Useful states and entanglement distillation

0.64 MB·

by  Felix Leditzky

Useful states and entanglement distillation
Felix Leditzky∗a, Nilanjana Datta†b, and Graeme Smith‡a,c
aJILA, University of Colorado/NIST, 440 UCB, Boulder, CO 80309, USA
bStatistical Laboratory, Centre for Mathematical Sciences, University of Cambridge,
Cambridge CB3 0WB, UK
cDepartment of Physics, University of Colorado, 390 UCB, Boulder, CO 80309, USA
April 5, 2017
7
1
0
2
Abstract
r
p
We derive general upper bounds on the distillable entanglement of a mixed state under
A
one-way and two-way LOCC. In both cases, the upper bound is based on a convex decom-

4 position of the state into ‘useful’ and ‘useless’ quantum states. By ‘useful’, we mean a state
whose distillable entanglement is non-negative and equal to its coherent information (and

] thus given by a single-letter, tractable formula). On the other hand, ‘useless’ states are
h
p undistillable, i.e., their distillable entanglement is zero. We prove that in both settings the
- distillable entanglement is convex on such decompositions. Hence, an upper bound on the
t
n distillable entanglement is obtained from the contributions of the useful states alone, being
a equal to the convex combination of their coherent informations. Optimizing over all such
u
decompositions of the input state yields our upper bound. The useful and useless states
q
are given by degradable and antidegradable states in the one-way LOCC setting, and by
[
maximally correlated and PPT states in the two-way LOCC setting, respectively. We also
2
extendourmethodtoquantumchannels, obtaininganupperboundonthequantumcapac-
v
ity based on a decomposition of the quantum channel into degradable and antidegradable
1
8 completely positive maps.
0 Interpreting our upper bound as a convex roof extension, we show that it reduces to
3 a particularly simple, non-convex optimization problem for the classes of isotropic states
0
and Werner states. In the one-way LOCC setting, this non-convex optimization yields an
.
1 upper bound on the quantum capacity of the qubit depolarizing channel that is strictly
0 tighterthanpreviouslyknownboundsforlargevaluesofthedepolarizingparameter. Inthe
7
two-way LOCC setting, the non-convex optimization achieves the PPT-relative entropy of
1
entanglement for both isotropic and Werner states.
:
v
i
X
1 Introduction
r
a
1.1 Entanglement distillation
Entanglement is an integral part of quantum information theory and quantum mechanics, act-
ing as an indispensable resource for quantum information protocols such as teleportation [5],
superdense coding [1], or entanglement-assisted classical [2] and quantum [10] communication
through quantum channels. In these protocols, the entanglement resource is usually assumed
∗Email: [email protected]
†Email: [email protected]
‡Email: [email protected]
1
to have the special form of independent and identically distributed (i.i.d.) copies of an ebit
|Φ+(cid:105) := √1 (|00(cid:105)+|11(cid:105)), that is, a pure maximally entangled state between two qubits. This
2
assumptionsimplifiestheaforementionedprotocolsandmakesthemamenabletoadetailedthe-
oretical analysis as well as experimental realization in the laboratory. It is therefore important
to find entanglement distillation protocols, which convert n copies of a noisy or mixed bipartite
entangled state into m ebits Φ to arbitrary precision with increasing n.
n +
In a general entanglement distillation protocol, two parties (say, Alice and Bob) are allowed
touselocaloperationsandclassicalcommunication(LOCC).Oneusuallydistinguishesbetween
the following two settings: either the classical communication is restricted to only one-way
communicationfromAlicetoBob,ortwo-waycommunicationbetweenAliceandBobispossible.
In both settings, Alice and Bob initially share n copies of a mixed bipartite state ρ , and their
AB
goal is to obtain, via one-way or two-way LOCC, a state that is close to Φ⊗mn with respect
+
to a suitable distance measure (such as the purified distance [29]). If the distance between
the final and the target state vanishes asymptotically, then the asymptotic rate at which ebits
are generated, lim m /n, is called an achievable rate for one-way (two-way) entanglement
n→∞ n
distillation. Theone-way distillable entanglement D (ρ )isdefinedasthesupremumoverall
→ AB
achievableratesunderone-wayLOCC.Likewise,thetwo-way distillable entanglement D (ρ )
↔ AB
is defined as the supremum over all achievable rates under two-way LOCC. Since every one-way
LOCC operation is also a two-way LOCC operation, we have for all bipartite states ρ that
AB
D (ρ ) ≤ D (ρ ).
