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Trabajo de Fin de Grado en Fı́sica
Quantum Teleportation
and
Quantum Cryptography
Urtzi Las Heras
Director:
Prof. Enrique Solano
Departamento de Quı́mica Fı́sica
Facultad de Ciencia y Tecnologı́a
Universidad del Paı́s Vasco UPV/EHU
Leioa, Junio del 2012
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Acknowledgments
En el momento en que el Prof. Enrique Solano, mi director de tesis, me dijo que debı́a introducir
una sección de agradecimientos en el trabajo, sólo puedo decir que entré en pánico. No obstante,
las caras de las personas que me han acompañado en esta aventura han ido apareciendo por mi
mente y no puedo hacer menos que plasmar en papel lo mucho que agradezco su apoyo.
En primer lugar, mi más sincero agradecimiento al Prof. Enrique Solano por revivir mi curiosidad y entusiasmo por la fı́sica que tanto han mermado a causa de los agotadores exámenes.
Gracias por el esfuerzo y dedicación en este trabajo y por haberme dado la oportunidad de
unirme a este estupendo grupo de investigación en el que no he podido sentirme más integrado.
Al Dr. Lucas Lamata, que sin su inestimable colaboración este trabajo no serı́a ni sombra
de lo que es, gracias por su infinita paciencia y por cada una de sus enseñanzas.
Del mismo modo, gracias a todos los compañeros; Laura, Unai, Julen, Antonio, Roberto,
Simone, Jorge, Daniel y Guillermo por su predisposición a ayudarme tantas y tantas veces.
A mis compañeros de Jiu Jitsu que con tantas ganas han entrenado conmigo haciéndome
olvidar las tensiones del dı́a a dı́a.
A mis amigos Yelco y Eneko, por haberme sacado más sonrisas que nadie, y en especial a
Lara por haber sido uno de los pilares en los que apoyarme cada vez que la he necesitado y que
me ha brindado más cariño que nadie.
Gracias a mis padres y a mi hermano, que son verdaderamente los que han lidiado con mis
enfados cada vez que las cuentas no me han salido y aun ası́ siempre han tenido palabras de
ánimo para mı́.
Finalmente, gracias también a mi abuela, que nunca ha dejado de decirme que estudie y
que el esfuerzo trae su recompensa.
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4
Contents
Contents
5
1 Introduction
7
2 Theory of quantum teleportation
9
3 Experimental aspects of quantum teleportation
15
3.1
The Innsbruck photon experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2
The NIST trapped-ion experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Theory of quantum cryptography
23
4.1
BB84 protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2
E91 protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Experimental aspects of quantum cryptography
31
6 Quantum Hacking
33
7 Conclusions
35
References
37
5
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6
1
Introduction
Quantum mechanics was discovered and formalized during the last century. So far, the predictions of quantum theory have led to a whole range of prospective applications inconceivable
until last decade that even today have not yet been developed to its full potentially. Combining quantum properties such as linear superposition and entanglement with information theory,
would allow one to realize enhanced computation and communication protocols unfeasible with
classical means. In this work, we review two of them, namely, quantum teleportation and
quantum cryptography.
Quantum teleportation is based on the transfer of a quantum state from one point to another
while destroying the original state. To accomplish it, one needs a quantum channel, which
consists of two entangled particles, and a classical one. We review Bennett et al. theoretical
proposal and we include an original calculation considering a non-perfect quantum channel.
The experiments made in Innsbruck and at NIST, making use of photons and trapped ions
respectively, are shown subsequently. For this, we introduce the basic physics of photons and
trapped ions.
Quantum cryptography consists in the study of secure communication making use of quantum properties. In this section we show how the classical cryptography is less secure than the
quantum one, and we review two of the best known quantum-cryptography protocols, namely,
BB84 and E91. In both protocols we study the case of an attack from an eavesdropper and
its consequences. We also add an original calculation in E91 protocol assuming an imperfect
communication between the legitimate users. Experimental aspects of quantum cryptography
are analyzed reviewing two of the experiments made in Innsbruck and in Los Alamos. In both
cases, they were accomplished using polarization-entangled photons.
Furthermore, we comment on the basic ideas of quantum hacking, that means obtaining
information from a message encoded with quantum cryptography techniques. The experiment
made in Singapore is reviewed. Here, the eavesdropper obtained the whole information without
being detected.
Finally, we present our conclusions.
7
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8
2
Theory of quantum teleportation
Quantum teleportation [1] consists in the transfer of a quantum state | i from one point to
another while destroying the original state [2] . This process takes place in two di↵erent
locations. Alice has the original state which will be teleported with the help of an entangled
state to Bob. Moreover, Alice and Bob need a classical channel for sending classical information.
Both features are essential to make quantum teleportation with 100% fidelity.
Alice and Bob share a maximally entangled state, known as an Einstein-Podolsky-Rosen
(EPR) [3] pair. Alice keeps one of the EPR particles with her, and Bob makes the same with
the other one. The state of the EPR pair is:
|
( )
23 i
1
= p (| "2 i| #3 i
2
| #2 i| "3 i).
(1)
The particles composing the pair can be of any kind, photons, atoms, e.g.. The only
requirement is to have two degrees of freedom, able to encode a quantum bit (qubit). Alice’s
original particle state is unknown,
| 1 i = a| "1 i + b| #1 i.
(2)
Alice makes a Bell measurement of the particles 1 and 2. The complete state of three
particles before Alice’s measurement is:
|
123 i
= | 1i ⌦ |
( )
23 i
=
a
p (| "1 i| "2 i| #3 i | "1 i| #2 i| "3 i) +
2
b
p (| #1 i| "2 i| #3 i | #1 i| #2 i| "3 i).
2
(3)
Using the Bell basis [4] of particles 1 and 2,
1
= p (| "1 i| #2 i ± | #1 i| "2 i)
2
1
(±)
| 12 i = p (| "1 i| "2 i ± | #1 i| #2 i)
2
(±)
12 i
|
(4)
(5)
we obtain
|
123 i
1
= [|
2
( )
12 i(
+|
a| "3 i
( )
12 i(a|
b| #3 i) + |
#3 i + b| "3 i) + |
(+)
12 i(
a| "3 i + b| #3 i)
(+)
12 i(a|
#3 i
b| "3 i)].
(6)
Once the Bell measurement is made by Alice, particle 3 is projected onto a pure state. As
there are four possible Bell states, particle 3 can be projected in four states. According to the
9
result of the measurement, these states will be:
a| "3 i b| #3 i ⌘ | 3 i,
a| "3 i + b| #3 i,
a| #3 i + b| "3 i,
a| #3 i b| "3 i,
where the Bell states measured are |
also be written as follows:
( )
12 i,
|
(+)
12 i,
|
z|
x|
i y|
where
x,
y
and
z
|
( )
12 i,
|
(7)
(+)
12 i,
respectively. These states can
3 i,
3 i,
3 i,
3 i,
(8)
are the Pauli unitary operators.
Hence, Bob only has to apply the corresponding unitary operation to the state of particle
3 to obtain the original state. For this, Alice must send through a classical channel the result
of the Bell measurement. The time of sending classical information is higher than the time it
takes light to travel from Alice to Bob, so there is no causality violation.
Alice
quantum channel
|
Bob
i
classical channel
Figure 1: Scheme of teleportation protocol.
It is possible to question whether the state has been teleported despite not having sent
information about Alice’s measurement, that is, whether there is superluminal information
sending. Although Bob has just four possible states, there is no way to find the original state
10
in case Alice does not send the result of the Bell measurement. Tracing out the Bell states, the
density matrix of the particle 3 results
X
X
h'| 123 ih 123 |'i =
h'|⇢123 |'i
|'i
( )
12 |⇢123 |
✓
2
( )
12 i +
◆
⇤
=h
1
|a| ab
=
a⇤ b |b|2
4
✓
◆
1/2 0
=
.
