Quantum Teleportation and Multi-Photon Entanglement

Quantum Teleportation and Multi-photon Entanglement

Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften eingereicht von M.Sc. Jian-Wei Pan University of Science and Technology of China

Durchgeführt am Institut für Experimentalphysik der Universität Wien bei o.Univ.Prof.Dr. Anton Zeilinger

Gefördert vom Fonds zur Förderung der wissenschaftlichen Forschung,Projekte S6502 und F1506 und durch das TMR-Netzwerk The Physics of Quantum Information der Europäischen Kommission. Contents

1 Introduction 5

2 Manipulation of Entangled States 10 2.1 Quantum network and its applications ...... 11 2.2 Practical schemes for entangled-state analysis ...... 15 2.2.1 Bell-state analysis ...... 16 2.2.2 GHZ-state analyzer ...... 20 2.3 Polarization-entangled photon pairs ...... 29

3 Quantum Teleportation 33 3.1 Introduction ...... 33 3.2 Quantum teleportation–the idea ...... 34 3.2.1 The problem ...... 34 3.2.2 The concept of quantum teleportation ...... 35 3.3 Experimental teleportation ...... 40 3.3.1 Experimental scheme ...... 40 3.3.2 Results ...... 43 3.4 Discussion ...... 50

4 Entanglement Swapping 53 4.1 Introduction ...... 53 4.2 Theoretical scheme ...... 54

i ii CONTENTS

4.3 Experimental entanglement swapping ...... 56 4.4 Generalization and applications ...... 61

5 Three-photon GHZ entanglement 64 5.1 Introduction ...... 64 5.2 Experimental Set-up ...... 65 5.3 Observation of three-photon entanglement ...... 72 5.4 Discussion and conclusion ...... 75

6 Experimental tests of the GHZ theorem 76 6.1 Introduction ...... 76 6.2 The conﬂict with local realism ...... 77 6.2.1 GHZtheorem ...... 77 6.2.2 Generalization to conditional GHZstate ...... 81 6.3 Experimental results ...... 84 6.4 Discussion and Prospects ...... 90

7 Conclusions and outlook 92 Zusammenfassung

Die vorliegende Dissertation ist das Ergebnis theoretischer und experimenteller Arbeitenuber ¨ die Physik von Mehrteilcheninterferenz. Die theoretischen Ergebnisse zeigen, daß man Quantenverschränkung mit einem Quantennet- zwerk aus einfachen Quantenlogikgattern und einer kleinen Anzahl von Qubits kontrollieren und manipulieren kann. Da es bis jetzt keine experimentelle Durchführung von Quantengattern für zwei unabhängig erzeugte Photonen gibt, präsentieren wir hier eine realisierbare Methode, verschränkte Viel- teilchenzustände zu erzeugen und zu identifizieren.

In der experimentellen Arbeit wurden die zum Studium von neuarti- gen Vielteilcheninterferenzphänomenen nötigen Techniken von Grund auf en- twickelt. Wir berichten in dieser Arbeituber ¨ die erstmalige experimentelle Realisierung von Quantenteleportation, ’Entanglement Swapping’ und der Erzeugung von Dreiteilchenverschränkung mithilfe einer gepulsten Quelle für polarisationsverschränkte Photonen. Mit der Quelle für Dreiteilchen- verschränkung wurde das erste Experiment zum Test von lokalrealistischen Theorien ohne Ungleichungen durchgeführt.

Die in diesen Experimenten entwickelten Methoden sind von großer Be- deutung für Forschungen auf dem Gebiet der Quanteninformation und für zukünftige fundamentale Experimente der Quantenmechanik.

1 Abstract

The present thesis is the result of theoretical and experimental work on the physics of multiparticle interference. The theoretical results show that a quantum network with simple quantum logic gates and a handful of qubits enables one to control and manipulate quantum entanglement. Because of the present absence of quantum gate for two independently produced photons, in the mean time we also present a practical way to generate and identify multiparticle entangled state.

The experimental work has thoroughly developed the necessary techniques to study novel multiparticle interference phenomena. By making use of the pulsed source for polarization entangled photon pairs, in this thesis we report for the ﬁrst time the experimental realization of quantum teleportation, of entanglement swapping and of production of three-particle entanglement. Using the three-particle entanglement source, here we also present the ﬁrst experimental realization of a test of local realism without inequalities.

The methods developed in these experiments are of great signiﬁcance both for exploring the ﬁeld of quantum information and for future experiments on the fundamental tests of quantum mechanics.

2 Acknowledgements

I am indebted to my advisor, Professor Anton Zeilinger, for his guidance and support throughout the course of my work leading to this thesis. He taught me with wisdom, encouragement, rich knowledge, insight and a deep understanding of physics, and more importantly the way to conduct scientiﬁc research. I am very grateful that he has been always available to discuss the many problems and questions I brought him. I would also like to thank him for critically reviewing this thesis.

I am very grateful to my second advisor, Professor Helmut Rauch, who warmly recommended and supported my application for the Austrian Chan- cellor Fellowship from the Austrian Academic Exchange Service, which en- abled my continuation of physics study in Austria.

I would like to express my deep appreciation to my former and present colleagues in the Quantum Optics and Foundations of Physics research group. Special thanks go to my friend and permanent colleague, Dr. Dik Bouwmeester, who introduced me to the subject of quantum optics and taught me many details of the experiment; he also impressed me with his devotion and ini- tiative. I also especially thank Professor Harald Weinfurter and Matthew Daniell, with whom I collaborated on most of the work. Matthew also has carefully read through some of the chapters in this thesis. Thanks also to the other colleagues in the photon laboratory, Dr. Birgit Dopfer, Dr. Klaus Mattle, Dr. Markus Michler, Dr. Michael Reck, Dr. Surasak Chiangga, Dr. Gregor Weihs, Thomas Jennewein, Alois Mair, Markus Oberparleitner and Christoph Simon. To the people in the atom laboratory, Professor Jörg Schmiedmayer, Dr. Markus Arndt, Dr. Stefan Bernet, Dr. Johannes Den- schlag, Dr. Sonja Frank, Donatella Cassettari, Claudia Keller, Olaf Nairz, and Gerbrand von der Zouw, whose instruments I sometimes stole. And to Mrs. Christine Obmascher, Professor Zeilinger’s secretary, for her quiet efficiency in the office, and for her continuous help on various matters throughout

3 the years.

Many helpful discussions and much of my knowledge about Bell’s inequalities are due to my friend and colleague Professor Marek Zukowski, the permanent visitor to our group.

