Pas -Selec ion Fil a ions (PSF):
Uni ying Quan um His o y wi h Ope a o -Algeb aic Founda ions
and Mac o-Reco d Insigh s
Pe e James Ba a o
No embe 22, 2025
Abs ac
I o mula e Pas -Selec ion Fil a ions (PSF): a il a ion o on Neumann subalgeb as (
A ,E
)
wi h ai h ul, no mal
φ
-p ese ing condi ional expec a ions ha cap u e “ eco ds o he pas ” in
an ope a o -algeb aic se ing. Using oo ideli y as ope a ional isibili y
V
(
ρ, σ
) =
pF(ρ, σ)
, I
p o e
V
(
)is noninc easing along he il a ion and ela e isibili y o he quan um Che no
exponen ia sha p wo-sided bounds.
I cons uc an explici dila ion/ins umen o
E
, es ablish delayed-choice in a iance unde
a
φ
-in a ian no malize hypo hesis and gi e a wo-qubi coun e example beyond i . Fo
agmen ed en i onmen s I de i e a edundancy
⇒
e asu e lowe bound and, in he independen -
p oduc case, ob ain exponen ial isibili y supp ession wi h a dimension-independen cons an
c
(
δ
)
≥ −1
2log
(1
−δ2
). I ex end PSF o algeb aic QFT; in his e ision I es ic claims o se ings
whe e exis ence o
E
is gua an eed (e.g., wedges in Minkowski acuum ia Bisognano–Wichmann
o hal -sided modula inclusions), and I p o ide a conc e e o mula ion o such cases. I de ine
nonnega i e “pas a es” om Bu es–Fishe and A aki ela i e en opy and in oduce a “pas
cu a u e” diagnos ic.
Con en s
1 No a ion glossa y (selec ed symbols) 3
2 P elimina ies and co e iden i ies 3
2.1 Fai h ul, phi-p ese ing condi ional expec a ions and he p edual ........... 3
3 S uc u al esul s o PSF 6
3.1 App oxima e delayed choice ( ini e-dimensional L2
φbound) ............... 7
4 Mac o- eco d limi s: edundancy and e asu e 7
5 Co ela ed en i onmen s: gua an eed bounds and a conjec u e 8
6 Rela ion o Quan um Da winism and Spec um B oadcas S uc u es 9
7 Rela i is ic gene aliza ions: PSF in algeb aic QFT 9
8 Double-sli and delayed choice (equa ional) 10
1
9 In o ma ion geome y o he pas 10
9.1 Pas cu a u e ....................................... 11
9.2 L2 phi ma ingale con e gence and li e a u e links ................... 11
10 Consis en his o ies ia PSF (wi h compa ibili y condi ions) 11
11 Ope a ional p o ocols and ini e-sample guidance 12
12 Classical educ ion and commu a i e limi 13
13 Coun e example wi hou phi-in a ian no malize 13
Ca model A: agmen ed qubi eco de ( edundan pu e eco ds) 13
Ca model B: bosonic poin e ia cohe en s a es 14
Re e ee-F iendly Summa y (assump ions, no el y, ep oducibili y)
•
Scope. Sepa able Hilbe spaces, wi h ex ensions o ype III on Neumann algeb as in AQFT.
I assume on Neumann subalgeb as
A ⊆ B
(
HR
)equipped wi h ai h ul, no mal,
φ
-p ese ing
condi ional expec a ions
E
:
B
(
HR
)
→A
o ming a il a ion (
1≤ 2⇒A 1⊆A 2
and
E 1
=
E 1◦E 2
). He e “
φ
-p ese ing” means
φ◦E
=
φ
. Fo QFT, I es ic o egimes whe e
E
is known o exis : (a) wedge algeb as in Minkowski acuum unde Bisognano–Wichmann; o
(b) inc easing amilies ob ained om hal -sided modula inclusions (HSMI) so ha
σφ
-in a iance
holds (Theo em 7.2).
•
Known ing edien s. Uhlmann ideli y F[
1
], ace dis ance D, Fuchs– an de G aa [
2
], quan um
Che no bound/exponen [
3
], S inesp ing/Naima k [
4
,
5
], SLD–QFI [
6
], Tomi a–Takesaki [
7
,
8
],
Tomiyama/Choi–E os [
23
,
24
], noncommu a i e il a ions [
35
,
36
], eco e abili y/Ma ko [
39
],
ope a o -algeb a backg ound [
42
], noncommu a i e ma ingales [
25
,
26
], modula inclusions in
AQFT [28,29].
