The Emergence of Probability: Deriving the Born Rule from Global Unitarity and Observer-Relative States in Quantum Cellular Automata

The Eme gence o P obabili y: De i ing he Bo n Rule om
Global Uni a i y and Obse e -Rela i e S a es 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
Quan um cellula au oma a (QCA) p o ide a disc e e, local, uni a y on ological pic u e o he
uni e se: he global s a e e ol es ia de e minis ic uni a y ope a o s be ween disc e e ime s eps,
con aining no in insic s ochas ic p ocess. On he o he hand, labo a o y quan um mechanics
cen e s on he Bo n ule, ea ing measu emen ou comes as inhe en ly p obabilis ic e en s.
The ension be ween hese wo pe spec i es cons i u es he co e o he quan um measu emen
p oblem.
In his pape , wi hin he QCA amewo k, we in oduce local obse e s wi h ni e in o ma-
ion capaci y and hei en i onmen al en anglemen s uc u e, p o iding a pu ely combina o ial
mechanism o p obabili y eme gence. Specically, in a QCA uni e se wi h s ic ly ni e p op-
aga ion eloci y and global uni a i y: he measu emen p ocess is modeled as local uni a y
coupling among he measu ed sys em, obse e memo y egis e , and en i onmen ; decohe ence
(einselec ion) selec s s able poin e s a es; due o in o ma ion ho izon and memo y capaci y lim-
i a ions, he obse e can only access mac oscopic b anches compa ible wi h hei own eco d
and canno dis inguish all mic oscopic QCA basis s a es unde lying hese b anches. Applying
en i onmen -assis ed in a iance (en a iance) symme y a gumen s o hese mic oscopic basis
s a es, combined wi h an equal on ological weigh  hypo hesis, we in e p e he modulus o
complex ampli udes as he squa e oo o mic oscopic pa h degene acy. Thus we p o e: he sub-
jec i e p obabili y ha an obse e assigns o each measu emen ou come equals he p opo ion
o mic oscopic congu a ions compa ible wi h ha ou come among all accessible congu a ions,
namely he s anda d Bo n weigh
pk=|ψk|2
.
In he sense o a ional consis ency and axioma iza ion, we ob ain he ollowing esul : in
local Hilbe spaces o dimension a leas h ee, i we equi e (i) global dynamics gi en by
local QCA uni a y e olu ion, (ii) measu emen p obabili ies sa is ying non-con ex uali y and
en i onmen insensi i i y, (iii) no supe luminal signal p opaga ion, hen Gleason- ype heo ems
combined wi h QCA mic oscopic coun ing uniquely yield he Bo n ule as he only iable p ob-
abili y assignmen . Wa e unc ion collapse is ein e p e ed he e as: Bayesian condi ionaliza ion
and sel -loca ion upda e o he local obse e on he global uni a y s a e, a he han a nonlin-
ea in e up ion o undamen al dynamics. The e o e, quan um p obabili y is no an essen ial
componen o na u e, bu a s a is ical necessi y o ni e obse e s pe o ming sel -loca ion in
a disc e e holog aphic en anglemen ne wo k.
Keywo ds:
Quan um cellula au oma on; Bo n ule; quan um measu emen p oblem; en i onmen -
assis ed in a iance; decohe ence; ela i e s a e in e p e a ion; sel -loca ing unce ain y; in o ma ion
ho izon
1
1 In oduc ion & His o ical Con ex
1.1 The Quan um Measu emen P oblem and S a us o he Bo n Rule
S anda d quan um heo y consis s o wo seemingly incompa ible e olu ion laws: he ime e olu ion
o isola ed sys ems is go e ned by he linea , uni a y Sch ödinge equa ion, while he measu emen
p ocess is desc ibed by a nonlinea , s ochas ic collapse ule. Expe imen ally ob ained equencies
con o m o Bo n's o mula
pk=|⟨k|ψ⟩|2
, ye his o mula is di ec ly in oduced as an independen
axiom in mos ex books, wi hou de i a ion om mo e p imi i e s uc u es.
