Asymptotic Convertibility of Entanglement: A General Approach to Entanglement Concentration and Dilution

0.19 MB·

by  Yong Jiao

Asymptotic Convertibility of Entanglement:
A General Approach to Entanglement Concentration
and Dilution
Yong Jiao∗1, Eyuri Wakakuwa†2, and Tomohiro Ogawa‡2
1 Graduate School of Information Systems, University of Electro-Communications, 1-5-1
Chofugaoka, Chofu-shi, Tokyo, 182-8585, Japan.
2 Graduate School of Informatics and Engineering, University of Electro-Communications,
7
1-5-1 Chofugaoka, Chofu-shi, Tokyo, 182-8585, Japan.
1
0
2
February 1, 2017
n
a
J

1
3
Abstract

] We consider asymptotic convertibility of an arbitrary sequence of bipartite pure states into an-
h
p otherbylocaloperationsandclassicalcommunication(LOCC).Weadoptaninformation-spectrum
- approachtoaddresscaseswhereeachelementofthe sequencesis notnecessarilyintensorpowerof
t
n abipartitepurestate. WederivenecessaryandsufficientconditionsfortheLOCCconvertibilityof
a onesequencetoanotherintermsofspectralentropyratesofentanglementofthe sequences. Based
u
onthese results,we alsoprovidesimple proofs for previously knownresults onthe optimalrates of
q
entanglement concentration and dilution of general sequences of pure states.
[

1
v 1 Introduction
0
5
Anentangledquantumstatesharedbetweentwodistantpartiesisusedasaresourceforperforming
0
nonlocal quantum information processing. When a state is not in a desired form as a resource,we
9
0 may need to transform it by local operations and classical communication (LOCC) to a state
. with the desired form. Well-known examples of such tasks are entanglement concentration and
1
dilution [1]. Entanglement concentration is a task to obtain a maximally entangled state from
0
7 manycopiesofanon-maximallyentangledstatebyLOCC,andentanglementdilutionisitsinverse
1 process. Whentheinitialstateiscopiesofabipartitepurestate,theoptimalratesofentanglement
v: concentration and dilution are asymptotically equal to the entanglement entropy [1].
i For cases where the initial and target states are not necessarily in tensor power of a bipartite
X
state, the information-spectrum method has been applied to analyze entanglement concentration
r
[2,3]andentanglementdilution[3]. Originally,the information-spectrummethodwasdevelopedin
a
classicalinformationtheorybyHanandVerdu´[4–6]andhasbeenextendedtoquantuminformation
theory by Nagaoka and Hayashi [7–9]. In the setting of the information-spectrum method, the
optimal rates of entanglement concentration and dilution are obtained in terms of spectral entropy
rates [2,3].
In this paper, we consider a more general situation in which an arbitrary sequence of bipartite
pure states ψAB = ψAB ∞ is converted into another φAB = φAB ∞ asymptotically by a
{ n }n=1 { n }n=1
sequence of LOCC protocols = ∞ . We require that the trace distance between the final
b L {Ln}n=1 b
state (ψAB) and the targetstate φAB vanishes in the limit of n . We address conditions in
Ln n b n →∞
which such a conversion is possible. Contrary to the previous approaches [2,3], we do not assume
that the target state or the initial state is a maximally entangled state.
The main results of this paper are as follows. As a direct part of the convertibility, it is proved
thatthe initialsequence ψAB canbe convertedto the targetφAB asymptoticallyif the inf-spectral
∗[email protected] b b
†[email protected]
‡[email protected]
1
entropy of entanglement of ψAB is larger than the sup-spectral entropy of entanglement of φAB.
As a converse part, we prove that if ψAB is convertible to φAB, the inf-/sup-spectral entropy of
b b
entanglementof ψAB must be largerthan those of φAB, respectively. If we restrictφAB or ψAB to
b b
beasequenceofmaximallyentanglestates,ourresultsturnouttobethoseobtainedbyHayashi[2]
b b b b
and Bowen-Datta [3], regarding the optimal rates of entanglement concentration and dilution.