0 1/2
|'i
h
+
(+)
✓12
(+)
( )
( )
(+)
(+)
|⇢123 | 12 i + h 12 |⇢123 | 12 i + h 12 |⇢123 | 12 i
◆ ✓ 2 ⇤ ◆ ✓
◆
|a|2
ab⇤
|b| a b
|b|2
a⇤ b
+
+
a⇤ b |b|2
ab⇤ |a|2
ab⇤ |a|2
(9)
As can be seen, Bob obtains a density matrix of the third particle completely depolarized. This implies that the information sent by Alice classically is essential to e↵ectuate the
teleportation of the original state, giving to the experiment a fidelity of 100 %.
It is important to note that in this process the initial state is destroyed, thus fulfilling the
no-cloning theorem. This ensures that there is no procedure by which an unknown quantum
state can be copied from one system to another. For this reason, the procedure is called
”teleportation” and not ”copy” of quantum states.
Now, we discuss the possibility of teleporting a state through a quantum channel which is
composed of a partially entangled state, with an original calculation. In this problem, we want
to see how large is the fidelity using a Werner state [5]. So, particles 2 and 3 are in the mixed
state
⇢23 = p|
( )
23 ih
( )
23 |
+
1
p
3
[|
(+)
23 ih
(+)
23 |
+|
( )
23 ih
( )
23 |
+|
(+)
23 ih
(+)
23 |],
(10)
where p goes from 0 to 1. Using the basis {"1 , #1 } and {"2 "3 , "2 #3 , #2 "3 , #2 #3 } respectively, the
density matrices of particles 1, ⇢1 and particles 2 and 3, ⇢23 , result
0
1
2(1 p)
0
0
0
✓ 2
◆
C
1B
|a| ab⇤
0
1 + 2p 1 4p
0
C,
⇢1 =
, ⇢23 = B
(11)
⇤
2
A
a b |b|
0
1 4p 1 + 2p
0
6@
0
0
0
2(1 p)
performing the tensorial product of these two matrices we obtain the total state that can be
written
0 2(1 p)|a|2
1
0
0
0
2(1 p)ab⇤
0
0
0
⇢123
B
1B
= B
6B
@
0
0
0
2(1 p)a⇤ b
0
0
0
(1 + 2p)|a|2
(1 4p)|a|2
0
0
(1 + 2p)a⇤ b
(1 4p)a⇤ b
0
(1 4p)|a|2
(1 + 2p)|a|2
0
0
(1 4p)a⇤ b
(1 + 2p)a⇤ b
0
0
0
2(1 p)|a|2
0
0
0
2(1 p)a⇤ b
0
0
0
2(1 p)|b|2
0
0
0
(1 + 2p)ab⇤
(1 4p)ab⇤
0
0
(1 + 2p)|b|2
(1 4p)|b|2
0
(1 4p)ab⇤
(1 + 2p)ab⇤
0
0
(1 4p)|b|2
(1 + 2p)|b|2
0
whose basis is {"1 "2 "3 , "1 "2 #3 , "1 #2 "3 , "1 #2 #3 , #1 "2 "3 , #1 "2 #3 , #1 #2 "3 , #1 #2 #3 }.
11
0
0
2(1 p)ab⇤
0
0
0
2(1 p)|b|2
C
C
C,(12)
C
A
From this density matrix we can calculate which state results for the particle 3 in case Alice
obtains anyone of the Bell states in her measurement. These matrices are
✓
◆
1 (1 + 2p)|a|2 + 2(1 p)|b|2
(1 4p)ab⇤
⌘ ⇢3 ,
(1 4p)a⇤ b
2(1 p)|a|2 + (1 + 2p)|b|2
3
✓
◆
1 (1 + 2p)|a|2 + 2(1 p)|b|2
(1 4p)ab⇤
,
(1 4p)a⇤ b
2(1 p)|a|2 + (1 + 2p)|b|2
3
✓
◆
1 2(1 p)|a|2 + (1 + 2p)|b|2
(1 4p)a⇤ b
,
(1 4p)ab⇤
(1 + 2p)|a|2 + 2(1 p)|b|2
3
✓
◆
1 2(1 p)|a|2 + (1 + 2p)|b|2
(1 4p)a⇤ b
,
(1 4p)ab⇤
(1 + 2p)|a|2 + 2(1 p)|b|2
3
(13)
( )
(+)
( )
(+)
where the Bell states measured are | 12 i, | 12 i, | 12 i, | 12 i respectively. Note that these
states can also be written applying the Pauli unitary operator in this way:
⇢3
†
3
⇢3 3 ,
( 1 )† ⇢3 ( 1 ),
(i 2 )† ⇢3 (i 2 ).
(14)
To obtain the fidelity of this teleportation protocol as a function of p, we trace onto the
product of the ideal state and the obtained state:
Tr(⇢1 · ⇢3 )
= Tr
=
✓✓
|a|2 ab⇤
a⇤ b |b|2
1 + 2p
.
3
◆
1
·
3
✓
(1 + 2p)|a|2 + 2(1 p)|b|2
(1 4p)a⇤ b
2(1
(1 4p)ab⇤
p)|a|2 + (1 + 2p)|b|2
◆◆
(15)
As can be seen, the fidelity is of 100% when p goes to 1, that is, when the quantum-channel
particles are on the EPR state.
Let us see for which p the mixed state ⇢23 is a classical state or it has a entangled character.
For this, we make the partial transposition, that is, transposing only one of the qubits. Then
we obtain the eigenvalues and we study them. If the eigenvalues are all positive, the state is
classical but if at least one eigenvalue is negative, it means that the state is entangled [6].
(⇢23 )pt =
0
1B
@
6
2(1
p)
0
0
0
1pt
0
0
0
1 + 2p 1 4p
0
C
A
1 4p 1 + 2p
0
0
0
2(1 p)
12
=
0
1B
@
6
2(1
p)
0
0
1
4p
1
0
0
1 4p
1 + 2p
0
0
C
A.
0
1 + 2p
0
0
0
2(1 p)
(16)
The eigenvalues of this matrix are 1 = 2 = 1 3 p , 3 = 1+2p
and 4 = 1 22p . Only the last
6
can be negative in case p > 12 and it becomes zero when p = 12 . As can be seen in figure 2, for
that value of p the fidelity is about 66%, which coincides with the classical limit for the fidelity
of the teleportation protocol.
Figure 2: Plot of the fidelity as a function of p.
13
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14
3
Experimental aspects of quantum teleportation
In this section, we review some of the most relevant experiments in the field of quantum
teleportation. Zeilinger [7] and De Martini [8] made their own experiments at the same time
using polarized photons. De Martini et al. achieved the teleportation but the initial state was
taken from the pair of particles that compose the quantum channel. Although he succeded in
teleporting this state in a 100 % of the times, this protocol is not entirely faithful to the original
of Bennett et al., where the initial state was a qubit independent of the quantum channel. The
experiment of Zeilinger et al. is showed in detail below. We also review experiment of Barrett
et al. [9], who made use of trapped ions as did Riebe et al. [10].
In following we will show which techniques are used to reduce a physical system to two
degrees of freedom, generate entangled states, make the Bell measurements and the quantum
state reconstruction with the help of the information sent by Alice.
3.1
The Innsbruck photon experiment
This experiment was made by Zeilinger’s group using the polarization states of the photons
as the degrees of freedom. Using the notation of Section 2, the state | "i corresponds to the
vertical polarization and equivalently | #i to the horizontal polarization. Therefore, the original
state of photon 1 is a linear combination of vertical and horizontal polarization.
The EPR state and the photon 1 are produced by type II parametric down-conversion(PDC).
With this technique, inside a nonlinear crystal, an incoming photon decays spontaneously in
two entangled photons that compose the quantum information channel.
In the experiment a UV radiation pulse is pumped through a nonlinear crystal, emitting
the EPR state. The rest of the pulse goes through the crystal and reflects in a mobile mirror to
come into the crystal again emitting photon 1 and a control photon. Photon 1 is polarized as
desired, because this is the state which will be teleported, and then is combined with photon
2, one of the entangled photons joined in the first PDC process. In this way, photons 1 and 2
are together to make the Bell measurement.
The device which makes the Bell measurement is called beam-splitter. It consists on a semi
reflective mirror. In ideal conditions the probability that a photon crosses the mirror is of 50%
and being reflected is the other 50%.