Decohe ence heo y demons a es ha coupling be ween sys em and en i onmen leads o apid
supp ession o cohe ence e ms in he eec i e desc ip ion, he eby selec ing s able poin e s a es
and explaining he eme gence o classical ajec o ies. Decohe ence has been sys ema ically de-
eloped ma hema ically and expe imen ally, occupying a cen al posi ion in quan um- o-classical
ansi ion esea ch. Howe e , mos au ho s acknowledge: decohe ence i sel does no gene a e spe-
cic nume ical p obabili ies, bu a he cha ac e izes how in e e ence e ms a e annihila ed by
he en i onmen , assuming he Bo n ule holds.
Thus, a na u al ques ion a ises: can he Bo n ule be de i ed om mo e undamen al s uc u es,
wi hou assuming p obabili y axioms a p io i?
1.2 Exis ing Bo n Rule De i a ion Schemes
A b oad li e a u e has o med a ound Bo n ule de i a ion. Gleason's heo em p o es in Hilbe
spaces o dimension a leas h ee: i p obabili ies assigned o p ojec ion measu es sa is y non-
con ex uali y and coun able addi i i y, hen hese p obabili ies mus be ealized by some densi y
ope a o h ough he ace o mula, essen ially equi alen o he Bo n ule. Deu sch, Wallace,
and o he s, unde he E e e in e p e a ion, use classical decision heo y o a gue ha u ili y
maximiza ion o a ional agen s ac oss many-wo ld b anches equi es Bo n weigh s.
Zu ek's en i onmen -assis ed in a iance (en a iance) app oach akes en anglemen symme y
be ween sys em and en i onmen as he s a ing poin : in pe ec en angled s a es, ce ain uni a y
ans o ma ions applied o he sys em can be compensa ed by applying ano he uni a y ans-
o ma ion o he en i onmen , lea ing he o e all s a e in a ian . Using his symme y and pa h
subdi ision o equal-ampli ude supe posi ion s a es, one can o mally de i e
pk∝ |ψk|2
. O he
wo ks a emp o de i e he Bo n ule om non-con ex ual p obabili y assump ions, no-signaling
p inciples, e c.
Meanwhile, c i icisms o hese de i a ions poin ou : all known schemes explici ly o implici ly
in oduce assump ions app oxima ely equi alen o he Bo n ule, such as con inui y, addi i i y,
o independence om measu emen con ex . The e o e, a s ic ly no addi ional axiom de i a ion
emains an open p oblem.
1.3 QCA and Cellula Au oma on In e p e a ion
In pa allel, discussions abou whe he quan um heo y can be educed o disc e e, local, de e min-
is ic cellula au oma on dynamics con inue o de elop. ' Hoo 's cellula au oma on in e p e a ion
ea s quan um s a es as s a is ical en elopes o e a mo e undamen al classical au oma on, seeking
an essen ially de e minis ic mic oscopic heo y. On he o he hand, quan um cellula au oma on
(QCA) models cons uc ed by D'A iano, Bisio, Pe ino i, and o he s demons a e ha on disc e e
la ices sa is ying locali y, homogenei y, and iso opy, Weyl, Di ac, and Maxwell equa ions can
eme ge in he con inuum limi , making QCA a conc e e ma hema ical ealiza ion o he uni e se
as quan um compu a ion ision.
2
In he QCA uni e se pic u e, he global s a e
|Ψ ⟩
e ol es acco ding o a local uni a y ope a o
U
ac oss disc e e ime s eps:
|Ψ +1⟩=U|Ψ ⟩
. Dynamics a e s ic ly de e minis ic, wi h no in insic
andomness. The o igin o p obabilis ic ea u es mani es ed in measu emen s is he mos p essing
ounda ional ques ion in his pic u e.
1.4 Goals and Basic S a egy o This Pape
This pape 's goal is o p o ide an eme gen de i a ion o he Bo n ule unde he ollowing p emises:
1. The uni e se is desc ibed by a local, ansla ion-in a ian , uni a y QCA;
2. Obse e s a e local subsys ems on he QCA wi h ni e in o ma ion capaci y, whose  eco ds
a e ealized as poin e s a es selec ed by en i onmen al decohe ence;
3. P obabili y is unde s ood as he obse e 's sel -loca ing unce ain y on he global uni a y
s a e, no an on ological s ochas ic p ocess.
