Tools for quantum repeaters: quantum teleportation, independent sources of entangled photons and entanglement purifcation

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Thesis

Reference

Tools for quantum repeaters: quantum teleportation, independent sources of entangled photons and entanglement purifcation

LANDRY, Olivier

Abstract

Dans cette thèse, après avoir décrit les bases de la communication quantique et le fonctionnement des répéteurs quantiques, nous décrivons trois expériences qui démontrent la faisabilité de trois éléments important pour la fabrication éventuelle de répéteurs quantiques et pour la communication quantique en général.

LANDRY, Olivier. Tools for quantum repeaters: quantum teleportation, independent sources of entangled photons and entanglement purifcation. Thèse de doctorat : Univ.

Genève, 2010, no. Sc. 4163

URN : urn:nbn:ch:unige-68279

DOI : 10.13097/archive-ouverte/unige:6827

Available at:

http://archive-ouverte.unige.ch/unige:6827

Disclaimer: layout of this document may differ from the published version.

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UNIVERSIT´E DE GEN`EVE

Groupe de Physique Appliqu´ee - Optique

FACULT´E DES SCIENCES Professeur Nicolas Gisin

Tools for Quantum Repeaters:

Quantum Teleportation, Independent Sources of Entangled Photons and Entanglement Purification

TH` ESE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention physique

par

Olivier Landry Canada

Th` ese N

^◦

4163

GEN`EVE

Atelier de reprographie ReproMail 2010

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Abstract

Quantum cryptography is the most developped application in quantum information science, being commercially available since 2001. It is however still limited to short distances and to point-to-point operation. The development of quantum repeaters would allow us to build quantum networks, which would allow quantum cryptography to be deployed on a much larger scale, but would also allow potentially unseen applications to emerge.

In this thesis, after an introduction on quantum communication, we describe simple forms of quantum relays and quantum repeaters. More importantly, we describe experimental implementations of three tools of quantum communication that will eventually be needed to build quantum repeaters.

We implement quantum teleportation over 800 m of commercially installed optical fiber. For the first time the teleported qubit is created only after the carrier qubit has left the laboratory. As a basic communication tool and as a quantum non-demolition measurement, quantum teleportation will be an important feature of any quantum network.

We develop synchronized sources of picosecond duration entangled pulsed photons using periodically poled lithium niobate waveguides. Picosecond sources allow us to tolerate a larger time of arrival jitter, necessary for applications where the fiber length is not stabilized. Independent sources are also necessary for actual quantum networks where the different nodes will be separated by kilometers. We demonstrate synchronization and undistinguishability of the photons produced by this source.

Finally, we implement a purification protocol for single photon entanglement.

Even taking into account advances in equipment efficiency and design, it is likely the rate of error or decoherence will never be diminished to zero. Such error-correction and purification protocols are essential to allow the use of many tools without error accumulation.

The advances described in this thesis, combined with advances in quantum memory technology, means that the building of a true quantum repeater in the near future is possible.

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R´ esum´ e en fran¸cais de la th` ese

“Dieu ne joue pas aux dés”, disait Albert Einstein. Il ne croyait en effet pas que la mécanique quantique puisse être une théorie complète, c’est-à-dire qu’il croyait qu’elle

était, au mieux, une bonne description statistique du monde réel, mais qu’une théorie ultime permettrait de l’“expliquer”[1].

Pendant longtemps on a cru que ces considérations étaient plus philosophiques que physiques. Pourtant, en 1964, John Bell découvrit qu’il était possible de différencier expérimentalement une théorie à variables cachées telles que celle préconisée par Einstein d’une théorie complète. Les premières démonstrations expérimentales de l’inégalité de Bell furent réalisées en 1972 [2], puis en 1982 [3] et démontrèrent hors de doute qu’Einstein avait tort: la mécanique quantique est bel et bien une théorie complète, et non pas simplement une description statistique. Cela signifie que les résultats les plus curieux de la mécanique quantique, telles que les corrélations en- tre les paires d’états intriqués, sont bels et bien des manifestations d’un monde in- trinsèquement non local.

Au même moment, en 1981, Richard Feynman proposa d’utiliser certaines de ces propriétés quantiques afin de réaliser des calculs, ce qui marqua le début de l’informatique quantique. Très rapidement, différentes applications furent découvertes, les plus connues étant certainement l’algorithme de Shor, qui permet de factoriser les nombres rapidement et donc de briser le modèle d’encryption à clé publique largement utilisé aujourd’hui, et la cryptographie quantique qui permet justement de baser la distribution sécuritaire de clé sur des principes physiques et donc inviolables même par les ordinateurs les plus puissants.

Parallèlement, une autre application, plus marquante pour les esprits même si apparemment moins utile, est apparue: la téléportation quantique. La possibilité de transférer l’information d’un point à un autre est une illustration parfaite de la nature non-locale de la mécanique quantique, et en même temps le fait que ce transfert ne

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peut se faire plus rapidement qu’`a la vitesse de la lumi`ere nous rassure que les limites que nous connaissons du monde ne sont pas toute fausses.

Plus qu’une simple curiosité, pourtant, la téléportation est un élément de base du champ d’étude des communications quantiques. Ce qui fait qu’un état quantique est plus puissant qu’un état classique dans le sens où il permet certaines actions interdites classiquement, soit la superposition et l’intrication, est très fragile. Toute interaction avec l’environnement, toute mesure d’un état quantique le projettera dans un état classique souvent inutile. De plus, l’intrication de deux particules nécessite une forme de contact: une particule à New York et une autre à Genève ne seront pas intriquées par hasard ou par magie. La communication quantique est l’ensemble des techniques qui permet de distribuer des états quantiques, en particulier des états intriqués, à grande distance.

Ce domaine est en plein essor. Certaines applications, telle la cryptographie quantique, sont disponibles sur une base commerciale et ce depuis 2001. Sa distance d’application reste tout de même limitée à moins de 100 km. L’avènement de répéteurs quantiques pourrait augmenter largement cette distance, mais plusieurs défis sont à relever avant d’atteindre cet objectif.

Dans cette thèse, après avoir décrit les bases de la communication quantique et le fonctionnement des répéteurs quantiques, nous décrivons trois expériences qui démontrent la faisabilité de trois éléments important pour la fabrication éventuelle de répéteurs quantiques et pour la communication quantique en général.

Premièrement, nous avons effectué une expérience de téléportation quantique sur une distance de 800 m à travers le réseau commercial de fibres optiques de la com- pagnie Swisscom. Pour la première fois dans cette expérience, les données téléportées ont été créées après la distribution de l’intrication.

