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BACHELOR’S THESIS
Quantum Channel Capacities
Author:
Alessandro Fasse
Supervisors:
Prof. Dr. Tobias J. Osborne
Dr. Ciara Morgan
Matriculation Number:
2782510
October 4, 2013
Institute for Theoretical Physics
Quantum Information Group
Abstract
Calculating the quantum capacity of the depolarizing channel is one of the longest
standing open problems in quantum information theory and therefore one of the
most interesting ones. The reason why it is so difficult to get the solution is that
in general the coherent information for a channel is non-additive, and this makes it
nearly impossible to find the exact value for the quantum capacity which is itself
defined as a regularization over the coherent information. But Devetak and Shor [4]
showed that there is a class of channels, known as degradable channels, which have
additive coherent information and it is therefore relatively easy to find the exact
solutions for corresponding capacities.
Smith et.al [16] use this class of channel to express the depolarizing channel as a
combination of degradable channels, in particular amplitude-damping channels, to
give tight bounds to the exact and at this point unknown, quantum capacity of the
channel.
In this thesis we will explore these bounds, the ideas they introduced and in the end
we will present some relations between the dimension of the environment and the
Kraus operators of a channel. The last pages include some exact calculation of the
outputs of Bob and Eve using the formulas previously introduced.
Contents
1 Introduction
2 The
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
basic concepts of quantum information theory
The qubit . . . . . . . . . . . . . . . . . . . . . . .
Hilbert space . . . . . . . . . . . . . . . . . . . . .
Density operators . . . . . . . . . . . . . . . . . . .
2.3.1 The qubit as a density operator . . . . . . .
Quantum channels . . . . . . . . . . . . . . . . . .
2.4.1 The reference system . . . . . . . . . . . . .
2.4.2 The Kraus operators . . . . . . . . . . . . .
Measures of distance . . . . . . . . . . . . . . . . .
2.5.1 Trace distance . . . . . . . . . . . . . . . . .
2.5.2 Fidelity . . . . . . . . . . . . . . . . . . . .
2.5.3 Relation between trace distance and fidelity
Purification . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Degradable channels . . . . . . . . . . . . .
Quantum information measures . . . . . . . . . . .
2.7.1 Conditional quantum entropy . . . . . . . .
2.7.2 Coherent information . . . . . . . . . . . . .
2.7.3 Quantum mutual information . . . . . . . .
Quantum capacity theorem . . . . . . . . . . . . .
Superactivation of quantum capacity . . . . . . . .
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3 The depolarizing channel
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3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 The classical capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Actual bounds of the quantum capacity of the depolarizing
4.1 Symmetric side channel assisted capacity . . . . . . . . .
4.2 Relations between the unassisted and the assisted cases .
4.3 An upper bound via amplitude-damping channels . . . .
4.4 A lower bound via hashing . . . . . . . . . . . . . . . . .
channel
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19
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5 Relation between the dimension of Eve and the Kraus operators
24
5.1 The amplitude-damping channel . . . . . . . . . . . . . . . . . . . . . 25
5.2 The depolarizing channel . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 Conclusion
27
References
28
3
1 Introduction
The foundation of quantum mechanics brought a lot of new observations, like entanglement and uncertainty which seem to be incompatible with our classical intuition,
but if we use quantum mechanics as a resource for our purposes we can explore some
useful and impressive phenomena.
To do so is the goal of quantum information theory, where physicists try to combine
quantum mechanics with information theory to develop a new picture of communication or to build the theory for some completely new computer-systems, so-called
quantum computer which could be faster than the classical pendant in some special
cases. The first steps in this direction are already established, namely the Shor
algorithm for prime factorization and the Grover algorithm as a searching routine.
Exactly like quantum mechanics promises us some new approaches it also gives us
a lot of problems, mostly because our intuition fails in several cases. For example
not every wire is ideal, i.e. if we are trying to send some, in this case, classical
information through the channel not every message will reach the end of that wire
without any issues or failures. In the real world we can not send every message
reliably. This is a problem in the classical case as well as in the quantum case
and in the quantum case the systems are the most time a lot more susceptible to
influences of the environment. Therefore we need some techniques to create a reliable transmission, i.e. to find some ways to communicate without errors or to
find some codes that can correct the intermittent failures. Shannon published in
the year 1948 this groundwork for the classical information theory [14]. His ideas
where developed over the years and attempts are being made to generalize this work
to the quantum case. In his paper he introduced the capacity of a channel which
gives us the maximum rate, where we can communicate through a classical channel
without any loss of information. This concept also exists in quantum theory but we
have different errors and different resources of information. The standard victims
for communication via a channel are our friends Alice and Bob (A and B), where
Alice wants to send information to Bob by encoding her information in some way,
for example as quantum states, to send it through a channel. Then Bob decodes
this message and hopefully he will get the initial message or is able to recover it.
The question now is, how we could do so? This was, more or less, the quantum
capacity theorem spoken in some simple words, i.e. the question how Alice and Bob
can manage the reliable transmission of information by using quantum information
instead of classical information.
Later we will see, that there is a formula for the quantum capacity, the so-called
LSD-formula1 , to calculate this quantity. But to handle the formula is not that nice
as it might seem at first glance. Only for some channels the exact quantum capacity
is known, in general we are not able to calculate it. But mathematics gives us some
1
The name LSD honors the innovators Lloyd [11], Shor [15] and Devetak [3] which have done
some excellent work in the topic of quantum information theory.
1 Introduction
4
tools to approximate the exact solutions2 . As you can see, quantum information
theory is a field of active research with many open fields and unanswered questions.
