THE INFORMATION-CARRYING CAPACITY OF CERTAIN QUANTUM CHANNELS
The thesis is submitted to
University College Dublin
for the degree of PhD
in the College of
Engineering, Mathematical and Physical Sciences
Based on research conducted in the
Dublin Institute for Advanced Studies
and the School of Mathematical Sciences,
University College Dublin
(Head of School: Dr. Micheál Ó Searcóid )
under the supervision of
Prof. Tony Dorlas & Prof. Joe Pulé
- 1 Introduction
- 2.1 Quantum states and quantum channels
- 2.2 Quantum channel
- 2.3 Positive operator-valued measure
- 2.4 Quantum entanglement
- 2.5 Classical information over a quantum channel
- 2.6 The Holevo-Schumacher-Westmoreland theorem
- 2.7 Models for quantum memory channels
- 2.8 Markov processes and channel memory
3 Deriving a minimal ensemble for the quantum amplitude damping channel
- 3.1 The amplitude-damping channel
- 3.2 The generalised amplitude-damping channel
- 3.3 The depolarising channel and the Holevo quantity
- 3.4 Summary
4 The classical capacity of two quantum channels with memory
- 4.1 Introduction
- 4.2 Classical capacity
- 4.3 The periodic channel
- 4.4 The classical capacity of a convex combination of memoryless channels
- 4.5 Convex combinations of two memoryless channels
- 4.6 The periodic channel with amplitude-damping channel branches
- 4.7 Summary
- 5 Strong converse to the channel coding theorem for a periodic quantum channel
- A Carathéodory’s Theorem & an application to minimal optimal-ensembles
- B Product-state capacity of a periodic quantum channel
It is a very sad thing that nowadays there is so little useless information.
Ba mhaith liom an tráchtas seo a tiomnú do mo mháthair, Helen
(I dedicate this thesis to my mother, Helen)
I thank my supervisor Tony Dorlas for giving me the opportunity to carrying out research in quantum information theory. I am grateful for his support and encouragement. I thank Matthias Christandl for reading and examining the thesis and for his comments and suggestions. I would also like to acknowledge and thank Andreas Winter for numerous interesting discussions over the last number of years, from which I learned many interesting and useful things, and also for the invitation to visit the University of Bristol and the National University of Singapore. I thank Nilanjana Datta for her helpful comments. I would also like to thank Christopher King for his encouragement.
I am very fortunate to have such a close and supportive family. I have dedicated this thesis to my Mom, who instilled in me a love of learning and who taught me the value of perseverance. My Dad, Brian, and Uncle Kieron, though no longer with us, are a constant source of inspiration to me. I thank my brother Brian and his wife Niamh for their steadfast support and for their help with the execution of the thesis. I really could not have wished for a more caring and protective brother. I also thank the O’ Reilly and Whale families for their encouragement.
My time as a PhD student has been very enjoyable. I am fortunate to have made wonderful friends both in DIAS and in UCD and I would like to thank them for their camaraderie. I would especially like to thank Sinéad Keegan for her advice and support. I also thank my oldest and closest friend Jonathan Tynan.
Above all, I could not have completed this thesis without the constant friendship, encouragement, love and support of my partner Mark Whale. Is tusa mo chroí.
In this thesis we consider the classical capacity of certain quantum channels, that is, the maximum rate at which classical information, encoded as quantum states, can be transmitted reliably over a quantum channel.
We first concentrate on the product-state capacity of a particular quantum channel, that is, the capacity which is achieved by encoding the output states from a source into codewords comprising of states taken from ensembles of non-entangled (i.e. separable) states and sending them over copies of the quantum channel. Using the “single-letter” formula proved by Holevo  and Schumacher and Westmoreland  we obtain the product-state capacity of the qubit quantum amplitude-damping channel, which is determined by a transcendental equation in a single real variable and can be solved numerically. We demonstrate that the product-state capacity of this channel can be achieved using a minimal ensemble of non-orthogonal pure states. We also consider the generalised amplitude-damping channel and show that the technique used to calculate the product-state capacity for the “traditional” amplitude damping channel also holds for this channel.
