G a i y as Local In o ma ion, Quan um as Blu
Black Holes, Speed Limi s, and he Two Poles o Reali y
Aleksanda Pe išić
Sep embe 2025
Abs ac
We p opose a uni ied, ope a ional b idge be ween g a i y and quan um heo y based on an
in o ma ion–ene ge ics p inciple: no ee in o ma ion. G a i y is ea ed as a local cons ain
on accessible in o ma ion densi y, while quan um heo y supplies he i educible blu ha
g a i y canno esol e. A emp s o ex ac a bi a ily p ecise p edic i e s uc u e wi hin
a ini e egion equi e ene gy ha g a i a es; beyond a h eshold, ho izons o m, sealing
he in o ma ion behind black holes. In his iew, black holes do no b eak he ligh -speed
limi — hey en o ce i : you may “know” mo e locally a he p ice o losing he abili y o
communica e i ou wa d. We o malize he adeo s wi h clean bounds (Bekens ein- ype
in o ma ion capaci y [
1
,
2
,
3
,
4
], Schwa zschild collapse c i e ion, Landaue e asu e cos [
7
,
8
],
quan um speed limi s [
9
,
10
]), in oduce a wo-pole en elope o scale (quan um mic o-pole
and g a i a ional mac o-pole), and s a e an ope a ional B idge P inciple: you canno lea n
inwa d mo e inely han in o ma ion can low ou wa d. The middle o he scale-space is
e ec i ely Gaussian; he ex emes ac as poles o igno ance. This a icle can be ead as a
g a i y/quan um case s udy wi hin a b oade blu –based epis emic amewo k de eloped
in [19].
1 P og am in one page
P1.
G a i y
≡
local in o ma ion budge . Fo a labo a o y o adius
R
and o al ene gy
E
inside, he accessible in o ma ion Iobeys he Bekens ein- ype capaci y bound
S≤2πkB
ℏcR E, I :=S
kBln 2 ≤2π
ln 2
R E
ℏc,(1)
while comp essing Ein o Rpushes he Schwa zschild adius
s=2GE
c4=2GM
c2,(E=Mc2)(2)
owa ds R. C ossing s=R o ms a ho izon: ou wa d channels a e se e ed.
P2.
Black holes en o ce
c
.A a ho izon, he en opy sa u a es an a ea law
SBH
=
kBA
4ℓ2
P
wi h
ℓ2
P
=
Gℏ
c3
[
1
,
5
,
6
]; no in o ma ion escapes a a ini e a e o in ini y. A emp s o “ ead he
u u e” by concen a ing esou ces canno c ea e communicable supe luminal ad an age:
he p ice is isola ion.
P3.
Quan um as i educible blu . To e ine p edic ions you mus pay Landaue (ene gy
pe e ased bi ) and obey quan um speed limi s ( ime o c ea e o hogonal s a es). G a i y
only sees coa se s ess-ene gy; he un e ealed ine s uc u e is he quan um blu . In he
language o he gene al blu amewo k [
19
], quan um beha iou is he mic o-pole whe e
his i educible blu is codi ied by ℏ.
1
2 Axioms and bounds: one law, h ee aces
Fix a sphe ical egion o adius Rcon aining o al ene gy Ea empe a u e T.
Capaci y / Bekens ein- ype
S≤2πkB
ℏcRE, (3a)
I:=S
kBln 2 ≤2π
ln 2
RE
ℏc,(3b)
Collapse / Schwa zschild
s=2GE
c4,(3c)
E≥c4
2GR=⇒ho izon,(3d)
Landaue e asu e (min. ene gy o e ase ∆Ibi s a empe a u e T)
∆E≥kBTln 2 ·∆I, (3e)
Quan um speed limi s (Mandels am–Tamm; Ma golus–Le i in)
τ≥maxπℏ
2 ∆E,πℏ
2 (⟨H⟩−E0).(3 )
P inciple 2.1 (No ee in o ma ion).Fo ixed (
R, T
), he ou wa d communicable in o ma ion
a e
˙
Iou
is bounded by a unc ion o (
R, E, T
) ha sa u a es a ho izon o ma ion. Inc easing
inwa d esolu ion (pushing blu scale
σ↓
0) a ixed (
R, E
)e en ually dec eases
˙
Iou
by igge ing
collapse.