In case two photons arrive to the beam splitter at the same time, one on each side, they
emerge in both sides only if both are reflected or transmitted. However, this process is only
possible for certain states of the photons 1 and 2.
The probability amplitude of this process is given by the coherent addition of the amplitudes
of reflection or transmission of both particles. It is important to add a minus sign to the wave
function due to the phase shift gained in reflections. That way, if the photons are in a state
such as | "1 "2 i the amplitude of finding the photons on both sides is zero.
On the other hand, the Bell state |
( )
12 i,
being antisymmetric, has an additional minus sign,
15
so the interference is constructive and the probability of finding a photon in each side of beam
splitter is di↵erent from zero.
( )
Consequently, to know that the particles 1 and 2 are in the Bell state | 12 i, placing two
detectors on both sides of beam splitter, the experiment has to be repeated until there is a
coincidence.
Photons 1 and 2 must arrive at the same time to the beam splitter because they must
interfere. For this, some techniques must be applied to make the arrival times indistinguishable.
In this experiment, beam pulses that pass through low bandwidth filters are used. Therefore,
coherence time increases until 520 fs, much longer than the length of the pulse, of 200 fs with a
frequency of 76 MHz. Furthermore, given that during the creation of photon 1 another photon
is also created, it can be used to know whether photon 1 is emitted.
To check that the teleportation happens for any unknown state, the authors used linearly
polarized states at 45 and 45 for photon 1, that form a rotated base with regard to the
base of the states polarized vertically and horizontally, that is, the base of photons 2 and 3.
Next, the teleportation of a linear superposition of these states was checked, equivalently for
circularly polarized.
In the first case, the photon 1 was polarized at 45 . When the detectors f1 and f2 ,located
behind the beam splitter where photons 1 and 2 interfere, detect a coincidence it means that
( )
the state of photons 1 and 2 is projected onto the state | 12 i, making the Bell measurement
in this way. Consequently, the photon 3 must be in the original state, polarized at 45 .
Figure 3: Graphic scheme of the experiment. Taken from [7].
16
The polarization of photon 3 is analyzed by a beam splitter that separates the polarizations
of 45 and 45 . These detectors are named d2 and d1 respectively. When the detectors f1
and f2 coincide, d2 always detects a photon. Moreover, d1 never coincides with f1 and f2, so
the photon 3 in all the experiments gets the state of polarization of photon 1, at 45 .
To get the highest accuracy in the interference of photons 1 and 2, the delay between the
first and second down conversion emissions was modified moving the mirror where the beam
reflects. In this way, modifying the delay continuously, it can be seen in detail in which time
overlap does the teleportation happen.
Making the experiment with the optimal delay, it was observed that the coincidences of
detectors f1f2d1 ( 45 ) had a dip. Furthermore, there was no dip for the coincidences of f1f2d2
(45 ), so it confirms the teleportation of the original state. In this analysis the possible spurious
matches were considered.
The experiment was repeated changing the polarization of photon 1 to linearly polarized
at 45 , 0 and 90 and also circularly polarized, obtaining similar results. Below, the results
are shown in a list that expresses the visibility of the dip in triple coincidences in detection of
orthogonal states to the polarization of photon 1.
Polarization
+45
45
0
90
Circular
Visibility
0.63 ± 0.02
0.64 ± 0.02
0.66 ± 0.02
0.61 ± 0.02
0.57 ± 0.02
(17)
Hence, this experiment checks the teleportation in a basis of states vertical and horizontal,
being the state to teleport a superposition of states of the base. To make the Bell measurement,
( )
the states of photons 1 and 2 have always been projected to the state | 12 i. Oppositely, as
the experiment consists in the search of coincidences of three detectors, there is no possibility
to find a coincidence for another Bell state, because f1 and f2 detectors only coincide when the
state of photons 1 and 2 is antisymmetric. For this reason, the experiment only works in a 25%
of times. In case it would be possible to make the Bell measurement for any state, and the
corresponding unitary operation could also be applied onto the state of particle 3, the copy of
the state with maximum probability would be achieved in a 100% of times.
In 2012, Ma et al. [11] succeeded in teleporting a qubit between two Canary Islands using
photons. The distance traveled is 143 km and the average fidelity achieved is f = 0.863(38).
This procedure opens a window into the long-distance teleportation where the next goal is the
teleportation of qubits sending photons between a satellite and ground.
17
3.2
The NIST trapped-ion experiment
This experiment was made at NIST, in Boulder (Colorado), in collaboration with the University
of Otago, New Zealand. In contrast with the previous experiment, this was made with massive
9
Figure
10.1:
Penning
trap,
cyclotron
and
motion
of them
the
particles.
TheyThe
confined
Be+ ions
in awith
segmented
ion trap
[12,magentron
13] where they
could use
as qubits
with a total
control.
ion
illustrated
below.
.
The traps used by the authors are Paul traps. They provide 2D confinement using highfrequency electric fields to simulate an electric field minimum, using four axial electrodes. These
electrodes are connected ,as shown in the next figure, to a variable potential that repels the ion
when it is near the center of trap and it becomes attractive when it moves away. For confining in
the axial direction positively charged endcaps are added. In this way, ions are totally confined
in three dimensions.
a
b
J. Appl. Phys., Vol. 83, No. 10, 15 May 1998
Figure 4: Side(a) and axial(b) view of the Paul trap. Taken from [13].
The control electrodes of the trap are segmented into eight sections, providing six trapping
zones, centered on electrode Figure
segments 10.2:
2 to 7 as
shown
in the
next figure. Potentials applied to
The
Paul
trap.
these electrodes can be varied in time to separate ions and move them to di↵erent segments of
.
the trap.
It was necessary to reduce all the possible states of the ions to only two, ground and excited
states. These states necessarily had to be metastable states because, in other case, the excited
would decay to ground state spontaneously. This happens for dipole allowed transitions, so
a good choice would be two states separated with a quadruple transition, whose lifetimes are
around a second. Other possibility would be210
two hyperfine ground states.
In this experiment, authors used qubits composed of the ground state hyperfine levels | "i ⌘
|F = 1, m = 1i and | #i ⌘ |F = 2, m = 2i which are separated by a frequency !0 ⇡ 2⇡⇥1.25
GHz.
To produce entanglement between ions, they are confined in the same segment of trap, that
18
making appropriate simplifications acts as a quantum harmonic oscillator. In this way, the
movement of the particles can be decomposed into normal modes of vibration and it is easy to
see how the movement of one a↵ects the others.
To describe the amount of movement of ions so-called Fock levels are used. Such levels are
given by the states |ni with an energy E = h̄!t ( 12 + n) where !t is the frequency of the trap.
Equivalently to what happens with the electromagnetic field in a cavity, Fock states describe
the eigenstates of the vibrational Hamiltonian.
Thus, there are two electronic states and n vibrational states, where n goes from 0 to 1.
So any state can be described in this way:
| i = A| #i ⌦
1
X
n=0
Cn |ni + B| "i ⌦
1
X
n=0
Dn |ni
(18)
It is necessary to use a laser beam to control the state of particles. Applying the rotating
wave approximation that assumes the laser detuning and Rabi frequency are much smaller than
optical frequencies, the Hamiltonian of the system becomes:
H = h̄⌦{
+e
i( t ')
exp(i⌘[ae
i!t
+ a† ei!t ])} + H.C.
(19)
where + is the atomic raising operator, a† and a denote the creation and annihilation operator
for a motional quantum, respectively. ⌦ characterizes the strength of the laser field in terms
of the so-called Rabi frequency, ' denotes the phase of the field with respect to the atomic
polarization and
is the laser-atom detuning. !t denotes the trap frequency, ⌘ = kz z0 is the
Lamb-Dicke parameter
p with kz being the projection of the laser fields wavevector along the z
direction and z0 = h̄/(2m!t ) is the spatial extension of the ion ground state wave function
in the harmonic oscillator being m the ion mass.
p
Using then the Lamb-Dicke approximation, ⌘ ((a + a† )2 )i ⌧ 1, that is valid for cold ion
strings, the Hamiltonian can be rewritten as follows:
H = h̄⌦{
+e
i( t ')
+
ei(
t ')
+ i⌘(
+e
i( t ')
i( t ')
)(ae i!t t
e
+ a† ei!t t )}
(20)
There are three cases of particular interest:
= 0 and
= ±!t . The first describes
the carrier transition, thus only electronic states | #i and | "iare changed. However, when
= +!t simultaneously to exciting the electronic state of ion, a motional quantum, that
is, a phonon is created. This is named blue sideband transition.
p The Rabi frequency in the
transition of two vibrational levels n and n + 1 is ⌦n,n+1 = n + 1⌘⌦, which describes the
floppy frequency between the states | #, ni and | ", n + 1i. Finally, in case
= !t a red
sideband transition occurs. As in the previous transition, the motional state
p changes but now
the phonon is annihilated. The Rabi frequency is at this time ⌦n,n 1 = n⌘⌦. Applying this
to the laser, a state | #, ni becomes | ", n 1i.