→ AB ↔ AB
Devetak and Winter [12] proved the hashing bound, establishing the coherent information as
anachievablerateforone-wayentanglementdistillation(andthusalsofortwo-wayentanglement
distillation):
D (ρ ) ≥ I(A(cid:105)B) , (1.1)
→ AB ρ
where the coherent information is defined as I(A(cid:105)B) := S(B) −S(AB) , with the von Neu-
ρ ρ ρ
mann entropy S(A) := −Tr(ρ logρ ). Furthermore, they derived the following regularized
ρ A A
formulae for the distillable entanglement under one-way and two-way LOCC [12]:
1
D (ρ ) = lim D(1)(ρ⊗n) (1.2)
→ AB n→∞ n → AB
1
D (ρ ) = lim D(1)(ρ⊗n). (1.3)
↔ AB n→∞ n ↔ AB
(1)
Here, D (·) for ∗ ∈ {→,↔} is defined as
∗
D(1)(ρ ) := max I(A(cid:48)(cid:105)B(cid:48)) ,
∗ AB Λ(ρ)
Λ:AB→A(cid:48)B(cid:48)
where the maximization is over one-way and two-way LOCC operations Λ: AB → A(cid:48)B(cid:48), re-
spectively.
Similar to the quantum capacity, the regularizations in (1.2) and (1.3) render the distillable
entanglement intractable to compute in most cases. Hence, it is desirable to identify classes of
bipartite states for which the formulae in (1.2) and (1.3) reduce to single-letter formulae that
can be easily computed. Moreover, we are interested in computable upper bounds on D (ρ )
→ AB
and D (ρ ) for arbitrary bipartite states. We address both problems in the present paper.
↔ AB
2
1.2 Method and main results
Toobtaincomputableupperboundsontheregularizedformulae(1.2)and(1.3)forthedistillable
entanglementunderone-wayandtwo-wayLOCC,wefirstidentifyclassesof‘useful’and‘useless’
(1)
states in both settings. Here, we call a state ρ useful, if D (ρ ) is equal to the coherent
AB ∗ AB
information I(A(cid:105)B) for ∗ ∈ {→,↔}, and thus additive on tensor products ρ⊗n. It then
ρ AB
follows immediately from (1.2) and (1.3) that also D (ρ ) = I(A(cid:105)B) . In the one-way setting,
∗ AB ρ
the useful states are degradable states (DEG) (cf. Definition 2.2), while in the two-way setting
the useful states are maximally correlated states (MC) (cf. Definition 3.1). Note that we have
MC ⊆ DEG.
On the other hand, useless states σ are such that D(1)(σ⊗n) is zero for all n ∈ N,
AB ∗ AB
from which D (σ ) = 0 follows. The class of useless states is given by antidegradable states
∗ AB
(cf.Definition2.2)intheone-waysetting, andbystateswithpositivepartialtranspose(orPPT
states for short) in the two-way setting. We list the four classes of states in Table 1 below.
useful useless DEG degradable
ADG antidegradable
1-way DEG ADG
MC maximally correlated
2-way MC PPT PPT positive partial transpose
Table 1: Useful and useless states for one-way and two-way entanglement distillation.
The crucial step in proving our main results is to observe that D (·) is convex on convex
∗
combinations of the corresponding useful and useless states. This is proved in Proposition 2.7
for the one-way setting by adapting an argument by Wolf and P´erez-Garc´ıa [39], and in Propo-
sition 3.6 for the two-way setting inspired by an argument by Rains [22]. Together with the
knownvaluesofthedistillableentanglementonusefulanduselessstates(givenbytheircoherent
information and 0, respectively), this proves the upper bounds on the one-way distillable en-
tanglement in Theorem 2.8 and on the two-way distillable entanglement in Theorem 3.7, which
constitute our main result. We note that in both settings the class of useful states includes all
pure quantum states (that is, every pure state is both degradable and maximally correlated).
Hence, any pure-state ensemble of a bipartite state yields a decomposition into useful states.
In both settings, the optimal such pure-state ensemble yields the entanglement of formation,
and our upper bounds can be understood as an improvement over the latter. Moreover, in
the one-way setting we extend our result to quantum channels, obtaining in Theorem 2.14 an
analogous upper bound on the quantum capacity of a quantum channel.
