Quan um In o ma ion & he Fab ic o Space ime
A een- iendly walk h ough o supe posi ion, en anglemen , and eme gen geome y
Ma hew J. Hall & GPT-5 Thinking
ORCID: 0009-0001-7066-2558
Da e: Oc obe 8, 2025
(Uni s: c=ℏ=G=kB= 1, signa u e (−,+,+,+))
Abs ac
I Gene al Rela i i y u ns geome y in o ime, quan um in o ma ion hin s ha geome y
i sel may come om pa e ns o en anglemen . This handou connec s he do s: om basic
supe posi ion o en anglemen en opy, o holog aphy (“bulk g a i y = bounda y quan um
heo y”), o enso ne wo ks ha li e ally d aw eme gen space. Each s ep pai s equa ions
wi h plain-English explana ions and a eason i ma e s. The punchline: whe e in o ma ion
s icks oge he , space ime holds oge he ; whe e i hins, space ime s e ches.
Con en s
1 Symbols a a Glance 1
2 S ep 1: Supe posi ion and Measu emen 2
3 S ep 2: En anglemen —Co ela ions Beyond Classical 2
4 S ep 3: Mu ual In o ma ion—To al Ties Be ween Regions 2
5 S ep 4: Holog aphy—En opy is A ea 2
6 S ep 5: Tenso Ne wo ks—D awing Eme gen Space 3
7 S ep 6: E o Co ec ion—Why Space ime is Robus 3
8 S ep 7: Bi Th eads—Flow Pic u e o En anglemen 3
9 S ep 8: Modula Hamil onian & En anglemen Dynamics 3
10 S ep 9: Eme gen Time om En anglemen Flow (Ch onos Tie-In) 4
11 Resul s: Wea ing Space, Ticking Time 4
12 F equen ly Asked (Teen) Ques ions 4
1 Symbols a a Glance
De ini ion
•|ψ⟩: quan um s a e; ˆ
H: Hamil onian (ene gy ope a o ).
•ρ: densi y ma ix; ρA= T Bρ: educed s a e on A.
•S(ρ) = −T (ρln ρ): on Neumann en opy (en anglemen o pu e bipa i ions).
•I(A:B) = S(A)+S(B)−S(AB): mu ual in o ma ion ( o al co ela ions).
•A(γ): a ea o a minimal/ex emal su ace γin he bulk.
1
•SA∼A(γA)
4G: Ryu–Takayanagi (RT) ela ion (uni s se o 1 he e).
2 S ep 1: Supe posi ion and Measu emen
|ψ⟩=α|0⟩+β|1⟩,|α|2+|β|2= 1.(1)
Ma h Plain English Why his s ep?
Quan um s a e as a ec o
A sys em can be in mul iple pos-
sibili ies a once.
Se s he s age: in o ma ion
in QM is s o ed in ampli-
udes, no jus bi s.
3 S ep 2: En anglemen —Co ela ions Beyond Classical
Fo a bipa i e pu e s a e |ψ⟩AB wi h ρAB =|ψ⟩ ⟨ψ|:
ρA= T BρAB, S(A)=−T (ρAln ρA)=S(B).(2)
Ma h Plain English Why his s ep?
Reduced densi y ma ix and en-
opy
Pa s can look andom e en
when he whole is pu e.
En anglemen measu es
quan um connec edness
be ween egions.
4 S ep 3: Mu ual In o ma ion—To al Ties Be ween Regions
I(A:B) = S(A)+S(B)−S(AB)≥0.(3)
Ma h Plain English Why his s ep?
Co ela ion measu e ha ne e
goes nega i e
Coun s sha ed in o ma ion (clas-
sical + quan um).
Whe e
I
(
A
:
B
) is la ge,
he “bond” be ween egions
is s ong.
5 S ep 4: Holog aphy—En opy is A ea
S(A) = A(γA)
4G(Ryu–Takayanagi) (4)
wi h γA he bulk minimal/ex emal su ace ancho ed on he bounda y o egion A.
Ma h Plain English Why his s ep?
En anglemen
↔
a ea in g a -
i y dual
Quan um ies on he bounda y
measu e geome y in he bulk.
