Trinity Theorem for Windowed Measurement\\[10pt] \large Born = Information Projection (iff),\\ Pointer = Spectral Minimum (iff),\\ Windows = Minimax Optimal

T ini y Theo em o Windowed Measu emen
Bo n = In o ma ion P ojec ion (i ),
Poin e = Spec al Minimum (i ),
Windows = Minimax Op imal
Au ic
Ve sion 1.1 (P ep in , No a ion Co ec ed)
Da e: Oc obe 25, 2025 (Re ised)
No embe 24, 2025
Abs ac
Unde uni ied amewo k o de B anges–K e˘ın (DBK) canonical sys em,sca e ing–
unc ional equa ion dic iona y and B egman/in o ma ion geome y, his pape
es ablishes “T ini y Theo em” o windowed measu emen . Conclusions in h ee ie s:
(I) Bo n = In o ma ion P ojec ion (i ): Unde o hogonal p ojec ion mea-
su emen (and i s gene aliza ion o POVM), op imal p obabili y ec o induced by
windowed eadou equi alen o I-p ojec ion (minimal KL/B egman cos ) o e
amily o linea alignmen cons ain s i and only i i equals Bo n p obabili y.
(II) Poin e = Spec al Minimum (i ): Fo any dis inguishable window am-
ily, le Ky Fan pa ial sum o “windowed ace quad a ic o m” minimize o e all
o hono mal bases, hen i and only i ha basis is spec al eigenbasis o measu ed
obse able (modulo degene acy).
(III) Windows = Minimax Op imal: Unde cons ain o bandlimi ed e en
window wi h no maliza ion w(0) = 1, aking Nyquis –Poisson–Eule –Maclau in
“alias + Be noulli laye + unca ion” non-asymp o ic e o uppe bound as ad e sa y,
op imal window exis s and (unde Hilbe s ongly con ex p oxy) unique, sa is ying
equency-domain p ojec ion KKT condi ion.
Key b idge is Bi man–K e˘ın o mula and Wigne –Smi h delay gi ing phase
de i a i e = spec al densi y, p ecisely connec ing windowed eadou wi h ela i e
s a e densi y (LDOS).
1 No a ion, Con en ions and Basic Se up
1.1 Hilbe Space and Measu emen
Hsepa able; pu e s a e ψ∈ H,|ψ|= 1. PVM case ake mu ually exclusi e comple e
p ojec ions {Pi}(PiPj=δijPi,PiPi=I), measu emen p obabili y pi=⟨ψ, Piψ⟩. POVM
case wi h {Ei⪰0},PiEi=I,pi=⟨ψ, Eiψ⟩.
1.2 DBK Canonical Sys em and Weyl–Ti chma sh
Fo one-dimensional channel, Weyl–Ti chma sh unc ion m(z) is He glo z–Ne anlinna
unc ion (analy ic in uppe hal -plane wi h non-nega i e imagina y pa ), bounda y alue
1
imagina y pa gi es spec al measu e dρ:ℑm(E+i0+) = π dρ/dE (almos e e ywhe e). de
B anges–K e˘ın (DBK) heo y gi es one- o-one co espondence be ween He glo z class
and canonical sys ems: each He glo z unc ion uniquely co esponds o canonical sys em
( ans e ma ix M( , z) sa is ying J-uni a i y), es ablishing spec al ep esen a ion and e al-
ua ion embedding.
1.3 Sca e ing–Func ional Equa ion Dic iona y and Phase–Spec al
Shi
de S(E) = exp
−2πi ξ(E),d
dE a g de S(E) = −2π ξ′(E).
I no malized de S(E) = c0e−2iφ(E), hen φ(E) = π ξ(E) (up o cons an ), hus
φ′(E) = π ξ′(E) = −π ρ el(E).
Con en ion s a emen : We ix de S(E) = c0e−2iφ(E)wi h |c0|= 1; acco ding o
Bi man–K e˘ın, de S=e−2πiξ, hus φ′=πξ′; by Q=−iS†S′and d
dE a g de S= Q,
ge Q=−2φ′=−2πξ′.
1.4 Wigne –Smi h Ma ix
Q(E) = −i S†(E)dS
dE (E) sel -adjoin ,1
2π Q(E) = ρ el(E) = −ξ′(E).
Compa ible wi h de S(E) = e−2πiξ(E)o
§
0.3, ob ained om Q=d
dE a g de S. Key
iden i y: ace o −iS†∂ESequals ∂Ea g de S(F iedel phase de i a i e), s anda d Wigne –
Smi h delay heo y.
1.5 Paley–Wiene Bandlimi ed E en Window and Fou ie Con en-
ion
PWe en
Ω={w: supp bw⊂[−Ω,Ω],w(E) = w(−E)}, w(0) = 1.
This pape adop s non-angula equency con en ion, o ene gy a iable E ake
Fou ie ans o m ( equency deno ed ξ):
b
(ξ) = ZR
e−iEξ (E)dE, (E) = 1
2πZR
eiEξ b
(ξ)dξ.
Scaled window de ini ion: Le bandlimi ed e en window w∈PWe en