Our proof of the direct part is based on the theory of classical random number generation and
muchsimpler than those of [2,3]. It has been pointed out by KumagaiandHayashi[10] that there
are close relations between convertibility of entanglement and classicalrandomnumber generation
mainly on the second order analysis of convertibility of entanglement in the i.i.d. setting. In this
paper, we pursue generality of such an idea in the information-spectrum setting and provide a
simple argument for the asymptotic convertibility of entanglement.
This paper isorganizedasfollows. Insection2, weprovidedefinitions ofthe problemandstate
the main results. Proofs of the main results are presented in Section 3 and Section 4. Conclusions
are given in section 5.
2 Main Results
In this section, we present definitions of the problem and state the main results of this paper. Let
A and B (n = 1,2,...) be arbitrary finite-dimensional Hilbert spaces and consider a general
Hn Hn
sequence of bipartite systems AB = A B (n = 1,2,...) composed of them. Let ψAB and
φAB in AB be arbitrary puHrenstateHs fnor⊗eHacnh n N. For simplicity of the notation,|wendienote
| n i Hn ∈
density operators by ψAB = ψAB ψAB and φAB = φAB φAB . For arbitrary density operators
n | n ih n | n | n ih n |
ψAB, the reduced density operators are written as ψA =Tr [ψAB] and ψB =Tr [ψAB].
n n B n n A n
2.1 Necessary and Sufficient Conditions for LOCC Convertibility
For arbitrary sequences of bipartite pure states ψAB = ψAB ∞ and φAB = φAB ∞ , we seek
{ n }n=1 { n }n=1
for conditions under which ψAB can be converted into φAB by LOCC for each n, up to a certain
n b n b
error that vanishes in the limit of n .
→∞
Definition 1. We say that ψAB can beasymptotically converted into φAB by LOCC, if thereexists
a sequence of LOCC (n=1,2,...) such that
Ln b b
lim (ψAB) φAB =0, (1)
n→∞kLn n − n k1
where is the trace norm defined by A =TrA for a operator A.
1 1
k·k k k | |
Inthis paper,weprovidenecessaryandsufficientconditionsfortheasymptoticconvertibilityof
two sequences of bipartite pure states in terms of spectral entropy rates, which are key ingredients
ininformation-spectrummethodsanddefinedasfollows. Letρ= ρ ∞ beanarbitrarysequence
{ n}n=1
of density operators and σ = σ ∞ be an arbitrary sequence of Hermitian operators. Then, for
each ε [0,1], the spectral div{erng}enn=c1e rates [9] are defined byb
∈ b
D(ερ σ):=sup a liminfTrρ ρ enaσ >0 1 ε , (2)
n n n
| || n (cid:12) n→∞ { − }≥ − o
D(ερb σb):=inf a (cid:12)limsupTrρ ρ enaσ >0 ε . (3)
n n n
| || (cid:26) (cid:12) n→∞ { − }≤ (cid:27)
b b (cid:12)
Here, A > 0 denotes the spectral projection corresponding to the positive part of a Hermitian
{ }
operator A. Specifically, using the spectral decomposition A= a E , A>0 is defined by
k k k { }
P
A>0 = E .
k
{ }
k:Xak>0
With the spectral divergence rates, the spectral entropy rates [9,13] are defined by
H(ερ):= D(ερ I), H(ερ):= D(ερ I) (4)
| − | || | − | ||
for ε [0,1], where I = I ∞b is the sebqubence of idebntity operabtorbs. Especially, for ε = 0 we
∈ { n}n=1
write
b
H(ρ)=H(0ρ), H(ρ)=H(0ρ).
| |
b b b b
2
For any general sequences of bipartite pure states ψAB, consider sequences of reduced states
ψA = ψA ∞ and ψB = ψB ∞ . Then it is clear that ψA and ψB have the same entropy
{ n}n=1 { n}n=1 b
spectral rates, i.e.,
b b b b
H(ψA)=H(ψB), H(ψA)=H(ψB).
