1
A Survey on Quantum Channel Capacities
Laszlo Gyongyosi,
1,2,3,∗
Member, IEEE, Sandor Imre,
2
Senior Member, IEEE, and Hung Viet
Nguyen,
1
Member, IEEE
1
School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK
2
Department of Networked Systems and Services, Budapest University of Technology and Economics, Budapest,
H-1117 Hungary
3
MTA-BME Information Systems Research Group, Hungarian Academy of Sciences, Budapest, H-1051 Hungary
Abstract—Quantum information processing exploits the quan-
tum nature of information. It offers fundamentally new solutions
in the field of computer science and extends the possibilities
to a level that cannot be imagined in classical communica-
tion systems. For quantum communication channels, many new
capacity definitions were developed in comparison to classical
counterparts. A quantum channel can be used to realize classical
information transmission or to deliver quantum information,
such as quantum entanglement. Here we review the properties
of the quantum communication channel, the various capacity
measures and the fundamental differences between the classical
and quantum channels.
Index Terms—Quantum communication, quantum channels,
quantum information, quantum entanglement, quantum Shannon
theory.
I. INTRODUCTION
According to Moore’s Law [326], the physical limitations of
classical semiconductor-based technologies could be reached
within the next few years. We will then step into the age of
quantum information. When first quantum computers become
available on the shelf, today’s encrypted information will
not remain secure. Classical computational complexity will
no longer guard this information. Quantum communication
systems exploit the quantum nature of information offering
new possibilities and limitations for engineers when designing
protocols. Quantum communication systems face two major
challenges.
First, available point-to-point communication link should be
connected on one hand to cover large distances an on the
other hand to reach huge number of users in the form of
a network. Thus, the quantum Internet [267], [304] requires
quantum repeaters and quantum switches/routers. Because of
the so called no-cloning theorem [551], which is the simple
consequence of the postulates of the quantum mechanics, the
construction of these network entities proves to be very hard
[523].
The other challenge – this paper focuses on – is the
amount of information which can be transmitted over quantum
channels, i.e. the capacity. The capacity of a communication
This work was partially supported by the European Research Council
through the Advanced Fellow Grant, in part by the Royal Societys Wolfson
Research Merit Award, in part by the Engineering and Physical Sciences
Research Council under Grant EP/L018659/1, by the Hungarian Scientific Re-
search Fund - OTKA K-112125, and by the National Research Development
and Innovation Office of Hungary (Project No. 2017-1.2.1-NKP-2017-00001).
*Email: l.gyongyosi@soton.ac.uk
channel describes the capability of the channel for delivering
information from the sender to the receiver, in a faithful and
recoverable way. Thanks to Shannon we can calculate the
capacity of classical channels within the frames of classical
information theory
1
[477]. However, the different capacities
of quantum channels have been discovered just in the ‘90s,
and there are still many open questions about the different
capacity measures.
Many new capacity definitions exist for quantum channels in
comparison to a classical communication channel. In the case
of a classical channel, we can send only classical information
while quantum channels extend the possibilities, and besides
the classical information we can deliver entanglement-assisted
classical information, private classical information, and of
course, quantum information [54], [136]. On the other hand,
the elements of classical information theory cannot be applied
in general for quantum information –in other words, they
can be used only in some special cases. There is no general
formula to describe the capacity of every quantum channel
model, but one of the main results of the recent researches
was a simplified picture in which various capacities of a
quantum channel (i.e., the classical, private, quantum) are all
non-additive [245].
In possession of admitted capacity definitions they have to
be calculated for various channel models. Channels behave in
very different ways in free-space or in optical fibers and these
two main categories divides into many subclasses and special
cases [178], [181], [567].
Since capacity shows only the theoretically achievable trans-
mission rate and gives no construction rules how to reach or
near them, therefore quantum channel/error correction coding
has similar importance from practical implementation point of
view as in case of classical information theory [171].
This paper is organized as follows. In Section II, prelimi-
naries are summarized. In Section III, we study the classical
information transmission capability of quantum channels. In
Section IV, we discuss the quantum capacity. Numerical
examples are included in Section V. Section VI focuses on
the practical implementations of quantum channels. Finally,
Section VII concludes the paper. Supplementary material is
included in the Appendix.
1
Quantum Shannon theory has deep relevance concerning the information
transmission and storage in quantum systems. It can be regarded as a natural
generalization of classical Shannon theory. Classical information theory
represents an orthogonality-restricted case of quantum information theory.
arXiv:1801.02019v1 [quant-ph] 6 Jan 2018
2
II. PRELIMINARIES
A. Applications and Gains of Quantum Communications
Before discussing the modeling, characteristics and capaci-
ties of quantum channels we present their potential to improve
state-of the-art communication and computing systems.