Deuxièmement, nous avons développé des sources indépendantes de photons in- triqués et non distinguables, nécessaires pour toute implémentation à longue distance de répéteurs quantiques.

Finalement, nous avons implémenté un protocole de purification de l’intrication qui permet de corriger les erreurs qui se glissent immanquablement dans toute implé- mentation et qui sont particulièrement nuisibles aux communications quantiques qui ne peuvent être amplifiées.

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List of Figures

2.1 Time-bin qubit encodes information in the time of arrival of a photon. 19

2.2 Conversion efficiency of QPM . . . 22

3.1 Bell State Analyzer . . . 26

3.2 Graphic representation of entanglement swapping . . . 30

3.3 Ekert protocol . . . 32

4.1 BB84 key rate . . . 38

4.2 Basic repeater link . . . 42

4.3 Two basic repeaters links . . . 43

5.1 The teleportation protocol . . . 46

5.2 Teleportation setup . . . 47

5.3 Mandel dip . . . 49

5.4 Dip stability . . . 51

5.5 Decision circuit . . . 52

5.6 An aerial view of the Plainpalais neighborhood of Geneva . . . 54

5.7 Teleportation result . . . 56

5.8 4-photons teleportation . . . 58

6.1 The source of entangled pairs . . . 61

6.2 The PicoQuant’s output spectrum under experimental parameters . . 62

6.3 Spectrum after pre-amplifier . . . 63

6.4 Spectrum after IPG amplifier . . . 64

6.5 Spectrum after FBG . . . 65

6.6 Spectrum after Keopsys amplifier . . . 66 11

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12 LIST OF FIGURES

6.7 Frequency doubled spectrum. . . 67

6.8 Jitter measurement . . . 68

6.9 Autocorrelation measurement . . . 69

6.10 Cross-correlation measurement . . . 70

6.11 Experimental setup used to measure a Mandel dip . . . 71

6.12 Non-degenerate spectrum . . . 72

6.13 Degenerate spectrum . . . 73

6.14 Visibility as a function of jitter . . . 74

6.15 Ppair measurement . . . 75

6.16 Modes in and out of a beamsplitter. . . 76

6.17 TDC histogram . . . 78

6.18 Mandel dip . . . 80

6.19 Mandel dip with a common source . . . 81

7.1 Combining modes to purify entanglement . . . 84

7.2 Purified fidelity . . . 86

7.3 Experimental setup for single photon entanglement purification . . . 87

7.4 Experimental setup without purification. . . 89

7.5 Initial fidelity measurement . . . 89

7.6 Fidelity distribution . . . 90

B.1 Monte-Carlo variables definition . . . 109

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Chapter 1 Introduction

When Richard Feynman teased scientists about the possibilities of quantum computers [4] in a keynote talk in 1981, few people took him seriously. Yet, already in 1935, the EPR paradox showed that there was more to quantum mechanics than just a different kind of mechanics. In 1964, John Bell derived the theorem that bears his name [5] which shows that it is possible to demonstrate experimentally a discrepancy between quantum mechanics and locality, an experiment that was realized for the first time by Freedman and Clauser in 1972 [2] and repeated later by Alain Aspect and coworkers in 1982 [3] and many others.

Soon thereafter, the first algorithm to leverage quantum mechanics to practical use, the BB84 quantum key distribution protocol [6], appeared. It quickly became more than a theoretical curiosity when Shor [7] discovered in 1994 an algorithm to find the prime factors of numbers, a fast algorithm that will invalidate the most widely used cryptography protocols once quantum computers become scalable, but leaves the BB84 protocol secure.

In the meantime, in 1993, Charles Bennet and coworkers developed a new protocol that allowed for the teleportation of information between separate locations [8]. More than a boon for Star Trek fans and public relations, this protocol allows us to highlight the many mysterious ways in which the quantum world functions. Even though it is the Bell experiment that formally put an end to the local realist world-view, it could be argued the reality of teleportation did more to drive the point home in the minds of many. Furthermore, it has practical uses as a form of non-demolition measurement

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14 CHAPTER 1. INTRODUCTION and as a part of quantum repeaters, a crucial feature of any future quantum network, which ensured its importance in modern research.

In this thesis we will focus on a selection of tools of quantum communication. We will first outline the basics of quantum communication, the different types of photonic qubits, with a focus on entangled qubits and their creation in non-linear crystals.

We will explain some of the protocols that serve as building blocks for higher level communication tools, namely Bell state measurements, teleportation, entanglement swapping and purification. We will describe a cryptography protocol, namely the Ek- ert protocol, for its general importance, because it is a good example of an application of the Bell theorem and because the main motivation of building quantum repeaters is to extend the reach of such protocols.

We will give examples of quantum relays and quantum repeaters, devices that promise to extend the reach of quantum communication and entanglement distribution, with immediate applications to quantum cryptography. Quantum repeaters are complex devices that couple many basic tools such as entanglement sources, entanglement swapping and quantum memories. Some of these tools are developped further in this thesis, other in laboratories around the world.

We will describe in details three experiments that were performed in this thesis.

The first is the teleportation of information from our laboratory to a location 800 m away using commercially installed fiber. This implementation was the first to wait until after the carrying photon left the laboratory to create the photon to be teleported. It also innovates in using different pulses of the same laser, a prefiguration of independent sources.

The second is the creation of independent synchronized sources of entangled photons. We developped a triggerable source of short pulses of light intense enough to serve as a pump for the creation of entangled pairs of photons. We demonstrated picosecond synchronization of this source with a cavity laser. We demonstrated undistinguishability of the photons pairs produced by these different sources. Independent sources will be important for any realistic implementation of quantum repeaters. Fur- thermore, this source is the first to use photons of picosecond coherence time, which allows relaxed alignment conditions on fibers outside the laboratory.

The third experiment is one of purification of entanglement. We create entengled

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modes of a single photon state. We artificially decrease the fidelity of this state by adding phase noise. Then we implement a protocol that combines two corrupted states into one with higher fidelity. Purification is necessary to build quantum repeaters consisting of an important number of imperfect links.

Each one of these implementations has a character that makes them new and brings us a step closer to a real world implementation of quantum repeaters. Inde- pendent sources of entanglement coupled with quantum memories would form the basic link of quantum repeaters. We have shown in the teleportation experiment that it is possible to perform these experiments outside the laboratory using commercially installed optical fibers. Finally, purification would allow us to correct any imperfection remaining.