The main results of this paper arebas followbs. b b
Theorem 1 (direct part). Let ψAB = ψAB ∞ and φAB = φAB ∞ be general sequences
{ n }n=1 { n }n=1
of bipartite pure states on AB (n = 1,2,...). If H(ψA) > H(φA) holds, then ψAB can be
Hn b b
asymptotically converted into φAB by LOCC.
b b b
Theorem 2 (conversepart). bLet ψAB = ψAB ∞ and φAB = φAB ∞ be general sequences of
{ n }n=1 { n }n=1
bipartite pure states on AB (n = 1,2,...). If ψAB can be asymptotically converted into φAB by
Hn b b
LOCC, it must hold that H(εψA) H(εφA) and H(εψA) H(εφA) for every ε [0,1].
| ≥ | b | ≥ | ∈ b
b b b b
2.2 Entanglement concentration and dilution
Inthissection,weusetheabovetheoremstoprovidesimpleproofsofknownresultsontheoptimal
rates of entanglement concentration and entanglement dilution for general sequences of bipartite
pure states.
Let M ∞ beanarbitrarysequenceofnaturalnumbers,andlet Φ AB beamaximally
{ n}n=1 | Mni∈Hn
entangled state with Schmidt rank M for each n. As a shorthand notation, we denote ΦAB =
n Mn
Φ Φ . Noting that ΦA =Tr [ΦAB] andΦB =Tr [ΦAB] are the maximally mixed states
| Mnih Mn| Mn B Mn Mn A Mn
with Schmidt rank M , it is straightforwardto verify that
n
1 1
H(ΦA)=liminf logM , H(ΦA)=limsup logM (5)
n n
n→∞ n n→∞ n
b b
for ΦA = ΦA ∞ .
{ Mn}n=1
b
2.2.1 Entanglement Concentration
Entanglement concentration is a task for two distant parties to obtain a sequence of maximally
entangled states ΦAB from a sequence of bipartite pure states ψAB by LOCC.
Definition 2 (Ebntanglement concentration rate). For a sequenbce ψAB = ψAB ∞ , a rate R is
{ n }n=1
said to be achievable if there exists a sequence of natural numbers M ∞ such that ψAB can be
{b n}n=1
asymptotically converted into ΦAB = ΦAB ∞ by LOCC and
{ Mn}n=1 b
b 1
liminf logM R
n
n→∞ n ≥
holds. The entanglement concentration rate, or distillable entanglement [3], of a sequence ψAB is
defined by
b
R(ψAB):=sup R R is achievable . (6)
{ | }
Proposition 1 (Hayashi[2,Theobrem1],Bowen-Datta[3,Theorem3]). For a sequenceof bipartite
pure states ψAB = ψAB ∞ , we have
{ n }n=1
b R(ψAB)=H(ψA). (7)
Proof. We apply Theorem 1 and Theoremb2 regardinbg the target state φAB as Φ . From
| n i | Mni
Theorem 1 and (5), ψAB can be asymptotically converted into ΦAB by LOCC if M = enR and
n
H(ψA)>H(ΦA)=R. Thus a rate R is achievable if H(ψA)>R. Conversely,suppose that a rate
b b
R is achievable. By Definition 2, there exists a sequence ΦAB = ΦAB ∞ such that ψAB can be
b b b { Mn}n=1
1
asymptotically converted into ΦAB and liminf logMn b R. Then from Theorem 2band (5), it
n→∞ n ≥
must hold that
b
1
H(ψA) H(ΦA)=liminf logM R.
n
≥ n→∞ n ≥
b b
Thus we obtain (7).
3
2.2.2 Entanglement Dilution
Entanglementdilutionisataskfortwodistantpartiestoconvertasequenceofmaximallyentangled
states ΦAB into a sequence of bipartite pure states φAB asymptotically by LOCC.
Definibtion 3 (Entanglement dilution rate). For a sbequence φAB = φAB ∞ , a rate R is said to
{ n }n=1
be achievable if there exists a sequence of natural numbers M ∞ such that ΦAB = ΦAB ∞
{bn}n=1 { Mn}n=1
can be asymptotically converted into φAB by LOCC and
b
b 1
limsup logM R
n
n→∞ n ≤
holds. The entanglement dilution rate, or entanglement cost [3], of a sequence φAB is defined by
R∗(φAB):=inf R R is achievable . b (8)
{ | }
b
Proposition 2 (Bowen-Datta [3, Theorem 4 ]). For a sequence of bipartite pure states φAB =
φAB ∞ , we have
{ n }n=1 b
R∗(φAB)=H(φA). (9)
Proof. We apply Theorem 1 and Theoremb2 by takinbg the initial state ψAB as Φ . From
| n i | Mni
Theorem1and(5),ΦAB canbeasymptoticallyconvertedintoφAB ifM =enR andR=H(ΦA)>
n
H(φA). ThusarateRisachievableifR>H(φA). Conversely,supposethatarateRisachievable.