Finally, we focus on the distillable entanglement of isotropic states and Werner states. In-
terpreting our upper bounds on the distillable entanglement as convex roof extensions allows
us to use a result by Vollbrecht and Werner [32] that exploits the symmetries of isotropic states
and Werner states to facilitate the computation of the convex roof extension. The result is a
simplification of our upper bound to a (non-convex) optimization problem that can be solved
numerically for small dimensions. In particular, this yields an upper bound on the quantum
capacity of the qubit depolarizing channel that is tighter than the best previously known upper
bound for large values of the depolarizing parameter.
The rest of this paper is structured as follows. We first fix some notation in Section 1.3. We
then dedicate Section 2 to developing the method outlined above for one-way entanglement dis-
tillation. Furthermore, we introduce and discuss the notion of approximately (anti)degradable
states in Section 2.5, which is inspired by and analogous to the notion of approximately degrad-
able quantum channels in [25]. The derivation of our main result for two-way entanglement
distillation is carried out in Section 3. Apart from the results mentioned above, we also discuss
3
a method for constructing decompositions into maximally correlated states via the generalized
Bell basis in Section 3.4. In Section 4 we derive the non-convex optimization form of our upper
bounds for isotropic and Werner states. Finally, we give some concluding remarks in Section 5.
In Appendix A we derive a ‘pretty strong’ converse for the one-way distillable entanglement
of a degradable state. This is the analogue for entanglement distillation of the pretty strong
converse for entanglement generation using a degradable quantum channel obtained by Morgan
and Winter [18]. Appendix B contains a discussion of antidegradable states and their maximal
overlap with maximally entangled states.
1.3 Notation
Throughout the paper we only consider finite-dimensional Hilbert spaces. For Hilbert spaces
H andH , wedenotebyB(H ,H )thesetoflinearmapsfromH toH , andwewriteB(H) =
1 2 1 2 1 2
B(H,H) for the algebra of linear operators on a single Hilbert space H. Upper-case indices are
used to label quantum systems: for a Hilbert space H corresponding to a quantum system A,
A
we write |ψ(cid:105) ∈ H and ρ ∈ B(H ), and we use the notation H := H ⊗H ⊗.... We
A A A A A1A2... A1 A2
write |A| := dimH for the dimension of a quantum system A with associated Hilbert space
A
∼
H , and rkρ for the rank of the operator ρ . We use the shorthand A = B to indicate that
A A A
∼
the Hilbert spaces associated to A and B are isomorphic, H = H . A quantum state (or
A B
simply state) is an operator ρ ∈ B(H ) with ρ ≥ 0 and Trρ = 1, and we denote the set of
A A A A
states on H by D(H ). We write ψ ≡ |ψ(cid:105)(cid:104)ψ| ∈ B(H ) for the rank-1 projector associated
A A A A A
to the pure state |ψ(cid:105) ∈ H .
A A
Thevon Neumann entropy ofastateρ isdefinedbyS(A) := −Tr(ρ logρ ), thecoherent
A ρ A A
information of a bipartite state ρ by I(A(cid:105)B) := S(B) − S(AB) , and the conditional
AB ρ ρ ρ
entropy by S(A|B) := −I(A(cid:105)B) . For a probability distribution {p } , the Shannon entropy
ρ ρ i i
(cid:80)
is defined by H({p } ) := − p logp . For p ∈ [0,1], the binary entropy is defined by h(p) :=
i i i i i
−plogp−(1−p)log(1−p). All exponentials and logarithms are taken to base 2.
AquantumchannelN: B(H) → B(K)isalinear, completelypositive(CP),trace-preserving
(TP) map between the algebras B(H) and B(K) of linear operators on Hilbert spaces H and K.
We write N: A → B for a quantum channel from B(H ) to B(H ). Let N(ρ ) = Tr (Vρ V†)
A B A E A
be the Stinespring representation of N with the isometry V : H → H ⊗ H . Then the
A B E
complementary channel Nc: A → E is defined by Nc(ρ ) := Tr (Vρ V†). We often omit the
A B A
identity map denoted by id, i.e., for a map T: A → A(cid:48) acting on the A part of a state ρ , we
AB
also write T(ρ ) instead of (T ⊗id )(ρ ).