Dr. Ramon Risco Delgado and Bjorn Hessmo are always remembered. I enjoyed our discussions about physics, the meaning of life, and all the rest, especially the delicious ﬁsh cooked by Bjorn.

This work would have been impossible without the patience and understanding of my wife Xiao-qing, who supported me during the ups and downs that are inevitable in such a major undertaking. My parents have provided me with invaluable support through my entire life and education. They have encouraged and supported me both for starting my undergraduate study in a distant city, and for continuing my doctorate study in another country far away from my homeland.

Among the many very good teachers I met throughout my academic career, I especially thank Professor Yong-de Zhang, my undergraduate and graduate advisor, for his outstanding guidance and continuous concern about my career.

I gratefully acknowledge the support of the Austrian Academic Exchange Service during my study. The ﬁnancial support of the research in this thesis was partially from the Austrian Fonds zur F¨orderung der Wissenschaftlichen Forschung who with the Schwerpunkt Quantenoptik (Project No. S06502), Project No. F1506 and the TMR-Network ”The Physics of Quantum Infor- mation” of the European Commission.

The attentive reader might notice that a number of text paragraphs were taken from joint papers of our group because the formulations found there are diﬃcult to improve.

Finally I sincerely thank all my friends, from all over the world, who made my years in Innsbruck and Vienna so delightful. Chapter 1

Introduction

Superposition, one of the most distinct features of the quantum theory, has been demonstrated in numerous particle analogs of Young’s classic double- slit interference experiment, such as in electron interferometer [Marton et al., 1954], neutron interferometer [Rauch et al., 1974] and atom interferometer [Carnal and Mlynek, 1991; Keith et al., 1991]. However, in multiparticle systems the superposition principle yields phenomena that are much richer and more interesting than anything that can be seen in one-particle systems.

Quantum Entanglement, a simple name for superposition in a multiparticle system, was first noticed by Schrödinger [Schrödinger, 1935] and since then it has baffled generations of physicists. It is at the heart of the discussions of the Einstein-Podolsky-Rosen (EPR) paradox, of Bell’s inequality, and of the non-locality of quantum mechanics [Einstein et al., 1935; Bell, 1964]. In recent years, entanglement has become a new focus of activity in quantum physics because of immense theoretical and experimental progress both in the foundation of quantum mechanics and in the new field of quantum information science.

On the theoretical side, while the discovery of the conﬂict with local realism following from Greenberger-Horne-Zeilinger (GHZ) entanglement of three- or more particles [Greenberger et al., 1989; 1990] allows us to per-

5 6 CHAPTER 1. INTRODUCTION form novel and completely new tests of local realism without inequalities, the resource of entanglement has also many useful applications in quantum information processing [Bennett, 1995], including quantum computation and quantum communication. On the one hand, quantum computation, based on a controlled manipulation of entangled states of quantum bits, might allow us to build a new generation of computers which promise to be more pow- erful than their classical counterparts [Deutsch, 1985], for example, Shor’s discovery of quantum algorithms [Shor, 1994] enables us to factorize large integers exponentially faster than the best known classical algorithms. On the other hand, quantum communication schemes, such as quantum cryptography [Bennett et al, 1992a], dense coding [Bennett and Wiesner, 1992] and teleportation [Bennett et al., 1993], oﬀer more eﬃcient and secure ways for the exchange of information in a network.

On the experimental side, the current technology is beginning to allow us to manipulate rather than just observe individual quantum phenomena. This opens up the possibility of realizing these above proposals in real experiments. However, although there is fast progress in the theoretical description of quantum information processing, the difficulties in handling quantum systems have not yet allowed an equal advance in the experimental realization of the new proposals. Besides the promising developments of quantum cryptography (the first provably secure way to send secret messages), peoples have only recently succeeded in demonstrating the possibility of quantum dense coding [Mattle et al., 1996], a way to quantum mechanically enhance data compression. The main reason for this slow experimental progress is that, although there exist methods to produce pairs of entangled photons [Kwiat et al., 1995], entanglement has been demonstrated for atoms [Hagley et al., 1997] only very recently and it has not been possible thus far to produce entangled states of more than two quanta. Yet, all known methods of quantum computation are applications of entanglement. The present experimental challenge is therefore not to build a full-fledged universal quantum computer straight away but rather to progress from experiments in which we merely observe quantum interference and entanglement to experiments in which we 7 can control those quantum phenomena in the required way. All this above leads to the main motivation of our experimental efforts in this dissertation.

Late in 1997, our group successfully achieved the first experimental demonstration of quantum teleportation [Bouwmeester, Pan et al., 1997] in which we disembodied the polarization state of a photon into classical data and EPR correlations, and then used these ingredients to reincarnate the state in another photon which has never been anywhere near the first photon. Due to our first realization of quantum teleportation, experimental research in the field of quantum information is now attracting increasing attention from both academia and industry. In the year of 1998, several groups in the world achieved a series of important advances involving quantum computation and teleportation: In February, De Martini’s group [Boschi et al., 1998] reported an optical realization for Popescu’s scheme–a variant of the original teleportation proposal. In May, we [Pan et al., 1998b] experimentally realized entanglement swapping, that is, teleportation of completely undefined quantum state; Chuang and his coworkers, meanwhile, reported the first experimental realization of the Deutsch-Jozsa quantum algorithm using a bulk nuclear magnetic resonance (NMR) technique [Chuang et al., 1998]. Then in October, Kimble’s group [Furusawa et al., 1998] succeeded in teleporting information on the amplitude and phase of an entire light beam to another beam. In November, Nielsen et al. [Nielsen et al., 1998] teleported quantum information from the nucleus of carbon atom to that of a neighboring hydrogen atom.

Recently, according to the proposal for production of GHZentanglement out of entangled pairs [Zeilinger et al., 1997], we implemented a source of GHZ entanglement for three spatially separated photons [Bouwmeester, Pan et al., 1999], which is a further development of the technique that has been used in our previous experiments for teleportation and entanglement swapping. Such a source, for the first time, opens the door to demonstrate the GHZ theorem. The first three-particle test of local realism without inequalities has been done most recently [Pan et al., 1999a]. All these significant advances 8 CHAPTER 1. INTRODUCTION greatly promote experimental research both in the foundations of quantum mechanics and in the field of quantum information.

The aim of the thesis is to report the first experimental realization of teleportation for arbitrary quantum states, to report the first observation of GHZentanglement for three spatially separated photons, and to report the first demonstration of non-locality of quantum mechanics for nonstatistical predictions of the theory. The main contents of the dissertation are organized as follow:

Chapter 1 is a concise introduction to the applications of quantum entanglement. We brieﬂy review the current theoretical and experimental advances both in the foundation of quantum mechanics and in the new ﬁeld of quantum information science.

