Universal Conservation of Information Rate: From Quantum Cellular Automata to Unified Framework of Relativity, Mass, and Gravity

Uni e sal Conse a ion o In o ma ion Ra e: F om Quan um
Cellula Au oma a o he Unica ion o Rela i i y, Mass, and
G a i y
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
In he amewo k o Quan um Cellula Au oma a (QCA) and ni e in o ma ion on ol-
ogy, we cons uc an eec i e desc ip ion o single-pa icle long-wa eleng h exci a ions and
p o e a co e esul based on Hilbe space geome y and uni a i y: o any disc e e quan-
um walk/QCA dened by local uni a y e olu ion and ansla ion in a iance ha eme ges
a one-dimensional Di ac- ype Hamil onian in he con inuous limi , he ex e nal g oup e-
loci y (
ex
) and in e nal s a e e olu ion eloci y (
in
) o i s long-wa eleng h single-pa icle
eigenmodes mus sa is y he In o ma ion Ra e Conse a ion Theo em:
2
ex + 2
in =c2,
whe e
c
is he maximum causal p opaga ion speed o he la ice sys em. This heo em is
no an addi ional axiom bu a geome ic esul en o ced by he local uni a i y o QCA and
he o hogonal decomposi ion o he an i-commu ing algeb a o in e nal deg ees o eedom
unde he FubiniS udy p ojec i e me ic.
By dening p ope ime (
τ
) wi h he in e nal e olu ion pa ame e , special ela i i y's ime
dila ion, ou - eloci y no maliza ion, and Minkowski line elemen can be di ec ly de i ed
om he In o ma ion Ra e Conse a ion Theo em. In he con inuous limi o Di ac- ype
QCA, he in e nal Hamil onian (
Hin
) gi es he in e nal equency (
ωin
), and mass ob ains
an in o ma ion- heo e ic deni ion:
mc2=ℏωin ,
sa is ying he Zi e bewegung equency ela ion:
ωZB = 2ωin .
Combined wi h he winding numbe and index in a ian s o QCA, massi e exci a ions can
be in e p e ed as op ical pa h quo as bound in opologically non- i ial sel - e e en ial loops.
A he many-body le el, we in oduce local in o ma ion p ocessing densi y (
ρin o(x)
) and
de i e he op ical me ic om local conse a ion o in o ma ion olume:
ds2=−η2(x)c2d 2+η−2(x)γij(x)dxidxj,
whe e
η(x)
de e mines he local eec i e speed o ligh :
ce (x) = η2(x)c,
and he e ac i e index:
n(x) = η−2(x).
In he weak-eld limi , his s uc u e eco e s he  s -o de expansion o he Schwa zschild
me ic and he s anda d ligh deec ion angle, and eld equa ions o mally equi alen o
Eins ein's equa ions can be ob ained h ough an in o ma ion-g a i y a ia ional p inciple.
Fu he mo e, we in oduce in o ma ion mass (
MI
) and, combining wi h Landaue 's p in-
ciple, analyze he asymp o ic s a iona y beha io and minimum dissipa ion powe o high-
in o ma ion-mass subjec s, p o iding a unied in o ma ion- heo e ic cha ac e iza ion o mass,
g a i y, and complex ene ge ic s uc u es, and p opose es able p edic ions based on supe -
conduc ing quan um ci cui s and quan um simula ion pla o ms.
1
Keywo ds:
Quan um Cellula Au oma a; Conse a ion o In o ma ion Ra e; FubiniS udy
Me ic; Op ical Me ic; Special Rela i i y; Gene al Rela i i y; Topological Mass; Zi e bewe-
gung; In o ma ion Mass; Landaue 's P inciple
1 In oduc ion & His o ical Con ex
Special and Gene al Rela i i y desc ibe he physical wo ld as a ou -dimensional mani old
(M, gµν)
wi h Lo en zian signa u e. The me ic enso
gµν
de e mines he causal s uc u e and geodesics,
and he eld equa ions
Rµν −1
2Rgµν = 8πGTµν
ela e he s ess-ene gy enso
Tµν
o cu a u e. Expe imen al es s such as g a i a ional ed-
shi , ligh deec ion, bina y pulsa iming, and g a i a ional wa e de ec ion highly suppo his
geome ic na a i e. Rela i i y is axioma ically cons uc ed on he cons ancy o he speed o
ligh and he p inciple o ela i i y, in oducing he Minkowski line elemen and Lo en z ans-
o ma ions, wi h i s geome ic s uc u e ypically ega ded as an a p io i backg ound.
