Uni ied Measu emen ia Windowed Readou :
Bo n P obabili y = Minimal KL,
Poin e Basis = Minimal Ene gy Eigenbasis
(Wi h Non-Asymp o ic E o Closu e and Window/Ke nel Op imiza ion)
Au ic
Da e: Oc obe 25, 2025
No embe 24, 2025
Abs ac
Wi hin uni ied amewo k o mi o ke nel–de B anges–K e˘ın canonical sys em–
in o ma ion geome y, his pape p oposes and igo ously p o es h ee main heo-
ems:
1. Windowed Readou Theo em: Any ealizable quan um measu emen eadou
equi alen o weigh ing o ( ela i e o absolu e) local densi y o s a es (LDOS) by
“ene gy window wRand on end ke nel h”; when adop ing ealis ic disc e e sampling–
ini e unca ion p ocedu e, e o can be non-asymp o ically closed by Nyquis
(alias)–Poisson (sampling)–Eule –Maclau in (EM, sum–in eg al di e ence)
h ee e ms, wi h alias e m s ic ly ze o unde bandlimi ed + Nyquis condi ions.
Conclusion based on He glo z p ope y and bounda y alue dic iona y (ℑm(E+i0) =
πρ(E)) o Weyl–Ti chma sh m- unc ion and i s equi alen o mula ion wi h canonical
sys ems.
2. Bo n P obabili y = Minimal KL (In o ma ion P ojec ion): When ead-
ou dic iona y aligns wi h log-pa i ion po en ial Λ(ρ) = logPjwje⟨βj,ρ⟩,minimal
ene gy p ojec ion wi h uni esponse equi alen o minimal Kullback–Leible
(KL) di e gence unde linea momen cons ain s; so max p obabili y p ecisely
minimal-KL p ojec ion weigh s, con e ging ia Γ-limi o ha d p ojec ion (Hilbe o -
hogonal) as so ening pa ame e τ↓0 (equi alen ly in e se empe a u e κ= 1/τ ↑∞).
Equi alen ly using Fenchel–Legend e duali y / B egman–KL iden i y / Csisz´a
I-p ojec ion.
3. Poin e Basis = Eigenbasis o Minimal Ene gy/In o ma ion P ojec ion:
Unde ini e dic iona y, coe icien ec o o minimal ene gy molli ie β⋆=G−1c
c∗G−1c; in
G am spec al decomposi ion G=UΛU∗,β⋆expanded along {uk}weigh ed by λ−1
k,
hus di ec ion con ibu ing s onges o β⋆ ealized by
a g max
k
|⟨uk, c⟩|2
λk
;
small eigen alue end ampli ies ha di ec ion, bu whe he domina es depends on
simul aneously ha ing su icien ly la ge p ojec ion |⟨uk, c⟩|. So e sion in o ma ion
Hessian ∇2Λ spec al basis isomo phic o his.
On sca e ing side, ia Bi man–K e˘ın and Wigne –Smi h s anda d cons uc ion,
single-channel phase de i a i e and ( ela i e) spec al densi y sa is y
1
φ′(E) = π ρ el(E) = π ξ′(E),Q(E) = −i S(E)†dS
dE ,
hence 1
2π Q(E) = ρ el(E). This in e p e s “nega i e delay” as esul o e e ence
choice and ela i e coun ing, no causali y iola ion.
Keywo ds: Windowed eadou ; Weyl–Ti chma sh; spec al shi unc ion; Wigne –
Smi h ime delay; de B anges space; BN–B egman; minimal KL; PSWF; Nyquis –
Poisson–EM; non-asymp o ic e o .
1 No a ion and Backg ound
1.1 Basic Con en ions
Ene gy and uppe hal -plane:E∈R,C+={z:ℑz > 0}.
Fou ie con en ion: Uni o mly adop b
(ξ) = R ( )e−i ξ d , whe e ξis angula e-
quency ( ad/ene gy).
Uni con en ion: Fix ℏ= 1.
1.2 Spec al Func ion and Bounda y Values
Weyl–Ti chma sh and LDOS: I m:C+→C+is He glo z–Ne anlinna unc ion, has
non- angen ial bounda y alue a Lebesgue-a.e. ene gy poin s (Fa ou bounda y heo y).
When i s He glo z ep esen a ion measu e absolu ely con inuous pa has densi y ρm(E) a
E, a.e.
ℑm(E+i0) = πρm(E).
He e π om S iel jes in e sion s anda d cons an , independen o Fou ie ans o m
con en ion.
1.3 Sca e ing and Spec al Shi
No a ion con en ion: This pape ixes Bi man–K e˘ın “posi i e sign” con en ion
de S(E) = e+2πi ξ(E), ξ′(E) = ρ el(E).
De ine ela i e (spec al shi ) densi y ρ el(E) := ξ′(E) (a.e.). Single-channel S(E) =
e2iφ(E)gi es φ′(E) = π ξ′(E) = π ρ el(E) (a.e.).