In the following chapter we consider the classical capacity of two quantum channels with memory namely, a periodic channel with quantum depolarising channel branches and a convex combination of quantum channels. The classical capacity is defined as the limit of the capacity of a channel, using a block of states which are permitted to be entangled over channel uses and divided by , as tends to infinity.
We prove that the classical capacity for each of the classical memory channels mentioned above is, in fact, equal to the respective product-state capacities. For those channels this means that the classical capacity is achieved without the use of entangled input-states. We also demonstrate that the method used in the proof of the classical capacity of a periodic channel with depolarising channels does not hold for a periodic channel with amplitude-damping channel branches. This is due to the fact that, unlike the depolarising channel, the maximising ensemble for a qubit amplitude-damping channel is not the same for all amplitude-damping channels.
We also investigate the product-state capacity of a convex combination of two memoryless channels, which was shown in  to be given by the supremum of the minimum of the corresponding Holevo quantities, and we show in particular that the product-state capacity of a convex combination of a depolarising and an amplitude-damping channel, is not equal to the minimum of their product-state capacities.
Next we introduce the channel coding theorem for memoryless quantum channels, providing a known proof  for the strong converse of the theorem. We then consider the strong converse to the channel coding theorem for a periodic quantum channel.
|Conjugate transpose of|
|Trace of operator|
|Trace-norm of operator , given by|
|Shannon entropy of probability distribution|
|Quantum state (density operator)|
|von Neumann entropy of state|
|Set of quantum states on Hilbert space|
|Holevo quantity with respect to the ensemble|
|Product-state capacity of , given by|
Chapter 1 Introduction
Informally, the capacity of a channel can be considered to be a measure of the channels usefulness for sending information faithfully from source (or sender) to receiver. The capacity of a quantum channel, can be thought of as a measure of the closeness of that channel to the quantum identity channel, which itself sends quantum information with perfect fidelity. Throughout the thesis we concentrate on the case where classical messages (or output from a classical information source) are encoded into quantum states and sent over quantum channels.
Classical information can be encoded into different types of quantum states, i.e. orthogonal or non-orthogonal, separable or entangled. Note that the latter is a purely quantum mechanical phenomenon. We are interested in the type of states and ensembles which achieve the capacity of certain quantum channels and pay particular attention to the capacity of noisy quantum channels with memory, i.e. channels which have correlations between successive uses.
We first introduce the concept of classical information entropy and classical channel capacity (see  and , for example). We do so because a great deal of what has been achieved in the field of quantum information theory to has date been inspired by results in classical information theory, most notably Claude Shannon’s seminal article  on classical channel capacity, published in 1948. The brief review of Shannon’s work on channel capacities is also justified in order to demonstrate that not all of his results on classical channels have been successfully generalised to the quantum setting. Unlike classical channels, there are a number of different types of quantum channel capacities, namely, the classical capacity, quantum capacity and the private capacity. These capacities have not yet been fully resolved. Moreover, entangled input states, mentioned above, have recently been shown to improve the classical capacity of quantum channels. Hastings , building on a result by Hayden and Winter , recently presented a violation of one of the longest standing conjectures in quantum information theory, namely the additivity conjecture involving the Holevo quantity [1, 2]. This counterexample implies that the conjectured formula for the classical capacity of a quantum channel is disproved and that a simple “single-letter” formula for the capacity remains to be discovered. The classical capacity of a quantum channel can therefore only be determined asymptotically. The question of whether this is an intrinsic property of the classical capacity of a quantum channel or whether there is some missing element which has not yet been understood remains open.
Smith and Yard  also recently proved the non-additivity of the quantum channel capacity, disproving the operational interpretation of the additivity the of quantum capacity of quantum channels, by showing that two channels, each with zero quantum capacity, when used together can give rise to a non-zero capacity. This is known as “superactivation” of channel capacity. Cubitt, Chen and Harrow  have also demonstrated a similar result for the zero-error classical capacity of a quantum channel. Smith and Smolin  and Li, Winter, Zou and Guo  have proved the non-additivity of the private capacity for a family of quantum channels.
1.1 Classical information theory
In information theory, entropy measures the amount of uncertainty in the state of a system before measurement. Shannon entropy measures the entropy of a variable associated with a classical probability distribution. More formally, the Shannon entropy of a random variable with probability distribution is given by . As entropy is measured in bits, is taken to the base , and .