This thesis is organized as follows. At the beginning in chapter 2 we introduce the
fundamental concepts and tools for quantum information theory. Once we have defined everything we will head on to the depolarizing channel, its definition in chapter
3 and in chapter 4 we will deal with the actual bounds of the capacity for the depolarizing channel. In the end of the thesis in chapter 5 we will find a relation between
the number of the so-called Kraus operators and the dimension of the corresponding
environment.
2
One of these approximations we will present in this thesis.
5
2 The basic concepts of quantum information theory
In this section we will introduce some concepts which are fundamental to understand
quantum information theory, like density matrices, channels, fidelity, trace distance,
the depolarizing channel and many more. In the end we will have a look at the main
task of this thesis: The quantum capacity of the depolarizing channel.
Let us start with the most basic idea of quantum information theory, the so-called
qubit:
2.1 The qubit
Before we define the quantum bit (qubit) we will have a look at the classical bit
(cbit), which is used inside of every computer to represent information. A cbit is
characterized by the fact that it has only two possible states. Let us call a classical
bit c and to allowed states we label3 with 0 and 1, therefore we can form a word by
composing bits, e.g. a possible word would be 01001011.
From this knowledge we can highlight the main difference between a classical- and
a quantum bit. The cbit is restricted to be only 1 or 0. A qubit can be a so-called
superposition of both states which comes from the linearity of the Schroedinger
equation. To formulate this in the concepts of quantum mechanics we will use the
conventional Dirac notation in which a state is referred by a vector inside of ket |·i.
Knowing this notation and the properties of a classical bit we can say, that a 0 or a
1 is a state of the bit, what leads us to the map:
0 → |0i
,
1 → |1i .
(2.1)
And with this we can write a state of a qubit |ψi by superpositioning these two
states together with two complex numbers α, β to form any combination :
|ψi = α |0i + β |1i .
(2.2)
It is easy to see that a qubit is a generalization of the cbit, because if we set for
example α = 1 and β = 0 we have
|ψi = 1 · |0i + 0 · |1i = |0i ,
(2.3)
which is indeed one of the two states of a classical bit, to get the other case we have
to exchange α and β.
The postulates of quantum mechanics claim also that a physical state has always
3
That we denote these states by 0 and 1 comes from the technical implementation inside of a
computer where a zero stands for off (no voltage) and 1 for on (some voltage).
2 The basic concepts of quantum information theory
6
to be normalized, this can be written that a state ψ has to satisfy the following
condition:
hψ|ψi = 1.
(2.4)
If we now insert equation ( 2.2 ) in ( 2.4 ) we get:
(α∗ h0| + β ∗ h1|)(α |0i + β |1i) = α∗ α h0|0i + α∗ β h0|1i + β ∗ α h1|0i + β ∗ β h1|1i , (2.5)
where the ∗ denotes the complex conjugated of the subscribed number.
To simplify this equation we have to assume that these two states |0i and |1i form
an orthonormal basis which satisfies the following equation:
hi|ji = δij
with i, j ∈ { 0, 1 } ,
(2.6)
where δij denotes the Kronecker delta. Using this gives us:
hψ|ψi = α∗ α + β ∗ β = |α|2 + |β|2 = 1
(2.7)
and leads to an equation for α and β, which ensures that every state written as (
2.2 ), together with the condition ( 2.7 ), is normalized to 1.
This normalization gives us also the possibility to represent a qubit in spherical
coordinates [12], in the so-called Bloch Sphere ( Figure 2.1 ), by setting:
!
θ
θ
|ψi = cos |0i + eiϕ sin |1i .
2
2
(2.8)
z
|ψi
θ
ϕ
y
x
Figure 2.1: The representation of a qubit in the model of a Bloch sphere.
2.2 Hilbert space
7
2.2 Hilbert space
Before we can continue defining operations and concepts with qubits we have to go
one step backwards. In the previous section we were a little bit sloppy in equation
( 2.4 ), because we did not speak about the existence and the properties of an inner
product h·|·i of our |ψi. To ensure this fact we introduce the Hilbert space H.
A space H is called a Hilbert space when two elements |ψi , |ϕi ∈ H together with
the inner product, denoted by h·|·i and h·|·i : H2 → C, of H satisfying the following
conditions:
1. If we swap the inner product of two elements, the result should be the complex
conjugation of the initial one:
hψ|ϕi = hϕ|ψi∗ .
(2.9)
2. The inner product has to be linear in the second argument:
hψ|αϕ1 + βϕ2 i = α hψ|ϕ1 i + β hψ|ϕ2 i .
(2.10)
3. The inner product of a vector ψ ∈ H with itself has to be positive definite:
hψ|ψi ≥ 0 ∀ψ ∈ H.
(2.11)
4. H is complete, what means that every Cauchy sequence (|ψi = limn |ψn i) converges in H, that is:
∀ε > 0 ∃n(ε) ∈ N :
n > n(ε) ⇒ || |ψn i − |ψi || ≤ ε.
(2.12)
2.3 Density operators
Most times the world is not that nice as presented on the previous pages. The
exact knowledge of the coefficients α and β is nearly impossible or too hard to get.
Therefore we will use the concept of density operators to capture the uncertainty.
We will call a density matrix ρ a state of a system instead of |ψi.
If we do not know the state of our system but we say that with some probability pk ,
P
with k pk = 1, it will be in the state |ψk i, with k ∈ N, then we can express this
ensemble { pi , | ψk i hψk | } as a density matrix
ρ=
X
k
pk |ψk i hψk | .
