According to Moore’s Law , 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 ,  requires quantum repeaters and quantum switches/routers. Because of the so called no-cloning theorem , which is the simple consequence of the postulates of the quantum mechanics, the construction of these network entities proves to be very hard .
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 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 theory111Quantum 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. . 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 , . 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 .
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 , , .
Since capacity shows only the theoretically achievable transmission 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 .
This paper is organized as follows. In Section II, preliminaries 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.
Ii-a Applications and Gains of Quantum Communications
Before discussing the modeling, characteristics and capacities 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 transmission 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 . The appealing gains and the associated application scenarios that we may expect from quantum communications are as follows.
The general existence of a qubitin a superposition state (see the next sub-sections of Section II) of two pure quantum states and can be represented by
where and are complex number. If a qubit is measured by and bases, the measurement result is randomly obtained in the state of or
with the corresponding probability ofor . This random nature of quantum measure have been favourably used for providing high quality random number generator [249, 265], . It is important to note that along with the measurement randomness, no-cloning theorem  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 , , , ,  where a popular example of mature applications is quantum key distribution (QKD) , .
Quantum entanglement is a unique characteristic of quantum mechanics, which is another valuable foundation for provisioning the absolute secure communication. Let us consider a two qubit system represented by
where are complex numbers having . If the system is prepared in one of the four states (see Appendix), for example
where , the measurement result of the two qubits is in either or 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 distance  or earth-station to satellite 1200 km distance , 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 , ,  or for transmitting qubits using the quantum teleportation protocol , .
Classical channels handle classical information i.e. orthogonal (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 measurement is needed to perform conversion between the quantum 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 information only.
The most discussed practical application of this approach is QKD. Optical fiber based , , ,  ground-ground  and ground-space  systems have already been demonstrated. These protocols – independently whether they are first-generation single photon systems or second-generation multi photon solutions – exchange classical sequences between Alice and Bob over the quantum channel being encoded in non-orthogonal quantum states. Since the no-cloning theorem ,  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 transmission rates of certain channels. For example the well-known bit-flip channel inverts the incoming bit value by probability and leaves it unchanged by . Classically this type of channel can not transmit any information at all if 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 without 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 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 , ,  we assume that Alice and Bob have already shared an entangled Bell-pair, such as (see Appendix), expressed as
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  widely used as a reference. Slotted Aloha can deliver
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 entangledstates are generated as
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 quantum computers where quantum information has to be delivered between modules over quantum connections. Similarly quantum memories are practically quantum channels – of course with different characteristics compared to communication channels – which store and read back quantum information.
Ii-B Privacy and Performance Gains of Quantum Channels
Due to the inherent no-cloning theory of quantum mechanics, the random nature of quantum measurement as well as to the unique entanglement phenomenon of quantum mechanics, 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 channels is capable of carrying much more information in comparison to the current classical binary based systems. Let us have a closer look at Eq. (1), where obviously one qubit contains superpositioned distinct states or values, which is equivalent to at least 2 bits. In the case of using two qubits in Eq. (3), distinct states or values are simultaneously conveyed by two qubits, meaning at least bits are carried by 2 qubits. Generally, qubits can carry up to states, which corresponds to 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 . 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.
Ii-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 , 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 carried 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 elements 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 transformations 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) 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 information 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 information over a quantum channel. Then, the interaction between quantum channel output and environment will be described.
Ii-C1 The Quantum Channel Map
From algebraic point of view, quantum channels are linear CPTP maps, while from a geometrical viewpoint, the quantum channel
is an affine transformation. While, from the algebraic view the transformations 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 channelalso can be analyzed in the geometrical picture.
To preserve the condition for a density matrix , the noise on the quantum channel must be trace-preserving (TP), i.e.,
and it must be Completely Positive (CP), i.e., for any identity map I, the map maps a semi-positive Hermitian matrix to a semi-positive Hermitian matrix.
For a unital quantum channel , the channel map transforms the I identity transformation to the I identity transformation, while this condition does not hold for a non-unital channel. To express it, for a unital quantum channel, we have
while for a non-unital quantum channel,
Focusing on a qubit channel, the image of the quantum channel’s linear transform is anellipsoid 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 .
