Holographic Entropy from Discrete Causal Horizons: A Unitary Solution to the Black Hole Information Paradox in Quantum Cellular Automata

Holog aphic En opy om Disc e e Causal Ho izons: A Uni a y
Solu ion o he Black Hole In o ma ion Pa adox in Quan um
Cellula Au oma a
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
The black hole in o ma ion pa adox o igina es om he ension be ween he causal isola ion
o e en ho izons in gene al ela i i y and he global uni a i y o quan um heo y. In he
semiclassical amewo k, black holes a e ea ed as he modynamic sys ems wi h empe a u e
TH
and en opy
SBH =kBA/(4ℓ2
P)
, whe e
A
is he ho izon a ea. Hawking adia ion exhibi s
a nea ly he mal spec um in he semiclassical limi , implying ha a black hole o med om
he collapse o a pu e s a e would e ol e in o a mixed s a e upon comple e e apo a ion, hus
mani es ing an appa en iola ion o uni a i y.
This pape p esen s and analyzes a mic oscopic model o black holes wi hin he disc e e on-
ology o quan um cellula au oma a (QCA) and he amewo k o in o ma ion a e conse a ion
2
ex + 2
in =c2
. The co e idea is ha he ho izon is no a geome ically absolu e causal bounda y,
bu a he a c i ical laye in he QCA ne wo k whe e he ex e nal in o ma ion anspo a e
ex
is supp essed and he in e nal phase e olu ion a e
in
sa u a es, e med he in o ma ion
eezing laye . On his disc e e ho izon, local connec ions (links) c ossing he su ace ca y all
en anglemen and in o ma ion channels be ween he in e io and ex e io egions.
A e explici ly dening he QCA uni e se model, op ical me ic, and disc e e ho izon, we
p o e:
(1)
Fo any QCA ne wo k sa is ying local ni e deg ee and Planck-scale la ice spacing
a∼ℓP
, he numbe o links c ossing he ho izon
Nlink
scales wi h a ea as
Nlink ∝A/ℓ2
P
.
(2)
I each link is app oxima ely in a maximally en angled s a e a black hole equilib ium,
he on Neumann en opy seen by ex e nal obse e s is
SBH = ln 2 ·Nlink ∝A
ℓ2
P
.
(3)
Iden i ying links as punc u es in a spin ne wo k o
SU(2)
, adop ing he loop quan um
g a i y a ea spec um
Aj= 8πγℓ2
Ppj(j+ 1)
wi h
j= 1/2
and aking he Immi zi pa ame e
γ= ln 2/(π√3)
eco e s he Bekens einHawking en opy o mula
SBH =kBA/(4ℓ2
P)
, consis en
wi h exis ing black hole en opy coun ing esul s.
Fu he mo e, ea ing he black hole and adia ion as a pu e-s a e sys em on a ni e-
dimensional Hilbe space, and assuming QCA e olu ion is a s ic ly uni a y disc e e- ime e o-
lu ion ope a o
U
, we u ilize he Page heo em and he as sc ambling hypo hesis o de i e he
Page cu e o adia ion en opy: he adia ion en anglemen en opy inc eases mono onically
wi h ime in he ea ly s age, eaches a peak when he accessible Hilbe space dimensions o
he black hole and adia ion become compa able, and subsequen ly declines o ze o as e apo a-
ion con inues, ensu ing he nal adia ion s a e is pu e. This esol es he in o ma ion pa adox
wi hin his amewo k.
We embed his QCA black hole model in o he unied scheme o op ical me ics and in o -
ma ion olume conse a ion, demons a ing ha he holog aphic en opy law can be iewed as
1
he con inuum ex apola ion o an en anglemen -coun ing heo em on disc e e causal ho izons in
QCA. We conclude by discussing connec ions o loop quan um g a i y, AdS/CFT, and analog
black hole expe imen s, and p opose se e al es able obse a ions and simula ions, including
sea ches o non- he mal co ela ions in Hawking adia ion and measu emen s o en anglemen
s uc u e on analog black hole pla o ms.
Keywo ds:
Black hole in o ma ion pa adox; Quan um cellula au oma on; Holog aphic p in-
ciple; Bekens einHawking en opy; En anglemen en opy; Page cu e; Loop quan um g a i y
1 In oduc ion and His o ical Con ex
Black hole he modynamics es ablishes a p ecise connec ion be ween geome ic a ea and en opy.
Bekens ein  s p oposed ha black holes mus ca y en opy p opo ional o hei ho izon a ea o
p ese e he gene alized second law du ing ma e abso p ion, and es ima ed he o de o magni ude
o he en opya ea ela ion. Subsequen ly, Hawking calcula ed in quan um eld heo y on cu ed
backg ounds ha a s a ic black hole should adia e a empe a u e
TH=ℏκ
2πkBc,
whe e
κ
is he su ace g a i y. Combined wi h he  s and second laws o he modynamics, his
uniquely de e mines he black hole en opy as
SBH =kBA
4ℓ2
P
,
whe e
ℓP=pGℏ/c3
is he Planck leng h.