•
Con en ion. Visibili y is he oo ideli y V(
ρ, σ
) :=
pF(ρ, σ)∈
[0
,
1], op imized o e e e sible
eco d ope a ions on
R
(isome ies/uni a ies); ideli y in a iance unde isome ies gi es V
max
=
V(Theo em 2.5). Complemen a i y is V
2
+ D
2≤
1(Theo em 2.6). Ope a ionally: F ia
swap es s/o e lap es ima ion o di ec ideli y es ima ion (DFE) [
30
,
31
]; D ia Hels om
disc imina ion;
ξQCB
ia mul i-copy e o exponen s; Vcoincides wi h in e e ome ic inge
isibili y when pa hs a e balanced (Sec ion 8).
•New con ibu ions.
(a)
PSF dila ion/ins umen wi h explici expec a ion s uc u e (Theo em 3.1). Heisenbe g/
Sch ödinge pic u es a e linked by he p edual (E )∗.
(b)
Delayed-choice in a iance wi h
φ
-in a ian no malize (Theo em 3.4) and an explici
wo-qubi coun e example ou side he hypo hesis (Sec ion 13).
(c) Redundancy ⇒e asu e (lowe bound; equali y in p oduc case) (Theo ems 4.2 and 4.3).
(d)
P oduc -case supp ession wi h Che no – isibili y bounds. I p o e a uni e sal uppe bound
V(
ρL
R, ρR
R
)
≤exp−1
2PmξQCB
(
ρL
m, ρR
m
)
, and equali y o Che no exponen s (hence he
2
wo-sided bounds
e−PmξQCB(ρL
m,ρR
m)≤
V
≤e−1
2PmξQCB(ρL
m,ρR
m)
) holds when all agmen
pai s sha e a common Che no minimize (e.g., i.i.d.). (Theo ems 4.4 and 4.5.)
(e)
Co ela ed en i onmen s: sa e bounds (Theo em 5.1) and a co ela ion-penalized subaddi i i y
conjec u e (Theo em 5.2) in o med by sandwiched Rényi eco e abili y [20–22].
( )
Rela i is ic PSF es ic ed o cases wi h gua an eed expec a ions: wedges o HSMI-gene a ed
nes s (Theo ems 7.1 o 7.3).
(g)
Ra es. Bu es–Fishe pas a e
γ
(
)and A aki-en opy pas a e
γA
(
) :=
1
2
d
d S
(
ωL
∥ωR
)
≥
0
(Theo em 9.2); pas cu a u e diagnos ics (Sec ion 9.1).
(h) L2
φ
ma ingale con e gence (Theo em 9.6) and app oxima e delayed-choice wi h ull p oo
(Theo em 3.6).
(i) Classical educ ion ( il a ions o σ-algeb as; Bha acha yya/Che no ) (Sec ion 12).
(j) Expe imen al p o ocols wi h ini e-sample guidance (Sec ion 11).
•
Wha is es able.
F
(hence
V
=
√F
) ia wo-copy o e lap/DFE es ima o s; bounds on
ξQCB
om
V
ia he sandwich (Eq.
(3)
); mul i-copy es s i one wan s asymp o ic
ξQCB
di ec ly;
Rδ
ia local omog aphy; V ia e e sible e asu e ope a ions; a es ia ime- esol ed omog aphy.
•
Limi a ions. Co ela ion-sensi i e uppe bounds beyond he p oduc case emain open; ou
subaddi i i y is conjec u al. AQFT claims a e limi ed o wedges o HSMI nes s whe e
σφ
-
in a iance is gua an eed.
1 No a ion glossa y (selec ed symbols)
Symbol Meaning
F(ρ, σ)Uhlmann ideli y; V(ρ, σ) = pF(ρ, σ)( isibili y).
D(ρ, σ) = 1
2∥ρ−σ∥1T ace dis ance; V2+ D2≤1.
Q(ρ, σ)Che no o e lap; Q(ρ, σ) = mins∈[0,1] T (ρsσ1−s),
ξQCB =−log Q.
(A , E )PSF algeb a and φ-p ese ing condi ional expec a ion.
N(A )No malize o A in U(HR).
RδRedundancy: |{m:dm=1
2∥ρL
m−ρR
m∥1≥δ}|.
dPSF
E asu e cos : minimal numbe o agmen s o ace o equalize
ma ginals.
