Entanglement Cost for Sequences of Arbitrary
Quantum States
Garry Bowen ∗and Nilanjana Datta†
January 10, 2019
Abstract
The entanglement cost of arbitrary sequences of bipartite states is
shown to be expressible as the minimization of a conditional spectral
entropy rate over sequences of separable extensions of the states in
the sequence. The expression is shown to reduce to the regularized
entanglement of formation when the nth state in the sequence consists
of n copies of a single bipartite state.
1 Introduction
A fundamental problem in entanglement theory is to determine how to op-
timally convert entanglement, shared between two distant parties Alice and
Bob, from one form to another. Entanglement manipulation is the process
by which Alice and Bob convert an initial bipartite state ρAB which they
share, to a required target state σAB using local operations and classical
communication (LOCC). If the target state σAB is a maximally entangled
state, then the protocol is called entanglement distillation, whereas if the
initial state ρAB is a maximally entangled state, then the protocol is called
entanglement dilution. Optimal rates of these protocols were originally eval-
uated under the assumption that the entanglement resource accessible to
Alice and Bob consist of multiple copies, i.e., tensor products ρ⊗nAB , of the
initial bipartite state ρAB, and the requirement that the final state of the
protocol is equal to n copies of the desired target state σ⊗nAB with asymptot-
ically vanishing error in the limit n→ ∞. The distillable entanglement and
entanglement cost computed in this manner are two asymptotic measures
of entanglement of the state ρAB . Moreover, in the case in which ρAB is
pure, these two measures of entanglement coincide and are equal to the von
∗Centre for Quantum Computation, DAMTP, University of Cambridge, Cambridge
CB3 0WA, UK
†Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge CB3
0WB, UK (e-mail:n.datta@statslab.cam.ac.uk)
http://arxiv.org/abs/0704.1957v3
Neumann entropy of the reduced state on any one of the subsystems, A or
In this paper we focus on entanglement dilution, which, as mentioned
earlier, is the entanglement manipulation process by which two distant par-
ties, say Alice and Bob, create a desired bipartite target state from a maxi-
mally entangled state which they initially share, using LOCC. In [6, 7] the
optimal rate of entanglement dilution, namely, the entanglement cost, was
evaluated in the case in which Alice and Bob created multiple copies of a
desired target state ρAB , with asymptotically vanishing error, from a shared
resource of singlets, using local operations and classical communication. In
particular, for the case of a pure target state ρAB , the entanglement cost
was shown [7] to be equal to the entropy of entanglement of the state, i.e.,
the von Neumann entropy of the reduced state on any one of the two sub-
systems A and B. Moreover, in [8] it was shown that for an arbitrary mixed
state ρAB, the entanglement cost is equal to the regularized entanglement
of formation of the state (see (10) for its definition).
The practical ability to transform entanglement from one form to another
is useful for many applications in quantum information theory. However, it
is not always justified to assume that the entanglement resource available
consists of states which are multiple copies (and hence tensor products) of
a given entangled state, or to require that the final state of the protocol
is of the tensor product form. More generally an entanglement resource
is characterized by an arbitrary sequence of bipartite states which are not
necessarily of the tensor product form. Sequences of bipartite states on AB
are considered to exist on Hilbert spaces H⊗nA ⊗H
B for n ∈ {1, 2, 3 . . .}.
A useful tool for the study of entanglement manipulation in this general
scenario is provided by the Information Spectrum method. The information
spectrum method, introduced in classical information theory by Verdu &
Han [1, 2], has been extended into quantum information theory by Hayashi,
Ogawa and Nagaoka [3, 4, 5]. The power of the information spectrum ap-
proach comes from the fact that it does not depend on the specific structure
of sources, channels or entanglement resources employed in information the-
oretical protocols.
In this paper we evaluate the optimal asymptotic rate of entanglement
dilution for an arbitrary sequence of bipartite states. The case of an arbitrary
sequence of pure bipartite state was studied in [9]. The paper is organized as
follows. In Section 2 we introduce the necessary notations and definitions.
Section 3 contains the statement and proof of the main result, stated as
Theorem 1. Note that if we consider the sequence of bipartite states to
consist of tensor products of a given bipartite state, then our main result
reduces to the known results obtained in [6, 8] (see the discussion after
Theorem 1 of Section 3). Finally, in Section 4 we show how Theorem 1 yields
an alternative proof of the equivalence of the asymptotic entanglement cost
and the regularised entanglement of formation [8].
2 Notations and definitions
LetB(H) denote the algebra of linear operators acting on a finite–dimensional
Hilbert space H of dimension d and let D(H) denote the set of states (or
density operators, i. e. positive operators of unit trace) acting on H. Fur-
ther, let H(n) denote the Hilbert space H⊗n. For any state ρ ∈ D(H), the
von Neumann entropy is defined as S(ρ) := −Tr (ρ log ρ).
Let Λn be a quantum operation used for the transformation of an initial
bipartite state ωn to a bipartite state ρn, with ωn, ρn ∈ D ((HA ⊗HB)⊗n).
For the entanglement manipulation processes considered in this paper, Λn
either consists of local operations (LO) alone or LO with one-way or two-
way classical communication. We define the efficacy of any entanglement
manipulation process in terms of the fidelity Fn := Tr
ρnΛn(ωn)
between the output state Λn(ωn) and the target state ρn. An entanglement
manipulation process is said to be reliable if the asymptotic fidelity F :=
lim infn→∞ Fn = 1.
For given orthonormal bases {|χiA〉}
i=1 and {|χiB〉}
i=1 in Hilbert spaces
A andH
B , of dimensions d
A and d
B respectively, we define the canonical
maximally entangled state of Schmidt rank Mn ≤ min{dnA, dnB} to be
|ΨMnAB〉 =
|χiA〉 ⊗ |χiB〉. (1)
In fact, in the following, we consider HA ≃ HB , for simplicity, so that
dnA = d
B . Here and henceforth, the explicit n-dependence of the basis states
|χiA〉 and |χiB〉 has been suppressed for notational simplicity.
The quantum information spectrum approach requires the extensive use
of spectral projections. Any self-adjoint operator A acting on a finite di-
mensional Hilbert space may be written in its spectral decomposition A =
i λiπi, where πi denotes the operator which projects onto the eigenspace
corresponding to the eigenvalue λi. We define the positive spectral projec-
tion on A as {A ≥ 0} =
πi, i.e., the projector onto the eigenspace of
positive eigenvalues of A. For two operators A and B, we can then define
{A ≥ B} as {A−B ≥ 0}. The following key lemmas are used repeatedly in
the paper. For their proofs see [3, 4].
Lemma 1. For self-adjoint operators A, B and any positive operator 0 ≤
P ≤ I the inequality
P (A−B)
A ≥ B
(A−B)
holds.
Lemma 2. Given a state ρn ∈ D(H⊗n) and a self-adjoint operator ωn ∈
B(H⊗n), we have
{ρn ≥ enγωn}ωn
≤ e−nγ . (3)
for any real number γ.