In he semiclassical amewo k, Hawking adia ion in scena ios wi hou back- eac ion o eec -
ing walls app oxima ely akes he o m o blackbody adia ion, wi h a densi y ma ix ha appea s
o ex e nal obse e s as a he mal mixed s a e. I a black hole o igina es om he collapse o an
ini ial pu e s a e and e apo a es comple ely, semiclassical calcula ions sugges ha he pu e s a e
would i e e sibly e ol e in o a mixed s a e, iola ing quan um uni a i y. This ension is known as
he black hole in o ma ion pa adox.
Nume ous candida e solu ions ha e been p oposed: black hole complemen a i y emphasizes
complemen a y desc ip ions o e en s by die en obse e s o p e en inconsis ency; he  ewall
p oposal sugges s a iolen high-ene gy wall a he ho izon o main ain en anglemen mono onici y;
ER=EPR conjec u es ha en angled pai s ac oss he ho izon a e ealized by non i ial Eins ein
Rosen b idges; and ecen island o mula and quan um ex emal su ace app oaches eco e he
Page cu e h ough semiclassical g a i a ional pa h in eg als. Howe e , hese p oposals ypically
es on an in insically con inuous space ime mani old.
On he o he hand, quan um cellula au oma a and disc e e quan um walks p o ide disc e e
amewo ks o desc ibing quan um elds and ela i is ic dynamics. In hese amewo ks, he uni-
e se is modeled as local uni a y upda e ules ac ing on spa ial la ice si es, and equa ions such as
Di ac and Maxwell can eme ge in he long-wa eleng h limi om QCA. This na u ally sugges s a
disc e e on ology: a he Planck scale, space ime is no a smoo h mani old bu a ne wo k o coupled
quan um uni s.
In such disc e e models, he geome ic concep s o e en ho izons and singula i ies equi e ein-
e p e a ion. I he unde lying dynamics a e s ic ly uni a y local upda es, he e is no ma hema ical
in o ma ion abso p ion endpoin ; any appa en i e e sibili y mus a ise om igno ing pa ial de-
g ees o eedom. Meanwhile, he black hole en opya ea law and he holog aphic p inciple hin
2
ha he in e io olume deg ees o eedom a e, in some sense, comp essed on o he wo-dimensional
su ace o he ho izon.
This pape builds upon he p e iously p oposed in o ma ion a e conse a ion and op ical
me ic amewo ks o in oduce a disc e e-ho izon QCA model p o iding a unied desc ip ion o
black hole en opy and Hawking adia ion. The co e s eps include:
(1)
Dening ex e nal g oup eloci y
ex
and in e nal phase e olu ion eloci y
in
in QCA, and
adop ing he mo he -scale iden i y
2
ex + 2
in =c2
as he axioma ic exp ession o in o ma ion a e
conse a ion.
(2)
Cons uc ing an op ical e ac i e index eld
n(x)
ia local in o ma ion p ocessing densi y
ρin o
and ela ing
ex
o
n(x)
, he eby dening he disc e e ho izon as an isosu ace whe e
n(x)
exceeds a c i ical alue.
(3)
P o ing ha he numbe o disc e e connec ions c ossing he ho izon is p opo ional o
a ea, and iden i ying black hole en opy as he en anglemen en opy o his connec ion se unde a
maximal en anglemen assump ion.
(4)
Viewing connec ions as spin ne wo k punc u es, bo owing es ablished loop quan um g a i y
esul s o a ea spec um and Immi zi pa ame e o eco e he
1/4
coecien .
(5)
Unde he assump ion o ni e-dimensional Hilbe space and as sc ambling, analyzing
adia ion en anglemen en opy e olu ion o e ime using he Page heo em, ob aining a Page cu e
consis en wi h uni a i y equi emen s, and showing ha he in o ma ion pa adox no longe a ises
in his disc e e model.
The g ea es die ence om exis ing con inuum amewo ks is ha his pape undamen ally
models he black hole ho izon as an in o ma ion eezing laye  in he QCA ne wo k, a he han
a ze o- hickness smoo h su ace in con inuum geome y. Black hole en opy is no longe an ancil-
la y p ope y o geome ic quan i ies, bu he s a is ical ou come o he coun o c oss-bounda y
en anglemen channels in he disc e e ne wo k.
2 Model and Assump ions
This sec ion p o ides igo ous deni ions o he QCA uni e se model, op ical me ic, and disc e e
ho izon used in his pape , and lis s key assump ions.
2.1 QCA Uni e se and In o ma ion Ra e Conse a ion
Le space consis o a h ee-dimensional egula la ice
Λ∼
=aZ3
wi h la ice spacing
a
o o de
Planck scale
a∼ℓP
. Each la ice si e
x∈Λ
is associa ed wi h a ni e-dimensional Hilbe space
Hx∼
=Cd
, and he global Hilbe space is he enso p oduc
H=O
x∈ΛHx.