Ii-C2 Steps of the Communication
The transmission of information through classical channels and quantum channels differs in many ways. If we would like to describe the process of information transmission through a quantum communication 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 (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 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 decoding and the extraction of the encoded information. The previous phase determines the success probability of the recovery of the original information. If the channel 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 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 .
Ii-D Formal Model
As shown in Fig. 3, the information transmission through the quantum channel is defined by the input quantum state and the initial state of the environment . In the initial phase, the environment is assumed to be in the pure state . The system state which consist of the input quantum state and the environment , is called the composite state .
If the quantum channel is used for information transmission, then the state of the composite system changes unitarily, as follows:
where is a unitary transformation, and . After the quantum state has been sent over the quantum channel , the output state can be expressed as:
where traces out the environment E from the joint state. Assuming the environment E in the pure state , , the noisy evolution of the channel can be expressed as:
while the post-state of the environment after the transmission is
where traces out the output system B. In general, the i-th input quantum state is prepared with probability , which describes the ensemble . The average of the input quantum system is
The average (or the mixture) of the output of the quantum channel is denoted by
Ii-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 , we have to make a distinction between the various capacities of a quantum channel. The encoded quantum states can carry classical messages or quantum messages. 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 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 which can be transmitted in a reliable form using the quantum channel infinitely many times. The encoding and the decoding functions mathematically can be described by the operators and , realized on the blocks of quantum states. These superoperators describe unitary transformations on the input states together with the environment of the quantum system. The model of communication through noisy quantum channel with encoding, delivery and decoding phases is illustrated in Fig. 4.
We note, in our paper we will use the terms classical quantity and quantum quantity with relation to the quantum channel as follows:
classical quantity: it is a measure of the classical transmission capabilities of a quantum channel. (See later: Holevo information, quantum mutual information, etc., in Section III)
quantum quantity: it is a measure of the quantum transmission capabilities of a quantum channel (See later: quantum coherent information,etc., in Section IV)
If we mention classical quantity we will do this with relation to the quantum channel , i.e., for example the Holevo information is also not a typical’ classical quantity since it is describes a quantum system not a classical one, but with relation to the quantum channel we can use the classical mark. The historical background with the description of the most relevant works can be found in the Related Work part of each section. For detailed information see .
Quantum information theory also has relevance to the discussion of the capacity of quantum channels and to information transmission and storage in quantum systems. As we will see in this section, while the transmission of product states can be described similar to classical information, on the other hand, the properties of quantum entanglement cannot be handled by the elements of classical information theory. Of course, the elements of classical information theory can be viewed as a subset of the larger and more complex quantum information theory .
First, we summarize the basic definitions and formulas of quantum information theory. We introduce the reader to the description of a noisy quantum channel, purification, isometric extension, Kraus representation and the von Neumann entropy. Next, we describe the encoding of quantum states and the meaning of Holevo information, the quantum mutual information and quantum conditional entropy.
Before starting the discussion on various capacities of quantum channels and the related consequences we summarize the basic definitions and formulas of quantum information theory intended to represent the information stored in quantum states. Those readers who are familiar with density matrices, entropies etc. may run through this section.
The world of quantum information processing (QIP) is describable with the help of quantum information theory (QIT), which is the main subject of this section. We will provide an overview of the most important differences between the compressibility of classical bits and quantum bits, and between the capacities of classical and quantum communication channels. To represent classical information with quantum states, we might use pure orthogonal states. In this case there is no difference between the compressibility of classical and quantum bits.
Similarly, a quantum channel can be used with pure orthogonal states to realize classical information transmission, or it can be used to transmit non-orthogonal states or even quantum entanglement. Information transmission also can be approached using the question, whether the input consists of unentangled or entangled quantum states. This leads us to say that for quantum channels many new capacity definitions exist in comparison to a classical communication channel.
Quantum information theory also has relevance to the discussion of the capacity of quantum channels and to information transmission and storage in quantum systems. While the transmission of product states can be described similar to classical information, on the other hand, the properties of quantum entanglement cannot be handled by the elements of classical information theory. Of course, the elements of classical information theory can be viewed as a subset of the larger and more complex quantum information theory.