b b b
By Definition 3, there exists a sequence ΦAB = Φ ∞ such that ΦAB can be asymptotically
b 1 b { Mn}n=1
converted into φAB and limsup logMnb R. From Theorem 2 and (5b), it must hold that
n→∞ n ≤
b
1
R limsup logM =H(ΦA) H(φA).
≥ n→∞ n n ≥
b b
Thus we obtain (9).
3 Direct Part
In this section, we give a proof of Theorem 1 using known results on classical random number
generations.
3.1 Random Number Generation and Majorization
Let us first review the information-spectrum approach for random number generation [6], intro-
ducing the spectral entropy rates of classical random variables. For an arbitrary sequence of real
valued random variables Z ∞ , we define the limit superior and inferior in probability by
{ n}n=1
p-limsupZ :=inf α lim Pr Z >α =0 ,
n n
n→∞ n (cid:12)n→∞ { } o
(cid:12)
p-liminfZ :=sup α(cid:12) lim Pr Z <α =0 .
n n
n→∞ n (cid:12)n→∞ { } o
(cid:12)
LetX= Xn ∞ anarbitrarysequenceofrando(cid:12)mvariables,calledageneralsource,takingvalues
{ }n=1
in arbitrary countable sets n (n=1,2,...), and PXn(xn) (xn n) be the probability function
of X for each n. Then theXspectral entropy rates of X is defined∈bXy
n
1 1 1 1
H(X):=p-liminf log , H(X):=p-limsup log . (10)
n→∞ n PXn(Xn) n→∞ n PXn(Xn)
Let Y and Y˜ be random valuables on a countable set and let q(y) and q˜(y) (y ) be the
corresponding probability functions, respectively. Then theYvariational distance betwe∈enYY and Y˜
is defined by
d(Y,Y˜):= q(y) q˜(y). (11)
| − |
yX∈Y
4
Proposition 3 (Nagaoka [6, Theorem 2.1.1]). Let X = Xn ∞ and Y = Yn ∞ be arbitrary
generalsources. IfH(Y)<H(X),thenthereexistsaseque{nceo}fnm=1apsϕ : n{ }nn=(1n=1,2,...)
n
X →Y
such that
lim d(Yn,ϕ (Xn))=0.
n
n→∞
Next, wetreatarelationbetweenrandomnumbergenerationandmajorization. Fora sequence
a = a m (m N), let a↓ = a↓ m denotes the sequence rearranged in decreasing order. We
{ i}i=1 ∈ { i}i=1
say a= a m is majorized by b= b m and write a b if we have
{ i}i=1 { i}i=1 ≺
k k
a↓ b↓ (k =1,2,...,m)
i ≤ i
Xi=1 Xi=1
and the equality for k = m. Note that the majorization relation a b can be defined even when
≺
the numbers of elements in a and b differs, by including zero if necessary. When both a b and
b a hold, or equivalently a↓ =b↓, we wirte a∼b. ≺
≺
The following fact is given by Kumagai-Hayashi[10]. We show a proof here for readers’ conve-
nience since we can not find a proof in the literature.
Lemma 1 (Kumagai-Hayashi [10, Section 3.2]). Given a map ϕ : from a finite set to
X → Y X
, and a probability function p:x p(x) [0,1] on , let
Y ∈X 7→ ∈ X
q(y)= p(x)
x∈ϕX−1({y})
be the induced probability function on . Then we have p q.
