Quantum Computation and Communication with Qubit Systems

Chapter 3
Quantum communication in quantum networks
3.1 Introduction
O
ne of the problems arising in the design of a feasible quantum computer is how to
build up quantum wires, i.e. wires able to transmit quantum information from one
end to another. The quantum data bus design involves the choice of the technology to
be used in order to achieve an optimal data transfer. In this chapter we will consider
channels composedbya sequenceofinteracting electron spins, usuallycalled spin chains
in short. A spin chain is a permanently coupled 1D system of electron spins. When a
quantum state is placed on one end of it, the state will be dynamically (due to quantum
time evolution, see appendix for the details) transmitted to the other end with some
efficiency if spins are coupled by an electromagnetic exchange interaction (further de-
tails in the appendix). However, such a data-bus requires the ability to modulate the
strength or nature of interactions between pairs of adjacent spins in time. For the typi-
cal use in quantum computer we should utilize systems that require a very low control,
as suitable candidates for connecting quantum registers. This is because in the normal
use of “data-buses”, such as to denote a cable connecting two computers, we mostly let
the information flow through it in its own natural way, thanks to the metal conduction
properties. A spin chain in which inter-spin interactions are permanent, i.e. the inter-
action don’t vary with time, is an example of a spin chain, that does not require a large
amount of control to be used. In the following we will define the spin chain model, the
natural extension to spin network and the conditions for having an high quality transfer
on quantum communication devices.
3.1.1 Spin chains and the exchange interaction
In quantum mechanics, spins are systems characterized by small quantized magnetic
moments. The mutual interactions of these spins makes them prefer alignment or anti-
alignment with respect to each other, resulting in diverse phenomena such as ferromag-
netism and anti-ferromagnetism. A spin chain models a large class of such materials
in which the spins are arranged in a one dimensional lattice and permanently coupled
to each other, usually with an interaction strength decreasing with distance. A com-
mon form of the Hamiltonian, i.e. the matrix (or in general the mathematical operator)
54 3. Quantum communication in quantum networks
Figure 3.1: This figure depicts the possibility of transporting quantum information from one
quantum processor to another through a line of stationary qubits.
that describes the total energy of the system (see appendix for further details), for the
interaction between the ith and the jth spin is written as
H
XYZ
ij
=J
ij
null
S
i
.
null
S
j
, (3.1)
where
null
S
i
.
null
S
j
≡S
x
i
S
x
j
+S
y
i
S
y
j
+S
z
i
S
z
j
andS
x
i
,S
y
i
,S
z
i
aretheoperatorsforthecomponentof
theith spin along thex,y and z directions respectively andJ
ij
is the coupling constant
between two interacting spins (normalized to 1). In particular, when all the spins form
spin-half systems, S
x
,S
y
and S
z
stand for the familiar Pauli matrices σ
x
,σ
y
and σ
z
. A
Hamiltonian of the above form is termed as exchange interaction as it can arise in from
the pure exchange of electrons between neighbor ions in a metal (details can be found
in appendix). It is also called the Heisenberg interaction after the name of the scientist
who studied this kind of interaction. In particular, the specific Hamiltonian we have
written above is called the isotropic exchange interaction because the interactions are
considered along the three Euclidean axes. Later on, we will also encounter a variant of
the above interaction
H
XY
ij
=J
ij
(S
x
i
S
x
j
+S
y
i
S
y
j
), (3.2)
which is called the XY interaction. We will be primarily concerned with chains and
networks of interacting spin-half systems in this thesis. A graphic representations of
such systems is shown in Fig.3.2
3.2. A spin chain quantum communication protocol 55
Figure 3.2: The figure shows a spin chain: a system of spins perpetually coupled to each other
with an interaction strength which generally decreases with distance. The double arrowed lines
depict interactions with the dotted line denoting a weaker strength than the solid line.