Before we would start to our introduction to quantum information theory, we have to make a clear distinction between quantum information theory and quantum information processing. Quantum information theory is rather a generalization of the elements and functions of classical information theory to describe the properties of quantum systems, storage of information in quantum systems and the various quantum phenomena of quantum mechanics. While quantum information theory aims to provide a stable theoretical background, quantum information processing is a more general and rather experimental field: it answers what can be achieved in engineering with the help of quantum information. Quantum information processing includes the computing, error-correcting schemes, quantum communication protocols, field of communication complexity, etc.
The character of classical information and quantum information is significantly different. There are many phenomena in quantum systems which cannot be described classically, such as entanglement, which makes it possible to store quantum information in the correlation of quantum states. Entangled quantum states are named to EPR states after Einstein, Podolsky and Rosen, or Bell states, after J. Bell. Quantum entanglement was discovered in the 1930s, and it may still yield many surprises in the future. Currently it is clear that entanglement has many classically indescribable properties and many new communication approaches based on it. Quantum entanglement plays a fundamental role in advanced quantum communications, such as teleportation, quantum cryptography etc.
The elements of quantum information theory are based on the laws of quantum mechanics. The main results of quantum information processing were laid down during the end of the twentieth century, the most important results being stated by Feynman, Bennett, DiVincenzo, Devetak, Deutsch, Holevo, Lloyd, Schumacher, Shor and many more. After the basic concepts of quantum information processing had been stated, researchers started to look for efficient quantum error correction schemes and codes, and started to develop the theoretical background of fault-tolerant quantum computation. The main results from this field were presented by Bennett, Schumacher, Gottesman, Calderbank, Preskill, Knill, and Kerckhoff. On the other hand, there are still many open questions about quantum computation. The theoretical limits of quantum computers were discovered by Bennett, Bernstein, Brassard and Vazirani: quantum computers can provide at best a quadratic reduction in the complexity of search-based problems, hence if we give an NP-complete problem to quantum computer, it still cannot solve it. Recently, the complexity classes of quantum information processing have been investigated, and many new classes and lower bounds have been found.
By the end of the twentieth century, many advanced and interesting properties of quantum information theory had been discovered, and many possible applications of these results in future communication had been developed. One of the most interesting revealed connections was that between quantum information theory and the elements of geometry. The space of quantum states can be modeled as a convex set which contains points with different probability distributions, and the geometrical distance between these probability distributions can be measured by the elementary functions of quantum information theory, such as the von Neumann entropy or the quantum relative entropy function. The connection between the elements of quantum information theory and geometry leads us to the application of advanced computational geometrical algorithms to quantum space, to reveal the still undiscovered properties of quantum information processing, such as the open questions on the capacities of the quantum channels or their additivity properties. The connection between the Hilbert space of quantum states and the geometrical distance can help us to reveal the fantastic properties of quantum bits and quantum state space.
Several functions have been defined in quantum information theory to describe the statistical distances between the states in the quantum space: one of the most important is the quantum relative entropy function which plays a key role in the description of entanglement, too. This function has many different applications, and maybe this function plays the most important role in the questions regarding the capacity of quantum channels. The possible applications of the quantum relative entropy function have been studied by Schumacher and Westmoreland and by Vedral.
Quantum information theory plays fundamental role in the description of the data transmission through quantum communication channels. At the dawn of this millennium new problems have arisen, whose solutions are still not known, and which have opened the door to many new promising results such as the superactivation of zero-capacity quantum channels in 2008, and then the superactivation of the zero-error capacities of the quantum channels in 2009 and 2010. One of the earliest works on the capacities of quantum communication channels was published in the early 1970s. Along with other researchers, Holevo was showed that there are many differences between the properties of classical and quantum communication channels, and illustrated this with the benefits of using entangled input states. Later, he also stated that quantum communication channels can be used to transmit both classical and quantum information. Next, many new quantum protocols were developed, such as teleportation or superdense coding. After Alexander Holevo published his work, about thirty years later, he, with Benjamin Schumacher and Michael Westmoreland presented one of the most important result in quantum information theory, called the Holevo-Schumacher-Westmoreland (HSW) theorem , . The HSW-theorem is a generalization of the classical noisy channel coding theorem from classical information theory to a noisy quantum channel. The HSW theorem is also called the product-state classical channel capacity theorem of a noisy quantum channel. The understanding of the classical capacity of a quantum channel was completed by 1997 by Schumacher and Westmoreland, and by 1998 by Holevo, and it has tremendous relevance in quantum information theory, since it was the first to give a mathematical proof that a noisy quantum channel can be used to transmit classical information in a reliable form. The HSW theorem was a very important result in the history of quantum information theory, on the other hand it raised a lot of questions regarding the transmission of classical information over general quantum channels.