3 G a i y as a local in o ma ional ield
P oposi ion 3.1 (In o ma ion–g a i y adeo ).Le
R
be ixed, and le
P
deno e he a e age
dissipa ed powe o e a ime window
τ
. Any p o ocol ha achie es inwa d in o ma ion gain ∆
I
wi hin ime τinside he lab obeys
∆I≤min2π
ln 2
RE
ℏc,P τ
kBTln 2,wi h E≤c4
2GR, (4)
else a ho izon o ms. Thus, pushing ∆
I
a ixed
R
o ces
E
upwa d un il communica ion o
in ini y is cu .
Idea.
Combine he capaci y bound
(1)
–
(3b)
and Landaue
(3e)
wi h he collapse h eshold
(3d)
.
The minimum o he wo in o ma ion ceilings applies p io o collapse; beyond
E
= (
c4/
2
G
)
R
he achie able ou wa d a e app oaches ze o.
4 Black holes as speed-limi en o ce s
Lemma 4.1 (No as - o wa d wi hou isola ion).I a p ocedu e a emp s o ou pace causal
signaling by inc easing local ene gy densi y (e.g. o o ecas many bi s o he “ u u e” o a sys em),
hen, as
E↗c4
2GR
and
s↗R
, he asymp o ic edshi di e ges and he achie able ou wa d
in o ma ion a e o in ini y ends o ze o. The agen may s o e mo e local bi s (a ea law), bu
canno ansmi hem o he ex e io a ini e a e.
Idea.
Nea
s→R
, null ays om he in e io su e unbounded edshi / ime dila ion o dis an
obse e s; ope a ional ou wa d capaci y collapses (c .
(3b)
,
(3d)
, and black-hole he modynam-
ics [1,5,6,14]).
2
5 Quan um as blu : wo poles wi h a Gaussian middle
G a i y esponds o coa se s ess-ene gy; phases and ine en anglemen a e la gely in isible o
classical cu a u e. We model he accessible scale p o ile wi h a wo-pole en elope on adius
∈[0, R∗]:
wα,β( ) = 1
B(α, β)
R∗α−11−
R∗β−1,0< α, β < 1,(5)
whe e
B
is he Eule be a unc ion. The mesoscopic egime is cap u ed by a Gaussian ke nel
Gσ( )cen e ed a 0:
K( ) = wα,β( )·Gσ( ), Gσ( ) = 1
√2πσ exp−( − 0)2/(2σ2).(6)
Rema k 5.1 (In e p e a ion).Nea
0
(ou e e yday scales), e ec i e ield heo y is well-beha ed
(Gaussian co e). App oaching ei he pole, he ke nel weigh ises while con ol anishes: a
↓
0
quan um inde e minacy domina es; a
↑R∗
g a i a ional back eac ion domina es. Ho izons a e
he ope a ional bounda y whe e inwa d p obing canno exceed ou wa d low. In he abs ac blu
language o [
19
], his wo-pole pic u e is one conc e e ins ance o a gene al me a-phenomenon:
o well-posed lows on compac egions, each ixed blu scale admi s only ini ely many e ec i e
his o ies ( ini e blu - amilies), and he poles a e whe e such blu -desc ip ions necessa ily b eak
down.