AB B AB
Let {|Φ (cid:105)} be the generalized Bell basis defined as follows. We define the
n,m n,m=0,...,d−1
generalized Pauli operators X and Z via their action on a fixed basis {|k(cid:105)}d−1 of Cd,
k=0
X|k(cid:105) := |k+1(mod d)(cid:105) Z|k(cid:105) := ωk|k(cid:105), (1.4)
where ω := exp(2πi/d) is a d-th root of unity. The generalized Pauli operators satisfy XZ =
ωZX. Setting |Φ+(cid:105) := √1d (cid:80)di=−01|ii(cid:105), we define
|Φ (cid:105) := (1 ⊗XmZn)|Φ (cid:105), (1.5)
n,m d +
which satisfy (cid:104)Φ |Φ (cid:105) = δ δ .
n,m n(cid:48),m(cid:48) n,n(cid:48) m,m(cid:48)
(cid:80)
Finally, for a vector |ψ(cid:105) = λ |i(cid:105) ⊗|j(cid:105) ∈ H ⊗H , we define an associated operator
i,j ij A B A B
op(ψ ) ∈ B(H ,H ) by
AB B A
(cid:88)
op(ψ ) := λ |i(cid:105) (cid:104)j| . (1.6)
AB ij A B
i,j
4
2 One-way entanglement distillation
2.1 Operational setting
Given a mixed bipartite state ρ , the one-way distillable entanglement D (ρ ) is defined as
AB → AB
the optimal rate of distilling ebits from many copies of ρ via local operations and forward
AB
(or one-way) classical communication (LOCC) from Alice to Bob.
Ageneralone-wayLOCCoperationcanbemodeledasaquantum instrument T: A → A(cid:48)M,
defined by
(cid:88)
T(θ ) := T (θ )⊗|m(cid:105)(cid:104)m| ,
A m A M
m
where {|m(cid:105)} is an orthonormal basis for the classical register M, and for each m the map
m
T : A → A(cid:48) is CP such that (cid:80) T is TP.
m m m
Asmentionedintheintroduction, DevetakandWinter[12]derivedthefollowingregularized
formula for the one-way distillable entanglement:
1
D (ρ ) = lim D(1)(ρ⊗n),
→ AB n→∞ n → AB
(1)
where D (ρ ) can be expressed as
→ AB
(cid:88)
D(1)(ρ ) := max λ I(A(cid:48)(cid:105)B) . (2.1)
→ AB T m m ρm
Here, the maximization is over instruments T: A → A(cid:48)M, and we set ρ := 1 T (ρ ), with
m λm m AB
λ := Tr(T (ρ )) denoting the probability of obtaining the outcome m of T. Equivalently,
m m AB
(2.1) can be written as
D(1)(ρ ) = maxI(A(cid:48)(cid:105)BM) . (2.2)
→ AB T T(ρAB)
Instead of maximizing the coherent information in (2.2) over instruments T, it can be more
convenienttoconsideramaximizationoverisometricextensionsofaninstrumentinthefollowing
(cid:80)
way. First, we note that it suffices to consider instruments T = T ⊗|m(cid:105)(cid:104)m| where each of
m m
the CP maps T has only one Kraus operator, i.e., T (·) = K ·K† for each m and operators
m m m m
K : A → A(cid:48) [12]. In this case, an isometric extension V : A → A(cid:48)MN can be defined as1
m
(cid:88)
V := K ⊗|m(cid:105) ⊗|m(cid:105) (2.3)
m M N
m
for a classical register N ∼= M. Since (cid:80) T = (cid:80) K ·K† is TP by definition of a quantum
m m m m m
instrument, we have (cid:80) K† K = 1 , which implies V†V = 1 . Hence, V is indeed an
m m m A A
isometry, and we have T(ρ ) = Tr (Vρ V†) for all ρ . Using (2.3), we can write (2.2) as
A N A A
D(1)(ρ ) = maxI(A(cid:48)(cid:105)BM) , (2.4)
→ AB ω
V
where ω = Tr (Vρ V†).
A(cid:48)BM N AB
(1)
Lemma 2.1. D (ρ ) ≥ 0 for all bipartite states ρ .
→ AB AB
1In the general case where each TP map T might have more than one Kraus operator, say {K } , an
m m,j j
isometric extension can be defined by including an additional system F with orthonormal basis {|φ (cid:105) }
m,j F m,j
that acts as the environment for each T .
m
5
(1)
Proof. Since D (ρ ) can be expressed as a maximization over all instrument isometries V of
→ AB
the form (2.3) as stated in (2.4), the lemma is proved by constructing a particular V for which
we obtain I(A(cid:48)(cid:105)BM) = 0 with ω = Tr (Vρ V†).