The quantum capacity of a quantum channel was firstly formulated by Seth Lloyd in 1996, then by Peter Shor in 2002, finally it was completed by Igor Devetak in 2003, - the result is known as the LSD channel capacity , , . While the classical capacity of a quantum channel is described by the maximum of quantum mutual information and the Holevo information, the quantum capacity of the quantum channels is described by a completely different correlation measure: called the quantum coherent information. The concept of quantum coherent information plays a fundamental role in the computation of the quantum capacity which measures the asymptotic quantum capacity of the quantum capacity in general. For the complete historical background with the references see the Related Works.
Ii-F2 Density Matrix and Trace Operator
In this section we introduce a basic concept of quantum information theory, called the density matrix.
Before we start to discuss the density matrix, we introduce some terms. An square matrix A is called positive-semidefinite if is a non-negative real number for every vector . If , i.e., A has Hermitian matrix and the eigenvalues of A are all non-negative real numbers then it is positive-semidefinite. This definition has important role in quantum information theory, since every density matrix is positive-semidefinite. It means, for any vector the positive-semidefinite property says that
In (15) we used, the density matrix is denoted by , and it describes the system by the classical probability weighted sum of possible states
where is the i-th system state occurring with classical probability . As can be seen, this density matrix describes the system as a probabilistic mixture of the possible known states the so called pure states. For pure state the density matrix is and the rank of the matrix is equal to one. Trivially, classical states e.g. and are pure, however, if we know that our system is prepared to the superposition then this state is pure, too. Clearly speaking, while superposition is a quantum linear combination of orthonormal basis states weighted by probability amplitudes, mixed states are classical linear combination of pure superpositions (quantum states) weighted by classical probabilities.
The density matrix contains all the possible information that can be extracted from the quantum system. It is possible that two quantum systems possess the same density matrices: in this case, these quantum systems are called indistinguishable, since it is not possible to construct a measurement setting, which can distinguish between the two systems.
The density matrix of a simple pure quantum system which can be given in the state vector representation can be expressed as the outer product of the ket and bra vectors, where bra is the transposed complex conjugate of ket, hence for the density matrix is
The density matrix contains the probabilistic mixture of different pure states, which representation is based on the fact that the mixed states can be decomposed into weighted sum of pure states .
To reveal important properties of the density matrix, we introduce the concept of the trace operation. The trace of a density matrix is equal to the sum of its diagonal entries. For an square matrix , the trace operator is defined as
where are the elements of the main diagonal. The trace of the matrix A is also equal to the sum of the eigenvalues of its matrix. The eigenvalue is the factor by which the eigenvector changes if it is multiplied by the matrix A
, for each eigenvectors. Theeigenvectors of the square matrix A are those non-zero vectors, whose direction remain the same to the original vector after multiplied by the matrix A. It means, the eigenvectors remain proportional to the original vector. For square matrix A, the non-zero vector is called eigenvector of A, if there is a scalar for which
where is the eigenvalue of A corresponding to the eigenvector .
The trace operation gives us the sum of the eigenvalues of positive-semidefinite A, for each eigenvectors, hence , and . Using the eigenvalues, the spectral decomposition of density matrix can be expressed as
are orthonormal vectors.
The trace is a linear map, hence for square matrices A and B
where s is a scalar. Another useful formula, that for matrix A and matrix B,
which holds for any matrices A and B for which the product matrix AB is a square matrix, since
Finally, we mention that the trace of a matrix A and the trace of its transpose are equal, hence
If we take the conjugate transpose of the matrix A, then we will find that
which will be denoted by and it is called the inner product. For matrices A and B, the inner product , which can be used to define the angle between the two vectors. The inner product of two vectors will be zero if and only if the vectors are orthogonal.