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16 CHAPTER 1. INTRODUCTION

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Chapter 2 Quantum Communication

2.1 Qubits

A qubit is the quantum version of a classical bit.

A bit is a binary number, and it can be 0 or 1 (or true or false, or black or white, or any other combination). The physical representation of a bit is therefore any object that can be in two distinguishable states. For example, on the hard disk of a computer, bits are represented by the direction of the magnetization of a small patch of ferromagnetic material that is read by a needle utilizing the Giant Magnetoresistance Effect [9].

In the same manner, a physical qubit is any physical system that can be in quantum states |0i or |1i. However, the system being quantum, it can also be in any (normalized) superposition state |Ψi=α|0i+β|1i.

There are many specific proposals to physically realize qubits. We will focus on two particular ones, phase qubits and time-bin qubits, which are used in experiments described in this thesis. There are many others, some which also use single photons such as polarization qubits [10], some which use different photon states such as squeezed states [11] or two-pulse sequences [12], and some which are not photon-based such as plasmons [13], ions [14], ions on optical lattices [15], Josephson junctions [16]

and a plethora of others.

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18 CHAPTER 2. QUANTUM COMMUNICATION

2.1.1 Dual-rail qubits

Take a single photon and send it through a beamsplitter. The first mode is the transmitted mode, the second mode the reflected one. Call each mode |0i if it is empty and |1iif it contains a photon, the overall state is therefore

|Ψi= 1

√2(|10i+i|01i) (2.1) This is a qubit: the photon is in a supersposition state of having been reflected (|10i) or transmitted (|01i). A simple detection apparatus consists in a single photon detector for each mode: whichever one clicks collapses the wavefunction into one definite state. A more interesting apparatus is to collect both modes on a second beamsplitter, thereby creating a Mach-Zehnder interferometer. By adding a phase modulator in one arm of the interferometer, we can choose which basis to use to perform the measurement. We will use a related model for the purification measurement in §7.

2.1.2 Time-bin qubits

A problem of the phase qubits is that the environment will influence differently both paths the photon can take. For optical fibers at telecommunication wavelengths, for example, the thermal expansion will be on the order of 10⁻⁵ K⁻¹. For 1 meter long fibers and 1550 nm light, this means that the relative phase between the two paths will change by 1.03^rad_K , limiting these setups to short distances in temperature-controlled environments.

A way to improve this is to combine both paths on the same fiber. This way, any environmental change will act equally on both locations of the wavefunction.

In practice, we send a photon in a stable unbalanced interferometer as in Fig. 2.1.

The photon can then take either the short path or the long path. It therefore has two possible output states, early or late, which we call |0i and |1i respectively, and the phase difference between them corresponds to the phase difference between the arms of the interferometer. After exiting the interferometer, the delocalized photon travels along the fiber without modifications.

To measure the state in the {|0i,|1i} basis, we can simply measure its time of

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2.2. ENTANGLEMENT 19 arrival. To measure it in a different basis, we send the light through a second equally unbalanced interferometer. If the photon takes twice the short path or twice the long path, its time of arrival will allow us to discriminate which path it has taken. However if it alternatively takes both path, we cannot differentiate between a photon that first took the short path and then the long one, or the inverse. Therefore these two possible paths will interfere. The photons will come out of the second interferometer in one of two detectors, and which detectors clicks corresponds to a measurement in the {^√¹₂(|0i+e^i∆|1i),^√¹₂(|0i −e^i∆|1i)}basis where ∆ is the phase difference between the two interferometers.

Figure 2.1: Time-bin qubit encodes information in the time of arrival of a photon.

2.2 Entanglement

Two objects are said to be entangled if their combined state cannot be described as two separate states. In the case of qubits, two qubits are entangled if there are no {α, β, γ, δ}such that¹

1In this thesis, we will consistently use the following notation: |xi_A⊗ |yi_B=|xi_A|yi_B=|xi|yi=

|xyi, where the order in which we write in indices is meaningful, i.e. in |xyi x always refers to photon A andy to photon B.

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|Ψi= (α|0i+β|1i)⊗(γ|0i+δ|1i) (2.2) The canonical examples are the Bell states, which form a complete set of states for pairs of qubits:

|Φ⁺i= 1

√2(|00i+|11i) (2.3)

|Φ⁻i= 1

√2(|00i − |11i)

|Ψ⁺i= 1

√2(|01i+|10i)

|Ψ⁻i= 1

√2(|01i − |10i)

2.2.1 Creating entangled qubits using non-linear crystal

Entangled states do not come easily in nature, especially for photons. Two photons which have no common source and have never interacted will of course have completely separable states. To create entangled pairs, we use spontaneous parametric down- conversion (SPDC) [17, 18].

2.2.1.1 Spontaneous Parametric Down-Conversion

A photon with energy ωp could, in theory, decay into two photons of energy ωs

and ωi (p,s and i stand for pump, signal and idler for historical reasons) as long as conservation of energy, or ωp = ωs +ωi is respected. The three-fields interaction necessary can be provided by the second order polarization tensor χ²_ijk as in the following formula [19], wherePi are the coordinates of the polarization of the medium.

Pi(t) = X

j

χ¹_ijEj(t) +X

j,k

χ²_ijkEj(t)Ek(t) +O(E³) (2.4)

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2.2. ENTANGLEMENT 21 It can be shown [20] that χ² and all other even-order contribution to polarization (corresponding to odd-order interaction) are identically zero in all centrosymmetric materials, which is why this phenomenon is not commonly observed. In non-centrosymmetric crystals (e.g., all piezoelectric materials), however, such phenomenon, along with related ones such as sum-frequency generation, are possible.

2.2.1.2 Phase Matching

Energy conservation for photons implies, through ω=kc/n, the following:

kp

np

= ks

ns

+ ki

ni

(2.5) , which is equivalent to the law of momentum conservation in the Abraham con- vention [21].

In a dispersive material, the indices of refractions np, ns and ni will not be equal, the phases of the electromagnetic wavesφj(t) =kjx(t)−ωjtwill not stay aligned and the different modes will see a phase mismatch

∆φ= (kp−ks−ki)∗x (2.6)

This will result in alternate constructive and destructive interference, as shown in Fig. 2.2. This is mitigated in birefringent material by cutting the non-linear crystal at an angle such that the phase mismatch is minimized and at a length where the conversion efficiency is maximal.