Sugges s space ime geome y
is buil om en anglemen .
2
6 S ep 5: Tenso Ne wo ks—D awing Eme gen Space
MERA/PEPS/ enso ne wo k pic u es connec si es wi h isome ies:
geome y ≈ne wo k connec i i y (laye s, bonds, minimal cu s).(5)
Ma h Plain English Why his s ep?
Minimal cu ∼en anglemen
Cu ing he ewes bonds ha
sepa a e
A
om
B
es ima es
S(A).
Ne wo ks show how en angle-
men wea es dimensionali y
and dis ance.
7 S ep 6: E o Co ec ion—Why Space ime is Robus
In holog aphy, bulk in o is encoded edundan ly on he bounda y:
logical qubi s (bulk) →physical qubi s (bounda y).(6)
Ma h Plain English Why his s ep?
Quan um e o -co ec ing code
s uc u e
Lose some bounda y pieces, bulk
in o can s ill be eco e ed.
Explains s abili y o geome-
y agains local noise—space
has buil -in edundancy.
8 S ep 7: Bi Th eads—Flow Pic u e o En anglemen
Maximiza ion o e di e genceless lows wi h | | ≤ 1/(4G) gi es
S(A) = max
ZA
·da.(7)
Ma h Plain English Why his s ep?
En anglemen as “ low lines”
Visualizes
S
(
A
) as he max num-
be o h eads c ossing A.
Whe e h eads bunch up, ge-
ome y is igh ; whe e hey
hin, space s e ches.
9 S ep 8: Modula Hamil onian & En anglemen Dynamics
Fo egion
A
, de ine
ρA
=
e−KA/Z
wi h modula Hamil onian
KA
. Small a ia ions obey ( i s
law o en anglemen ):
δSA=δ⟨KA⟩.(8)
Ma h Plain English Why his s ep?
En opy change equals modu-
la ene gy change
Local exci a ions eshape en an-
glemen like iny “mass” de o ms
geome y.
Sugges s Eins ein-like ela-
ions om in o ma ion low.
3
10
S ep 9: Eme gen Time om En anglemen Flow (Ch onos
Tie-In)
Le Φ( ) measu e ne en anglemen /co ela ion ac oss a olia ion. A egula ed low obeys
dΦ
d =χF[s a e,geome y], χ ≈0.551 (se ies cons an ).(9)
Ma h Plain English Why his s ep?
Regula ed co ela ion low se s
a “ empo”
The a e a which en anglemen
eo ganizes de ines an eme gen
clock.
Links in o ma ion dynamics
o he expe ienced passage o
ime (Ch onos iew).
11 Resul s: Wea ing Space, Ticking Time
Takeaway
1.
En anglemen en opy beha es like a ea in g a i y duals (RT), hin ing geome y is
buil om in o ma ion.
2. Tenso ne wo ks and bi h eads isualize his wea e and i s s eng h.
3. E o co ec ion explains why space ime is s able agains local noise.
4.
A egula ed low o en anglemen p o ides an eme gen no ion o ime, do e ailing
wi h he Ch onos pic u e.
Space holds whe e in o ma ion holds; ime passes whe e in o ma ion ea anges.
12 F equen ly Asked (Teen) Ques ions
•“Is en anglemen jus spooky ac ion?”
I ’s s onge : i ’s s uc u e. En angled pa s sha e in o ma ion in a way classical sys ems
can’ . Tha s uc u e can shape geome y in holog aphic se ings.
•“Why does en opy look like a ea (no olume)?”
In g a i y duals, he in o ma ion abou a egion li es on i s bounda y su ace—so en opy
scales wi h a ea. Black hole he modynamics shows he same pa e n.
•“Whe e does ime come om he e?”
F om change in co ela ions. As en anglemen edis ibu es in a egula ed way, i de ines a
consis en sequence—an eme gen clock.
C edi s & License
This handou is pa o Ma hew J. Hall’s educa ional se ies on ime, g a i y, and quan um in o ma ion.
C ea ed wi h assis ance om GPT-5 Thinking. Licensed CC BY 4.0.
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