As we have seen, the trace operation gives the sum of the eigenvalues of matrix A, this property can be extended to the density matrix, hence for each eigenvectors of density matrix
Now, having introduced the trace operation, we apply it to a density matrix. If we have an n-qubit system being in the state , then
where we exploited the relation for unit-length vectors
Thus the trace of any density matrix is equal to one
The trace operation can help to distinguish pure and mixed states since for a given pure state
while for a mixed state ,
where and , where are the eigenvalues of density matrix .
Similarly, for a pure entangled system
while for any mixed subsystem of the entangled state (i.e., for a half-pair of the entangled state), we will have
The density matrix also can be used to describe the effect of a unitary transform on the probability distribution of the system. The probability that the whole quantum system is in can be calculated by the trace operation. If we apply unitary transform to the state , the effect can be expressed as follows:
If the applied transformation is not unitary, a more general operator denoted by is introduced, and with the help of this operator the transform can be written as
where for every matrices . In this sense, operator describes the physically admissible or Completely Positive Trace Preserving (CPTP) operations. The application of a CPTP operator on density matrix will result in a matrix , which in this case is still a density matrix.
Ii-F3 Quantum Measurement
Now, let us turn to measurements and their relation to density matrices. Assuming a projective measurement device, defined by measurement operators - i.e., projectors . The projector is a Hermitian matrix, for which and . According to the 3 Postulate of Quantum Mechanics the trace operator can be used to give the probability of outcome j belonging to the operator in the following way
After the measurement, the measurement operator leaves the system in a post measurement state
If we have a pure quantum state , where and . Using the trace operator, the measurement probabilities of and can be expressed as
in accordance with our expectations. Let us assume we have an orthonormal basis and an arbitrary (i.e., non-diagonal) density matrix . The set of Hermitian operators satisfies the completeness relation, where is the projector over , i.e., quantum measurement operator is a valid measurement operator. The measurement operator projects the input quantum system to the pure state from the orthonormal basis . Now, the probability that the quantum state is after the measurement in basis state can be expressed as
In the computational basis , the state of the quantum system after the measurement can be expressed as
and the matrix of the quantum state will be diagonal in the computational basis , and can be given by
To illustrate it, let assume we have an initial (not diagonal) density matrix in the computational basis e.g. with and as
and we have orthonormal basis . In this case, the after-measurement state can be expressed as
As it can be seen, the matrix of is a diagonal matrix in the computational basis . Eq. (44) and (45) highlights the difference between quantum superpositions (probability amplitude weighted sum) and classical probabilistic mixtures of quantum states.
Now, let us see the result of the measurement on the input quantum system
For the measurement operators and the completeness relation holds
Using input system , where , the state after the measurement operation is
As we have found, after the measurement operation , the off-diagonal entries will have zero values, and they have no relevance. As follows, the initial input system after operation M becomes
Orthonormal Basis Decomposition
Let assume we have orthonormal basis , which basis can be used to rewrite the quantum system in a unique decomposition
with complex . Since , we can express it in the form
where is the complex conjugate of probability amplitude , thus is the probability of measuring the quantum system in the given basis state , i.e.,
This density matrix is a diagonal matrix with the probabilities in the diagonal entries
The Projective and POVM Measurement
The projective measurement is also known as the von Neumann measurement is formally can be described by the Hermitian operator , which has the spectral decomposition
is a projector to the eigenspace ofwith eigenvalue . For the projectors
and they are pairwise orthogonal. The measurement outcome m corresponds to the eigenvalue , with measurement probability
When a quantum system is measured in an orthonormal basis , then we make a projective measurement with projector , thus (56) can be rewritten as
The POVM (Positive Operator Valued Measurement) is intended to select among the non-orthogonal states and defined by a set of POVM operators , where
and since we are not interested in the post-measurement state the exact knowledge about measurement operator is not required. For POVM operators the completeness relation holds,
For the POVM the probability of a given outcome n for the state can be expressed as
The POVM also can be imagined as a ‘black-box’, which outputs a number from 1 to m for the given input quantum state , using the set of operators
where are responsible to distinguish m different typically non-orthogonal states i.e., if we observe on the display of the measurement device we can be sure, that the result is correct. However, because unknown non-orthogonal states can not be distinguished with probability 1, we have to introduce an extra measurement operator, , as the price of the distinguishability of the m different states and if we obtain m+1 as measurement results we can say nothing about . This operator can be expressed as
Such can be always constructed if the states in are linearly independent. We note, we will omit listing operator in further parts of the paper. The POVM measurement apparatus will be a key ingredient to distinguish quantum codewords with zero-error, and to reach the zero-error capacity of quantum channels.
