantum’ entropy of a 2-dimensional radiation bath
subsystem $(B)$ can be expressed as

$\displaystyle S_{quantum}[B]=[ext\\{{Area(\partial I)\over
4G_{N}}+S[B~{}U~{}I]\\}]_{min}$ (1)

This model uses a hybrid type ‘gravity plus gauge theory’ holographic model.
It includes contribution of (gravitational) island entropy to the bath
entanglement entropy. The islands are usually disconnected surfaces inside
bulk JT gravity. The JT (conformally or nearly AdS) gravity is a dual theory
of ‘dot’ like quatum system on the boundary. The same dot lives on the
boundary of an infinte $CFT_{2}$ bath system. In the low energy description
the JT gravity is treated as a system which is in contact with a 2-dim
radiation bath over a flat Minkowski coordinate patch. So there are both field
theoretic ($S[B~{}U~{}I]$) as well as gravitational contributions
(${Area(\partial I)\over 4G_{N}}$) present in the formula (1). Secondly there
is a need to pick the lowest contribution out of a set of many such possible
extremas, which may include entropy contributions of islands and the
radiation. 222In other extensions of the hybrid models one also includes
wormhole contributions, see [5]. Although complicated looking, the expression
in (1) seemingly reproduces a Page-curve for the bath radiation entropy [4].
However AMM proposal ignores contributions of several subleading terms which
we shall altogether pronounce as ’icebergs’ terms. Here it should be clear
that the icebergs also are contributions of various disconnected elements to
the bath entropy, similar to the islands entropy.

The important feature of AMM-proposal is that it highlights the appearance of
island inside bulk gravity. The islands are normally situated outside the
black hole horizon. These arise by means of a dynamical principle, and usually
the islands are associated with the presence of a bath system when in contact
with the quantum-dot. We highlight in present work that other subleading
(disconnected) contributions of ‘icebergs’ entropies would have to be properly
accounted for in the bath entropy. Here we will be extending our 2-dimensional
proposal [18] to the case of higher dimensional CFTs. We provide the picture
that no doubt there would exist island contributions but we should not ignore
(subleading) icebergs contributions to the bath entropy. If these are ignored
we only end up with incomplete interpretation of the Page curve for quantum
entropy.

We clearly show that when island, icebergs and leading (pure) bath entropy
once added together as a series, that gives rise to a system dependent
constant contribution to the entropy of bath system. These contributions are
naturally inseparable from each other as they compensate each other perfectly
well no matter how large or small their individual contributions could be. We
have explicitly shown this phenomenon for limiting case of a quantum dot like
system located at the interface of symmetrical CFT bath system [18].
Correspondingly we find that there are gravitational contributions to the bath
entropy arising from island and the subleading icebergs, for system-A attached
to large bath (subsystem-B). We must note that the entanglement entropy of two
such systems together $(AUB)$ can be expanded as

$S_{total}[AUB]=S_{pure~{}bath}+S_{island}+S_{icebergs}\equiv S_{l}$

This expansion can always be done whenever system-B is sufficiently large
compared to system-A, e.g. see drawing in lower figure in (2).

Figure 1:  A symmetrical arrangement of CFT system-A in the center (along
$x_{1}$-axis) and CFT bath subsystem-B (of size $b$) situated on either side.
The transverse spatial directions other than $x_{1}$ (if any) are all
suppressed. The complimentary bath system $(B^{c})$ is also drawn.

Note $S_{l}\gg 0$ is the entropy contained in the total system ($AUB$). It is
a constant quantity once the total system size is fixed, and if the system
being conserved.

Furthermore, it is proposed that the quantum entropy of entanglement of a
large bath subsystem-B should be obtained from local minimality principle

$S[B]_{quantum}=\\{S[A],~{}S_{l}+S[A]\\}_{min}=S[A]$

where $S[A]$ is the entropy of system-A. To make it further clear the above
minimality principle works because both $S[A]$ and $S[B]$, as systems being in
contact, involve identical local entropies, corresponding to common RT
surfaces they do share. Note two systems in contact will have common
interfaces fig.(1). The complementary bath system $(B^{c})$ which is semi-
infinite on both sides does not play direct role here. For non-contact type
systems one should study them separately. In conclusion, the above two
equations reproduce the Page curve for the entropy of a large bath system,
including for finite temperature systems. The formula is definitely valid at
least in static (equillibrium) cases. For time dependent processes involving
black hole evaporation, or if there is a continuous change in the total system
size (e.g. due to change in mass, energy, or horizon size), the scenario is
expected to be similar at any point in time, hence it might be applicable for
slow enough processes.

The rest of the article is organized as follows. In section-2 we explain the
island and icebergs contributions and define the generalized entropy
formulation for pure $AdS$ case. On the boundary we take a finite size system
in contact with a symmetrical bath. In section-3 we also discuss a limiting
case when system-A becomes very small and appears as point like (where we
would use Kaluza-Klein scenario) and is in contact with large size CFT bath.
Under Kaluza-Klein perspective the situation emerges similar to that of $2n$
parallel quantum (strip) systems attached to some large bath. The results for
the AdS black holes cases are covered in section-4. The section-5 contains a
summary.