ω A(cid:48)BM N AB
To this end, let |φ(cid:105) be a purification of ρ , and consider a Schmidt decomposition of
ABE AB
|φ(cid:105) with respect to the bipartition A|BE,
ABE
(cid:88)
|φ(cid:105) := λ |i(cid:105) |i(cid:105) ,
ABE i A BE
i
where the Schmidt coefficients λ ≥ 0 for all i. We define the instrument isometry V : A →
i
A(cid:48)MN,
(cid:88)
V := |i(cid:105) (cid:104)i| ⊗|i(cid:105) ⊗|i(cid:105) ,
A(cid:48) A M N
i
where A(cid:48) ∼= A. Applying V to the purification |φ(cid:105) of ρ , we obtain the pure state
ABE AB
(cid:88)
|ω(cid:105) = V|φ(cid:105) = λ |iii(cid:105) |i(cid:105) ,
A(cid:48)MNBE ABE i A(cid:48)MN BE
i
whose marginals ω and ω are given by
BM EN
(cid:88) (cid:88)
ω = λ2|i(cid:105)(cid:104)i| ⊗Tr |i(cid:105)(cid:104)i| ω = λ2|i(cid:105)(cid:104)i| ⊗Tr |i(cid:105)(cid:104)i| .
BM i M E BE EN i N B BE
i i
Evaluating the coherent information of the state ω yields
A(cid:48)BM
I(A(cid:48)(cid:105)BM) = S(BM) −S(A(cid:48)BM)
ω ω ω
= S(BM) −S(EN)
ω ω
(cid:88)
= λ2(S(Tr |i(cid:105)(cid:104)i| )−S(Tr |i(cid:105)(cid:104)i| ))
i E BE B BE
i
= 0,
which proves the claim.
2.2 (Conjugate) degradable and antidegradable states
We now define the classes of ‘useful’ and ‘useless’ states for one-way entanglement distillation,
as explained in Section 1.2.
Definition 2.2. Let ρ be a bipartite state with purification |φ(cid:105) . The state ρ is called:
AB ABE AB
(i) degradable, if there is an isometry U: B → E(cid:48)G with E(cid:48) ∼= E such that for the state
|ϕ(cid:105) = U|φ(cid:105) we have
AE(cid:48)GE ABE
ϕ = ϕ = φ ; (2.5)
AE AE(cid:48) AE
(ii) conjugate degradable, if (2.5) holds up to complex conjugation, that is,
ϕ = C(ϕ ) = φ ,
AE AE(cid:48) AE
where C denotes entry-wise complex conjugation with respect to a fixed basis of E(cid:48) ∼= E;
(iii) antidegradable, if there is an isometry V : E → B(cid:48)F with B(cid:48) ∼= B such that for the state
|ψ(cid:105) = V|φ(cid:105) we have
ABB(cid:48)F ABE
ψ = ψ = φ .
AB(cid:48) AB AB
6
We note that Definition 2.2 is independent of the chosen purification of ρ , since any two
AB
purifications of ρ are related by an isometry acting only on the purifying systems. We can
AB
then compose the (conjugate) (anti)degrading isometries from Definition 2.2 with the isometry
relating the different purifications.
The coherent information of a degradable state ρ is non-negative, I(A(cid:105)B) ≥ 0, since
AB ρ
I(A(cid:105)B) = I(A(cid:105)E(cid:48)G) ≥ I(A(cid:105)E(cid:48)) = I(A(cid:105)E) = −I(A(cid:105)B) ,
ρ ϕ ϕ φ ρ
whereweusedthedataprocessinginequalityforthecoherentinformationinthefirstinequality,
and the duality relation I(A(cid:105)B) = −I(A(cid:105)E) for a pure state |ψ(cid:105) in the second inequality.
ψ ψ ABE
Using a similar argument, an antidegradable state σ has non-positive coherent information,
AB
I(A(cid:105)B) ≤ 0. Symmetric states, which are both degradable and antidegradable, therefore have
σ
zero coherent information.
Everypurestate|ψ(cid:105) isdegradable,whichcanbeseenbychoosinganarbitrarypurification
AB
|φ(cid:105) = |ψ(cid:105) ⊗|χ(cid:105) with some pure state |χ(cid:105) , and considering the isometry U defined by
ABE AB E E
U|θ(cid:105) := |θ(cid:105) ⊗|χ(cid:105) . Alargeclassofmixed(conjugate)(anti)degradablestatescanbeobtained
B B E
from (conjugate) (anti)degradable quantum channels. We call a quantum channel N: A → B
degradable, if there exists a quantum channel D: B → E (called a degrading map) such that
Nc = D◦N. (2.6)
The channel N is conjugate degradable [7], if instead of (2.6) we have
C ◦Nc = D◦N,
whereC denotesentry-wisecomplexconjugationwithrespecttoafixedbasisasinDefinition2.2.