In the quantum information spectrum approach one defines spectral di-
vergence rates, which can be viewed as generalizations of the quantum rel-
ative entropy. The spectral generalizations of the von Neumann entropy,
the conditional entropy and the mutual information can all be expressed as
spectral divergence rates.
Definition 1. Given a sequence of states ρ̂ = {ρn}∞n=1, with ρn ∈ D(H⊗n),
and a sequence of positive operators ω̂ = {ωn}∞n=1, with ωn ∈ B(H⊗n),
the quantum spectral sup-(inf-) divergence rates are defined in terms of the
difference operators Πn(γ) = ρn − enγωn, for any arbitrary real number γ,
D(ρ̂‖ω̂) = inf
γ : lim sup
{Πn(γ) ≥ 0}Πn(γ)
D(ρ̂‖ω̂) = sup
γ : lim inf
{Πn(γ) ≥ 0}Πn(γ)
respectively.
The spectral entropy rates and the conditional spectral entropy rates
can be expressed as divergence rates with appropriate substitutions for the
sequence of operators ω̂ = {ωn}∞n=1. These are S(ρ̂) = −D(ρ̂‖Î) and S(ρ̂) =
−D(ρ̂‖Î), where Î = {In}∞n=1, with In being the identity operator acting
on the Hilbert space H⊗n. Further, for sequences of bipartite states ρ̂AB =
{ρnAB}∞n=1,
S(A|B) = −D(ρ̂AB‖ÎA ⊗ ρ̂B) (4)
S(A|B) = −D(ρ̂AB‖ÎA ⊗ ρ̂B) . (5)
In the above, ÎA = {InA}∞n=1 and ρ̂B = {ρnB}∞n=1, with InA being the identity
operator in B(H
A ) and ρ
B = TrAρ
AB, the partial trace being taken on the
Hilbert space H
A . Various properties of these quantities, and relationships
between them, are explored in [10].
For sequences of states ρ̂ = {ρ⊗n} and ω̂ = {ω⊗n}, with ρ, ω ∈ D(H), it
has been proved [5] that
D(ρ̂‖ω̂) = D(ρ̂‖ω̂) = S(ρ‖ω), (6)
where S(ρ‖ω) := Tr ρ log ρ− Tr ρ logω, is the quantum relative entropy.
Two parties, Alice and Bob, share a sequence of maximally entangled
states {|ΨMnAB〉}
n=1, and wish to convert them into a sequence of given bi-
partite states {ρnAB}∞n=1, with ρn ∈ D(H⊗n) and |Ψ
AB〉 ∈ H
B . The
protocol used for this conversion is known as entanglement dilution. The
concept of reliable entanglement manipulation may then be used to define
an asymptotic entanglement measure, namely the entanglement cost.
Definition 2. A real-valued number R is said to be an achievable dilu-
tion rate for a sequence of states ρ̂AB = {ρnAB}, with ρnAB ∈ D((HA ⊗
⊗n), if ∀ε > 0, ∃N such that ∀n ≥ N a transformation exists that takes
|ΨMnAB〉〈Ψ
AB | → ρnAB with fidelity F 2n ≥ 1− ε and
logMn ≤ R.
Definition 3. The entanglement cost of the sequence ρ̂AB is the infimum
of all achievable dilution rates:
EC(ρ̂AB) = inf R (7)
To simplify the expressions representing the entanglement cost, we define
the following sets of sequences of states. Firstly, given a sequence of target
states ρ̂AB = {ρnAB}∞n=1, define the set Dcq(ρ̂AB) as the set of sequences
of tripartite states ˆ̺RAB = {̺nRAB}∞n=1 such that each ̺nRAB is a classical-
quantum state (cq-state) of the form
̺nRAB =
R〉〈inR| ⊗ |φ
AB〉〈φ
AB |, (8)
where ρnAB =
AB〉〈φ
AB | and the set of pure states {|i
R〉} form an
orthonormal basis of H⊗nR . We refer to the state ̺
RAB as a cq-extension
of the bipartite state ρnAB. Let D
AB) denote the set of all possible cq-
extensions of ρnAB .
The entanglement of formation of the bipartite state ρnAB ∈ D((HA ⊗
⊗n) is defined as
EF (ρ
AB) := min
i S(ρ
where ρ
A = TrB |φ
AB〉〈φ
AB |, the partial trace being taken over the Hilbert
space H⊗nB , and the minimization is over all possible ensemble decomposi-
tions of the state ρnAB. Alternatively, the entanglement of formation of the
state ρnAB can be expressed as
EF (ρ
AB) = min
Dncq(ρ
S(A|R)̺n
, (9)
where S(A|R)̺n
denotes the conditional entropy
S(A|R)̺n
= S(̺nRA)− S(̺nR),
with ̺nRA = TrB ̺
RAB and ̺
R = TrA ̺
RA, the state ̺
RAB being a cq-
extension of the state ρnAB.
The regularized entanglement of formation of a bipartite state ρAB ∈
D(HA ⊗HB) is defined as
E∞F (ρAB) := lim
EF (ρ
AB) (10)
The sup-conditional entropy rate S(A|R) of the sequence ˆ̺RA := {̺nRA}∞n=1,
defined as
S(A|R) = −D(ˆ̺RA‖ÎA ⊗ ˆ̺R), (11)
where ˆ̺R := {̺nR}∞n=1, will be of particular significance in this paper.
Note: For notational simplicity, the explicit n-dependence of quantities are
suppressed in the rest of the paper, wherever there is no scope of any ambi-
guity.
3 Entanglement Dilution for Mixed States
The asymptotic optimization over entanglement dilution protocols leads to
the following theorem.
Theorem 1. The entanglement cost of a sequence of bipartite target states
ρ̂AB = {ρnAB}∞n=1, is given by
EC(ρ̂AB) = min
Dcq(ρ̂AB)
S(A|R), (12)
or equivalently minDcq(ρ̂AB) S(B|R), where Dcq(ρ̂AB) is the set of sequences
of tripartite states ˆ̺RAB = {̺nRAB}∞n=1 defined above.
The proof of Theorem 1 is contained in the following two lemmas. How-
ever, before going over to the proof, we would first like to point out that
previously known results on entanglement dilution [7, 8] can be recovered
from the above theorem. In [8] it was proved that the entanglement cost of
an arbitrary (mixed) bipartite state ρAB , evaluated in the case in which Alice
and Bob create multiple copies (i.e., tensor products) of ρAB (with asymp-
totically vanishing error, from a shared resource of singlets, using LOCC)
is given by the regularized entanglement of formation E∞F (ρAB) (10). In
Section 4 we prove how this result can be recovered from Theorem 1. As
regards the entanglement cost of pure states, in [9] we obtained an expres-
sion for the entanglement cost of an arbitrary sequence of pure states and
we proved that this expression reduced to the entropy of entanglement of a
given pure state (say, |ψAB〉), if the sequence consisted of tensor products
of this state – thus recovering the result first proved in [6].
Lemma 3. (Coding) For any sequence ρ̂AB = {ρnAB}∞n=1 and δ > 0, the
dilution rate
R = S(A|R) + δ, (13)
where S(A|R) is the sup-conditional spectral rate given by (11), is achievable.