6 B idge P inciple (ope a ional)
Theo em 6.1 (Inwa d lea ning
⇒
ou wa d cos ).Fo any p o ocol Πope a ing wi hin adius
R
and ene gy
E
o e ime
τ
, wi h blu scale
σ
(smalle
σ
means ine inwa d esolu ion), he
ou wa d communicable in o ma ion h oughpu sa is ies
˙
Iou (Π) ≤Φ
R, E, T, τ, σ,Φ(R, E, T, τ, σ)−−−→
σ↓00whene e E↗c4
2GR. (7)
Equi alen ly: you canno lea n inwa d ( educe
σ
) beyond he a e a which in o ma ion can low
ou wa d; a emp s o do so con e communicabili y in o isola ion.
Ske ch.
Bound he a ainable ∆
I
by
(4)
, con e o a a e ia
τ
, and accoun o channel capaci y
deg ada ion as edshi g ows nea
s→R
. The limi Φ
→
0 e lec s he anishing ex e io
ligh like channels in he ho izon limi .
7 Though expe imen : o ecas ing by ene gy concen a ion
An agen inside adius
R
builds a de ice o “ ead he u u e” by (i) inc easing measu emen
bandwid h and (ii) accele a ing compu a ion. Each addi ional bi o eliable p edic ion cos s a
leas
kBTln
2in ene gy and Ω(
ℏ/
∆
E
)in ime, while inc easing
E
cu es space ime. I he agen
pushes a enough, a ho izon o ms: hey may know mo e, bu nobody ou side can e e lea n ha
hey know. Thus, black holes a e no loopholes o c; hey a e he mechanism ha p ese es i .
8 Consequences and es s (quali a i e)
•
Design bound. Any a chi ec u e p omising unbounded p edic i e h oughpu om a
bounded egion mus ei he (a) adia e was e hea ( aising
T
) and slow down (QSL), o (b)
inc ease Eand sel -isola e (ho izon endency).
3
•
Dual edges. Phenomena a ul a-small and ul a-la ge scales will emain go e ned by
pole-like igno ance (quan um andomness / g a i a ional opaci y). Mid-scale Gaussian
models emain obus .
•
Holog aphic budge ing. P o ocols ha appea o su pass blu limi s mus e eal ei he
hidden ese oi s (la ge e ec i e
R
o
T
) o ade ou wa d communicabili y o inwa d
s o age; c . Bekens ein/holog aphic bounds and QNEC [2,12,4,13,3].
8.1 Time as Causal Residue Be ween Two Unce ain y P inciples
We si be ween wo ope a ional “unce ain ies”: he small-scale (quan um) limi
∆E∆ ≳ℏ
2(heu is ic ene gy– ime ela ion; ope a i e bounds ia QSLs (3 )),(8)
and he la ge-scale (g a i a ional/causal) limi ha caps communicable in o ma ion om a
bounded egion o adius Rand powe P:
˙
I≤min
P
kBTln 2
| {z }
Landaue -limi ed e asu e a e
,2πR
ℏcln 2 P
| {z }
Bekens ein en elope
,
E≤c4
2GR(no-collapse a ixed R; hence o e a window τ,P≤E/τ).
(9)
These ela ions o malize ha causali y is no a ee lunch: sus aining any nonze o ou wa d
a e
˙
I
equi es ini e ene gy lux
P
o e a nonze o dwell ime
τ
, and sh inking
τ↓
0a ixed
R
ei he demands
P→∞
(which back eac s up o collapse) o o ces
˙
I→
0. In his sense, ime is
he esidual budge o e which dynamics un olds: he slack a iable ha econciles quan um
kinema ics wi h g a i a ional causali y.
Ope a ional co olla y ( ime cos pe bi ). Le
τbi
be he minimal la ency o communica e
one eliable bi om he egion o in ini y. Combining (3 ) and (9),
τbi ≥maxπℏ
2 ∆E,kBTln 2
P,ℏcln 2
2πR
1
P,(10)
wi h
P
i sel bounded by he no-collapse condi ion a adius
R
. Thus any a emp o “speed
up causali y” by educing
τbi
mus ei he aise ∆
E
(quan um-limi ed) o
P
(g a i y-limi ed);
pushing oo a isola es he sys em (ho izon o ma ion), while s a ing he budge s sends
˙
I→
0.