Y ≺
Proof. For each y , let n(y) = ϕ−1( y ) and ϕ−1( y ) = x ,x ,...,x , and define
y,1 y,2 y,n(y)
∈ Y | { } | { } { }
n(y)-dimensional real vectors by
t
α := p(x ),p(x ),...,p(x ) ,
y y,1 y,2 y,n(y)
β :=((cid:0)q(y),0,...,0)t, (cid:1)
y
where (...)t denotes the transposition of the vector. It is straightforward to verify that α β
y y
≺
holds. Thus there exists a doubly stochastic matrix such that α = β . Indeed, letting
y y y y
D D
n(y)
p(x )
= y,j U (12)
Dy q(y) n(y),j
Xj=1
gives the relation α = β , where U is a n(y) dimensional permutation matrix transposing
y y y n(y),j
D
the 1st and j-th elements. Since is a convex combination of permutation matrices, it is doubly
y
stochastic. Now let us introduceDa notation for the direct sum of vectors u Rn and v Rm, and
the corresponding direct sum of matrices A Rn×n and B Rm×m, by ∈ ∈
∈ ∈
u A 0
u v = , A B = .
⊕ (cid:18)v(cid:19) ⊕ (cid:18)0 B(cid:19)
Then we have p∼ α and q ∼ β , and hence,
y∈Y y y∈Y y
L L
p∼ α = D β = D β β ∼q,
y y y y y y
(cid:18) (cid:19)(cid:18) (cid:19)≺
yM∈Y yM∈Y yM∈Y yM∈Y yM∈Y
where the majorization follows from the fact that is a doubly stochastic matrix.
≺ y∈YDy
L
We note that the above lemma and the proof are valid for countable sets and .
X Y
3.2 Proof of Theorem 1
Let ψAB and φAB (n=1,2,...) be the initial and target states, respectively, and
| n i | n i
ψAB = p (xn) eA eB ,
| n i n | xni⊗| xni
xnX∈Xnp
φAB = q (yn) fA fB
| n i n | yni⊗| yni
ynX∈Ynp
5
be their Schmidt decompositions. Then their reduced density operators are given by
ψnA =TrB ψnAB = pn(xn)|exnihexn|,
(cid:2) (cid:3) xnX∈Xn
φAn =TrB φAnB = qn(yn)|fynihfyn|.
(cid:2) (cid:3) ynX∈Yn
From the Schmidt coefficients we can define random variables Xn and Yn subject to probability
functions p (xn) (xn n) and q (yn) (yn n), and general sources X = Xn ∞ and Y =
n ∈ X n ∈ Y { }n=1
Yn ∞ composedofthem. ForsequencesofdensityoperatorsψA = ψA ∞ andφA = φA ∞ ,
{ }n=1 { n}n=1 { n}n=1
it is straightforwardto verify that
b b
H(X)=H(ψA), H(Y)=H(φA). (13)
b b
SupposethatH(ψA)>H(φA),orequivalentlyH(X)>H(Y). FromProposition3,thereexists
a sequence of maps ϕ : n n (n = 1,2,...) such that the variational distance between
b n X b→ Y
q˜ (yn)=p(ϕ−1( yn )) and q (yn) (yn ) goes to zero asymptotically, i.e.,
n n { } n ∈Y
lim d(Yn,Y˜n)=0, (14)
n→∞
whereY˜n isarandomvariablesubjecttotheprobabilityfunctionq˜ (yn). FromLemma1itimplies
n
that p q˜ .
n n
≺
Consider a state
φ˜AB := q˜ (yn) fA fB .
| n i n | yni⊗| yni
ynX∈Ynp
Due to Nielsen’s theorem [11], ψAB can be deterministically converted to φ˜AB by LOCC for
| n i | n i
each n.
To complete the proof,we verifythat the state φ˜AB is equalto the targetstate φAB asymp-
| n i | n i
totically. LetF(ρ,σ)bethefidelitybetweenstateρandσ,definedbyF(ρ,σ):=Tr√ρ√σ . Noting
| |
that
φ˜An =TrBhφ˜AnBi=ynX∈Ynq˜n(yn)|fynihfyn|,
we have
F(φ˜AB,φAB)= φ˜ ,φ = q˜ (yn)q (yn)=F(φ˜A,φA). (15)
n n |h n ni| n n n n
ynX∈Ynp
It is well known [12] that the trace distance and the fidelity are related as
1 F(ρ,σ) ρ σ 1 F(ρ,σ)2. (16)
1
− ≤k − k ≤ −
p
Noting φ˜A φA =d(Yn,Y˜n), from (14) we have
k n − nk1
lim φ˜A φA =0, (17)
n→∞k n − nk1
which implies
lim F(φ˜A,φA)=1 (18)
n n
n→∞
from the first inequality of (16). From (15), it implies
lim F(φ˜AB,φAB)=1, (19)
n n
n→∞
which leads to
lim φ˜AB φAB =0 (20)
n→∞k n − n k1
due to the second inequality of (16). (cid:3)
6
4 Converse Part
In this section, we prove Theorem 2 after reviewing properties of spectral divergence rates.