The disc e e ime s ep is
∆
, and one s ep o QCA e olu ion is ealized by a uni a y ope a o
U:H → H
sa is ying s ic locali y: he e exis s a ni e adius
R
such ha any local ope a o
Ox
e ol ed as
U†OxU
is suppo ed only wi hin a ni e neighbo hood o adius
R
cen e ed a
x
. This
locali y denes a ni e p opaga ion speed
c=Ra
∆ ,
which can be iden ied wi h he speed o ligh in he con inuum limi .
3
Conside a local exci a ion desc ibed as a wa epacke in he long-wa eleng h limi , whose g oup
eloci y in coa se-g ained coo dina es is deno ed
ex
, while he phase p ecession a e in in e nal
deg ees o eedom (coin o in e nal spin space) is deno ed
in
. In p e ious wo k, i has been
shown ha in Di ac- ype QCA one can cons uc an in o ma ion a e ec o
( ex , in )
sa is ying
2
ex + 2
in =c2,
and by using
τ
as he in e nal phase e olu ion pa ame e , s anda d special- ela i is ic p ope ime
and ou - eloci y no maliza ion can be eco e ed. We ake his ela ion as he axioma ic s a emen
o in o ma ion a e conse a ion:
Axiom 1
(In o ma ion Ra e Conse a ion)
.
Fo any dis inguishable local exci a ion, he ex e nal
p opaga ion a e
ex
and in e nal phase a e
in
sa is y
2
ex + 2
in =c2.
In ui i ely, i
ex
app oaches
c
, he exci a ion is app oxima ely massless and p opaga es nea ly
a he speed o ligh be ween la ice si es; i
ex
app oaches
0
, he exci a ion ha dly p opaga es in
posi ion space bu unde goes phase ips and sel - e e en ial e olu ion in he in e nal Hilbe space
a a a e app oaching
c
.
2.2 Local In o ma ion Densi y and Op ical Re ac i e Index
Dene he local in o ma ion p ocessing densi y
ρin o(x)
a each la ice si e as he numbe o eec i e
uni a y deg ees o eedom ha can be implemen ed pe uni ime on ha cell, no malized o a
dimensionless quan i y, and assume an uppe bound
ρmax
, co esponding o he case whe e he cell
is lled wi h local maximal en anglemen .
We adop a simple bu sucien ly exp essi e e ac i e index model ela ing g a i a ional ed-
shi , dening he eec i e e ac i e index
n(x)
as
n(x) = 1
1−ρin o(x)/ρmax
.
When
ρin o ≪ρmax
,
n(x)≈1
; when
ρin o →ρmax
,
n(x)→ ∞
. Op ical pa h conse a ion and
s anda d op ical me ic heo y imply ha he local eec i e speed o ligh sa ises
ce (x) = c
n(x).
In he long-wa eleng h con inuum limi o QCA, one can cons uc an op ical me ic
ds2=−η2(x)c2d 2+η−2(x)γij(x)dxidxj,
whe e
η(x)
is equi alen o
n(x)
and
γij
is he h ee-dimensional spa ial me ic. He e we e ain only
he ela ion
ce (x) = c/n(x)
o cha ac e ize he supp ession o ex e nal g oup eloci y.
Combining Axiom
??
, we can w i e he local ex e nal g oup eloci y as
| ex (x)|=c
n(x)=c1−ρin o(x)
ρmax .
When
ρin o(x)→ρmax
,
| ex (x)| → 0
and
in (x)→c
.
4
2.3 Disc e e Ho izon and In o ma ion F eezing Laye
Unde he abo e se up, we gi e he deni ion o he black hole ho izon in his pape .
Deni ion 2
(Disc e e Ho izon)
.
Le
Nc i >1
be a gi en c i ical e ac i e index. The disc e e
ho izon
H
is dened as he app oxima e disc e e se on la ice si es o closed isosu ace amilies
sa is ying
n(x)≥Nc i
in he con inuum limi .
On he QCA ne wo k, he ho izon co esponds o a shell egion o hickness abou one o se e al
la ice spacings:
H∆={x∈Λ : Nc i ≤n(x)< Nc i +δ},
whe e
δ≪Nc i
.
Deni ion 3
(In o ma ion F eezing Laye )
.
When
n(x)
is sucien ly la ge ha
| ex (x)| ≪ c
and
in (x)≈c
, he shell is called he in o ma ion eezing laye . In his egion, spa ial mig a ion o
exci a ions is nea ly blocked, while phase and en anglemen e olu ion in he in e nal Hilbe space
p oceeds a a a e app oaching
c
.
Assump ion 4
(Volume F eezing and Bounda y S o age)
.
Fo egions sa is ying
n(x)≥Nc i ,
he eec i e dynamics o he QCA sa is y:
(i)
In o ma ion p opaga ing om he ex e nal egion along he adial di ec ion o he in o ma ion
eezing laye nds ou wa d g oup eloci y app oaching ze o, making i dicul o p opaga e u he
in o he in e io olume egion.
(ii)
Con e sely, cells inside he eezing laye can unde go high- equency en anglemen ea -
angemen and ene gyin o ma ion edis ibu ion h ough local uni a y coupling.