Quan um heo y is o mula ed in Hilbe space
H
, whe e s a es a e ec o s o densi y op-
e a o s, obse ables a e sel -adjoin ope a o s, and ime e olu ion is gene a ed by he uni a y
g oup. S a is ical in e p e a ion is buil on he Bo n ule, wi h supe posi ion, in insic phase,
and en anglemen cons i u ing co e s uc u es. The wo heo ies a e s i ched oge he in Quan-
um Field Theo y by dening "eld ope a o s on a backg ound mani old," bu hei on ological
s a ing poin s emain sepa a ed: one side is a con inuous, cu able space ime mani old, and he
o he is an abs ac linea Hilbe space.
When app oaching he Planck scale, he assump ions o con inuous mani olds and classical
me ics lose empi ical suppo , while he Hilbe space s uc u e i sel does no depend on con-
inuous space ime. Quan um Cellula Au oma a (QCA) p o ide an al e na i e o mula ion wi h
disc e e s uc u e as on ology: dening ni e-dimensional local Hilbe spaces and local uni a y
e olu ion on coun able la ices, equi ing s ic causali y and ni e p opaga ion adius. Exis ing
esea ch has shown ha in app op ia e con inuous limi s, Di ac, Weyl, and Maxwell equa ions
can eme ge om local uni a y e olu ion o QCA, and QCA possesses sys ema ic opological
classica ion and index heo y.
On he o he hand, Hilbe space i sel has a na u al p ojec i e geome ic s uc u e. The
p ojec i e Hilbe space
CPn
is equipped wi h he FubiniS udy me ic, whose a c leng h gi es
he na u al dis ance be ween quan um s a es. Fo uni a y e olu ion d i en by a ime-independen
Hamil onian
H
, he " eloci y" o he s a e ec o unde he FubiniS udy me ic is de e mined
by he ene gy unce ain y
∆H
, and he "pa h leng h" o quan um e olu ion can be iewed as a
measu e o in o ma ion upda e.
This pape a emp s o uni y he abo e h ee h eads unde an in o ma ion- heo e ic pe -
spec i e:
1. Assume he uni e se is mic oscopically desc ibed by a local uni a y, ansla ion-in a ian
QCA wi h a maximum p opaga ion speed
c
; 2. T ea single-pa icle long-wa eleng h exci a ions
as a class o eec i e modes in QCA, whose ex e nal mo ion is desc ibed by g oup eloci y
(
ex
) and in e nal s a e sel - e e en ial e olu ion is desc ibed by geome ic eloci y (
in
) in
p ojec i e Hilbe space; 3. P o e ha in he con inuous limi o Di ac- ype QCA, he o hogonal
decomposi ion induced by he an i-commu ing s uc u e o he Hamil onian and he Fubini
S udy me ic necessa ily yields he In o ma ion Ra e Conse a ion Theo em:
2
ex + 2
in =c2,
he eby ele a ing "conse a ion o op ical pa h leng h" om an assump ion o a heo em.
On his basis, special ela i i y can no longe be iewed as an independen axiom bu as an
eme gen esul o QCA uni a i y and Hilbe geome y; mass can be in e p e ed as he coecien
2
o in e nal equency
ωin
; g a i a ional geome y can be in e p e ed as a mani es a ion o local
in o ma ion p ocessing densi y and op ical me ic s uc u e; and he "in o ma ion mass" o
complex ene ge ic sys ems can be linked o Landaue 's p inciple, p o iding a unied pic u e o
mass, g a i y, and complexi y.
2 Model Assump ions
2.1 QCA Uni e se and Local Uni a i y
Le
Λ
be a coun able connec ed g aph, whose nodes ep esen "spa ial cells." Each cell
x∈Λ
ca ies a ni e-dimensional Hilbe space
Hx≃Cd
. Fo any ni e subse
F⋐Λ
, dene he local
Hilbe space
HF=O
x∈FHx,
and he local ope a o algeb a
B(HF)
. The global quasi-local
C∗
-algeb a is
A=[
F⋐ΛB(HF).
A Quan um Cellula Au oma on is specied by a
∗
-au omo phism
α:A→A
, equi ing he
exis ence o a uni a y ope a o
U
such ha
α(A) = U†AU, A ∈ A,
and he exis ence o a ni e p opaga ion adius
R < ∞
, such ha o any local ope a o
A
suppo ed on
F
,
supp α(A)⊂BR(F),
whe e
BR(F)
is he
R
-neighbo hood o
F
in he sense o g aph dis ance. Gi en an ini ial s a e
ω0
, he disc e e ime e olu ion is
ωn=ω0◦αn, n ∈Z.
Assume
Λ
can be embedded in h ee-dimensional Euclidean space wi h eec i e la ice spacing
a
, and a single s ep o e olu ion co esponds o physical ime
∆
. I
R= 1
, he maximum
p opaga ion speed is
c=a
∆ .