As all applications of entanglement necessitate both preparation and measurement of entangled states, in Chapter 2, we shall theoretically describe how quantum networks with simple quantum logic gates and a handful of qubits allow us to control and manipulate quantum entanglement. Due to the present absence of a general quantum gate, meanwhile we also present a practical way to generate and identify entangled states, which constitutes the basis of all experiments in the dissertation.

Quantum teleportation, the transmission and reconstruction over arbitrary distances of the state of a quantum system, is experimentally demonstrated in Chapter 3. We describe in detail the theoretical and experimental schemes of quantum teleportation. During teleportation, an initial photon which carries the polarization that is to be transferred and one of a pair of entangled photons are subjected to a measurement such that the second photon of the entangled pair acquires the polarization of the initial photon. Quantum teleportation will be a critical ingredient for future quantum computation networks.

Entanglement swapping enables one to entangle particles that never phys- 9 ically interacted with one another or which have never been dynamically cou- pled by any other means. Chapter 4 reports in detail an optical experimental realization of entanglement swapping. As we will see below, entanglement swapping, besides its interest to fundamental physics, will have a number of important applications in quantum communication.

In Chapter 5, we report the ﬁrst observation of polarization entanglement of three spatially separated photons. Such an entangled state is the long-coveted GHZstate. In addition to facilitating more advanced forms of quantum cryptography, our GHZstate will help provide a non-statistical test of the foundations of quantum physics.

Chapter 6 is concerned with the test of local realism via a GHZstate. Though previous experiments, based on observation of the entangled state of two photons, have provided highly convincing evidence against local realism, these ”Bell’s inequalities” tests require the measurement of many pairs of entangled photons to build up a body of statistical evidence against the idea. In contrast, the GHZtheorem described in the chapter shows that in principle studying a single set of properties in the GHZphotons could verify the predictions of quantum mechanics while contradicting those of local realism. Using the source exploited in Chapter 5, we present here the ﬁrst experimental demonstration of the GHZtheorem.

Chapter 7 involves conclusions and outlooks. In the chapter, we brieﬂy summarize the results in the thesis, and discuss the diﬀerences among those recent experimental advances of quantum teleportation. Finally, we also give some prospects for future experiments. Chapter 2

Manipulation of Entangled States

It is well known that the preparation and measurement of Bell states is essential for quantum dense coding [Bennett and Wiesner, 1992], for quantum teleportation [Bennett et al., 1993], and for entanglement swapping [Zukowski et al., 1993], the extension to the GHZsituation allows us to generalize the two to the multi-particle case. This chapter consists of three sections. First, we shall theoretically describe how using only single-quantum-bit (qubit) operations and controlled-NOT gates [Barenco et al., 1995a; 1995b] one can construct a suitable quantum network to produce and identify any of the maximally entangled states for any number of particles [Bruss et al., 1997]. Second, since until now such quantum networks have not yet been built in the laboratory, we shall also describe the existing Bell-state analyzer [Weinfurter, 1994; Braustein and Mann, 1995] and then present a universal scheme and practically realizable procedures, by which one can readily identify two of the maximally entangled states of any number of photons [Pan and Zeilinger, 1998]. At the end of this chapter a high intensity source of polarization-entangled photon pairs [Kwiat et al., 1995], which has been used in all experiments of the dissertation, is described, and some basic alignment procedures are discussed brieﬂy.

10 2.1. QUANTUM NETWORK AND ITS APPLICATIONS 11

Figure 2.1: Graphical representations of Hadamard and the quantum controlled-NOT gates. Here, a + b denotes addition modulo 2.

2.1 Quantum network and its applications

Generally, a bit is a classical system with two Boolean states 0 and 1, while a qubit means a generic two-state quantum system with a chosen ”computa- {| | } 1 tional basis” 0 , 1 (e.g. the polarization of a photon or a spin- 2 particle). A quantum logic gate is an elementary device which performs a ﬁxed unitary operation on selected qubits in a ﬁxed period of time. Single-qubit quantum logic gates are rather trivial and can be implemented, for example, by ex- citing selected atomic transitions with laser pulses of controllable frequency, intensity and duration. In fact, using a simple Hadamard gate and the quantum controlled-NOT gates, one can prepare and identify any of N-particle entangled states. The action of a Hadamard gate (Fig. 2.1a) is equivalent to the following unitary transformation:

|0→ √1 (|0 + |1) 2 (2.1) |1→ √1 (|0−|1) 2 12 CHAPTER 2. MANIPULATION OF ENTANGLED STATES and the controlled-NOT gate(Fig. 2.1b) ﬂips the second of two qubits if and only if the ﬁrst is |1,namely

|0|0→|0|0 |0|1→|0|1 (2.2) |1|0→|1|1 |1|1→|1|0

Consider now the network shown in Figure 2.2a. Under the action of the gates on the left-hand side of the network, the input two-particle states will undergo a series of unitary transformation. For example, if the input state is |0|0, after passing through the two gates it will be transformed into:

Hadmard |0|0 −−−−→ √1 (|0|0 + |1|0) 2 C−NOT 12 (2.3) −−−−→ √1 (|0|0| + |1|1) 2 which is one of the four maximally entangled Bell states,

|Ψ± = √1 (|0|1± |1|0) 2 (2.4) |Φ± = √1 (|0|0±|1|1) . 2

Correspondingly, the network could also prepare the two qubits in one of the remaining three Bell states:

1 |1|0−→√ (|0|0| − |1|1) (2.5) 2

1 |0|1−→√ (|0|1| + |1|0) (2.6) 2

1 |1|1−→√ (|0|1| − |1|0) (2.7) 2

It is easy to verify that the reversed quantum network (right-hand side of Figure 2.2a) can be used to implement the so-called Bell measurement on the 2.1. QUANTUM NETWORK AND ITS APPLICATIONS 13

Figure 2.2: (a) The Bell measurement: the gates on the left hand side allow us to generate the four Bell states from the four possible diﬀerent inputs. Reversing the order of the gates (right-hand side of the diagram) corresponds to a Bell measurement. (b) GHZ measurement: the same as in (a) for the eight GHZ states. 14 CHAPTER 2. MANIPULATION OF ENTANGLED STATES two qubits by disentangling the Bell states. In this way, the Bell measurement is reduced to two single-particle measurements. The method can be directly extended to a three-qubit case. Figure 2.2b shows how to prepare eight maximally entangled three-particle states, known as the GHZstates [Greenberger 1989,1990]. Reversing the procedure we obtain the unitary transformation which reduces the GHZmeasurement to the three single-particle measurements.