The POVM can be viewed as the most general formula from among of any possible measurements in quantum mechanics. Therefore the effect of a projective measurement can be described by POVM operators, too. Or with other words, the projective measurements are the special case POVM measurement . The elements of the POVM are not necessarily orthogonal, and the number of the elements can be larger than the dimension of the Hilbert space which they are originally used in.
Ii-G Geometrical Interpretation of the Density Matrices
While the wavefunction
representation is the full physical description of a quantum system in the space-time, the tensor product of multiple copies of two dimensional Hilbert spaces is its discrete version, with discrete finite-dimensional Hilbert spaces. The geometrical representation also can be extended to analyze the geometrical structure of the transmission of information though a quantum channel, and it also provides a very useful tool to analyze the capacities of different quantum channel models.
As it has been mentioned, the Bloch sphere is a geometrical conception, constructed to represent two-level quantum systems in a more expressive way than is possible with algebraic tools. The Bloch sphere has unit radius and is defined in a three-dimensional real vector space. The pure states are on the surface of the Bloch sphere, while the mixed states are in the interior of the original sphere. In the Bloch sphere representation, the state of a single qubit can be expressed as
where is the global phase factor, which can be ignored from the computations, hence the state in the terms of the angle and can be expressed as
The Bloch sphere is a very useful tool, since it makes possible to describe various, physically realized one-qubit quantum systems, such as the photon polarization, spins or the energy levels of an atom. Moreover, if we would like to compute the various channel capacities of the quantum channel, the geometrical expression of the channel capacity also can be represented by the Bloch sphere. Before we would introduce the geometrical calculation of the channel capacities, we have to start from the geometrical interpretation of density matrices. The density matrix can then be expressed using the Pauli matrices (a set of three complex matrices which are Hermitian and unitary) , and as
where is the Bloch vector, , and . In the vector representation, the previously shown formula can be expressed as
In conclusion, every state can be expressed as linear combinations of the Pauli matrices and according to these Pauli matrices every state can be interpreted as a point in the three-dimensional real vector space. If we apply a unitary transformation to the density matrix , then it can be expressed as
and realizes a unitary transformation on as a rotation.
A density matrix can be expressed in a ‘weighted form’ of density matrices and as follows:
where , and and are pure states, and lie on a line segment connecting the density matrices in the Bloch sphere representation. Using probabilistic mixtures of the pure density matrices, any quantum state which lies between the two states can be expressed as a convex combination
This remains true for an arbitrary number of quantum states, hence this result can be expressed for arbitrary number of density matrices. Mixed quantum states can be represented as statistical mixtures of pure quantum states. The statistical representation of a pure state is unique. On the other hand we note that the decomposition of a mixed quantum state is not unique. In the geometrical interpretation a pure state is on the surface of the Bloch sphere, while the mixed state is inside. A maximally mixed quantum state, , can be found in the center of the Bloch sphere. The mixed state can be expressed as probabilistic mixture of pure states and . As it has been stated by von Neumann, the decomposition of a mixed state is not unique, since it can be expressed as a mixture of or equivalently of .
One can use a pure state to recover mixed state from it, after the effects of environment () are traced out. With the help of the partial trace operator, Bob, the receiver, can decouple the environment from his mixed state, and the original state can be recovered by discarding the effects of the environment. If Bob’s state is a probabilistic mixture , then a global pure purification state exists, which from Bob’s state can be expressed as
Note, density matrix can be recovered from after discarding the environment. The decoupling of the environment can be achieved with the operator. For any unitary transformation of the environment, the pure state is a unique state.
We have seen, that the decomposition of mixed quantum states into pure quantum states is not unique, hence for example, it can be easily verified by the reader, that the decomposition of a mixed state can be made with pure states , and also can be given with pure states . Here, we have just changed the basis from rectilinear to diagonal, and we have used just pure states - and it resulted in the same mixed quantum state.