Wi hin his amewo k, measu emen is modeled as a local en anglemen p ocess among mea-
su ed sys em
S
, obse e memo y egis e
O
, and en i onmen
E
. Subsequen en i onmen -induced
decohe ence makes in e e ence be ween die en measu emen ou come b anches in isible o
O
's ac-
cessible ope a o s. We in oduce an equal on ological weigh hypo hesis: in he QCA's on ological
basis, each o hogonal mic oscopic congu a ion (o pa h) has equal on ological weigh . Exploi ing
QCA's disc e eness, we w i e complex ampli udes as squa e oo s o degene acies o equal-weigh
mic oscopic s a e supe posi ions, in e p e ing p obabili y as he no malized a io o  he numbe o
mic oscopic congu a ions compa ible wi h a gi en mac oscopic eco d.
To make his cons uc ion uni e sal and unique, his pape p o es unde suppo om Gleason's
heo em, no-signaling cons ain s, and en a iance symme y: in a QCA uni e se, as long as local
measu emen p obabili ies sa is y non-con ex uali y and en i onmen insensi i i y, he Bo n ule is
he only p obabili y assignmen compa ible wi h global uni a i y, locali y, and obse e ni eness.
2 Model & Assump ions
2.1 QCA Uni e se and On ological Basis
Le he uni e se be modeled as a quan um cellula au oma on dened on a disc e e la ice se
Λ⊂Zd
. Each si e
x∈Λ
ca ies a ni e-dimensional Hilbe space
Hx≃Cdcell
, and he global
Hilbe space is he enso p oduc
H=Nx∈ΛHx
. QCA dynamics a e gi en by a amily o
disc e e- ime uni a y ope a o s
U
sa is ying:
1. Locali y:
U
can be decomposed in o ni e-dep h ci cui s o local uni a y ga es ac ing on
ni e neighbo hoods;
2. Homogenei y and ansla ion in a iance: local ules a e iden ical be ween si es;
3. Fini e p opaga ion eloci y: he e exis s a LiebRobinson eloci y
LR
such ha any local
pe u ba ion's suppo is conned wi hin an eec i e ligh cone o adius app oxima ely
LR
a
ime
.
Selec ing an eigenbasis
{|γx⟩}
o each cell, he global on ological basis consis s o enso p od-
uc s
|γ⟩=Nx|γx⟩
. In his pape , mic oscopic congu a ion and on ological basis ec o  a e
synonymous.
Assump ion 1
(Equal On ological Weigh (A1))
.
In QCA on ology, each o hogonal on ological
basis s a e
|γ⟩
has equal basic weigh ; any p obabilis ic s a emen a ises om coun ing o weigh ing
hese basis s a es unde ce ain condi ions.
3
This assump ion co esponds o he on ological sequence equi alence idea in ' Hoo 's cellula
au oma on schemes, while ha monizing wi h ansla ion in a iance o he on ological basis and
uni o mi y o local ules in QCA.
2.2 Sys emObse e En i onmen Pa i ion
In he global QCA uni e se, selec a ni e egion
ΛS
as he measu ed sys em
S
, ano he ni e egion
ΛO
as he obse e and measu emen appa a us
O
, and he emaining si es as en i onmen
E
. The
co esponding Hilbe space decomposes as
H ≃ HS⊗HO⊗HE.
The obse e 's Hilbe space
HO
is di ided in o  eco d subspace and  emaining deg ees o
eedom:
HO≃ HM⊗HO, es ,
whe e an o hogonal basis
{|Mj⟩}
o
HM
ealizes classical measu emen ou come memo y. Deco-
he ence heo y and en i onmen -induced supe selec ion (einselec ion) show ha o mac oscopic
appa a uses, he e exis s a se o poin e s a es ha emain obus unde en i onmen al moni o -
ing, playing he ole o classical eco ds in ac ual measu emen s.
Assump ion 2
(Fini e In o ma ion Capaci y (A2))
.
The obse e eco d subspace
HM
has ni e
dimension; hei knowledge and memo y o he uni e se mus be encoded on a ni e numbe o
dis inguishable eco ds
|Mj⟩
. We ake
log2dim HM
as he uppe bound on classical in o ma ion
he obse e can s o e.