2.2.1.3 Waveguides and Quasi-Phase Matching

To increase overall efficiency of the down-conversion process, we want our focused beam to stay as long as possible in the non-linear crystal. In a Gaussian beam, the Rayleigh length and beam waist are limited by each other [19]. By using a waveguide, we can focus the beam over a small interaction area and keep it confined over a distance of several centimeters.

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22 CHAPTER 2. QUANTUM COMMUNICATION However, as we have seen, phase mismatch between the different travelling waves means that the energy flow will alternate between positive and negative along the crystal. In practice, this means crystals must be cut at an optimal thickness; using a longer crystal would mean converting the signal and idler back into the pump. A way out of this is to use quasi-phase-matching (QPM). The orientation of the nonlinearity in these materials is periodically reversed. In this way, the direction of the energy flow is reversed at points where it would start to flow in the wrong direction, as shown in Fig. 2.2. This inversion of non-linearity is often achieved by periodic poling of a ferroelectric material, such as lithium niobate.

0 5 10 15 20

0 2 4 6 8 10 12

Conversion efficiency (arb. units)

Crystal length (arb. units)

Figure 2.2: Typical conversion efficiency for perfect phase matching (dashed line), phase mismatched material (dot-dashed line) and quasi-phase-matched material (solid line)

SPDC in a typical bulk crystal such as the one we use in§5 will have an efficiency of∼10⁻¹², while typical SPDC waveguides made of periodically poled lithium niobate

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2.2. ENTANGLEMENT 23

(PPLN), such as the ones we use in § 6, can reach efficiencies of ∼10⁻⁶ [17].

2.2.2 Entanglement as a ressource

Entangled quantum states have a very interesting property: they are inherently non- local. By this we mean that a measurement on one part of the state will have an effect on the whole. For example, take the second photon of a pair originally in the

|Φ⁺i= ^√¹₂(|0iA|0iB+|1iA|1iB) state. If we consider it in isolation, it is a completely mixed state, as shown in eqn. 2.7.

ρB =T rA(|Φ⁺ihΦ⁺|)

= 1

2(|0ih0|+|1ih1|) (2.7) However, if the first photon is measured to be in the state |0i, then the state of the second photon becomes definite.

h0|A|Φ⁺i=|0iB (2.8)

This change occurs even if the two photons are space-like separated. Experiments have been made that tried to measure the speed at which this change occurs and all have shown it must be faster than the speed of light [22], or even outside the bounds of space-time [23]. This change, however, does not allow any information transfer, and is therefore non-signaling, as will be revisited in more detail in§3.2.2. It is in fact theorized that the condition of non trivial signalling could be a fundamental limit of nature [24].

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Chapter 3 Communication Protocols

We present here the theoretical underpinnings of a selection of protocols widely used in quantum communication: Bell state measurement, quantum teleportation, entanglement swapping and a brief description of purification, as we will implement specific versions of these protocols, and quantum cryptography, due to its general importance.

3.1 Bell state Measurement

A Bell-State Analyzer (BSA) is a device that performs a state measurement in the basis of the Bell states (eq. (2.3)), also called Bell-state measurement (BSM). In general, BSAs are not easy to design as they must operate a joint measurement. In fact, one can show that it is impossible to design a BSA with an efficiency over 50%

using only linear optics [25, 26].

In our experiments, we use BSA designed for time-bin qubits that discriminates only the |Ψ⁻i states, so that it has an efficiency of 25%.

Take a pair of qubits and mix them using a 50/50 beamsplitter (BS), and put detectors on the BS output as shown in Fig. 3.1. If the incoming pair was in the

|Φ⁺i or |Φ⁻i state, then both photons will arrive on the detectors at the same time.

We can exclude these states by excluding all cases except those where both detectors click with a time difference of one time-bin.

If the incoming photons are in a|Ψ⁺istate, then we can describe the state before the BS as follows, where a^†andb^†are the left and right-hand side modes respectively,

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26 CHAPTER 3. COMMUNICATION PROTOCOLS

Figure 3.1: Bell State Analyzer

and subscript e and l denote the time of arrival of the photon, early or late.

|Ψ⁺i= 1

√2(|01i+|10i)

= 1

√2(a^†_eb^†_l +a^†_lb^†_e)|0i (3.1)

The BS has the following effect, where c^† and d^† are the left and right-hand side modes after the BS respectively.

a^† → 1

√2(ic^†+d^†) (3.2)

b^† → 1

√2(c^†+id^†)

So that the state after the BS is

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3.2. TELEPORTATION 27

|ψ^′i= 1 2√

2((ic^†_e+d^†_e)(c^†_l +id^†_l) + (ic^†_l +d^†_l)(c^†_e+id^†_e))|0i

= i

√2(c^†_ec^†_l +d^†_ed^†_l)|0i (3.3)

so that the photons will bunch and always exit the BS on the same side (although not at the same time). A similar bunching effect will be exploited in §6.4.2.

On the other hand, if a|Ψ⁻iis incoming, then the resulting state after the beamsplitter will be

|ψ^′i= 1 2√

2((ic^†_e+d^†_e)(c^†_l +id^†_l)−(ic^†_l +d^†_l)(c^†_e+id^†_e))

= 1

√2(c^†_ld^†_e−c^†_ed^†_l)|0i (3.4) And photons will always appear on different sides. Therefore if we detect clicks on both detectors with a time difference corresponding to the time-bin delay, then we have measured the input state to be in the |Ψ⁻i state. All other combinations of clicks are inconclusive.

Note that the |Ψ⁺i would be detectable if our detectors were fast enough to measure two closely separated clicks. Unfortunately, detectors always have a dead time after a click, usually much longer than the time-bin time difference, which makes a |Ψ⁺i detection impossible in this scheme.

3.2 Teleportation

Although all the necessary mechanics has been known since the 1920’s, it is only in 1993 that Bennet and Brassard [8] realized that the properties of entangled states allowed for teleportation of information.

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3.2.1 Teleportation Protocol

Start with an arbitrary state|ψi=a|0i+b|1iand the Bell state|Φ⁺i= ^√¹₂(|00i+|11i).

Any other Bell state would lead to a similar result. The total three-qubit state can be written as eqn. (3.5). We can rewrite the same as eqn. (3.6) using simple algebra, where σi are the Pauli operators. In that way, all the information from the original state |ψi, which is the data we want to transmit, is now contained in the 3^rd photon.