Proof. Let the target bipartite state ρnAB have a decomposition given by
ρnAB =
pi|φiAB〉〈φiAB |, (14)
where the Schmidt decomposition of |φiAB〉 is given by
|φiAB〉 =
λi,k|ψi,kA 〉|ψ
B 〉, (15)
with the Schmidt coefficients λik being arranged in non-increasing order, i.e.,
λi1 ≥ λi2 . . . ≥ λidn , for dn = dimH
Alice locally prepares the classical-quantum state (cq-state) ρnRAA′ =
i pi|iR〉〈iR| ⊗ |φiAA′〉〈φiAA′ | ∈ D ((HR ⊗HA ⊗HA′)⊗n). She then does a
unitary operation on the system RAA′ given by
InA ⊗ΘnRA′
ρnRAA′
InA ⊗ΘnRA′
where
ΘnRA′ :=
|jR〉〈jR| ⊗
|χlA′〉〈ψ
|, (16)
with {|χlA′〉}
l=1 being a fixed orthonormal basis in H
A′ . This results in the
state
pi|iR〉〈iR| ⊗
λi,k λi,k
′ |ψi,kA 〉〈ψ
A | ⊗ |χ
A′〉〈χk
A′ |, (17)
where, once again, the explicit n-dependence of the terms has been sup-
pressed for notational simplicity. Note that Alice’s operation amounts to a
coherent implimentation of a projective measurement on R with rank one
projections |jR〉〈jR|, followed by a unitary Uj =
l |χlA′〉〈ψ
A′ | on A
′, condi-
tional on the outcome j. Alice teleports the A′ state to Bob. The resultant
shared state is
νnRAB :=
pi|iR〉〈iR|⊗
k,k′=1
λi,k λi,k
′|ψi,kA 〉〈ψ
A |⊗|χ
B〉〈χk
B |+σnRAB (18)
where σnRAB is an unnormalized error state. Note that the sum over the
index k is truncated to Mn. This truncation occurs due to the so-called
quantum scissors effect [11], i.e., if the quantum state to be teleported lives
in a space of dimension higher than the rank Mn of the shared entangled
state (used by the two parties for teleportation), then all higher-dimensional
terms in the expansion of the original state are cut off. Moreover the system
A′ is now referred to as B, since it is now in Bob’s possession.
Alice also sends the “classical” state R to Bob through a classical chan-
nel. Bob then acts on the system RB, which is now in his possession, with
the unitary operator (ΘnRB)
†. The final shared state can therefore be ex-
pressed as
InA ⊗ (ΘnRB)†
νnRAB
InA ⊗ΘnRB
= ωnRAB + σ̃
pi|iR〉〈iR| ⊗ |φ̂iAB〉〈φ̂iAB |+ σ̃nRAB ,
where
|φ̂iAB〉 := (Q
A ⊗ I
B)|φiAB〉 , (20)
with Q
A being the orthogonal projector onto span of the Schmidt vectors
corresponding to theMn largest Schmidt coefficients of |φiAB〉, and σ̃nRAB :=
(ΘnRB)
†σnRABΘ
By Uhlmann’s theorem (see [12]) it follows that F (ωnAB + σ̃
AB , ρ
AB) ≥
F (ωnAB, ρ
1 and the fidelity between the state ωnAB of the entanglement
dilution protocol and the target state ρnAB is bounded below by
Fn ≥ max
∣〈ρnABC |ωnABC〉
∣, (21)
where |ωnABC〉 is any fixed purification of the final state ωnAB and the maxi-
mization is taken over all purifications of ρnAB.
By choosing purifications |ωnCAB〉 =
pi|iC〉|φ̂iAB〉 and |ρnCAB〉 =
pi|iC〉|φiAB〉, we obtain the following lower bound to F 2n(ωnAB, ρnAB).
Let QnRA :=
i |iR〉〈iR| ⊗ Q
A and ρ
RA :=
i pi|iR〉〈iR| ⊗ ρ
A , where
A = TrB |φ
AB〉〈φiAB |. Then
F 2n ≥
∣〈ωnCAB|ρnCAB〉
QnRAρ
Explicitly examining the projection operator PnRA := {
i pi|iR〉〈iR|⊗ρ
e−nαρnR⊗InA}, where α is a real number, we can express it in the form PnRA =
i pi|iR〉〈iR|⊗
A − e
−nαInA
≥ 0} =
i |iR〉〈iR|⊗ {ρ
A ≥ e
−nαIA}. The
rank of each of the projectors {ρn,iA ≥ e−nαIA} is then bounded by Tr[{ρ
e−nαIA}] ≤ enα by Lemma 2, and hence by comparing PnRA with QnRA we
can see that Mn = ⌈enα⌉ implies that Tr[QnRAρnRA] ≥ Tr[PnRAρnRA]. For any
δ > 0 we can always choose a positive integer N such that for all n ≥ N
there is an integer Mn satisfying S(A|R) + δ ≥ 1n logMn > S(A|R). Thus,
using a sequence of maximally entangled states {|ΨMnAB〉}
n=1 of Schmidt rank
Mn, from the definition of S(A|R) it follows that
2 ≥ lim
QnRAρ
≥ lim
PnRAρ
= 1. (23)
and entanglement dilution at the rate R = S(A|R) + δ is achievable.
Lemma 4. (Weak Converse) For any arbitrary sequence of states ρ̂AB, any
entanglement dilution protocol with a rate
∗ < min
S(A|R), (24)
1Take a purification |ω〉 such that F (ω, ρ) = 〈ω|ρ〉. Then utilize purifications |ω〉|0〉,
|σ〉|1〉, and |ρ〉|0〉, which along with Uhlmann’s theorem implies F (ω + σ, ρ) ≥ 〈ω|ρ〉 =
F (ω, ρ).
where S(A|R) is the sup-conditional spectral rate given by (11), is not reli-
able.
Proof. Let TnAB denote any LOCC operation used for transforming the max-
imally entangled state |ΨMnAB〉 ∈ H
B to the target state ρ
AB in this
Hilbert space, such that F
|ΨMnAB〉〈Ψ
, ρnAB
→ 1 as n → ∞. Em-
ploying the Lo & Popescu theorem [13], the final state of the protocol is
expressible as
ωnAB := T
|ΨMnAB〉〈Ψ
A ⊗ U
B )|Ψ
AB〉〈Ψ
AB |(K
A ⊗ U
† (25)
A = I
A, and U
B is unitary.
Let |ωnCAB〉 :=
k |knC〉 ⊗ (K
A ⊗ U
B )|Ψ
AB〉, denote a purification of
the final state, ωnAB, of the entanglement dilution protocol, with C denoting
a reference system, and {|knC〉} denoting an orthonormal basis in its Hilbert
Space H⊗nC . By Uhlmann’s theorem, for this fixed purification |ω
CAB〉, the
fidelity is given by
AB , ω
AB) = max
|〈ρnCAB |ωnCAB〉|, (26)
where the maximization is over all purifications |ρnCAB〉 of the target state
ρnAB. However, this maximization is equivalent to a maximization over all
possible unitary transformations acting on the reference system C. This
in turn corresponds to a particular decomposition of the purification of the
target state ρnAB with respect to a fixed reference system [14]. Explicitly we
then have |ρnCAB〉 =
|knC〉|φ
AB〉, where
|φn,kAB〉〈φ
AB | is the
given decomposition of ρnAB obtained from the maximization.