4.1 Properties of Spectral Divergence Rates
It is proved in [13] that spectral divergence rates have properties of monotonicity and continuity
for ε = 0. We extend these results to any ε [0,1]. We also prove an inequality for the spectral
∈
entropy rates of a sequence of product states.
4.1.1 Prerequisites
Let A be a Hermitian operator, and let A = a E be the spectral decomposition. Then the
k k k
positive and negative parts of A are, respectivePly, defined by
A := a E , A := ( a )E .
+ k k − k k
−
k:Xak>0 k:Xak≤0
Following [7,9], we denote the corresponding projections by
A>0 := E , A 0 := E .
k k
{ } { ≤ }
k:Xak>0 k:Xak≤0
With the above notations, we have A =A A>0 and A = A A 0 . Note that
+ −
{ } − { ≤ }
A=A A , A =A +A (21)
+ − + −
− | |
are,respectively,theJordandecompositionandtheabsolutevalueoftheoperatorA. Thefollowing
lemma is essential in information-spectrum methods.
Lemma 2. For any 0 T I, we have
≤ ≤
TrA =TrA A>0 TrAT, (22)
+
{ }≥
or equivalently,
TrA = max TrAT. (23)
+
T:0≤T≤I
It is also useful to note the relation with the trace norm:
1 1
TrA = TrA +TrA , TrA = TrA TrA ,
+ −
2{ | | } 2{ | |− }
which follows from (21). Especially, if TrA=0
TrA =2TrA =2TrA , (24)
+ −
| |
TrA =2 max TrAT. (25)
| | T:0≤T≤I
It should also be noted that from Tr(A B) =Tr(A B) A B >0 0, we have
+
− − { − }≥
TrA A B >0 TrB A B >0 . (26)
{ − }≥ { − }
and it obviously holds that
Tr(A B) =Tr(A B) A B >0 TrA A B >0 . (27)
+
− − { − }≥ { − }
ThefollowinglemmawaspoitedoutbyBowen-Datta[3]forcompletelypositiveandtracepreserving
maps, It should be noted that is no need to be complete positive map.
F
Lemma 3. Let A and B be Hermitian operators. For any trace preserving (TP) maps , we have
F
TrA Tr (A) .
+ +
≥ F
7
4.1.2 Monotonicity
ThereisanalternativeexpressionforthespectraldivergenceratesintroducedbyBowen-Datta[13].
For each ε [0,1], let
∈
C(ερ σ):=sup a liminfTr(ρ enaσ ) 1 ε ,
n n +
| || n (cid:12) n→∞ − ≥ − o
C(ερb σb):=inf a (cid:12)limsupTr(ρ enaσ ) ε .
n n +
| || (cid:26) (cid:12) n→∞ − ≤ (cid:27)
b b (cid:12)
It can be shown that these apparently different definitions yield the same quantities [13].
Lemma 4. For any ε [0,1], we have
∈
C(ερ σ)=D(ερ σ), (28)
| || | ||
C(ερ σ)=D(ερ σ). (29)
|b||b |b||b
The proof is given in Appendix. b b b b
We use this lemma to prove the monotonicity of spectral divergence rates as follows.