Fini e local dimension and ni e p opaga ion adius imply ha he numbe o dis inguishable
physical s a es in any ni e space ime window is ni e, and he in o ma ion capaci y o he
uni e se in any ni e egion has an uppe bound.
2.2 Eec i e Space o Single Exci a ions and Ex e nal Veloci y
Conside a local "single-exci a ion" mode, whose eec i e Hilbe space unde app op ia e ap-
p oxima ion can be ep esen ed as
He ≃ HCOM ⊗Hin ,
whe e
HCOM
desc ibes he cen e -o -mass coo dina e o wa e packe en elope, and
Hin
desc ibes
in e nal deg ees o eedom.
In he con inuous limi , app oxima e posi ion ope a o
X
and momen um ope a o
P
exis
on
HCOM
, and he eec i e Hamil onian
He
gene a es coa se-g ained ime e olu ion. Dene
he ex e nal (g oup) eloci y
ex =d
d ⟨X⟩=1
iℏ⟨[X, He ]⟩.
3
In symme ic cases, long-wa eleng h single-pa icle eigenmodes can be labeled by momen um,
|ψp⟩
sa is ying
He |ψp⟩=E(p)|ψp⟩,
and he g oup eloci y o his mode is
ex (p) = dE
dp .
2.3 In e nal Hilbe Space and FubiniS udy Me ic
The in e nal s a e
|ψin ( )⟩ ∈ Hin
can be iewed as a poin on he p ojec i e space
CPDin −1
.
The FubiniS udy me ic
ds2
FS = 41−|⟨ψ|ψ+dψ⟩|2
gi es he na u al dis ance be ween wo s a es in p ojec i e Hilbe space. Fo uni a y e olu ion
d i en by a ime-independen Hamil onian
H
:
iℏ∂ |ψ( )⟩=H|ψ( )⟩,
he FubiniS udy eloci y can be dened as
FS := dsFS
d .
Fo a gene al s a e,
FS
is ela ed o ene gy unce ain y
∆H
, while o ene gy eigens a es,
FS = 0
.
In he amewo k o his pape , he ocus is no on
FS
on he global
H
, bu on decomposing
H
in o wo mu ually o hogonal gene a o s co esponding o ex e nal ansla ion and in e nal
sel - e e ence, he eby dening "in e nal e olu ion eloci y" on he in e nal p ojec i e space:
in := ds(in )
FS
d ≥0.
This eloci y cha ac e izes he geome ic mo ion a e o he in e nal s a e in
CPDin −1
, and i s
deni ion depends on he o hogonal decomposi ion o he Hamil onian.
2.4 Di ac-Type QCA and O hogonal Decomposi ion o Hamil onian
Take he one-dimensional Di ac- ype QCA as a conc e e model. In he long-wa eleng h limi ,
i s eec i e Hamil onian can be w i en as
He (p) = cˆp σz+mc2σx,
whe e
σx, σz
a e Pauli ma ices,
ˆp=−iℏ∂x
, and
m
is he eec i e mass pa ame e .
Decompose i in o
HT=cˆp σz, HM=mc2σx, H =HT+HM.
HT
gene a es ex e nal ansla ion, and
HM
gene a es in e nal sel - e e en ial o a ion. Pauli
ma ices sa is y he an i-commu a ion ela ion
{σz, σx}=σzσx+σxσz= 0,
and
σ2
x=σ2
z=I
. The e o e
H2=H2
T+H2
M= (c2ˆp2+m2c4)I.
4
This gi es he ope a o o igin o he ela i is ic ene gy-momen um ela ion
E2=p2c2+m2c4.
In he Bloch sphe e desc ip ion, in e nal s a es co espond o uni ec o s on
S2
, and he
Hamil onian
He (p)
co esponds o he angula eloci y ec o on he Bloch sphe e:
Ω(p) = 2
ℏmc2,0, cp,
whose magni ude
|Ω(p)|=2E(p)
ℏ
gi es he o al geome ic eloci y in he in e nal p ojec i e space. Due o he o hogonali y o
he commu a o and an i-commu a o s uc u es o
σx
and
σz
in he Lie algeb a, he " eloci y
componen s" co esponding o
HT
and
HM
can be unde s ood as wo mu ually o hogonal
di ec ions, he sum o whose squa es gi es he squa e o he o al speed.
This s uc u e is he algeb aic and geome ic basis o he In o ma ion Ra e Conse a ion
Theo em de i ed la e .
3 Main Resul s (Theo ems and Alignmen s)
Unde he abo e model amewo k, his pape p esen s he ollowing main esul s.