Mutual information, , measures the amount of information two random variables and have in common,
where, is the joint entropy .
Let and be the respective input and output alphabets for a classical channel and let and both be sequences of random variables such that and . A channel can be described in terms of the conditional probabilities i.e. the probabilities of obtaining different outcomes, , given the input variable .
1.1.1 Shannon’s noisy channel coding theorem
The capacity of a classical channel provides a limit on the number of classical bits which can be transmitted reliably per channel use. The direct part of the classical channel coding theorem  states that using copies of the channel, bits of information can be sent reliably over the channel at a rate if and only if in the asymptotic limit.
The strong converse of the channel coding theorem states that if the rate at which classical information is transmitted over a classical channel exceeds the capacity of the channel, i.e. if , then the probability of decoding the information correctly goes to zero in the number of channel uses.
The capacity of a noisy classical channel, , is given by the maximum of the mutual information obtained over all possible input distributions, , for ,
where is given by Equation 1.1.1.
The first proof of Shannon’s coding theorem is due to Feinstein .
1.2 Quantum information theory
Quantum communication promises to allow unconditionally secure communication . Techniques to protect quantum information from noise are therefore of great importance. A simple “single-letter” formula which could be used to calculate the classical capacity of a quantum channel, would lead to a better understanding of optimal encodings used to protect quantum information from errors. Whether such a formula can be found remains an open question.
The quantum analogue of Shannon entropy is von Neumann entropy. It is defined as follows. The entropy of a quantum state, , is given by the von Neumann entropy, . If has eigenvalues , then .
1.2.1 Noisy quantum channel coding
The sender encodes their message into a block of quantum states. This codeword can then be transmitted over copies of a quantum channel, , see Figure 1.2.
The above encodings and will be important in a later chapter when we investigate the channel coding theorem for quantum channels.
1.3 Thesis layout
In Chapter 2 we introduce some mathematical preliminaries and discuss concepts fundamental to the understanding of quantum information theory.
We obtain a maximiser for the quantum mutual information for classical information sent over the qubit amplitude-damping and depolarising channels in Chapter 3. This is achieved by limiting the ensemble of input states to antipodal states, in the calculation of the product state capacity for the channels. In Section 3.1 we evaluate the capacity of the amplitude-damping channel and plot a graph of this capacity versus the damping parameter. We discuss the “generalised” amplitude damping channel in Section 3.2, and show that the approach taken to calculate the product state capacity of the conventional amplitude damping channel can also be taken for this channel. We introduce the depolarising channel in Section 3.3 and discuss the maximising ensemble of the corresponding Holevo quantity. The contents of this chapter have been published by T.C. Dorlas and the Author in .
Chapter 4 is based on an article published by Dorlas together with the Author . Here we investigate the classical capacity of two quantum channels with memory, that is, a periodic channel with depolarising channel branches, and a convex combination of depolarising channels. We prove that the capacity is additive in both cases. As a result, the channel capacity is achieved without the use of entangled input states. In the case of a convex combination of depolarising channels the proof provided can be extended to other quantum channels whose classical capacity has been proved to be additive in the memoryless case.
In Section 4.3 we introduce the periodic channel and investigate the product-state capacity of the channel with depolarising channel branches. We derive a result based on the invariance of the maximising ensemble of the depolarising channel, which enables us to prove that the capacity of such a periodic channel is additive. In Section 4.4 the additivity of the classical capacity of a convex combination of depolarising channels is proved. This is done independently of the result derived in Section 4.3 and can therefore be generalised to a class of other quantum channels.
In Section 4.5 we state the theorem proved by Datta and Dorlas in  concerning the product-state capacity of a convex combination of memoryless channels and we show that in the case of two (or more) depolarising channels or two (or more) amplitude-damping channels, this is in fact equal to the minimum of the individual capacities. We show however in the case of a depolarising channel and an amplitude-damping channel, that this is not the case.
The channel coding theorem and strong converse is discussed in Chapter 5 and we provide the proof by Winter . We then consider the strong converse for a periodic quantum channel, in light of a result shown in Section 4.6 for the periodic channel with amplitude damping channel branches.