We highlight the fact that from application point of view
the concept of channel can represent any medium possessing
an input to receive information and an output to give back a
modified version of this information. This simplified definition
highlights the fact that not only an optical fiber, a copper cable
or a free-space link can be regarded as channel but a computer
memory, too.
Quantum communication systems are capable of providing
absolute randomness, absolute security, of improving trans-
mission quality as well as of bearing much more information
in comparison to the current classical binary based systems.
Moreover, when the benefits of quantum computing power are
properly employed, the quantum based solutions are capable
of supporting the execution of tasks much faster or beyond
the capability of the current binary based systems [131]. The
appealing gains and the associated application scenarios that
we may expect from quantum communications are as follows.
The general existence of a qubit ψ in a superposition state
(see the next sub-sections of Section II) of two pure quantum
states |0i and |1i can be represented by
|ψi = α|0i + β|1i, (1)
where α and β are complex number. If a qubit ψ is measured
by |0i and |1i bases, the measurement result is randomly
obtained in the state of |0i or |1i with the corresponding
probability of |α|
2
or |β|
2
. This random nature of quantum
measure have been favourably used for providing high quality
random number generator [249, 265], [316]. It is important
to note that along with the measurement randomness, no-
cloning theorem [551] of qubit says that it is not possible
to clone a qubit. This characteristics allow quantum based
solutions to support absolute security, to which there have
been abundant examples of quantum based solutions [176],
[300], [302], [569], [553] where a popular example of mature
applications is quantum key distribution (QKD) [53], [68].
Quantum entanglement is a unique characteristic of quan-
tum mechanics, which is another valuable foundation for pro-
visioning the absolute secure communication. Let us consider
a two qubit system σ represented by
|σi = α
00
|0i|0i + α
01
|0i|1i + α
10
|1i|0i + α
11
|1i|1i,(2)
where α
00
, α
01
, α
10
, α
11
are complex numbers having
|α
00
|
2
+|α
01
|
2
+|α
10
|+|α
11
|
2
= 1. If the system σ is prepared
in one of the four states (see Appendix), for example
|σi = α
00
|0i|0i + α
11
|1i|1i, (3)
where |α
00
|
2
+|α
11
|
2
= 1, the measurement result of the two
qubits is in either |00i or |11i state. In this state, the two qubits
are entangled, meaning that having the measurement result
of either of the two is sufficient to know the measurement
result of the other. As a result, if the two entangled qubits are
separated in the distance, for example 144 km terrestial dis-
tance [158] or earth-station to satellite 1200 km distance [561],
information can be secretly transmitted over two locations,
where there exists entanglement between the two locations.
The entanglement based transmission can be employed for
transmitting classical bits by using the superdense coding
protocol [1], [33], [242] or for transmitting qubits using the
quantum teleportation protocol [55], [226].
Classical channels handle classical information i.e. orthog-
onal (distinguishable) basis states while quantum channels
may deliver superposition states (linear combination of basis
states). Of course, since quantum mechanic is more complete
than classical information theory classical information and
classical channels can be regarded as special cases of quantum
information and channels. Keeping in mind the application
scenarios, there is a major difference between classical and
quantum information. Human beings due to their limited
senses can perceive only classical information; therefore mea-
surement is needed to perform conversion between the quan-
tum and classical world.
From the above considerations, quantum channels can be
applied in several different ways for information transmission.
If classical information is encoded to quantum states, the
quantum channel delivers this information between its input
and output and finally a measurement device converts the
information back to the classical world. In many practical
settings, quantum channels are used to transfer classical in-
formation only.
The most discussed practical application of this approach is
QKD. Optical fiber based [243], [255], [282], [511] ground-
ground [565] and ground-space [301] systems have already
been demonstrated. These protocols independently whether
they are first-generation single photon systems or second-
generation multi photon solutions exchange classical se-
quences between Alice and Bob over the quantum channel
being encoded in non-orthogonal quantum states. Since the
no-cloning theorem [244], [551] makes no possible to copy
(to eavesdrop) the quantum states without error, symmetric
ciphering keys can be established for both parties. In this case
quantum channel is used to create a new quality instead of
improving the performance of classical communication.