3.1 Theo em 1 (In o ma ion Ra e Conse a ion Theo em)
In any disc e e quan um walk/QCA sys em sa is ying local uni a i y and ansla ion in a iance
ha eme ges a one-dimensional Di ac- ype eec i e Hamil onian in he long-wa eleng h limi ,
o any posi i e-ene gy single-pa icle eigenmode, deno e he ex e nal g oup eloci y as
ex (p) = dE
dp ,
and dene he in e nal e olu ion eloci y in he in e nal p ojec i e Hilbe space as
in (p) := cmc2
E(p),
hen i mus hold ha
2
ex (p) + 2
in (p) = c2,
whe e
c
is he maximum causal p opaga ion speed o he QCA.
This heo em is gua an eed join ly by he an i-commu ing decomposi ion o he Hamil onian
and he o hogonali y o gene a o s unde he FubiniS udy me ic, and is an ine i able esul
o local uni a i y and Di ac s uc u e, no an addi ional assump ion.
3.2 Co olla y 1 (Eme gence o Special Rela i i y)
Dene p ope ime
τ
using he in e nal e olu ion pa ame e such ha
in d =c dτ.
F om Theo em 1, we ob ain
dτ
d 2= 1 − 2
c2, := ex .
5

Dening ou - eloci y
uµ=dxµ
dτ =γ( ) (c, ), γ( ) = 1
p1− 2/c2,
hen unde he Minkowski me ic
ηµν = diag(−1,1,1,1)
, he no maliza ion condi ion holds:
uµuµ=−c2,
and he co esponding line elemen is
ds2=−c2dτ2=−c2d 2+dx2.
Time dila ion and eloci y no maliza ion o special ela i i y eme ge di ec ly om conse a ion
o in o ma ion a e.
3.3 Theo em 2 (Mass as In e nal F equency)
In oduce a Hamil onian on he in e nal Hilbe space
Hin
:
iℏ∂τ|ψin (τ)⟩=Hin |ψin (τ)⟩.
I he e exis s a s a iona y s a e
|ψin ⟩
sa is ying
Hin |ψin ⟩=E0|ψin ⟩,
he in e nal s a e e ol es as
|ψin (τ)⟩= e−iE0τ/ℏ|ψin ⟩.
Dene in e nal equency
ωin =E0
ℏ.
Iden i ying
E0
as he es ene gy
mc2
, we ob ain
m=ℏωin
c2.
Mass is gi en by he in e nal equency, exp essing he ex en o which he in e nal sel - e e en ial
s uc u e occupies he op ical pa h quo a.
3.4 P oposi ion 1 (Zi e bewegung F equency and In e nal F equency)
In he con inuous limi o one-dimensional Di ac- ype QCA, he eec i e Hamil onian is
He (k) = cℏkσz+mc2σx,
eigen alues a e
E±(k) = ±p(cℏk)2+m2c4.
In he Heisenbe g pic u e, he e olu ion o he posi ion ope a o
X( )
con ains a apidly oscil-
la ing e m wi h equency
ωZB(k) = 2E+(k)
ℏ.
In he es limi (
k= 0
),
E+(0) = mc2
, so
ωZB(0) = 2mc2
ℏ= 2ωin .
The Zi e bewegung equency is wice he in e nal equency.
6
3.5 Theo em 3 (Topological S abili y and Non-Ze o In o ma ion Phase An-
gle)
Conside a one-dimensional ansla ion-in a ian QCA whose single-s ep uni a y ope a o
U(k)∈
U(N)
denes a closed cu e in momen um space. The winding numbe
W[U] = 1
2πiZπ/a
−π/a
∂klog de U(k)dk ∈Z
emains in a ian unde ni e-dep h local uni a y ans o ma ions.
I
W[U]= 0
, he e exis local exci a ions ca ying non-ze o opological cha ge, which canno
be con inuously de o med o he opologically i ial acuum by any ni e-dep h local uni a y
ans o ma ion. To main ain opological phase winding, he in e nal Hamil onian o such exci-
a ions mus ha e a non-ze o eigen equency
ωin >0
, hus
in >0
, and he in o ma ion phase
angle
θ= a c an( in / ex )
is non-ze o. The exis ence o mass is he e o e s abilized by he
opological s uc u e o he QCA.
3.6 Theo em 4 (Op ical Me ic and Weak-Field G a i y)
In he many-body case, in oduce coa se-g ained local in o ma ion p ocessing densi y
ρin o(x)
,
ep esen ing he a e age pa h leng h a e sed by he in e nal Hilbe space unde he Fubini
S udy me ic pe uni ime pe uni olume. Allow local escaling o ime and spa ial scales in
he coo dina e sys em
( , xi)
, in oducing a scale ac o
η(x)
such ha he line elemen can be
w i en as
ds2=−η2(x)c2d 2+η−2(x)γij(x)dxidxj,
whe e
γij(x)
is he h ee-dimensional spa ial me ic. Dene coo dina e speed o ligh
ce (x) := 
dx
d =η2(x)c,
and e ac i e index
n(x) := c
ce (x)=η−2(x).