2 P elimina ies and co e iden i ies
Le
HS,HR
be sepa able;
B
(
H
)bounded ope a o s;
D
(
H
)densi y ope a o s. CPTP maps Λ(
X
) =
PkKkXK†
k,PkK†
kKk=1.
2.1 Fai h ul, φ-p ese ing condi ional expec a ions and he p edual
Theo em 2.1 (Takesaki).Le
A⊆ M
be on Neumann algeb as and le
φ
be a ai h ul no mal
s a e on
M
. The e exis s a ai h ul no mal condi ional expec a ion
E
:
M→A
wi h
φ◦E
=
φ
i
A
is in a ian unde he modula g oup σφ
[7,8].
3
Rema k 2.2 (Uniqueness o
φ
-p ese ing expec a ions).I
A⊂M
is in a ian unde
σφ
, hen he
φ
-p ese ing condi ional expec a ion
E
:
M→A
gi en by Takesaki is unique. I use his uniqueness
in Theo em 3.4 when concluding AdUR◦E ◦Ad−1
UR=E .
Rema k 2.3 (Tomiyama/Choi–E os con ex ).No m-one p ojec ions on o on Neumann subalgeb as
a e condi ional expec a ions (Tomiyama) and uni al comple ely posi i e idempo en maps on o
ope a o sys ems ca y a C∗-algeb a s uc u e (Choi–E os); see [23,24].
De ini ion 2.4 (P edual map).Fo a no mal CP map
E
:
B
(
HR
)
→ B
(
HR
)(Heisenbe g pic u e),
i s p edual
E∗
:
T1
(
HR
)
→ T1
(
HR
)(Sch ödinge pic u e) is he unique no mal CP map sa is ying
T [E(X)ρ] = T [X E∗(ρ)] o all X∈ B(HR)and ace-class ρ. I w i e (E )∗ o he p edual o E .
Reco d pai and isibili y. Gi en a balanced wo-pa h p epa a ion, he eco ds induce (
ρL
R, ρR
R
)
∈
D
(
HR
)
2
. W i e Uhlmann ideli y
F
(
ρ, σ
) :=
∥√ρ√σ∥2
1
and ace dis ance
D
(
ρ, σ
) :=
1
2∥ρ−σ∥1
.
De ine isibili y
V
(
ρ, σ
) :=
pF(ρ, σ)
and (ope a ionally) op imize only o e e e sible eco d
ope a ions on R(isome ies/uni a ies); he op imum equals V(ρ, σ).
Lemma 2.5 (Visibili y as oo ideli y).Wi h he ope a ional class es ic ed o isome ies/uni a ies
on R,
Vmax(ρL
R, ρR
R)=V(ρL
R, ρR
R) = qF(ρL
R, ρR
R).(1)
P oo .
Uhlmann’s heo em gi es
pF(ρ, σ)
=
maxU∈U(HR)T √ρ√σ U
. In he in e e ome ic
se ing, e e sible ope a ions on Rac as Uon one a m, so he achie able inge ampli ude equals
he RHS; he maximize is he pa ial isome y om he pola decomposi ion
√σ√ρ
=
W|√σ√ρ|
,
i.e. U=W†.
Lemma 2.6 (Complemen a i y).
V(ρ, σ)2+D(ρ, σ)2≤1,1−D(ρ, σ)≤V(ρ, σ)≤q1−D(ρ, σ)2.
(Fuchs– an de G aa [2].)
PSF il a ion and owe p ope y. A PSF is a amily (
A ,E
)wi h ai h ul no mal
φ
and
φ◦E
=
φ
such ha
1≤ 2⇒ A 1⊆ A 2
and
E 1
=
E 1◦E 2
=
E 2◦E 1
. Taking p eduals e e ses
he composi ion o de .
Rema k 2.7 (Analy ic assump ions o in ini e-dimensional se ings).Unless s a ed o he wise,
HR
is
sepa able and all algeb as a e on Neumann algeb as equipped wi h no mal, ai h ul s a es. The
condi ional expec a ions
E
a e he unique
φ
-p ese ing expec a ions gua an eed by Takesaki, hence
no mal, comple ely posi i e, con ac i e, and idempo en . These assump ions ensu e ha he usual
da a-p ocessing inequali ies, con inui y o ideli y and ace dis ance unde no mal maps, and he
s anda d limi heo ems o inc easing amilies o no mal p ojec ions (Kadison–Ring ose II §6.5)
apply e ba im in he in ini e-dimensional case.
