Vergleich von Methoden zur Rekonstruktion von Quantenzuständen

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414 ≈3·108
pi
|Ψii
ρ=X
i
pi|ΨiihΨi|

{Mm}
X
m
MmMm=1
m
ρ
m
p(m) = tr(MmMmρ)
m
ρ0=MmρMm
tr(MmMmρ)
Em:= MmMm
Em
{Mm}

p(m) = tr(Emρ)

ρ
ρ

x
∆x
0,1
Sd
Hd
Sd=ρ:Hd→ Hd|ρ , ρ =ρ
hρ, σi:= tr(ρ·σ)
RSd
kρk2=hρ, ρi=tr(ρ2)
Sdd2
d
Sd≡ S

N
H2⊗H2⊗..⊗H22N
dimR(S) = (2N)2= 4N
4N4N−1
N
M V
a, b ∈M0≤λ≤1
λa + (1 −λ)b∈M
S
N
N:= {ρ∈ S | tr(ρ)=1}⊂S
N
S
P
P:= {ρ∈ S | ρ≥0} ⊂ S
P
S
D
D:= {ρ∈ S | tr(ρ)=1, ρ ≥0}=N ∩ P
N P P =N ∩ P

S
ρtomo
ρ
A B
tr(A·B) = hA, Bi ≥ 0
A≥0⇒A=A⇒
A|ali=al|ali,hal|aki=δlk, al≥0
⇒tr(A·B) = tr(B·A) = tr BX
l
al|alihal|!
=X
l
altr(B|alihal|) = X
l
alX
jhaj|B|alihal|aji
=X
lX
j
alhaj|B|aliδlj =X
l
alhal|B|ali
| {z }
≥0
≥0

ρ∈ D E≥0
tr(ρE) = hρ, Ei ≥ 0
S
ηE:Sd→Rkρ
E
ηE(ρ) := 





tr(E1ρ)
..
tr(Eiρ)
..
tr(Ekρ)





=





hE1, ρi
..
hEi, ρi
..
hEk, ρi






ηE≡η
~
P= (p1, p2, .., pk)∈Rk
Θ := (~
P= (p1, p2, .., pk)∈Rk|
k
X
i=1
pi= 1)⊂Rk
Θ
Ω := n~
P= (p1, p2, .., pk)∈Rk|pi≥0∀io⊂Rk
Ω
W:= (~
P= (p1, p2, .., pk)∈Rk|
k
X
i=1
pi= 1, pi≥0)= Θ ∩Ω⊂Rk
W
Θ Ω Rk
W= Θ ∩Ω
η(N)=Θ
η(P)⊆Ω

η(D)⊆ W
ρ≥0η(ρ)i=tr(Eiρ)≥0
~
P(ρ) = ηE(ρ)=(p1(ρ), .., pk(ρ))
ρ E
k
X
i=1
pi(ρ) = tr(ρ)
tr(ρ) = 1 1
k
X
i=1
pi=
k
X
i=1
tr(Eiρ) = tr k
X
i=1
Eiρ!=tr ρ
k
X
i=1
Ei!=tr(ρ1) = tr(ρ)
E={E1, E2, .., Ek}Ei≥0PiEi=1
ME=hE1, E2, .., EkiR⊆ S
ES
ME≡ M
S
S=M⊕M⊥,M∩M⊥={0}
M⊥M
M⊥:= {ρ∈ S | ∀ρM∈ M :hρ, ρMi= 0}
ηESd

ηE(λρ1+ρ2)i=tr(Ei(λρ1+ρ2)) = tr(λEiρ1+Eiρ2)
=λ tr(Eiρ1) + tr(Eiρ2)
=ληE(ρ1)i+ηE(ρ2)i∀ρ1, ρ2∈ Sd,∀λ∈R
ηEM⊥
E0∈Rk
ρ⊥∈ M⊥
E
ηE(ρ⊥)i=hEi, ρ⊥i= 0
Ei∈ MEρ⊥∈ M⊥
EηE(ρ⊥)=0
M⊥
E
tr(ρ⊥)=0
~
P=ηE(ρ⊥)=0⇒Pipi= 0 ⇒
tr(ρ⊥)=0
M⊥
ρ0∈ D ρ1=ρ0+ ∆ρ, ∆ρ∈ M⊥
tr(ρ1) = tr(ρ0) + tr(∆ρ) = tr(ρ0)=1
η(ρ1) = η(ρ0+ ∆ρ) = η(ρ0) + η(∆ρ) = η(ρ0)
∆ρ ρ1≥0ρ0ρ1
E
E
ρ⊥∈ M⊥\{0}
tr(ρ⊥)=0 ρ⊥
ρ⊥=X
i
λi|φiihφi|,X
i
λi= 0
λj6= 0