3.2 A spin chain quantum communication protocol
We want to start our protocol with the spin chain initialized in a very simple state, such
as one in which all spins are in the same pure state, say |0null. We will have to choose
the couplings J
ij
in Eq.(3.1) in such a manner that initialization of the spin chain to
such a state is easy. For example, all spins could be pointing down in the conventional
state denoted as |0null. In the simplest quantum communication protocol, let us assume
two users conventionally named Alice and Bob interact via a spin chain. To start the
protocol, Alice places an arbitrary quantum state at one end of the spin chain in such a
“all down” state. This is depicted in the upper part of Fig.3.3, where Alice has placed
an arbitrary state on the first spin of the chain, while all the other spins are still in the
down state. Due to the natural time evolution of the chain, this state both disperses
and propagates along the chain. As a result of this evolution, the state of the spin at
Bob’s end of the chain will vary with time. Bob now chooses an optimal moment of
time in a suitable interval depending on the network features, to receive Alice’s state.
Since the transmitted state evolves with time, this moment is carefully chosen in order
to maximize the ”similarity” between the received state and the one that Alice intended
to transmit. At this optimum time, Bob simply picks up (formally, measures the state)
the spin at his end of the chain, to conclude the communication protocol.
56 3. Quantum communication in quantum networks
Figure 3.3: The simplest spin chain communication protocol. A spin-1/2 ferromagnetic spin
chain with all spins facing down is the quantum channel. Alice simply places a quantum state at
one end of the chain and Bob simply picks up a close approximation of this state from his end
of the spin chain after waiting an optimal time.
3.2.1 Fidelity and entanglement as figures of merit
In order to judge how well a quantum state is transferred by a spin chain from one end
to another it is useful to introduce a figure of merit. Suppose that the state placed into
the spin at the end of the chain is depicted by|ψ
in
nulland the state at the other end of the
spinat the optimum timet
opt
is depicted in general by the density operatorρ
j
(t
opt
) (the
output state is depicted by a density operator to allow for the possibility for it to be a
mixed state). Then a measure of the quality of the transfer is defined by the fidelity
f(i,j,t) =nullψ
i
|ρ
j
(t)|ψ
i
null, (3.3)
which is always between 0 and 1, with higher value meaning a higher quality transfer.
When f(i,j,t) is unity we have an optimal transfer, known with the term Perfect State
Transfer (PST). In generalf(i,j,t) needs to be greater than 2/3 in a quantum commu-
nication scheme to be better than straightforward classical communication. A fidelity
of 2/3, in fact, can already be obtained if Alice simply measured her state, communi-
cated the results classically to Bob and Bob simply reconstructed the state from this
data. Later, we will introduce the use of another figure of merit, namely the amount of
entanglement that can be transmitted by a spin chain or a spin network. In particular,
we will look at the transmission of the state of one member of a pair of particles in the
entangled state |ψ
+
null through the spin chain channel. This is the usual procedure for
sharing entanglement between separated parties through any channel and it is named
3.3. Perfect State transfer in Spin networks 57
Figure 3.4: The mechanism of transferring entanglement down a spin chain. The state of one
member of a pair of qubits in a maximally entangled state is placed on the spin at Alice’s end of
the chain, while the other member is held by Alice. After a while, the spin at Bob’s end of the
chain will be entangled with the qubit held by Alice.
Entanglement transfer. A sketch of the problem can be seen as Alice preparing two
qubits in the state|ψ
+
null, holding one of them (say,A) in her hand and placing the other
on the first node of the chain. On the other side, Bob receives one particle of the entan-
gled pairs and after a time t
opt
he measures the entanglement degree after the transfer.
We will show that the final correlation between the two particles transfer depends only
on the network topology and on the fidelity reached during the transfer. This procedure
is illustrated in Fig.3.4. Furthermore in the final part of this chapter we will see how
to enhance the entanglement shared by two particles of two end users,until obtaining a
singlet state |ψ
+
nullshared between them. This procedure requires only local actions, i.e.
projective measurements by the end users and classical communication between them.
The performance of such protocol will be evaluated through a specific figure of merit
called Efficiency.
