Superadditivity of Private Information for Any Number of Uses of the Channel
David Elkouss
1,2
and Sergii Strelchuk
3
1
Departamento de Análisis Matemático and Instituto de Matemática Interdisciplinar,
Universidad Complutense de Madrid, 28040 Madrid, Spain
2
QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, Netherlands
3
Department of Applied Mathematics and Theoretical Physics, University of Cambridge,
Cambridge CB3 0WA, United Kingdom
(Received 4 February 2015; revised manuscript received 16 May 2015; published 20 July 2015)
The quantum capacity of a quantum channel is always smaller than the capacity of the channel for
private communication. Both quantities are given by the infinite regularization of the coherent and the
private information, respectively, which makes their evaluation very difficult. Here, we construct a family
of channels for which the private and coherent information can remain strictly superadditive for unbou nded
number of uses, thus demonstrating that the regularization is necessary. We prove this by showing that
the coherent information is strictly larger than the private information of a smaller number of uses of the
channel. This implies that even though the quantum capacity is upper bounded by the private capacity, the
nonregularized quantities can be interleaved.
DOI: 10.1103/PhysRevLett.115.040501 PACS numbers: 03.67.Hk, 03.67.Dd
Efficient information transmission is the cornerstone of
all information processing tasks in our interconnected
world. In the most basic scenario, two parties, linked by
a fixed communication channel wish to exchange messages
with each other. What is the maximum rate at which they
can reliably transmit information?
Classical information theory gives an exhaustive answer
to this question [1]. There exists an efficient convex
optimization algorithm which takes the description of a
channel and calculates its capacity to convey information.
This is the consequence of a particularly simple analytic
expression for the classical capacity of a channel. Our
world is inherently quantum and when we turn to the
channels that transmit quantum information we are able to
perform many novel information processing tasks which
are impossible in the classical theory, such as establishing
entanglement between sender and receiver. Presently, when
confronted with the above question for the quantum
channels, there is no known efficient algorithm that takes
the description of an arbitrary channel and calculates its
capacity. Different types of capacity of the quantum
channel are defined as regularized quantities [2–9], which
implies that in order to compute them it is necessary to
perform an unbounded optimization over the number of
the copies of the channel. In practice it means that to estimate
the capacity for n uses of the channel the dimension
of the state space which one has to optimize ov er may
increase exponentially in n.
Arguably, the biggest practical success of quantum
information theory t o date is the possibility of quantum
key distribution (QKD) [10–12]. QKD allows two distant
parties to agree on a secret key i ndependent of any
eavesdropper. The required assump tions are access to
a quantum channel with positive private capacity and
the validity of quantum physics. However, in practice
one does not know the quantum channel exactly, and to
characterize it one uses a public authentic classical
channel. On the other hand, key distribution is a primitive
that can only be implemented with classical resources if
one is willing to constrain the power of the eavesdropper.
Even though there exist practical QKD schemes which
enable secure communication over large distances with
high key rates [1 3– 16], some of the fundamental questions
about the capacity to transmit secure correlations remain
unanswered.
There are essentially two quantities that describe the
ability of the channel to send secure messages to the
receiver, and consequently, generate secret keys. The first
one is called private capacity P [6,17]. It can be viewed as
the optimal rate at which the sender, Alice, can send
classical communication to the receiver, Bob, while keep-
ing Eve in a product state with Alice and Bob. For a
quantum channel, which is a completely positive trace-
preserving map N ,itisgivenby
PðN Þ¼ lim
n→∞
1
n
P
ð1Þ
ðN
⊗n
Þ; ð1Þ
The private capacity is given by the regularization of
P
ð1Þ
ðN Þ, the private information of the channel
P
ð1Þ
ðN Þ¼max
ρ∈R
IðX; BÞ − IðX; EÞ; ð2Þ
where the maximum is taken over the set of classical-
quantum states R of the form ρ
XA
¼
P
x
p
x
jxihxj
X
⊗ ρ
A
x
,
with X being an auxiliary classical register, and IðX; BÞ the
quantum mutual information [18].
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This capacity also characterizes the optimal rates for key
distribution [6,17]. A better understanding of this quantity
would allow us to evaluate precisely the usefulness of
communications channels for practical QKD links.
In the case of private capacity, the eavesdropper, Eve, is
given a purification of the channel output which means that
she is as powerful as is allowed by quantum mechanics.