PM:S → M
hρ, σi=tr(ρ·σ)
PM(ρ) =
k0
X
i=1 ρ, E0
iE0
i
E0={E0
1, .., E0
k0} MEdimR(ME) = k0≤
k
ρ1, ρ2∈ S
η(ρ1) = η(ρ2)⇔PM(ρ1) = PM(ρ2)
⇐PM(ρ1) = PM(ρ2)
ρ1=ρM+ρ⊥,1
ρ2=ρM+ρ⊥,2
ρM∈ M ρ⊥,1, ρ⊥,2∈ M⊥
η(ρ1) = η(ρ2) = η(ρM)
⇒η(ρ1) = η(ρ2)
hρ1, Eii=hρ2, Eii ∀i
E0={E0
1, .., E0
k0} ME
PM(ρ1) = PM(ρ2)
⇔
k0
X
i=1 ρ1, E0
iE0
i=
k0
X
i=1 ρ2, E0
iE0
i
⇔X
iρ1, E0
i−ρ2, E0
iE0
i= 0
⇔ρ1, E0
i−ρ2, E0
i= 0 ∀i
⇔ρ1, E0
i=ρ2, E0
i∀i
E0MEρ1, ρ2
M M⊥
ρ1=ρM,1+ρ⊥,1
ρ2=ρM,2+ρ⊥,2

η(ρ1) = η(ρ2)⇔ hρM,1, Eii=hρM,2, Eii ∀i
ρ1, E0
i=ρ2, E0
i∀i⇔ρM,1, E0
i=ρM,2, E0
i∀i⇔ρM,1=ρM,2
hρM,1, Eii=hρM,2, Eii ∀i⇒ρM,1=ρM,2
hρM,1, Eii=hρM,2, Eii ∀i
ρM,1=
k0
X
l=1
c1lE0
l, c1l∈R
ρM,2=
k0
X
l=1
c2lE0
l, c2l∈R
ρM,16=ρM,2n≥1ρM,1ρM,2
n
dl:= c1l−c2l6= 0
1≤l≤n c1l=c2l≡cln+ 1 ≤l≤k0
ρM,1=
n
X
l=1
c1lE0
l+
k0
X
l=n+1
clE0
l
ρM,2=
n
X
l=1
c2lE0
l+
k0
X
l=n+1
clE0
l
⇒ρM,1−ρM,2=
n
X
l=1
(c1l−c2l)E0
l≡
n
X
l=1
dlE0
l, d 6= 0
hρM,1, Eii=hρM,2, Eii ∀i⇔ hρM,1−ρM,2, Eii= 0 ∀i
⇔*n
X
l=1
dlE0
l, Ei+= 0 ∀i⇒06=
n
X
l=1
dlE0
l∈ M⊥
ρM,16=ρM,2

~
f
~
P=η(ρ)≈~
f
ρ
η
ρSST := η−1(~
f)
η
3.1
ρSST =η−1
M(~
f) + ρ⊥=: ρ∗
SST +ρ⊥
ρ⊥M⊥ρ∗
SST
S
ME=SηES
ρSST =η−1(~
f) = ρ∗
SST
η
Sd2
~
f=η(ρSST )
X
i
fi=tr(ρSST )=1

ρSST
∆ρtomo
S~
f∈Rk~
f η
ρ1, ρ2∈ S ~
f
η(ρ1) = η(ρ2) = ~
f
ρ1ρ2M
PM(ρ1) = PM(ρ2) = ρM
⇒ρ1=ρM+ρ⊥1, ρ2=ρM+ρ⊥2
⇒λρ1+ (1 −λ)ρ2=ρM+λρ⊥1+ (1 −λ)ρ⊥2
⇒PM(λρ1+ (1 −λ)ρ2) = ρM
⇒η(λρ1+ (1 −λ)ρ2) = ~
f
⇒λρ1+ (1 −λ)ρ2∈η−1~
f
~
P
D(~
P) := nρ∈ D | η(ρ) = ~
P)o=η−1~
P∩ D
nρ=ρM+ρ⊥|ρM=η−1
M(~
P), ρ⊥∈ M⊥:ρ≥0o
D(~
P)Dη−1~
P
ηMρ∈ D(~
P)
PM(ρ) = η−1
M(~
P)