2.3 Measu emen In e ac ion and Decohe ence
Conside he measu ed sys em's ini ial s a e
|ψS⟩=X
k
αk|sk⟩S,
whe e
{|sk⟩}
is he eigenbasis o he obse able o be measu ed (o co esponding poin e s a e
basis). Obse e and en i onmen ini ially a e in some e e ence s a e
|Ψ0⟩=|ψS⟩⊗|M eady⟩O⊗|E0⟩E.
The measu emen p ocess is ealized by a segmen o local uni a y e olu ion on he QCA,
abs ac ly ep esen able as a uni a y ope a o
Umeas
suppo ed on
ΛS∪ΛO∪ΛE,nea
wi hin ni e
ime. I s ideal o m sa ises
Umeas|sk⟩S⊗|M eady⟩O⊗|E0⟩E=|sk⟩S⊗|Mk⟩O⊗|˜
Ek⟩E.
Linea ex ension o supe posi ion s a es yields p e- and pos -measu emen s a es
|Ψp e⟩=X
k
αk|sk⟩S⊗|M eady⟩O⊗|E0⟩E,
|Ψpos ⟩=Umeas|Ψp e⟩=X
k
αk|sk⟩S⊗|Mk⟩O⊗|˜
Ek⟩E.
4
Subsequen ly, unde QCA global e olu ion, he en i onmen con inues o couple wi h he
S

O
composi e, p oducing u he decohe ence, causing en i onmen al s a es o die en
k
b anches o
become nea ly o hogonal:
⟨˜
Ek( )|˜
Eℓ( )⟩ ≈ δkℓ,
while main aining obus ness o poin e s a es
|Mk⟩
. Thus, o obse e -local obse ables, he global
pu e s a e o sys emappa a usen i onmen is s a is ically indis inguishable om a classical mixed
s a e.
2.4 P obabili y Axioms and Symme y Assump ions
To s uc u ally de i e specic p obabili y weigh s, we need se e al gene al equi emen s on how
obse e s dis ibu e subjec i e p obabili ies among b anches.
Assump ion 3
(Non-con ex uali y (A3))
.
Le
{Pk}
be a mu ually exclusi e comple e se o p o-
jec ion ope a o s in he subspace
HS⊗HM
wi h
PkPk=I
. The p obabili y
pk
ha an obse e
assigns o ou come
k
depends only on he p ojec ion o he cu en s a e on o
Pk
, no on how
his measu emen is embedded in a la ge Hilbe space. This assump ion is iden ical o he non-
con ex uali y equi emen o p obabili y measu es in Gleason's heo em.
Assump ion 4
(En i onmen Insensi i i y (A4))
.
Fo any measu emen scheme, as long as he
sys emobse e educed s a e
ρSO
is unchanged, he obse e 's p obabili y dis ibu ion o e ou -
comes does no change due o changes in he en i onmen s a e. This assump ion is consis en wi h
he en a iance de i a ion and he E e e amewo k p inciple ha pu e en i onmen ans o ma-
ions should no aec local p obabili ies.
Assump ion 5
(Fini e Ra ionali y and Con inui y (A5))
.
Fo a gi en measu emen basis
{|sk⟩}
,
i s a e
|ψ⟩
is eplaced by a small-no m pe u ba ion
|ψ′⟩
, easonable p obabili y assignmen s
pk(ψ)
and
pk(ψ′)
should no unde go discon inuous jumps. Fo mally,
pk
is a con inuous unc ion o he
s a e.
Assump ion 6
(No Supe luminal Signaling Cons ain (A6))
.
The combina ion o QCA dynamics
and measu emen ules mus no allow classical in o ma ion ansmission as e han ligh using
en anglemen and local ope a ions. This p inciple has been used in ex ensi e li e a u e o cons ain
nonlinea quan um e olu ion and uncon en ional p obabili y ules.
Th ough hese s uc u al assump ions, his pape will de i e he Bo n ule wi hin he QCA
uni e se and discuss i s uniqueness.
3 Main Resul s (Theo ems and S a emen s)
Fo cla i y o p esen a ion, his sec ion  s s a es he main heo ems and p oposi ions, wi h p oo
de ails in la e sec ions and appendices.