3.3 Perfect State transfer in Spin networks
Networks of interacting qubits are a generalization of spin chains. Such networks are
of theoretical importance for the study multi-particles quantum systems and could con-
stitute a good test ground for technologies spanning from quantum key distribution to
multi-usernetworks. Therapidgrowthoftheareaofquantuminformationhasledtocon-
siderthe idea of multi usersquantumnetworks withthe finalgoal of realizing nano-scale
devices and communication protocols [107]. In the last few years, this kind of networks
have been specifically considered to be good candidates for engineering perfect quantum
channels andallowing information transferbetween distant locations [101, 104, 109] (see
also [102], for a review). Such networks appear to be useful for the implementation of
data buses in quantum mechanical devices, in particular because the communication
protocol is based on the free dynamics of the system after an initial set-up, that is only
thenaturaltimeevolutionofthequantumsystemisexploitedtoimplementtheprotocol.
Since the first works [104, 101, 109] (see also [102], for a review), networks of interacting
58 3. Quantum communication in quantum networks
qubits are considered to be good candidates for engineering perfect quantum channels
and allow information transfer between distant particles. One of the problems arising in
the scenario is given by natural dispersion effects, which determine a loss of information
often proportional to the distance (i.e., the number of spins) between communicating
sites. Ways to circumvent the issue are based on a local tuning of the exchange interac-
tion among quantum spins [104] or on increase of kinetic energy to the particles as we
will see in the next section of this chapter.
The first approach used is based on the study of which network topologies guarantee
a small decay of fidelity, when the couplings are homogeneous and constant (i.e. J
ij
is constant and normalized during the entire evolution of the system). Results in this
direction have provided examples of perfect state transfer, in relation to combinatorial
properties of the graphs modeling the networks (a list of references on this area is in
[100]). In the XY model, when considering a single excitation, it has been shown that
PSTdependsessentially ontheeigensystem oftheadjacency matrixofthegraph. Inthe
cases analyzed so far, even if PSToccurs between two specificsites ofa network, routing
arbitrarily over the entire network still requires an external controller to switch off the
desired couplings. One of the possible applications is routing a qubit (i.e., transferring
the qubit between any two vertices of the network arbitrarily. The protocol involves an
external agent, whose role is to switch OFF (by varying, for example, the distance, as
discussed before) only the link between the sites that we wish to put in communication
[139].
In the next sections we will consider networks of spin-half particles with the XYZ
interaction. Specifically, wewilldescribethestructureoftheHamiltonian whichgoverns
the single excitation setting, i.e. a network where only one spin can be up or down at a
fixed time t.
Furthermore we will study quantum time evolution in a network in which all sites
can communicate between each other (in both directions). Such a network is modeled
by the complete graph on n vertices K
n
and it is called all-to-all network. The number
of links in K
n
is n(n−1)/2. It turns out that K
n
does not exhibit PST. However, we
will show that deleting an edge inK
n
will allow PSTbetween the two non-adjacent sites
in the resulting network, when n is a multiple of 4 (apart the trivial case n = 2). This
is modeled by a graph usually denoted byK
−
n
, an all-to-all network with a missing link.
This counterintuitive phenomenon is due to interference effects.
3.4 Graph Theory and Spin networks
Inthissectionwewillanalyzehowthestateandtheinteractionofanetworkofelectronic
spinsevolveovertimeusingtheconceptsofgraphtheoryinordertofindtheconditionsto
achievePSTbytheanalysisofitsunderlying graph inotherwordsagraphrepresentation
3.4. Graph Theory and Spin networks 59
of the spatial distribution of the spins is used in order to exploit the mathematical
properties deriving from graph theory. In order to describe the connection between
graph theory and networks of interacting spins, let us recall the concepts that will be
used in the following sections.
3.4.1 Problem set up
Let G = (V,E) be a simple undirected graph (that is, without loops or parallel edges),
with set of vertices V(G) and set of edgesE(G). We take V(G) ={1,...,n} and assume
that |E(G)| =m. The degree d(i) of a vertex i is the number of edges incident with i.
Theadjacency matrix ofGisdenotedbyA(G)anddefinedby[A(G)]
ij
=1,ifij ∈E(G);
[A(G)]
ij
=0 if ij / ∈E(G).