AB , ω
AB) = |〈ρnCAB |ωnCAB〉| = |
pk〈φn,kAB |K
A ⊗ U
AB〉|, (27)
Note that
AB〉 =
|χjA〉U
|χjA〉U
B〉, (28)
where Nn = dimH
A and P
k=1 |χkA〉〈χkA|.
For simplicity, let us consider HA ≃ HB ≃ H, and let the state |φn,kAB〉 ∈
(HA⊗HB)⊗n ≃ H⊗2n have a Schmidt decomposition |φn,kAB〉 =
A,i 〉⊗
|ψn,kB,i〉. Further, let W kA and W kB be unitary operators in B(H⊗n) such that
W k|ψn,kA,j〉 = |χ
A〉 and Wk|ψ
B,j〉 = |χ
B〉. Then from (28) it follows that
AB〉 =
W kA|ψ
A,j〉U
B,j〉,
A,j〉|ψ
B,j〉 (29)
where V
A := (U
TW kA. Here we have used the relation
j |j〉 ⊗
U |j〉 =
T |j〉 ⊗ |j〉 for U unitary and {|j〉} an orthonormal basis in
〈φn,kAB |K
A ⊗ U
AB〉 = Tr
where ρ
A = TrB |φ
AB〉〈φ
AB | =
A,i 〉〈ψ
A,i |. Then from (27), using
the Cauchy Schwarz inequality we then obtain
pn,kρ
A · P
pn,kρ
PMnA (K
pn,kρ
pn,kTr
where P
A = (V
†PMnA V
A . The third inequality follows by the following
argument. Express the third line as
qkπk with qk =
PMnA (K
and πk = Tr
A pkρ
≥ 0. From the properties Tr[PMnA ] = Mn and
A = I
A it follows that
k qk = 1 and qk ≥ 0 for all k. Then
using the concavity of the map x 7→
x, we have that
k : qk>0
k : qk>0
πk, (31)
yielding the inequality in the last line of (30).
Defining the projection operator
PnRA :=
|jnR〉〈jnR| ⊗ P
and the state
ρnRA :=
pn,k|knR〉〈knR| ⊗ ρ
the square of the fidelity can then be bounded by
F 2n ≤ Tr
PnRAρ
pn,kTr
, (33)
whereQ
A is the orthogonal projector onto the span of the Schmidt vectors
corresponding to the Mn largest Schmidt coefficients of |φn,kAB〉.
Note that eqs. (22) and (33) yields an alternative proof of the following
lemma stated in [15]:
Lemma 5. The entanglement dilution fidelity for a given bipartite state
ρnAB :=
i pi|φiAB〉〈φiAB |, under an LOCC transformation Λn is given by
F 2(Λn(ΨMnAB), ρ
AB) =
λij , (34)
where λij , j = 1, . . . ,Mn denote the Mn largest Schmidt coefficients of |φiAB〉.
From (32), using Lemma 1, with Πn(γ) := ρnRA − e−nγρnR ⊗ InA,
PnRAρ
PnRAΠ
+ e−nγ
pkTr[P
{Πn(γ) ≥ 0}Πn(γ)
since Tr[Pn,k] = Tr[PMn ] =Mn. Hence for Mn ≤ enR we have
F 2n ≤ Tr
{Πn(γ) ≥ 0}Πn(γ)
+ e−n(γ−R). (35)
Choosing a number γ and δ > 0 such that R+ δ = γ < S(A|R), the second
term on RHS of (35) tends to zero as n → ∞. However, since γ < S(A|R)
the first term on RHS of (35) does not converge to 1 as n→ ∞. Hence, the
asymptotic fidelity F is not equal to 1.
It is then straightfoward to show that the particular choice of decom-
position of each ρnAB imposed by the fidelity criterion gives a minimization
over possible cq-sequences. Suppose there exists a cq-sequence σ̂RAB with
Sσ(A|R) = Sρ(A|R) − ε for some ε > 0. It then follows from the cod-
ing theorem that the rate R = Sσ(A|R) + ε/2 is asymptotically attainable.
However, if we take F ′n = |〈σnRR′AB |ωnRR′AB〉| then this is less than the max-
imization over all possible purifications, bounding the asymptotic fidelity
below 1, giving a contradiction.
4 The regularized entanglement of formation
The application of the main result to the case of multiple copies of a single
bipartite state provides a new proof of the equivalence [8] between the reg-
ularized entanglement of formation E∞F (ρAB) (10), of a bipartite state ρAB,
and its entanglement cost EC(ρAB).
First note that as the entanglement of formation is a bounded non-
increasing function of n we have infn
EF (ρ
AB) = limn→∞
EF (ρ
AB) =
E∞F (ρAB). Consider a sequence ρ̂AB = {ρnAB}∞n=1 of a bipartite states. For
any state ρnAB in the sequence, let S(An|Bn)ρnAB denote the conditional en-
tropy:
S(An|Bn)ρn
= S(ρnAB)− S(ρnB).
From results in [5] it can be shown that the conditional entropy rate of the
sequence is bounded above by the sup-conditional spectral entropy rate:
E∞F (ρAB) = lim sup
S(An|Bn) ≤ S(A|B) (36)
Thus, for any sequence of cq-states ˆ̺RAB = {̺nRAB}∞n=1 on RAB, which
reduce to product sequences ρ̂AB = {̺⊗n}∞n=1 on AB, we have from (9) and
EF (̺
⊗n) ≤ lim inf
S(An|Rn)̺n
≤ S(A|R),
where S(A|R) denotes the sup-conditional spectral entropy rate defined in
(11).
For the reverse inequality we simply construct states of block size m on
RAB such that ωmn = (
i |iR〉〈iR|⊗|φmi 〉〈φmi |AB)⊗⌊n/m⌋⊗σRAB , where
σ is an asymptotically irrelevant buffer state whenever m does not divide n.
Using the chain rule [16] S(A|R) ≤ S(RA)−S(R), the definitions of S(RA)
and S(R), and (6), we obtain
S(A|R) ≤
S(ωmmRA )− S(ωmmR )
i S(ω
for ωmA,i = TrB |φmi 〉〈φmi |AB . Taking the infimum over both m and decompo-
sitions then implies
EC(ρAB) = E
F (ρAB) (37)
for product sequences, and hence the regularized entanglement of formation
for a bipartite state is equal to its entanglement cost.
Acknowledgments
This work is part of the QIP-IRC supported by the European Community’s
Seventh Framework Programme (FP7/2007-2013) under grant agreement
number 213681.