However, this setting may be too restrictive for practical
applications given the current state of the art in quantum
information processing. A natural relaxation of this strong
security requirement is to assume that Eve obtains infor-
mation about the key by performing a measurement on her
state. This security requirement is reflected in the second
quantity, locking capacity L.ByL we denote all the
recently introduced locking capacities [9] of a quantum
channel. They are defined by the optimal rate of reliable
classical communication requiring Eve to have vanishing
accessible information about the message. This difference
in the security criterion has striking consequences. For
instance, it implies that some channels that have no private
capacity have close to maximum locking capacity [19], and
for some relevant classes of channels locked communica-
tion can be performed at almost the classical capacity rate
[20]. The following upper bound is known for the locking
capacities:
LðN Þ ≤ L
u
ðN Þ¼sup
n
1
n
L
ð1Þ
u
ðN
⊗n
Þ; ð3Þ
where L
ð1Þ
u
, which we will call the locking information, is
given by
L
ð1Þ
u
ðN Þ¼max
ρ∈R
IðX; BÞ − I
acc
ðX; EÞ: ð4Þ
The accessible information I
acc
ðX; EÞ¼max
Γ
IðX; YÞ,
where Γ is the set of all POVMs on E.
Two other important types of capacity of a quantum
channel are the quantum [2,5,6] and classical capacity [3,4]
given by
QðN Þ¼ lim
n→∞
1
n
Q
ð1Þ
ðN
⊗n
Þ; ð5Þ
CðN Þ¼ lim
n→∞
1
n
C
ð1Þ
ðN
⊗n
Þ; ð6Þ
where
Q
ð1Þ
ðN Þ¼max
ρ
A
HðBÞ − HðEÞ; ð7Þ
C
ð1Þ
ðN Þ¼max
ρ∈R
IðX; BÞ: ð8Þ
The optimization of the quantum capacity is performed
over all valid states on the input register A while the
optimization of the classical capacity is performed over the
set R as in Eq. (1), and H is the von Neumann entropy.
The form of the expression for the capacities in Eqs. (1),
(3), (5), and (6) contains the optimization over an infinite
number of copies of the channel. This is not at all
computationally feasible. Do we have to resort to the
infinite regularization, or, perhaps, we can stop the regu-
larization after a constant number of uses? It has recently
been shown that at least in the case of the quantum capacity
the calculation cannot involve a fixed number of channel
uses even when we attempt to answer the question whether
the channel has any capacity at all [21]. For the classical
capacity, which is known to be superadditive for two uses
of the channel [22], there is some evidence that ultimately
the regularization might not be required [23,24].
Despite the significance of the private and locking
information, we still understand very little about its
behavior when the communication channel is used many
times. Authors in Refs. [25,26] provide evidence that
P
ð1Þ
ðN Þ is superadditive for a small finite number of
channel uses, although the magnitude of this effect is
quantitatively very small. Recently, the existence of two
quantum channels N
1
; N
2
with CðN
1
Þ ≤ 2; PðN
2
Þ¼0
for which PðN
1
⊗ N
2
Þ ≥ 1=2 log d, where d is the
dimension of the output of the joint channel, has been
shown [27]. This example shows that the private capacity is
a superadditive quantity (this was also proved in Ref. [28]
using a different construction).
Even less is known about the locking capacity. It follows
trivially that L
ð1Þ
u
is sandwiched between the classical
information and the private information [9]:
Q
ð1Þ
ðN Þ ≤ P
ð1Þ
ðN Þ ≤ L
ð1Þ
u
ðN Þ ≤ C
ð1Þ
ðN Þ: ð9Þ
Here we show that private information can be strictly
superadditive for an arbitrarily large number of uses of the
channel. More precisely, we prove the following theorem:
Theorem 1.—For any n there exists a triple ðn; p; dÞ and
a quantum channel N
n;p;d
such that for n>k≥ 1
1
k
P
ð1Þ
ðN
⊗k
n;p;d
Þ <
1
k þ 1
Q
ð1Þ
ðN
⊗kþ1
n;p;d
Þ: ð10Þ
This proves that entangled inputs increase the private
information of a quantum channel and this effect persists
for an arbitrary number of channel uses. Furthermore,
since Q
ð1Þ
ðN
n;p;d
Þ ≤ P
ð1Þ
ðN
n;p;d
Þ < Q
ð1Þ
ðN
⊗2
n;p;d
Þ=2 ≤
P
ð1Þ
ðN
⊗2
n;p;d
Þ=2 < … < Q
ð1Þ
ðN
⊗n
n;p;d
Þ=n ≤ P
ð1Þ
ðN
⊗n
n;p;d
Þ=n
follows from Theorem 1, it turns out that even though the
quantum capacity is upper bounded by the private capacity,
the nonregularized quantities can be interleaved. As a
bonus, we obtain a qualitatively different proof for the
unbounded superadditivity of the coherent information
[21]. The construction of the latter exhibits a jump from
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zero coherent information to positive coherent information
between n
0
and n
1
uses, with n
0
≪ n
1
; here, we obtain a
jump in the coherent information (also in the private
information) between consecutive uses for each of the first
n uses of the channel for any fixed n>1.