Using these transitions the states of ions can be fully manipulated. As in this experiment
the electronic levels are hyperfine ground states such that the so-called Raman transitions are
19
used. They consist in the excitement to a virtual electronic level, and then the decay to the
first-wanted level.
By means of two laser beams, single qubit rotations are implemented
R(✓, ) = cos(✓/2)I + i sin(✓/2) cos( )
x
+ i sin(✓/2)
y
(21)
where I is the identity operator and x , y and z denote the Pauli matrices in the base
{| "i, | #i}. ✓ is proportional to the duration of the Raman pulse and is the relative phase
between the Raman beams at the position of the ion. Raman beams are also used to generate
entanglement between two ions by implementing the phase gate
a| ""i + b| "#i + c| #"i + d| ##i ! a| ""i
ib| "#i
ic| #"i + d| ##i.
(22)
Raman beams are also used to generate spin-echo pulses (R(⇡, SE )), which are applied in
the sixth trapping zone. These pulses are necessary to prevent dephasing due to variations in
the magnetic field. The duration between spin-echo pulses is lower than the timescale of such
variations. Accordingly, with an appropriate choice of SE , the dephasing can be compensated
if it is caused by a static magnetic field gradient. As spin-echo pulses do not fundamentally
a↵ect the teleportation, their e↵ects are omitted in the next discussion.
To begin the experiment, the initial state | 23 i ⌦ | #1 i has to be prepared. First, the
system is initialized to | #1 #2 #3 i by optical pumping, that is cooling the system until it is in
the minimum energy level. Applying the phase gate shown in (22) combined with rotations
to the ions 2 and 3, the state (| #2 #3 i i| "2 "3 i) is achieved. Then applying some individual
rotations to this state the singlet state is finally obtained. First, using a ⇡/2 pulse it becomes
(| "2 #3 i + | #2 "3 i) and a ⇡ pulse, with a ⇡/2 phase di↵erence yields the state (| "2 #3 i | #2 "3 i).
It should be noted that the normalization factors have been removed to simplify the notation.
After obtaining the singlet state, | 23 i, given that is invariant under global rotations
R(✓, )123 upon all three ions, the ion 1 rotates while ions 2 and 3 are not a↵ected. Choosing
the correct ✓ and , this global rotation allows to produce the state | 1 i = a| "1 i + b| #1 i for
any a and b.
When the system is ready to begin the teleportation, the Bell measurement should be
performed. It is needed more than one step to project the state of ions 1 and 2 onto one of the
four possible Bell states. First, three ions come into the trap 6 and then separated, with ions 1
and 2 going to trap 5 and ion 3 to trap 7. In trap 5, a phase gate (22) followed by a ⇡/2 pulse
R(⇡/2, 0) is applied to ions 1 and 2.
In the separation process, normally it was shown a significant amount of motional-mode
heating, and it was achieved with 95% probability. However, the authors of this experiment
used a smaller separation electrode in the current trap, separating the ions with no detectable
failure rate. Furthermore, the heating was extremely reduced so far the stretch mode of the
two ions in trap 5 is in number of about 1. This allows to implement the phase gate (22)
between ions 1 and 2 with fidelity greater than 90%, and more importantly, with no necessity
of sympathetic recooling. Ideally, and considering the result of spin-echo pulses insignificant,
20
this leaves the ions in the state
| "1 "2 i ⌦ R(⇡/2, ⇡/2) x | 3 i + | "1 #2 i ⌦ R(⇡/2, ⇡/2)I| 3 i
+| #1 "2 i ⌦ R(⇡/2, ⇡/2) y | 3 i | #1 #2 i ⌦ R(⇡/2, ⇡/2) z | 3 i
(23)
where | 3 i = a| "3 i + b| #3 i.
To complete the Bell measurement it is necessary to detect the ions one by one. It is
important to note that in this equation ions are not in the Bell state basis. Given that the
measurement of ions is individual, the basis required is the decoupled one. Therefore, three
ions are recombined in the trap 6 and then separated again, moving the ion 2 to trap 5 and
the ions 1 and 3 to trap 7. Detection of the state of ion 2 is achieved through state-dependent
resonance fluorescence measurements. The state | #2 i fluoresces strongly whereas | "2 i does
not. Once the measure is made, the ion 2 is optically pumped back to the state | #2 i.
After this, all three ions are recombined in trap 6 and separated again. In this case, ions 1
and 2 are transferred to trap 5 and ion 3 returns to trap 7. As the last spin-echo pulse applied
in trap 6 changes the state of ion 2 to | "2 i, a subsequent simultaneous detection of both ions
in trap 5 determines the state of the ion1 with a failure rate less than 1% due to presence of
ion 2.
Figure 5: Schematic representation of the teleportation protocol. Taken from [14, 9].
At this point the Bell measurement is made, and classical information has been extracted.
To complete the teleportation, one only has to apply the unitary operation that reconstructs
21
the state of the ion 1 in the ion 3. First, ions 1 and 2 are transported to trap 2 while ion
3 goes to trap 5. Here a ⇡/2 pulse (R(⇡/2, ⇡/2)) is applied followed by the corresponding
unitary operation x , I, y , z for the measurement outcomes | "1 "2 i, | "1 #2 i, | #1 "2 i, | #1 #2 i
respectively. Although the spin-echo pulses do not a↵ect the teleportation protocol, one has
to take care of the phase shift introduced. In case SE = ⇡/2, the unitary operations must be
reordered after the ⇡/2 pulse to y , z , x , I respectively.
The teleportation protocol was tested using six di↵erent states for the ion 1. Concretely,
they used the eigenstates of Pauli operators z , x and y . Assuming that the particle is in the
state | #i in z direction by default, it is easy to apply a Raman beam to transform the original
state. To obtain | "i it is just needed to excite the ion. The fidelity of the experiment was of
80 %, 78 ± 3% for | "i and 84 ± 2% for | #i.
Di↵erently, to accomplish the teleportation with the eigenstates of x and y , | ± Xi and
| ± Y i respectively,and to get them from | #i, it is needed a ⇡/2 pulse with a relative phase of
0 (R(⇡/2, 0)) for | ± Xi and ⇡/2 (R(⇡/2, ⇡/2)) for | ± Y i. To measure the final state of ion
3 once the teleportation protocol has ended, it only has to do the opposite unitary operation
to pass from the states | ± Xi and | ± Y i to | #i. This transformation is needed because the
fluorescence measurement is made in the basis {| #i, | "i}.
The average fidelity achieved was of 78 ± 2%. The authors studied the causes that limited
the fidelity and found three significant mechanisms; imperfect preparation of the initial state
| 23 i ⌦ | #1 i, imperfections in the second phase gate due to heating during the separation
process, and dephasing of the teleported state due to variations of magnetic fields. Studying
these issues in independent experiments, it is observed a loss of 8 ± 3% in the fidelity of the
final state which is consistent with the results obtained in the complete experiment of the
teleportation. Although the fidelity is not of a 100% it exceeds the 66%, which is essential to
beat in order to ensure the presence of entanglement.
22
4
Theory of quantum cryptography
Cryptography [15, 16] has been one of the most important aspects of the theory of information
since 1949. Shannon [17] proved that there exist unbreakable codes or perfectly secret systems.
In fact, this field is responsible for the study of algorithms, protocols and systems used to
protect information and providing secure communications between the communicating entities.