Furthermore, quantum encoding can improve the transmis-
sion rates of certain channels. For example the well-known bit-
flip channel inverts the incoming bit value by probability p and
leaves it unchanged by (1−p). Classically this type of channel
can not transmit any information at all if p = 0.5 even if we
apply redundancy for error correction. However, if classical
bits are encoded into appropriate quantum bits one-by-one, i.e.,
no redundancy is used, the information will be delivered with-
out error. This means that quantum communication improves
the classical information transmission capability of the bit-
flip channel form 0 to the maximum 1. The different models
of classical information transmission over a quantum channel
will be detailed in Section III (particulary in Section III-C-
Section III-G).
The second approach applies quantum channels to deliver
quantum information and this information is used to improve
the performance of classical communication systems. The
3
detailed discussion of the transmission of quantum information
is the subject of Section IV. These protocols exploit over-
quantum-channel-shared entangled states, i.e. entanglement
assisted communications is considered. In case of quantum
superdense coding [58], [70], [244] we assume that Alice and
Bob have already shared an entangled Bell-pair, such as |β
00
i
(see Appendix), expressed as
|β
00
i =
1
√
2
(|00i + |11i) .
(4)
When Alice wants to communicate with Bob, she encodes
two classical bits into the half pair she possesses and sends
this quantum bit to Bob over the quantum channel. Finally,
Bob leads his own qubit together the received one to a
measuring device which decodes the original two classical bits.
Practically 2 classical bits have been transferred at the expense
of 1 quantum bit, i.e., the entanglement assisted quantum
channels can outperform classical ones.
Another practical example of this approach is distributed
medium access control. In this case a classical communication
channel is supported by pre-shared entanglement. It is well-
known that WiFi and other systems can be derived from
the Slotted Aloha protocol [2] widely used as a reference.
Slotted Aloha can deliver [0.5/e, 1/e] packets in average in
each timeslot if the number of nodes is known for everyone,
and optimal access strategy is used by everyone. This is
because of collisions and unused timeslots. Practically the
size of the population can be only estimated which decreases
the efficiency. However, if special entangled |w
n
i states are
generated as
|w
n
i =
1
√
n
n
X
i=1
|2
(n−i)
i. (5)
and used to coordinate the channel access in a distributed way
the timeslot usage will improve to 100% and there is no need
to know the number of users.
Further important application scenarios are related to quan-
tum computers where quantum information has to be delivered
between modules over quantum connections. Similarly quan-
tum memories are practically quantum channels of course
with different characteristics compared to communication
channels which store and read back quantum information.
B. Privacy and Performance Gains of Quantum Channels
Due to the inherent no-cloning theory of quantum mechan-
ics, the random nature of quantum measurement as well as
to the unique entanglement phenomenon of quantum mechan-
ics, secure communications can be guaranteed by quantum
communications. The private classical capacity of a quantum
channel is detailed in Section III-C.
Moreover, quantum communications using quantum chan-
nels is capable of carrying much more information in compar-
ison to the current classical binary based systems. Let us have
a closer look at Eq. (1), where obviously one qubit contains
superpositioned 2
1
distinct states or values, which is equivalent
to at least 2 bits. In the case of using two qubits in Eq. (3),
2
2
distinct states or values are simultaneously conveyed by
two qubits, meaning at least 2
2
× 2 bits are carried by 2
qubits. Generally, n qubits can carry up to 2
n
states, which
corresponds to 2
n
×n bits. The superposition nature of qubits
leads to the advent of powerful quantum computing, which
is in some cases proved be 100 millions times faster than
the classical computer [131]. Moreover, in theory quantum
computer is capable of providing the computing power that is
beyond the capability of its classical counterpart. Importantly,
in order to realise such supreme computing power, the crucial
part is quantum communications, which has to be used for
transmitting qubits within the quantum processor as well as
between distributed quantum processors.
Additionally, quantum receivers [49] relying on quantum
communications principle has proved to outperform classical
homodyne or heterodyne receiver in the context of optical
communications. For the sake of brevity, please allow us to
refer interested readers to the references [49], [516].
C. Communication over a Quantum Channel
Communication through a quantum channel cannot be
described by the results of classical information theory; it
requires the generalization of classical information theory
by quantum perception of the world. In the general model
of communication over a quantum channel N, the encoder
encodes the message in some coded form, and the receiver
decodes it, however in this case, the whole communication is
realized through a quantum system.
The information sent through quantum channels is car-
ried by quantum states, hence the encoding is fundamentally
different from any classical encoder scheme. The encoding
here means the preparation of a quantum system, according
to the probability distribution of the classical message being
encoded. Similarly, the decoding process is also different:
here it means the measurement of the received quantum state.
The properties of quantum communication channel, and the
fundamental differences between the classical and quantum
communication channel cannot be described without the ele-
ments of quantum information theory.