We can write this in the following compact form, where a and b can each take the values 0 and 1 anda ¯ and ¯b denote NOT-a and NOT-b, respectively:

1 |0|a|b⇐⇒√ |0|a||b + |1|a¯|¯b (2.8) 2

1 |1|a|b⇐⇒√ |0|a||b−|1|a¯|¯b (2.9) 2

Let us also mention that the GHZmeasurement provides an interesting possibility of labeling the GHZstates via the corresponding binary output. The three output bits then have the following meanings:

(ii) The second output bit indicates whether the ﬁrst two bits in the GHZ superposition are the same or diﬀerent. If the second output bit is |0,they are the same.

(iii) The third output bit provides the same information with respect to the ﬁrst and third bit of the GHZsuperposition.

Finally, we would like to emphasize that, in general, any measurement on any number of qubits can be implemented using only single-qubit operations 2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 15

Figure 2.3: The beam splitter coherently transforms two input spatial modes (a, b) into two output spatial modes (c, d)

and the quantum controlled-NOT gates.

This follows from the fact that the quantum controlled-NOT gate, together with relatively trivial single-qubit operations, forms an adequate set of quantum gates, i.e., the set from which any unitary operation may be built [Barenco et al., 1995a; 1995b]. Thus if we want to measure observable A pertaining to n qubits, we could construct a compensating unitary trans- n formation U which maps 2 states of the form |a1|a2...|an,whereai =0, 1, into the eigenstates of A. This allows both to prepare the eigenstates of A, which in general can be highly entangled, and to reduce the measurement described by A to n simple, single-qubit measurements.

2.2 Practical schemes for entangled-state analysis

Though quantum networks present a novel way to manipulate entangled states, their experimental realization remains to be a challenge for the future due to the lack of a quantum controlled-NOT gate. However, while no complete Bell-state measurement procedure exists, we can already experimentally identify two of the maximally entangled states of N photons. 16 CHAPTER 2. MANIPULATION OF ENTANGLED STATES

2.2.1 Bell-state analysis

The Bell-state analyzer ﬁrst suggested by Weinfurter et al. isbasedonthe two-photon interference eﬀect at a 50:50 standard beam splitter. The beam splitter has two spatial input modes a and b and two output modes c and d (Fig. 2.3). Quantum mechanically, the action of the beam splitter on the input modes can be written as

|a−→ √i |c + √1 |d 2 2 (2.10) |b−→ √1 |c + √i |d 2 2 where, e.g. |a describes the spatial quantum state of the particle in input beam a.Eq.2.10 describes the fact that the particle can be found with equal probability (50%) in either of the output modes c and d, no matter through which input beam it came. Here the factor i in Eq. 2.10 is a consequence of unitarity. It corresponds physically to a phase jump upon reﬂection at the semi-transparent mirror [Zeilinger, 1981]. Note that a standard beam splitter is polarization independent, and thus has no eﬀect on the polarization state of the photon.

Let us now consider our beam splitter with two incident photons, 1 and 2, photon 1 in input beam a, and photon 2 in input beam b. Suppose that photon 1 is in polarization state α|H1+β|V 1, and photon 2 is in polarization state γ|H2 + δ|V 2 (hereH and V denote horizontal and vertical linear polarizations, and |α|2 + |β|2 =1,|γ|2 + |δ|2 = 1). They each have the same probability p =0.5 to transmit the beam splitter or be reﬂected. Thus, four diﬀerent possibilities arise (Fig. 2.4).

(1) Both particles are reflected, (2) both particles are transmitted, (3) the upper particle is reflected, the lower one is transmitted, and (4) the upper one is transmitted and the lower one is reflected. Each of the four occurs with the same probability, and one has to investigate now whether any interference between these processes is possible. For distinguishable particles, for example for classical ones, no interference arises and we thus arrive at the prediction 2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 17

Figure 2.4: Two particles incident onto a beam splitter, one from each side. Four possibilities exist how the two particles can leave the beam splitter. 18 CHAPTER 2. MANIPULATION OF ENTANGLED STATES that in two of the cases, that is, with total probability p =0.5, the two particles end up in diﬀerent output ports and, with probability p =0.25, both particles end up in the upper output beam and, with the same probability p =0.25, they end up in the lower output beam.

Let us now assume that the two photons have the same frequency and arrive at the beam splitter simultaneously. As a result they are quantum mechanically indistinguishable. In this case it is not possible, not even in principle, to decide which of the incident particles ended up in a given output port, we therefore have to consider coherent superpositions of the amplitudes for these diﬀerent possibilities. To show how the Bell-state analyzer works, consider the input state

|ψ =(α|H + β|V )|a · i 1 1 1 (2.11) (γ|H2 + δ|V 2)|b2. where, for example, the ﬁrst term in the equation indicates photon 1 with a polarization state α|H1 + β|V 1 is in input mode a.

AsshowninEq.2.10, for photons 1 and 2 passing through the beam splitter their spatial modes will undergo a corresponding unitary transformation. The state in Eq. 2.11 thus evolves into

|ψ = √1 (α|H + β|V )(i|c + |d )· f 12 2 1 1 1 1 (2.12) √1 (γ|H + δ|V )(|c + i|d ). 2 2 2 2 2

It should be noted that photons 1 and 2 are not distinguishable anymore after passing through the beam splitter. The total two-photon state including both the spatial and the spin part, therefore, has to obey bosonic quantum statistics. This implies that the outgoing physical state must be symmetric under exchange of labels 1 and 2. To do so, one should symmetrize the state

|ψf 12, that is, also include its exchange wave-function

|ψ = √1 (α|H + β|V )(i|c + |d )· f 21 2 2 2 2 2 (2.13) √1 (γ|H + δ|V )(|c + i|d ). 2 1 1 1 1 2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 19

The ﬁnal outgoing state therefore reads

√1 |ψf = (|ψf 12 + |ψf 21), (2.14) 2 and consequently we have

|ψ = √1 [(αγ + βδ)(|H |H + |V |V ) · i(|c |c + |d |d ) f 2 2 1 2 1 2 1 2 1 2 +(αγ − βδ)(|H |H −|V |V ) · i(|c |c + |d |d ) 1 2 1 2 1 2 1 2 (2.15) +(αδ + βγ)(|H1|V 2 + |V 1|H2) · i(|c1|c2 + |d1|d2) +(αδ − βγ)(|H1|V 2 −|V 1|H2) · (|d1|c2 −|c1|d2)].