Appendix A.1 states Carathéodory’s Theorem which is used in Chapter 3. A proof provided by N. Datta, which states that it is sufficient to consider ensembles consisting of at most pure states in the maximisation of the Holevo quantity for a CPT map, is given in Appendix A.2.
The proof of the product-state capacity of a periodic quantum channel, provided by Datta and Dorlas, is given in Appendix B.1. The periodic channel is a special case of a channel with arbitrary Markovian noise correlations. The proof of the formula for the product-state capacity of such a channel, i.e. one with noise given by arbitrary Markovian correlations, is given by Datta and Dorlas in .
Chapter 2 Preliminaries
We begin by establishing some concepts fundamental to the study of quantum information theory. We build on the framework of quantum information transmission introduced in the previous chapter by making these ideas mathematically concrete. We introduce quantum states and channels and describe the operator sum representation, a tool widely used in quantum information theory to describe the behavior of an input state with a given quantum channel. The definition of quantum entanglement is provided and we discuss the mutual information between different quantum states and the Holevo-Schumacher-Westmoreland theorem, which is used to determine the product-state capacity for classical information sent over quantum channels. The theory of Markov processes is introduced, providing definitions necessary for Chapter 4, where we introduce two channels which have memory that can be described using Markov chains.
2.1 Quantum states and quantum channels
We now provide definitions for quantum states and quantum channels.
2.1.1 Quantum state
A quantum state is given by a positive semi-definite Hermitian operator of unit trace on a Hilbert space. We now define the terms used in this definition.
A Hilbert space is a complex vector space equipped with an inner product. Note that we will only consider finite dimensional Hilbert spaces. An element of a Hilbert space, known as a vector, is denoted . An element of the dual space , the conjugate transpose of , is denoted , where
and is the complex conjugate of . In fact, due to the correspondence between and its dual space , given by the inner product, we can consider each element as an element of .
The norm of the vector is defined as follows,
Note that positive operators are a subclass of Hermitian operators, both are defined below. An operator is Hermitian if where is the adjoint of the operator , defined by , for all vectors and in the state space of .
An operator is called positive (semi-definite), if for all vectors , the following holds .
Since a density operator is defined to be a positive (Hermitian) operator with trace one on a Hilbert space a quantum state can be represented by a density operator. We now define a quantum state to be a positive operator of unit trace , where denotes the algebra of linear operators acting on a finite dimensional Hilbert space .
184.108.40.206 Pure and mixed states
According to the first postulate of quantum mechanics, a quantum system is completely described by its state vector, . A state vector is a unit vector in the state space of the system. A system whose state is completely known is said to be in a pure state. The density operator for that system is given by the projection . If, however a system is in one of a number of states, then the system is said to be in a mixed state. If a system is in one of the states with respective probabilities , then is called an ensemble of pure states. The corresponding density operator is given by .
220.127.116.11 Composite quantum systems
A composite of two quantum systems and can be described by the tensor product of the two Hilbert spaces . Note that . The state of one of the Hilbert spaces can be extracted from the product state of the two Hilbert spaces by performing the partial trace on the composite system.
The tensor product and partial trace are both defined in the following section.
18.104.22.168 The tensor product and partial trace
The tensor product of two vectors, and , is defined as
where and represent Hilbert spaces with respective bases and .
The following demonstrates how the tensor product of two vectors and two matrices is computed, respectively
Properties of the tensor product include,
The trace of an operator , with the orthonormal basis , is given by
We now introduce partial trace. Let and represent two Hilbert spaces with orthonormal bases and , respectively. Let be a state defined on the composite system such that . The state of the subsystem is given by the reduced density operator and is defined by
where and are states on . Inserting the basis elements for and , the partial trace can be calculated as follows
where is the partial trace operation from onto . The state of the subsystem is similarly defined.
2.2 Quantum channel
A map is said to be completely positive if
where , is an operator defined on the Hilbert space , where is an arbitrary space, is the Hilbert space of the output state and is the identity operator.
A quantum channel is defined as a completely positive, trace preserving map, which maps density operators from one Hilbert space to another.
In general, when a pure input state is transmitted through a noisy quantum channel, the output state is not known with absolute certainty i.e. it is no longer a pure state. The corresponding state is said to be mixed.