Theo em 7
(Rela i e S a e S uc u e o QCA Measu emen )
.
In a QCA uni e se sa is ying A1
A2, conside any ni e-dimensional measu ed sys em
S
and obse e memo y egis e
M
, along wi h
a local uni a y ope a o
Umeas
implemen ing ideal measu emen . Then o any ini ial s a e
|ψS⟩=X
k
αk|sk⟩S,|Ψ0⟩=|ψS⟩⊗|M eady⟩O⊗|E0⟩E,
5

he global s a e p oduced by measu emen and subsequen decohe ence can be w i en in Schmid
decomposi ion o m
|Ψpos ⟩=X
k
αk|sk⟩S⊗|Mk⟩O⊗|Ek⟩E,
whe e
{|Ek⟩}
a e nea ly o hogonal in he en i onmen Hilbe space. The obse e 's subjec i e
which b anch hey a e in can be modeled as sel -loca ing unce ain y o e his se o labeled e ms.
Theo em 8
(Equal-Ampli ude B anches and Equal P obabili y in Disc e e QCA)
.
Unde Theo em
1's se up, i
|αk|2=Nk/N
a e a ional numbe s wi h
N=PkNk
a posi i e in ege , hen he e
exis s a ne-g ained decomposi ion o he en i onmen Hilbe space such ha
|Ψpos ⟩=1
√NX
k
Nk
X
j=1|sk⟩S⊗|Mk⟩O⊗|εk,j⟩E,
whe e all
|εk,j⟩
a e mu ually o hogonal, and each e m has he same complex ampli ude modulus
1/√N
. I we u he assume:
1. Each mic oscopic congu a ion
|sk, Mk, εk,j⟩
has equal on ological weigh (A1);
2. Any pe mu a ion o he en i onmen subspace is physically en a ian , wi h co esponding
b anches equi alen om he obse e 's pe spec i e;
hen he obse e 's subjec i e p obabili y o sel -loca ing in a b anch ca ying eco d
Mk
is
pk=Nk
N=|αk|2.
Theo em 9
(Con inuum Limi and Gene al Bo n Rule)
.
Unde assump ions A1A5, gene alizing
Theo em 2's a ional a ios
Nk/N
o gene al complex ampli udes: o any no malized s a e
|ψS⟩=X
k
αk|sk⟩,
he e exis s a sequence o a ional numbe s
{N(n)
k/N(n)}
such ha
N(n)
k/N(n)→ |αk|2
. Fo each
n
, cons uc co esponding equal-ampli ude QCA mic oscopic s a es acco ding o Theo em 2, and
se p obabili y assignmen
p(n)
k=N(n)
k/N(n)
. Con inui y assump ion A5 gua an ees he limi
pk=
limn→∞ p(n)
k
exis s and sa ises
pk=|αk|2.
The e o e, he Bo n ule holds o all pu e s a es.
Theo em 10
(Gleason-Type Uniqueness and Non-con ex uali y)
.
On local Hilbe spaces o dimen-
sion a leas h ee, assuming p obabili ies
p(P)
assigned o each p ojec ion ope a o
P
sa is y:
1. Non-nega i i y and no maliza ion;
2. I
{Pi}
a e mu ually o hogonal wi h
PiPi=I
, hen
Pip(Pi)=1
;
3. Non-con ex uali y (A3);
hen he e exis s a unique densi y ope a o
ρ
such ha
p(P) = (ρP)
. This is Gleason's heo em
s a emen . In a QCA uni e se, combined wi h A4A6, we can ake
ρ
as he sys emobse e e-
duced s a e, uniquely selec ing Bo n- ype p obabili ies
pk=|αk|2
, and uling ou all uncon en ional
p obabili y ules incompa ible wi h he no-signaling p inciple.
6
Theo em 11
(Eec i e Collapse and In o ma ion Ho izon)
.