##  2 Islands and Icebergs and bath subsystem entropy

Let us consider a system (A) in contact with a bath subsystem (B) in fully
symmetrical set up, and both having finite sizes, and living on the boundary
of the $AdS_{d+1}$ spacetime. Thus it is assumed that both systems A and B are
made of identical field species, i.e. described by same field contents, for
simplicity of the problem. The pure $AdS_{d+1}$ spacetime geometry is
described by following line element

$\displaystyle ds^{2}={L^{2}\over
z^{2}}(-dt^{2}+dx_{1}^{2}+\cdots+dx_{d-1}^{2}+dz^{2})$ (2)

where constant $L$ represents a large radius of curvature (in string length
units) of spacetime. The coordinate ranges lie
$-\infty\leq(t,~{}x_{i})\leq\infty$ and $0\leq z\leq\infty$, where coordinate
$z$ represents the holographic range of boundary theory.333 The Kaluza-Klein
compactification over a circle (say $x_{d-1}\simeq x_{d-1}+2\pi R$) produces a
conformally  anti-de Sitter solution in lower dimensional gravity.
Particularly, for $d=2$ case one obtains Jackiw-Teitelboim type 2-dim
dilatonic background [12, 13],  $\displaystyle ds^{2}_{JT}={L^{2}\over
z^{2}}(-dt^{2}+dz^{2})$ $\displaystyle
e^{-2(\phi-\phi_{0})}=\sqrt{g_{xx}}={L\over z}$ (3) where $\phi$ is a
2-dimensional dilaton field of the effective bulk gravity theory, all written
in the standard convention (effective string coupling vanishing near the
boundary). The two Newton’s constants get related as ${2\pi R\over
G_{d+1}}\equiv{1\over G_{d}}$, with $G_{2}$ being dimensionless for $AdS_{2}$.

Figure 2:  Two different extremal situations are drawn schemetically. The
extremal RT surfaces when bath-B size is small compared to the system-A ($b\ll
a$), we have all disconnected minimal area surfaces. The lower graph for $b\gg
a$ instead has connected surfaces. We keep $l$ same in the both the graphs.

The $CFT_{d}$ theory lives on $d$-dimensional $(t,\vec{x})$ flat Minkowski
spacetime describing boundary dynamics of $AdS_{d+1}$ bulk geometry. Consider
a set up in which a finite size bath subsystem-B lives on the coordinate
patches $[-(b+a),-a]$ & $[a,(b+a)]$ along $x_{d-1}$ direction, whereas
system-A, sandwiched in between the bath, lives over the patch $[-a,a]$. The
sketches are provided in figures (1), (2) for clarity. The entire systems set-
up is arranged in a symmetrical way for the convenience. The states of the
system-A and the bath subsystem-B are obviously entangled. The complementary
bath system $B^{c}$ lives over the coordinate patches $[b+a,\infty]$ and
$[-(b+a),-\infty]$. The $B^{c}$ is semi-infinite on either sides.

From Ryu-Takayanagi holographic prescription the entanglement entropy of a
$d$-dimensional $CFT$ strip-shape subsystem of width $l~{}(l=2a+2b)$ is given
by

$\displaystyle
S_{total}[AUB]={L^{d-1}V^{(d-2)}\over(2d-4)G_{d+1}}\left({1\over\epsilon^{d-2}}-{b_{0}^{d-1}\over(a+b)^{d-2}}\right)\equiv
S_{l}$ (4)

where $\epsilon\simeq 0$ is the UV cut-off of the $CFT_{d}$ (we shall only
consider examples where $d>2$). $V^{(d-2)}$ is the volume of $(d-2)$ spatial
directions perpedicular to the strip width direction $x_{1}$.444 Note
$b_{0}={1\over 2(d-1)}B({d\over 2d-2},{1\over 2})$ is specific dimension
dependent coefficients involving explicit Beta-functions, see more in appendix
of [17]. We take $l$ sufficiently large but fixed, so that $S_{l}$ has a
constant value. Our aim is to determine entanglement entropy where size $b$ is
varied from $b\simeq 0$ to $b\simeq l/2$, by hand. We simply assume local
conservation laws, so that the net gain (loss) of system-A is compensated by
equal loss (gain) in the size of bath subsystem-B and vice-versa. Note that
such a process will keep $l$ fixed. Especially for explicit time dependent
cases one may have some definite rate of change, while $\dot{a}=-\dot{b}$ is
true due to conservation of energy. The exact rate of loss or gain and the
mechanism by which it may happen is not important here and the actual details
of the physical process is also not required here. All we are considering is
that local conservation laws are at work for complete system within total box
size, $l$.555 Any explicit time dependent processes are not studied here.
Obviously we are assuming here that the system and the bath are made up of
identical (CFT) field content. Let us consider two extreme cases below.

Case-1: Independent entropies

When $b\ll a$, for the bath subsystem-B being very small in size, the
entanglement entropy of the larger system-A can be found by its extremal
surface area as [14, 15]

$\displaystyle S[A]$ $\displaystyle=$
$\displaystyle{L^{d-1}V^{(d-2)}\over(2d-4)G_{d+1}}\left({1\over\epsilon^{d-2}}-{b_{0}^{d-1}\over
a^{d-2}}\right)$

While the extremal surfaces of bath subsytem on both sides become
disconnected. The entropy of small subsystem-B become independently,

$\displaystyle
S[B]={L^{d-1}V^{(d-2)}\over(d-2)G_{d+1}}\left({1\over\epsilon^{d-2}}-{2^{d-2}b_{0}^{d-1}\over
b^{d-2}}\right)$ (6)

Eq.(6) involves area contributions from two disconnected but identical
extremal surfaces, which contribute to the bath entropy, see the upper graph
as in fig.(2). Note that $S[A]$ and $S[B]$ have the local parts which depend
on individual strip parameters $a$ and $b$, respectively, Thus both systems
entropies are completely independent even though systems are in contact with
each other. The entropy of the small bath system-B would be defined by
equation (6) until the crossover point is reached. After the crossover new
extremal (connected) surfaces will emerge as drawn in lower graph of fig.(2).
We will discuss it next.