ρM∈ M tr(ρM)=1
D(ρM) := nρ=ρM+ρ⊥|ρ⊥∈ M⊥, ρ ∈ Do
~
f∈ W
D(~
f) = ∅
WD
ρ∈ D
WD:= n~
P∈ W | ∃ρ∈ D :η(ρ) = ~
Po⊆ W
WDW
~
P1,~
P2∈ WD
∃ρ1∈ D :~
P1=η(ρ1)
∃ρ2∈ D :~
P2=η(ρ2)
~
Pλ:= λ~
P1+ (1 −λ)~
P2∈ WD∀0≤λ≤1
~
Pλ=λ~
P1+ (1 −λ)~
P2∈ W
Wρλ∈ D
η(ρλ) = ~
Pλ=λ~
P1+ (1 −λ)~
P2
ρλ:= λρ1+ (1 −λ)ρ2
Dρλ∈ D
η(ρλ) = η(λρ1+ (1 −λ)ρ2) = λη(ρ1) + (1 −λ)η(ρ2) = λ~
P1+ (1 −λ)~
P2=~
Pλ

S
MD⊆ M
MD:= nρM∈ M | tr(ρM)=1,∃ρ⊥∈ M⊥:ρM+ρ⊥≥0o
MD
MD≡PM(D)
S
W Mw:= η−1(W)
MwM∩N
MD
ρ1
M, ρ2
M∈ Mw~
P1=η(ρ1
M)∈ W ~
P2=
η(ρ2
M)∈ W 0≤λ≤1
η(λρ1
M+ (1 −λ)ρ2
M) = λ~
P1+ (1 −λ)~
P2∈ W
MD⊆ MwWD⊆ W

WDMD
MDWD
η:MD→ WD
ρM7→ η(ρM)
η η MDWD
η(MD)⊆ WDρ∈ MD
ρ⊥∈ M⊥ρ+ρ⊥∈ D WD3η(ρ+ρ⊥) = η(ρ)
η~
P∈ WD
ρ∈ D η(ρ) = ~
P ρM=PM(ρ)ρM∈ MD
η(ρM) = ~
P
η η M MD⊆ M
ηMD
η
η
ρM∈ M MD
ρ⊥∈ M⊥ρM+ρ⊥≥0
ρM+ρ⊥

ρM∈ MDkρMk2=tr(ρ2
M)≤1
ρM∈ MDρ⊥∈ M⊥ρ:= ρM+ρ⊥∈ D
1≥ kρk2=tr(ρ2) = tr((ρM+ρ⊥)2)
=tr(ρ2
M+ρMρ⊥+ρ⊥ρM+ρ2
⊥)
=tr(ρ2
M)+2tr(ρMρ⊥)
| {z }
= 0
+tr(ρ2
⊥)
| {z }
≥0
=tr(ρ2
M) + tr(ρ2
⊥)
1≥1−tr(ρ2
⊥)≥tr(ρ2
M)
ρSST
ρ0
≡ρ
ρAdhoc =ρ0
tr(ρ0)
ρSST

ρSST
~
P=η(ρ)≈~
f
m p(m) = tr(Emρ)
N
ρ N
~
f
E ρ
~
f
L(~
f , ρ, E) = N!
(Nf1)!(N f2)!..(N fk)!
k
Y
j=1
pfjN
j
=N!
(Nf1)!(N f2)!..(N fk)!
k
Y
j=1
tr(Ejρ)fjN
ρML ∈ D
L(~
f , ρML, E ) = max
ρ∈D L(~
f , ρ, E)
~
f E
L(~
f , ρ, E)