As we will see below, Eq. 2.15 allows us to readily project the two-photon state into two of the four maximally polarization-entangled states:

|Ψ± = √1 (|H |V ±|V |H ) 12 2 1 2 1 2 (2.16) |Φ± = √1 (|H |H ±|V |V ) . 12 2 1 2 1 2

From Eq. 2.15, it is easy to verify that the two photons proceed after the beam splitter in different emerging beams if, and only if, their polarization − state is in the state |Ψ 12 (refer to the fourth term of Eq. 2.15). Thus − we arrived at a possibility to identify one of the four Bell states, |Ψ 12, uniquely on the basis that it is the only one which gives rise to a detection of one photon in each of the outgoing beams of the beam splitter. For a full analysis, we further need a way to distinguish between the other three + + − states, |Ψ 12, |Φ 12,and|Φ 12. Again, one can easily find that it is only in + the state |Ψ 12 that the two emerging photons have different polarizations. ± In the two |Φ 12 states, they always share the same polarization. Thus, a further step in Bell-state analysis implies that one inserts a two-channel polarizer into each of the output ports of the beam splitter. Then, only the + state |Ψ 12 will give a coincidence count between the two output ports of ± the polarizer on either side of the beam splitter. Yet, both |Φ 12 states will give rise to the same joint detection of the two photons in either detector after the final polarizer. By making use of two-particle interference effects at a beam splitter, we can thus distinguish two of the four Bell states via two-fold coincidence analysis. 20 CHAPTER 2. MANIPULATION OF ENTANGLED STATES

2.2.2 GHZ-state analyzer

As an application of the concept of quantum erasure, we present in this part a practical GHZ-state analyzer for identifying two of the N-particle entangled states [Pan 1998a]. The basic elements of the experimental setup are just polarizing beam splitters(PBS) and half-wave plates(HWP).

Before starting to discuss the GHZ-state analyzer, let’s first give a mod- ified version of the Bell-state analyzer, which is similar to but different from the former one shown in Fig. 2.3. Consider the arrangement of Fig. 2.5. Two identical photons enter our Bell-state analyzer from modes A and B respectively. Suppose they are in the most general polarization-superposition state

|ψin = α|HA|HB + β|HA|VB + γ|VA|HB + δ|VA|VB (2.17)

Because the polarizing beam splitter PBS transmits only the horizontal polarization component and reﬂects the vertical component, after passing through

PBSAB the incident state will evolve into

|ψin−→α|HA2|HB1 + β|HA2|VB2 + γ|VA1|HB1 + δ|VA1|VB2 (2.18)

Where subscript ij(i = A, B, j =1, 2) denotes the transformation from input mode i to output mode j. Suppose that these two photons A and B arrive at the polarizing beam splitter PBSAB simultaneously, and therefore their spatial wavefuctions overlap each other. Then, according to the indistinguishability of identical particles, we can directly denote HA2 as H2, VA1 as V1, HB1 as H1,andVB2 as V2.Thus,Eq.2.18 reads

α|H1|H2 + β|H2|V2 + γ|V1|H1 + δ|V1|V2 (2.19)

Note that for the terms |H1|H2 and |V1|V2 the two photons are in diﬀerent output ports, and while for the terms |H2|V2 and |V1|H1 both 2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 21

Figure 2.5: A modiﬁed Bell-state analyzer. Two indistinguishable photons enter the Bell-state analyzer from input ports A and B. PBSAB,PBS1and PBS2 are three polarizing beam splitters, which transmit the horizontal polarization component and reﬂect the vertical component. DH1,DV 1,DH2,and DV 2 are four photon-counting detectors. 22 CHAPTER 2. MANIPULATION OF ENTANGLED STATES

photons are in the same output port. Therefore, we can identify states |ψd =

α|H1|H2+δ|V1|V2 and |ψs = β|H2|V2+γ|H1|V1 using the coincidence between detectors in mode 1 and in mode 2. Obviously, in terms of the Bell states, ψd can be rewritten as

√1 + √1 − |ψd = (α + δ)|Φ 12 + (α − δ)|Φ 12 (2.20) 2 2

Thus, in order to ﬁnish the Bell-state measurement we now only need to + − identify states |Φ 12 and |Φ 12, that is , we have to determine the relative phase between terms of |H1|H2 and |V1|V2. Let the angle between the HWP axis and the horizontal direction be 22.50 such that it corresponds to a450 rotation of the polarization. Therefore, for a photon passing the HWP, its polarization state will undergo the following unitary transformation: |H −→ √1 (|H + |V ) i 2 i i |V −→ √1 (|H −|V ) i 2 i i

+ − where i =1, 2. Finally, |Φ 12 and |Φ 12 will thus be transformed into

+ + √1 |Φ 12 →|Φ 12 = (|H1|H2 + |V1|V2) (2.21) 2

− + √1 |Φ 12 →|Ψ 12 = (|H1|V2 + |V1|H2) (2.22) 2

The above analysis show that we can easily identify two of the four incident Bell states. Speciﬁcally, if we observe a coincidence either between detectors D and D or D and D , then the incident state was √1 (|H |H + H1 H2 V 1 V 2 2 A B |VA|VB). On the other hand if we observe coincidence between detectors D and D or D and D , then the incident state was √1 (|H |H − H1 V 2 V 1 H2 2 A B |VA|VB). The other two incident Bell states will lead to no coincidence between detectors in mode 1 and in mode 2. Such states are signiﬁed by some kind of superposition of |HA|VB and |VA|HB. This concludes our demonstration that we can identify two of the four Bell states using the coincidence between modes 1 and 2. 2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 23

Figure 2.6: A GHZ-state analyzer. Three photons incident one each in modes A, B, and C will give rise to distinct 3-fold coincidence if they are in the GHZ- states |Φ+ or |Φ−. All the notations are the same as those in Fig. 2.5.

The reason why we discuss the modiﬁed version of Bell-state analyzer is that the above scheme can directly be generalized to the N-particle case. Making use of its basic idea, one can easily construct a type of GHZ-state analyzer by which one can immediately identify two of the 2N maximally entangled GHZstates.