Case-2: Entropies with identical local components

When $b\gg a$, in this regime of large bath system the entanglement entropy of
system-B is given by the equation

$\displaystyle
S[B]={L^{d-1}V^{(d-2)}\over(2d-4)G_{d+1}}\left({1\over\epsilon^{d-2}}-{2^{d-2}b_{0}^{d-1}\over(l-2b)^{d-2}}\right)+S_{l}$
(7)

where $S_{l}$ is total entropy of the systems together, which is a fixed
quantity. $S_{l}$ gets the contribution from the outer RT surface connecting
two farther ends of the symmetrical bath on either side, follow the lower
graph in fig(2). Note $S_{l}$ is independent of individual sizes $b$ or $a$,
and it is a fixed quantity for given $l$. We might still vary individual
system sizes such that we keep $l$ fixed. Now for smaller system-A of strip
width $2a$ the entropy is given by

$\displaystyle
S[A]={L^{d-1}V^{(d-2)}\over(2d-4)G_{d+1}}\left({1\over\epsilon^{d-2}}-{2^{d-2}b_{0}^{d-1}\over(l-2b)^{d-2}}\right)$
(8)

where we have simply used $2a={l}-2b$.

From eqs. (7) and (8), we observe that the $S[A]$ and $S[B]$ differ only by an
overall constant, $S_{l}$, otherwise they have the same type of local
dependence on $b$. It also implies that under small change in the system sizes

$\displaystyle{\partial S[A]\over\partial b}={\partial S[B]\over\partial b}$
$\displaystyle{\partial S[A]\over\partial a}={\partial S[B]\over\partial a}$

Put in other words, $S[A]$ and $S[B]$ represent two independent extrema of the
same observable and only differ up to a constant. Classically the areas of
these extremal surfaces is such that $S[B]>S[A]$, but quantum entropy of large
bath instead would be governed by the minimization

$\displaystyle S^{quantum}_{b\gg a}[B]=\\{S[A],S[A]+S_{l}\\}_{min}=S[A]\ .$
(10)

This states that the quantum entropy of a large bath is the same as the
entropy of smaller system-A. Furthermore we note that the quantum entropy of
bath decreases as $b$ gets larger and larger but at the same rate as that of
system-A. So the Page-curve for entropy of large bath system ($b\gg a$)
follows from the principle of minimum entropy, if there exist multiple extrema
separated by constants, like $S_{l}$ here. The system-A and the bath
subsystem-B entropy otherwise have identical local dependences. This is the
net conclusion of the proposal given in eq.(10). Although we might still
wonder that the bath entropy ought to have been taken simply as $S[B]$ given
in (7), which is net classical area of bulk extremal surface. Instead, as per
quantum minimality proposal (10), the bath entropy has to be given by smaller
quantity $S[A]$, mainly because both have the same local dependence. The
latter is in agreement with the Page curve expectation and unitarity for
quantum systems that follows from conservation laws. The quantum entropy
proposal shound be taken as complete result for extremal CFTs (at zero
temperature) being in contact with symmetrical bath.

Figure 3:  Entropy plots for the values $l=10,~{}\epsilon=.1$, for $CFT_{3}$
system. The lower falling curve (in yellow) is preferable for quantum entropy
of large size bath. The rising (green) curve is good for entropy in the small
bath region only. The topmost graph is classical entropy of large bath and is
not the physical one. We have set ${L^{2}l_{2}\over 2G_{4}}=1$ for the graphs.

Island and Icebergs: full system entropy

We did not explicitly encounter any island like isolated (disconnected)
surface area contributions to the entropy so far.666In higher dimensional bulk
theories the ’islands’ are multidimensional (co-dim-2) surfaces. These are no
longer point objects as in JT gravity and radiation bath models. So where are
these contributions hidden in the above analysis? We have got the Page curve
using quantum minimality principle for large bath system without knowing about
the island (fragments) entropy contributions. To understand it we need to
dissect the total entropy of system-A and system-B. It is vital to understand
first what $S[AUB]$ teaches us, even though it is only some system parameter
and a fixed quantity. We explore this systematically for $d=3,~{}4$ and $5$
dimensions. (For $d=2$ CFT case, this analysis has been presented in [18].)

i) For $d=3$ CFT: The total system entropy $S_{l}$ is given by ($l=2a+2b$)

$\displaystyle S_{l}=S[AUB]$ $\displaystyle=$ $\displaystyle{L^{2}l_{2}\over
2G_{4}}\left({1\over\epsilon}-{b_{0}^{2}\over a+b}\right)$ (11)

where $l_{2}$ is the length of the strip. Making an expansion on the r.h.s. in
small ratio $s\ll 1$ $(~{}s={a\over b})$, one can find

$\displaystyle S_{l}$ $\displaystyle=$ $\displaystyle{L^{2}l_{2}\over
2G_{4}}({1\over\epsilon}-{b_{0}^{2}\over b})+{L^{2}l_{2}\over
2G_{4}}{b_{0}^{2}\over b}(s-s^{2}+s^{3}-\cdots)$ (12) $\displaystyle\equiv$
$\displaystyle S^{(0)}_{bath}+S_{Island}+S_{Icebergs}$

where expressions in the last line have been identified as: 1) entropy of pure
bath:

$\displaystyle S^{(0)}_{bath}={L^{2}l_{2}\over
2G_{4}}({1\over\epsilon}-{b_{0}^{2}\over b}),~{}~{}~{}~{}~{}$ (13)

2) subleading gravitational entropy of island-ic boundary:

$\displaystyle S_{island}={L^{2}l_{2}a\over 2G_{4}}{b_{0}^{2}\over
b^{2}},~{}~{}~{}$ (14)

3) rest all sub-subleading (icebergs) entropies together as:

$\displaystyle S_{icebergs}=-{L^{2}l_{2}b_{0}^{2}\over
2G_{4}}\left({a^{2}\over b^{3}}-{a^{3}\over b^{4}}+\cdots\right).$ (15)

Note $S^{(0)}_{bath}$ in eq.(13) represents the HEE of $CFT_{3}$ strip system
having width $2b$ (net size of the bath subsystem on both sides). Note it is
entropy of pure bath system without the presence of system-A. Whereas
$S_{island}$, which genuinely represents the interactions between bath-B and
system-A, is the ‘gravitational’ entropy of island-like boundary situated at
$z=b$ (e.g. corresponding to an island-like region lying in between $z=\infty$
and $z=b$ inside the $AdS_{4}$). The codim-2 island boundary located at $z=b$
has a geometrical area:

${\cal{A}}_{island}={2L^{2}l_{2}a\over b^{2}}$

It is proportional to actual size $2l_{2}a$ of boundary CFT system-A. Thus
geometric entropy of island eq.(14) may be expressed as

$S_{island}={{\cal{A}}_{island}\over 4\tilde{G}_{4}}$

Here $\tilde{G}_{4}=G_{4}/b_{0}^{2}$ is treated as effective Newton’s
constant. Other sub-subleading terms, which are altogether named as icebergs’
entropies, include contributions from remaining terms in small $s$ expansion.
Thus by doing a series expansion it gets revealed that the various terms in
the series (12) although may have got distinct interpretations, but they are
actually inseparable from each other. No matter what these individual values
might be, all the terms are important because these add up nicely to
constitute $S_{l}$, i.e. the total entropy. The total entropy, including
island and icebergs, thus has a constant value, for given $l$. (The $S_{l}$
depends only on the parameter $l$, which is measure of total systems size. In
this sense $S_{l}$ is actually a global quantity.) One could still vary $a$
and $b$ individually but keeping $l(=2a+2b)$ fixed. That means the bath size
could grow at the cost of size of system-A and vice versa, under mutual local
exchanges or those processes which may lead to shift in the mutual interface
of systems A and B. (This appears akin to what might happen in black hole
evoparation processes also, e.g. through Hawking radiation, where the Hawking
radiation is treated as bath.) Perhaps simple CFT models may teach us
something about the black hole evoparation process! The relevant entropy
graphs are plotted in the figure (3) with a discussion in the caption for
small and large bath cases.

Unitarity and the locality of entropy

The question still arises what would happen if we ignored the subleading
(icebergs) contributions in eq.(12), due to their smallness, as being
subleading $O(s^{2})$ and even smaller. Although one is free to do so but we
immediately find that the r.h.s. of $S_{l}$ will now start depending on $a$
and $b$ in independent ways! This will lead to varied conclusions regarding
the Page curve, unitarity and about the information content of the systems.
Alternatively we may decide that $S_{icebergs}$ terms should not be dropped
from leading terms in any situation. In other words precise knowledge of
isolated contributions is vital for the unitarity! Furthermore, the island and
the icebergs would remain invisible, as these contribute to eq. (7), which is
an higher entropy extrema and hence unphysical as per ‘quantum entropy’
proposal of bath-B. The latter reason solely arises from the quantum entropy
principle that for large size bath entropy should be taken with smaller value
amongst $(S_{l}+S[A])$ and $S[A]$, where $S_{l}\gg 0$ is total entropy.

ii) $d=4$ case: For $CFT_{4}$ the total system entropy $S[AUB]$ can be written
as

$\displaystyle S_{l}$ $\displaystyle=$ $\displaystyle{L^{3}l_{3}l_{2}\over
4G_{5}}\left({1\over\epsilon^{2}}-{b_{0}^{3}\over(a+b)^{2}}\right)$ (16)

where $l_{2}\&l_{3}$ are the sizes of two transverse spatial coordinates of
the $CFT_{4}$ on the boundary of $AdS_{5}$ geometry. The systems are separated
along $x_{1}$ direction. By making an expansion on the r.h.s. in (16), for
small $s$ we get

$\displaystyle S_{l}$ $\displaystyle=$ $\displaystyle{L^{3}l_{3}l_{2}\over
4G_{5}}\left({1\over\epsilon^{2}}-{b_{0}^{3}\over b^{2}}+{b_{0}^{3}\over
b^{2}}s(1+s)^{-2}\right)$ (17) $\displaystyle=$
$\displaystyle{L^{3}l_{3}l_{2}\over
4G_{5}}\left({1\over\epsilon^{2}}-{b_{0}^{3}\over b^{2}}+{b_{0}^{3}\over
b^{2}}(s-2s^{2}+O(s^{3}))\right)$ $\displaystyle\equiv$ $\displaystyle
S^{(0)}_{B}+S_{Island}+S_{Icebergs}$

where break up of various expressions in the last line is as: the leading pure
bath entropy,

$\displaystyle S^{(0)}_{B}={L^{3}l_{3}l_{2}\over
4G_{5}}({1\over\epsilon^{2}}-{b_{0}^{3}\over b^{2}}),~{}~{}~{}~{}~{}$ (18)

the island-ic (gravitational) entropy,

$\displaystyle S_{island}={L^{3}l_{3}l_{2}b_{0}^{3}a\over
4G_{5}b^{3}}\equiv{{\cal{A}}_{island}\over 4\tilde{G}_{5}},~{}~{}~{}$ (19)

and other subleading icebergs entropies:

$\displaystyle S_{icebergs}=-{L^{3}b_{0}^{3}l_{3}l_{2}\over
2G_{5}}\big{[}{a^{2}\over b^{4}}-{a^{3}\over 3b^{5}}+\cdots\big{]}.$ (20)

Note again $S_{island}$ is proportional to ${2L^{3}l_{3}l_{2}a\over b^{3}}$
which is the geometrical area of extended 3-dim island boundary located at
$z=b$ inside $AdS_{5}$ spacetime, with a redefined 5-dim Newton’s constant as
$\tilde{G}_{5}=G_{5}/b_{0}^{3}$. The related entropies are plotted in the
figure (5). They exhibit similar properties as $d=3$ case for small and large
bath cases.

iii) For $d=5$ case: Above results clearly tell us that for $CFT_{5}$ the
subsystem entanglement properties would also be similar to $d=3,~{}4$ cases.
We may convince ourselves that it will be a common occuring phenomenon
whenever there are two systems (system and the bath) in contact, i.e.
separated by interfaces. The islands and icebergs typically arise as a result
of interactions and information sharing between the dofs of the systems.