P oposi ion 2.8 (Mono onici y along he il a ion).Fo 1≤ 2,
F(E 1)∗ρL
R,(E 1)∗ρR
R≥F(E 2)∗ρL
R,(E 2)∗ρR
R≥F(ρL
R, ρR
R),(2)
hence V( )is noninc easing in .
4
P oo .
Le
1≤ 2
. The owe p ope y implies
E 1◦E 2
=
E 1
=
E 2◦E 1
. Passing o p eduals
and using (E 2◦E 1)∗= (E 1)∗gi es
(E 1)∗◦(E 2)∗= (E 1)∗.
. Apply ideli y mono onici y unde he CPTP pos
-
p ocessing (
E 1
)
∗
o he pai ((
E 2
)
∗ρL
R,
(
E 2
)
∗ρR
R
):
F(E 1)∗(E 2)∗ρL
R,(E 1)∗(E 2)∗ρR
R≥F(E 2)∗ρL
R,(E 2)∗ρR
R.
Since (E 1)∗◦(E 2)∗= (E 1)∗, he le -hand side equals F((E 1)∗ρL
R,(E 1)∗ρR
R), p o ing
F(E 1)∗ρL
R,(E 1)∗ρR
R≥F(E 2)∗ρL
R,(E 2)∗ρR
R.
The second inequali y
F
((
E 2
)
∗ρL
R,
(
E 2
)
∗ρR
R
)
≥F
(
ρL
R, ρR
R
)is he usual da a
-
p ocessing inequali y
unde (E 2)∗. Hence he claim.
Theo em 2.9 (Da a p ocessing o isibili y).Fo any CPTP map Λon
R
,V(Λ
ρL
R,
Λ
ρR
R
)
≥
V(ρL
R, ρR
R).
P oo .
Immedia e om mono onici y o Uhlmann ideli y
F
(Λ
ρ,
Λ
σ
)
≥F
(
ρ, σ
); aking squa e oo s
p ese es he inequali y. Equali y condi ion. Equali y holds o a gi en CPTP map Λi Λis
su icien o he pai (
ρL
R, ρR
R
)in he sense o Pe z—i.e., he e exis s a CPTP eco e y map
R
wi h
R◦Λ(ρL/R
R)=ρL/R
R[13,14].
Theo em 2.10 (Che no – isibili y sandwich).Le
Q
(
ρ, σ
) :=
min0≤s≤1T ρsσ1−s
and
ξQCB
:=
−log Q. Then
V(ρ, σ)2≤e−ξQCB(ρ,σ)≤V(ρ, σ).(3)
P oo .
By Audenae e al.
[3]
,F(
ρ, σ
)
≤Q
(
ρ, σ
)
≤pF(ρ, σ)
. Taking
−log
and no ing V =
√F
yields he s a ed inequali ies.
Theo em 2.11 (QFI– isibili y law).Fo
ρS
(
θ
) =
1
21 Veiθ
Ve−iθ 1
, he single-copy SLD quan um
Fishe in o ma ion is FQ= V2and o Ncopies F(N)
Q=NV2[6].
P oo .
W i e
ρ
(
θ
) =
1
2I
+ V(
cos θ σx
+
sin θ σy
)
, i.e., a uni a y encoding
ρ
(
θ
) =
e−iθσz/2ρ
(0)
eiθσz/2
wi h gene a o
H
=
σz/
2. Le
{λ±,|±⟩}
be he spec al decomposi ion o
ρ
(0), whe e
λ±
= (1
±
V)
/
2
and {|±⟩} a e eigens a es o σx. The s anda d uni a y- amily o mula gi es
FQ= 2 X
i,j
(λi−λj)2
λi+λj|⟨i|H|j⟩|2.
Only he (+
,−
)and (
−,
+) e ms con ibu e; he e (
λ+−λ−
)
2/
(
λ+
+
λ−
) = V
2
and
|⟨
+
|H|−⟩|2
=
1
4
(since
⟨
+
|σz|−⟩
= 1). Hence
FQ
= 2
·
V
2·1
4
+ 2
·
V
2·1
4
= V
2
. Addi i i y o e i.i.d. copies yields
F(N)
Q=NV2.