ρ
N L N
L
L(~
f , ρ, E) :=
k
Y
j=1
tr(Ejρ)fj
max
ρ∈D L(~
f , ρ, E)b= max
ρ∈D L(~
f , ρ, E) = max
ρ∈D
k
Y
j=1
tr(Ejρ)fj
L
ρ ρ ∈ D
~x = (x1, .., xk), ~y = (y1, .., yk)∈Rk
xi, yi≥0∀iPk
i=1 yi= 1
k
Y
i=1
xyi
i≤
k
X
i=1
yixi
k
Y
i=1
xyi
i=
k
X
i=1
yixi⇔xi≡x0∀i
k
Y
i=1
xyi
i≤
k
X
i=1
yixi
⇔ln Y
i
xyi
i!≤ln X
i
yixi!
⇔X
i
yiln(xi)≤ln X
i
yixi!

xi≡x0∀i
L~
f(~
P) :=
k
Y
i=1
pfi
i
W
WD⊆ W
L
g(~
P) := ln k
Y
i=1
pfi
i!=
k
X
i=1
ln pfi
i=
k
X
i=1
piln(fi)
L
ln(L)~x 6=~y ∈ W
0<λ<1
g(λ~x + (1 −λ)~y)> λg(~x) + (1 −λ)g(~y)
⇔X
i
piln(λxi+ (1 −λ)yi)> λ X
i
piln(xi) + (1 −λ)X
i
piln(yi))
⇔X
i
piln(λxi+ (1 −λ)yi)>X
i
pi(λln(xi) + (1 −λ) ln(yi))
xi6=yi
ln(λxi+ (1 −λ)yi)> λ ln(xi) + (1 −λ) ln(yi)
~
P∈ W pi≥0pl>0
X
i
piln(λxi+ (1 −λ)yi)>X
i
pi(λln(xi) + (1 −λ) ln(yi))
~x 6=~y ∈ W 0<λ<1
ln(L~
f(~
P)) = ln k
Y
i=1
pfi
i!=
k
X
i=1
piln(fi)
D(~q, ~p) := X
i
qiln qi
pi=X
i
qiln qi−X
i
qiln pi

ln(L~
f(~
P)) = −D(~
f, ~
P) + X
i
filn fi
~
P~
fWD
max
~
P∈WDL~
f(~
P)b= max
~
P∈WD
ln(L~
f(~
P)) = max
~
P∈WD−D(~
f, ~
P) + X
i
filn fib= min
~
P∈WD
D(~
f, ~
P)
W
D(~q, ~p)≥0∀~q, ~p ∈ W
D(~q, ~p) = 0 ⇔~p =~q
L~
f(~
P) :=
k
Y
i=1
pfi
i
WDρ
η(ρ) = max
~
P∈WDL~
f(~
P) := ~
PML
L~
fWD
ρ1, ρ2∈ D L(~
f , ρ)D
PM(ρ1) = PM(ρ2)
η(ρ1) = η(ρ2) = ~
PML
PM(ρ1) = PM(ρ2)

~
f∈ W
E= (E1, .., Ek)
L(~
f , ρ, E) =
k
Y
i=1
tr(Eiρ)fi
D
Γ = nρ=ρM+ρ⊥|ρ⊥∈ M⊥:ρ≥0o=D(ρM)⊆ D
ρM∈ MDL(~
f , ρM, E) = maxρ∈MDL(~
f , ρ, E)
Γ
L(~
f , ρ)D~
PML L~
f
WDWDMD
MD
ρM∈ MDη(ρM) = ~
PML Γ = D(~
PML)
ρ0
Γ = nρ=PM(ρ0) + ρ⊥|ρ⊥∈ M⊥:ρ≥0o
PM(ρ0)
dimR(M) = d2
M⊥={0}
Γ = nρ=ρM+ρ⊥|ρ⊥∈ M⊥={0}:ρ≥0o={ρM}
ρ∗
SST =η−1
M(~
f)) MD

ρ∗
SST MD
ρ∗
SST
Mw
max
ρ∈MwL(~
f , ρ) = L(~
f , ρ∗
SST ) = L(~
f , η−1
M(~
f))
max
ρ∈MwL(~
f , ρ)b= max
ρ∈MwX
i
tr(Eiρ) ln(fi)b= min
ρ∈Mw
D(~
f , η(ρ))
D(~
f , η(ρ)) = 0 ⇔~
f=η(ρ)⇔ρ=η−1
M(~
f) = ρ∗
SST
ρ∗
SST ∈ MD
ρ∗
SST =η−1
M(~
f)∈ MD⇒ L(~
f , ρ∗
SST ) = max
ρ∈MDL(~
f , ρ)
MD⊆ Mw
ρ∗
SST =η−1
M(~
f)MD
ρML MD≡∂MD
MDWDL
MDρML MD
MwρML L Mw
ρML
ρ∗
SST /∈ MD
MD
R(ρ)ρ=ρ
R(ρ) := Pk
i=1
fi
tr(Eiρ)Ei∈ S