In he s a ic, sphe ically symme ic, weak-eld limi
η(x) = 1 + ϵ(x),|ϵ(x)| ≪ 1,
aking
ϵ( ) = ϕ( )/c2
, whe e
ϕ( ) = −GM/
is he New onian po en ial, we ob ain
g00 ≃ −(1 + 2ϕ/c2)c2, gij ≃(1 −2ϕ/c2)δij,
consis en wi h he  s -o de expansion o he Schwa zschild me ic in iso opic coo dina es.
The e ac i e index
n( )≃1−2ϕ( )
c2≃1 + 2GM
c2 >1,
gi es he ligh deec ion angle in g a i a ional lensing heo y based on he GaussBonne heo-
em:
∆θ=4GM
c2b,
whe e
b
is he impac pa ame e , consis en wi h he s anda d esul o Gene al Rela i i y.
7
3.7 Theo em 5 (In o ma ion-G a i y Va ia ional P inciple)
Conside he ac ion
S o [g, ρin o] = 1
16πG ZM
√−g R[g]d4x+ZM
√−gLin o[ρin o, g]d4x.
Va ying wi h espec o
gµν
(igno ing bounda y e ms) yields
Rµν −1
2Rgµν = 8πG T (in o)
µν ,
whe e
T(in o)
µν := −2
√−g
δ(√−gLin o)
δgµν .
I in he low-ene gy limi he choice o
Lin o
makes
T(in o)
µν
consis en wi h he s ess-ene gy enso
o s anda d ma e , his equa ion is o mally equi alen o Eins ein's eld equa ions.
3.8 Theo em 6 (In o ma ion Mass and Asymp o ic S a iona i y)
Fo sys ems wi h in e nal models and sel - e e en ial mechanisms, in oduce in o ma ion mass
MI(σ) = K(σ), D(σ), Sen (σ),
whe e
K
is Kolmogo o complexi y,
D
is logical dep h,
Sen
is in e nal en anglemen en opy, and
is a mono onically inc easing unc ion. Assume he a e age in e nal in o ma ion a e
in (MI)
equi ed o main ain a gi en
MI
is mono onically inc easing and bounded by
c
. F om Theo em
1, we ha e
2
ex (MI) = c2− 2
in (MI).
I
lim
MI→∞ in (MI) = c,
hen
lim
MI→∞ ex (MI)=0,
meaning high-in o ma ion-mass subjec s end o be asymp o ically s a iona y in ex e nal geom-
e y.
3.9 P oposi ion 2 (Landaue Cos o Main aining In o ma ion Mass)
Assume a sys em upda es i s in e nal model a a a e
Rupd
, e asing
∆I
bi s o old in o ma ion
on a e age pe upda e. The a e o in o ma ion e asu e pe uni ime is
˙
Ie ase =Rupd∆I.
In a hea ba h a empe a u e
T
, acco ding o Landaue 's p inciple, e asing one bi o in o ma ion
dissipa es a leas
kBTln 2
hea . The minimum powe consump ion is
Pmin =kBTln 2 ˙
Ie ase =kBTln 2 Rupd∆I.
The con inued exis ence o high-in o ma ion-mass sys ems is necessa ily accompanied by non-
ze o minimum powe consump ion and en opy ux ou pu .
4 P oo s
This sec ion p o ides p oo s o p oo ideas o he main heo ems. De ailed calcula ions and
model de ails a e placed in he appendices.
8
4.1 P oo o Theo em 1 (In o ma ion Ra e Conse a ion Theo em)
Conside he long-wa eleng h limi o a one-dimensional Di ac- ype QCA wi h eec i e Hamil-
onian
H(p) = HT(p) + HM, HT(p) = c p σz, HM=mc2σx,
whe e
p
is momen um. Pauli ma ices sa is y
σ2
x=σ2
z=I,{σz, σx}= 0.
The e o e
H2(p) = H2
T(p) + H2
M= (c2p2+m2c4)I.
Fo a posi i e-ene gy eigenmode
|ψp⟩
, we ha e
H(p)|ψp⟩=E(p)|ψp⟩, E2(p) = c2p2+m2c4.
The ex e nal g oup eloci y is dened as
ex (p) = dE
dp .
F om
E2=c2p2+m2c4
, we ge
2EdE
dp = 2c2p,
so
ex (p) = dE
dp =c2p
E(p).
The in e nal pa is gi en by
HM=mc2σx
. I s co esponding "ene gy sha e" ela i e o o al
ene gy
E(p)
is
EM
E=mc2
E(p).
Dene in e nal eloci y as
in (p) := cEM
E=cmc2
E(p).