The adjacency matrix is a useful tool to describe mathematically the interaction
among a network of n spin-half quantum particles. The particles are usually associated
with the vertices of G, while the edges of G represent their allowed couplings. If one
considers the XYZ interaction model (isotropic Heisenberg model, see appendix), then
{i,j} ∈E(G) means that the particles i and j interact by the Hamiltonian [H(G)]
ij
=
(X
i
X
j
+Y
i
Y
j
+Z
i
Z
j
),whereX
i
,Y
i
andZ
i
arethePaulioperatorsσ
i
onthei-thparticle
(here we consider unit coupling constant,see appendix for further details). Thus, the
Hamiltonian of the whole network reads
H
XYZ
(G) =
1
2
null inull=j
[A(G)]
ij
(X
i
X
j
+Y
i
Y
j
+Z
i
Z
j
), (3.4)
and it acts on the Hilbert space K =
null C
2
null ⊗n
. The notation X
i
X
j
means that the
operatorσ
x
hasto beappliedon boththe particles withlabelsi andj respectively while
the Identity operators acts on the remaining particles. More explicitly X
i
X
j
stands for
I ⊗...⊗I ⊗σ
i
x
⊗I ⊗...⊗I ⊗σ
j
x
⊗I ...⊗I
Let us now restrict our attention to the single excitation subspaceH
∼
=C
n
, i.e., the
subspace of dimension n spanned by the vectors {|1null,...,|nnull}. A vector |jnull indicates
the presence of the excitation on the j-th site and the absence on all the others. This
is equivalent to the following tensor product of the Z eigenstates |0...010...0
null nullnull null n
null, being
1 in thej-th position. Following the results given in [38], in the basis{|1null,...,|nnull}, the
Hamiltonian coming from Eq. (3.10) has the following entries
[H
XY
(G)]
ij
=2[A(G)]
ij
, inull =j (3.5)
and
[H
Z
(G)]
ii
=
1
2
null i,j
[A(G)]
ij
−2
null j
[A(G)]
ij
(3.6)
where the subscripts indicate the contributions to the hamiltonian given by each pauli
operators. Eq. (3.5) comes from the term X
i
X
j
+Y
i
Y
j
and Eq. (3.6) comes from the
60 3. Quantum communication in quantum networks
termZ
i
Z
j
. Then it is clear that in theXY model the relation between the Hamiltonian
and adjacency matrix simply reduces to [H
XY
(G)]
ij
= 2[A(G)]
ij
. Here, H
XYZ
(G) =
H
XY
(G)+H
Z
(G). By associating the vertex i∈V(G) to the vector |inull∈{|1null,...,|nnull},
we can introduce the following modified version of the adjacency matrix:
3.1. Definition. The XYZ adjacency matrix of a graph G is denoted by H(G) and
defined by
[H
XYZ
(G)]
ij
=





2, if ij ∈E(G);
0, if ij / ∈E(G);
m−2d(i), if i =j.
The matrix H(G) has a neat relation to the Laplacian of G. Let Δ(G) be an n×n
diagonal matrix such that [Δ(G)]
ii
= d(i). The (combinatorial) Laplacian of G is the
matrix L(G) :=Δ(G)−A(G). Let I
n
be the n×n identity matrix.
3.1. Lemma. The XYZ adjacency matrix of a graph G is H(G) =mI
n
−2L(G).
Proof. By Definition 3.1,H(G) =2A(G)+mI
n
−2Δ(G). By combining this fact with
the definition of Laplacian, we obtain the desired expression.
3.1. Corollary. Given any graph G, the matrices L(G) and H(G) have a common
set of eigenvectors. Moreover, if null is an eigenvalue of L(G) then λ = m− 2null is an
eigenvalue of H(G).
Let {|1null,...,|nnull} be the standard basis of an Hilbert space H
∼
= C
n
. As before, we
associate the vertex i ∈ V(G) to the vector |inull ∈ {|1null,...,|nnull}. Let ι =
√
−1. In this
context, given a graph G, two of its vertices i and j, and a real number 0<t<∞, the
fidelity at timetbetween vertexi andvertexj is definedbyf
G
(i,j,t) :=|nullj|e
−ιH(G)t
|inull|.