We now introduce the key components of our construc-
tion which are required to prove Theorem 1.
Main construction: Switch channel.—The action of a
channel N
A→B
can be defined via an isometry V
A→BE
:
N
A→B
ðρÞ¼tr
E
VρV
, and its complementary channel is
N
A→E
c
ðρÞ¼tr
B
VρV
. Register superscripts are omitted
when they do not add to clarity.
We first introduce switch channels:
N
SA→SB
ðρ
SA
Þ¼
X
i
P
S→S
i
⊗ N
A→B
i
ðρ
SA
Þ: ð11Þ
A switch channel consists of two input registers S and A of
dimensions d and n, respectively. Register S is measured in
the standard basis and conditioned on the measurement
outcome i;acomponent channel N
i
is applied to the
second register. The computation of P
ð1Þ
ðN Þ and L
ð1Þ
u
ðN Þ
when N is of the form (11) can be simplified; it suffices to
restrict inputs to a special form. The equivalent result for
the quantum capacity was proved in Ref. [29].
Lemma 1.—Consider a switch channel N
SA→SB
and let T ¼fρ∶ρ ¼
P
x
p
x
jxihxj
X
⊗ jsihsj
S
⊗ ρ
A
x
g.
Then (1) P
ð1Þ
ðN Þ¼max
1≤s<n
P
ð1Þ
ðN
s
Þ, (2) L
ð1Þ
u
ðN Þ¼
max
1≤s<n
L
ð1Þ
u
ðN
s
Þ. Both P
ð1Þ
ðN Þ and L
ð1Þ
u
ðN Þ can be
achieved by some ρ ∈ T .
The proof of Lemma 1 is located in the Supplemental
Material [30].
There are two types of channels which we will use in
place of N
i
. The first channel is the erasure channel:
E
A→B
p;d
ðρ
A
Þ¼ð1 − pÞρ
B
þ pjeihej
B
; ð12Þ
where jeihej is the erasure flag and d the dimension of the
input register A.Forp ≤ 1=2 the erasure channel is degrad-
able and QðE
p;d
Þ¼PðE
p;d
Þ¼maxf0; ð1 − 2pÞ log dg,
and CðE
p;d
Þ¼ð1 − pÞ log d [31].
For any quantum channel N used alongside E
p;d
the
classical information is additive:
Lemma 2.—For all quantum channels N
C
ð1Þ
ðN ⊗ E
⊗n
p;d
Þ¼C
ð1Þ
ðN ÞþnC
ð1Þ
ðE
p;d
Þ: ð13Þ
The proof of Lemma 2 is located in the Supplemental
Material [30].
Intuitively, Lemma 2 states that the erasure channel
cannot convey more information than an identity channel of
dimension d
1−p
, even in the presence of other channels.
Furthermore, we can use the classical capacity to obtain a
trivial bound for the locking and private information.
The second channel that we use alongside E
p;d
is a
d-dimensional “rocket” channel, R
d
[27]. It consists of
two d-dimensional input registers A
1
and A
2
and a
d-dimensional output register B. A
1
and A
2
are first subject
to a random unitary and then jointly decoupled with a
controlled dephasing gate. Then, the contents of A
1
becomes the output of the channel and the contents of
A
2
is traced out. Bob also receives the classical description
of the unitaries which acted on A
1
and A
2
. Since dephasing
occurs after the input registers have been scrambled by a
random unitary, it is very hard for Alice to code for such
a channel; hence, it has a very low classical capacity:
CðR
d
Þ ≤ 2.