The initial state to a channel is given by the tensor product of the information state, , defined on the Hilbert space and the initial state of the environment , assumed to be in a pure state and defined on the Hilbert space .
It may be assumed that the initial state of a system is in a pure state as the state of the system can always be defined in terms of a larger composite system which can be chosen to be in a pure state. This is known as state purification.
During transmission over a channel, the composite state, will evolve unitarily such that , for the unitary operator on .
After the interaction of the channel with the state , the output state is given by
This corresponds to a measurement operator on the information state alone after it has evolved in interaction with the environment.
Note that, a unitary operator is defined as . A unital channel is a channel where the following identity holds,
Next we introduce operator sum representation, a way of describing the action of a quantum channel on an input state.
2.2.1 Operator sum representation
Quantum channels can be represented using operator-sum, or Kraus representation. By tracing over the state space of the environment, the dynamics of the principal system alone are extracted and represented explicitly. We will now show that this representation is a re-statement of equation (2.2.2).
where , an orthonormal basis for and . The operators are known as operation elements.
The operator represents the output state, and therefore must satisfy a completeness relation such that . Using the operation elements defined above,
Using the cyclic property of trace,
since this must hold for all , it follows that .
A map is completely positive and trace preserving if it admits a Kraus Representation 
A memoryless channel is given by a completely positive map , where and denote the states on the input and output Hilbert spaces and .
2.3 Positive operator-valued measure
If measurement, described by the set , is performed on a system in a state , then the probability of obtaining outcome label is given by .
2.4 Quantum entanglement
A state is said to be separable if it can be written as a probabilistic mixture of product states
where , , , and . Otherwise the state is said to be an entangled state.
Entanglement is an important resource in quantum information processing and plays an essential role in quantum teleportation, quantum cryptography, quantum computation, quantum error correction .
2.5 Classical information over a quantum channel
The transmission of classical information over a quantum channel is achieved by encoding the information into quantum states. To accomplish this, a set of possible input states with probabilities are prepared, describing the ensemble . The average input state to the channel is expressed as The average output state is .
2.5.1 Holevo bound
When a state is sent through a noisy quantum channel, the amount of information about the input state that can be inferred from the output state is called the accessible information. For any ensemble , the Holevo quantity is defined as
The Holevo bound  provides an upper bound on the accessible information and is given by,
where is the Holevo quantity of channel . Here is the random variable representing the classical input to the channel. The possible values are mapped to states which are transformed to by the channel. Then, a generalised measurement with corresponding POVM allows the determination of the output random variable with conditional probability distribution given by
The second term in the Holevo bound is often referred to as the output entropy. This term represents the joint entropy of the system after evolution and can be interpreted as the final entropy of the environment, assuming that the environment was initially in a pure state. This is the amount of information that the information state, or principal system, has exchanged with the environment. We therefore want to minimize the output entropy and maximise the entropy of the expected state. This justifies the definition of the capacity of the channel as the maximum of the mutual information. When quantum information is sent down a noisy quantum channel, the output entropy is known as entropy exchange .
Holevo  has introduced a measure of the amount of classical information remaining in a state that has been sent over a noisy quantum channel. The product-state capacity of a channel is given by the maximisation of this Holevo quantity over an ensemble of input states, and can be interpreted as the amount of information that can be sent, in the form of product-states, reliably over the channel.
In this case the fact that the capacity is given by the maximum of the Holevo quantity is known as the Holevo-Schumacher-Westmoreland (HSW) Theorem.
2.6 The Holevo-Schumacher-Westmoreland theorem
If the possible input states to a channel are prepared as product states of the form , then the associated capacity is known as the product state capacity. This implies that the input states have not been entangled over multiple uses of the channel. The capacity for channels with entangled input states has been studied , and it has been shown that for certain channels the use of entangled states can enhance the inference of the output state and increase the capacity (e.g. ).
The HSW theorem, proved independently by Holevo  and by Schumacher and Westmoreland , provides an expression to calculate the product state capacity for classical information sent through a quantum channel, , and can be calculated using the following expression,
where is the von Neumann entropy, . If has eigenvalues , then . The capacity is given by the maximum mutual information calculated over all ensembles . Properties characterising optimal input ensembles for have been studied .