Ii-H Channel System Description
If we are interested in the origin of noise (randomness) in the quantum channel the model should be refined in the following way: Alice’s register X, the purification state P, channel input A, channel output B, and the environment state E. The input system A is described by a quantum system , which occurs on the input with probability . They together form an ensemble denoted by , where x is a classical variable from the register X. In the preparation process, Alice generates pure states
according to random variablex, i.e., the input density operator can be expressed as , where the classical states form an orthonormal basis. According to the elements of Alice’s register X, the input system can be characterized by the quantum system
The system description is illustrated in Fig. 5.
The system state with the corresponding probability distribution can be indentified by a set of measurement operators . If the density operators in are mixed, the probability distribution and the classical variable x from the register X cannot be indentified by the measurement operators , since the system state is assumed to be a mixed or in a non-orthonormal state. Alice’s register X and the quantum system A can be viewed as a tensor product system as
where the classical variable x is correlated with the quantum system , using orthonormal basis . Alice’s register X represents a classical variable, the channel input system is generated corresponding to the register X in the form of a quantum state, and it is described by the density operator . The input system A with respect to the register X, is described by the density operator
where is the density matrix representation of Alice’s input state .
The purification gives us a new viewpoint on the noise of the quantum channel. Assuming Alice’s side A and Alice’s register X, the spectral decomposition of the density operator can be expressed as
where is the probability of variable x in Alice’s register X. The together is called an ensemble, where is a quantum state according to classical variable x. Using the set of orthonormal basis vectors of the purification system P, the purification of (76) can be given in the following way:
From the purified system state , the original system state can be expressed with the partial trace operator (see Appendix) , which operator traces out the purification state from the system
where is the joint system of purification state P and channel input A , which represents a noisy state.
Ii-H2 Isometric Extension
Isometric extension has utmost importance, because it helps us to understand what happens between the quantum channel and its environment whenever a quantum state is transmitted from Alice to Bob. Since the channel and the environment together form a closed physical system the isometric extension of the quantum channel is the unitary representation of the channel
enabling the ‘one-sender and two-receiver’ view: beside Alice the sender, both Bob and the environment of the channel are playing the receivers. In other words, the output of the noisy quantum channel can be described only after the environment of the channel is traced out
Ii-H3 Kraus Representation
The map of the quantum channel can also be expressed by means of a special tool called the Kraus Representation. For a given input system and quantum channel , this representation can be expressed as
where are the Kraus operators, and . The isometric extension of by means of the Kraus Representation can be expressed as
The action of the quantum channel on an operator , where form an orthonormal basis also can be given in operator form using the Kraus operator . By exploiting the property , for the input quantum system
If we trace out the environment, we get the equivalence of the two representations
Ii-H4 The von Neumann Entropy
Quantum information processing exploits the quantum 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 communication systems. On the other hand, it requires the generalization of classical information theory through a quantum perception of the world. As Shannon entropy plays fundamental role in classical information theory, the von Neumann entropy does the same for quantum information. The von Neumann entropy of quantum state can be viewed as an extension of classical entropy for quantum systems. It measures the information of the quantum states in the form of the uncertainty of a quantum state. The classical Shannon entropy of a variable X with probability distribution can be defined as
with , where is the cardinality of the set X.
The von Neumann entropy
measures the information contained in the quantum system . Furthermore can be expressed by means of the Shannon entropy for the eigenvalue distribution
where d is the level of the quantum system and are the eigenvalues of density matrix .
Ii-H5 The Holevo Quantity
The Holevo bound determines the amount of information that can be extracted from a single qubit state. If Alice sends a quantum state with probability over an ideal quantum channel, then at Bob’s receiver a mixed state
appears. Bob constructs a measurement to extract the information encoded in the quantum states. If he applies the measurement to , the probability distribution of Bob’s classical symbol B will be . As had been shown by Holevo , the bound for the maximal classical mutual information between Alice and Bob is
where is called the Holevo quantity, and (90) known as the Holevo bound.