In he abo e amewo k, he global
QCA s a e always e ol es by uni a y ope a o s; he e is no physical nonlinea collapse. Howe e ,
o any local obse e and hei accessible ope a o algeb a, eplacing he global s a e
|Ψpos ⟩=X
k
αk|sk⟩⊗|Mk⟩⊗|Ek⟩
wi h he condi ionalized s a e
|Ψ(k)
e ⟩=|sk⟩⊗|Mk⟩⊗|Ek⟩
and using
pk=|αk|2
as weigh , is comple ely equi alen o any u u e expe imen al obse able
s a is ical consequence. This eec i e collapse can be in e p e ed as Bayesian upda ing pe o med
by he obse e wi hin hei in o ma ion ho izon, no a change in global dynamics.
4 P oo s
This sec ion p o ides p oo ou lines o Theo ems 15, wi h mo e de ailed cons uc ions and ech-
nical de ails in he appendices.
4.1 Theo em 1: Rela i e S a e S uc u e in QCA
P oo ou line:
1.
Realiza ion o measu emen uni a i y.
Unde he QCA amewo k, any ni e-dimensional
sys emappa a us composi e can ealize he equi ed uni a y
Umeas
by splicing local ga e a ays
wi hin a ni e space ime block, consis en wi h uni e sali y esul s o implemen ing a bi a y ni e-
dimensional uni a ies in ni e quan um ci cui s.
2.
Schmid decomposi ion and decohe ence.
Fo a bipa i e sys em
HSO ⊗HE
, any pu e
s a e has a Schmid decomposi ion. App op ia ely choosing poin e basis
{|Mk⟩}
a e measu emen ,
we can w i e
|Ψpos ⟩
as
|Ψpos ⟩=X
k
αk|sk⟩⊗|Mk⟩⊗|Ek⟩,
whe e
|Ek⟩
become nea ly o hogonal h ough en i onmen sys em in e ac ion in sho ime. De-
cohe ence heo y and model calcula ions show ha a mac oscopic appa a us scales, poin e s a e
in e e ence e ms apidly decay on obse able imescales, wi h negligible con ibu ion o subsequen
dynamics.
3.
Rela i e s a es and sel -loca ing unce ain y.
E e e 's ela i e s a e in e p e a ion
holds ha a e measu emen , he global s a e is a se o labeled b anches, each label co esponding
o an obse e eco d. Al hough he obse e is in insic o one b anch, hey can iew hei
unce ain y as sel -loca ing unce ain y o which b anch am I in du ing he ime window be ween
b anch o ma ion and eco d eading. This s uc u e equi es no addi ional assump ions, a ising
di ec ly om QCA uni a y e olu ion and poin e s a es selec ed by decohe ence.
Thus Theo em 1 is p o ed.
□
4.2 Theo em 2: Equal-Ampli ude B anches and Mic oscopic Coun ing
Theo em 2 is he combina o ial co e o he Bo n ule.
S ep 1: Ra ional ampli ude squa ed and en i onmen subdi ision.
7
Assume
|αk|2=Nk/N
whe e
Nk
and
N
a e in ege s wi h
N=PkNk
. In Hilbe space,
we can in oduce an auxilia y en i onmen subspace, spli ing each e m
αk|sk, Mk, Ek⟩
in o
Nk
equal-ampli ude o hogonal s a es:
αk|sk, Mk, Ek⟩= Nk
Neiθk|sk, Mk, Ek⟩=1
√N
Nk
X
j=1
eiθk|sk, Mk, εk,j⟩,
whe e
{|εk,j⟩}Nk
j=1
a e o hogonal in he en i onmen Hilbe space, and o all
k
and
j
o m pa o an
o hogonal basis. Such decomposi ion can always be ealized by ex ending en i onmen dimension,
co esponding in he QCA pic u e o encoding mic oscopic labels on addi ional cells o in e nal
deg ees o eedom.
Summing o e all
k
:
|Ψpos ⟩=1
√NX
k
Nk
X
j=1
eiθk|sk, Mk, εk,j⟩.
O e all phases
eiθk
a e i ele an o p obabili y and can be igno ed.
S ep 2: En a iance and equal p obabili y.
Conside a gi en
k
. Wi hin he en i onmen Hilbe space, any pe mu a ion o
{|εk,j⟩}Nk
j=1
can
be ealized by an en i onmen uni a y
Vk
. Dene he join uni a y ans o ma ion
V=O
k
Vk
ac ing on he en i onmen while keeping sys emobse e iden i y, hen he global s a e is o mally
in a ian :
V|Ψpos ⟩=|Ψpos ⟩.