(2.13)
2 The basic concepts of quantum information theory
8
There is a special case where only one pk = 1 and the others are equal to zero, then
the state
ρ = |ψk i hψk |
(2.14)
is called a pure state, because we can know that our system is with certainty in |ψk i,
all other cases we call mixed states. Therefore we can see that every mixed stated
is a mixture of pure states |ψk i hψk |.
The reason why we call this as a density operators is based on the fact that ρ is a
bounded operator of H, i.e. ρ ∈ B(H). That is that there exists a constant C ∈ R
such that
||ρ |ψi || < C|| |ψi || ∀ |ψi ∈ H.
(2.15)
Because a density operator describes a given ensemble it has some nice properties:
For the trace of ρ we have:
X
tr (ρ) =
λi = 1,
(2.16)
i
where the λi are the eigenvalues of ρ. In addition all these eigenvalues are bigger
than or equal to zero. In some literature the fact is denoted by the sentence, that ρ
is a positive semi-definite operator:
ρ ≥ 0.
(2.17)
2.3.1 The qubit as a density operator
The reason why we need density operators is the fact that we can write a qubit as
a density matrix by:
ρ = |ψi hψ| = |α|2 |0i h0| + αβ ∗ |0i h1| + α∗ β |1i h0| + |β|2 |1i h1| .
(2.18)
Now we can simplify this expression by setting
|α|2 = p,
(2.19)
and together with equation 2.7 we have
|β|2 = 1 − |α|2 = 1 − p.
(2.20)
Furthermore, we can see that expect of complex conjugation the coefficents of |1i h0|
and |0i h1| are the same. Therefore we can set αβ ∗ = c ∈ C. This characterizes a
qubit with only two parameters
!
p
c∗
ρ = p |0i h0| + c |0i h1| + c |1i h0| + (1 − p) |1i h1| =
,
c 1−p
∗
(2.21)
with p ∈ [0, 1] and c ∈ C. The last equation is the term before written in the
computation basis because we will use this matrix form to do some calculations.
2.4 Quantum channels
9
2.4 Quantum channels
In information theory we are not only interested in the information itself, but also
interested in manipulating, sending or storing information through some "wires" for
our purposes. This leads to the concept of a channel.
Before we define what we mean by a channel, we start with a simple example. The
simplest channel we can imagine is the "flip channel", denoted by X : H → H which
flips a |0i to |1i and |1i to |0i. If this channel acts on a state ψ we have:
X(|ψi) = X(α |0i + β |1i) = α |1i + β |0i .
(2.22)
Of course there are a lot of other channels for a single qubit. We do not want to
focus explicitly on the exact mechanisms of channels, more on the mathematical
properties a quantum channel has to satisfy. Due to the common notations in the
literature we will denote a channel by N , which takes density operators to density
operators, that is defined by a completely-positive trace-preserving map (CPTP
map).
This CPTP formalism ensures that the channel really takes density matrices to
density matrices, that is
N (ρ) ≥ 0 ,
tr(N (ρ)) = 1.
(2.23)
The most common way to visualize channels is via Alice A0 and Bob B, where Alice
tries to send information through a channel to Bob. That means that Alice sends a
0
0
density operator ρA to Bob with ρB = N (ρA ). The notation of superscripting the
A0 and B should denote that the corresponding density matrix is an operator of the
bounded operator spaces of the Hilbert spaces of Alice and Bob.
A′
′→B
A
N
B
Figure 2.2: A schematic representation of the action of a channel between Alice A0
and Bob B.
We can write this property of N as
N : B(HA0 ) → B(HB ).
(2.24)
2 The basic concepts of quantum information theory
10
2.4.1 The reference system
Some could wonder now why we denote Alice by A0 instead of A. The A is here
reserved for a referee which is related to Alice. Actually, we need this system in
order to compare the results of the transmission with the initial input to decide if
the channel works in that way, that the output is close to the initially send one. To
realize this we
let Alice A0 and the reference system A share an maximally entangled
0
qubit |ΦiAA , Figure 2.3 represent this principle.
A
E
AA′
Φ
A′
Figure 2.3: The schematic of a reference system or sometimes called reference frame
0
where Alice and the referee share a maximally entangled state |ΦAA i.
This allows us to find out "how close" the outcome after a transmission
over a
AA0
channel N is in relation to the input by comparing trA0 (|Φi hΦ| ) with ρB =
0
N (trA (|Φi hΦ|AA )). This closeness relations can be evaluated by using the definitions given in section 2.5.
2.4.2 The Kraus operators
Every quantum channel can be represented in terms of Kraus operators [18]. Suppose
we have a channel N and a set of operators Ak with
X
A†k Ak = 1
(2.25)
k
and the operators correspond to the channel N a the way, that they obey the
following property:
X
N (ρ) =
Ak ρA†k .
(2.26)
k
That means, that the Kraus operators completely characterize a channel. This
representation is really powerful to do calculations, because these operators allow us
2.5 Measures of distance
11
to write down exact matrices with which we can handle inside a computer program
or on paper.
2.5 Measures of distance
We need some tools to measure how reliable a channel sends information, because
we want to know how "good" our method is or the channel itself. In fact, there are
two important "distance" measures in quantum information theory, the fidelity and
the trace distance. The word distance in our case is not a real distance but the word
gives a useful interpretation of these two measures, that is "how far" two states are
apart.
2.5.1 Trace distance
The trace distance ||· − ·||1 between two states ρ and σ is given [18] by
||ρ − σ||1 = tr
q
(ρ −
σ)† (ρ
− σ) .
(2.27)
A notable property of the trace distance is that for the input of two density matrices
its value is bounded by:
0 ≤ ||ρ − σ||1 ≤ 2.