This deni ion can be unde s ood as: in he o al in o ma ion a e budge
c
, he componen
occupied by in e nal e olu ion is weigh ed by he "mass ene gy sha e." Thus we ha e
2
ex (p)
c2=c4p2
c2E2(p)=c2p2
E2(p), 2
in (p)
c2=m2c4
E2(p).
Adding hem gi es
2
ex (p)
c2+ 2
in (p)
c2=c2p2+m2c4
E2(p)= 1,
i.e.,
2
ex (p) + 2
in (p) = c2.
Algeb aically, his is a di ec esul o he Hamil onian decomposing in o wo an i-commu ing
ope a o s, leading o he Py hago ean o m o ene gy squa ed. Geome ically, on he in e nal
wo-dimensional Hilbe space, he Hamil onian can be w i en as
H(p) = n(p)·σ,n(p) = (mc2,0, cp),
co esponding o he angula eloci y ec o on he Bloch sphe e
Ω(p) = 2
ℏn(p) = 2
ℏmc2,0, cp.
9
Re e ences
[1] T. Fa elly, "A Re iew o Quan um Cellula Au oma a", Quan um 4, 368 (2020).
[2] A. Bisio, G. M. D'A iano, A. Tosini, "Di ac Quan um Cellula Au oma on in One Dimen-
sion: Zi e bewegung and Sca e ing om Po en ial", Phys. Re . A 88, 032301 (2013).
[3] G. W. Gibbons, M. C. We ne , "Applica ions o he GaussBonne Theo em o G a i a ional
Lensing", Class. Quan um G a . 25, 235009 (2008).
[4] M. Halla, "Applica ion o he GaussBonne Theo em o Lensing in S a ic Sphe ically Sym-
me ic Space imes", Gen. Rela i . G a i . 52, 95 (2020).
[5] R. Landaue , "I e e sibili y and Hea Gene a ion in he Compu ing P ocess", IBM J. Res.
De . 5, 183191 (1961).
[6] Fo o he li e a u e e iews on QCA opological classica ion, quan um simula ion pla -
o ms, and op ical me ic me hods, see [14] and e e ences he ein.
A Appendix A: Con inuous Limi and In o ma ion Ra e Conse -
a ion o One-Dimensional Di ac-QCA
A.1 A.1 Model Deni ion
Conside a 1D la ice
Λ = aZ
, whe e each si e
x
ca ies a wo-componen spin
ψx=ψx,L
ψx,R.
Dene he condi ional shi ope a o
S
as
(Sψ)x,L=ψx+a,L,(Sψ)x,R=ψx−a,R,
In e nal o a ion
W(θ) = cos θ−i sin θ
−i sin θcos θ.
Single-s ep QCA e olu ion is
ψ( + ∆ ) = U(θ)ψ( ), U(θ) := W(θ)S.
In momen um ep esen a ion, dening
ψk=X
x
e−ikxψx,
we ha e
S(k) = eikaσz, U(θ;k) = W(θ)S(k).
A.2 A.2 Eec i e Hamil onian and Di ac Equa ion
Taking
a, ∆ , θ
o be small simul aneously, dene eec i e Hamil onian
He (k) = iℏ
∆ log U(θ;k).
Expanding o small pa ame e s gi es
log U(θ;k)≃ikaσz−iθσx,
16

hus
He (k)≃ℏ
∆ (kaσz+θσx).
Le ing
c=a
∆ , mc2=ℏθ
∆ ,
we ob ain
He (k)≃cℏkσz+mc2σx,
whose posi ion space o m is he one-dimensional Di ac equa ion
iℏ∂ ψ(x, ) = −iℏcσz∂x+mc2σxψ(x, ).
Dispe sion ela ion is
E±(k) = ±p(cℏk)2+m2c4.
G oup eloci y
ex (k) = 1
ℏ
∂E+
∂k =c2k
pk2+ (mc/ℏ)2< c.
A.3 A.3 Explici Realiza ion o In e nal Veloci y and In o ma ion Ra e Con-
se a ion
Fo xed
k
, he in e nal s a e is an eigens a e
|u+(k)⟩
in he 2D spin space, whose in e nal phase
e ol es wi h equency
E+(k)/ℏ
. In Bloch sphe e ep esen a ion, he qubi s a e co esponds o
a uni ec o
∈S2
, and Hamil onian
He (k) = n(k)·σ
gene a es uni o m o a ion a ound
n(k)
wi h angula speed magni ude
|Ω(k)|=2|n(k)|
ℏ=2E+(k)
ℏ.
The FubiniS udy me ic on
CP1
is equi alen o he s anda d me ic on he Bloch sphe e, so
in e nal geome ic eloci y is p opo ional o
|Ω(k)|
.
In Di ac- ype QCA
n(k)=(mc2,0, cℏk).