5
3 S uc u al esul s o PSF
Theo em 3.1 (PSF dila ion, ins umen , and expec a ion s uc u e).Le
A⊆ B
(
HR
)admi a
ai h ul no mal
φ
-p ese ing condi ional expec a ion
E
:
B
(
HR
)
→A
. Then he e exis a Hilbe
space
K
, an isome y
V
:
HR→ HR⊗K
, and a PVM
{P }
on
K
such ha in he Heisenbe g
pic u e
E(X)=V†(X⊗IK)V, X ∈ B(HR),(4)
and he associa ed ins umen on s a es (Sch ödinge pic u e) is
I (ρ)=(I⊗P )V ρV †(I⊗P ), E∗(ρ) = T KV ρV †=X
V ρ V †
, V := (I⊗⟨ |)V. (5)
Mo eo e ,
E
is idempo en wi h ange
A
and sa is ies he
A
-bimodule p ope y
E
(
aXb
) =
a E
(
X
)
b
o all
a, b ∈A
and
X∈ B
(
HR
); con e sely, a no mal uni al CP idempo en map on o a on
Neumann subalgeb a wi h he bimodule p ope y is a condi ional expec a ion [13,14].
Rema k 3.2 (Expec a ion s uc u e; Naima k/K aus in he commu an ).W i ing
V
=
P V ⊗| ⟩
( o any ONB
{| ⟩}⊂K
) de ines a POVM
{M
:=
V†
V }
wi h
P M
=
I
. By Naima k, he e is a
PVM {P }on Ksuch ha I (ρ)=(I⊗P )V ρV †(I⊗P )implemen s {M }on R.
K aus ope a o s in he commu an . When
E
is a condi ional expec a ion on o a on Neumann
subalgeb a
A⊂B
(
HR
)(Tomiyama), one can choose a S inesp ing dila ion wi h K aus ope a o s
in he commu an
A′
, i.e.,
E
(
X
) =
PjK†
jXKj
wi h each
Kj∈A′
and
PjK†
jKj
= 1. This
makes he
A
-bimodule p ope y immedia e:
E
(
aXb
) =
PjK†
jaXbKj
=
a E
(
X
)
b
o all
a, b ∈A
.
See Tomiyama’s heo em and ela ed s uc u e esul s (e.g., Kawahigashi [
27
]) o expec a ions
commu ing wi h subalgeb as.
De ini ion 3.3 (No malize ).Fo a on Neumann subalgeb a
A ⊆ B
(
HR
), i s no malize is
N(A )={UR∈ U(HR) : URA U†
R=A }.
Theo em 3.4 (Delayed-choice in a iance:
φ
-in a ian no malize c i e ion).Le
E
:
B
(
HR
)
→A
be he
φ
-p ese ing condi ional expec a ion (Takesaki), and le
U
=
US⊗UR
wi h
UR∈ N
(
A
)and
φ◦AdUR=φ. Then he Heisenbe g maps (id ⊗E )and AdUcommu e:
(id ⊗E )◦AdU= AdU◦(id ⊗E ).(6)
Consequen ly, in he Sch ödinge pic u e
(id ⊗(E )∗)◦AdU= AdU◦(id ⊗(E )∗).(7)
P oo ske ch.
Since
URA U†
R
=
A
and
φ◦AdUR
=
φ
, he map
AdUR◦E ◦Ad−1
UR
is a
φ
-p ese ing
condi ional expec a ion on o
A
. By Takesaki’s uniqueness, i equals
E
. Tenso ing wi h
id
on
S
yields he claim.
Lemma 3.5 (S a e-dependen inne -p oduc s abili y).Le
φ
(
X
) =
T
(
ρX
)be ai h ul on a
ini e-dimensional B(HR)and se κ2
φ:= ∥ρ−1∥∞= 1/λmin(ρ). Then o all X, Y ,
⟨U†X, U†Y⟩φ−⟨X, Y ⟩φ=T (ρ−UρU†)X†Y≤2ε κ2
φ∥X∥2,φ∥Y∥2,φ,
whe e ε := 1
2∥φ−φ◦AdUR∥1and ∥Z∥2
2,φ = T (ρZ†Z).
6
3.1 App oxima e delayed choice ( ini e-dimensional L2
φbound)
I quan i y de ia ions when he no malize /
φ
-in a iance hypo heses a e only app oxima e. Le
⟨X, Y ⟩φ
:=
φ
(
X†Y
)and
∥X∥2,φ
:=
q⟨X, X⟩φ
. W i e
P
o he o hogonal p ojec ion on he
φ
-GNS
Hilbe space on o he closu e o A ; o a φ-p ese ing expec a ion E ,P is implemen ed by E .