R(ρ)ρ
R(ρ)ρ=ρ⇔(R(ρ)ρ) = ρ
⇔ρR(ρ) = ρ⇒R(ρ)ρR(ρ) = R(ρ)ρ=ρ
⇔R(ρ)ρR(ρ) = ρ
ρi+1 = Φ(ρi) = (1 −α)ρi+αR(ρi)ρiR(ρi)
tr(R(ρi)ρiR(ρi))
0< α ≤1
ρi∈ D ⇒ Φ(ρi)∈ D
R(ρi)ρi=ρiα
S
MD
∂MD≡∂D
|ΨihΨ|ρ
p0
ρtomo

D
x2+y2+z2≤1
ρ=1
21 + z x −iy
x+iy 1−z
σxσyN= 2k k ∈N
k k
ρ
L=(2k)!
k!k!2
pk
x(1 −px)kpk
y(1 −py)k
px:= tr(ρ|1xih1x|)py:= tr(ρ|1yi h1y|)
1px=
py= 1/2ρML =1/2
Lmax =(2k)!
k!k!21
24k
(x, y) = (0,0)
¯
L
¯
L:= L
Lmax
= 24kpk
x(1 −px)kpk
y(1 −py)k
pxpyx, y, z
¯
L=L
Lmax
= 4k(x+ 1)k1−x+ 1
2k
(y+ 1)k1−y+ 1
2k
=4(x+ 1) 1−x+ 1
2(y+ 1) 1−y+ 1
2N/2
xy
N
¯
L

N
x y b1/2
x=rcos(φ), y =rsin(φ),0≤r≤1
¯
L(r, φ) = (1 −r2−r4cos2(φ) sin2(φ))N/2≥(1 −r2)N/2
¯
L≥(1 −r2)N/2≥1/2
(1 −r2)N/2≥1
2⇔r≤p1−2−2/N =: b1/2(N)
2
N
limN→∞ b1/2(N) = 2√1−1=0
N

ρML
ρBM E := ZD
Π( ~
f , ρ)ρ dρ
Π( ~
f , ρ)dρ ∝ L(ρ)dρ Π Π( ~
f , ρ)≥0
RDΠ( ~
f , ρ)dρ = 1
D
ρBM E ∈ D
ρBM E
Γ
Γ
Γ

M⊥
I(λ, ρ) := λS(ρ) + 1
Nlog L(ρ)
λ > 0S(ρ)
λ
S(ρ) Γ
ρMLME
M
LH(~
f , ρ) := det(ρ)βL(~
f , ρ)
β≥0
β→0
ρ
LH
det(ρ)βρHM L
β > 0ρM L
ρ−1
LHρ
β

N r
m∝dr log2d
d= 2N
w=
N
O
i=1
wi
wi∈ {1, σx, σy, σz}
d2
dr d2dr log2d
ρ
ρSST
ρSST
ρAdhoc

k
kd2
ρML
ρ∗
ML ∈ MD
L(~
f , ρ∗
ML) = max
ρ∈MDL(~
f , ρ)
ρ
ρi+1 = (1 −α)ρi+αR(ρi)ρiR(ρi)
tr(R(ρi)ρiR(ρi)) 0< α ≤1
ρi+1 = (1 −α)ρi+α PMR(ρi)ρiR(ρi)
tr(R(ρi)ρiR(ρi)) 0< α ≤1
ρ0=1
d
ρi∈ MD
i
i= 0 ρ0∈ M