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	1 Introduction
	2 Notations and definitions
	3 Entanglement Dilution for Mixed States
	4 The regularized entanglement of formation

  The entanglement cost of arbitrary sequences of bipartite states is shown to
be expressible as the minimization of a conditional spectral entropy rate over
sequences of separable extensions of the states in the sequence. The expression
is shown to reduce to the regularized entanglement of formation when the n-th
state in the sequence consists of n copies of a single bipartite state.

Introduction
A fundamental problem in entanglement theory is to determine how to op-
timally convert entanglement, shared between two distant parties Alice and
Bob, from one form to another. Entanglement manipulation is the process
by which Alice and Bob convert an initial bipartite state ρAB which they
share, to a required target state σAB using local operations and classical
communication (LOCC). If the target state σAB is a maximally entangled
state, then the protocol is called entanglement distillation, whereas if the
initial state ρAB is a maximally entangled state, then the protocol is called
entanglement dilution. Optimal rates of these protocols were originally eval-
uated under the assumption that the entanglement resource accessible to
Alice and Bob consist of multiple copies, i.e., tensor products ρ⊗nAB , of the
initial bipartite state ρAB, and the requirement that the final state of the
protocol is equal to n copies of the desired target state σ⊗nAB with asymptot-
ically vanishing error in the limit n→ ∞. The distillable entanglement and
entanglement cost computed in this manner are two asymptotic measures
of entanglement of the state ρAB . Moreover, in the case in which ρAB is
pure, these two measures of entanglement coincide and are equal to the von
∗Centre for Quantum Computation, DAMTP, University of Cambridge, Cambridge
CB3 0WA, UK
†Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge CB3
0WB, UK (e-mail:n.datta@statslab.cam.ac.uk)
http://arxiv.org/abs/0704.1957v3
Neumann entropy of the reduced state on any one of the subsystems, A or
In this paper we focus on entanglement dilution, which, as mentioned
earlier, is the entanglement manipulation process by which two distant par-
ties, say Alice and Bob, create a desired bipartite target state from a maxi-
mally entangled state which they initially share, using LOCC. In [6, 7] the
optimal rate of entanglement dilution, namely, the entanglement cost, was
evaluated in the case in which Alice and Bob created multiple copies of a
desired target state ρAB , with asymptotically vanishing error, from a shared
resource of singlets, using local operations and classical communication. In
particular, for the case of a pure target state ρAB , the entanglement cost
was shown [7] to be equal to the entropy of entanglement of the state, i.e.,
the von Neumann entropy of the reduced state on any one of the two sub-
systems A and B. Moreover, in [8] it was shown that for an arbitrary mixed
state ρAB, the entanglement cost is equal to the regularized entanglement
of formation of the state (see (10) for its definition).
The practical ability to transform entanglement from one form to another
is useful for many applications in quantum information theory. However, it
is not always justified to assume that the entanglement resource available
consists of states which are multiple copies (and hence tensor products) of
a given entangled state, or to require that the final state of the protocol
is of the tensor product form. More generally an entanglement resource
is characterized by an arbitrary sequence of bipartite states which are not
necessarily of the tensor product form. Sequences of bipartite states on AB
are considered to exist on Hilbert spaces H⊗nA ⊗H
B for n ∈ {1, 2, 3 . . .}.
A useful tool for the study of entanglement manipulation in this general
scenario is provided by the Information Spectrum method. The information
spectrum method, introduced in classical information theory by Verdu &
Han [1, 2], has been extended into quantum information theory by Hayashi,
Ogawa and Nagaoka [3, 4, 5]. The power of the information spectrum ap-
proach comes from the fact that it does not depend on the specific structure
of sources, channels or entanglement resources employed in information the-
oretical protocols.
In this paper we evaluate the optimal asymptotic rate of entanglement
dilution for an arbitrary sequence of bipartite states. The case of an arbitrary
sequence of pure bipartite state was studied in [9]. The paper is organized as
follows. In Section 2 we introduce the necessary notations and definitions.
Section 3 contains the statement and proof of the main result, stated as
Theorem 1. Note that if we consider the sequence of bipartite states to
consist of tensor products of a given bipartite state, then our main result
reduces to the known results obtained in [6, 8] (see the discussion after
Theorem 1 of Section 3). Finally, in Section 4 we show how Theorem 1 yields
an alternative proof of the equivalence of the asymptotic entanglement cost
and the regularised entanglement of formation [8].
2 Notations and definitions
LetB(H) denote the algebra of linear operators acting on a finite–dimensional
Hilbert space H of dimension d and let D(H) denote the set of states (or
density operators, i. e. positive operators of unit trace) acting on H. Fur-
ther, let H(n) denote the Hilbert space H⊗n. For any state ρ ∈ D(H), the
von Neumann entropy is defined as S(ρ) := −Tr (ρ log ρ).
Let Λn be a quantum operation used for the transformation of an initial
bipartite state ωn to a bipartite state ρn, with ωn, ρn ∈ D ((HA ⊗HB)⊗n).
For the entanglement manipulation processes considered in this paper, Λn
either consists of local operations (LO) alone or LO with one-way or two-
way classical communication. We define the efficacy of any entanglement
manipulation process in terms of the fidelity Fn := Tr
ρnΛn(ωn)
between the output state Λn(ωn) and the target state ρn. An entanglement
manipulation process is said to be reliable if the asymptotic fidelity F :=
lim infn→∞ Fn = 1.
For given orthonormal bases {|χiA〉}
i=1 and {|χiB〉}
i=1 in Hilbert spaces
A andH
B , of dimensions d
A and d
B respectively, we define the canonical
maximally entangled state of Schmidt rank Mn ≤ min{dnA, dnB} to be
|ΨMnAB〉 =
|χiA〉 ⊗ |χiB〉. (1)
In fact, in the following, we consider HA ≃ HB , for simplicity, so that
dnA = d
B . Here and henceforth, the explicit n-dependence of the basis states
|χiA〉 and |χiB〉 has been suppressed for notational simplicity.
The quantum information spectrum approach requires the extensive use
of spectral projections. Any self-adjoint operator A acting on a finite di-
mensional Hilbert space may be written in its spectral decomposition A =
i λiπi, where πi denotes the operator which projects onto the eigenspace
corresponding to the eigenvalue λi. We define the positive spectral projec-
tion on A as {A ≥ 0} =
πi, i.e., the projector onto the eigenspace of
positive eigenvalues of A. For two operators A and B, we can then define
{A ≥ B} as {A−B ≥ 0}. The following key lemmas are used repeatedly in
the paper. For their proofs see [3, 4].
Lemma 1. For self-adjoint operators A, B and any positive operator 0 ≤
P ≤ I the inequality
P (A−B)
A ≥ B
(A−B)
holds.
Lemma 2. Given a state ρn ∈ D(H⊗n) and a self-adjoint operator ωn ∈
B(H⊗n), we have
{ρn ≥ enγωn}ωn
≤ e−nγ . (3)
for any real number γ.