##  3 Lower dimensional Kaluza-Klein perspective: Multiple thin strips

Figure 4:  An arrangement of system-A (in the middle on $x_{1}$-axis) and CFT
bath subsystem-B of size $b$ each lie on either side. All transverse spatial
directions (if any) except $x_{1}$ direction are suppressed. The complimentary
bath system $(B^{c})$ is also drawn.

It is important to discuss special case of the systems described by eq.(7).
Consider very small size for system-A, such that $a\simeq R$, where $R$ is
Kaluza-Klein scale of the theory (analogous to JT-gravity and near $CFT_{1}$
case). (We expect that for such small size system, with a narrow width,
system-A could be effectively treated as being compactified on $S^{1}$, with a
compactification radius $R$, but $R\gg\epsilon$.) In that case we can safely
take $a\simeq\pi nR$, with $n$ being a positive integer. 777Note that it is
our assumption that there is an intermediate Kaluza-Klein compactification on
$S^{1}$ when size $a$ becomes approximately $\simeq R$ at shorter scales. Then
it becomes plausible to study a lower dimensional dual (gravitational)
description for system-A (viewed as mutiple strips of narrow width $\pi R$
wrapped along circle (KK direction) but fully extended along other transverse
directions). For simplificity we shall consider only small $n$ values, also
$\epsilon\ll\pi R\ll b$ (If there is any difficulty one can simply take
$n=1$). The system-A can essentially be treated as assembly of $2n$ narrow
(parallel) strips sandwiched between symmetrical bath system-B (on either
sides of total size $2b$). We have a situation as depicted in the figure (4),
where transverse directions $x_{2}$ and $x_{3}$ are suppressed. An expansion
on the r.h.s of (7), for small ${n\ll{b\over\pi R}}$, gives bath entanglement
entropy (for $d=3$)

$\displaystyle S[B]$ $\displaystyle=S_{l}+{L^{2}l_{2}\over
2G_{4}}({1\over\epsilon}-{b_{0}^{2}\over n\pi R})$ (21)
$\displaystyle\equiv(S^{(0)}_{bath}+S_{island}+S_{icebergs})+S_{2n-strips}[R]$

where leading term $S^{(0)}_{bath}$ is same as described earlier, and other
expressions are

$\displaystyle S_{island}={L^{2}l_{2}\over 4G_{3}}{n\over b^{2}},~{}~{}~{}$
$\displaystyle S_{icebergs}=-{L^{2}l_{2}\over 4G_{3}}({n^{2}\pi
Rb_{0}^{2}\over b^{3}})+O(n^{3})$ (22)

The $G_{3}=G_{4}/(2\pi R)$ is 3-dimensional Newton’s constant. The islandic
contribution, especially for $n=1$, is similar to the gravitational entropy of
the island boundary situated at $z=b$ inside bulk. The $S_{icebergs}$ includes
rest all subleading contributions. The eq.(21) involves an infinite series
which is perturbative. Actually it will not be wise to separate icebergs from
first two leading terms in (21) at all! They all remain important as the total
sum of these terms (within the parenthesis) sum up to $l$-dependent entropy
$S_{l}$. It is quite clear from the starting line of the perturbative
expansion in (21).

While the entanglement entropy of the $2n$ parallel strips, assembled side by
side, and in contact with bath, is simply (for $d=3$)

$\displaystyle S_{2n-strips}[A]={L^{2}l_{2}\over
2G_{4}}({1\over\epsilon}-{b_{0}^{2}\over n\pi R})$ (23)

and the bath system entropy is

$\displaystyle S[B]=S_{l}+S_{2n-strips}[R]$ (24)

The local $R$-dependent terms in eqs. (23) and (24) are identical, and since
the two expressions differ only by an overall constant $S_{l}$, the actual
‘quantum entropy’ of bath would be given by the local entropy contained in
system-A (multiple narrow strips) only. So under minimality selection rule
quantum entropy of bath should simply be stated as

$\displaystyle
S^{quantum}[B]=\\{S_{2n-strips},S_{l}+S_{2n-strips}\\}_{min}=S_{2n-strips}$
(25)

where entopy of all $2n$ parallel strips arranged together is given above.
Note $l_{2}$ is length of these strips, and the individual strip width is
taken as $\pi R$. This is the net entanglement entropy of a large bath system,
when $b\gg R$, i.e. towards the end of the Page curve. It entirely gets its
contribution from single RT surface homologous to central $2n$ strips system.
Note the narrow strips have width $\pi R$ $(\ll b)$. It is important to note
that the bath entropy is discretized since KK-level $n$ takes integral values
($n\in Z$). Obviously we would trust these results for small $n$ values only.
For large $n$ it would be good to use noncompact continuum description in one
higher dimensions.