The e o e, om he iewpoin o ex e nal obse e s, he in e io olume deg ees o eedom a e
dynamically p ojec ed on o a wo-dimensional in o ma ion s o age a ay on he eezing laye .
This assump ion is quali a i ely consis en wi h he holog aphic p inciple.
2.4 C oss-Ho izon Connec ions and Hilbe Space Decomposi ion
Pa i ion he la ice se along he eezing laye su ace
Σ
in o he ex e io egion
Ou
, in e io egion
In
, and eezing laye
H∆
. The Hilbe spaces o he ex e io and eezing laye can be w i en as
Hou =O
x∈Ou Hx,HH=O
x∈H∆Hx.
Conside he local adjacency g aph o he QCA, and deno e by
L
he se o all nea es -neighbo
connec ions c ossing he eezing laye , whe e each connec ion
ℓ∈ L
connec s one ex e io la ice
si e o one eezing-laye la ice si e.
Deni ion 5
(Ho izon Connec ions)
.
The ho izon connec ion se
L
is dened as all edges sa is ying:
(i)
One end belongs o
Ou
, he o he o
H∆
.
(ii)
The wo endpoin s ha e a di ec coupling e m in he adjacency g aph.
5

The connec ion coun
Nlink =|L|
will be he key coun ing objec o black hole en opy.
In he  eezingbounda y s o age limi , he ele an pa o he Hilbe space can be w i en
as
HH⊗Hou ∼
=O
ℓ∈L H(in)
ℓ⊗H(ou )
ℓ⊗H es ,
whe e
H(in)
ℓ
and
H(ou )
ℓ
a e he local subspaces a he wo ends o he connec ion, and
H es
con ains
all deg ees o eedom i ele an o c oss-bounda y en anglemen .
This pape will ocus p ima ily on he en anglemen s uc u e on
L
and p o e ha black hole
en opy can be gi en by he en anglemen on Neumann en opy o hese connec ions.
3 Main Resul s: Theo ems and Alignmen s
Unde he abo e model and assump ions, he co e esul s o his pape can be summa ized in he
ollowing ou heo ems.
Theo em 6
(Holog aphic S o age on Disc e e Ho izon)
.
In a QCA uni e se wi h bounded local
deg ee, ni e la ice spacing, and sa is ying he in o ma ion a e conse a ion axiom, suppose he e
exis s a closed eezing laye
H∆
dening he black hole ho izon, and p opaga ion modes in he
in e io egion a e s ongly supp essed dynamically. Then, o he ope a o algeb a accessible o
ex e nal obse e s, all in o ma ion in he black hole in e io can equi alen ly be encoded in he
deg ees o eedom suppo ed by he connec ion se
L
on he eezing laye .
Tha is, he e exis s an isomo phism
HBH ∼
=O
ℓ∈L H(in)
ℓ⊗Haux,
whe e
HBH
is he eec i e black hole Hilbe space isible o ex e nal obse e s, and
Haux
does no
di ec ly en angle wi h he ex e io .
Theo em 7
(Connec ion Coun and A ea Law)
.
Suppose he QCA ne wo k nea he eezing laye
has an app oxima ely iso opic connec ion g aph on la ge scales and has ni e a e age deg ee. Then,
o a sucien ly la ge ho izon su ace
Σ
, he numbe o connec ions c ossing he eezing laye
sa ises
Nlink =ηA
a2+oA
a2,
whe e
A
is he con inuum-limi su ace a ea o
Σ
, and
η
is a dimensionless geome ic ac o depend-
ing only on he local la ice s uc u e.
Theo em 8
(En anglemen O igin o Black Hole En opy)
.
Unde he condi ions o Theo ems
??
and
??
, i a equilib ium each connec ion in he eezing laye is app oxima ely in a maximally
en angled s a e, and co ela ions among connec ions can be ega ded as diso de ed on la ge scales,
hen he on Neumann en opy o he black hole as seen by ex e nal obse e s is
SBH =kBln 2 ·Nlink ≈kBηln 2 A
a2.
6
Iden i ying he la ice spacing wi h he Planck leng h
a=ℓP
and choosing he geome ic ac o
η
o
ma ch he s anda d alue o he Immi zi pa ame e in loop quan um g a i y, we eco e
SBH =kBA
4ℓ2
P
.
Theo em 9
(Uni a y E apo a ion and Page Cu e)
.
Suppose he black hole adia ion join sys em
a any ime
can be desc ibed by a pu e s a e
|Ψ( )⟩
on a ni e-dimensional Hilbe space
H o =HBH( )⊗H ad( ),
gi en by uni a y i e a ion o he global QCA e olu ion ope a o . I he in e nal dynamics on he
eezing laye a e as sc ambling, and assuming a each s age
|Ψ( )⟩
is close o a Haa - andom pu e
s a e o he gi en dimension pai
(dBH( ), d ad( ))
, hen he on Neumann en opy o he adia ion
app oxima ely sa ises
S ad( )≈kBmin (ln dBH( ),ln d ad( )) ,
hus yielding a Page cu e ha  s ises, hen alls, ul ima ely e u ning o ze o. When he black
hole ully e apo a es,
dBH →1
, adia ion en opy
S ad →0
, and he join sys em emains in a pu e
s a e, so he in o ma ion pa adox no longe appea s in his amewo k.