In the quantum information spectrum approach one defines spectral di-
vergence rates, which can be viewed as generalizations of the quantum rel-
ative entropy. The spectral generalizations of the von Neumann entropy,
the conditional entropy and the mutual information can all be expressed as
spectral divergence rates.
Definition 1. Given a sequence of states ρ̂ = {ρn}∞n=1, with ρn ∈ D(H⊗n),
and a sequence of positive operators ω̂ = {ωn}∞n=1, with ωn ∈ B(H⊗n),
the quantum spectral sup-(inf-) divergence rates are defined in terms of the
difference operators Πn(γ) = ρn − enγωn, for any arbitrary real number γ,
D(ρ̂‖ω̂) = inf
γ : lim sup
{Πn(γ) ≥ 0}Πn(γ)
D(ρ̂‖ω̂) = sup
γ : lim inf
{Πn(γ) ≥ 0}Πn(γ)
respectively.
The spectral entropy rates and the conditional spectral entropy rates
can be expressed as divergence rates with appropriate substitutions for the
sequence of operators ω̂ = {ωn}∞n=1. These are S(ρ̂) = −D(ρ̂‖Î) and S(ρ̂) =
−D(ρ̂‖Î), where Î = {In}∞n=1, with In being the identity operator acting
on the Hilbert space H⊗n. Further, for sequences of bipartite states ρ̂AB =
{ρnAB}∞n=1,
S(A|B) = −D(ρ̂AB‖ÎA ⊗ ρ̂B) (4)
S(A|B) = −D(ρ̂AB‖ÎA ⊗ ρ̂B) . (5)
In the above, ÎA = {InA}∞n=1 and ρ̂B = {ρnB}∞n=1, with InA being the identity
operator in B(H
A ) and ρ
B = TrAρ
AB, the partial trace being taken on the
Hilbert space H
A . Various properties of these quantities, and relationships
between them, are explored in [10].
For sequences of states ρ̂ = {ρ⊗n} and ω̂ = {ω⊗n}, with ρ, ω ∈ D(H), it
has been proved [5] that
D(ρ̂‖ω̂) = D(ρ̂‖ω̂) = S(ρ‖ω), (6)
where S(ρ‖ω) := Tr ρ log ρ− Tr ρ logω, is the quantum relative entropy.
Two parties, Alice and Bob, share a sequence of maximally entangled
states {|ΨMnAB〉}
n=1, and wish to convert them into a sequence of given bi-
partite states {ρnAB}∞n=1, with ρn ∈ D(H⊗n) and |Ψ
AB〉 ∈ H
B . The
protocol used for this conversion is known as entanglement dilution. The
concept of reliable entanglement manipulation may then be used to define
an asymptotic entanglement measure, namely the entanglement cost.
Definition 2. A real-valued number R is said to be an achievable dilu-
tion rate for a sequence of states ρ̂AB = {ρnAB}, with ρnAB ∈ D((HA ⊗
⊗n), if ∀ε > 0, ∃N such that ∀n ≥ N a transformation exists that takes
|ΨMnAB〉〈Ψ
AB | → ρnAB with fidelity F 2n ≥ 1− ε and
logMn ≤ R.
Definition 3. The entanglement cost of the sequence ρ̂AB is the infimum
of all achievable dilution rates:
EC(ρ̂AB) = inf R (7)
To simplify the expressions representing the entanglement cost, we define
the following sets of sequences of states. Firstly, given a sequence of target
states ρ̂AB = {ρnAB}∞n=1, define the set Dcq(ρ̂AB) as the set of sequences
of tripartite states ˆ̺RAB = {̺nRAB}∞n=1 such that each ̺nRAB is a classical-
quantum state (cq-state) of the form
̺nRAB =
R〉〈inR| ⊗ |φ
AB〉〈φ
AB |, (8)
where ρnAB =
AB〉〈φ
AB | and the set of pure states {|i
R〉} form an
orthonormal basis of H⊗nR . We refer to the state ̺
RAB as a cq-extension
of the bipartite state ρnAB. Let D
AB) denote the set of all possible cq-
extensions of ρnAB .
The entanglement of formation of the bipartite state ρnAB ∈ D((HA ⊗
⊗n) is defined as
EF (ρ
AB) := min
i S(ρ
where ρ
A = TrB |φ
AB〉〈φ
AB |, the partial trace being taken over the Hilbert
space H⊗nB , and the minimization is over all possible ensemble decomposi-
tions of the state ρnAB. Alternatively, the entanglement of formation of the
state ρnAB can be expressed as
EF (ρ
AB) = min
Dncq(ρ
S(A|R)̺n
, (9)
where S(A|R)̺n
denotes the conditional entropy
S(A|R)̺n
= S(̺nRA)− S(̺nR),
with ̺nRA = TrB ̺
RAB and ̺
R = TrA ̺
RA, the state ̺
RAB being a cq-
extension of the state ρnAB.
The regularized entanglement of formation of a bipartite state ρAB ∈
D(HA ⊗HB) is defined as
E∞F (ρAB) := lim
EF (ρ
AB) (10)
The sup-conditional entropy rate S(A|R) of the sequence ˆ̺RA := {̺nRA}∞n=1,
defined as
S(A|R) = −D(ˆ̺RA‖ÎA ⊗ ˆ̺R), (11)
where ˆ̺R := {̺nR}∞n=1, will be of particular significance in this paper.
Note: For notational simplicity, the explicit n-dependence of quantities are
suppressed in the rest of the paper, wherever there is no scope of any ambi-
guity.
3 Entanglement Dilution for Mixed States
The asymptotic optimization over entanglement dilution protocols leads to
the following theorem.
Theorem 1. The entanglement cost of a sequence of bipartite target states
ρ̂AB = {ρnAB}∞n=1, is given by
EC(ρ̂AB) = min
Dcq(ρ̂AB)
S(A|R), (12)
or equivalently minDcq(ρ̂AB) S(B|R), where Dcq(ρ̂AB) is the set of sequences
of tripartite states ˆ̺RAB = {̺nRAB}∞n=1 defined above.
The proof of Theorem 1 is contained in the following two lemmas. How-
ever, before going over to the proof, we would first like to point out that
previously known results on entanglement dilution [7, 8] can be recovered
from the above theorem. In [8] it was proved that the entanglement cost of
an arbitrary (mixed) bipartite state ρAB , evaluated in the case in which Alice
and Bob create multiple copies (i.e., tensor products) of ρAB (with asymp-
totically vanishing error, from a shared resource of singlets, using LOCC)
is given by the regularized entanglement of formation E∞F (ρAB) (10). In
Section 4 we prove how this result can be recovered from Theorem 1. As
regards the entanglement cost of pure states, in [9] we obtained an expres-
sion for the entanglement cost of an arbitrary sequence of pure states and
we proved that this expression reduced to the entropy of entanglement of a
given pure state (say, |ψAB〉), if the sequence consisted of tensor products
of this state – thus recovering the result first proved in [6].
Lemma 3. (Coding) For any sequence ρ̂AB = {ρnAB}∞n=1 and δ > 0, the
dilution rate
R = S(A|R) + δ, (13)
where S(A|R) is the sup-conditional spectral rate given by (11), is achievable.