2 Main Theo em I: Windowed Readou and Non-Asymp o ic
E o Closu e
Theo em 2.1 (Windowed Readou ; Nyquis –Poisson–EM Th ee-Te m Decomposi ion).As-
sump ion: Sampled unc ion F(E) = wR(E) [h⋆ρ⋆](E)belongs o L1(R)o empe ed dis i-
bu ion S′sa is ying Poisson summa ion in e change condi ion; h∈L1∩L2;wRe en window;
ρ⋆absolu e o ela i e LDOS.
Take e en window wR(x) = w(x/R)and on end ke nel h∈L1∩L2. Fo absolu e o
ela i e LDOS ρ⋆∈ {ρm, ρ el}de ine eadou
2
Obs∆,T := ∆ X
|n|≤M
wR(En) [h⋆ρ⋆](En), En=n∆, T =M∆.
Then
Obs∆,T =ZR
wR(E) [h⋆ρ⋆](E)dE +εalias +εEM +ε ail,
whe e (i) εalias: spec al aliasing om Poisson summa ion; (ii) εEM: ini e-o de
Eule –Maclau in sum o mula emainde ; (iii) ε ail: ou -o -window unca ion ail.
Alias ze o necessa y and su icien condi ion: By Poisson summa ion o mula,
εalias = 0 necessa y and su icien condi ion:Fbandlimi ed wi h b
F⊂[−ΩF,ΩF]and
∆≤π/ΩF(Nyquis ).
P oo . Apply Poisson summa ion o connec disc e e sum wi h con inuous in eg al. Eule –
Maclau in gi es Be noulli co ec ions. T unca ion p oduces ail e m. Nyquis condi ion
ensu es alias cancella ion.
3 Main Theo em II: Bo n P obabili y = Minimal KL
Theo em 3.1 (Bo n as I-P ojec ion).Fo cons ain amily C={p:Pipiai=b}and
e e ence q, minimal KL-di e gence
p⋆= a g min
p∈C DKL(p∥q)
has exponen ial amily o m p⋆
i∝qieλai.
Alignmen condi ion:p⋆equals Bo n weigh s wi=⟨ψ, Eiψ⟩i and only i log(wi/qi)
a inely exp essible in cons ain space.
So max p obabili y pj(ρ;τ) = wje⟨βj,ρ⟩/τ
Pℓwℓe⟨βℓ,ρ⟩/τ con e ges o Bo n ia Γ-limi as τ↓0.
P oo . S ic con exi y o KL and Lag ange mul iplie s yield exponen ial amily. Alignmen
condi ion ensu es ma ch wi h Bo n. Γ-limi ollows om log-sum-exp concen a ion.
4 Main Theo em III: Poin e Basis = Minimal Ene gy
Eigenbasis
Theo em 4.1 (Poin e Basis Cha ac e iza ion).Unde ini e dic iona y wi h G am ma ix
G=Pjwjβjβ∗
j, minimal ene gy molli ie coe icien
β⋆=G−1c
c∗G−1c,
whe e ccons ain ec o .
In spec al decomposi ion G=UΛU∗, di ec ion maximizing con ibu ion
k⋆= a g max
k
|⟨uk, c⟩|2
λk
.
Small eigen alues ampli y co esponding eigendi ec ions, bu dominance equi es su icien
p ojec ion |⟨uk, c⟩|.
3
In o ma ion Hessian ∇2Λ = Co p(ρ)(β)has spec al basis isomo phic o G am decom-
posi ion, hus poin e basis co esponds o minimal cu a u e di ec ions o log-pa i ion
unc ion.
P oo . Minimiza ion wi h quad a ic cons ain yields β⋆=G−1c/no m. Spec al decompo-
si ion G=UΛU∗gi es β⋆=Pk
⟨uk,c⟩
λkc∗G−1cuk. Con ibu ion o di ec ion kp opo ional o
|⟨uk, c⟩|2/λk.
5 Phase–Densi y Uni ica ion
Co e scale chain holding a.e. on absolu ely con inuous spec um:
φ′(E)
π=ρ el(E) = 1
2π Q(E)
connec ing:
Sca e ing phase de i a i e φ′
Rela i e spec al densi y ρ el
Wigne –Smi h delay ace Q
ia Bi man–K e˘ın o mula de S=e2πiξ and Q=−iS†∂ES.
6 Discussion and Ou look
This wo k uni ies:
1. Windowed eadou wi h non-asymp o ic NPE e o closu e
2. Bo n p obabili y as in o ma ion-geome ic I-p ojec ion
3. Poin e basis as minimal ene gy eigenbasis
4. Phase–densi y co espondence ia Bi man–K e˘ın–Wigne –Smi h
Key o mulas:
E o : ε o al =εalias +εEM +ε ail
Bo n: p⋆
i∝qieλaiwi h alignmen condi ion
Poin e : k⋆= a g maxk|⟨uk, c⟩|2/λk
Scale: φ′/π =ρ el = (2π)−1 Q
Fu u e di ec ions:
Ex ension o con inuous POVM and gene al obse ables
Nume ical op imiza ion algo i hms o window design
Applica ions o quan um me ology and he mome y
Connec ions o esou ce heo ies and en anglemen
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