We say that two vertices i and j of a graph G admits perfect state transfer (w.r.t. the
XYZadjacencymatrix)ifthereexiststforwhichf
G
(i,j,t) =1. LetU
t
(G) =e
−ιH(G)t
be
theunitarymatrixassociated toH(G)asafunctionoft. Therefore,wehavef
G
(i,j,t) =
1, if|[U
t
(G)]
i,j
| =1.
3.4.2 Perfect State Transfer in all-to-all networks
In this section, we consider a network in which all nodes interact directly. This network
is modeled by the complete graph K
n
on n vertices. In K
n
, we have d(i) = n−1, for
every i∈V(G). The principal result of the section is the following statement:
3.2. Theorem. LetK
n
be the complete graph onn vertices. For every vertexi∈V(K
n
)
and value n∈N, we have the following:
3.4. Graph Theory and Spin networks 61
• min
t
f
Kn
(i,i,t) =1−2/n and t=
π
2n
+
πk
n
, for k≥0;
• max
t
f
Kn
(i,i,t) =1 and t=πk/n, for k≥1.
For every two distinct vertices i,j ∈V(K
n
) and value n∈N, we have the following:
• min
t
f
Kn
(i,j,t) =0 is attained when f
Kn
(i,i,t) is maximum;
• max
t
f
Kn
(i,j,t) =2/n is attained when f
Kn
(i,i,t) is minimum.
Proof. Since d(i) = n−1, for every i ∈ V(K
n
), we have A(K
n
) = J
n
−I
n
, where J
n
is the n×n all-ones matrix. Then, L(K
n
) = (n−1)I
n
−(J
n
−I
n
) =nI
n
−J
n
. Given
|E(K
n
)| =
null n
2
null , it follows that
H(K
n
) =
null n
2
null I
n
−2(nI
n
−J
n
) =
n(n−5)
2
I
n
+2J
n
.
TheeigenvaluesofH(K
n
)arethenλ
[1]
1
=n(n−1)/2andλ
[n−1]
2
([H(K
n
)]
ii
−[H(K
n
)]
ij
) =
n(n−5)/2. Letusdenoteby
− →
J
n
theall-onesvectorofdimensionn. Since[H(K
n
),J
n
]=
0, these two matrices share a common set of eigenvectors. The matrix J
n
is diag-
onalized by F
n
, the Fourier transform over the group Z
n
: [F
n
]
ij
= e
ι2πij/n
≡ ω
ij
.
As a consequence, the single eigenvector corresponding to the eigenvalue λ
1
is of the
form |λ
1
null = n
−1/2
− →
J
n
; the n − 1 eigenvectors corresponding to λ
2
are of the form
|λ
i
2
null:=n
−1/2
null n
j=1
ω
ij
|jnull,foreveryi =1,...,n−1. Now,wecanreadU
t
(K
n
)≡e
−ιH(Kn)t
in its spectral decomposition:
U
t
(K
n
) =n
−1


e
−ιλ
1
t
J
n
+e
−ιλ
2
t
n−1;n
null i;j,k=1
ω
ij
ω
ik
|jnullnullk|


.
Then
[U
t
(K
n
)]
ii
=n
−1
e
−ι(
n
2
)t
+
n−1
n
e
−ι
n(n−5)
2
t
for every i and
[U
t
(K
n
)]
ij
=n
−1
null e
−ι(
n
2
)t
−e
−ι
n(n−5)
2
t
null ifinull =j. Thus, we have 1−2/n≤|[U
t
(K
n
)]
ii
|≤1, for everyi. The minimum is attained
when t =
πk
n
+
π
2n
. The maximum is attained for each t=πk/n. So, 0≤|[U
t
(K
n
)]
ij
|≤
2/n. For these entries, the minimum value is attained for each t such that |[U
t
(K
n
)]
ii
|
is maximum and viz.