Finally,achannelN iscalledantidegradable,ifthereexistsaquantumchannelA: E → B (called
an antidegrading map) such that
N = A◦Nc.
Let now |Φ(cid:105) be a maximally entangled state between A(cid:48) ∼= A and A, then the Choi state
A(cid:48)A
τ of N: A → B is defined as
A(cid:48)B
τ := N(Φ ). (2.7)
A(cid:48)B A(cid:48)A
Similarly, we define the Choi state τ = Nc(Φ ) of the complementary channel Nc. The
A(cid:48)E A(cid:48)A
following result is obvious:
Lemma 2.3. Let N: A → B be a quantum channel. Then the Choi state τ as defined in
A(cid:48)B
(2.7) is (conjugate) (anti)degradable if and only if N is (conjugate) (anti)degradable.
Finally, we note that we occasionally simplify Definition 2.2 to the following (equivalent)
form, whichiscloselyrelatedtothechannelpictureabove: astateρ withpurification|φ(cid:105)
AB ABE
and‘complementarystate’ρ := Tr φ isdegradableifthereexistsaCPTPdegradingmap
AE B ABE
D: B → E such that ρ = D(ρ ). In this case, the degrading isometry U from Definition 2.2
AE AB
can be chosen as the Stinespring isometry (cf. Section 1.3) of D. Conversely, every degrading
isometry as in Definition 2.2 gives rise to a degrading map by defining D(·) := Tr (V · V†)
G
and identifying E with E(cid:48). We also use analogous simplifications in the case of conjugate
degradability and antidegradability.
7
2.3 Upper bounds on the one-way distillable entanglement
The hashing bound (1.1) states that for any state ρ the coherent information I(A(cid:105)B) is an
AB ρ
achievable rate for one-way entanglement distillation. The first result of this section shows that
for (conjugate) degradable states the coherent information is the optimal rate for entanglement
distillation:
(1)
Proposition 2.4. Let ρ be a (conjugate) degradable state. Then D (ρ ) is equal to the
AB → AB
coherent information I(A(cid:105)B) and thus additive: for all n ∈ N,
ρ
D(1)(ρ⊗n) = nD(1)(ρ ) = nI(A(cid:105)B) .
→ AB → AB ρ
Hence, the one-way distillable entanglement of ρ is equal to the coherent information,
AB
D (ρ ) = I(A(cid:105)B) . (2.8)
→ AB ρ
Proof. Let us first assume that ρ is degradable, that is, we have
AB
ϕ = ϕ (2.9)
AE AE(cid:48)
where |ϕ(cid:105) = W|φ(cid:105) and W: B → E(cid:48)G is a degrading isometry. Let us furthermore
AE(cid:48)GE ABE
define the following pure states:
|ω(cid:105) := V|φ(cid:105)
A(cid:48)MNBE ABE
|σ(cid:105) := W|ω(cid:105) = V|ϕ(cid:105) ,
A(cid:48)MNE(cid:48)GE A(cid:48)MNBE AE(cid:48)GE
where V : A → A(cid:48)MN is given as in (2.3). Consider now the following steps:
I(A(cid:48)(cid:105)BM) = I(A(cid:48)(cid:105)E(cid:48)GM)
ω σ
= S(E(cid:48)GM) −S(A(cid:48)E(cid:48)GM)
σ σ
= S(E(cid:48)GM) −S(NE)
σ σ
= S(E(cid:48)GM) −S(ME)
σ σ
= S(E(cid:48)GM) −S(ME(cid:48)) (2.10)
σ σ
= S(G|E(cid:48)M)
σ
≤ S(G|E(cid:48))
σ
= S(GE(cid:48)) −S(E(cid:48))
σ σ
= S(GE(cid:48)) −S(E) (2.11)
σ σ
= S(GE(cid:48)) −S(A(cid:48)MNE(cid:48)G)
σ σ
= S(B) −S(AB)
ρ ρ
= I(A(cid:105)B)
ρ
where the third line follows from the fact that σ is a pure state, the fourth line follows
A(cid:48)MNE(cid:48)GE
from the symmetry in M and N (which is evident from the definition (2.3) of the isometry V),
the fifth line follows from the degradability (2.9) of the state ρ , the seventh line follows from
AB
the fact that conditioning reduces entropy, the ninth line follows again from the degradability
of ρ , and the tenth line follows from the fact that σ is pure.