Two de ia ion pa ame e s. Fo UR∈ U(HR), de ine
η+
:= sup
A∈A
∥A∥2,φ≤1
∥AdUR(A)−A∥2,φ and η−
:= sup
B∈A
⊥
∥B∥2,φ≤1
∥P AdUR(B)∥2,φ .(8)
Thus
η+
=
∥
(
I−P
)
UP ∥
and
η−
=
∥P U
(
I−P
)
∥
as ope a o no ms on
L2
φ
, whe e
U
deno es he
(bounded) ope a o on L2
φimplemen ing AdUR.
Theo em 3.6 (App oxima e delayed choice in
L2
φ
).Assume ini e-dimensional
HR
. Fo e e y
Y∈ B(HR),
E (AdUR(Y)) −AdUR(E (Y))
2,φ ≤max{η+
, η−
} ∥Y∥2,φ.(9)
Equi alen ly, a he ope a o le el on L2
φone has he exac iden i y
∥P U−UP ∥= max
∥P U(I−P )∥,∥(I−P )UP ∥= max{η−
, η+
}.(10)
I mo eo e
φ◦AdUR
=
φ
(so ha
U
ac s uni a ily on
L2
φ
), he iden i y
(10)
s ill holds wi h he
same cons an .
Wo k on L2
φ, whe e E is he o hogonal p ojec ion P on o A . Then
E ◦AdUR−AdUR◦E ↔P U−UP ="0P U(I−P )
−(I−P )UP 0#
in he o hogonal decomposi ion L2
φ=A ⊕A ⊥.
Rema k 3.7 (Connec ing o s a e-d i pa ame e s).The de ia ion pai (
η+
, η−
)is in insic and
equi es no auxilia y assump ions. I one addi ionally quan i ies depa u e om
φ
-in a iance by
ε
:=
1
2∥φ−φ◦AdUR∥1
and se s
κ2
φ
:=
∥ρ−1∥∞
o
φ
(
X
) =
T
(
ρX
)( ini e
B
(
HR
)), hen Lemma 3.5
implies ha
U
isa(1 +
O
(
κ2
φε
)) nea
-
isome y on
L2
φ
. In symme ic nea
-
no malize egimes
(empi ically,
η−
≈η+
), he bound
(9)
simpli ies o
E
(
AdURY
)
−AdUR
(
E Y
)
2,φ ≲η+
∥Y∥2,φ
up o an O(κ2
φε ) ac o .
4 Mac o- eco d limi s: edundancy and e asu e
Assume HR=Nn
m=1 Hm,ρL/R
m= T =mρL/R
R, and dm:= 1
2∥ρL
m−ρR
m∥1.
De ini ion 4.1 (PSF e asu e cos and edundancy).Fo
U⊆ {
1
, . . . , n}
w i e
T U
:=
T Nm∈UHm
.
The e asu e cos is
dPSF(ρL
R, ρR
R) := min|U|: T UρL
R= T UρR
R,(11)
and, o δ∈[0,1], he edundancy is Rδ:= {m:dm≥δ}.
7
Theo em 4.2 (Redundancy lowe -bounds e asu e).Any e asu e se
U
ha equalizes he ma ginals
mus include all indices mwi h dm≥δ. Consequen ly,
dPSF(ρL, ρR)≥Rδ.(12)
This bound is igh in he p oduc /independence case (see Theo em 4.3).
P oo .
I some
m
wi h
dm≥δ
is no aced, hen
ρL
m
=
ρR
m
pe sis s in he ma ginal, con adic ing
equali y o T UρLand T UρR.
Co olla y 4.3 (Equali y condi ion in he p oduc case).I
ρL/R
R
=
NmρL/R
m
and
S
:=
{m
:
dm>
0
}
,
hen dPSF =|S|. Hence dPSF =Rδ o any 0< δ ≤minm∈Sdm.