The result is that there is no PST in K
n
(but for the trivial cases n null = 1,2). This
is equivalent to say that a qubit can not be accurately transferred between two sites
in a network in which all sites are connected to each other. The fidelity given by the
evolution in K
n
, with n=4,8 is illustrated in the Fig.3.5
62 3. Quantum communication in quantum networks
0.5 1.0 1.5 2.0 2.5 3.0
0.2
0.4
0.6
0.8
1.0
0.5 1.0 1.5 2.0 2.5 3.0
0.2
0.4
0.6
0.8
1.0
Figure 3.5: The fidelity given by the evolution in K
n
, with n = 4,8 is illustrated (left and
right, resp.) for t ∈ [0,π]. The dashed lines represent |[U
t
(K
n
)]
ij
| with i null = j; the solid ones,
|[U
t
(K
n
)]
ii
|.
For t=
πk
n
+
π
2n
, Theorem 3.4 prompts to the following facts. If n is odd then
[U
t
(K
n
)]
ij
=
null δ
ij
−2ι/n, if n=4k−1;
δ
ij
−2/n, if n=4k+1.
If n is even, we need to distinguish two cases:
• n=4k,
[U
t
(K
n
)]
ij
=
null −(δ
ij
−2/n)
√
i, if k even;
(δ
ij
−2/n)
√
i, if k odd;
• n=4k+2,
[U
t
(K
n
)]
ij
=
null −(δ
ij
−2/n)
√
i, if k even;
(δ
ij
−2/n)
√
i, if k odd.
where the overline stands for the complex conjugate of the term. Note that [U
t
(K
n
)]
ij
representsforanyntheGrover operatorwithanoverall phase. Recall thatthisoperator
playsacentralroleintheGroveralgorithmfordatabasesearch[21]. TheunitaryU
t
(K
n
)
induces a continuous-time quantum walk on K
n
. For a review on quantum walks see
[169]
3.4.3 Perfect State Transfer in all-to-ll networks with a missing link
In this section, we consider a network in which all sites but two interact directly. This
network is modeled byK
−
n
, the complete graph minus an edge onn vertices. The graph
K
−
n
is obtained from K
n
by deleting an arbitrary edge.
3.3. Theorem. Let K
−
n
be the complete graph minus an edge on n vertices. For every
vertex i∈V(K
−
n
) and value n=4k, with k∈N, we have the following statements:
If i=1,n then
• min
t
f
K
−
n
(i,i,t) =0 and t=
π
4
+
πk
2
, for k≥0;
3.4. Graph Theory and Spin networks 63
• max
t
f
K
−
n
(i,i,t) =1 and t=πk/2, for k≥1.
If inull =1,n then
• min
t
f
K
−
n
(i,i,t) =1−2/n and t=
π
2n
+
πk
n
, for k≥0;
• max
t
f
K
−
n
(i,i,t) =1 and t=πk/n, for k≥1.
If i=1 and j =n then
• min
t
f
K
−
n
(1,n,t) =0 is attained when f
Kn
(1,1,t) is maximum;
• max
t
f
K
−
n
(1,n,t) =1 is attained when f
Kn
(1,1,t) is minimum.
In all other cases, when inull =j (and inull =1,n)
• min
t
f
K
−
n
(i,j,t) =0 is attained when f
Kn
(i,i,t) is maximum;
• max
t
f
K
−
n
(i,j,t) =2/n is attained when f
Kn
(i,i,t) is minimum.
Proof. Let us define the n×n matrix P
n
such that [P
n
]
1,1
= [P
n
]
n,n
= 1, [P
n
]
1,n
=
[P
n
]
n,1
=−1, and [P
n
]
ij
=0, otherwise. The Laplacian of K
−
n
can be written as
L(K
−
n
) =








n−2 −1 nullnullnull − 1 0
−1 n−1 −1 nullnullnull − 1
.
.
. −1
.
.
. −1
.
.
.
−1 nullnullnull − 1 n−1 −1
0 −1 nullnullnull − 1 n−2








=L(K
n
)−P
n
.
The XYZ adjacency matrix of K
−
n
has then the form
H(K
−
n
)=
nullnull n
2
null −1
null I
n
−2L(K
−
n
)
=
nullnull n
2
null −1
null I
n
−2(nI
n
−J
n
−P
n
).