If we were to simply measure this third photon, the information would be mixed and unrecoverable; however it is entangled with the state of the first two photons.

|Ψi=|ψi|Φ⁺i

= (a|0i+b|1i)( 1

√2(|00i+|11i) (3.5)

= 1

2√

2((|00i+|11i)(a|0i+b|1i) +(|00i − |11i)(a|0i −b|1i) +(|01i+|10i)(a|1i+b|0i) +(|01i − |10i)(a|1i −b|0i)

= 1

2(|Φ⁺i|ψi+|Φ⁻iσZ|ψi +|Ψ⁺iσX|ψi+|Ψ⁻iσXσZ|ψi)

(3.6)

For example, if the first pair of qubits is measured to be in the state |Φ⁺i, which occurs with 25% probability, then the third photon is known to be in the same state as the original ψ; we say then that a qubit of information has been teleported from the first to the third qubit. The three other Bell states are equally probable; in those cases a unitary transformation can be applied to recover the original state.

3.2.2 No faster-than-light signaling

It is important to note that this protocol does not allow for faster than light signaling.

The only way for the recipient to recover the information contained in the third photon is to receive the classical information contained in the joint measurement

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3.3. ENTANGLEMENT SWAPPING 29 of the first two photons. Without this information, and without the corresponding unitary transformation, the third photon is in a completely mixed state and contains no information.

At the same time, the classical information sent has no correlation to the qubit being transffered. Therefore the information that interests us, while unrecoverable for an amount of time corresponding, at a minimum, to the distance between the point, cannot be said to travel between those two points. Rather, it disappears at one point to reappear at the other. However, since the information is unrecoverable for a time equivalent to the travel time at the speed of light, causality still holds.

3.3 Entanglement Swapping

Entanglement swapping is a natural extension of quantum teleportation, in fact it is often called teleportation of entanglement. Imagine you start with two pairs of entangled photons instead of one, for example both in the |Φ⁺i state. Here again, the argument would be similar with a different Bell state.

|Ψi=|Φ⁺i12|Φ⁺i34

=1 2

|Φ⁺i14|Φ⁺i23+|Φ⁻i14|Φ⁻i23

+|Ψ⁺i14|Ψ⁺i23+|Ψ⁻i14|Ψ⁻i23

(3.7)

We can rewrite this state in terms of Bell states of the 2^ndand 3^rdphotons, as shown in eqn. (3.7). Of course, these photons are not entangled since they have no common history. However, by measuring them in the basis of the Bell states, the 1^stand 4^th photons will be projected into an entangled states, even though they have no common history themselves. By sending classically the information concerning which state photons 2 and 3 were found in, the receivers of photons 1 and 4 can then use local operations to manipulate their common entangled state to any Bell state they wish. The final result is to create an entangled state with two photons that have no common source, as shown in Fig. 3.2

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Figure 3.2: Graphic representation of entanglement swapping, for the case where the BSA measurement yields the fourth term of eqn. (3.7)

3.4 Purification

Both teleportation and entanglement swapping make use of entangled states as their main ressource. Noise-free distribution of entanglement is a difficult task. In general, for a given entangled state |Ψi being created and distributed, the received state’s corrupted density matrix will be of the form

ρ=F|ΨihΨ|+ (1−F)A (3.8) where F∈[¹₂,1] is called the fidelity and A could be arbitrary. For many applications, particular sources of noise dominate and a given form forA is assumed. If F is far from 1, then the teleportation or swapping will fail or yield a corrupted result.

Fortunately, it is possible to purify a quantum state. This means that given a noisy channel that outputs states with partial entanglement, it is possible to combine several such states into one with better entanglement, and to asymptotically reach perfect entanglement.

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3.5. CRYPTOGRAPHY 31 Bennet was the first to propose a purification protocol [27]. Different protocols exist for different types of noise or forms of qubits [28]. In §7, we will demonstrate a purification protocol for single-photon qubits [29].

3.5 Cryptography

Quantum Cryptography is the art of leveraging properties of quantum states, in particular the fact that an unknown state cannot be observed or cloned without leaving a trace, to create secure communication channels. It is arguably the most widely used application in quantum information. It was first discovered in 1984 [6]

and commercially available since 2001 [30].

Many different protocols exist, which are optimized for different systems or partic- ularly resilient against particular attack. A common feature is that these system are used to share a secret key across an unsecure channel to two actors, canonically called Alice and Bob, which can then use this secret key to encrypt their communication.

As such, quantum cryptography should really be called quantum key distribution (QKD), as the actual encryption takes place classically, using a one-time pad for perfect security or more often a protocol such as AES-128 [31]. If a spy (whom we will call Eve) eavesdrops on the secure channel, this will be discovered by Alice and Bob and no usable key will be produced.

We will describe here the Ekert protocol [32]. This choice is motivated by the fact that this protocol is well suited to a description of quantum repeaters based on entanglement swapping as will be described in §4. However, all cryptographic protocols can make use of specifically designed quantum repeaters.

3.5.1 Ekert’s protocol

A source of entangled photons sends each photon from a pair in the singlet state|Ψ⁻i to two distant locations, Alice and Bob, as in Fig. 3.3

We have used entangled time-bin qubits (§2.1.2) while the original description used spin-¹₂ particles; the protocol is exactly the same.

Alice and Bob receive one qubit each and measure it in some basis. The statis- tics for each individual qubit is entirely random, since each individual qubit is in a

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Figure 3.3: In the Ekert protocol, a pair of entangled photons is sent to Alice and Bob.

completely mixed state (§2.2.2). There will be, however, correlation between their results.

Alice and Bob each chose the phase of their interferometer among the possibilities ai ∈ {0,^π₄,^π₂} for Alice and bi ∈ {^π4,^π₂,^3π₄ } for Bob. The measurement results are called + and -, whichever basis is used, and correspond to which detector clicks as shown in §2.1.2.

The choice of basis should be done randomly and not decided in advance. If Alice and Bob chose the same basis, then their result will be perfectly anti-correlated.

After the measurement, Alice and Bob need to share the basis they have chosen.

They also reveal the results they have obtained when they happen to have chosen different basis of measurement, but not when they have chosen the same basis. They can use this data to verify the security of the line through the CHSH inequality (named after its inventors John Clauser, Michael Horne, Abner Shimony and Richard Holt [33]). Even if Eve intercepts this publically shared data, the best she will achieve is to verify for herself the security of the line.

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3.5. CRYPTOGRAPHY 33 3.5.1.1 Correlations

Define the correlation coefficients Eaibj, where i, j refers to the basis chosen and Pkl

refers to the probability of obtaining results k for Alice and l for Bob in this basis.

Eaibj =P+++P₋₋−P₊₋−P₋₊ (3.9) We can compute that (the derivation is shown in appendix A)

Eaibj =−cos(ai−bj) (3.10) When the basis are identical the results are perfectly anti-correlated, as expected.