(2.28)
Therefore we will call two states identical if their trace distance is equal to zero,
that is
||ρ − σ||1 = 0 ⇒ ρ = σ.
(2.29)
2.5.2 Fidelity
Another way is the so-called fidelity, which we can express by the Uhlmann’s Theorem [18], as
√ √ (2.30)
F (ρ, σ) = ρ σ .
1
2.5.3 Relation between trace distance and fidelity
There is one really interesting relation between these two distance measures, which
is given by the inequality
1−
q
F (ρ, σ) ≤
q
1
||ρ − σ||1 ≤ 1 − F (ρ, σ).
2
(2.31)
2 The basic concepts of quantum information theory
12
2.6 Purification
In this section we will have a look at the theorem of purification, which provides
the possibility to consider the noise of a channel N : A0 → B as the influence of an
inaccessible environment E. That means, that there exists an isometric extension
or Stinespring dilation U with
0
0
N A →B (ρ) = trE UNA →BE (ρ) = trE UN ρUN† .
(2.32)
We claimed that UN forms an isometry and therefore possesses the properties:
UN† UN = 1A
0
UN UN† = ΠBE ,
,
(2.33)
where 1A is the unitary operator in the system of Alice and ΠBE is a projector
operator of the joint system BE of Bob and Eve. This principle of purification is
shown in the figures 2.4 and 2.5.
0
A′
′
N A →B
B
Figure 2.4: The initial concept of a channel N , which maps from A0 to B, is shown.
A′
B
E
′→BE
A
UN
Figure 2.5: Now we have the principle of purification shown in this scheme. The
0
isometry UNA →BE maps from A0 to B together with the inaccessible environment E. We can get N by tracing out E.
In addition we can write down the environment channel by tracing out B instead
of E and we will call this channel the complementary channel of N denoted by N c
which maps from Alice A0 to Eve E:
N c (ρ) = trB UN ρUN† .
(2.34)
2.7 Quantum information measures
13
2.6.1 Degradable channels
The principle of purification brings one very important class of channels, the degradable channels. As explained on earlier pages there is for every channel an isometric
0
extension UNA →BE which gives us, by tracing out Eve, the channel N itself and
the corresponding complementary channel N c by tracing out Bob. Then we will
call a channel degradable if there exists a completely positive trace preserving map
D : B(B) → B(E) such that:
D ◦ N (ρ) = N c (ρ),
(2.35)
In addition we will call a channel bidegradable if there are two CPTP-maps D :
B(B) → B(E) and D̂ : B(E) → B(B) with:
D ◦ N (ρ) = N c (ρ) ,
D̂ ◦ N c (ρ) = N (ρ).
(2.36)
Also a channel can be anti-degradable if the complement is degradable, i.e. there is
one completely positive trace preserving map D̂ : B(E) → B(B) with
D̂ ◦ N c (ρ) = N (ρ).
(2.37)
The reason why degradable channels are so valuable in quantum information theory
is because the corresponding coherent information is additive [2] [4], that is:
Q(1) (N ⊗n ) =
max
I(AiB ⊗n ) = n · max0 I(AiB) = n · Q(1) (N ).
0 ⊗n
|ψiA(A
)
(2.38)
|ψiAA
For a degradable channel we have a nice, so-called "single-letter" formula because
the capacity of the channel is the same as using the channel a single time:
Q(N ) = Q(1) (N ) = max0 I(AiB),
(2.39)
|ψiAA
for every degradable N . Later we will show why additivity admits a single-letter
formula.
2.7 Quantum information measures
On the following pages we will introduce the most important concepts, which are
needed for the informational approach to quantum Shannon theory. The basic concepts are based on the von Neumann entropy which admits an intuitive interpretation for quantum entropy.
0
Let us assume that Alice generates a quantum state given by a density operator ρA ,
then the von Neumann entropy is given by:
0
0
H(A0 ) = −tr(ρA log ρA ).
(2.40)
Using this entropy we can form a lot of other different information measures which
are useful to later form the quantum capacity theorem.
2 The basic concepts of quantum information theory
14
2.7.1 Conditional quantum entropy
The most useful quantum entropy is the conditional entropy. Again we have two
systems Alice A0 and Bob B, then
H(A0 |B)ρ = H(A0 B)ρ − H(B)ρ .
(2.41)
This entropy measures how much quantum communication we need in order to
transfer a full state from A0 to B. In the classical case this measure can only be
positive, but the quantum case is quite different. A negative magnitude is also
possible. At first this sounds a little bit weird and impossible, but this case also
allows an interpretation. Horodecki et. al. have shown that a negative coherent
information means that the sender and the receiver gain a potential for future quantum communication [9]. Bob is then able to send the full state only using classical
communication.
Because the conditional entropy can also be negative, it makes sense to define another quantum measure, the coherent information, which is the negative of the
conditional one.
2.7.2 Coherent information
We assume that Alice A0 wants to send information over to Bob B. Because Alice
shares, together with a reference frame a state we have after the transmission, a bipartite state ρAB between the reference system and Bob which allows us to calculate
the coherent information I(AiB) as
I(AiB) := H(B)ρ − H(A0 B)ρ = −C(A0 |B),
(2.42)
where ρB is the state of Bobs system. As you can see the coherent information gives
some measure between of the correlations of the reference frame and Bob after the
transmission of Alice’s state, which is before the transmission is maximally entangled to the referee.
2.7.3 Quantum mutual information
Obviously are we interested in measuring correlations between density operators,
which lead to the really useful concept of the mutual information. We assume that
there is a joint operator ρA0 B and there are the local operators ρA0 and ρB . Then
we can define the mutual information I(A0 ; B) between A0 and B as
I(A0 ; B) := H(A0 ) + H(B)ρ − H(A0 B)ρ .