This matrix elements are of the form [H(K
−
n
)]
1,n
= [H(K
−
n
)]
n,1
= 0, [H(K
−
n
)]
i,i
=
m−2d(i) and[H(K
−
n
)]
i,j
=2. The eigenvalues ofH(K
−
n
) are thenλ
[1]
1
=n(n−1)/2−1,
λ
[1]
2
= [H(K
−
n
)]
1,1
= n(n− 1)/2− 2n + 3 and λ
[n−2]
3
= [H(K
−
n
)]
2,2
− [H(K
−
n
)]
1,2
=
n(n−5)/2−1. We can chose an orthonormal basis of eigenvectors such that, from the
spectraldecompositionoftheunitarymatrixU
t
(K
−
n
)≡e
−ιH(K
−
n
)t
, wehavethefollowing
diagonal entries:
64 3. Quantum communication in quantum networks
[U
t
(K
−
n
)]
ii
=







n
−1
null n
2
e
−ι[(
n
2
)−2n+3]t
+e
−ι[(
n
2
)−1]t
+(
n
2
−1)e
−ι
null n(n−5)
2
−1
null t
null , if i=1,n;
n
−1
null e
−ι[(
n
2
)−1]t
+(n−1)e
−ι
null n(n−5)
2
−1
null t
null , otherwise.
Let us first consider the minimum value of f
K
−
n
(i,i,t). In general, we would need
to distinguish two cases depending on the parity of n. Here we take n to be a multiple
of 4. This implies that n is always even. A simple calculation shows the next facts.
It follows that min
t
f
K
−
n
(1,1,t) = 0 and max
t
f
K
−
n
(1,1,t) = 1 if t =
π
4
+
πk
2
(k ≥ 0),
and t = πk/2 (k ≥ 1), respectively. The same holds for i = n. If i null = 1,n, it follows
that min
t
f
K
−
n
(i,i,t) = 1−2/n and max
t
f
K
−
n
(i,i,t) = 1 if t =
π
2n
+
πk
n
and t = πk/n,
respectively.
The off-diagonal entries of U
t
(K
−
n
) are
[U
t
(K
−
n
)]
ij
=













n
−1
null e
−ι[(
n
2
)−1]t
−
n
2
e
−ι[(
n
2
)−2n+3]t
+(
n
2
−1)e
−ι
null n(n−5)
2
−1
null t
null ,
if i=1 and j =n;
n
−1
null e
−ι[(
n
2
)−1]t
−e
−ι[
n(n−5)
2
−1]t
null ,
otherwise.
Given this equation, for i = 1 and j = n, we have min
t
f
K
−
n
(1,n,t) = 0 when
t=πk/2 and max
t
f
K
−
n
(1,n,t) =1 whent=
π
4
+
πk
2
, i.e.,f
K
−
n
(1,n,t) is minimum when
f
K
−
n
(1,1,t) is maximum and viz. The last case to describe is i null = j and i null = 1,j null = n:
min
t
f
K
−
n
(i,j,t) = 0 and max
t
f
K
−
n
(i,j,t) = 2/n if t = πk/n (k ≥ 0) and t =
π
2n
+
πk
n
(k≥1), respectively.
ThefidelitygivenbytheevolutioninK
−
n
,withn=4,8,16,40 isillustratedinFig.3.6
When n is a multiple of 4, the unitary U
t
(K
−
n
) can be used to route a qubit over n
sites. To put in communication site i and site j, we need to let the system evolve for a
time t=
π
4
+
πk
2
, once we have deleted fromK
n
exactly the edge ij.
The setting described above can be generalized by deleting more than a single edge
fromK
n
. In fact, as far as we delete edges without common vertices (or, in other words,
vertex disjoint edges), we still can observe that PST between the end points of deleted
edges, whenever the number of vertices in the graph is a multiple of 4. The maximum
numberofedgesthatcanbedeletedisthenn/2. Inthiscase, thedeletedsetcorresponds
to a perfect matching, that is, a set of vertex disjoint edges including all the vertices of
the graph. As it is for K
−
n
, PST occurs when t =
π
4
+
πk
2
. Figure 3.8 describes an
example.