Proof. Let the target bipartite state ρnAB have a decomposition given by
ρnAB =
pi|φiAB〉〈φiAB |, (14)
where the Schmidt decomposition of |φiAB〉 is given by
|φiAB〉 =
λi,k|ψi,kA 〉|ψ
B 〉, (15)
with the Schmidt coefficients λik being arranged in non-increasing order, i.e.,
λi1 ≥ λi2 . . . ≥ λidn , for dn = dimH
Alice locally prepares the classical-quantum state (cq-state) ρnRAA′ =
i pi|iR〉〈iR| ⊗ |φiAA′〉〈φiAA′ | ∈ D ((HR ⊗HA ⊗HA′)⊗n). She then does a
unitary operation on the system RAA′ given by
InA ⊗ΘnRA′
ρnRAA′
InA ⊗ΘnRA′
where
ΘnRA′ :=
|jR〉〈jR| ⊗
|χlA′〉〈ψ
|, (16)
with {|χlA′〉}
l=1 being a fixed orthonormal basis in H
A′ . This results in the
state
pi|iR〉〈iR| ⊗
λi,k λi,k
′ |ψi,kA 〉〈ψ
A | ⊗ |χ
A′〉〈χk
A′ |, (17)
where, once again, the explicit n-dependence of the terms has been sup-
pressed for notational simplicity. Note that Alice’s operation amounts to a
coherent implimentation of a projective measurement on R with rank one
projections |jR〉〈jR|, followed by a unitary Uj =
l |χlA′〉〈ψ
A′ | on A
′, condi-
tional on the outcome j. Alice teleports the A′ state to Bob. The resultant
shared state is
νnRAB :=
pi|iR〉〈iR|⊗
k,k′=1
λi,k λi,k
′|ψi,kA 〉〈ψ
A |⊗|χ
B〉〈χk
B |+σnRAB (18)
where σnRAB is an unnormalized error state. Note that the sum over the
index k is truncated to Mn. This truncation occurs due to the so-called
quantum scissors effect [11], i.e., if the quantum state to be teleported lives
in a space of dimension higher than the rank Mn of the shared entangled
state (used by the two parties for teleportation), then all higher-dimensional
terms in the expansion of the original state are cut off. Moreover the system
A′ is now referred to as B, since it is now in Bob’s possession.
Alice also sends the “classical” state R to Bob through a classical chan-
nel. Bob then acts on the system RB, which is now in his possession, with
the unitary operator (ΘnRB)
†. The final shared state can therefore be ex-
pressed as
InA ⊗ (ΘnRB)†
νnRAB
InA ⊗ΘnRB
= ωnRAB + σ̃
pi|iR〉〈iR| ⊗ |φ̂iAB〉〈φ̂iAB |+ σ̃nRAB ,
where
|φ̂iAB〉 := (Q
A ⊗ I
B)|φiAB〉 , (20)
with Q
A being the orthogonal projector onto span of the Schmidt vectors
corresponding to theMn largest Schmidt coefficients of |φiAB〉, and σ̃nRAB :=
(ΘnRB)
†σnRABΘ
By Uhlmann’s theorem (see [12]) it follows that F (ωnAB + σ̃
AB , ρ
AB) ≥
F (ωnAB, ρ
1 and the fidelity between the state ωnAB of the entanglement
dilution protocol and the target state ρnAB is bounded below by
Fn ≥ max
∣〈ρnABC |ωnABC〉
∣, (21)
where |ωnABC〉 is any fixed purification of the final state ωnAB and the maxi-
mization is taken over all purifications of ρnAB.
By choosing purifications |ωnCAB〉 =
pi|iC〉|φ̂iAB〉 and |ρnCAB〉 =
pi|iC〉|φiAB〉, we obtain the following lower bound to F 2n(ωnAB, ρnAB).
Let QnRA :=
i |iR〉〈iR| ⊗ Q
A and ρ
RA :=
i pi|iR〉〈iR| ⊗ ρ
A , where
A = TrB |φ
AB〉〈φiAB |. Then
F 2n ≥
∣〈ωnCAB|ρnCAB〉
QnRAρ
Explicitly examining the projection operator PnRA := {
i pi|iR〉〈iR|⊗ρ
e−nαρnR⊗InA}, where α is a real number, we can express it in the form PnRA =
i pi|iR〉〈iR|⊗
A − e
−nαInA
≥ 0} =
i |iR〉〈iR|⊗ {ρ
A ≥ e
−nαIA}. The
rank of each of the projectors {ρn,iA ≥ e−nαIA} is then bounded by Tr[{ρ
e−nαIA}] ≤ enα by Lemma 2, and hence by comparing PnRA with QnRA we
can see that Mn = ⌈enα⌉ implies that Tr[QnRAρnRA] ≥ Tr[PnRAρnRA]. For any
δ > 0 we can always choose a positive integer N such that for all n ≥ N
there is an integer Mn satisfying S(A|R) + δ ≥ 1n logMn > S(A|R). Thus,
using a sequence of maximally entangled states {|ΨMnAB〉}
n=1 of Schmidt rank
Mn, from the definition of S(A|R) it follows that
2 ≥ lim
QnRAρ
≥ lim
PnRAρ
= 1. (23)
and entanglement dilution at the rate R = S(A|R) + δ is achievable.
Lemma 4. (Weak Converse) For any arbitrary sequence of states ρ̂AB, any
entanglement dilution protocol with a rate
∗ < min
S(A|R), (24)
1Take a purification |ω〉 such that F (ω, ρ) = 〈ω|ρ〉. Then utilize purifications |ω〉|0〉,
|σ〉|1〉, and |ρ〉|0〉, which along with Uhlmann’s theorem implies F (ω + σ, ρ) ≥ 〈ω|ρ〉 =
F (ω, ρ).
where S(A|R) is the sup-conditional spectral rate given by (11), is not reli-
able.
Proof. Let TnAB denote any LOCC operation used for transforming the max-
imally entangled state |ΨMnAB〉 ∈ H
B to the target state ρ
AB in this
Hilbert space, such that F
|ΨMnAB〉〈Ψ
, ρnAB
→ 1 as n → ∞. Em-
ploying the Lo & Popescu theorem [13], the final state of the protocol is
expressible as
ωnAB := T
|ΨMnAB〉〈Ψ
A ⊗ U
B )|Ψ
AB〉〈Ψ
AB |(K
A ⊗ U
† (25)
A = I
A, and U
B is unitary.
Let |ωnCAB〉 :=
k |knC〉 ⊗ (K
A ⊗ U
B )|Ψ
AB〉, denote a purification of
the final state, ωnAB, of the entanglement dilution protocol, with C denoting
a reference system, and {|knC〉} denoting an orthonormal basis in its Hilbert
Space H⊗nC . By Uhlmann’s theorem, for this fixed purification |ω
CAB〉, the
fidelity is given by
AB , ω
AB) = max
|〈ρnCAB |ωnCAB〉|, (26)
where the maximization is over all purifications |ρnCAB〉 of the target state
ρnAB. However, this maximization is equivalent to a maximization over all
possible unitary transformations acting on the reference system C. This
in turn corresponds to a particular decomposition of the purification of the
target state ρnAB with respect to a fixed reference system [14]. Explicitly we
then have |ρnCAB〉 =
|knC〉|φ
AB〉, where
|φn,kAB〉〈φ
AB | is the
given decomposition of ρnAB obtained from the maximization.