There are di↵erent ways of encrypting a message. The most famous protocols are the
One-time pad and the Public-key cryptographic system. In addition, we review the RivestShamir-Adleman system [18], which is used nowadays to protect electronic bank accounts.
The one-time pad consists in ciphering a message, writing it as a series of bits. Then a key
randomly chosen is combined with the plain text, adding one to one each bit. Given that the
sum of a number with a random number is also a random number, the cyphered text can only
be decrypted by someone who knows the key. If the key is only used once, it is impossible for
the eavesdropper to obtain correlations. However, it is necessary to use a totally random key
with a greater length than the text, because if the key is used cyclically it is possible to extract
information from the encoded text.
In the public-key cryptographic system two keys are involved. There is a public key which
can be used by any sender and the person who receives the coded message has a secret private
key, the inverse of the public one, which decodes the encrypted message. This system is based
on trapdoor one-way functions, that are computationally tractable functions whose inverse
functions can not be solved within a reasonable time. In this way, any sender can encrypt a
message but only the receiver can decode it due to the fact that he already knows the private
key, that is, the inverse function which decrypts the message.
Rivest, Shamir and Adleman created in 1977 the RSA method implementing the public-key
cryptographic system. This method is based on the difficulty of factoring large integer numbers.
Using a computer, it is needed one second to factorize a number of order 1012 , at least a year
for a number of order 1020 and it would be needed more than the age of universe to factorize a
number of 60 digits.
The idea of using quantum properties to obtain secure methods for coding was suggested
in 1969 by Stephen Wiesner [20]. He defended that quantum cryptography would rely for the
first time on laws of physics and not on mathematical conjectures. Since then, quantum key
distribution protocols (QKD) have been gaining importance and nowadays there are companies
fully dedicated to the study and sale of this kind of technology.
QKD methods consist in creating a key which can be known only by Alice and Bob and
then use it to apply the one-time pad protocol. It can be demonstrated that this procedure
is much safer than any public-key cryptographic system. For example, in case we could use a
quantum computer, applying the Shor’s algorithm [19], any large integer could be factorized in
a short enough time. Consequently, the RSA system would not be valid anymore for encoding
messages.
23
Below, we review the BB84 [21] and E91 [22] protocols, based on the use of quantum
properties of linear superposition and entanglement.
4.1
BB84 protocol
Also known as four-state scheme, this was the first protocol devised in quantum cryptography.
By this method, Alice and Bob get a private key which can be used in the one-time pad due
to its high security.
In this procedure, Alice and Bob make use of a quantum channel and another classical one
A.public
Galindo
M.quantum
A. Martı́n-Delgado:
and
quantum
which isA.
too.and
The
one, that isClassical
usually an
optical
fiber,information
is used for sending
Galindo
and
M. A. Martı́n-Delgado:
Classical
and
quantum
information
photons one by one. On the other hand, the public channel can be accessed by anybody.
clearly random. For instance, denoting by H,
el, while keeping
a record
the sequence
The protocol
can of
be described
in four of
clear steps:
clearly random. For instance, denoting by H,
el, while keeping
a record
of
the sequence
of
ed states andStep
of the
associated
sequence
of 0’spolarized
A the
and 135° pola
1.
Alice
prepares
photons
linearly
at 0horizontal,
, 90 , 45 andvertical,
135 , that45°,
are eigened states and of the associated sequence of 0’s
A the
horizontal,
vertical,
45°,
and 135° pol
ained representing
by
0 the
choices
of 0itand
respectively,
and
bythrough
!, " the
thequantum
polarization ba
states
of
the
bases
(+,
⇥).
She
does
randomly
and
she
sends
them
ained representing by 0 the choices of 0 and
respectively, and by !, " the polarization ba
and by 1channel
otherwise.
This
is states.
recording
thesequence
sequence ofof
thebits
prepared
Denoting
thepossible
states horizontal,
45 , ver!D,A",
Alice’s
sequences
are
and by 1tical
otherwise.
This
sequence
of
bits
is
!D,A",
Alice’s
possible
sequences
are
and 135 by H,D,V and A respectively, she gives the values 0 for the first two states and 1
for the lasts ones. In this way, Alice achieves a sequence of bits totally random. Morever, she
writes down three di↵erent sequences: one for the polarization bases, other for the polarization
states and the last made of bits. Alice’s sequences are:
b has two analyzers, one ‘‘rectangular’’ (!type), the other ‘‘diagonal’’ ("type). Upon receivin
b has two analyzers,
one
(!type),
the
other
‘‘diagonal’’
("type).
Upon
receivin
Step 2.
can‘‘rectangular’’
analyze
the photons
Alice
making
use ofthe
twoaleatory
devices,
one
in
the of
tons, he decides
at Bob
random
what analyzer
tosent
use,byand
writes
down
sequence
analy
tons, he decides
at the
random
what
analyzer
to use, which
and writes
down
the
aleatory
sequence
of anal
base
+
and
other
in
⇥.
He
chooses
randomly
analyzer
to
use
to
measure
each
photon,
the result of each measurement. He also produces a bit sequence associating 0 to the cases in
he result writing
of each
measurement.
alsowhich
produces
a used
bit sequence
associating
0 to
the cases in
down
the sequence of He
the basis
has been
for the measurement
of each
photon
ent produces
a
0°
or
45°
photon,
and
1
in
cases
of
90°
or
135°.
With
the
following
analyzers
and the
result
thisphoton,
measurements
as Alice,
Bob
0 thethe
states
H and 45 ,analyzers
ent produces
a 0°
or of
45°
and 1too.
in Just
cases
of 90°
ordenotes
135°. byWith
following
possible result
of
action
on way,
Alice’s
sequence
is
and 1 of
to VBob’s
and 135
. In this
Bob previous
could obtain
the next sequences:
possible result
Bob’s
action
on Alice’s
previous
sequence
is
xt they communicate
other
the public
channel
the sequences
polarization
Note that ifwith
Aliceeach
and Bob
use through
the same basis,
the result
of the measurement
is theofsame
xt they communicate
with
each
other
through
the public
channel
the sequences
of polarization
1
In Bob’s
other case,
they have
a probability
of never
of obtaining
the same
result.
mployed, for
as both.
well as
failures
in detection,
but
the specific
states
prepared by Alice in
2
mployed, as well as Bob’s failures in detection, but never
the specific states prepared by Alice in
ulting states obtained by Bob upon measuring.
ulting states obtained by Bob upon measuring.
24
ext they communicate with each other through the public channel the sequences of polarization b
Using
the public
channel,
and Bob
put inthe
common
the sequences
ext
they communicate
with
each information
other
through
theAlice
public
channel
sequences
of polarization
b
employed,
asStep
well3. as
Bob’s
failures
in detection,
but
never
the
specific
states
prepared
byofAlice in ea
polarization
andfailures
analyzers
It is
important
to note
that they
never
communicate
employed,
as well
asbases
Bob’s
inemployed.
detection,
but
never the
specific
states
prepared
by Alice in e
esulting
states
obtained
by
Bob
upon
measuring.
each
other
which
ones
are
the
prepared
states
nor
the
resulting
states
measured
by
Bob.
sulting states obtained by Bob upon measuring.
ey discard those
in which
photons,
andhad
alsoa those
which the
Step cases
4. They
discardBob
thosedetects
cases innowhich
Bob has
failure cases
in theindetection,
or preparat
ey
discard
those
cases
in
which
Bob
detects
no
photons,
and
also
those
cases
in
which
the
preparat
Alice and the
analyzer
type usedbasis
by Bob
differ.
this distillation,
both
left with
same
in which
the polarization
used by
AliceAfter
to prepare
the photons and
theare
analyzer
basis the
Alice
and
the
analyzer
type
used
by
Bob
differ.