Can be w i en as
|n(k)|2= (mc2)2+ (cℏk)2.
Decomposing he o al "angula eloci y ec o " in o
Ω(k) = ΩM(k) + ΩT(k),ΩM(k) = 2
ℏ(mc2,0,0),ΩT(k) = 2
ℏ(0,0, cℏk).
The wo componen s a e o hogonal,
|Ω(k)|2=|ΩM(k)|2+|ΩT(k)|2
.
No malizing he in e nal eloci y
in (k)
as
in (k)
c=|ΩM(k)|
|Ω(k)|=mc2
E+(k),
and ex e nal eloci y
ex (k)
gi en by g oup eloci y
ex (k)
c=cℏk
E+(k).
Then
2
ex (k)
c2+ 2
in (k)
c2=c2ℏ2k2+m2c4
E2
+(k)= 1,
i.e.,
2
ex (k) + 2
in (k) = c2.
This explici ly ealizes he In o ma ion Ra e Conse a ion desc ibed in Theo em 1 in he Di ac-
QCA model.
17
B Appendix B: Op ical Me ic, Local Volume Conse a ion, and
Ligh Deec ion
B.1 B.1 Local Volume Elemen and Scale Fac o Cons ain s
Unde he me ic
ds2=−η2(x)c2d 2+η−2(x)γij(x)dxidxj
he ou - olume elemen is
dV4=√−g d4x.
Unde iso opy assump ion, he 3D spa ial me ic can be w i en as
γijdxidxj= Ψ4(x)dx2,
hen
√−g∝η(x)η−3(x)Ψ6(x) = η−2(x)Ψ6(x).
Local Hilbe olume conse a ion can be simplied as "physical Hilbe olume co esponding
o uni coo dina e olume is in a ian ," exp essed by cons ain
η (x)η3
x(x)=1.
Unde iso opy, aking
η (x) = η(x), ηx(x) = η−1(x),
yields he op ical me ic o m
ds2=−η2(x)c2d 2+η−2(x)γij(x)dxidxj.
B.2 B.2 Weak-Field Expansion and Schwa zschild Me ic
In s a ic, sphe ically symme ic case, Schwa zschild me ic in iso opic coo dina es
( , , θ, φ)
is
ds2=−h1−GM
2c2
1 + GM
2c2 i2c2d 2+1 + GM
2c2 4(d 2+ 2dΩ2).
Expanding unde weak-eld app oxima ion
GM
c2 ≪1
gi es
g00 ≃ −(1 −2GM
c2 )c2=−(1 + 2ϕ/c2)c2,
gij ≃(1 + 2GM
c2 )δij = (1 −2ϕ/c2)δij,
whe e
ϕ( ) = −GM/
is he New onian po en ial.
S a ing om he op ical me ic, aking
η( ) = 1 + ϕ( )
c2, γij =δij,
expanding gi es
g00 =−η2c2≃ −(1 + 2ϕ/c2)c2,
gij =η−2δij ≃(1 −2ϕ/c2)δij,
consis en wi h he weak-eld expansion o he Schwa zschild me ic.
18
B.3 B.3 Re ac i e Index and Ligh Deec ion Angle
Fo null geodesics
ds2= 0
unde he op ical me ic,
η2( )c2d 2=η−2( )dx2,
Coo dina e speed o ligh
ce ( ) = 
dx
d =η2( )c.
Re ac i e index
n( ) := c
ce ( )=η−2( ).
In he weak-eld limi
η( ) = 1 + ϕ( )
c2,
ϕ( )
c2≪1,
 s -o de expansion gi es
n( ) = η−2( )≃1−2ϕ( )
c2.
Taking
ϕ( ) = −GM/
gi es
n( )≃1 + 2GM
c2 >1, ce ( ) = c
n( )< c,
consis en wi h he physical pic u e o ligh slowing down in weak g a i y.
In he GibbonsWe ne me hod, he op ical me ic can be iewed as a 2D Riemann su ace,
and ligh ajec o ies as geodesics on his su ace. Calcula ing he deec ion angle ia he Gauss
Bonne heo em in he weak deec ion limi yields
∆θ≃4GM
c2b,
consis en wi h Eins ein's p edic ion. I only
g00
is modied while keeping spa ial me ic a ,
he co esponding e ac i e index is only
n( )≃1−ϕ( )
c2,
and he deec ion angle would be hal o he abo e alue, showing ha simul aneously de o ming
ime and spa ial scales (i.e., in oducing op ical me ic) is c ucial o eco e ing he co ec
deec ion ac o .