Ω,scaled window
de ined as wR(E) := w(E/R). Then cwR(ξ) = Rbw(Rξ), suppo loca ed in [−Ω/R, Ω/R].
2 Bo n = In o ma ion P ojec ion (I and Only I )
Theo em 2.1 (Bo n P obabili y as I-P ojec ion).Se PVM elemen s Pi, deno e
ϕi:= Piψ
|Piψ|(|Piψ|>0), wi:= ⟨ψ, Piψ⟩=|Piψ|2.
Fo linea cons ain amily C={p:Pipiai=b}and e e ence dis ibu ion q, i e e ence
suppo con ains Bo n suppo (supp w⊆supp q), hen I-p ojec ion
2
p⋆= a g min
p∈C DKL(p∥q)
has exponen ial amily o m p⋆
i∝qieλai.
Alignmen condi ion (necessa y and su icien ):p⋆=w(Bo n) i and only i on
{i:wi>0}exis s λsuch ha log(wi/qi)a inely exp essible in cons ain space spanned by
{ai}.
P oo . S ic con exi y o KL and Lag ange mul iplie s gi e exponen ial amily and unique-
ness. Alignmen condi ion ensu es Bo n weigh s ma ch I-p ojec ion solu ion. POVM case by
Naima k dila ion.
3 Poin e = Spec al Minimum (I and Only I )
Theo em 3.1 (Poin e Basis as Ky Fan Minimum).Fo sel -adjoin obse able Aand dis in-
guishable window amily {w, h}, de ine windowed ace ope a o s Kw,h. Le {ek}m
k=1 o hono -
mal sys em. Then
m
X
k=1
⟨ek, Kw,hek⟩ ≥
m
X
k=1
λ↑
k(Kw,h),
equali y i and only i {ek}spans minimal eigensubspace o Kw,h (Ky Fan minimum sum).
Unde window dis inguishabili y, Kw,h commu e wi h spec al p ojec ion o A, hus minimal
eigensubspace equals (modulo degene acy) spec al eigenbasis o A.
P oo . S anda d Ky Fan a ia ional p inciple. Window dis inguishabili y ensu es commu a-
i i y ia S one–Weie s ass and di ec in eg al decomposi ion.
4 Windows = Minimax Op imal
Theo em 4.1 (Window Minimax Op imali y wi h KKT Condi ion).On PWe en
Ωwi h no -
maliza ion w(0) = 1, conside s ongly con ex p oxy
J(w) =
M−1
X
j=1
γj∥w(2j)
R∥2
L2+λ∥1{|E|>T }wR∥2
L2.
Exis s unique minimize w⋆sa is ying equency-domain bandlimi ed p ojec ion–KKT
equa ion:
P(ξ)
BM−1
X
j=1
γjξ4jc
w⋆
R+λc
w⋆
R−λ
π(T·)∗c
w⋆
R=ηc
w⋆
R
whe e P(ξ)
B equency p ojec ion o B= [−Ω/R, Ω/R],ηno maliza ion mul iplie .
P oo . S ong con exi y ensu es unique minimize . F eche de i a i e wi h cons ain gi es
KKT condi ion in equency domain.
3
5 E o Closu e: Nyquis –Poisson–EM Decomposi ion
Theo em 5.1 (NPE Th ee-Te m E o Decomposi ion).Fo windowed eadou disc e iza ion
wi h s ep ∆and unca ion N, ha e
E o =εalias
|{z}
Poisson
+R2M
|{z}
EM
+ε ail
|{z}
unca ion
.
When Fbandlimi ed wi h supp b
F⊂[−ΩF,ΩF]and ∆≤π/ΩF(Nyquis ), alias e m
εalias = 0.
EM emainde sa is ies |R2M| ≤ 2ζ(2M)
(2π)2MR|F(2M)|.
Tail con olled by unc ion decay: |ε ail| ≤ R|E|>N∆|F|.
P oo . Poisson summa ion + Eule –Maclau in expansion + unca ion analysis.
6 Uni ied Scale Chain
The ini y heo em uni ied by scale chain holding a.e. on absolu ely con inuous spec um:
φ′(E)
π=ρ el(E) = 1
2π Q(E)
connec ing:

Bo n: In o ma ion p ojec ion op imal unde alignmen

Poin e : Ky Fan minimum o windowed ope a o s

Windows: Minimax op imal unde NPE e o

Phase–Densi y: Bi man–K e˘ın–Wigne –Smi h b idge
7 Discussion and Ou look
This wo k es ablishes ini y o windowed measu emen :
1. Bo n p obabili y as in o ma ion-geome ic op imum
2. Poin e basis as spec al-geome ic minimum
3. Window design as minimax-op imal unde ini e-sample e o
Key con ibu ions:

Rigo ous i -and-only-i cha ac e iza ions

Uni ied ia Bi man–K e˘ın–Wigne –Smi h scale chain

Non-asymp o ic e o bounds ia NPE decomposi ion

DBK canonical sys em heo e ical ounda ion
Fu u e di ec ions:
4

Ex ension o con inuous POVM and gene al obse ables

Nume ical op imiza ion o window amilies

Applica ions o quan um me ology and sensing

Connec ions o quan um he modynamics
5