After
this
distillation,
both
are
left
with
the
same
by Bob
in will
the measurement
do shared
not coincide.
nce of bitsemployed
0, 1, which
they
adopt as the
secretAfter
key:the discarding, both obtain the
nce of bitssame
0, 1,subsequence
which they
adopt
theasshared
secret
key:
of will
bits that
theyastake
the secret
shared
key:
that Eve ‘‘taps’’ the quantum channel and that
the distilledTherefore
key is the key is
thatsame
Eve ‘‘taps’’
the quantum
channel and
tha
the distilled key is
the
equipment
as Bob, analyzes
the pol
1011110000011001001010...
the same
equipment
as Bob, analyzes
thetopol
110000011001001010¯
state
of each
photon, forwarding
them next
B
110000011001001010¯
state
of
each
photon,
forwarding
them
next
to
B
much
like Bob,
ignorant
of theofstate
of each pho
supposing
that
Bob detects
the photons sent
by Alice,
on average,
the length
the key
ngth is, onand
average,
and
assuming
no all
detection
much
like will
Bob,use
ignorantwrong
of the state of each
pho
by
Alice,
ngth
is, on
nosequence.
detection
one-half
ofand
the
first
This is due to the
probability
of both the
Alice and Bobanalyzer
making with pro
ne-half
ofisaverage,
the
length
of assuming
eachsequence’s.
initial
by Alice, will use the wrong analyzer with pr
ne-half ofuse
theof length
eachwhich
initial
sequence.
the sameofbasis,
is exactly
a half. 1/2 and will replace Alice’s photon by another
1/2 and
will
replace Alice’s
another
that
upon
measurement
Bobphoton
will getbyAlice’s
st
Now Alice only has to write the message to Bob
as
a
series
of
bits
and
then
add
the
shared
that upon measurement
Bobofwill
get
Alice’s s1
probability
only
3/8,
instead
the
probability
ropping effects
secret key, making the message random. Then sheprobability
has to communicate
it through
the
only 3/8,
instead
of classical
the probability
1
absence
of
eavesdropping.
Therefore
this interve
ropping effects
channel and finally Bob can decode the message absence
detractingof
theeavesdropping.
key.
Therefore
this interv
his holds in the ideal case in which there are no
Eve induces on each photon
a probability
of e
his
holds
in
the
ideal
case
in
which
there
are
no
All
this
is
written
in
the
ideal
case
that
there
are
no
defects
in
the
transmission
of
photons,
Eve
induces
on
each
photon
a
probability
of e
ppers, no noises in the transmission, and no deReturning to the previous example, let us assu
pers,
no noises
in and
the no
transmission,
and
noanalyze
de- theEve’s
no noises
eavesdropper.
Let’s
case that
the
isexample,
perfect
butletproduce
Returning
tocommunication
the previous
us assu
he
production,
reception,
or analysis:
the
dismeasurements
on Alice’s
photons
there
is
an
eavesdropper,
Eve,
who
intercepts
the
quantum
channel
and
also
has
the
same
he
reception,
or But
analysis:
disEve’s measurements
on Alice’s photons produc
s ofproduction,
Alice and Bob
coincide.
let us the
assume
lowing
results:
equipment
as
Bob.
s of Alice and Bob coincide. But let us assume
lowing results:
Supposing Eve detects Alice’s photons and then she replaces and sends them to Bob, she
hys., Vol. 74,can
No.induce
2, April an
2002
error. If the analyzer’s basis is the same as Alice’s polarization basis, Eve can
hys., Vol. 74, No. 2, April 2002
create an equal photon. Even so, if the basis are not the same, whose probability is 1/2,
the reproduced photon may have the same polarization as Alice’s photon and also can be 90
25
shifted. In this way, Eve can induce an error with a probability of 1/4. Let’s suppose that Eve
obtainsA.
theGalindo
next sequence
the photons sent Classical
by Alice: and quantum information
and M.from
A. Martı́n-Delgado:
A.
Classical and
andquantum
quantuminformation
information
A. Galindo
Galindo and
and M.
M. A.
A. Martı́n-Delgado:
Martı́n-Delgado: Classical
es are now those
reaching
who
with
his
sequence
of analyzers
will
obtain,
for instance,
Eve
sends theBob,
photons
which
has his
measured
to Bob,
writes down
the
next sequence
tes are
those
reaching
Bob,
who
his
sequence
ofwho
analyzers
will
obtain,
forinstance,
instance,
ates
arenow
nowThen
those
reaching
Bob,
with
sequence
of
analyzers
will
obtain,
for
using his analyzers:
ging
asasasinininStep
4:4:4: Alice and Bob communicate their sequences of basis to obtain the key.
Step
Finally,
ng
Step
As can be seen, in one out of four cases Eve’s induced error appears and the key sequence
does not coincide
Alice and
used the
basisbut
to prepare/measure
notsame
know,
that Alice canthelater show
that the coincidences
in although
the distilled
listsBob
gethave does
does
not
know,
that
can
hat
thecoincidences
coincidences
in the
the distilled
distilled
lists the
getpresence
does
not
know, but
but
thatinAlice
Alice
can
latersho
sh
at the
in
photons.
Accordingly,
they
can
verify
of
an
eavesdropper
putting
common
a later
when he claims it. Resorting
to entangled
EP
in one out of four cases, the coincidence disof thecases,
key that
they
can not use anymore.
Sacrificing
aclaims
piece of it.
the
key whose length
is
when
he
Resorting
totoentangled
E
none
oneout
outpiece
of four
four
cases,
the
coincidence
diswhen
he
claims
it.
Resorting
entangled
n
of
the
coincidence
makes
it possible
for
either
party
of
the coupl
Sacrificing
for
verification
a piece
of
the lists
N
N
bits,
the
probability
of
not
detecting
eavesdropper’s
e↵ects
is
(3/4)
.
Using
a
piece
of
the
makes
itit possible
the
acrificing
for verification
verification
piece ofAlice
makes
possible for
for either
eitherparty
partyofof
thecou
co
crificing
for
aa piece
the lists
have
dishonestly
Alice
random from
theenough,
final sequences,
and
key long
if they not find
any discrepancies
they
can discard (a
the cheating
possibility of
beingcould cha
have
Alice
could
ch
andom
from
the final
final sequences,
sequences,
have dishonestly
dishonestly
(a cheating
cheating
Alice
could
ndom from
the
Alice and
publicly
compare
their differences
commitment
at the (a
end
without Bob’s
being
awa
interceptedthem,
by an and
eavesdropper.
ublicly
compare
them, Ifand
and
their
commitment
the
end
Bob’s
being
blicly
them,
differences
commitmentat
at
thecould
endwithout
without
Bob’sinformat
beingaw
a
t Eve’scompare
intervention.
thetheir
length
ofprotocol
that butuntrustworthy
Bob
obtain
some
This
is
just
an
ideal
case
of
the
in
practice
one
has
to
take
care
of
the
noise
Eve’s
intervention.
If probability
the length
lengththat
untrustworthy
Bob
obtain
some
Eve’s
intervention.
If
the
of Eve’s
that
untrustworthy
Bobcould
could
obtain
some
inform
artial
sequence
is N, the
without
asking equipment.
Alice;
Mayers,
1996,
1997;informa
Brassa
of the emitting
source, the transmission
channel
or
the
receiving
This
can
create
N
rtial
sequence
is
N,
the
probability
that
Eve’s
without
Alice;
Mayers,
tial
sequence
is N,
the
probability
as not
produced
discrepancies
(3/4) N bit
andsequences
is
without
asking
Alice;
Mayers,
1996,
1997;Bras
Bra
1997).
discrepancies
between
theisdistilled
by Alice asking
and Bob,
but not
as much1996,
as the1997;
N and is
s
not
produced
discrepancies
is
(3/4)
1997).
not produced
discrepancies
is (3/4) should
and is
gible
for eavesdropper
N large
enough.
Therefore,
introduces.
1997).
There exists a proof of the unconditional se
ible any
for N
N large
large enough.
enough.
Therefore,
should
find
discordance,
they
can
feel
safe
about
There
exists
aa proof
the
ble
for
Therefore,
should
There key
exists
proof of
of
theunconditional
unconditional
quantum
distribution
through
noisy channels
Sequences have been taken from [15].
nd
any
discordance, But
theythey
canmust
feel clearly
safe about
about
ce
of
eavesdroppers.
disd any
discordance,
they
can
feel
safe
quantum
key
through
noisy
chann
to
any distance,
by means of
a protocol
based
quantum
key distribution
distribution
through
noisy
chanu
eof
ofeavesdroppers.
eavesdroppers.