C Appendix C: Zi e bewegung and In e nal F equency in Di ac
Theo y
C.1 C.1 Di ac Equa ion and Plane Wa e Solu ion
Conside 1D Di ac equa ion
iℏ∂ ψ= (cαp +βmc2)ψ,
aking ep esen a ion
α=σz
,
β=σx
,
p=−iℏ∂x
. Plane wa e solu ions a e
ψk,±(x, ) = u±(k) ei(kx−ω± ),
ω±=± (ck)2+m2c4
ℏ2.
Gene al wa e packe can be w i en as
ψ(x, ) = Za+(k)ψk,+(x, ) + a−(k)ψk,−(x, )dk.
19
C.2 C.2 Posi ion Ope a o in Heisenbe g Pic u e
In Heisenbe g pic u e, posi ion ope a o e ol es as
X( )=eiH /ℏX(0)e−iH /ℏ, H =cαp +βmc2.
Heisenbe g equa ions gi e
dX
d =i
ℏ[H, X] = cα, dα
d =i
ℏ[H, α].
Sol ing o
α( )
and subs i u ing back in o
X( )
gi es
X( ) = X(0) + c2H−1P +iℏc
2H−1e−2iH /ℏ−1α(0) −cH−1P,
whe e he second e m is uni o m mo ion, and he hi d e m is a apidly oscilla ing e m wi h
equency
2E/ℏ
, i.e., Zi e bewegung, whe e
E=p(cP)2+m2c4
is he posi i e ene gy b anch.
In he es limi (
P= 0
),
E=mc2
, oscilla ion equency
ωZB(0) = 2mc2
ℏ.
In he amewo k o his pape , in e nal equency is dened as
ωin =mc2
ℏ,
hus
ωZB(0) = 2ωin .
D Appendix D: In o ma ion Mass, Landaue 's P inciple, and
Minimum Powe Consump ion
D.1 D.1 In o ma ion E asu e and Minimum Dissipa ion
Assume a sys em's in e nal s a e is
σ
, and i s in o ma ion mass
MI(σ)
is ela ed o he scale and
complexi y o i s in e nal model. To main ain model alidi y, he sys em mus egula ly upda e
in e nal ep esen a ions wi h new obse a ions, a p ocess necessa ily in ol ing e asu e o some
old in o ma ion.
Assume he sys em upda es he model a a e
Rupd
, e asing
∆I
bi s o old in o ma ion on
a e age pe upda e. The amoun o in o ma ion e ased pe uni ime is
˙
Ie ase =Rupd∆I.
In a hea ba h a empe a u e
T
, acco ding o Landaue 's p inciple, e asing one bi o in o ma ion
mus dissipa e a leas
kBTln 2
hea in o he en i onmen ; minimum dissipa ed powe is
Pmin =kBTln 2 ˙
Ie ase =kBTln 2 Rupd∆I.
This esul is independen o he specic physical implemen a ion o he sys em, depending only
on he numbe o bi s e ased and en i onmen al empe a u e, se ing as a uni e sal lowe bound
o powe consump ion equi ed o implemen any high-in o ma ion-mass sys em.
20
D.2 D.2 High In o ma ion Mass and High Dissipa ion
I in o ma ion mass
MI(σ)
inc eases wi h he scale, s uc u al complexi y, and upda e equency
o he in e nal model, hen la ge
MI
ypically equi es la ge
Rupd
and
∆I
o con inuously
disca d obsole e in o ma ion and in oduce new in o ma ion. The e o e, in gene al
Pmin(MI) = kBTln 2 Rupd(MI) ∆I(MI)
will inc ease wi h inc easing
MI
.
Combining wi h he In o ma ion Ra e Conse a ion Theo em o his pape , we a i e a a
unied pic u e: 1. To main ain high
MI
, he sys em mus alloca e a la ge amoun o op ical pa h
quo a o in e nal e olu ion (la ge
in
), he eby limi ing ex e nal mo ion speed
ex
, mani es ing as
asymp o ic s a iona i y; 2. Meanwhile, equen in e nal upda es and e asu es lead o con inuous
en opy ow o he ou side, making he sys em a signican hea sou ce, mani es ing as s ong
dissipa i e cha ac e is ics; 3. Mac oscopically, hese wo eec s o en appea in egions wi h
deep g a i a ional po en ials: s ella in e io s main ain high in o ma ion densi y and complex
s uc u e ia nuclea eac ions while emi ing massi e adia ion; biological and neu al sys ems
main ain low en opy s uc u es ia me abolism while ou pu ing hea o he en i onmen .
The e o e, om he pe spec i e o In o ma ion Ra e Conse a ion, he connec ion be ween
mass, g a i y, and complex ene ge ic s uc u es can be uniedly unde s ood as: o main ain a
highly o de ed, opologically s able s uc u e in e nally, one mus con inuously consume op ical
pa h quo a and ou pu en opy and ene gy, and his p ocess mani es s geome ically as cu a u e
and dynamically as g a i y.
21