It can be concluded that the entropy for a large bath will necessarily show
discrete jumps as and when KK level changes. This is an example of strongly
coupled system of $2n$ parallel sheets (system-A) placed in the middle of
large symmetrical CFT baths on either side. The conclusions shall remain
unchanged even under infinite size limit $(b\to\infty,~{}l\to\infty)$ and thus
should be treated universal. We conclude that we would not be able to see
island and icebergs physically, as their net contribution to entropy always
results in a fixed constant only.

###  3.1 Entropy spectrum of strip systems

We are interested in exchanging small number of strips between bath (B) and
the multi-strip system (A). This will lead to changes in the systems
entanglement entropy. Note a small number of strip exchanges between B and A
will not change the total entropy $S_{l}$. Thus net change in entropy of
system-A with KK-strips number $n_{2}$ and with KK-strips number $n_{1}$ can
be found to be $(n_{1}<n_{2})$

$\displaystyle\bigtriangleup S_{1\to 2}$
$\displaystyle=S_{strips}[n_{2}]-S_{strips}[n_{1}]$ (26)
$\displaystyle={L^{2}l_{2}\over 2G_{4}}({1\over n_{1}}-{1\over
n_{2}}){b_{0}^{2}\over\pi R}$ $\displaystyle\equiv{1\over T_{E}}\bigtriangleup
E_{1\to 2}$

In the last line the entanglement temperature of bath can be taken as
$T_{E}\simeq{1\over l}$ [16]. Empirically we may determine that typical change
in energy density of systems (energy per unit strip-length)

$\displaystyle{\bigtriangleup{\cal E}_{1\to 2}}={\bigtriangleup E_{1\to
2}\over l_{2}}={L^{2}b_{0}^{2}\over 2\pi G_{4}lR}({1\over n_{1}}-{1\over
n_{2}})={L^{2}b_{0}^{2}\over 4\pi^{2}G_{3}lR^{2}}({1\over n_{1}}-{1\over
n_{2}})$ (27)

This energy spectrum is obviously discrete in nature! A fixed quantum of
energy exchanges is required to take place between bath-B and multi-strip
system-A during an exchange of the strips. Note we are discussing the CFT
ground state (zero temperature) only. The spectrum appears analogous to the
atomic spectrum, but system-A here is made up of discrete number of strips
whereas a finite number of strips (quantum matter and not the photons) are
exchanged between various KK levels. This exchange process entails KK-level
‘jumps’. Here $n_{1}=1$ may be treated as the lowest level (for smallest size
single strip system) while $n_{2}>1$ will correspond to higher levels (more
than one strip cases). Note a level jump in the strip number is necessarily
associated with discrete (quantum) CFT matter exchanges between system and its
surrounding bath. The relevant physical scale is compactification radius $R$.

A plausible interpretation of above energy spectrum may be given as follows.
The discrete KK modes have typical momentum $\sim{1\over R}$. (The strip
length $l_{2}$ is some large quantity (fixed), $L\gg 1$ is $AdS$ radius of
curvature, $G_{3}\equiv G_{4}/(2\pi R)$ is 3-dim Newton’s constant.) Thus

$\bigtriangleup E_{1\to 2}\propto{1\over R^{2}}$

appears primerily due to the KK momentum modes in this case. The energy-matter
exchanges between central strip system-A and the bath are precise and
discrete! We emphasize that with more careful analysis of subleading terms in
the entropy, we might be able to see winding modes contributions! We hope to
report about it in subsequent communications.

A similar analysis for 4-dimensional $CFT_{4}$ would provide energy density in
exchanges

$\displaystyle{\bigtriangleup E_{1\to 2}\over l_{3}l_{2}}={L^{3}b_{0}^{3}\over
8\pi^{3}G_{4}lR^{3}}({1\over n_{1}^{2}}-{1\over n_{2}^{2}})$ (28)

where $(l_{3}l_{2})$ is the transverse size of the $3d$ strips, where the
individual width of strips is $\sim\pi R$.

We will show that change in entanglement entropy under strip exchange (or KK
level jumps) can be determined for finite temperature cases as well. There the
$|\bigtriangleup E|$ will involve thermal corrections. Perturbatively we shall
estimate and find that the leading thermal correction grows proportional to
$R$, the width of the strips. We guess these corrections involve string
winding modes.

##  4 Finite temperature systems

The previous exercise can be extended to the case of CFT at finite temperature
as well. The whole process goes in parallel for any CFT system which is in a
mixed state. The limitation is that for $d>2$ case the HEE at finite
temperature can only be estimated by using perturbative methods. Alternative
options would be resort to the numerical approach. Consider the asymptotically
$AdS_{d+1}$ geometry that has Schwarzschild black hole at the center

$\displaystyle ds^{2}={L^{2}\over z^{2}}(-f(z)dt^{2}+{dz^{2}\over
f(z)}+dx_{1}^{2}+\cdots+dx_{d-1}^{2})$ (29)

where $f(z)=(1-{z^{d}\over z_{0}^{d}})$ with $z=z_{0}$ being the location of
black hole horizon. There is a finite temperature in the field theory on AdS
boundary. Assume now that the strip shaped system-A with strip width $2a$ is
taken in thermal equillibrium with symmetrical bath system-B on either sides,
so that system-A is located in the middle of bath (net width of bath system
being $2b$). Other transverse directions of the systems are infinitely
extended. Here both systems $A$ and $B$ have same temperatures. We only
discuss the case when $a\ll b$, because $a\gg b$ case is rather straight
forward. The entanglement entropy of the strip like bath system-B on the
boundary of (29) can be written as

$\displaystyle
S[B]=S_{l}({l,z_{0}})+S({2a,z_{0}})=S_{l}({l,z_{0}})+S({l-2b,z_{0}})$ (30)

In our notation a functional $S(x,z_{0})$ represents the HEE for strip system
having width $x$ obtained from area of extremal surfaces in a black-brane AdS
geometry [17]. The $z_{0}$ dependences indicate there is finite temperature
effect (horizon dependence). Note the first term on the r.h.s. of (30)
involves those constants which treat system-A and bath-B together as single
entity. It measures a global information involving $A$ and $B$. Only second
term has the local entropy information regarding central system-A like the
size and location of interface boundaries between system-A and bath-B.