We can then measure the quantity S, defined [33] as

S =E(a1, b1)−E(a1, b3) +E(a3, b1) +E(a3, b3)

=−2√

2 (3.11)

This result assumes everything went smoothly in the transmission. What if Eve measured the state of the photons on the way? Then the states Alice and Bob receive will not be the entangled state, rather they will have been measured, and therefore contain what Einstein called “elements of reality” [33, 1].

3.5.1.2 CHSH inequality derivation

If the states received by Alice and Bob have an independent reality, then we can compute a fundamental limit to their degree of correlation. The correlation coefficients can be written as

Eij = Z

ρ(λ)Ai(λ)Bj(λ)dλ (3.12)

Where A and B are values between -1 and 1 (perfect anti-correlation and perfect

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34 CHAPTER 3. COMMUNICATION PROTOCOLS correlation respectively), λ is some random variable which can be taken to be any- thing and unknowable, and ρ is a probability distribution of that variable such that R ρ(λ)dλ = 1. This equality basically assumes the existence of definite values for A and B, chosen from some probability distribution, while quantum mechanics does not assume any value exists before the measurement process. Theories that obey this equality are called hidden variable theories.

We have that

kEij ±Eij^′k= Z

ρ(λ)kAi(λ)kkBj(λ)±Bj^′(λ)kdλ (3.13) or, since kAk ≤1,

kEij ±Eij^′k ≤ Z

ρ(λ)kBj(λ)±Bj^′(λ)kdλ (3.14) independently of i.

From kBk ≤1 we can show that

kBj(λ)−Bj^′(λ)k+kBj(λ) +Bj^′(λ)k ≤2 (3.15) Plugging (3.15) into (3.14) for two different values of i and carrying the resulting integral over the probability density we get

kEij −Eij^′k+kEi^′j +Ei^′j^′k ≤2 (3.16) or, through the triangle inequality,

kEij −Eij^′ +Ei^′j +Ei^′j^′k ≤2 (3.17)

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3.5. CRYPTOGRAPHY 35 which is violated by (3.11). This violation is actually a particular case of a violation of a Bell inequality [5] and is of fundamental interest; here we simply use it to prove the security of our algorithm.

3.5.1.3 Security check and key generation

Alice and Bob publicly share the result they have recorded in the cases where their detectors used the settings {a1, b1}, {a1, b3}, {a3, b1} or {a3, b3}. They can then verify that they obtain the result of (3.11). If they do not, then either the state was imperfectly prepared, the line is noisy or an eavesdropper listened in.

If the line is secure, then Alice and Bob keep the result they have obtained when they used the same setting for their measurements and discard the others. The resulting string of bits will be secret and perfectly anti-correlated, and can be used to encrypt a message.

It can be shown (see e.g. [34]) that as long as (3.17) is violated, then it is possible, through error correction and privacy amplification, to extract from an imperfectly anti-correlated string of bits a perfect, shorter one such that the eavesdropper retains none of the partial information it possibly obtained, and the protocol remains secure.

If the eavesdropper detects so much information that privacy amplification becomes impossible, then this will be detected and the resulting presumably anti-correlated bits will be thrown out and not used, so that the eavesdropper cannot use that information.

Of course, for unconditional security, one also has to prove that even if the eavesdropper possesses, for example, a quantum memory or an optimal cloning machine she cannot extract information. Such unconditional security proofs (see e.g. [35]) are outside the scope of this thesis.

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Chapter 4 Quantum repeaters

The main limitation of quantum cryptography is that it is limited by qubit loss.

Photons in fibers are attenuated at a rate of 0.17 dB/km (that is, the probability of loss of any single photon is 3.8% for each kilometer of fiber) for the best fibers available [36], not taking into account the additional losses caused by connections.

In classical communication each bit can be represented by a bunch containing millions of photons and retains its value even if most of them are absorbed before being detected. However in quantum communication each qubit is encoded in a single photon, and that qubit is lost when the photon is lost. This means the rate at which we can transmit information is exponentially dependent on the distance at which we want to send it: it halves at every 17.7 km.

Additionally, errors in the detection of the qubits are often undistinguishable from errors caused by an intervening eavesdropper. To create secure communication, we must therefore assume all errors are caused by an intervening spy. These errors do not drop at the same rate as the actual rate of communication. For example, one of the most common source of errors are dark counts in the detectors: a pulse is created in the semi-conductor photodiode but is caused by thermal fluctuations instead of an incoming photon. The dark-count rates are fixed, typically at around 10⁻⁵ ns⁻¹ in InGaAs Avalanche Photodiodes (APD). When the rate of incoming photons gets so low that it is comparable to the error rate, no secure communication is possible, as seen in Fig. 4.1

Additionally, we cannot use amplifiers to periodically boost our signals as they 37

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38 CHAPTER 4. QUANTUM REPEATERS

Figure 4.1: This figure taken from [37], illustrates the rate at which secure key is created by a quantum cryptographic protocol, here the BB84 protocol. The detector used was a superconductor single photon detector (SSPD) which has in perfect conditions a negligible dark count rate[38] of <0.001 Hz, compared to ∼ 10⁴ Hz for InGaAs APDs. The upper blue line uses a 10 GHz photon emission rate and the lower red line only 1 GHz. The filled points correspond to experiments with actual fibers while the hollow points use an optical attenuator and assume a corresponding fiber length with an attenuation of 0.2 dB/km. We can see even a 10 times increase in speed does not allow for communication at a larger distance.

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4.1. QUANTUM RELAY 39 would not revive already lost qubits, and furthermore would not create good copies of the still-existing photons because of the no-cloning theorem [39]. This means that we cannot increase the rate of communication except by running our algorithms faster and getting more transparent optical fibers. However, we can increase the distance we can go before we hit the wall of too much noise using quantum relays, which we will describe here.

4.1 Quantum Relay

The Ekert protocol (Fig. 3.3) uses a distributed entangled pair of photons; what if we used entanglement swapping to distribute this entanglement?

First, the rate would decrease. If we send pump pulses in a non-linear crystal at rate Rclock, with each pulse having a probability Pp of creating a pair, which it sends in fibers of length l/2 (such that Alice and Bob are separated by a total distance l) with losses α = 0.17dB/km to two lossless interferometers fitted with detectors with efficiency η, we will measure these pairs at a rate

REkert=RclockPp(10¹⁰¹ ^−αl² )²η² (4.1) However, if we use the same equipment with the same parameters in the setup of Fig. 3.2, with each fiber arm having a length of l/4 for a total length of l again, we get the following rate.