ρi∈ MDρi+1
MD
ρ⊥
i∈ M⊥σi:= ρi+ρ⊥
i≥0
α= 1 MD
ρi+1 =PMR(ρi)ρiR(ρi)
tr(R(ρi)ρiR(ρi)) 
4R(σi)σiR(σi)σi
R(σi)σiR(σi) = R(ρi+ρ⊥
i)(ρi+ρ⊥
i)R(ρi+ρ⊥
i)≥0
R
R(ρi+ρ⊥
i) =
k
X
j=1
fj
tr((ρi+ρ⊥
i)Ej)Ej=
k
X
j=1
fj
tr(ρiEj) + tr(ρ⊥
iEj)Ej
=
k
X
j=1
fj
hρi, Eji+Dρ⊥
i, EjE
| {z }
≡0
Ej=
k
X
j=1
fj
tr(ρiEj)Ej=R(ρi)
˜ρ:= R(ρi)ρiR(ρi) ˆρ:= R(ρi)ρ⊥
iR(ρi)
R(σi)σiR(σi) = R(ρi)(ρi+ρ⊥
i)R(ρi)
=R(ρi)ρiR(ρi) + R(ρi)ρ⊥
iR(ρi)
= ˜ρ+ ˆρ≡˜ρM+ ˜ρ⊥+ ˆρM+ ˆρ⊥≥0
˜ρM:= PM(˜ρ) ˆρM:= PM(ˆρ)
PM(R(σi)σiR(σi)) = PM(˜ρM+ ˜ρ⊥+ ˆρM+ ˆρ⊥) = ˜ρM+ ˆρM∈ M
PMR(σi)σiR(σi)
tr(R(σi)σiR(σi)) =PM(R(σi)σiR(σi))
tr(R(σi)σiR(σi)) =˜ρM+ ˆρM
tr(˜ρM+ ˜ρ⊥+ ˆρM+ ˆρ⊥)
=˜ρM+ ˆρM
tr(˜ρM+ ˆρM)∈ MD
˜ρ⊥+ ˆρ⊥
tr(˜ρM+ ˆρM)∈ M⊥
˜ρM+ ˆρM
tr(˜ρM+ ˆρM)+˜ρ⊥+ ˆρ⊥
tr(˜ρM+ ˆρM)=˜ρM+ ˜ρ⊥+ ˆρM+ ˆρ⊥
tr(˜ρM+ ˆρM)≥0

tr ˜ρM+ ˜ρ⊥+ ˆρM+ ˆρ⊥
tr(˜ρM+ ˆρM)=tr( ˜ρM+ ˜ρ⊥+ ˆρM+ ˆρ⊥)
tr(˜ρM+ ˆρM)=tr( ˜ρM+ ˆρM)
tr(˜ρM+ ˆρM)= 1
ρi+1 MD
PMR(ρi)ρiR(ρi)
tr(R(ρi)ρiR(ρi)) =˜ρM
tr(˜ρM)∈ MD
˜ρM+ ˆρM
tr(˜ρM+ ˆρM)∈ MD?
⇒˜ρM
tr(˜ρM)∈ MD
16
ρ
E
ηE(ρ)
16 3 ·108

~v =~a +i~
b ~a ~
b
S
ρ= (1 −m)|Ψi hΨ|+m1
d
0≤m≤1
2 16
σi⊗σji, j = 0,1,2,3x, y, z
σ0⊗σ0=1
1
k k −1 (i, j)6= (0,0)
σi⊗σj
k
k−1
ηE(ρ)
[0,1] k
N[0,1]
α= 1
d(ρi+1, ρi) = ptr((ρi+1 −ρi)2)
S


d(ρi+1, ρi)2=tr((ρi+1 −ρi)2)!
≤2=: 0
0= 10−12
0
Nmax = 5 ·104
1%
R(ρi)ρiR(ρi)
M
Mk
ρ∗
ML ρML
< d(ρtomo, ρ∗
ML)2>
N
k
2000
k= 4,8,12
~
f /∈ WDm

k= 4 N= 16 0,6%
k= 8 N= 64 0,7%
k= 12 N= 144 0,3%

MD
ρM∈ MD
P
ρ=AA
A
log L(ρ=AA)Str(AA)=1
∇ k
X
i=1
filog(tr(EiAA)) + λ(tr(AA)−1)!!
= 0
MDρ=AA
ρ=PM(AA)
M M⊥
PM(AA)
| {z }
∈M
+ (AA −PM(AA))
| {z }
∈M⊥
=AA ≥0
MD
log L(ρ=PM(AA))
Str(PM(AA))) = 1
tr(PM(AA)) = tr(AA)=1
AA M⊥
∇ k
X
i=1
filog(tr(EiPM(AA))) + λ(tr(AA)−1)!!
= 0