Usually for small width $x$ one can estimate $S({x,z_{0}})$ using perturbative
tools as discussed in [17]. It is the best method so long as we have $x\ll
z_{0}$. Also this is what one would be requiring most when approaching the end
of the Page curve involving two systems A & B. Typically the perturbative
series looks like [17]

$\displaystyle S({x,z_{0}})=S_{0}(x)+S_{1}(x,z_{0})+S_{2}(x,z_{0})+\cdots$
(31)

where leading term $S_{0}(x)$ is the entropy of AdS ground state, see eqs.(2)
and (6), while the first order term is given by

$\displaystyle S_{1}({x,z_{0}})={L^{d-1}V^{(d-2)}\over
16G_{d+1}}{(d-1)a_{1}x^{2}\over(d+1)b_{0}^{2}z_{0}^{d}}$ (32)

note it has explicit horizon $(z_{0})$ dependence, and same is true for second
order term and so on. (Here $a_{1}$ and $b_{0}$ are specific dimension
dependent coefficients involving explicit Beta-functions, see appendix of
[17]).

At the same time the entanglement entropy of central system-A, having width
$2a$, is given by

$\displaystyle S[A]=S({2a,z_{0}})=S({l-2b,z_{0}})$ (33)

In second equality we used $l=2a+2b$, as the bath has size $2b$. Again one can
compare that the expressions in equations (30) and (33) differ only by an over
all constant (global) quantity $S_{l}({l,z_{0}})$, that depends only on full
size (system plus symmetrical bath) $l$. From the perspective of system-A
$S_{l}$ is some global fixed quantity. Therefore we conclude that these two
equations represent two different extrema of the same entropy observable,
primerily because both entropies contain identical local terms. The local
terms arise from extremal RT surface that connects common interfaces of
systems A and B. These local terms contain mutual entanglement information
which two systems share between them. Further eqs. (30) and (33) are telling
us that they contain identical entanglement entropy (through local exchanges
along common interfaces) for system $A$ and system $B$. Classically they
differ only up to overall constant. We conclude that ‘quantum’ entanglement
entropy measure at finite temperature for large bath subsystem-B (i.e. $b\gg
a$) ought to be taken as the smaller value between (30) and (33). Therefore
the quantum entropy of any large thermal bath system (in contact with smaller
system-A) would be

$\displaystyle S_{quantum}[B]=\\{S[A],S_{l}+S[A],\cdots\\}_{min}=S[A]\ .$ (34)

The above result in (34) is consistent with the expectations of the quantum
entropy Page curve when bath is very large. Note this conclusion is
independent of how large $l$ might be, so long as $b\gg a$ is obeyed! In
infinite bath limit $b\to\infty$ ( $l\to\infty$, $a=fixed$), the constant
$S_{l}$ has a limit:

$S_{l}(l,z_{0})\to S_{BH}(z_{0})$

where $S_{BH}$ is black hole entropy. So we will get

$\displaystyle\lim_{b\to\infty}S_{quantum}[B]=\\{S[A],S_{BH}+S[A],\cdots\\}_{min}\equiv
S[A]=S(2a,z_{0})\ .$ (35)

The right hand side of the eqs. (34) and (35) are identical. Hence for mixed
states irrespective of the overall system size, the entropy of a large bath
system is determined by the entropy of smaller system-A, under quantum
minimality principle. $S[A]$ again accounts for the smallest entanglement
entropy between two contact systems A & B, even in thermal case!

Figure 5:  Entropy plots for the values $l=10,~{}\epsilon=.3$, for the
$CFT_{4}$ system. The lower falling curve (yellow) is only preferred for
quantum entropy at large $b$. The rising (green) curve is good only for bath
entropy in small $b$ region only. The topmost graph is classical entropy for
large bath but it is unphysical. We have set ${L^{3}l_{3}l_{2}\over
4G_{5}}=1$.

Perhaps an interesting stage (during blackhole evaporation) can arise when
system-A size becomes very small such that it becomes comparable to
intermediate Kaluza-Klein scale of the theory, i.e. $a\simeq n\pi R$, and
$b\simeq{l\over 2}-\pi nR$, where $R$ is radius of Kaluza-Klein circle. Then
bath entropy may be expressed as

$\displaystyle S_{quantum}[B]=S({2\pi nR,z_{0}})$ (36)

which is the entropy of $2n$ independent strips (or sheets) put together. It
is the smallest quantum entropy for a large bath, and is quantized by the
virtue of the presence of KK scale in the (bulk) theory. However we should
trust this result for small $n\in Z$ values only. It is clear that any change
in large bath entropy tends to be discrete in nature, if there exists KK
scale. Similarly, one can deduce that during quantum evolution of a black hole
state after long times, i.e. towards the end of evaporation, when system-A
will shrink to zero size, the bath entropy

$S_{quantum}[B]\to 0\ .$

That means all the information which was kept within system-A has been
transferred to system-B (a very large bath). No loss of information can be
expected in this process even for thermal systems!