Theo em 4.4 (P oduc -case exponen ial supp ession).I
ρL/R
R
=
NmρL/R
m
, hen ideli y ac o izes
and
V(ρL
R, ρR
R) = Y
m
V(ρL
m, ρR
m).(13)
Fo each agmen he Che no – isibili y sandwich (3)gi es
e−ξQCB(ρL
m,ρR
m)≤
V(
ρL
m, ρR
m
)
≤e−1
2ξQCB(ρL
m,ρR
m)
. Mul iplying o e
m
yields he uncondi ional
wo-sided bounds
e−PmξQCB(ρL
m,ρR
m)≤V(ρL
R, ρR
R)≤e−1
2PmξQCB(ρL
m,ρR
m).(14)
Mo eo e , he o al Che no exponen sa is ies
ξQCB(ρL
R, ρR
R)≤X
m
ξQCB(ρL
m, ρR
m),(15)
wi h equali y i he minimizing
s⋆∈
[0
,
1] o
Q
(
ρL
m, ρR
m
;
s
)is common o all
m
(e.g., i.i.d.). In
pa icula , i dm≥δ o a leas Rδ agmen s, hen
V≤exp
−1
2Rδc(δ), c(δ) := in {ξQCB(ρ, σ) : 1
2∥ρ−σ∥1≥δ} ≥ −1
2log(1 −δ2).(16)
Co olla y 4.5 (Ca s abili y).Fo mac oscopic edundancy, V(ρL
R, ρR
R)→0exponen ially in Rδ.
5 Co ela ed en i onmen s: gua an eed bounds and a conjec u e
Theo em 5.1 (Mono onici y unde agmen selec ion).Fo any subse
S⊆ {
1
, . . . , n}
and pa ial
ace ΛS:= T Sc,
ξQCB(ρL
R, ρR
R)≥ξQCB
ρL
S, ρR
S,V(ρL
R, ρR
R)≤V
ρL
S, ρR
S.(17)
In pa icula , ξQCB(ρL
R, ρR
R)≥maxmξQCB(ρL
m, ρR
m).
Reduc ion o he i.i.d. case. When
R
=
NmRm
and (
ρL/R
R
) =
Nm
(
ρL/R
m
), he il a ion’s Che no –
isibili y mono onici y educes o he explici ac o iza ion and exponen ial decay de i ed in Sec ion 4;
see Theo ems 4.4 and 4.5. In co ela ed se ings, explici a es need no exis and only s uc u al
mono onici y su i es.
Conjec u e 5.2 (Co ela ion-awa e subaddi i i y).The e exis co ela ion unc ionals
C
(
ρL
R, ρR
R
)(e.g.,
based on sandwiched Rényi di e gences) such ha
ξQCB(ρL
R, ρR
R)≥X
m
ξQCB(ρL
m, ρR
m)− C(ρL
R, ρR
R),(18)
wi h
C
= 0 in he p oduc case and
C ≥
0o he wise; c . mono onici y/ eco e abili y esul s in F ank
and Lieb [20], Beigi [21], Mülle -Lenne e al. [22].
8
6
Rela ion o Quan um Da winism and Spec um B oadcas S uc-
u es
Con ex . Quan um Da winism (QD) explains objec i i y ia he p oli e a ion o edundan eco ds
in o many en i onmen agmen s [
15
,
16
]. Spec um B oadcas S uc u es (SBS) cap u e objec i i y
by equi ing agmen s a es condi ioned on poin e ou comes o commu e and be b oadcas in
many copies [18,19].
PSF’s addi ions. PSF packages edundancy and objec i i y wi hin a il a ion o
φ
-p ese ing
condi ional expec a ions, ying: (i) edundancy o a conc e e e asu e cos lowe bound
dPSF ≥Rδ
(Theo ems 4.2 and 4.3); (ii) p oduc -case scaling ia a Che no – isibili y uppe bound and, when
a common Che no minimize exis s (e.g., i.i.d.), he wo-sided bounds (Theo em 4.4); and (iii)
delayed-choice in a iance o an explici
φ
-in a ian no malize c i e ion (Theo em 3.4). The explici
c
(
δ
)
≥−1
2log
(1
−δ2
)bound links edundancy pla eaus om QD o exponen ial isibili y supp ession;
c . Zwolak, Riedel, and Zu ek [17] o QCB-based edundancy a es.
7 Rela i is ic gene aliza ions: PSF in algeb aic QFT
I wo k wi h a Haag–Kas le ne
O7→ A
(
O
)
⊆ B
(
H
)[
10
] and es ic o egimes whe e modula /spli
hypo heses ensu e he exis ence o φ-p ese ing condi ional expec a ions.
Why modula /spli hypo heses now? Unlike he ype I/ ini e-dimensional se ings o Sec-
ions 1–6 whe e
φ
-p ese ing expec a ions a e au oma ic, local algeb as in AQFT a e ype III and
φ
-p ese ing condi ional expec a ions ypically do no exis unless one assumes modula /spli con-
di ions ha gua an ee
σφ
-in a iance o he ele an inclusions. I he e o e es ic o egimes whe e
hese hold (e.g., wedges in acuum ia Bisognano–Wichmann and hal -sided modula inclusions),
ensu ing exis ence/uniqueness o
φ
-p ese ing expec a ions [
11
,
12
,
28
,
29
]. See also [
37
,
38
] o
ela ed s uc u al backg ound.