Prior to the HSW theorem, Holevo  developed a formula for calculating the product-state capacity of a quantum channel where a maximisation of the accessible information is taken explicitly over both the input ensemble and over product measurements performed on the output of the channel. It has been shown that, in certain cases, more information can be transmitted per use of a quantum channel using collective measurements rather than separable ones (see [1, 32, 33]).
2.6.1 Optimal input enembles
By concavity of the entropy, the maximum in Equation 2.6.1 is always attained for an ensemble of pure states . Indeed, we can decompose each as convex combinations of pure states: . This does not change the first term of (2.6.1), but by concavity of the entropy,
A statement of Carathéodory’s Theorem is provided in Appendix A.1 along with a proof by N. Datta in Appendix A.2 which states that it is sufficient to consider ensembles consisting of pure states in the maximisation of the Holevo quantity , for some CPT map .
Next we introduce two models for quantum memory.
2.7 Models for quantum memory channels
Bowen and Mancini [37, 38] introduced two models for quantum channels with memory. The model shown in Figure 2.1 depicts an interaction between each memory state and its environment . The environments are correlated, which leads to a memory effect at each stage of the evolution.
In contrast to the previous model, Figure 2.2 depicts the input state interacting with its own environment and with the memory state. The error operators at each stage of the evolution are correlated, and may be determined using the relevant unitary operator and the input state. Both process will be described in the following subsections.
2.7.1 Several uses of a memoryless quantum channel
Recall that it is known that any quantum channel, described by a completely positive trace preserving (CPT) map, can be represented by a unitary operation on the input state to the channel and the initial (known) state of the environment . The output state following a sequence of uses of the memoryless channel, is given by
where, is a (possibly entangled) input state codeword and
represents the (product) state of the environment. Note that the trace is taken over each state comprising the state of the environment.
2.7.2 Several uses of a quantum channel with memory
The action of a quantum memory channel on a sequence of input states can be viewed in the following two ways.
22.214.171.124 Model 1
126.96.36.199 Model 2
Each input state , to the channel will act with a unitary interaction on the channel memory state, denoted , and also an independent environment . This process is depicted in Figure 2.2.
The output state from such a quantum memory channel can be expressed as follows
Note that if the unitaries acting on the state memory and environment can be written as then the memory can be traced out and we recover the memoryless channel.
Quantum channels which have Markovian noise correlations are a class of channel which can be represented by the above model. This class of channel is of particular interest to us and is discussed below.
2.8 Markov processes and channel memory
Next, we provide definitions  needed to describe quantum channels with classical memory.
2.8.1 Definitions and basic properties
Let denote a countable set and let , where is a random variable taking values in the state space . Let denote a transition matrix, with entries labeled .
A Markov chain is given by a sequence of random variables with the following property,
Equivalently, a discrete time random process denoted can be considered to be a Markov chain with transition matrix and initial distribution , if and only if the following holds for , (see Norris  Theorem 1.1.1)
A state is said to be accessible from state , and can be written , if there exists such that
The state is said to communicate with state if and . This relation, denoted , partitions the state space into communicating classes. A Markov chain is said to be irreducible if the state space is a single class.
A state has period if any return to the state occurs in multiples of time steps, i.e.
Next we use the concepts to describe classical memory.
2.8.2 Classical memory
A channel, of length , with Markovian noise correlations can be described as follows 
where denotes the elements of the transition matrix of a discrete-time Markov chain, and represents an invariant distribution on the Markov chain.
In later chapters we analyse two particular channels with classical memory, the periodic channel and a convex combination of memoryless channels. Both are described below.
A periodic channel can be described as follows
where are CPT maps and the index is cyclic modulo the period . In this case , where
A convex combination of product channels is defined by the following channel
where is a probability distribution over channels
. The action of the channel can be interpreted as follows. With probability a given input state is transmitted through one of the memoryless channels. The corresponding Markov chain is aperiodic but not irreducible.