But they
they
must
clearly
disbinary
string they But
have
made
public
and
not
must clearly disto
any
distance,
by
means
of
a
protocol
based
sharing
of EPR pairs
and their
an
to any distance,
by means
of apurification,
protocol base
binary
string they
they have
have made
made public
public and
and not
coding.
inary string
not 26 the
sharing
of
EPR
pairs
and
their
purification,
hypothesis
that
bothand
parties
and Bo
sharing
of EPR
pairs
their(Alice
purification,
oding.
tice,
the
emitting
source,
the
transmission
ding.
the
that
parties
(Alice
fault-tolerant
quantum
computers
andand
Cha
the hypothesis
hypothesis
that both
both
parties(Lo
(Alice
andB
ice,
the
emitting
source,
the
transmission
nd
the
receiving
equipment
all
display
noise,
ce, the emitting source, the transmission
fault-tolerant
quantum
and
Ch
Likewise,
the unconditional
security(Lo
of
the
BB8
fault-tolerant
quantumcomputers
computers
(Lo
and
C
nd
the
receiving
equipment
all
display
noise,
ld spoil,
even
without
Eve’s
intervention,
the
the receiving equipment all display noise,
Likewise,
the
unconditional
security
of
the
BB
col
is
also
claimed
(Mayers,
1998).
Likewise, the unconditional security of the B
of the
bit sequences
distilled
by Alice and
spoil,
even
without Eve’s
Eve’s
intervention,
the
spoil,
even
without
intervention,
the
col is also claimed (Mayers, 1998).
4.2
E91 protocol
Ekert’s protocol, just as in the previous one, is based on obtaining a private key and using it
in the one-time pad. To obtain the key, Alice and Bob share spin 1/2 particles in a state of
maximum entanglement. Due to the correlations, they are able not only to get a secure key
but also to detect the presence of an eavesdropper.
The communication between Alice and Bob is provided by a classical channel and a quantum
one. We assume that Alice has a source that emits EPR pairs, from which one is measured by
Alice and the other travels to Bob’s system where it is also measured.
The detectors of both systems perform measurements on spin components along three directions perpendicular to the trajectory of the particles. Choosing the z axis as the trajectory
direction, the measurements are made in the x-y plane. Measurement directions are characterized by the azimuthal angles as follows: a1 = 0 , a2 = 45 , a3 = 90 and b1 = 45 , b2 = 90 ,
b
3 = 135 , where subscripts ”a” and ”b” refer to Alice and Bob respectively and the angle
starts in the x axis.
Alice chooses randomly one of the three orientations of the detector for each pair of entangled
particles, and Bob does the same. They obtain one bit of information in each measurement, 1
in case the result is spin up and 0 if it is spin down.
We define the correlation coefficient as
E(ai , bj ) = P++ (ai , bj ) + P
(ai , bj )
P+ (ai , bj )
P
+ (ai , bj )
(24)
where ai and bj are the directions of the Alice and Bob’s analyzers. P±± correspond to the
probability that the result of the measurement has been spin up or spin down along the direction
ai and bj respectively. In agreement with the quantum formalism
E(ai , bj ) =
ai · bj .
(25)
Consequently, in case the analyzers of both users have the same direction (a2 , b1 and a3 , b2 ) the
anticorrelation of the results obtained by Alice and Bob is total, E(a2 , b1 ) = E(a3 , b2 ) = 1.
Following the CHSH inequality [23], which is based on Bell’s theorem [24], we define a
quantity composed of the correlation coefficients for which the detector’s orientations, a and
b, are di↵erent:
S = E(a1 , b1 )
E(a1 , b3 ) + E(a3 , b1 ) + E(a3 , b3 ).
(26)
It can be easily seen that
S=
p
2 2,
which violates Bell’s inequality, |S|  2, due to the non-locality of the EPR state.
(27)
Once the transmission is completed, Alice and Bob communicate the sequence of the orientations of the analyzers employed. They discard the cases in which the detections have failed
27
and then separate in two groups, one for the cases they have used di↵erent orientation in their
detectors and another where the orientations coincide. They put in common the results of
the measurements of the first group, so they can calculate the value of S by averaging the
probabilities P±± for each combination
p of analyzer’s directions. In case the particles were not
perturbed, S should remain being 2 2, so the users can be sure that the quantum channel is
not being intercepted by an eavesdropper.
Given that there is a total anticorrelation if Alice and Bob use the same direction in their
analyzers, Alice’s second group results are the opposite of Bob’s. Consequently, Bob has to
change his results to obtain Alice’s sequence, which will be used as the key. Note that the
results of the second group has never been shared through the quantum channel, so only the
users have them and the one-time pad can be accomplished.
Besides, if there is an eavesdropper intercepting the quantum channel, he will measure the
EPR particles, and he also will try to replace them. However, the correlation coefficients will
be altered, which changes (26) as follows:
R
S = ⇢(na , nb )dna dnb [(a1 · na )(b1 · nb ) (a1 · na )(b3 · nb )
+(a3 · na )(b1 · nb ) + (a3 · na )(b3 · nb )],
(28)
where na and nb are the directions of the eavesdropper analyzer for the particles that will
be measured by Alice and Bob respectively. ⇢(na , nb ) can be considered as the strategy of
the eavesdropper, because is the probability of intercepting a spin component along the given
direction for a particular measurement.
In case Alice has the source of particles in the singlet state, the eavesdropper only could
intercept Bob’s particle the directions na and nb check the relation nb = na . In this way,
(26) can be simplified giving
Z
p
S = ⇢(na , nb )dna dnb [ 2na · nb ],
(29)
so the value S is restricted to
p
2S
p
2.
(30)
This result is contrary to (27) for any strategy ⇢(na , nb ). So then, the eavesdropper has no
way to escape from being detected.
Now, we show an original calculation in Ekert’s protocol assuming that the quantum channel
is composed of two particles in the Werner state that we introduced in (10). So, the density
matrix of particles 1 and 2 is
0
1
2(1 p)
0
0
0
C
1B
0
1 + 2p 1 4p
0
C.
⇢12 = B
(31)
A
0
1 4p 1 + 2p
0
6@
0
0
0
2(1 p)
28
First of all we calculate correlation coefficients. For this, we should calculate the probabilities
that the result of the measurement has been spin up or spin down along the directions ai and
bj . This is given by
P±± (ai , bj ) = Tr[(⇤± (ai ) ⌦ ⇤± (bj )) · ⇢12 ],
(32)
where ⇤± (v) is the projector of the state linearly polarized along the direction given by v with
eigenvalues +1 and -1 respectively. That projector can be constructed from
1
⇤± (v) = [ ±
2
being
=
x nx
+
y ny
+
· v],
(33)
z nz .
We analyze correlation coefficients in the cases that Alice and Bob’s detectors have the same
polarization direction
E(a2 , b1 ) = P++ (a2 , b1 ) + P (a2 , b1 ) P+
1
1
1
= (1 p) + (1 p)
(1 + 2p)
3
3
6
(a2 , b1 ) P + (a2 , b1 )
1
1
(1 + 2p) = (1 4p).
6
3
(34)
It is easy to see that E(a2 , b1 ) = E(a3 , b2 ) = 13 (1 4p). So if p = 1, the anticorrelation
becomes maximum, which is in agreement with the case of the singlet state.
Let us calculate S replacing the singlet state by the Werner one. In this case we obtain
S = E(a1 , b1 ) E(a1 , b3 ) + E(a3 , b1 ) + E(a3 , b3 )
1 4p 4p 1 1 4p 1 4p
4(1 4p)
p + p + p =
p
= p
.
3 2
3 2
3 2
3 2
3 2
(35)
p
p 2 , so security can not
As can be seen in Figure 6, S only violates Bell’s theorem if p > 3+
4 2
be assured in case p is less than this value. Moreover, the less is p the less is the probability
of establishing a perfect key between Alice and Bob due to the fact that the anticorrelation
coefficient for the same direction of measurement decreases proportionally to p.
29
Figure 6: Plot of the S as a function of p.