Proposition 4. For any sequence of TP maps = ∞ , the monotonicity of the spectral
F {Fn}n=1
divergence rates hold, that is,
b
D(ερ σ) D ε (ρ) (σ) , (30)
| || ≥ |F ||F
(cid:0) (cid:1)
D(ερb σb) D ε b(ρb) b(σb) , (31)
| || ≥ |F ||F
(cid:0) (cid:1)
for any ε [0,1]. b b b b b b
∈
Proof. For any γ >0, choose a=D ε (ρ) (σ) γ. From Lemma 3, we have
|F ||F −
(cid:0) (cid:1)
1 ε Tr (ρ )b benab b(σ ) Tr(ρ enaσ ) . (32)
− ≤ Fn n − Fn n + ≤ n− n +
(cid:0) (cid:1)
Taking liminf of (32), we have
n→∞
1 ε liminfTr (ρ ) ena (σ ) liminfTr(ρ enaσ ) . (33)
− ≤ n→∞ Fn n − Fn n + ≤ n→∞ n− n +
(cid:0) (cid:1)
From (28) and the definition of C, we obtain a = D ε (ρ) (σ) γ D(ερ σ) for all γ > 0,
|F ||F − ≤ | ||
which implies (30). (cid:0) (cid:1)
Similarly, if we choose a=D(ρ σ)+γ, and from Lebmmba 3b, wbe have b b
||
Tr (ρ )b bena (σ ) Tr(ρ enaσ ) ε, (34)
Fn n − Fn n + ≤ n− n + ≤
(cid:0) (cid:1)
Thus, taking limsup of (34), we have
n→∞
limsupTr (ρ ) ena (σ ) limsupTr(ρ enaσ ) ε. (35)
n→∞ (cid:0)Fn n − Fn n (cid:1)+ ≤ n→∞ n− n + ≤
From (29) and the definition of C, we have D ε (ρ) (σ) a = D(ερ σ)+γ for all γ > 0,
|F ||F ≤ | ||
which leads to (31). (cid:0) (cid:1)
b b b b b b
From (4), we have the following Corollary.
Corollary 1. For any sequence of unital TP maps = ∞ , the following inequalities hold
F {Fn}n=1
for any ε [0,1]:
∈ b
H(ερ) H ε (ρ) , (36)
| ≤ |F
(cid:0) (cid:1)
H(εbρ) H ε b(bρ) . (37)
| ≤ |F
(cid:0) (cid:1)
b b b
8
4.1.3 Continuity
Spectral divergence rates are “continuous” with respect to the sequences of density operators in
the first argument, that is, spectral divergence rates of two sequences coincide if the sequences are
asymptotically equal.
Lemma 5. Let ρ= ρ ∞ and ρ′ = ρ′ ∞ be sequences of density operators. If
{ n}n=1 { n}n=1
b b lim ρ ρ′ =0, (38)
n→∞|| n− n||1
then
D(ερ σ) = D(ερ′ σ), (39)
| || | ||
D(ερ σ) = D(ερ′ σ) (40)
|b||b |b||b
hold for any 0 ε 1 and any sequencebσb= σ ∞ obf Hbermitian operators.
≤ ≤ { n}n=1
b
Proof. From (25), we have
ρ ρ′ = (ρ enaσ ) (ρ′ enaσ )
k n− nk1 || n− n − n− n ||1
=Tr(ρ enaσ ) (ρ′ enaσ )
| n− n − n− n |
2Tr(ρ enaσ ) ρ enaσ >0
n n n n
≥ − { − }
2Tr(ρ′ enaσ ) ρ enaσ >0
− n− n { n− n }
2Tr(ρ enaσ ) 2Tr(ρ′ enaσ )
≥ n− n +− n− n +
where the last inequality follows from (22). Hence
1
Tr(ρ′ enaσ ) + ρ ρ′ Tr(ρ enaσ ) . (41)
n− n + 2|| n− n||1 ≥ n− n +
For any γ >0, let a=D(ερ σ) γ. Then from D(ερ σ)=C(ερ σ), we have
| || − | || | ||
b bliminfTr(ρ enaσ ) b b1 ε. b b (42)
n n +
n→∞ − ≥ −
Thus taking liminf of (41) gives
n→∞
liminfTr(ρ′ enaσ ) liminfTr(ρ enaσ )
n→∞ n− n + ≥ n→∞ n− n +
1 ε,
≥ −
which implies a = D(ερ σ) γ D(ερ′ σ). Since γ > 0 can be arbitrary, we have D(ερ σ)
| || − ≤ | || | || ≤
D(ερ′ σ). Interchanging the role of ρ and ρ′, we have the converse inequality D(ερ σ)
D(ε|ρ′||σ). Thus we havbeb(39). In the sabmebway, we have (40). | b|| b ≥
|b||b b b b b
Fbromb(4), we have the following corollary.