Chapter 3 Deriving a minimal ensemble for the quantum amplitude damping
In this chapter we focus on obtaining the maximiser for classical information transmitted in the form of product-states over a noisy quantum channel. We consider in particular the problem of determining this maximiser in the case of the amplitude damping channel. The amplitude damping channel models the loss of energy in a system and is an example of a non-unital channel (see Section 2.1). The effect of the qubit amplitude damping channel on the Bloch (Poincaré) sphere is to “squash” the sphere to the pole, resulting in an ellipsoid. The Bloch sphere will be discussed in more detail in Section 3.1.1 and the resulting space of the output states from an example amplitude-damping channel, using the optimal input ensemble for that channel, can be seen in Figure 3.7 of this chapter.
It is known in general that the maximising ensemble can always be assumed to consist of at most pure states if is the dimension of the state space, but we show that in the case of the qubit amplitude-damping channel, the maximum is in fact obtained for an ensemble of two pure states. Moreover, these states are in general not orthogonal. This result is rather surprising, since nonorthogonal quantum states cannot be distinguished with perfect reliability.
Note that Fuchs  has also described a particular channel, the so-called “splaying” channel, whose product state capacity is maximised using an ensemble of non-orthogonal states.
3.1 The amplitude-damping channel
The qubit amplitude-damping channel models the loss of energy in a qubit quantum system and is described, with error parameter , by the following operation elements 
Using the operation elements above, the qubit amplitude-damping channel can be expressed as follows
Note that since , the operator is a CPT map and therefore a legitimate quantum channel.
Acting on the general qubit state , given by
the output of the channel is given by
The amplitude-damping channel can be interpreted as follows. Evaluating
we can easily see that if the input state is given by , then the state is left unchanged by , with probability . However, if the input state is , the amplitude of the state is multiplied by a factor .
On the other hand,
In this case, the input state is replaced with the state with probability .
The eigenvalues of are easily found to be
Next we derive the product-state capacity of the qubit amplitude damping channel.
3.1.1 Product-state capacity of the qubit amplitude damping
Recall (Section 2.5.1) that the Holevo quantity for a channel is defined as
In the case of the amplitude-damping channel, given by Equation (3.1.4), the Holevo quantity is given as follows,
To maximise Equation (3.1.11) we will show that the first term is
keeping the second term fixed, if each pure state is replaced by itself and its mirror image in the real -axis. In other words, replacing associated with probability , with the states and , both with probabilities , will increase Equation (3.1.11).
The Bloch sphere (also known as the Poincaré sphere) is a representation of the state space of a two-level quantum system i.e. a qubit. Pure states (corresponding to the extreme points in the (convex) set of density operators) are given by points on the surface of the sphere.
An example of antipodal states is shown in Figure 3.1 below, which depicts a two-dimensional cross-section of the Block sphere. Here, the state has been replaced by itself and , similarly for .
As remarked above, the maximum in Equation (2.6.1) can be achieved by a pure state ensemble of (at most) states, where is the dimension of the input to the channel. In general, the states must lie inside the Poincaré sphere
and so, the pure states will lie on the boundary
We first show that the second term in Equation (3.1.10) remains unchanged when the states are replaced in the way described above. Indeed, since the eigenvalues (3.1.9) depend only on , we have and therefore,
Secondly, we prove that is in fact increased by replacing each state with itself and its mirror image, each with half their original weight. Indeed, as is a concave function,
and again, since ,
We can conclude that the first term in Equation (3.1.10) is increased with the second term fixed if each state is replaced by itself together with its mirror image.
It follows,in particular, that we can assume from now on that all are real as the average state has zero off-diagonal elements, whereas the eigenvalues of only depend on .
188.8.131.52 Convexity of the output entropy
We concentrate here on proving that, in the case of the amplitude- damping channel, the second term in the equation for the Holevo quantity is convex as a function of the parameters , when is taken to be a pure state, i.e. . Thus is a function of one variable only, i.e. . It is given by,
Inserting into Equation (3.1.9) the eigenvalues for the amplitude-damping channel can be written as
We prove that is positive. A straightforward calculation yields
Since the first term in the above equation is positive, the problem of proving the convexity of reduces to proving that,
This is easily shown. Note that . Both functions are plotted in Figure 3.2 below. We conclude that is positive and therefore is convex.
Writing , with and since the entropy function is convex in we have
The capacity is therefore given by
where is given by
We have proved that is convex. Therefore is concave. On the other hand, it follows from the concavity of that the first term is also a concave function of .
It follows that