Rswapping =Rclock P_p²

|{z}

2 pairs are created simultaneously

P

photons 2,3 are detected in the BSA

z }| {

(10¹⁰¹ ^−αl⁴ )²η² (10¹⁰¹ ^−αl⁴ )²η²

| {z }

P

photons 3,4 are detected by Alice and Bob

(4.2)

=RclockP_p²(10¹⁰¹ ^−αl⁴ )⁴η⁴

which is lower by a factor Ppη² ≈ 0.001, with realistic values Pp = η = 0.1.

However, how far can this rate extend? Let us assume the protocol breaks down when the rate of false positives is equal to the rate of true positives (this is of the

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40 CHAPTER 4. QUANTUM REPEATERS right order of magnitude, the exact value will depend on the particular protocol used).

If each detector has a dark count probability ofPdc= 10⁻⁵ per trial, then the rate of false positives will be in the first caseR^{f alse}_Ekert=Rclock(P_dc² + 2·10¹⁰¹ ^−αl² ηPdc) (assuming losses are large), which limits the length l.

REkert> R^{f alse}_Ekert

∴ lEkert<−20

α log Pp

η²+ 2η

<37.9 km (4.3)

∴ REkert(l^max_Ekert) = 4.8·10⁻⁴Rclock

(4.4) On the other hand, if we use an entanglement swapping protocol where each branch has a length ₄^l for a total distribution distance of l, we will only open the detector for photons 1 and 4 if the BSA has functioned correctly. We start by com- puting the dark counts if only one pair of photons is created, with a factor 2 since this photon could have been created on either side.

R_1pair^{f alse}= 2Rclock(Pp(10¹⁰¹ ^−αl⁴ )²η²P_dc² + 2·Pp·10¹⁰¹ ^−αl⁴ ηP_dc³ +P_dc⁴) (4.5) If both pairs are created, it could still be that some photons do not reach the detectors.

R_2pairs^{f alse} =RclockP_p²(4·(10¹⁰¹ ^−αl⁴ )³η³Pdc+ 6·(10¹⁰¹ ^−αl⁴ )²η²P_dc² + 4·10¹⁰¹ ^−αl⁴ ηP_dc³ +P_dc⁴) (4.6) And of course no pairs could be created, so that the total rate of false results is R_swapping^{f alse} =R_2pairs^{f alse} +R^{f alse}_1pair +RclockP_dc⁴.

We can now compute the new limit

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4.2. QUANTUM REPEATERS 41

Rswapping > R^{f alse}_swapping

(4.7) which is a 4^thorder polynomial that can be solved, but we’ll skip to the conclusion:

l_swapping^max = 701.4 km

Rswapping(l_swapping^max ) = 1.2·10⁻¹⁸Rclock (4.8) We can see that such a relay cannot increase the rate of key disribution over a direct qubit distribution protocol. However, the maximum distance at which a realistic key distribution program can be implemented is enlarged. Many such entanglement swapping links can be chained to increase the distance even further.

4.1.1 Working principle

The fact that the BSA functions heralds the arrival of a true entangled pair at Alice and Bob. They can then open their detectors only in those cases where they expect it is likely a pair will arrive, diminishing their likelihood of recording a dark count when there is no photon. In a sense, by performing an extra measurement, we increase the probability that the final measurement of interest will prove successful. For protocols that use single photons, such as BB84, teleportation can be used in the same manner.

4.2 Quantum repeaters

The very low rates achieved by quantum relays come from the fact that many things need to be successful at once, namely two pairs of photons need to be created and all four need to be detected. Detectors and entangled photon sources being what they are, probabilities are against us.

Furthermore, if we had perfect detectors (100% efficiency and no dark counts), then quantum relays would be useless, and the rate at which we can communicate

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42 CHAPTER 4. QUANTUM REPEATERS would still go down exponentially with distance. WithRclock = 10 GHz, the maximum distance at which our entanglement swapping would distribute at least 1 pair/sec is 235 km, far from the noise limit.

A better setup is that of the quantum repeater, which require a quantum memory.

4.2.1 Quantum memories

Quantum memories (QM) are devices that are able, with a finite efficiency and a small probability of error, to store a quantum state and release it on demand. The construction of quantum memories is a subject of ongoing research [40]. Quantum memories are able to store quantum states for a period of time, so that we do not need everything to work at once: we can store those parts of the repeaters that did work and wait for the other parts to work too.

Most proposals rely on absorbing photons in atoms. These memories can be read at will, have short lifetime (although we will neglect the lifetime in our analysis) but cannot by themselves announce their success. They have low efficiency but negligible decoherence.

4.2.2 Elementary link

Figure 4.2: A basic repeater link

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4.2. QUANTUM REPEATERS 43 The elementary link of a quantum repeater is shown in Fig. 4.2. A pair of entangled photons is created with a probability Pp. One of the photons is stored in a quantum memory with probability ηQM while the other travels a distance llink/2. Another source situated at a distance llink does the same and two photons meet mid-way. If a BSA is successful, the link announces its success, which happens at a rate

1 Tlink

=Rlink=RclockP_p²(10⁻¹¹⁰^αllink² )²η² (4.9) We can use many of these elementary links, as shown in Fig. 4.3, by considering them as independent entanglement sources and performing entanglement swapping, similarly to the previous quantum relay. If the swapping is successful, two repeater links each of length llink = l/2 will have distributed entangled states in quantum memories a length l apart.

Figure 4.3: Two basic repeaters linked increase the range of entanglement distribution

If we have two such links, then the average time to wait before any one of them is successful is ^T^link₂ , and the average time before the second link is also successful isTlink, for a total success rate of ²₃Rlink. We can generalize this forN links to RN = _H¹

NRlink

where HN =PN k=1 1

k are the harmonic numbers which grow as logN.

Once N links hold entangled pairs in their memories, the success rate of the repeater depends only on the success of entanglement swapping. Since the distance between the QM and the BSA is short, the overall rate is determined by the individual links and the detector efficiency.

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44 CHAPTER 4. QUANTUM REPEATERS

Rrepeater = 1 HN

Rlinkη^2·(N⁻¹⁾ (4.10)

The important point is that this only depends on Rlink/log(N), and not onR^N_link as in a the quantum relay picture. Therefore the distribution rate is no longer limited by the exponential losses in the fibers. Of course, sources of noise are also present.

Depending on the sources and detectors characteristics, an optimal number of links can always be found.