P oo s o he abo e heo ems a e gi en in subsequen sec ions and appendices.
4 P oo s
This sec ion p o ides p oo ou lines o each heo em, wi h echnical de ails and comple e de i a ions
o ela ed ma hema ical ools de e ed o appendices.
4.1 P oo o Theo em ??: Holog aphic S o age on F eezing Laye
The p oo elies on he locali y and causal s uc u e o QCA.
S ep 1. Causal cones and accessibili y
Fo any local ope a o
Oou
in he ex e io egion, i s Heisenbe g pic u e e olu ion
Oou ( ) =
U− Oou U
has suppo es ic ed o he causal cone based on he ini ial suppo o ha ope a-
o . The nea -ze o p opaga ion speed in he in o ma ion eezing laye means ha , on ni e ime
scales, he con ibu ion om he in e io olume egion o ex e nal obse a ional ope a o s can be
neglec ed, while con ibu ions om he eezing laye deg ees o eedom domina e.
S ep 2. Eec i e ope a o algeb a comp ession
Mo e p ecisely, conside he ope a o algeb a
Aou
accessible o ex e nal obse e s. Any
A∈ Aou
ime-e ol ed in he Heisenbe g pic u e can be w i en as
A( ) = U− AU =AH( )⊗Ibulk +
small e ms
,
whe e
AH( )
is suppo ed on he eezing laye and ex e io , wi h he eec o in e io olume deg ees
o eedom supp essed by a small pa ame e (con olled join ly by
ex /c
and e olu ion ime).
S ep 3. S inesp ing s uc u e and encoding map
In his limi , one can cons uc a CPTP map encoding he in e io olume Hilbe space
Hbulk
deg ees o eedom on o auxilia y deg ees o eedom
Haux ⊂ HH
on he eezing laye , such ha
expec a ion alues o ex e nal ope a o s emain unchanged:
bulk (ρBHAou ) = aux (˜ρHAou ).
7
By he S inesp ing dila ion heo em and ope a o algeb a isomo phism, he e exis s a Hilbe space
equi alence
HBH ∼
=O
ℓ∈L H(in)
ℓ⊗Haux,
yielding he conclusion o Theo em
??
.
De ails o he cons uc ion a e gi en in Appendix A.1.
4.2 P oo o Theo em ??: Connec ion Coun P opo ional o A ea
This heo em is essen ially a geome iccombina o ial s a emen holding on la ice ne wo ks wi h
ni e deg ee and uni o mi y.
S ep 1. Local iso opy assump ion
Assume he connec ion g aph nea he eezing laye is app oxima ely uni o mly iso opic on
sucien ly la ge scales, i.e., he a e age deg ee
deg(x)
o each la ice si e is s a is ically cons an
d0
, and connec ion di ec ions a e uni o mly dis ibu ed on he sphe e.
S ep 2. Su ace a ea disc e iza ion
View he app oxima e disc e iza ion o he con inuum su ace
Σ
on he la ice as he se o all
la ice si es wi h dis ance less han
a/2
om he su ace. The ca dinali y
NΣ
o his se sa ises
NΣ=ζA
a2+oA
a2,
whe e
ζ
is a cons an ela ed o he la ice ype (cubic, body-cen e ed cubic, e c.).
S ep 3. C oss-bounda y connec ion coun
Fo each la ice si e on
Σ
, coun he connec ions poin ing inwa d and ou wa d. Because a e -
age deg ee is ni e and he e a e no long- ange connec ions, he o al numbe o c oss-bounda y
connec ions is p opo ional o
NΣ
, i.e.,
Nlink =ηNΣ=ηζ A
a2+oA
a2.
Me ging
ηζ
in o
η
gi es he exp ession o Theo em
??
.
This easoning sha es i s o igin wi h he numbe o su ace a oms
∝
su ace a ea calcula ion
in solid-s a e physics. De ailed de i a ion is in Appendix A.2.
4.3 P oo o Theo em ??: En anglemen En opy and A ea Law
Based on Theo ems
??
and
??
, we iew c oss-bounda y connec ions as en angled wo-sys em sub-
spaces.
S ep 1. En opy con ibu ion om a single connec ion
Assume he Hilbe space on each connec ion
ℓ
is
C2⊗C2
, co esponding o a pai o qubi s,
na u ally in a Bell- ype maximally en angled s a e. T acing ou he ex e io , he educed densi y
ma ix o a single ex e io qubi is
ρ(ou )
ℓ=1
2I2,
wi h on Neumann en opy
Sℓ=−kB ρ(ou )
ℓln ρ(ou )
ℓ=kBln 2.
S ep 2. Mul i-connec ion enso p oduc and independence
8
In he as -sc ambling limi on he eezing laye , en anglemen on die en connec ions can
be app oxima ely ea ed as s a is ically independen , and hei join s a e as seen ex e nally is a
enso p oduc densi y ma ix
ρou =O
ℓ∈L
ρ(ou )
ℓ,
so o al en opy is
SBH =X
ℓ∈L
Sℓ=kBln 2 ·Nlink.