SB={B1, .., Bd2}
SB⊇B0={B1, .., Bk} M
A ci∈R
A=
d2
X
i=1
ciBi⇒A2=
d2
X
i,l=1
ciclBiBl∈ S
A2SB
AA =
d2
X
m=1 hBm, AAiBm=
d2
X
m=1 *Bm,
d2
X
i,l=1
ciclBiBl+Bm=:
d2
X
m=1
qmBm
AM
PM(AA) =
k
X
l=1 *Bl,
d2
X
m=1
qmBm+Bl=
k
X
l=1
d2
X
m=1
qmhBl, BmiBl
hBl, Bmi=(δlm 1≤l, m ≤k
0 1 ≤l≤k k + 1 ≤m≤d2
PM(AA)
PM(AA) =
k
X
m=1
qmBm
qm:=
d2
X
i,l=1
ciclhBm, BiBli
~q = (q1, .., qk)
∇~c F:= ∇~c k
X
i=1
filog k
X
m=1
qmtr(EiBm)!+λ k
X
m=1
qmtr(Bm)−1!! !
= 0
~c = (c1, .., cd2)A
F
∂F
∂cα
=
k
X
j=1
∂F
∂qj
∂qj
∂cα

∂F
∂qj
=∂
∂qj
k
X
i=1
filog k
X
m=1
qmtr(EiBm)!+λ∂
∂qj k
X
m=1
qmtr(Bm)−1!
=
k
X
i=1
fi
tr(EiBj)
Pk
m=1 qmtr(EiBm)+λtr(Bj)
~c
∂qj
cα
=∂
cα
d2
X
i,l=1
ciclhBj, BiBli
hBj, BiBli hjiliqj
qj=
d2
X
i=1
ci
d2
X
l=1
clhjili=X
i6=α
ciX
l
clhjili+cαX
lhjαli
=X
i6=α
ci
X
l6=α
clhjili+cαhjiαi
+cα
X
l6=α
clhjαli+cαhjααi

∂qj
∂cα
=∂
∂cα
X
i6=α
ci
X
l6=α
clhjili+cαhjiαi
+cα
X
l6=α
clhjαli+cαhjααi


=X
i6=α
cihjiαi+X
l6=α
clhjαli+ 2cαhjααi
=X
i6=α
cihjiαi+cαhjααi+X
l6=α
clhjαli+cαhjααi
=X
i
cihjiαi+X
l
clhjαli=X
l
cl(hjlαi+hjαli)
=
d2
X
l=1
clhBj, BlBα+BαBli
∂F
∂cα
=
k
X
j=1 k
X
i=1
fi
tr(EiBj)
Pk
m=1 qmtr(EiBm)+λtr(Bj)!d2
X
l=1
clhBj, BlBα+BαBli!
= 0

~c
∂qj
∂cα~q
qj=X
i,l
ciclhjili=X
i,l
ciclhjili+X
i,l
ciclhjlii − X
i,l
ciclhjlii
| {z }
=qj
⇒2qj=X
i,l
cicl(hjili+hjlii) = X
l
cl X
i
ci(hjili+hjlii)!=
d2
X
l=1
cl
∂qj
∂cl
6.4~c
d2
X
α=1
cα
∂F
∂cα
=
d2
X
α=1
cα
k
X
j=1 k
X
i=1
fi
tr(EiBj)
Pk
m=1 qmtr(EiBm)+λtr(Bj)!∂qj
∂cα
=
k
X
j=1 k
X
i=1
fi
tr(EiBj)
Pk
m=1 qmtr(EiBm)+λtr(Bj)!d2
X
α=1
cα
∂qj
∂cα
=
k
X
j=1 k
X
i=1
fi
tr(EiBj)
Pk
m=1 qmtr(EiBm)+λtr(Bj)!2qj
= 2 X
i
fiPjqjtr(EiBj)
Pmqmtr(EiBm
+ 2λX
j
qjtr(Bj)
= 2 X
i
fi+ 2λqjtr(Bj) = 2 + 2λX
j
qjtr(Bj)
d2
X
α=1
cα
∂F
∂cα
= 2 + 2λX
j
qjtr(Bj)
ρ=PM(AA)≡
PmqmBm∂F
∂cα=
0∀α
0 =
d2
X
α=1
cα
∂F
∂cα
= 2 + 2λX
j
qjtr(Bj)
λ
tr(ρ) = Pjqjtr(Bj)=1
0 = 2 + 2λX
j
qjtr(Bj)⇔ −1 = λX
j
qjtr(Bj)
⇔ −1
λ=X
j
qjtr(Bj)!
= 1 ⇔λ=−1