(2.43)
2.8 Quantum capacity theorem
15
Using the definition of the coherent information ( section 2.7.2 ) we can write it as
I(A0 ; B) = H(A0 ) + I(A0 iB).
(2.44)
2.8 Quantum capacity theorem
Now we head on to the most important theorem for this thesis, on which we want to
focus later on. Like in the classical case we are interested in "how much information"
we can send over the channel without losing any, or rather how we can decode the
information of the outcome. This means that we construct a code C ⊂ A⊗n and a
related decoding method Dn : B(B ⊗n ) → B(C) such that we have for all |ψi ∈ C:
Dn ◦ N ⊗n (|ψi hψ|) ≈ |ψi hψ| .
(2.45)
With this idea in mind we are able to formulate what is meant by the quantum
capacity Q(N ) of a quantum channel N .
To define what Q is, we need at first the notion of a rate R. We call R an achievable
rate, if for every ε > 0 and a sufficiently large n ∈ N there is a code Cn ⊂ A⊗n , with
log2 dim Cn ≤ R · n, and a decoding process Dn : B(B ⊗n ) → B(Cn ) which is satisfied
with
F |ψi hψ| , Dn ◦ N ⊗n (|ψi hψ|) ≥ 1 − ε
(2.46)
for all |ψi ∈ Cn .
If we now think of this problem by keeping the coherent information defined in
section 2.7.2 in mind, we can say, that for
R ≥ I(AiB)
(2.47)
we have an achievable rate for communication, because the coherent information is
a measure of correlations between the referee and Bob. Therefore we can define the
capacity of the channel as:
Q(1) (N ) = max0 I(AiB),
(2.48)
|ψiAA
0
where we are maximizing over all possible input states |ψiAA .
In the end,this capacity is not the "real" one since we know that coherent information is non-additive [5]. Therefore we can define quantum capacity over the so-called
LSD-formular, given by Lloyd [11], Shor [15] and Devetak [3], which is the regularization of this upper one by using the channel n times, dividing it by n and letting
n go to infinity, that is:
1
(2.49)
Q(N ) = lim Q(1) (N ⊗n ).
n→∞ n
2 The basic concepts of quantum information theory
16
A
A
A′
En
N
N
N
..
.
n-times
I(AiB)
Dn
B
N
Figure 2.6: This figure depicts the operation mode of the regularization to get the
quantum capacity of a channel N
2.9 Superactivation of quantum capacity
The possibility of some channels with no single-letter formula, i.e. their coherent
information is non-additive, admits an interesting possibility. Smith and Yard [17]
have shown that it is possible to communicate with a combination of channels, which
have on their own a zero capacity, i.e. if we have two zero capacity channels N1 and
N2 with
Q(N1 ) = Q(N2 ) = 0,
(2.50)
then it could be that:
Q(N1 ⊗ N2 ) > 0.
(2.51)
This might seem impossible, because there is absolutely no analog in the classical
case for this possibility and therefore we can not argue with our intuition.
It opens up new paths to think about communication. The bricks of the system are
not the crucial parts to determine whether it is possible to transmit information or
not. It is more the context of the system rather than the complete system itself.
17
3 The depolarizing channel
3.1 Definition
We are now approaching the key aspect of this thesis, the depolarizing channel
Γp : B(A0 ) → B(B) , which leaves ρ, the input state, with a probability 1 − p
untouched or otherwise destroys the state completely, mapping it to the maximally
mixed state of the same dimension, denoted by π = 1d . This mechanism gives us
Γp (ρ) = (1 − p)ρ + p · π.
(3.52)
As mentioned on the previous pages there is a way to write the depolarizing channel
with some Kraus operators which correspond to the depolarizing channel and gives
the form:
X
p
(3.53)
Γp (ρ) =
Ak ρA†k = (1 − p)1ρ1 + (XρX + Y ρY + ZρZ) ,
3
k
where X,Y and Z are the Pauli matrices and 1 is the identity matrix. And therefore
we have:
A0 = A†0 = 1 ,
A1 = A†1 = X
,
A2 = A†2 = Y
,
A3 = A†3 = Z
(3.54)
3.2 The classical capacity
One interesting fact is that in contrast to the quantum case the classical capacity
of the depolarizing channel is known and calculated. Classical capacity means Alice
transmits to Bob only classical information. Therefore she encodes the her message
into quantum states, sends them over the channel and Bob decodes this quantum
states to get the classical message. The simplest way to do so is that Alice only uses
tensor product states, so that there is absolutely no entanglement.
The Holevo-Schuhmacher-Westmoreland [8, 13] theorem allows to calculate the classical capacity with the Holevo quantity
!
χ∗ = sup S(ρ) −
ρ
X
px S(ρx ) ,
(3.55)
x
where ρ = x px ρx , ρx = |ψx i hψx |, |ψx i are the chosen tensor product states and
the supremum is taken over all possible input states.
Christopher King has shown that this quantity is additive4 for the depolarizing
P
4
Additivity was a long time an expected property of the Holevo-capacity for every channel, but in
the year 2009 Hastings found a counter example via a random unitary channel [7] that showed
that it is not self-evident.
3 The depolarizing channel
18
channel and therefore gives us the classical capacity [10]:
p
d−1
p
p
C(Γp ) = χ (Γp ) = log (d) +
p log
+ (1 − p + ) log 1 − p +
, (3.56)
d
d
d
d
∗
where C(Γp ) labels the classical capacity of the depolarizing channel and the log
is taken to the basis 2. Figure 3.7 shows the corresponding equation for the qubit
case, that means for d = 2.