###  4.1 Spectrum of narrow KK strips

When the size of the system-A becomes very small such that it is closer to a
Kaluza-Klein radius, $R$, we can set system size as $2a\simeq 2n\pi R$, for
some integer KK-level $n\geq 1$. ($n=0$ means there is no system-A.) We are
now interested in exchang of a small number of KK strips between bath (B) and
the strip system (A). We may call this process as ‘sheets of CFT matter’
exchange between bath and system-A at their mutual interfaces. This will
result in change of level $n$ for the system-A. The resulting change in
entanglement entropy from level $n_{2}$ to $n_{1}$ can be calculated
$(n_{1}<n_{2})$ as

$\displaystyle\bigtriangleup S_{1\to 2}$
$\displaystyle=S_{strips}[n_{2}]-S_{strips}[n_{1}]$ (37)
$\displaystyle={L^{2}l_{2}\over 2G_{4}}({1\over n_{1}}-{1\over
n_{2}}){1\over\pi R}+\bigtriangleup S_{thermal}$ $\displaystyle\equiv{1\over
T_{E}}\bigtriangleup E_{1\to 2}$

Assuming entanglement temperature can be set as $T_{E}={1\over l}$. So
empirically it can be determined that change in energy of strips is discrete,

$\displaystyle{\bigtriangleup E_{1\to 2}}={L^{2}l_{2}\over G_{3}l(2\pi
R)^{2}}({1\over n_{1}}-{1\over n_{2}})+{\bigtriangleup E_{1\to 2}^{thermal}}$
(38)

A conclusion is drawn from here that a discrete quantum of energy would have
to be exchanged between outer bath system-B and central (strip-like) system-A
for finite temperature CFT. The length of strips $l_{2}$ is some large value
(fixed), $L\gg 1$ is AdS radius of curvature, and $G_{3}$ is the Newton’s
constant.

The thermal part of entropy is difficult to estimate exactly. However in the
regime of our interest, when $a\ll z_{0}$, we can evaluate it perturbatively.
From (32) up to first order (say, for the $CFT_{3}$)

$\displaystyle\bigtriangleup E_{1\to 2}^{thermal}={L^{2}l_{2}\over
32G_{3}l}{a_{1}\pi(n_{2}^{2}-n_{1}^{2})R\over 2b_{0}^{2}z_{0}^{3}}$ (39)

Thus schematically we get

$|\bigtriangleup E_{1\to 2}|\propto{1\over
n_{1}n_{2}R^{2}}+\\#{(n_{2}+n_{1})R\over z_{0}^{3}}$

Indeed the first term appears primerily due to the KK momentum modes. While
the second term (due to thermal correction) grows linearly with $R$, which we
guess presumably is consequence of string winding (wrapping) modes. We
conclude that these energy and matter exchanges are all discrete! The same
analysis can also be done for other $CFT_{d}$.

##  5 Summary

We have proposed that quantum entropy of entanglement for a large bath
system-B (CFT) when in contact with small subsystem-A follows quantum
minimality principle

$S_{quantum}[B]=\\{S[A],S[A]+S_{l},\cdots\\}_{min}=S[A].$

where $S_{l}$ is full system $(AUB)$ entropy. Any small fluctuation in the
size of system-A (due to the matter exchange with bath) would not alter this
conclusion provided systems follow conservation laws. Thus the equation
realizes the Page curve for the entropy of quantum matter in contact with
sufficiently large bath system. This conclusion is based upon the observation
that for small subsystem-A and relatively large bath-B the respective
entanglement entropies differ only by an overall constant. The constant does
depend on the total systems size $(l)$, which is fixed. For this reason
$S_{l}$ is conserved and quantum information it contains is essentially
global.

We have explicitly shown that islands and subleading entropis (icebergs)
contribute to the unphysical extremum of bath entropy. Actually all these
contributions form various parts of $S_{l}$ only. The (physical) quantum
entropy of bath however does not get contribution from these fictitious parts!
In this light the entropy expression (1) is very close to our proposal, but it
is only approximate and may not cover full account of quantum entanglement
between systems, mainly it ignores vital subleading contributions beyond the
islands. On the contrary, we have shown that there will be an infinitum of
such subleading contributions. It is shown to be true for all $CFT_{d}$
systems in equillibrium. Furthermore we have analysed our results when
subsystem-A size becomes ‘point-like’, similar in size as Kaluza-Klein scale
of the theory, if there exist such an scale. As the small system size
approaches KK-scale, we find necessary discreteness in the entropy and the
energy spectrum due to existence of low lying KK towers. Should there be no
spontaneous compactification scale in the theory, the entropy of a large bath
would vanish smoothly as and when the subsystem-A disappears. In summary it is
indicated that the change in bath entropy does capture Kaluza-Klein
discreteness.

Acknowledgments: It is pleasure to thank Stefan Theisen for several insightful
discussions on this subject. I am thankful to MPI Golm for the kind
hospitality where part of this work was carried out. The financial support
from the Alexander-von-Humboldt foundation is also highly acknowledged.

##  Appendix A An effective construction of the hybrid gravity and CFT systems

For small size central CFT subsystem-A, such that its size $2a\approx 2\pi R$,
i.e. when the system size can approximately fit within the Kaluza-Klein
radius, the system-A may be treated as being point like. The symmetrical bath
subsystem-B on either side is being comparatively very large so it can
continue to be described by respective (noncompact) CFT. However, for all
practical purposes, with out any loss of physical picture, the system-A can
also be replaced by dual ‘near AdS’ geometry in one lower spacetime
dimensions. The Newton’s constant for near-AdS bulk geometry would become
$G_{d}=G_{d+1}/(2\pi R)$. We have tried to draw these situations in the figure
(6) for systems $A$, $B$ and the compliment system $B^{c}$. The mixed gravity
and CFT systems set up has been a favourable arrangement for an island
proposal [4].

Figure 6:  An hybrid arrangement of the gravitational and CFT systems.

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