AB A(cid:48)MNE(cid:48)GE
Hence, the trivial isometry achieves the maximum in max I(A(cid:48)(cid:105)BM) , and D(1)(ρ ) =
V ω → AB
I(A(cid:105)B) . Sincethecoherentinformationisadditiveontensorproducts,wethenhaveD(1)(ρ⊗n) =
ρ → AB
nI(A(cid:105)B) , and (2.8) follows from (1.2).
ρ
8
If ρ is only conjugate degradable, (2.9) is replaced by
AB
ϕ = C(ϕ ), (2.12)
AE AE(cid:48)
where C denotes entry-wise complex conjugation with respect to a fixed basis. Using (2.12)
instead of (2.9) in steps (2.10) and (2.11) above, together with the fact that the von Neumann
entropy is invariant under complex conjugation, we obtain the claim in the case of conjugate
degradability of ρ .
AB
The following lemma shows the well-known fact (see e.g. [3]) that the one-way distillable
entanglementofantidegradablestatesis0. Weprovideashortproofforthesakeofcompleteness.
(1)
Lemma 2.5. Let σ be an antidegradable state. Then D (σ ) = 0 and D (σ ) = 0.
AB → AB → AB
Proof. Let |ψσ(cid:105) be a purification of σ , and denote by A: E → B the antidegrading map
ABE AB
satisfying σ = A(σ ), where σ = Tr ψσ. Let V : A → A(cid:48)MN be an arbitrary isometry
AB AE AE B
of the form (2.3), and consider the following steps for the coherent information evaluated on
the state Vσ V†:
AB
I(A(cid:48)(cid:105)BM) ≤ I(A(cid:48)(cid:105)EM)
= −I(A(cid:48)(cid:105)BN)
= −I(A(cid:48)(cid:105)BM),
where we used data processing with respect to A in the inequality, duality for the coherent
information in the first equality, and symmetry in M ↔ N in the second inequality. It follows
that I(A(cid:48)(cid:105)BM) ≤ 0 for any instrument isometry V, and hence, D(1)(σ ) ≤ 0. Together
→ AB
with Lemma 2.1, this proves D(1)(σ ) = 0 for all antidegradable states σ . Since σ⊗n
→ AB AB AB
is antidegradable for all n ∈ N (with the antidegrading map given by A⊗n), we then have
D (σ ) = 0.
→ AB
We now derive a general upper bound on the one-way distillable entanglement of arbitrary,
notnecessarilydegradable,bipartitestates. Tothisend,wefirstprovethefollowingproposition,
(1)
which shows that we can ignore the contributions from antidegradable states for the D (·)
→
quantity.
Proposition 2.6. Let ρ be degradable and σ be antidegradable. Then
A1B1 A2B2
D(1)(ρ ⊗σ ) = D(1)(ρ ).
→ A1B1 A2B2 → A1B1
(1) (1)
Proof. We first observe that D (ρ ⊗σ ) ≥ D (ρ ) holds for any two states ρ
→ A1B1 A2B2 → A1B1 A1B1
and σ not necessarily degradable and antidegradable. This follows from extending an
A2B2
optimal instrument for ρ trivially to A and using the data processing inequality for the
A1B1 2
coherent information with respect to tracing out the B system.
2
To prove the other inequality, let ρ with purification |ψρ(cid:105) be degradable with
A1B1 A1B1E1
degrading map D: B → E , and let σ with purification |ψσ(cid:105) be antidegradable with
1 1 A2B2 A2B2E2
antidegrading isometry W: E → B(cid:48)E(cid:48) such that |τ(cid:105) := W|ψσ(cid:105) satisfies
2 2 2 A2B2B2(cid:48)E2(cid:48) A2B2E2
τ = τ = σ .