CHpL
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
p
Figure 3.7: The Figure depicts the classical capacity C of the depolarizing channel
depending on the parameter p for the qubit case, where d = 2.
19
4 Actual bounds of the quantum capacity of the
depolarizing channel
The paper by G. Smith et. al. [16], published 2008, introduces a new and tighter
bound for the quantum capacity for the depolarizing channel compared to previous
works. Within the next pages we will explain the concept of symmetric side channel
assisted capacity and how they use it to find the new bound. The unassisted principle
is already shown, in section 2.8 with the quantum capacity theorem and therefore
will directly have a look at the side channels assisted case.
4.1 Symmetric side channel assisted capacity
Suppose there are two d-dimensional spaces > and ⊥ and a subspace Wd ⊂ >⊗ ⊥
which is spanned by the following vectors:
|i, ji =

 √1
2
(|ii |ji + |ji |ji) i 6= j
|ii |ii
i=j
,
(4.57)
with i, j ∈ N and the |ii are an orthonormal basis for > and ⊥. Due to these vectors
we find that dim Wd = d(d + 1)/2. Now we consider a map Vd which maps from
Cd(d+1)/2 to Wd .
With this definitions we can define what the symmetric side channel Ad is:
Ad (ρ) = tr⊥ Vd ρVd† ,
(4.58)
that is a map from B(Cd(d+1/2) ) to B(>). With this knowledge we can define how
we can assist a channel with the symmetric side channel. Figure 4.8 shows how
we can assist a channel N , where we add Ad parallel to the n uses of the channel
itself.
4 Actual bounds of the quantum capacity of the depolarizing channel
20
A
A
A′
En
N
N
N
..
.
n-times
Dn
B
Ad
Figure 4.8: Shown is how the symmetric side channel Ad is included in the scheme
of the uses of the channel and the encoding - decoding process.
Like in the unassisted case we can formulate a capacity Q(1)
ss (N ) for the assisted case.
We have a channel N : B(Ã) → B(B) and we can define Q(1)
ss )(N ) as the supremum
of the coherent information of all input states |φiwhich are invariant under swapping
> and ⊥, that is
Q(1)
(4.59)
ss (N ) = sup I(AiB>)|ωi ,
|φiAÃ>⊥
with the output state |ωiAB>⊥ = (1A>⊥ ⊗N ) |φiAÃ>⊥ . It turns out that this capacity
has some nice properties. First of all it is additive [16], that is:
(1)
(1)
Q(1)
ss (N1 ⊗ N2 ) = Qss (N1 ) + Qss (N2 )
(4.60)
and the regularization Qss is given by [16]:
Qss (N ) = lim
n→∞
1 (1) ⊗n
Q (N ).
n ss
(4.61)
The combination of these two gives us, that:
Qss (N ) = Q(1)
ss (N ).
(4.62)
Another very important property of the assisted quantum capacities is that it is a
convex function of the channel N , that means:
Qss (pN1 + (1 − p)N2 ) ≤ pQss (N1 ) + (1 − p)Qss (N2 ),
(4.63)
for a p ∈ [0, 1] and two channels N1 and N2 . We will use this fact later on to show
that there is a maximum of capacities for a channel.
4.2 Relations between the unassisted and the assisted cases
21
4.2 Relations between the unassisted and the assisted cases
Let us consider the principle presented in section 2.8 where we defined the capacity
of a channel as:
1
(4.64)
Q(N ) = lim Q(1) (N ⊗n ).
n→∞ n
If we assist this a symmetric side channel we would expect, that the capacity would
be equal or bigger, because we could use this additional channel to send information
to Bob. This gives an inequality for the capacities:
Q(N ) ≤ Qss (N ).
(4.65)
As presented by Smith et. al. this equality holds for the class of degradable channels.
4.3 An upper bound via amplitude-damping channels
With all this introduced equations and principles Smith et. al. [16] were able to
calculate an upper bound for Q(N ) in the case of the depolarizing channel.
They used the fact that for a degradable channel it is indeed
Q(N ) = Qss (N ).
(4.66)
and they write the depolarizing channel as a convex combination of degradable
channels, because as shown in equation ( 4.63 ) is Qss a convex function in the
channel. Therefore, this method gives us an upper bound for Q(N ).
To do so they used the amplitude-damping channel, given in the Kraus operator
form by:
∆γ (ρ) =
1
X
Ak ρA†k ,
(4.67)
k=0
with
1 √ 0
A0 =
1−γ
0
!
,
0
A1 =
0
√ !
γ
.
0
(4.68)
If we set
N(q,q,pz ) (ρ) = (1 − 2q − pz )ρ + qXρX + qY ρY + pz ZρZ,
(4.69)
√
γ
γ
1
together with q = 4 and pz = 2 (1 − 2 − 1 − γ), then we are able to find that
1
1
N(q,q,pz ) (ρ) = ∆γ (ρ) + Y ∆γ (Y ρY )Y.
(4.70)
2
2
This equation looks similar to the one of the depolarizing channel and if we had
three of these together with different parameters we get:
1
N(q,q,pz ) (ρ) + N(q,pz ,q) (ρ) + N(pz ,q,q) (ρ) = N(D,2q+pz ) (ρ).