In classical information theory and classical communication systems, the mutual information is bounded only by the classical entropy of , hence . The mutual information is bounded by the classical entropy of , hence . On the other hand, for mixed states and pure non-orthogonal states the Holevo quantity can be greater than the mutual information , however, it is still bounded by , which is the bound for the pure orthogonal states
The Holevo bound highlights the important fact that one qubit can contain at most one classical bit i.e., cbit of information.
Ii-H6 Quantum Conditional Entropy
While the classical conditional entropy function is always takes a non negative value, the quantum conditional entropy can be negative. The quantum conditional entropy between quantum systems A and B is given by
If we have two uncorrelated subsystems and , then the information of the quantum system does not contain any information about , or reversely, thus
hence we get , and similarly . The negative property of conditional entropy can be demonstrated with an entangled state, since in this case, the joint quantum entropy of the joint state less than the sum of the von Neumann entropies of its individual components. For a pure entangled state, , while since the two qubits are in maximally mixed state, which is classically totally unimaginable. Thus, in this case
and and .
Ii-H7 Quantum Mutual Information
The classical mutual information measures the information correlation between random variables A and B. In analogue to classical information theory, can be described by the quantum entropies of individual states and the von Neumann entropy of the joint state as follows:
i.e., the quantum mutual information is always a non negative function. However, there is a distinction between classical and quantum systems, since the quantum mutual information can take its value above the maximum of the classical mutual information. This statement can be confirmed, if we take into account that for an pure entangled quantum system, the quantum mutual information is
and we can rewrite this equation as
For some pure joint system , the equation (97) can be satisfied such that and .
If we use entangled states, the quantum mutual information could be 2, while the quantum conditional entropies could be 2. In classical information theory, negative entropies can be obtained only in the case of mutual information of three or more systems. An important property of maximized quantum mutual information: it is always additive for a quantum channel.
The character of classical information and quantum information is significantly different. There are many phenomena in quantum systems which cannot be described classically, such as entanglement, which makes it possible to store quantum information in the correlation of quantum states. Similarly, a quantum channel can be used with pure orthogonal states to realize classical information transmission, or it can be used to transmit non-orthogonal states or even quantum entanglement. Information transmission also can be approached using the question, whether the input consists of unentangled or entangled quantum states. This leads us to say that for quantum channels many new capacity definitions exist in comparison to a classical communication channel. In possession of the general communication model and the quantities which are able to represent information content of quantum states we can begin to investigate the possibilities and limitations of information transmission through quantum channels .
Ii-H8 Quantum Relative Entropy
The quantum relative entropy measures the informational distance between quantum states, and introduces a deeper characterization of the quantum states than the von Neumann entropy. Similarly to the classical relative entropy, this quantity measures the distinguishability of the quantum states, in practice it can be realized by POVM measurements. The relative entropy classically is a measure that quantifies how close a probability distribution p is to a model or candidate probability distribution q. For probability distributions p and q, the classical relative entropy is given by
while the quantum relative entropy between quantum states and is
In the definition above, the term is finite only if for all diagonal matrix elements. If this condition is not satisfied, then could be infinite, since the trace of the second term could go to infinity.
The quantum informational distance (i.e., quantum relative entropy) has some distance-like properties (for example, the quantum relative entropy function between a maximally mixed state and an arbitrary quantum state is symmetric, hence in this case it is not just a pseudo distance), however it is not commutative, thus and iff and iff Note, if has zero eigenvalues, may diverge, otherwise it is a finite and continuous function. Furthermore, the quantum relative entropy function has another interesting property, since if we have two density matrices and , then the following property holds for the traces used in the expression of
The symmetric Kullback-Leibler distance is widely used in classical systems, for example in computer vision and sound processing. Quantum relative entropy reduces to the classical Kullback-Leibler relative entropy for simultaneously diagonalizable matrices.
We note, the quantum mutual information can be defined by quantum relative entropy . This quantity can be regarded as the informational distance between the tensor product of the individual subsystems , and the joint state as follows:
Ii-H9 Quantum Rényi-Entropy
As we have seen, the quantum informational entropy can be defined by the von Neumann entropy function. On the other hand, another entropy function can also be defined in the quantum domain, it is called the Rényi-entropy and denoted by . This function has relevance mainly in the description of quantum entanglement. The Rényi-entropy function is defined as follows