The T→0limi ( ozen channel, cohe en blu ). In he idealized c yogenic limi ,
T→0+, P bounded,∆Ebounded,(11)
a he mal e asu e channel is una ailable and elaxa ion imes di e ge ( hi d-law phenomenology);
quan um speed limi s impose a ini e lowe bound on s a e-change imes. Consequen ly, o any
ealizable p o ocol wi h bounded esou ces,
˙
I−−−−→
T→0+0,(12)
e en hough he mic oscopic s a e e ains cohe en s uc u e ( acuum/g ound-s a e co ela ions).
The acuum is hus no emp y—i ha bo s luc ua ions and en anglemen —ye , absen ca ie s
and budge s, i is ope a ionally in o ma ionless o he ou side.
4
A chimedean iew. Time he e plays he ole o an “A chimedean backg ound”: locally a
small cos pe causal s ep, bu in eg a ed globally an eno mous esou ce enabling his o ies o
exis . Dec easing ha esidue (s a ing he budge s) eezes dynamics; inc easing i ( unding
∆
E
o
P
) es o es low un il g a i y pushes back. In be ween lies he e ec i e, Gaussian-like
middle whe e ou physics li es.
9 Two appa en eali ies a a ho izon (ope a ional pic u e)
A a black-hole ho izon he e a e wo equally alid ope a ional desc ip ions, ied o whe e he
obse e ’s ou wa d channel e mina es.
(i) In alle ’s local ame (smoo hness). By he equi alence p inciple, a eely alling
a ele who su i es idal o ces expe iences no sha p d ama a he ho izon: local physics
emains c isp and low-blu . P ope imes and he mome e eadings along he wo ldline beha e
egula ly; any b eakdown is de e ed o deep in e io scales.
(ii) Dis an obse e ’s ame (opaci y). Fo an ex e io obse e , he in alle ’s ou wa d
channel collapses: signals edshi and ime-dila e owa d ze o a e as null ays skim he ho izon.
Ope a ionally, he a ele ’s s a e becomes in o ma ion- heo e ically blu ed—no because local
mic os uc u e anishes, bu because i s communicable in o ma ion o in ini y ends o ze o
wi hin ini e ex e io ime (c . P inciple 2.1). In his sense “disappea ance” means loss o
ex ac able in o ma ion, no necessa ily des uc ion o local s uc u e.
Inside: a g adien o an inne blu adius
Wi hin he ho izon, ou wa d-di ec ed null ays s ill mo e inwa d; causal cones ip so ha e e y
imelike wo ldline ends a he in e io pole. Communica ion among nea by laye s emains
possible o a while, bu ope a ional capaci y deg ades apidly. We o malize his wi h an
adjacen -laye capaci y.
De ini ion 9.1 (Adjacen -laye capaci y).Le
B
and
B −δℓ
be concen ic shells o p ope
sepa a ion δℓ inside a Schwa zschild black hole. O e a window τ, de ine
Cadj( ;δℓ, τ):= sup
p o ocols
I(B →B −δℓ;τ)
τ,
he maximal achie able classical in o ma ion a e om B o B −δℓ.
De ini ion 9.2 (Ope a ional inne blu adius).Fix small h esholds
ε, ℓc>
0. The blu adius
blu is he smalles adius o which
Cadj( ;δℓ, τ)< ε/τ o all 0< δℓ ≤ℓc.
Beyond
blu
, e en immedia ely adjacen laye s a e, o p ac ical pu poses, causally decoupled
a any ini e esou ce le el—ope a ionally a “soup o andomness” wi h no eco e able channel
s uc u e.