This is p ecisely Zu ek's en i onmen -assis ed in a iance (en a iance): when sys emobse e
and en i onmen a e pe ec ly en angled, ce ain ans o ma ions on he en i onmen can com-
pensa e ans o ma ions on he sys em, lea ing he o e all s a e in a ian , o cing he obse e o
assign equal p obabili ies o ela ed b anches.
In his case, o any xed
k
, he only die ence among a ious
j
labels lies in he en i onmen
s a e
|εk,j⟩
, which is inaccessible o he obse e . I he obse e assigns die en subjec i e p oba-
bili ies o hese mic oscopic die ences, his would cause p obabili y changes unde ope a ions ha
only al e he en i onmen wi hou changing sys emobse e obse ables, di ec ly con adic ing
A4 (en i onmen insensi i i y). The e o e, he only choice sa is ying en a iance and A4 is: assign
equal p obabili y ac oss all
j
unde he same
k
.
Le
qk,j
be he p obabili y o obse e sel -loca ion in mic oscopic b anch
(k, j)
. Then o any
xed
k
:
qk,1=qk,2=··· =qk,Nk≡qk.
The o e all no maliza ion condi ion is
X
k
Nk
X
j=1
qk,j = 1 ⇒X
k
Nkqk= 1.
By symme y and A1's equal on ological weigh , we can di ec ly ake
qk,j = 1/N
. Combining
hese:
qk=1
N, pk≡
Nk
X
j=1
qk,j =Nk
N.
8
Thus
pk=|αk|2.
Theo em 2 is p o ed.
□
4.3 Theo em 3: Con inuous Ex ension o Gene al Ampli udes
Fo any pu e s a e
|ψS⟩=Pkαk|sk⟩
, le
k=|αk|2
sa is ying
Pk k= 1
. Since a ionals a e dense
in eals, we can cons uc a a ional app oxima ion sequence
(n)
k=N(n)
k
N(n)→ k,X
k
(n)
k= 1
( o example, ake
(n)
k
as
k
ounded o
n
decimal places and eno malized). Fo each
n
, cons uc
equal-ampli ude mic oscopic s a es and p obabili y assignmen
p(n)
k= (n)
k
acco ding o Theo em 2.
Con inui y assump ion A5 equi es: when
∥ψ(n)−ψ∥ → 0
, we ha e
p(n)
k→pk
. Since
(n)
k→ k
:
pk= lim
n→∞ p(n)
k= lim
n→∞ (n)
k= k=|αk|2.
The e o e, he Bo n ule holds in gene al.
□
4.4 Theo em 4: Gleason-Type Uniqueness and No-Signaling
Theo ems 2 and 3 p o ide cons uc i e de i a ions o he Bo n ule o specic measu emen sce-
na ios. To demons a e uniqueness, we in oke Gleason- ype esul s and no-signaling cons ain s.
Gleason's heo em shows: in Hilbe spaces o dimension a leas h ee, i p obabili y measu e
µ(P)
assigned o p ojec ion ope a o
P
sa ises:
1.
µ(P)≥0
and
µ(I)=1
;
2. Fo any coun able amily o mu ually o hogonal p ojec ions
{Pi}
:
µ(PiPi) = Piµ(Pi)
;
hen he e mus exis a unique densi y ope a o
ρ
such ha
µ(P) = (ρP)
.
In he QCA uni e se, measu emen p obabili ies sa is ying non-con ex uali y and addi i i y a e
uniquely gi en by he Bo n ule ia Gleason's heo em. Combined wi h no-signaling cons ain s,
any al e na i e p obabili y ule iola es ei he non-con ex uali y o causali y.
□
4.5 Theo em 5: Eec i e Collapse
The a gumen o Theo em 5 is essen ially a igo ous o mula ion o he eec i e collapse idea om
decohe ence heo y in he QCA con ex . Due o decohe ence-induced o hogonali y o en i onmen
s a es in die en b anches, local obse ables canno dis inguish be ween he ull supe posi ion and
an eec i e mix u e. This jus ies ea ing collapse as Bayesian upda ing wi hin he obse e 's
in o ma ion ho izon.
□
9