AB , ω
AB) = |〈ρnCAB |ωnCAB〉| = |
pk〈φn,kAB |K
A ⊗ U
AB〉|, (27)
Note that
AB〉 =
|χjA〉U
|χjA〉U
B〉, (28)
where Nn = dimH
A and P
k=1 |χkA〉〈χkA|.
For simplicity, let us consider HA ≃ HB ≃ H, and let the state |φn,kAB〉 ∈
(HA⊗HB)⊗n ≃ H⊗2n have a Schmidt decomposition |φn,kAB〉 =
A,i 〉⊗
|ψn,kB,i〉. Further, let W kA and W kB be unitary operators in B(H⊗n) such that
W k|ψn,kA,j〉 = |χ
A〉 and Wk|ψ
B,j〉 = |χ
B〉. Then from (28) it follows that
AB〉 =
W kA|ψ
A,j〉U
B,j〉,
A,j〉|ψ
B,j〉 (29)
where V
A := (U
TW kA. Here we have used the relation
j |j〉 ⊗
U |j〉 =
T |j〉 ⊗ |j〉 for U unitary and {|j〉} an orthonormal basis in
〈φn,kAB |K
A ⊗ U
AB〉 = Tr
where ρ
A = TrB |φ
AB〉〈φ
AB | =
A,i 〉〈ψ
A,i |. Then from (27), using
the Cauchy Schwarz inequality we then obtain
pn,kρ
A · P
pn,kρ
PMnA (K
pn,kρ
pn,kTr
where P
A = (V
†PMnA V
A . The third inequality follows by the following
argument. Express the third line as
qkπk with qk =
PMnA (K
and πk = Tr
A pkρ
≥ 0. From the properties Tr[PMnA ] = Mn and
A = I
A it follows that
k qk = 1 and qk ≥ 0 for all k. Then
using the concavity of the map x 7→
x, we have that
k : qk>0
k : qk>0
πk, (31)
yielding the inequality in the last line of (30).
Defining the projection operator
PnRA :=
|jnR〉〈jnR| ⊗ P
and the state
ρnRA :=
pn,k|knR〉〈knR| ⊗ ρ
the square of the fidelity can then be bounded by
F 2n ≤ Tr
PnRAρ
pn,kTr
, (33)
whereQ
A is the orthogonal projector onto the span of the Schmidt vectors
corresponding to the Mn largest Schmidt coefficients of |φn,kAB〉.
Note that eqs. (22) and (33) yields an alternative proof of the following
lemma stated in [15]:
Lemma 5. The entanglement dilution fidelity for a given bipartite state
ρnAB :=
i pi|φiAB〉〈φiAB |, under an LOCC transformation Λn is given by
F 2(Λn(ΨMnAB), ρ
AB) =
λij , (34)
where λij , j = 1, . . . ,Mn denote the Mn largest Schmidt coefficients of |φiAB〉.
From (32), using Lemma 1, with Πn(γ) := ρnRA − e−nγρnR ⊗ InA,
PnRAρ
PnRAΠ
+ e−nγ
pkTr[P
{Πn(γ) ≥ 0}Πn(γ)
since Tr[Pn,k] = Tr[PMn ] =Mn. Hence for Mn ≤ enR we have
F 2n ≤ Tr
{Πn(γ) ≥ 0}Πn(γ)
+ e−n(γ−R). (35)
Choosing a number γ and δ > 0 such that R+ δ = γ < S(A|R), the second
term on RHS of (35) tends to zero as n → ∞. However, since γ < S(A|R)
the first term on RHS of (35) does not converge to 1 as n→ ∞. Hence, the
asymptotic fidelity F is not equal to 1.
It is then straightfoward to show that the particular choice of decom-
position of each ρnAB imposed by the fidelity criterion gives a minimization
over possible cq-sequences. Suppose there exists a cq-sequence σ̂RAB with
Sσ(A|R) = Sρ(A|R) − ε for some ε > 0. It then follows from the cod-
ing theorem that the rate R = Sσ(A|R) + ε/2 is asymptotically attainable.
However, if we take F ′n = |〈σnRR′AB |ωnRR′AB〉| then this is less than the max-
imization over all possible purifications, bounding the asymptotic fidelity
below 1, giving a contradiction.
4 The regularized entanglement of formation
The application of the main result to the case of multiple copies of a single
bipartite state provides a new proof of the equivalence [8] between the reg-
ularized entanglement of formation E∞F (ρAB) (10), of a bipartite state ρAB,
and its entanglement cost EC(ρAB).
First note that as the entanglement of formation is a bounded non-
increasing function of n we have infn
EF (ρ
AB) = limn→∞
EF (ρ
AB) =
E∞F (ρAB). Consider a sequence ρ̂AB = {ρnAB}∞n=1 of a bipartite states. For
any state ρnAB in the sequence, let S(An|Bn)ρnAB denote the conditional en-
tropy:
S(An|Bn)ρn
= S(ρnAB)− S(ρnB).
From results in [5] it can be shown that the conditional entropy rate of the
sequence is bounded above by the sup-conditional spectral entropy rate:
E∞F (ρAB) = lim sup
S(An|Bn) ≤ S(A|B) (36)
Thus, for any sequence of cq-states ˆ̺RAB = {̺nRAB}∞n=1 on RAB, which
reduce to product sequences ρ̂AB = {̺⊗n}∞n=1 on AB, we have from (9) and
EF (̺
⊗n) ≤ lim inf
S(An|Rn)̺n
≤ S(A|R),
where S(A|R) denotes the sup-conditional spectral entropy rate defined in
(11).
For the reverse inequality we simply construct states of block size m on
RAB such that ωmn = (
i |iR〉〈iR|⊗|φmi 〉〈φmi |AB)⊗⌊n/m⌋⊗σRAB , where
σ is an asymptotically irrelevant buffer state whenever m does not divide n.
Using the chain rule [16] S(A|R) ≤ S(RA)−S(R), the definitions of S(RA)
and S(R), and (6), we obtain
S(A|R) ≤
S(ωmmRA )− S(ωmmR )
i S(ω
for ωmA,i = TrB |φmi 〉〈φmi |AB . Taking the infimum over both m and decompo-
sitions then implies
EC(ρAB) = E
F (ρAB) (37)
for product sequences, and hence the regularized entanglement of formation
for a bipartite state is equal to its entanglement cost.
Acknowledgments
This work is part of the QIP-IRC supported by the European Community’s
Seventh Framework Programme (FP7/2007-2013) under grant agreement
number 213681.
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	1 Introduction
	2 Notations and definitions
	3 Entanglement Dilution for Mixed States
	4 The regularized entanglement of formation