Corollary 2. Let ρ= ρ ∞ and ρ′ = ρ′ ∞ be sequences of density operators. If
{ n}n=1 { n}n=1
b b lim ρ ρ′ =0, (43)
n→∞k n− nk1
then
H(ερ)=H(ερ′), (44)
| |
H(ερ)=H(ερ′) (45)
|b |b
hold for any 0 ε 1. b b
≤ ≤
9
4.1.4 Spectral Entropy of Product States
Lemma 6. For arbitrary sequences ρA = ρA ∞ and σB = σB ∞ , the followings hold for any
{ n}n=1 { n}n=1
ε [0,1]:
∈ b b
H(ερA σB) H(ερA), (46)
| ⊗ ≥ |
H(ερA σB) H(ερA). (47)
|b ⊗b ≥ |b
Proof. From (2), (3) and (4), we have b b b
H(ερA)=inf a liminfTrρA ρA >e−naI 1 ε , (48)
| n (cid:12) n→∞ n (cid:8) n n(cid:9)≥ − o
H(ερbA)=sup a(cid:12) limsupTrρA ρA >e−naI ε , (49)
| (cid:26) (cid:12) n→∞ n (cid:8) n n(cid:9)≤ (cid:27)
and b (cid:12) (50)
H(ερA σB)=inf a liminfTr(ρA σB) ρA σB >e−naI 1 ε , (51)
| ⊗ n (cid:12) n→∞ n ⊗ n (cid:8) n ⊗ n n(cid:9)≥ − o
H(ερbA σbB)=sup a(cid:12) limsupTr(ρA σB) ρA σB >e−naI ε . (52)
| ⊗ (cid:26) (cid:12) n→∞ n ⊗ n (cid:8) n ⊗ n n(cid:9)≤ (cid:27)
b b (cid:12)
Let
ρA = lA φA φA , (53)
n n,k| n,kih n,k|
Xk
σB = lB φB φB (54)
n n,l| n,lih n,l|
Xl
be spectral decompositions of ρA and ρB. Then (48), (49), (51) and (52) can be rewritten as,
n n
H(ερA)=inf a liminf lA 1 ε , (55)
| (cid:26) n→∞ n,k ≥ − (cid:27)
(cid:12) Xk
b (cid:12) −1 loglA ≤a
n n,k
H(ερA)=sup a limsup lA ε , (56)
| (cid:26) (cid:12) n→∞ Xk n,k ≤ (cid:27)
b (cid:12) −1 loglA ≤a
n n,k
H(ερA σB)=inf a liminf lA lB 1 ε , (57)
| ⊗ (cid:26) n→∞ n,k n,l ≥ − (cid:27)
(cid:12) Xk,l
b b (cid:12) −1 loglA −1 loglB ≤a
n n,k n n,l
H(ερA σB)=sup a limsup lA lB ε . (58)
| ⊗ (cid:26) (cid:12) n→∞ Xk,l n,k n,l ≤ (cid:27)
b b (cid:12) −1 loglA −1loglB ≤a
n n,k n n,l
First, we prove (46).
1 ε lA lB lA lB = lA , (59)
− ≤ n,k n,l ≤ n,k n,l n,k
Xk,l Xk,l Xk
−n1loglAn,k−n1 loglBn,l≤a −n1 loglAn,k≤a −n1 loglAn,k≤a
taking liminf of (59) gives,
n→∞
1 ε liminf lA lB liminf lA , (60)
− ≤ n→∞ n,k n,l ≤ n→∞ n,k
Xk,l Xk
−n1 loglAn,k−n1 loglBn,l≤a −n1loglAn,k≤a
which implies (46) from (55) and (57).
Next, we prove (47).
lA lB lA lB = lA ε, (61)
n,k n,l ≤ n,k n,l n,k ≤
Xk,l Xk,l Xk
−n1loglAn,k−n1 loglBn,l≤a −n1 loglAn,k≤a −n1loglAn,k≤a
10