∂F
∂cα
=
k
X
j=1 k
X
i=1
fi
tr(EiBj)
Pk
m=1 qmtr(EiBm)−tr(Bj)!∂qj
∂cα
!
= 0
~c
F
∆F=∇~cF·∆~c =
d2
X
α=1
∂F
∂cα
∆cα=
d2
X
α=1
k
X
j=1 k
X
i=1
fi
tr(EiBj)
Pk
m=1 qmtr(EiBm)−tr(Bj)!∂qj
∂cα
∆cα
=
k
X
j=1 k
X
i=1
fi
tr(EiBj)
Pk
m=1 qmtr(EiBm)−tr(Bj)!d2
X
α=1
∂qj
∂cα
∆cα
≡
k
X
j=1 k
X
i=1
fi
tr(EiBj)
Pk
m=1 qmtr(EiBm)−tr(Bj)!∆qj(∆~c)
=
k
X
j=1
∂F
∂qj
∆qj(∆~c) = ∇~qF·∆~q(∆~c)
∇~cF= 0
∆F=∇~cF·∆~c !
= 0 ∀∆~c
⇔ ∇~qF·∆~q(∆~c)!
= 0 ∀∆~c
∆~q ∆~c
∇~cF·∆~c !
= 0 ∀∆~c ⇔ ∇~cF= 0
∇~qF·∆~q(∆~c)!
= 0 ∀∆~c ⇔ ∇~q F= 0
MD(q1, .., qk)
∇F= 0
S
MD

ρtomo
L(ρtomo)maxρ∈D L(ρ)

N→ ∞
L/Lmax
lim
N→∞
L
Lmax
(ρ) = (1ρ=ρ0
0
W(~q)
L L0=W(~q)βL

α= 1
α
k= 8 α= 0.5
α= 1
k= 8 N= 64 α= 0.5
%

%
%
−%
−%
%
−
−
−−
−

−−
−%
%
−−
%
%
%
%−
−

−
−
%
%
%
%

−
%
%
%
%
%
%

−
%
%
%
%
%
%
%

%
%
−
%
−
−
%
%
−

k= 8
%
%
%
%
−
−
%
%
−

ρtomo
ρ∈ D
tr 1
d2!=1
d≤tr(ρ2) = kρk2≤1
dρ, 1
d2
=


ρ−1
d



2
=tr ρ−1
d2!=tr ρ2−2
d1ρ+12
d2
=tr(ρ2)−2
dtr(ρ) + d
d2=tr(ρ2)−2
d+1
d=tr(ρ2)−1
d
ρ∈ D
0≤dρ, 1
d2
≤1−1
d=: R(d)2
D
1
dR(d)
N
KrrN
ρ ρ ≥0
Kr: = ρ∈ N | d(ρ, 1
d)≤r=ρ∈ N | tr(ρ2)−1
d≤r2
=ρ∈ N | tr(ρ2)≤r2+1
d=: ρ∈ N | tr(ρ2)≤˜r2
tr(ρ) = 1 tr(ρ2)≤˜r2ρ≥0
λ1, .., λdρ tr(ρ) = Piλi= 1 tr(ρ2) = Piλ2
i≤
˜r2λi≥0∀i
λj<0
λj~
λ0= (λ1, .., λj−1, λj+1, .., λd)
X
i
λi=λj+X
i6=j
λi= 1

Pi6=jλi>1
k~
λ0k2
1=X
i6=j|λi| ≥ X
i6=j
λi>1
k~vkp:= (Pi|vi|p)
1
p
tr(ρ2) = X
i
λ2
i=λ2
j+X
i6=j
λ2
i
=λ2
j
|{z}
>0
+k~
λ0k2
2≤˜r2
k~
λ0k2≥˜r
α
k~
λ0k1≤αk~
λ0k2
√d−1
1<k~
λ0k1≤√d−1k~
λ0k2
⇔1
√d−1<k~
λ0k1
√d−1≤ k~
λ0k2
˜r:= 1
√d−1r2= ˜r2−1
d=
1
d−1−1
d=1
d(d−1)
Kr=(ρ∈ N | d(ρ, 1
d)≤r=1
pd(d−1))
d−1/2d
N
N
ρtomo