S anding modula -in a iance assump ion (made explici ). Fix a Rindle wedge
W
in
Minkowski space ime and le
φ
be he acuum s a e. Le
σφ
s
deno e he modula g oup o (
A
(
W
)
, φ
)
(Bisognano–Wichmann iden i ies
σφ
s
wi h he one-pa ame e boos subg oup p ese ing
W
). I
conside any inc easing amily {A } ∈Io on Neumann subalgeb as wi h
A ⊂A(W)and σφ
s(A )⊆A o all s∈Rand all ∈I. (19)
By Takesaki, o each he e is a unique φ-p ese ing condi ional expec a ion E :A(W)→A .
De ini ion 7.1 (Rela i is ic PSF il a ion: wedge case).Le
{A } ∈I⊂A
(
W
)sa is y
(19)
and be
inc easing in
. De ine he il a ion by (
A , E
)
wi h
E
as abo e. Fo wo no mal s a es
ωL/R
on
A
(
W
)se
V
:=
qF(ωL◦E , ωR◦E )
.Conc e e ( i ial) example. I
B0⊂A
(
W
)is ixed by
σφ
(e.g., he ixed-poin algeb a o a compac symme y along he wedge edge), hen he cons an
amily A ≡B0 o all sa is ies (19).
9
[29]
H.-W. Wiesb ock. Hal -sided modula inclusions o on Neumann algeb as. Comm. Ma h. Phys.
157:83–92, 1993.
[30]
S. T. Flammia and Y.-K. Liu. Di ec ideli y es ima ion om ew Pauli measu emen s. Phys.
Re . Le . 106:230501, 2011.
[31]
M. Fanizza, M. Rosa i, M. Sko inio is, J. Calsamiglia, and V. Gio anne i. Beyond he Swap
Tes : Op imal Es ima ion o Quan um S a e O e lap. Phys. Re . Le . 124:060503, 2020.
[32]
R. B. G i i hs. Consis en his o ies and he in e p e a ion o quan um mechanics. J. S a .
Phys. 36:219–272, 1984.
[33]
R. Omnès. Consis en in e p e a ions o quan um mechanics. Re . Mod. Phys. 64:339–382,
1992.
[34]
J. J. Halliwell. A e iew o he decohe en his o ies app oach o quan um mechanics. In
Fundamen al P oblems in Quan um Theo y, Ann. N.Y. Acad. Sci. 755:726–740, 1995.
[35]
Y. Ueda, “Amalgama ed ee p oduc o e Ca an subalgeb a,” Paci ic Jou nal o Ma hema ics,
ol. 191, no. 2, pp. 359–392, 1999.
[36]
P. Biane, “P ocesses wi h ee inc emen s,” Ma hema ische Zei sch i , ol. 227, no. 1, pp. 143–
174, 1998.
[37]
R. Longo, “Index o sub ac o s and s a is ics o quan um ields. I,” Communica ions in Ma he-
ma ical Physics, ol. 126, no. 2, pp. 217–247, 1989.
[38]
D. Buchholz and S. J. Summe s, “An algeb aic cha ac e iza ion o acuum s a es in Minkowski
space,” Communica ions in Ma hema ical Physics, ol. 155, no. 3, pp. 449–458, 1993.
[39]
O. Fawzi and R. Renne , “Quan um condi ional mu ual in o ma ion and app oxima e Ma ko
chains,” Communica ions in Ma hema ical Physics, ol. 340, no. 2, pp. 575–611, 2015.
[40]
A. Bha achayya, “On a measu e o di e gence be ween wo s a is ical popula ions de ined by
hei popula ion dis ibu ions,” Bulle in Calcu a Ma hema ical Socie y, ol. 35, no. 99–109,
p. 28, 1943.
[41]
H. Che no , “A measu e o asymp o ic e iciency o es s o a hypo hesis based on he sum o
obse a ions,” Annals o Ma hema ical S a is ics, 23(4):493–507, 1952.
[42]
R. V. Kadison and J. R. Ring ose, Fundamen als o he Theo y o Ope a o Algeb as. Volume
II: Ad anced Theo y. Academic P ess, New Yo k, 1986.
16