S ep 3. Ma ching wi h he a ea law
Subs i u ing he esul o Theo em
??
,
Nlink ≈ηA
a2,
we ob ain
SBH ≈kBηln 2 A
a2.
Taking he na u al la ice spacing
a=ℓP
, he specic alue o
η
can be xed by a mo e mic oscopic
spin ne wo k model.
S ep 4. Consis ency wi h loop quan um g a i y a ea spec um
In he loop quan um g a i y amewo k, he ho izon is punc u ed by a spin ne wo k, and he
a ea eigen alue is
Aj= 8πγℓ2
Ppj(j+ 1),
whe e
γ
is he Immi zi pa ame e . I we iden i y each qubi connec ion in QCA as a
j= 1/2
punc u e, he single-punc u e a ea is
A1/2= 4πγ√3ℓ2
P,
wi h spin s a e space dimension
2j+ 1 = 2
, co esponding o en opy
ln 2
. Coun ing he numbe o
punc u es
N=A/A1/2
, he o al en opy is
S=NkBln 2 = A
4ℓ2
P
kB
i and only i
γ=ln 2
π√3.
This alue is consis en wi h ecen de i a ions based on loop quan um g a i y and Landaue 's
p inciple, showing ha he p esen QCA model is compa ible wi h such wo k ega ding en opy
a ea ela ion.
This comple es he p oo o Theo em
??
. De ailed spin ne wo k coun ing and discussion o he
Immi zi pa ame e a e in Appendix B.
4.4 P oo o Theo em ??: Uni a y E apo a ion and Page Cu e
This heo em elies on wo elemen s: global uni a i y o QCA and a e age en opy esul s o
Haa - andom pu e s a es.
S ep 1. Fini e-dimensional pu e s a e and on Neumann en opy symme y
9
[16] A. Mallick, C. M. Chand asheka , Di ac cellula au oma on om spli -s ep quan um walk,
Sci. Rep.
6
, 25779 (2016).
[17] T. A. B un, M. C. Kimbe ley, Quan um cellula au oma a and quan um eld heo y in wo
spa ial dimensions,
Phys. Re . A
102
, 062222 (2020).
[18] C. Hue a Alde e e e al., Quan um walks and Di ac cellula au oma a on a p og ammable
apped-ion quan um compu e ,
Commun. Phys.
3
, 89 (2020).
[19] P. A ighi, S. Facchini, M. Fo e s, Disc e e Lo en z co a iance o quan um walks and quan um
cellula au oma a,
New J. Phys.
16
, 093007 (2014).
[20] J. S einhaue , Obse a ion o quan um Hawking adia ion and i s en anglemen in an analogue
black hole,
Na . Phys.
12
, 959 (2016); J. R. Muñoz de No a e al., Obse a ion o he mal
Hawking adia ion and i s empe a u e in an analogue black hole,
Na u e
569
, 688 (2019).
[21] W. G. Un uh, Expe imen al black-hole e apo a ion?,
Phys. Re . Le .
46
, 1351 (1981).
[22] M. Visse , Di y black holes: en opy e sus a ea,
Phys. Re . D
48
, 583 (1993).
[23] I. A e 'e a e al., Comple e e apo a ion o black holes and Page cu es,
Symme y
15
, 170
(2023).
[24] X. Wang, R. Li, J. Wang, Page cu es o a amily o exac ly sol able e apo a ing black holes,
Phys. Re . D
103
, 126026 (2021).
[25] M. Cadoni, E. F anzin, S. Mignemi, Uni a i y and Page cu e o e apo a ion o 2D AdS black
hole,
Phys. Re . D
105
, 024027 (2022).
A QCA Fo malism and Causal Comp ession
A.1 Ex e nal Ope a o Algeb a and Equi alen Encoding on F eezing Laye
This appendix p o ides a mo e de ailed cons uc ion o he Hilbe space equi alence in Theo em
??
.
Le
Aou
be he bounded ope a o algeb a on he ex e io egion, gene a ed by local ope a o s
suppo ed in he
Ou
egion and hei limi s. Conside he Heisenbe g e olu ion o QCA:
Φ :Aou → B(H),Φ (A) = U− AU .
Due o locali y, he e exis cons an s
C
and eloci y
c
(o o de
c
) such ha a LiebRobinson- ype
inequali y holds: o any ope a o s
A, B
wi h spa ial dis ance
dis (suppA, suppB)>0
, we ha e
∥[Φ (A), B]∥ ≤ C∥A∥∥B∥exp (−µ[dis (suppA, suppB)− c ]+),
whe e
[·,·]
is he commu a o ,
[·]+
is he posi i e pa , and
µ > 0
.
Inside he eezing laye , ex e nal g oup eloci y
| ex |
is supp essed, so signals p opaga ing om
he in e io olume egion o he ex e io equi e ime scales a longe han he obse a ion ime
T
. Thus, in he ime window
≤T
, he in e io olume egion can be iewed as causally silen ,
and i s eec on he ex e nal ope a o algeb a can be comp essed ia a CPTP map on o auxilia y
deg ees o eedom on he eezing laye .