4.2.3 Other protocols

We have presented a simple quantum repeater protocol that is useful to understand their properties. Other proposed designs are likely to be more efficient or use simpler technology than this one, such as for example the DLCZ protocol [41, 42] based on atomic ensembles or single-photon protocols [43].

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Chapter 5 Teleportation over 800 m with realistic experimental challenges

We have performed [44] a teleportation experiment using time-bin qubits over a distance of 800 m. Our goal was to show that teleportation could be performed in real-life conditions, so we imposed the following conditions:

1. We used only commercially installed fiber. We used photons at communication wavelengths (1310 nm and 1550 nm). We used time-bin qubits and not polarization encoded qubits as most commercially installed fiber is not polarization maintaining.

2. We used photons created from different laser pulses, instead of many photons coming from the same pulse as in all previous experiments.

3. We encoded the data qubit only after entanglement distribution took place.

Condition 2 is really a prefiguration of the independent sources that will be shown in §6. Condition 3 is the real novelty of this experiment.

5.1 Setup

The quantum teleportation protocol [8] (schematically described in Fig.5.1) requires that Bob (the receiver) and Charlie (a third party) share an entangled state, which

45

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46 CHAPTER 5. TELEPORTATION

Figure 5.1: The teleportation protocol. Quantum channels are in plain lines, the classical channel is in dashed line. The state that Bob measures is the same that Alice sent to Charlie up to a unitary transformation. EPR source is a source of entangled states, or Einstein-Podolsky-Rosen states.

in this case is a |φ⁺istate. Alice (the sender) needs to send a qubit over to Bob, but does not possess a direct quantum channel. She sends it to Charlie who performs a Bell State measurement [25] using a beamsplitter and classically announces the result to Bob. The experimental setup of the experiment is shown in Fig. 5.1.

A mode-locked titanium-sapphire laser or Ti:Sa (Mira Coherent, pumped using a Verdi laser) creates 185 fs pulses with a spectral width of 4 nm at a central wavelength of 711 nm, a mean power of 400 mW and a repetition rate of 75 MHz. This beam is split in two parts using a variable coupler (halfwaveplate and a polarization beamsplitter).

The transmitted light is sent through an unbalanced Michelson interferometer stabilized using a frequency-stabilized HeNe laser (Spectra Physics 117A) and then on a lithium triborate (LBO) non-linear crystal (NLC) cut for type-I phase-matching, which creates a time-bin entangled photon pair in the |φ⁺i state by SPDC. The created photons have wavelengths of 1310 and 1555 nm and are easily separated using a wavelength division multiplexer (WDM). A Si filter is used to remove the remaining

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5.1. SETUP 47

Figure 5.2: Teleportation setup. PC: Polarization controller. QM: rudimentary Quan- tum Memory (fiber spools)

711 nm light.

5.1.1 Charlie’s photon

The 1310 nm photon is sent in a 179.72 m spool of fiber. This spool serves as a rudimentary QM. While Charlie’s part of the entangled qubit pair is waiting in this spool, Bob’s part leaves the laboratory.

In all previous experiments, the Bell-State measurement (BSM) would have been done immediately after the creation of the photon pair, or after only a few meters of optical fibre. We argue that the moment when teleportation occurs is the moment the

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48 CHAPTER 5. TELEPORTATION BSM is performed. As such, an experiment that does not use a quantum memory to let Bob’s photon exit the laboratory by a significant distance does not perform a long- distance experiment, even if Bob is a long distance away from the laboratory. Rather, it performs a short distance experiment and wait for a long time before confirming its success.

5.1.2 Alice’s photon

Alice prepares her photon using the light reflected from the variable coupler. A pair of photons is created in the same type of crystal as above, then separated. The 1555 nm photon can either be discarded or detected by an InGaAs APD in order to herald the photon to be teleported [45]. If it is not detected, teleportation still occurs without other changes in the setup.

The 1310 nm photon is stored in a 177 m spool of fiber. The 2.72 m difference with Charlie’s QM corresponds exactly to the spacing between two subsequent pulses of the laser. This means that Alice’s photon upon which the qubit to be teleported will be encoded is produced from a different pulse of the laser than Charlie’s and Bob’s photons. This is a conceptually important step towards completely independent laser sources [46, 47], a challenge that will be accomplished in §6.

In order to encode a qubit on her photon, Alice sends it after the spool to an unbalanced fiber interferometer independently and actively stabilized [48] by a frequency stabilized laser at a wavelength of 1552 nm (Dicos OFS-2123). Only then is Alice’s qubit created. Note that at this point, Bob’s photon is already 177 m away from the laboratory.

Once Alice’s photon has been encoded, Charlie performs a BSM jointly with his photon and the photon Alice prepared.

5.1.3 Alignment and Stabilization

Alice’s and Charlie’s photons need to arrive at the beamsplitter within their coherence time and be undistinguishable for the Bell state measurement to be successful.

Charlie’s photon passes through a polarization controller to make both polarizations

Tools for quantum repeaters: quantum teleportation, independent sources of entangled photons and entanglement purifcation

Thesis

Reference

Tools for quantum repeaters: quantum teleportation, independent sources of entangled photons and entanglement purifcation

Tools for Quantum Repeaters:

Quantum Teleportation, Independent Sources of Entangled Photons and Entanglement Purification

TH` ESE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention physique

par

Olivier Landry Canada

Th` ese N

4163

Abstract

R´ esum´ e en fran¸cais de la th` ese

Contents

List of Figures

Chapter 1 Introduction

Chapter 2

Quantum Communication

2.1 Qubits

2.1.1 Dual-rail qubits

2.1.2 Time-bin qubits

2.2 Entanglement

2.2.1 Creating entangled qubits using non-linear crystal

2.2.2 Entanglement as a ressource

Chapter 3

Communication Protocols

3.1 Bell state Measurement

3.2 Teleportation

3.2.1 Teleportation Protocol

3.2.2 No faster-than-light signaling

3.3 Entanglement Swapping

3.4 Purification

3.5 Cryptography

3.5.1 Ekert’s protocol

Chapter 4

Quantum repeaters

4.1 Quantum Relay

P

P

4.1.1 Working principle

4.2 Quantum repeaters

4.2.1 Quantum memories

4.2.2 Elementary link

4.2.3 Other protocols

Chapter 5

Teleportation over 800 m with realistic experimental challenges

5.1 Setup

5.1.1 Charlie’s photon

5.1.2 Alice’s photon

5.1.3 Alignment and Stabilization