O de -o -magni ude es ima e (classical GR + Planck cu o ). Fo a Schwa zschild
mass M, he cu a u e in a ian
K( ) = RabcdRabcd =48 G2M2
c4 6
5
eaches he Planck scale when K∼ℓ−4
Pwi h ℓ2
P=Gℏ
c3. Sol ing gi es a quan um-g a i y adius
qg(M)≈(48)1/62−1/3
| {z }
≈1.5 s1/3ℓ2/3
P, s=2GM
c2.(13)
Thus o mac oscopic
M
,
qg ≪ s
: any undamen al b eakdown ( he “pole” o o al inde e mi-
na ion) is a inside he ho izon. In ou amewo k one may iden i y
blu ∈[ 0, qg(M) ],
wi h he p ecise alue con olled by mic oscopic sc ambling/mixing ha h o les
Cadj
be o e
Planckian cu a u e is eached. Equa ion
(13)
supplies a conse a i e inne bound ied o
cu a u e alone.
Ope a ional summa y. F om he in alle ’s iew he ho izon is mundane; om in ini y i is
an in o ma ion ba ie . Deepe in, he g adien o communicabili y s eepens un il, a
blu
, e en
nea es -neighbo laye s ail o sus ain a usable channel. Whe he his e ec i e blu pole si s
exac ly a he geome ic cen e (
= 0) o a a ini e
blu >
0is a ques ion o mic ophysics; ei he
way i lies well inside he known adius
s
. This pic u e meshes wi h he No-F ee-In o ma ion
law: pushing inwa d esolu ion beyond wha can be communica ed ou wa d ine i ably con e s
knowledge in o isola ion.
10 Re u a ion P obe and a Runnable Tes bed
Aim. P o ide (i) an in e nal consis ency p obe ha s ess- es s he No-F ee-In o ma ion (NFI)
p inciple agains s anda d physics, and (ii) a unnable es bed ( oy expe imen /simula ion) ha
a eade can implemen o seek a iola ion. We close wi h a c isp alsi iabili y checklis and ou
e dic .
A. In e nal Consis ency P obe (desk check)
We examine plausible “escape ha ches” and how known physics esponds.
1.
Re e sible compu ing s. Landaue . Re e sible ga es a oid dissipa ion pe logical s ep,
bu eliable p edic ion equi es e o co ec ion a nonze o noise. S abiliza ion in oduces
en opy expo ; a ixed
R
his ei he (a) aises
E
(pushing
s↑R
) o (b) slows he ne
in e ence a e by cycle ime and edundancy. Toge he wi h quan um speed limi s, his
p ese es he NFI en elope [7,8,9,10].
2.
Quan um me ology and squeezed s a es. Heisenbe g scaling aises Fishe in o ma ion
F
wi h pho on numbe /ene gy; a ixed
R
, Bekens ein and collapse bounds cap usable
E
,
and de ec o sa u a ion plus backac ion h o le ou wa d capaci y Cou [2,3].
3.
T a e sable wo mholes / wa p d i es. Known a e sable cons uc ions equi e uned
nega i e-ene gy luxes and a e cons ained by quan um inequali ies [
18
]; sus ained, b oadband,
supe luminal-capaci y channels a e no known o be possible.
4.
Black hole e apo a ion and in o ma ion e u n. Uni a y Page-cu e scena ios allow
e en ual in o ma ion eco e y ia Hawking adia ion, bu p ac ical decoding is conjec u ed
o be compu a ionally in ac able (Ha low–Hayden) and ou wa d a es a e iny (g ey-
body/ edshi ) [
14
,
16
,
15
,
17
]. NFI conce ns ini e- ime, ini e- esou ce communica ion; no
con adic ion a ises.
6
In e im e dic . Wi hin es ablished physics (GR, QFT, he modynamics, quan um in o), no
known mechanism simul aneously inc eases inwa d p ecision (sh inks blu
σ
) and main ains o
inc eases ou wa d channel capaci y
Cou
a ixed (
R, E, T
)wi hou igge ing back eac ion ha
educes Cou . The p inciple is cohe en ; ou s anding ca ea s a e no ed in Rema k 10.1.
B. Runnable Tes bed: Lieb–Robinson Lab P o ocol
A able op analogue wi h a buil -in “speed o ligh ” p o ides a di ec s ess es .