Mo e p ecisely, x ime window
[0, T]
and conside he e ol ed ope a o amily
Φ (Aou )
. Dene
He =Hou ⊗HH,
16

and p ojec ion
P:H → He ,
acing ou in e io olume deg ees o eedom. Since eedback om he in e io o he ex e io is
exponen ially supp essed in ime
T
, one can p o e he e exis s a CPTP map
E:B(He )→ B(He )
such ha o any
A∈ Aou
,

PΦ (A)P−E (PAP)
≤ϵ(T),
whe e
ϵ(T)
is con olled by
T
.
To an ex e nal obse e , wi hin accu acy
ϵ(T)
, he in e io olume can be eec i ely ea ed
as an en i onmen sys em
Haux
on he eezing laye . The S inesp ing ep esen a ion heo em
gua an ees he exis ence o
HBH ∼
=O
ℓ∈L H(in)
ℓ⊗Haux,
co esponding o he s a emen o Theo em
??
.
A.2 Geome ic Es ima ion o Connec ion Coun
In he case o cubic la ice s uc u e, la ice si es nea he eezing laye can be app oxima ely iewed
as a wo-dimensional disc e e mesh a ached o he con inuum su ace
Σ
, wi h la ice spacing
a
.
(1)
Pa i ion
Σ
in o a ea elemen s
∆A=a2
, each co esponding o one o se e al la ice si es.
(2)
Fo a simple cubic la ice, each la ice si e on a e age has one no mal connec ion c ossing he
su ace (poin ing inwa d o ou wa d) and se e al angen connec ions. C oss-bounda y connec ions
mainly come om no mal edges.
(3)
Fo sucien ly la ge
A
, edge eec s can be neglec ed, and he c oss-bounda y connec ion
coun is app oxima ely
Nlink ≈A
a2.
Mo e gene al la ice ypes only change he geome ic cons an p e ac o .
This es ima e is consis en wi h he s anda d esul numbe o su ace a oms
∼A/a2
 in solid-
s a e physics.
B Spin Ne wo k Pic u e and he 1/4 Coecien
This appendix connec s he QCA connec ion model o he loop quan um g a i y spin ne wo k
ho izon model, explaining why he
1/4
coecien can be eco e ed unde na u al assump ions.
B.1 A ea Spec um and Immi zi Pa ame e
In loop quan um g a i y, he eigen alue spec um o he a ea ope a o is
A= 8πγℓ2
PX
ipji(ji+ 1),
whe e he sum uns o e all spin ne wo k edges punc u ing he ho izon, and
ji
is he co esponding
spin quan um numbe .
17
I we assume ha in he la ge-a ea limi , he main con ibu ion comes om
j= 1/2
punc u es,
hen he single-punc u e a ea con ibu ion is
A1/2= 4πγ√3ℓ2
P.
The in e nal edge s a e space dimension co esponding o he punc u e is
dim H1/2= 2j+ 1 = 2,
so each punc u e can con ibu e en opy
kBln 2
.
W i ing he o al punc u e coun as
N=A
A1/2
=A
4πγ√3ℓ2
P
,
he o al en opy is
S=NkBln 2 = kBln 2
4πγ√3
A
ℓ2
P
.
Requi ing
S=kBA/(4ℓ2
P)
gi es
γ=ln 2
π√3.
Recen wo k shows his alue can be independen ly ob ained ia Landaue 's p inciple and in o ma-
ion e asu e cos , s eng hening i s physical in e p e a ion.
B.2 Iden ica ion o QCA Connec ions wi h Spin Punc u es
In he p esen QCA model, he Hilbe space on each connec ion
ℓ
is aken as
H(in)
ℓ⊗H(ou )
ℓ
, whe e
each ac o is a wo-le el sys em. Iden i ying
H(in)
ℓ∼
=H(j=1/2)
wi h he spin-
1/2
ep esen a ion, hen:
(1)
Each connec ion co esponds o one spin ne wo k edge punc u e on he ho izon, con ibu ing
a ea
A1/2
.
(2)
Each connec ion con ibu es one bi o maximal en anglemen en opy
kBln 2
.
Thus he QCA connec ion ne wo k on he ho izon is isomo phic in s a e coun ing o he loop
quan um g a i y spin ne wo k model. The black hole en opya ea ela ion can be iewed as wo
exp essions o he same coun ing p oblem in die en languages.
B.3 Non-
j= 1/2
Modes and Co ec ion Te ms
In mo e gene al cases, high-spin
j > 1/2
punc u es and many-body cons ain s b ing loga i hmic
co ec ions o he leading e m. Exis ing esea ch shows
−αln A
co ec ions appea in loop quan um
g a i y black hole en opy, wi h coecien depending on he mic oscopic model.
In he QCA connec ion model, hese co ec ions can be unde s ood as:
(1)
Some connec ions co espond o highe -dimensional local Hilbe spaces (mul i-le el sys ems
o mul iple line coincidences).