3
(4.71)
4 Actual bounds of the quantum capacity of the depolarizing channel
22
So is the depolarizing channel√Γp written√as a convex combination of amplitudedamping channels with γp = 4 1 − p(1 − 1 − p). This gives us the upper bound,
because the capacity of the amplitude damping channel is known [6]. This leads
to
Q(Γp ) ≤ Qss (Γp ) ≤ min Q(∆γp ), (1 − 4p) ,
(4.72)
with
Q(∆γp ) = max (H2 (t(1 − γp ) − H2 (tγp ))) ,
(4.73)
0≤t≤1
H2 (x) is the standard binary entropy and p ∈ [0, 0.25].
4.4 A lower bound via hashing
Smith et. al. introduced in their paper an equation for Q(1)
ss (N ) for a channel N
with Stinespring dilation UN : A0 → BE:
Q(1)
ss (N ) = sup
ρAA0 F
1
(I(AiBF )ω − I(AiEF )ω ) ,
2
(4.74)
with ωABEF = (1AF ⊗ UN )ρ(1AF ⊗ UN )† . If we now pick
1
X
|φi =
√
qst X s Z t ⊗ 1 |Φ+ iAA0 |stiF
(4.75)
s,t=0
as a state, where |Φ+ i = √12 |0i |0i + |1i |1i is the first Bell state, then we have the
possibility to give a lower bound with equation ( 4.74 ) :
1
I(AiBF )(1AF ⊗Γp )φ + I(AiB)1AF ⊗Γp )φ
2
Q(1)
ss (N ) ≥
for any choice of qst with
by choosing
P
st qst
(4.76)
= 1 with φ = |φi hφ|. Now we can maximize this
q q q
(4.77)
qst = (1 − q, , , ).
3 3 3
With this knowledge is it possible to calculate all the von Neumann entropies:
1 4pq
1 2pq
−
− 2ηp,q log
−
− ηp,q
2
9
4
9
1 2pq
1 4pq
−
+ 2ηp,q log
−
+ ηp,q
−
2
9 4
9
8pq
2pq
−
,
log
9
9
S(BF ) = −
4pq
4pq
S(AB) = − 1 − p − q +
log 1 − p − q +
3 3
4pq
p + q 4pq
− p+q−
log
−
,
3
3
9
(4.78)
(4.79)
4.4 A lower bound via hashing
with ηp.q =
we get:
1
36
q
23
S(B) = 1,
(4.80)
S(ABF ) = H2 (p) + p log(3),
(4.81)
81 − 720pq − 513p2 q 2 + 576pq(p + q). Fitting this in equation 4.76
1
(1 − H2 (p) − p log(3) + S(BF ) − S(AB)) .
(4.82)
2
The last step is now to optimize over q to get the biggest values for a tight bound.
Qss (Γp ) ≥
Q
1.0
0.8
0.6
0.4
0.2
0.05
0.10
0.15
0.20
Figure 4.9: The Figure depicts the upper and the lower bound for Qss .
0.25
p
5 Relation between the dimension of Eve and the Kraus operators
24
5 Relation between the dimension of Eve and the
Kraus operators
On the previous pages we presented results of other and now we want to focus on
some own concepts and calculations we made. An important question of the dimension of the environment and therefore the output that Eve gets.
Normally purification is the point of view where Alice transmits to Bob the information and the noise to Eve. Due to evolution of time the environment should already
exist before the transmission. Therefore we can extend A with E, that is:
A → AE.
(5.83)
A
N A→B
B
E
VNAE→BE
E
Figure 5.10: The figure shows the extension of A to AE and the evolutions of the
whole system under VN .
Also we require that E should be in a pure state, i.e. |0i h0|E , before the transmission
and these two system should also be separated, because that allows us writing them
as a tensor product:
ρAE = ρA ⊗ |0i h0|E .
(5.84)
The postulates of quantum mechanics tell us that there is a unitary time evolution
VN : AE → BE which is related to the original channel N with:
ρBE = (VN ) ρAE (VN )† .
(5.85)
Now we are interested in the local density operator of Bob ρB , which we get by
tracing out Eve:
(5.86)
ρB = trE (ρBE ) = trE (VN ) ρAE (VN )†
If we insert the initial state ( equation 5.84 ) we get::
B
A
ρ = trE VN (ρ ⊗
|0i h0|)VN†
=
dim
XE
k=1
hk| VN |0i ρA h0| VN† |ki .
(5.87)
5.1 The amplitude-damping channel
25
The third equation comes from the evaluation of the partial trace by summing up
over a orthonormal basis { | ki } of Eve with k ∈ N and k ≤ dim E. If we now look
at these equation we can define the Kraus operators:
Ak = hk| VN |0i
(5.88)
and get:
ρB =
dim
XE
Ak ρA A†k .
(5.89)
k=1
Now we can see that the number of Kraus operators is indeed the dimension of Eve,
that means on the other hand that if we know the number of Kraus operators, then
we know what kind of qudit(a qudit is a d-dimensional qubit) Eve gets. Another
important question is whether these Kraus operators are unique or not?
They are not, because if we look at equation ( 5.88 ) we can see that they depend
on the basis we choose for Eve. Another important thing to mention is, that if we
have a set of Kraus operators { Ak } it is possible to find the space of AE [1], the
corresponding vector |0i and VN like in equation ( 5.88 ).
5.1 The amplitude-damping channel
We have now built a way to calculate the outcomes of Bob and Eve explicitly by
using the purification approach and the Kraus operators of a channel. To give a
simple example we will calculate these for the amplitude-damping channel and of
course for the depolarizing channel as well. Let us start with the first one. The
amplitude damping channel is given by:
!
!
!
!
√
X
1−p
0
0
1
0
1
0
0
†
√ ρ
√ , (5.90)
ρ √
+
∆p (ρ) =
Ak ρAk =
0
0
1−p 0
0
p
0
p
k
with p ∈ [0, 1]. We can characterize a qubit by the formula ( 2.21 ) and here we
replace p with γ to distinguish it from the p in the definition of the channel, resulting
in:
!