30
5
Experimental aspects of quantum cryptography
Although quantum-cryptography (QC) protocols began to be published in 1984, the first experiments could not be accomplished until 1999. The high difficulty of changing rapidly the
basis of the analyzers, which was one of the requirements of QC protocols, was insurmountable
for more than a decade.
With the implementation of new technologies, the field of quantum communication made
a remarkable progress. In fact, in 2000 the result of three QC experiments from University
of Innsbruck [25], Los Alamos National Laboratory [26] and University of Geneva [27] were
published in the journal Physical Review Letters. These ones reproduced E91 protocol making
use of polarization-entangled photons in the first ones while the last consists in energy-time
entanglement. In the following we review the polarization-entanglement experiments.
Both groups used photon pairs at a wavelength of 700nm because this was the easiest way
to be detected by the single-photon detectors based on silicon avalanche photodiode’s. To
generate the entangled pair, the source consisted in one or two -BaB2 O4 crystals pumped by
an argon-ion laser. This is the same technique that was used in Innsbruck in the teleportation
experiment, which is known as parametric down-conversion.
Entangled photons go to Alice and Bob’s analyzers, which are composed of an active polarization rotator, a beam splitter and a an avalanche photodiode. The first element allows to the
user to rote, instead of beamsplitter’s polarization direction, photon’s one obtaining the same
result.
et al.: Quantum cryptography
nce of a
have a
to one.43
e singleobability.
of a pair
tly hears
21. Typical
for quantum
exploiting
ng multi-FigureFIG.
7: Graphic
scheme of system
the experiment.
PR, activecryptography
polarization rotator;
PBS, polarizing
APD,
avalanche
photo-diode.
Taken from [16].
photon
pairs
entangled
in polarization:
PR, active polarization
en meanbeamsplitter;
rotator; PBS, polarizing beamsplitter; APD, avalanche phototy pulse
diode.
he same
In Innsbruck, the distance between Alice and Bob analyzers had a length of 360m. ConseI.A.2). quently,
they needed to use a special optical fibers designed for guiding only a single mode at
photons700nm. Each
analyzer saved
the results of
the measurements and later systems
on that information
Generally
speaking,
entanglement-based
are was
n unused
far more complex than setups based on faint laser
31
bserving
pulses. They will most certainly not be used in the near
ivalently
future for the realization of industrial prototypes. In aded guardition, the current experimental key creation rates obtained with these systems are at least two orders of maghe differ-
put in common following the implemented protocols, one based on Wigner’s inequality [28],
which is a specific case of Bell’s inequalities, and the other based on BB84.
On the other hand, Los Alamos’ experiment was a table-top realization that spanned no
more than a few meters, so photons travelled a short free-space distance. In this case, they
employed the six-state protocol and the E91. For the last, they simulated di↵erent eavesdropping strategies and an increase in the discrepancies between Alice and Bob was clearly observed
when the information obtained by the eavesdropper increased.
32
6
Quantum Hacking
The best known attack on a cryptographic system may be the one that involves the Enigma
machine [29]. This system could encrypt any message but also decrypt it. The best minds, Alan
Turing among others, worked hardly to decode the machine. Their contribution was invaluable
to win World War II. This is a clear example of importance of cryptography in the modern
world.
Nowadays is common to hear about events where security of computer systems is broken
up. This is called hacking. As we have shown above, there are mechanisms to decode security
keys generated classically. Even though the one-time pad is 100% secure, it is not really useful
if the users are not able to create secret keys steadily. The public-key cryptographic system
may be more useful. However, the users should be aware that an eavesdropper can decrypt
the message in case he finds the way to obtain the inverse of the function that encrypts the
message.
Given that quantum cryptography is based on physical properties and not on mathematical
conjectures, the eavesdropper has to interact with the quantum channel to obtain the key. This
allows legitimate users to detect him, making the quantum key distribution protocols the most
secure ones to date. In fact, as we have shown above, Alice and Bob have mechanisms to detect
the presence of an eavesdropper although they never issue the key.
Until now, we have restricted to the ideal case that there are no noises in the transmission
and no defects in the production, reception or analysis, where quantum cryptography protocols
are perfect. On the contrary, implementations use imperfect devices available with current
technology and thus, the eavesdropper can access the information.
In 2011 a full-field implementation of a perfect eavesdropper on a quantum cryptography
system was accomplished in the National University of Singapore [30]. The authors succeeded
in obtaining the entire ”secret” key while Alice and Bob did not detect any secure breach.
In this experiment, the four-state protocol is realized with polarization encoding and passive
basis choice. Eavesdropper’s analyzers are the same as Bob’s, four linear polarizers in front
of avalanche photodiodes. He has to make Bob to measure in the same basis as him to avoid
being detected. Therefore, he makes use of a laser diode that blinds Bob’s detectors emitting
continuous-wave circularly polarized light. For each photon from Alice, he adds a linearly
polarized pulse of the same polarization that has measured when he intercepts Alice’s photon.
Only the analyzer chosen by the eavesdropper will detect the reproduced photon, achieving
that Bob obtains the same measurement as his own. In this way, the eavesdropper and Bob
obtain the same result sequence, and after Alice communicates her sequence of basis, both Bob
and the eavesdropper get the key.
This experiment confirms that quantum cryptography can be broken although theoretical
protocols might be perfect. Therefore, the field of cryptography is not already closed and it
has to be improved to become highly secure.
33
Pagina en blanco
34
7
Conclusions
This chapter lists the key results obtained during research conducted in this work. The study of
quantum theory in combination with the theory of information has been developed during the
last decades but even today it is still an open field that has not been fully exploited. This work
is just a sample of the wealth of this field, not only for its predictions in theoretical protocols
but also for the huge variety of applications in which it can be implemented.
In the following we show the conclusions of each section:
• A quantum channel and a classical one are necessary to teleport a quantum state from
one point to another. In case one of them is missing the teleportation protocol can not be
applied. The quantum channel has to consist of two entangled particles. As we have seen,
the fidelity of the protocol decreases proportionally to the degree of entanglement of the
state until 66% fidelity of teleportation protocol corresponding to the maximum fidelity of
classical state. We also demonstrate that the classical channel is essential to accomplish
this protocol because Bob does not obtain any information until Alice communicates the
result of the measurement to him, even if she has already made it.
• We have reviewed two of the most relevant experiments in the field of quantum teleportation. Innsbruck group only succeeded in a 25% of times due to the fact that the employed
devices only could detect the state | i making the Bell measurement. The experiment
was accomplished using photons polarized in several ways to get a high accuracy. On the
other hand, in the experiment realized at NIST, authors made use of trapped ions. They
demonstrated that this system can attain higher fidelities than one composed of photons,
achieving an accuracy of 78%.
• We have shown that classical cryptography is secure nowadays with some methods,
namely, the one-time pad and the public-key cryptographic system. However, quantum cryptography is based on laws of physics and not on mathematical conjectures. We
have reviewed the BB84 and the E91 protocols, which are based in quantum properties
of linear superposition and entanglement respectively. In both we have analyzed the case
that an eavesdropper intercepts the quantum communication and in both cases the legitimate users detect the presence of the eavesdropper. In addition, we have studied the
E91 protocol in case the quantum channel is not perfect, obtaining when the security is
assured as a function of the imperfection of the quantum channel.
• We have shortly reviewed two of the implementations of quantum-cryptography protocols.
In these experiments, the authors made use of polarization-entangled photons. In both,
two systems succeeded establishing the secure key. Furthermore, in the experiment based
on Ekert’s protocol, the authors introduced an eavesdropper and they observed an increase
of discrepancies in the key when the eavesdropper interacted with the quantum channel.
• Experimental quantum cryptography is not secure yet even though it is safe from a
theoretical point of view. The experiment made in Singapore shows how an eavesdropper
35
can obtain the key without being detected due to the imperfections of employed devices,
so it is possible to achieve a perfect attack on a current quantum cryptographic system.
Summarizing, quantum teleportation and quantum cryptography are two fundamental primitives of quantum communication that very likely will revolutionize future transmission of information. Framed inside the broad field of quantum information, these protocols, together
with quantum computation and simulation, are expected to shape the future technologies of
processing and transmission of information. This is likely just the beginning of a new era in
Information and Communication Technologies.
36
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