Model. A 1D spin chain o leng h
N
wi h nea es -neighbo coupling, local Hamil onian
ˆ
H
=
Pihi,i+1
o bounded no m. The Lieb–Robinson eloci y
LR
bounds in o ma ion p opaga ion [
11
].
Spli he chain in o:
• egion A: con iguous block o leng h R( he “lab”),
• egion B: a de ec o block a away a dis ance D≫R.
Knobs (hold geome y ixed).
1.
Blu
σ
: une con inuous-measu emen s eng h
γ
in
A
(weak
→
s ong), o igh en POVMs
o inc ease local Fishe in o ma ion abou a hidden pa ame e in
A
. Smalle
σ
co esponds
o la ge γ.
2.
Ene gy/ empe a u e: bound o al ene gy in
A
by ixing d i e ampli ude and du y cycle;
eco d local ene gy densi y εA.
Measu ed quan i ies.
1.
Inwa d p ecision: es ima e classical Fishe in o ma ion
F
(
σ
) om ou comes in
A
abou he
hidden pa ame e .
2.
Ou wa d capaci y: p epa e codewo ds in
A
and es ima e an achie able classical capaci y o
B
o e ime window
τ
ia he Hole o quan i y
χτ
(o mu ual in o ma ion om omog aphy).
De ine Cou (σ) := χτ/τ.
P edic ion ( ini e- esou ce speed limi ). Fo ixed (
R, εA, τ
)wi h
D > LRτ
and weak
leakage o B:
∂Cou
∂σ ≤0,∂Cou
∂σ <0once F(σ)exceeds he sho -noise scale.(14)
Ope a ionally: pushing
σ↓
0(s onge local eadou /p ocessing in
A
) educes cohe en signal
eaching
B
wi hin
τ
because (i) measu emen backac ion and con ol pulses decohe e ca ie s,
(ii) ini e
LR
caps usable modes, and (iii) ixed ene gy budge is ealloca ed om communica ion
o me ology.
How o un. Supe conduc ing qubi s o cold a oms su ice.
1. Calib a e LR by ligh -cone omog aphy.
2. Choose R, D, τ so ha D≳ LRτ.
3.
Sweep
γ
o a y
σ
; a each se ing, (i) es ima e
F
(
σ
)in
A
, (ii) es ima e
Cou
(
σ
) o
B
ia
epea ed codewo d ansmission and omog aphy.
4.
Plo
Cou
s.
F
(o
σ
). NFI p edic s a descending adeo cu e and a knee whe e s onge
me ology penalizes comms.
7
C. Falsi iabili y Checklis
A single empi ical/ heo e ical coun e example e u es NFI:
1.
A ixed (
R, E, T
)and ixed geome y/couplings, demons a e a egime whe e dec easing
σ
(inc easing
F
)inc eases o lea es unchanged
Cou
o e ixed
τ
o dis an
B
, wi hou hidden
esou ce injec ion o long- ange couplings.
2.
Exhibi a sus ainable supe luminal channel o a capaci y ha g ows as ene gy densi y
app oaches a collapse h eshold once back eac ion is included.
Rema k 10.1 (Ca ea s & scope).(i) Bekens ein- ype bounds a e widely suppo ed bu no
p o en in ull gene ali y; we use hem as an en elope (see also Casini’s ela i e-en opy p oo and
co a ian bounds [
3
,
4
]). (ii) Hawking adia ion suppo s he blu pic u e: emission is e ec i ely
sou ced in an ex ended nea -ho izon egion, so he quan um blu does no e mina e a a sha p
su ace. Ene gy—and, in uni a y accoun s, co ela ions/in o ma ion—can leak ou wa d, bu he
a e is ex emely small (g eybody il e ing, edshi ) and decoding is belie ed ha d [
14
,
15
,
16
,
17
].