A2B2(cid:48) A2B2 A2B2
Denoting by F the swap operator exchanging B and B(cid:48), we define the state
B2B2(cid:48) 2 2
1 (cid:16) (cid:17)
|Ω(cid:105) := √ |τ(cid:105) ⊗|0(cid:105) +F |τ(cid:105) ⊗|1(cid:105) ,
A2B2B2(cid:48)E2(cid:48)CB 2 A2B2B2(cid:48)E2(cid:48) CB B2B2(cid:48) A2B2B2(cid:48)E2(cid:48) CB
9
which satisfies |Ω(cid:105) = F ⊗X |Ω(cid:105) and Ω = σ . Here, X denotes the Pauli X
B2B2(cid:48) CB A2B2 A2B2 CB
operator on the C system. Let T: A A → A(cid:48)M be an arbitrary instrument with isometry
B 1 2
V : A A → A(cid:48)MN, then we have
1 2
I(B ;MB ) ≥ I(E ;NB(cid:48))
1 2 T(ρ⊗τ) 1 2 (D◦T)(ρ⊗τ)
⇐⇒ S(B )+S(MB )−S(E )−S(NB(cid:48)) ≥ S(MB B )−S(NE B(cid:48)) (2.13)
1 2 1 2 1 2 1 2
by the data processing inequality for the mutual information with respect to D, and because
S(MB ) = S(NB(cid:48)) . (2.14)
2 T(ρ⊗τ) 2 T(ρ⊗τ)
Consider now the following steps:
I(A(cid:48)(cid:105)MB B ) = S(MB B ) −S(A(cid:48)MB B )
1 2 T(ρ⊗σ) 1 2 T(ρ⊗σ) 1 2 T(ρ⊗σ)
= S(MB B )−S(NE B(cid:48)E(cid:48)C )
1 2 1 2 2 B T(ρ⊗τ)
= S(MB B )−S(NE B(cid:48))+S(NE B(cid:48))−S(NE B(cid:48)E(cid:48)C )
1 2 1 2 1 2 1 2 2 B
≤ S(B )−S(E )+S(MB )−S(NB(cid:48))+I(E(cid:48)C (cid:105)NE B(cid:48))
1 1 2 2 2 B 1 2
= I(A (cid:105)B )+I(E(cid:48)C (cid:105)NE B(cid:48)), (2.15)
1 1 2 B 1 2
whereweused(2.13)intheinequality, andonceagain(2.14)inthelastequality. Forthesecond
coherent information in (2.15), observe that
I(E(cid:48)C (cid:105)NE B(cid:48)) ≤ I(E(cid:48)C (cid:105)NB B(cid:48))
2 B 1 2 2 B 1 2
= I(E(cid:48)C (cid:105)MB B )
2 B 1 2
= −I(E(cid:48)C (cid:105)A(cid:48)NE B(cid:48))
2 B 1 2
≤ −I(E(cid:48)C (cid:105)NE B(cid:48)).
2 B 1 2
Here, the first and second inequality follow from the data processing inequality for the coherent
information with respect to D and partial trace over A(cid:48), respectively. The second line follows
fromsymmetryofV(|ψρ(cid:105)⊗|Ω(cid:105))inM ↔ N, andfromtheinvarianceofthecoherentinformation
under the local unitary F ⊗X . Hence, I(E(cid:48)C (cid:105)NE B(cid:48)) ≤ 0, and (2.15) yields
B2B2(cid:48) CB 2 B 1 2
D(1)(ρ ⊗σ ) = maxI(A(cid:48)(cid:105)MB B ) ≤ I(A (cid:105)B ) = D(1)(ρ ),
→ A1B1 A2B2 T 1 2 1 1 → A1B1
which we set out to prove.
The last ingredient for our general upper bound on the one-way distillable entanglement
D (·) is Proposition 2.7 below, which establishes that D (·) is convex on mixtures of states
→ →
(1)
whose tensor products have subadditive D (·). This result is analogous to the corresponding
→
property of the quantum capacity proved by Wolf and P´erez-Garc´ıa [39], and our proof of
Proposition 2.7 closely follows the one given in [39]. We introduce the following notation: For
a binary string wn = (w ,...,w ) ∈ {0,1}n, we denote by |wn| := |{i: w = 1}| the Hamming
1 n i
weight ofwn, i.e.thenumberof1’sinwn. Forstatesρ andρ andwn = (w ,...,w ) ∈ {0,1}n,
0 1 1 n
we set ρ := ρ ⊗...⊗ρ . We then have the following:
wn w1 wn
Proposition 2.7. Let ρ and ρ be bipartite states on AB satisfying
0 1
(cid:88)
D(1)(ρ ) ≤ D(1)(ρ ) = (n−|wn|)D(1)(ρ )+|wn|D(1)(ρ ) (2.16)
→ wn i → wi → 0 → 1
for all wn ∈ {0,1}n and n ∈ N. Then for all p ∈ [0,1],
D (pρ +(1−p)ρ ) ≤ pD (ρ )+(1−p)D (ρ ).
→ 0 1 → 0 → 1
10