γ
c∗
ρ(γ, c) =
.
(5.91)
c 1−γ
If we now insert this in the equation for Bobs outcome we get:
√ !
1 − p(1 − γ)
c∗ p
√
.
∆p (γ, c) =
c p
p(1 − γ)
(5.92)
In order to calculate the information that Eve gets we have to trace out Bob from
the purification of the system:
N c (ρ) = trB UN ρUN† ,
(5.93)
5 Relation between the dimension of Eve and the Kraus operators
26
where the purification is given by:
UN =
X
Ak ⊗ |ki .
(5.94)
k
Inserting this into equation ( 5.93 ) gives us for Eve:
N c (ρ) =
tr Ai ρA†k |ii hj| .
X
(5.95)
i,k
If we use this form of the complementary channel than Eves outcome is given as
follows:
!
√
∗
1
−
p
γ
+
p(1
−
γ)
c
√
(5.96)
∆cp (ρ) =
(1 − γ)(1 − p).
c 1−p
By comparing these two outcomes we can easily see the corresponding map, which
makes the amplitude-damping channel degradable, that is:
D : B(B) → B(E) ,
p 7→ 1 − p.
(5.97)
In addition this channel is bidegradable because the inverse D̂ of D exists:
D̂ : B(E) → B(B) ,
1 − p 7→ p.
(5.98)
5.2 The depolarizing channel
Now we want to do the same for the depolarizing channel. In equation ( 3.54 ) we
gave the Kraus operator form as:
Γp (ρ) =
X
Ak ρA†k = (1 − p)1ρ1 +
k
p
(XρX + Y ρY + ZρZ) .
3
(5.99)
Evaluating equation ( 5.99 ) by inserting the form for a qubit (equation ( 5.91 ))
gives us:
!
4c∗ p
2
∗
p(1
−
2γ)
+
γ
c
−
3
Γp (γ, c) = 3
.
(5.100)
p
c − 4cp
1
−
γ
+
(4γ
− 2)
3
3
And for the complementary channel we have to do the same as in equation ( 5.95 ),
that gives us:

c
ND,p
(γ, p) =
√
1−p
√
 (2<(c)) −(−1+p)p

√

 i(2=(c))√3−(−1+p)p
−
√
 √
3
−(−1+p)p(−1+2γ)
√
3
(2<(c))
−(−1+p)p
√
3
√
−
i(2=(c))
−(−1+p)p
√
3
√
−(−1+p)p(−1+2γ)
√
3
p
3
− 13 ip(−1+2γ)
1
(2=(c))p
3
1
ip(−1+2γ)
3
p
3
− 13 i(2<(c))p
1
(−2<(c))p
3
1
i(2<(c))p
3
p
3




.


(5.101)
As you can see Eve receives indeed a 4-dimensional qudit as presented in section
5.
27
6 Conclusion
The aim of this thesis is to explore the problem of calculating the quantum capacity
of the depolarizing channel. The problem is notoriously difficult since the quantum
capacity is only known for the class of degradable channels [4].
Here we have reviewed the work of Smith, Smolin and Winter [16] where they
write the depolarizing channel in terms of degradable channels, namely amplitudedamping channels, and use so-called symmetric side channel assistance. The symmetric side assisted capacity is a useful tool to find bounds for unknown capacities
for channels which are hard to handle. An interesting question would be if we could
use this principle to give bounds for other channels where the quantum capacity is
also unknown.
We think that to focus research on special types and classes of channels is an important field. Because there maybe other classes of channels which also have additive
coherent information and have therefore also an easy expression for the quantum capacity. Another promising approach could be the development of numerical methods
to calculate the quantum capacity. This could also be an interesting field of research
and could provide some good results.
A possible approach could be a so-called "gate-representation" of channels, i.e. one
could try to build channels as combinations of known quantum gates to approximate
the exact Hamiltonian which corresponds to the system.
In the end we were not able to improve the knowledge of the depolarizing channel
explicitly, we focused on discussions of degradable channels and the outputs because
these are crucial for the property of being degradable.
None the less quantum information theory is a really interesting field of research.
At one time it might be possible to build a quantum computer then the theory has
brought us the basics and algorithms to use this type of computer for our purposes.
References
28
References
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[3] I. Devetak. The private classical capacity and quantum capacity of a quantum
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[4] I. Devetak and P. W. Shor. The capacity of a quantum channel for
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[14] C. E. Shannon. A mathematical theory of communication. The Bell System
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Lecture notes, MSRI Workshop on Quantum Computation, San Francisco,
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[16] G. Smith, J. A. Smolin, and A. Winter. The quantum capacity with
symmetric side channels. IEEE Trans. Info. Theory, 54:4208–4217, 2008.
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Acknowledgment
At first I want to thank Dr. Ciara Morgan who supervised me through the whole
thesis and helped me out with every question and language problem I had.
Furthermore, I want to thank my girlfriend Karolin, who ever elated me to continue
and my parents who gave me the chance to study physics.
Selbstständigkeitserklärung
Hiermit erkläre ich, diese Arbeit selbständig und nur mit den angegebenen Hilfsmitteln angefertigt zu haben. Alle Stellen der Arbeit, die wörtlich oder sinngemäß
aus anderen Werken übernommen wurden, sind als solche kenntlich gemacht. Diese
Arbeit lag weder in dieser, noch in ähnlicher Form bereits einer Prüfungsbehörde
vor.
Hannover, October 4, 2013
————————————–