For example, in the case of three identical photons, the eight maximally entangled GHZstates are given by

1 |Φ± = √ (|H|H|H±|V |V |V ) (2.23) 2 ± √1 |Ψ1 = (|V |H|H±|H|V |V ) (2.24) 2 24 CHAPTER 2. MANIPULATION OF ENTANGLED STATES

± √1 |Ψ2 = (|H|V |H±|V |H|V ) (2.25) 2 ± √1 |Ψ3 = (|H|H|V ±|V |V |H) (2.26) 2

± Where |Ψi designates that GHZ-state where the polarization of photon i is diﬀerent from the other two. Consider now the setup of Fig. 2.6 and suppose that three photons enter the GHZanalyzer, each one from modes A, B and C respectively. A suitable arrangement can be realized such that the photon coming from mode A and the one coming from mode B overlap at

PBSAB and, thus they are correspondingly transformed into mode 1 and into mode BC. Let’s further suppose that the photons from mode BC and mode

C overlap each other at PBSBC. Thus, following the above demonstration for the case of the Bell-state analyzer, it is easy to ﬁnd that the eight GHZ states above will correspondingly evolve into

√1 (|H1|H2|H3±|V1|V2|V3) (2.27) 2 √1 (|H1|H3|V3±|V1|V2|H2) (2.28) 2 √1 (|H2|V2|H3±|H1|V1|V3) (2.29) 2 √1 (|H1|V1|H2±|V2|H3|V3) (2.30) 2

immediately after these three photons passed through PBSAB and PBSBC, and before they enter the half-wave plates(HWP). Here, e.g. Hi denotes a photon with polarization H in output mode i.

From Eqs. 2.27-2.30, it is evident that one can observe three-fold coincidence between modes 1, 2 and 3 only for the state of Eq. 2.27. For the other states, there are always two particles in the same mode. We can thus distinguish the two states √1 (|H|H|H±|V |V |V ) from the other six GHZ 2 states. Furthermore, after the states of Eq. 2.27 pass through the HWP, we ﬁnally obtain 2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 25

√1 (|H |H |H + |V |V |V ) 2 1 2 3 1 2 3 → 1 | | | | | | 2 ( H1 H2 H3 + H1 V2 V3 (2.31) +|V1|H2|V3 + |V1|V2|H3) √1 (|H |H |H −|V |V |V ) 2 1 2 3 1 2 3 → 1 | | | | | | (2.32) 2 ( H1 H2 V3 + H1 V2 H3 +|V1|H2|H3 + |V1|V2|V3)

Thus, using three-fold coincidence we can readily identify the relative phase between states |H|H|H and |V |V |V . This is because only the initial + state |Φ leads to coincidence between detectors DH1,DH2 and DH3 (or − H1V2V3, V1H2V3, V1V2H3). On the other hand, only the state |Φ leads to coincidence between detectors DH1,DH2 and DV 3 (or H1V2H3, V1H2H3, + − V1V2V3). Or, in conclusion, states |Φ and |Φ are identiﬁed by coincidences between all three output modes 1, 2, and 3. They can be distinguished because behind the half wave plates |Φ+ results in one or three horizontally, and zero or two vertically polarized photons, while |Φ− results in zero or two horizontally polarized photons and one or three vertically polarized ones.

Our GHZ-state analyzer has many possible applications. For example the three photons entering via the modes A, B and C respectively could each come from one entangled pair. Then projection of these three photons using the GHZ-state analyzer onto the GHZ state Φ+ or Φ− implies that the other three photons emerging from each pair will be prepared in a GHZstate. It is clear that our scheme can readily be generalized to analyze entangled states consisting of more than three photons by just adding more polarizing beam splitters and half wave plates. Also, identiﬁcation of analogs of our scheme for GHZ-state analysis of atoms or of mode entangled states instead of polarization entangled ones is straightforward.

Here, it is worth noting that an extension of the above scheme would provide us with a conditional GHZ-state source by which we can conveniently observe four-particle GHZcorrelations and further prepare three freely propagating particles in a GHZ state [Zeilinger et al, 1997; Pan and Zeilinger, 1998; 26 CHAPTER 2. MANIPULATION OF ENTANGLED STATES

Figure 2.7: A three- and four-photon polarization-entanglement source. The photon sources, A and B, pumped by short pulses, each one emits a photon pair in the superposition HH+VV. Then, one photon coming from source A and one coming from source B overlap at the polarizing beam splitter PBS1. F is a narrow ﬁlter, PBS is a special polarizing beam splitter which transmits ◦ ◦ 45 polarization and reﬂects −45 polarization. DT 1 and DT 2 are two single- photon detectors. 2.2. PRACTICAL SCHEMES FOR ENTANGLED-STATE ANALYSIS 27

Pan and Zeilinger, 1999b]. Consider two sources A and B (see Fig. 2.7)each one emitting a photon pair. Consider for simplicity that the photons emitted by sources A and B are both in the same entangled state √1 (|H|H+|V |V ). 2 Then, using very analogous arguments as above we ﬁnd that the state of the four particles immediately after passage through the polarizing beam splitter PBS1 will be the superposition

1 | | | | 2 ( H1 H2 H3 H4 +|V |V |V |V 1 2 3 4 (2.33) +|H1|H3|V3|V4 +|V1|V2|H2|H4)

Again, only for the superposition |H1|H2|H3|H4 + |V1|V2|V3|V4 we would observe four-fold coincidence. Therefore, we then know these four particles are in the superposition |H1|H2|H3|H4 + |V1|V2|V3|V4 as soon as we observe four-fold coincidence. Note that here the GHZstate is not directly prepared but we know that the four particles are in a GHZstate under the condition that one particle each is detected in each of the outgoing beams 1, 2, 3 and 4. This is a much weaker condition than any post-selection procedure which might be based on properties of the particles. In an experiment our case will not be distinguishable from the real situation occurring anyway because of finite detector efficiency. That is, from a practical point of view, even if one definitely prepares a full GHZstate one only will observe four- fold coincidence in a fraction of time anyway. Thus, we conclude that using our conditional GHZ-state one will be able to experimentally demonstrate all features of a 4-particle GHZstate.

Yet, in the meantime we would like to note that our scheme in Fig. 2.7 also allows us to generate unconditional three-particle GHZstates via so- called entangled entanglement [Krenn and Zeilinger, 1996]. For example, one could analyze the polarization state of photon 2 by passing it through a polarizing beam splitter PBS selecting 45◦ and −45◦) polarization. Then the polarization state of the remaining three photons 1, 3 and 4 will be projected 28 CHAPTER 2. MANIPULATION OF ENTANGLED STATES into

√1 (|H1|H3|H4 + |V1|V3|V4 2

if and only if detector DT 1 detects a single photon. Correspondingly, the state of photons 1, 3 and 4 will be projected into

√1 (|H1|H3|H4−|V1|V3|V4 2

if and only if detector DT 2 detects a single photon. In the scheme, the detection of photon 2 actually plays the double role of both getting rid of the last two terms in Eq. 2.33 and projecting the remaining three photons into a spatially separated and freely propagating GHZstate. Such a GHZ-state could be extremely useful both in further test of local realism versus quantum mechanics and in future application of third-man cryptography.