The model of the quantum channel represents the physically
allowed transformations which can occur on the sent quantum
system. The result of the channel transformation is another
quantum system, while the quantum states are represented
by matrices. The physically allowed channel transforma-
tions could be very different; nevertheless they are always
Completely Positive Trace Preserving (CPTP) transformations
(trace: the sum of the elements on the main diagonal of a
matrix). The trace preserving property therefore means that the
corresponding density matrices (density matrix: mathematical
description of a quantum system) at the input and output of
the channel have the same trace.
The input of a quantum channel is a quantum state, which
encodes information into a physical property. The quantum
state is sent through a quantum communication channel,
which in practice can be implemented e.g. by an optical-fiber
channel, or by a wireless quantum communication channel.
To extract any information from the quantum state, it has
to be measured at the receiver’s side. The outcome of the
measurement of the quantum state (which might be perturbed)
4
depends on the transformation of the quantum channel, since it
can be either totally probabilistic or deterministic. In contrast
to classical channels, a quantum channel transforms the infor-
mation coded into quantum states, which can be e.g. the spin
state of the particle, the ground and excited state of an atom, or
several other physical approaches. The classical capacity of a
quantum channel has relevance if the goal is transmit classical
information in a quantum state, or would like to send classical
information privately via quantum systems (private classical
capacity). The quantum capacity has relevance if one would
like to transmit quantum information such as superposed
quantum states or quantum entanglement.
First, we discuss the process of transmission of informa-
tion over a quantum channel. Then, the interaction between
quantum channel output and environment will be described.
1) The Quantum Channel Map: From algebraic point of
view, quantum channels are linear CPTP maps, while from a
geometrical viewpoint, the quantum channel N is an affine
transformation. While, from the algebraic view the transfor-
mations are defined on density matrices, in the geometrical
approach, the qubit transformations are also interpretable via
the Bloch sphere (a geometrical representation of the pure
state space of a qubit system) as Bloch vectors (vectors in
the Bloch sphere representation). Since, density matrices can
be expressed in terms of Bloch vectors, hence the map of a
quantum channel N also can be analyzed in the geometrical
picture.
To preserve the condition for a density matrix ρ, the noise
on the quantum channel N must be trace-preserving (TP), i.e.,
T r (ρ) =T r (N (ρ)) , (6)
and it must be Completely Positive (CP), i.e., for any identity
map I, the map I⊗N maps a semi-positive Hermitian matrix
to a semi-positive Hermitian matrix.
Fig. 1: Geometrical picture of a noisy qubit quantum channel
on the Bloch sphere [Imre13].
For a unital quantum channel N, the channel map trans-
forms the I identity transformation to the I identity transfor-
mation, while this condition does not hold for a non-unital
channel. To express it, for a unital quantum channel, we have
N (I) =I, (7)
while for a non-unital quantum channel,
N (I) 6=I. (8)
Focusing on a qubit channel, the image of the quantum
channel’s linear transform is an ellipsoid on the Bloch sphere,
as it is depicted in Fig. 1. For a unital quantum channel, the
center of the geometrical interpretation of the channel ellipsoid
is equal to the center of the Bloch sphere. This means that a
unital quantum channel preserves the average of the system
states. On the other hand, for a non-unital quantum channel,
the center of the channel ellipsoid will differ from the center of
the Bloch sphere. The main difference between unital and non-
unital channels is that the non-unital channels do not preserve
the average state in the center of the Bloch sphere. It follows
from this that the numerical and algebraic analysis of non-
unital quantum channels is more complicated than in the case
of unital ones. While unital channels shrink the Bloch sphere
in different directions with the center preserved, non-unital
quantum channels shrink both the original Bloch sphere and
move the center from the origin of the Bloch sphere. This fact
makes our analysis more complex, however, in many cases,
the physical systems cannot be described with unital quantum
channel maps. Since the unital channel maps can be expressed
as the convex combination of the basic unitary transformations,
the unital channel maps can be represented in the Bloch
sphere as different rotations with shrinking parameters. On
the other hand, for a non-unital quantum map, the map cannot
be decomposed into a convex combination of unitary rotations
[245].
2) Steps of the Communication: The transmission of in-
formation through classical channels and quantum channels
differs in many ways. If we would like to describe the process
of information transmission through a quantum communica-
tion channel, we have to introduce the three main phases
of quantum communication. In the first phase, the sender,
Alice, has to encode her information to compensate the noise
of the channel N (i.e., for error correction), according to
properties of the physical channel - this step is called channel
coding. After the sender has encoded the information into the
appropriate form, it has to be put on the quantum channel,
which transforms it according to its channel map - this second
phase is called the channel evolution. The quantum channel
N conveys the quantum state to the receiver, Bob; however
this state is still a superposed and probably mixed (according
to the noise of the channel) quantum state. To extract the
information which is encoded in the state, the receiver has to
make a measurement - this decoding process (with the error
correction procedure) is the third phase of the communication
over a quantum channel.