Our switch channel construction has the following form:
N
n;p;d
¼ P
0
⊗ R
n
d
þ P
1
⊗
~
E
n
p;d
: ð14Þ
That is, it allows Alice to choose between R
n
d
¼ R
⊗n
d
and
~
E
n
p;d
¼ E
p;d
⊗ E
1;d
2n−1
,ad-dimensional erasure channel
padded with a full erasure channel to match the input
dimension of R
n
d
.
Upper bound.—To upper bound the private information
of N
n;p;d
we only need to optimize over all the possible
different choices of R
n
d
and
~
E
n
p;d
. Thus, the upper bound for
P
ð1Þ
ðN
⊗k
n;p;d
Þ for k ≥ 1 reads
P
ð1Þ
ðN
⊗k
n;p;d
Þ¼max
0≤i≤k
P
ð1Þ
ðE
⊗i
p;d
⊗ ðR
n
d
Þ
⊗k−i
Þ
≤ max
8
>
>
>
<
>
>
>
:
C
ð1Þ
ððR
n
d
Þ
⊗k
Þ
max
1≤i≤k−1
C
ð1Þ
ðE
⊗i
p;d
⊗ ðR
n
d
Þ
⊗k−i
Þ;
P
ð1Þ
ðE
⊗k
p;d
Þ
≤ max
8
>
>
<
>
>
:
2kn;
ð2n þðk − 1Þð1 − pÞ log dÞ:
ð1 − 2pÞk log d
ð15Þ
Superadditivity of P
ð1Þ
.—We denote A
½k
xy
with superscript
½k to indicate the kth use of the channel and the subscript
xy to indicate the input register as pictured in Fig. 1.
Consider the following protocol for conveying quantum
information over j þ 1 > 1 uses: Alice chooses the rocket
channel for the first use and E
n
p;d
for the remaining j uses.
She prepares a maximally entangled state in the registers
R
z
A
½1
z1
and A
½1
z2
A
½zþ1
11
for z ∈ ½1;j (see Fig. 2). After the
first use of N
n;p;d
the registers A
½1
11
;A
½1
21
; …;A
½1
j1
get
completely dephased by R
n
d
. Without the auxiliary registers
A
½2
11
;A
½3
11
; …;A
½jþ1
11
Bob is unable to undo the dephasing
and thus establish maximally entangled states between the
registers R
1
;R
2
; …;R
j
and B
1
;B
2
; …;B
j
, respectively. So
Alice transmits the former registers using the erasure
channel. The input registers A
½1
ki
of the rocket channel
for k ≥ j, i ¼f1; 2g and the registers that pad the dimen-
sion of the erasure channel do not play any role in the
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protocol, so Alice can send any pure state through each of
them. The input state without the padding subsystems has
the form
ρ ¼ ⊗
j
z¼1
Φ
þ
R
z
A
½1
z1
⊗ Φ
þ
A
½1
z2
A
½zþ1
11
; ð16Þ
where Φ
þ
AB
¼ 1=d
P
d
i;j¼1
jiiihjjj
AB
.
We now analyze the coherent information established by
this protocol between Alice and Bob. For every use of the
rocket channel, if the auxiliary register gets erased the
coherent information is zero—the state is completely
dephased in a random basis. If the auxiliary register is
transmitted to Bob, he can reverse the action of the channel
and obtain a maximally entangled state [27]. This occurs
with probability 1 − p, in which case the coherent infor-
mation is log d. Since this process is repeated j times, the
regularized coherent information is
Q
ð1Þ
ðN
⊗jþ1
n;p;d
; ρÞ¼
j
j þ 1
ð1 − pÞ log d: ð17Þ
This immediately gives a lower bound for the locking
and private information. Now, we are ready to prove
Theorem 1.
Proof.—Fix d ¼ 2
4n
2
=ð1−2pÞ
and p ¼ð11=24Þ. Then
the regularized upper bounds (15) for P
ð1Þ
after k uses
of the channel have the form U
1
k
¼ð2n=kÞ, U
2
k
¼
f2n½13ðk − 1Þn þ 1=kg and U
3
k
¼ 4n
2
; the lower bound
(17) after k þ 1 uses of the channel has the form
L
kþ1
¼½26kn
2
=ðk þ 1Þ.