Ou capaci y no ion conce ns ini e- ime, ini e- esou ce communica ion; hus no con adic ion
wi h NFI a ises. (iii) Exo ic nega i e-ene gy e ec s a e cons ained by quan um inequali ies;
sus ained iola ions needed o aise Cou a e no known o be possible [18].
Rema k 10.2 (Compu a ional co olla y: link o
P
s.
NP
).In a physically local model
wi h ini e signal speed, “ e i ica ion” can be local while “disco e y” mus espec ligh -cone
p opaga ion. High blu can make
P
look like
NP
on noisy ins ances, bu unblu ing ine i ably
incu s ene gy/ ime/causal cos s. Ope a ionally, he p esen No-F ee-In o ma ion law is he
guiding p inciple behind ha compu a ional gap: making disco e y as cheap as e i ica ion
would equi e sh inking
σ
wi hou paying he capaci y/collapse cos s ha codi y he speed o
ligh .
Acknowledgmen
This is a concep ual syn hesis; o mulas a e used a he le el o guiding bounds a he han
p ecision equali ies. Fo mal de i a ions and cons an s can be uned wi hou al e ing he
ope a ional con en .
Re e ences
[1] J. D. Bekens ein, “Black holes and en opy,” Phys. Re . D 7, 2333–2346 (1973).
[2]
J. D. Bekens ein, “Uni e sal uppe bound on he en opy- o-ene gy a io o bounded
sys ems,” Phys. Re . D 23, 287 (1981).
[3]
H. Casini, “Rela i e en opy and he Bekens ein bound,” Class. Quan um G a . 25, 205021
(2008).
[4] R. Bousso, “A co a ian en opy conjec u e,” J. High Ene g. Phys. 07, 004 (1999).
[5]
S. W. Hawking, “Pa icle c ea ion by black holes,” Commun. Ma h. Phys. 43, 199–220
(1975).
[6]
J. M. Ba deen, B. Ca e , and S. W. Hawking, “The ou laws o black hole mechanics,”
Commun. Ma h. Phys. 31, 161–170 (1973).
[7]
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).
8
[8]
C. H. Benne , “Logical e e sibili y o compu a ion,” IBM J. Res. De . 17, 525–532 (1973).
[9]
L. Mandels am and I. Tamm, “The unce ain y ela ion be ween ene gy and ime in
non ela i is ic quan um mechanics,” J. Phys. (USSR) 9, 249 (1945).
[10]
N. Ma golus and L. B. Le i in, “The maximum speed o dynamical e olu ion,” Physica D
120, 188–195 (1998).
[11]
E. H. Lieb and D. W. Robinson, “The ini e g oup eloci y o quan um spin sys ems,”
Commun. Ma h. Phys. 28, 251–257 (1972).
[12] L. Susskind, “The wo ld as a holog am,” J. Ma h. Phys. 36, 6377–6396 (1995).
[13]
R. Bousso, Z. Fishe , J. Koelle , S. Leichenaue , and A. C. Wall, “P oo o he Quan um
Null Ene gy Condi ion,” Phys. Re . D 93, 024017 (2016).
[14]
D. N. Page, “Pa icle emission a es om a black hole: massless pa icles om an uncha ged,
non o a ing hole,” Phys. Re . D 13, 198–206 (1976).
[15] D. N. Page, “In o ma ion in black hole adia ion,” Phys. Re . Le . 71, 3743–3746 (1993).
[16]
P. Hayden and J. P eskill, “Black holes as mi o s: quan um in o ma ion in andom
subsys ems,” J. High Ene g. Phys. 09, 120 (2007).
[17]
D. Ha low and P. Hayden, “Quan um compu a ion s. i ewalls,” J. High Ene g. Phys. 06,
085 (2013).
[18]
L. H. Fo d and T. A. Roman, “Quan um ield heo y cons ains a e sable wo mhole
geome ies,” Phys. Re . D 53, 5496–5507 (1996).
[19] A. Pe išić, Blu , Quan a, and he Obse e , Zenodo, 2025.
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