The channel transformation represents the noise of the
quantum channel. Physically, the quantum channel is the
medium, which moves the particle from the sender to the
receiver. The noise disturbs the state of the particle, in the case
of a half-spin particle, it causes spin precession. The channel
evolution phase is illustrated in Fig. 2.
Finally, the measurement process responsible for the de-
coding and the extraction of the encoded information. The
5
Fig. 2: The channel evolution phase.
previous phase determines the success probability of the
recovery of the original information. If the channel N is
completely noisy, then the receiver will get a maximally mixed
quantum state. The output of the measurement of a maximally
mixed state is completely undeterministic: it tells us nothing
about the original information encoded by the sender. On the
other hand, if the quantum channel N is completely noiseless,
then the information which was encoded by the sender can be
recovered with probability 1: the result of the measurement
will be completely deterministic and completely correlated
with the original message. In practice, a quantum channel
realizes a map which is in between these two extreme cases. A
general quantum channel transforms the original pure quantum
state into a mixed quantum state, - but not into a maximally
mixed state - which makes it possible to recover the original
message with a high - or low - probability, depending on the
level of the noise of the quantum channel N.
D. Formal Model
As shown in Fig. 3, the information transmission through
the quantum channel N is defined by the ρ
in
input quantum
state and the initial state of the environment ρ
E
= |0i h0|. In
the initial phase, the environment is assumed to be in the
pure state |0i. The system state which consist of the input
quantum state ρ
in
and the environment ρ
E
= |0i h0|, is called
the composite state ρ
in
⊗ρ
E
.
Fig. 3: The general model of transmission of information over
a noisy quantum channel.
If the quantum channel N is used for information transmis-
sion, then the state of the composite system changes unitarily,
as follows:
U (ρ
in
⊗ρ
E
) U
†
, (9)
where U is a unitary transformation, and U
†
U=I. After the
quantum state has been sent over the quantum channel N, the
ρ
out
output state can be expressed as:
N (ρ
in
) =ρ
out
=T r
E
U (ρ
in
⊗ρ
E
) U
†
, (10)
where T r
E
traces out the environment E from the joint
state. Assuming the environment E in the pure state |0i,
ρ
E
= |0i h0|, the N (ρ
in
) noisy evolution of the channel N
can be expressed as:
N (ρ
in
) =ρ
out
=T r
E
Uρ
in
⊗|0i h0| U
†
, (11)
while the post-state ρ
E
of the environment after the transmis-
sion is
ρ
E
=T r
B
Uρ
in
⊗|0i h0| U
†
, (12)
where T r
B
traces out the output system B. In general, the
i-th input quantum state ρ
i
is prepared with probability p
i
,
which describes the ensemble {p
i
, ρ
i
}. The average of the
input quantum system is
σ
in
=
X
i
p
i
ρ
i
, (13)
The average (or the mixture) of the output of the quantum
channel is denoted by
σ
out
=N (σ
in
) =
X
i
p
i
N (ρ
i
). (14)
E. Quantum Channel Capacity
The capacity of a communication channel describes the
capability of the channel for sending information from the
sender to the receiver, in a faithful and recoverable way.
The perfect ideal communication channel realizes an identity
map. For a quantum communication channel, it means that
the channel can transmit the quantum states perfectly. Clearly
speaking, the capacity of the quantum channel measures the
closeness to the ideal identity transformation I.
To describe the information transmission capability of the
quantum channel N, we have to make a distinction between
the various capacities of a quantum channel. The encoded
quantum states can carry classical messages or quantum mes-
sages. In the case of classical messages, the quantum states
encode the output from a classical information source, while
in the latter the source is a quantum information source.
On one hand for classical communication channel N, only
one type of capacity measure can be defined, on the other
hand for a quantum communication channel N a number
of different types of quantum channel capacities can be
applied, with different characteristics. There are plenty of open
questions regarding these various capacities. In general, the
single-use capacity of a quantum channel is not equal to the
asymptotic capacity of the quantum channel (As we will see
later, it also depends on the type of the quantum channel).
The asymptotic capacity gives us the amount of information
![Fig. 27: The classical and quantum capacities of the erasure quantum channel as a function of erasure probability [Imre13].](/figures/fig-27-the-classical-and-quantum-capacities-of-the-erasure-6ujx7y99.webp)