Consider the differences D
i
k
¼ −U
i
k
þ L
kþ1
for i ¼
1; 2; 3. Then, a simple substitution shows that D
1
k
¼
½26kn
2
=ðkþ1Þ−ð2n=kÞ, D
2
k
¼−½2nðk−13nþ1Þ=kðkþ1Þ,
D
3
k
¼½2ð11k−2Þn
2
=ðkþ1Þ. All of the differences are
positive for n>k≥ 1. □
Superadditivity of L
ð1Þ
u
.—We now study the conditions
necessary to obtain a similar result for the locking infor-
mation of our channel construction. First, we need to
establish several bounds about the locking capacity of the
channels which are used in it. The locking information of
the erasure channel is currently unknown. An upper bound
is obtained in the following lemma:
Lemma 3.—Let p ≤ 1=2, the locking information of E
p;d
is upper bounded by
L
ð1Þ
u
ðE
p;d
Þ ≤ ð1 − pÞ log d − pγ
d
log e; ð18Þ
where γ
d
≔ ln d −
P
d
t¼2
t
−1
, and lim
d→∞
γ
d
¼ γ is Euler’s
constant.
The proof of Lemma 3 is located in the Supplemental
Material [30].
Some algebra shows that the upper bound given by
Lemma 3 combined with the lower bound given by Eq. (17)
does not yield superadditivity. Our upper bound is very
loose and might be improved: we show that if Eve applies
the trivial strategy and performs a random measurement
on her state, then she would be able to extract the amount
of information which is equal to subentropy [32]. The
maximum value of the latter is constant and is independent
of the dimension. It is natural to conjecture that Eve could
extract an amount of information which is proportional to
the dimension of her system by applying some other
strategy. The smallest bound on Eve’s accessible informa-
tion as a function of the dimension of her output which
leads to superadditivity of the locking information in our
construction is given below:
Conjecture 1.—[Sharper upper bound for L
ð1Þ
u
]
L
ð1Þ
u
ðE
p;d
Þ ≤ ð1 − pÞ log d − pϵ log d; ð19Þ
where ϵ > ½ð1 − pÞ=pðn − 1Þ.
The proof of the conjecture together with the techniques
used in the proof of Theorem 1 would allow us to prove
superadditivity.
FIG. 1. The channel has two input registers: the control register
S and the data register A ¼ A
11
A
12
A
21
…A
n2
. The control register
is measured in the computational basis and depending on the
output either the erasure channel
~
E
n
p;d
or n copies of the
d-dimensional rocket channel are applied. For each A
xy
, xy
enumerates the input register. In particular, when R
n
d
acts on A, x
denotes the input to the xth instance of R
d
. For each R
d
, while y
specifies one of the two inputs to R
d
.
FIG. 2. The first part of the protocol consists in sending a state
maximally entangled between the different inputs of the rocket
channel and an external reference.
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Discussion.—In this Letter we have constructed a family
of channels for which the private and coherent information
can remain strictly superadditive any number of uses of the
channel. We are able to prove this result by showing that the
private information of k uses of the channel is smaller than
the coherent information of k þ 1 uses. That is, both
quantities can be interleaved use after use for the first n
uses of the channel. This shows that even though the
quantum capacity is upper bounded by the infinite regu-
larization of the private information, the quantum capacity
can be larger than a finite regularization of the private
information.
Similarly, we expect weak locking information to be
superadditive. For this to be true with our channel con-
struction a tighter bound on the accessible information to
the environment would be necessary.
The results shown here raise questions about the proper-
ties that a channel has to verify such that its different
capacities can be computed exactly using only finitely
many (preferably only a few) copies of the channel.
We thank David Perez García and Māris Ozols for many
useful discussions and feedback. We also thank the referees
for the detailed comments. S. S. acknowledges the support
of Sidney Sussex College and the European Union under
project QALGO (Grant Agreement No. 600700). D. E.
acknowledges financial support from the European CHIST-
ERA project CQC (funded partially by MINECO Grant
No. PRI-PIMCHI-2011-1071) and from Comunidad de
Madrid (Grant QUITEMAD þ −CM, Ref. S2013/ICE-
2801). This work has been partially supported by STW,
QuTech and by the project HyQuNet (Grant No. TEC2012-
35673), funded by Ministerio de Economía y
Competitividad (MINECO), Spain. This work was made
possible through the support of Grant No. 48322 from the
John Templeton Foundation. The opinions expressed in this
publication are those of the authors and do not necessarily
reflect the views of the John Templeton Foundation.
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PRL 115, 040501 (2015)
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