Finally we would like to note that in all these schemes we have used the principle of quantum erasure in a way that behind our GHZ-state analyzer at least some of the photons registered cannot be identified anymore as to which source they came from. This implies very specific experimental schemes, because the particles might have been created at different times. One theoretical possibility is to apply the principle of ultra-coincidence [Zukowski et al., 1993]. This means that the photons must be registered within a time short compared to their coherence time. For practical reasons, i.e. the un- availability of sufficiently fast detectors, the scheme cannot be realized at present. Yet, alternatively, one can create the particles within a time interval small compared to their coherence times. This in practice implies the use of pulsed sources and of filters behind them which introduce coherence times larger than the pulse length [Zukowski et al., 1995]. Such a scheme has been successfully used in our following experiments on quantum teleportation, entanglement swapping and three-particle GHZentanglement. 2.3. POLARIZATION-ENTANGLED PHOTON PAIRS 29

2.3 Polarization-entangled photon pairs

Because of the same reason, i.e. the absence of a quantum logic gate, at the moment there are not so many possibilities to create states of entangled particles in the laboratory. Fortunately, the process of spontaneous parametric down-conversion provides mechanisms by which pairs of entangled photons can be produced with reasonable intensity and in good purity. In the down- conversion process, one uses a non-centrolsymmetric crystal with nonlinear electric susceptibility. In such a medium, an incoming photon can decay with relatively small probability into two photons in a way that energy and momentum inside the crystal are conserved.

Here we will describe a simple technique to produce polarization-entangled photon pairs using the process of noncollinear type-II parametric down- conversion [Kwiat et al., 1995]. In the experiment, the desired polarization- entangled state is produced directly out of a single nonlinear crystal [BBO (beta-barium-borate)]. In that process, the two photons are emitted with different polarizations(Fig. 2.8). Calculating the emission direction of the photons [Caro and Garuccio, 1994; Kwiat et al, 1995], one notices that photons of each polarization are emitted into one cone in such a way that momenta of two photons always add up to the momentum of the pump photon. Thus, the emission direction of each individual photon is completely uncertain within the cone, but once one photon is registered, and thus its emission direction is deﬁned, the other photon is found just exactly opposite from the pump beam on the other cone. The total quantum mechanical state is therefore extremely rich and is a superposition of all such pairs of emission modes.

The interesting point is now that the crystal can be cut and arranged such that the two cones intersect, as shown in Fig. 2.8. Then, along the lines of intersection, the polarization of neither photon is defined, but what is defined is the fact that the two photons have to have different polarizations. This contains all the necessary features of entanglement in a nutshell. Mea- surement on each of the photons separately is totally random and gives with 30 CHAPTER 2. MANIPULATION OF ENTANGLED STATES

Figure 2.8: Principle of type-II parametric down-conversion. Inside a nonlinear crystal(here, BBO), an incoming pump photon can decay spontaneously into two photons. Two down-converted photons arise polarized orthogonally to each other. Each photon is emitted into a cone, and the photon on the top cone is vertically polarized while its exactly opposite partner in the bottom cone is horizontally polarized. Along the directions where the two cones intersect, their polarizations are undeﬁned; all that is known is that they have to be diﬀerent, which results in polarization entanglement between the two photons in beams A and B. 2.3. POLARIZATION-ENTANGLED PHOTON PAIRS 31 equal probability vertical or horizontal polarization. But once one photon, for example photon A, is measured, the polarization of the other photon B is orthogonal! Choosing an appropriate basis, e.g. |H and |V ,thestate emerging through the two beams A and B thus is a superposition of |H|V and |V |H,say

√1 iα (|HA|V B + e |V A|HB) (2.34) 2 where the relative phase α arises from the crystal birefringence, and an overall phase shift is omitted.

Using an additional birefringent phase shifter (or even slightly rotating the down-conversion crystal itself), the value of α can be set as desired, e.g., to the values 0 or π. Somewhat surprisingly, a net phase shift of π may be obtained by a 90◦ rotation of a quarter wave plate in one of the paths. Similarly, a half wave plate in one path can be used to change horizontal polarization to vertical and vice versa. One can thus very easily produce any of the four EPR-Bell states in Eq. 2.16.

The birefringent nature of the down-conversion crystal complicates the actual entangled state produced, since the ordinary and extraordinary photons have different velocities inside the crystal, and propagate along different directions even though they become collinear outside the crystal (an effect well known from calcite prisms, for example). The resulting longitudinal and transverse walk-offs between the two terms in the state (2.34) are maximal for pairs created near the entrance face, which consequently acquire a relative time delay δT = L(1/uo − 1/ue)(L is the crystal length, and uo and ue are the ordinary and extraordinary group velocities, respectively) and a relative lateral displacement d = L tan ρ (ρ is the angle between the ordinary and extraordinary beams inside the crystal). If δT ≥ τc, the coherence time of the down-conversion light, then the terms in Eq. 2.34 become, in principle, distinguishable by the order in which the detectors would fire, and no interference will be observable. Similarly, if d is larger than the coherence width, 32 CHAPTER 2. MANIPULATION OF ENTANGLED STATES the terms can become partially labeled by their spatial location.

Because the photons are produced coherently along the entire length of the crystal, one can completely compensate for the longitudinal walk-off [Ru- bin et al., 1994]—after compensation, interference occurs pairwise between processes where the photon pair is created at distances ±x from the middle of the crystal. The ideal compensation is therefore to use two crystals, one in each path, which are identical to the down-conversion crystal, but only half as long. If the polarization of the light is first rotated by 90◦ (e.g., with a half wave plate), the retardations of the o and e components are exchanged and complete temporal indistinguishability is restored (δT =0).Thesame method provides optimal compensation for the transverse walk-off effect as well. Here, the compensation crystals were oriented along the same direction as that of the down-conversion crystal. In the following experiments we always slightly rotate the orientation of one of the compensation crystals to tune the relative phase α = π.