(2)
Global cons ain s exis inside he eezing laye (such as o al spin, opological numbe
conse a ion), educing he allowed en anglemen congu a ion coun .
These eec s do no change he leading a ea e m
A/(4ℓ2
P)
bu yield loga i hmic o powe
co ec ions a subleading o de , co esponding o signa u es o die en quan um g a i y schemes.
18
C Page Cu e in a Fini e-Dimensional QCA Toy Model
To conc e ely demons a e ealiza ion o he Page cu e in he QCA con ex , conside he ollowing
simple model:
(1)
Take he ini ial black hole Hilbe space dimension
dBH(0) = 2N
, co esponding o
N
connec ions o
N
eezing-laye bi s.
(2)
A each disc e e ime s ep, elease one qubi om he black hole o adia ion, sucien ly
en angling i wi h al eady adia ed qubi s and emaining black hole deg ees o eedom ia a andom
uni a y ga e.
C.1 I e a i e P ocess
A e he
k
- h s ep, he black hole has
N−k
emaining qubi s, and adia ion has
k
qubi s. Unde
he as sc ambling assump ion, he join s a e
|Ψk⟩ ∈ C2⊗(N−k)⊗C2⊗k
can be app oxima ely iewed as a Haa - andom pu e s a e.
By he Page heo em, i
k≤N/2
, hen adia ion en opy is app oxima ely
S ad(k)≈kBkln 2 −1
2,
while when
k≥N/2
, he smalle subsys em is he black hole, and adia ion en opy becomes
S ad(k)≈kB(N−k) ln 2 −1
2.
A e no maliza ion, a symme ic Page cu e is ob ained, eaching a peak
≈kB(Nln 2 −1/2)
a
k=N/2
, hen linea ly declining as
k
inc eases, wi h nal s a e
k=N
ha ing
S ad(N)≈0
.
C.2 Co espondence wi h Con inuous-Time E apo a ion
In con inuous- ime models, black hole mass
M( )
and ho izon a ea
A( )
slowly dec ease wi h Hawk-
ing adia ion. The Page cu e o adia ion en opy o e ime can be ob ained by iewing
k
as
he numbe o eleased eec i e deg ees o eedom and in e pola ing wi h con inuous pa ame e
. Recen wo k's nume ical simula ions show ha , unde he assump ion o uni a y e apo a ion
and as sc ambling, such simplied models yield Page cu es quali a i ely consis en wi h mo e
sophis ica ed g a i yquan um eld heo y models.
C.3 Realiza ion in QCA Con ex
In QCA, he abo e oy model can be ealized as ollows:
(1)
Ini ialize
N
cells in he eezing laye o a highly en angled s a e, isola ed om he ex e nal
en i onmen .
(2)
A each ime s ep, map one eezing-laye cell's deg ees o eedom o an ex e nal adia ion
link ia a local uni a y ga e, while applying sucien ly deep local andom ci cui s inside he eezing
laye o achie e as sc ambling.
(3)
A e each s ep, pe o m comple e omog aphy on he ex e nal adia ion subsys em, com-
pu ing on Neumann en opy o ob ain he disc e e Page cu e.
This p ocess can be ealized on any p og ammable quan um compu ing pla o m, p o iding a
easible pa h o es ing he quan i a i e ela ionship among  eezing laye connec ionPage cu e.
19
D Rema ks on Ex ensions and Open P oblems
(1) Ro a ing and cha ged black holes
Fo Ke and Reissne No ds öm black holes, ho izon s uc u e and empe a u eangula momen um
cha ge ela ions a e mo e complex. The QCA model can simula e hese eec s by in oducing
aniso opy in angula momen um and cha ge ow densi y on he eezing laye ; he connec ion
coun and en opya ea ela ion a e expec ed o e ain he leading o m.
(2) Mul iple ho izons and inne ho izon s abili y
In cases wi h inne ho izons (e.g., Reissne No ds öm, Ke Newman), mul iple eezing laye s
can be cons uc ed, and he opological s uc u e and in o ma ion anspo o connec ion ne wo ks
on hem can be analyzed. The disc e e model may p o ide a new pe spec i e on mic oscopic
mechanisms o mass ina ion and inne ho izon ins abili y.
(3) Rela ion o he island o mula
Recen island o mulas econs uc adia ion en opy calcula ions by in oducing quan um ex-
emal su aces. The eezing laye in QCA can be iewed as a kind o disc e e ex emal su ace,
whose posi ion is join ly de e mined by in o ma ion a e and en anglemen s uc u e. How o uni y
he a ia ional p oblems o he wo in o a disc e econ inuum hyb id ex emal p oblem is a di ec ion
wo h deep in es iga ion in he u u e.
(4) Obse a ional cons ain s and complexi y bounds
Al hough in o ma ion is s ic ly p ese ed in QCA, decoding in o ma ion in black hole adia ion
may equi e exponen ially la ge ci cui complexi y. How o in oduce complexi y geome y and
compu abili y bounda ies in his amewo k will be an impo an b idge connec ing he in o ma-
ion pa adox and compu a ional complexi y heo y.
20