The BBO crystal used in our experiments is 2.0mm long and was cut at ◦ θpm =43.5 (the angle between the crystal optic axis and the pump) in order to result in a well-defined the intersection between the two cones. The two cone-overlap directions, selected by irises before the detectors, were consequently separated by 5◦ when the pump beam is precisely orthogonal to the surface of the crystal. The transverse walk-off d (0.2mm) was small compared to the coherent pump beam width (2mm), so the associated labeling effect was minimal. However, it was necessary to compensate for longitudinal walk-off, since our 2.0mm BBO crystal produced δT = 260fs, while τc [determined by the collection irises and interference filters (centered at 788nm, 4.6nm FWHM)] was about of the same order. As discussed above, we used ◦ an additional BBO crystal (1.0mm thickness, θpm =43.5 )ineachofthe paths, preceded by a half wave plate to exchange the roles of the horizontal and vertical polarizations. Chapter 3

Quantum Teleportation

3.1 Introduction

The dream of teleportation is to be able to travel by simply reappearing at some distant location. An object to be teleported can be fully characterized by its properties, which in classical physics can be determined by measurement. To make a copy of that object at a distant location one does not need the original parts and pieces–all that is needed is to send the scanned information so that it can be used for reconstructing the object. But how precisely can this be a true copy of the original? What if these parts and pieces are electrons, atoms and molecules? What happens to their individual quantum properties, which according to Heisenberg’s uncertainty principle cannot be measured with arbitrary precision?

Bennett et al. [Bennett et al., 1993] have suggested that it is possible to transfer the quantum state of a particle onto another particle– the process of quantum teleportation–provided one does not get any information about the state in the course of this transformation. This requirement can be fulfilled by using entanglement, according to Schrödinger, the essential feature of quantum mechanics [Schrödinger, 1935]. It describes correlations between quantum systems much stronger than any classical correlation could be.

33 34 CHAPTER 3. QUANTUM TELEPORTATION

The possibility of transferring quantum information is one of the cor- nerstones of the emerging ﬁeld of quantum communication and quantum computation [Bennett, 1995]. As we will see below, quantum teleportation is indeed not only a critical ingredient for quantum computation and communication, its experimental realization will also allow new studies of the fundamentals of quantum theory.

In the present chapter, we report the first experimental verification of quantum teleportation [Bouwmeester, Pan et al., 1997]. By producing pairs of entangled photons by the process of parametric down-conversion and using two-photon interferometry for analyzing entanglement, one could transfer a quantum property (in our case the polarization state) from one photon to another. The methods developed for this experiment will be of great importance both for exploring the field of quantum information as well as for future experiments on the foundations of quantum mechanics.

3.2 Quantum teleportation–the idea

3.2.1 The problem

To make the problem of transferring quantum information clearer suppose that Alice has some particle in a certain quantum state |Ψ and she wants Bob, at a distant location, to have a particle in that state. There is certainly the possibility to send Bob the particle directly. But suppose that the communication channel between Alice and Bob is not good enough at the time of the procedure to preserve the necessary quantum coherence or suppose that this would take too much time, which could easily be the case if |Ψ is the state of a more complicated or massive object. Then, what strategy can Alice and Bob pursue?

As mentioned above, no measurement that Alice can perform on |Ψ will be suﬃcient for Bob to reconstruct the state because the state of a quantum 3.2. QUANTUM TELEPORTATION–THE IDEA 35 system cannot be fully determined by measurements. Quantum systems are so evasive because they can be in a superposition of several states at the same time. A measurement on the quantum system will force it into only one of these states; this is often referred to as the projection postulate. We can illustrate this important quantum feature by taking a single photon, which can be horizontally or vertically polarized, indicated by the states |H and |V . It can even be polarized in the general superposition of these two states

|Ψ = α |H + β |V , (3.1) were α and β are two complex numbers satisfying |α|2 + |β|2 =1.Toplace this example in a more general setting we can replace the states |H and |V in Eq.(3.1)by|0 and |1, which refer to the states of any two-state quantum system. Superpositions of |0 and |1 are called qubits to signify the new possibilities introduced by quantum physics into information science [Schumacher, 1995].

If a photon in state |Ψ passes through a polarizing beamsplitter, a device that reﬂects (transmits) horizontally (vertically) polarized photons, it will be found in the reﬂected (transmitted) beam with probability |α|2 (|β|2). Then the general state |Ψ has been projected either onto |H or onto |V by the action of the measurement. We conclude that the rules of quantum mechanics, in particular the projection postulate, make it impossible for Alicetoperformameasurementon|Ψ by which she would obtain all the information necessary to reconstruct the state.

3.2.2 The concept of quantum teleportation

Although the projection postulate in quantum mechanics seems to bring Alice’s attempts to provide Bob with the state |Ψ to a halt, it was realized by Bennett et al. [Bennett et al., 1993] that precisely this projection postulate enables teleportation of |Ψ from Alice to Bob. During teleportation Alice 36 CHAPTER 3. QUANTUM TELEPORTATION will destroy the quantum state at hand while Bob receives the quantum state, with neither Alice nor Bob obtaining information about the state |Ψ.Akey role in the teleportation scheme is played by an entangled ancillary pair of particles which will be initially shared by Alice and Bob.

teleported state

ALICE U classical information BSM BOB

entangled pair initial state EPR-source

Figure 3.1: Scheme showing principle of quantum teleportation. Alice has a quantum system, particle 1, in an initial state which she wants to teleport to Bob. Alice and Bob also share an ancillary entangled pair of particles 2 and 3 emitted by an Einstein-Podolsky-Rosen(EPR) source. Alice then performs a joint Bell-state measurement (BSM) on the initial particle and one of the ancillaries, projecting them also onto an entangled state. After she has sent the result of her measurement as classical information to Bob, he can perform a unitary transformation (U) on the other ancillary particle resulting in it being in the state of the original particle.

Suppose particle 1 which Alice wants to teleport is in the initial state | | | Ψ 1 = α H 1 + β V 1 (Fig. 3.1), and the entangled pair of particles 2 and 3sharedbyAliceandBobisinthestate:

1 Ψ− = √ (|H |V −|V |H ) . (3.2) 23 2 2 3 2 3 3.2. QUANTUM TELEPORTATION–THE IDEA 37

That entangled pair is a single quantum system in an equal superposition of | | | | the states H 2 V 3 and V 2 H 3. The entangled state contains no information on the individual particles; it only indicates that the two particles will be in opposite states. The important property of an entangled pair is that as soon as a measurement on one of the particles projects it, say, onto |H the state of the other one is determined to be |V , and vice versa. How could a measurement on one of the particles instantaneously inﬂuence the state of the other particle which, can be arbitrarily far away?! Einstein, among many other distinguished physicists, could simply not accept this ”Spooky action at a distance”. But this property of entangled states has now been demonstrated by numerous experiments (for reviews, see refs. [Clauser and Shimony, 1978; Greenberger et al., 1993].

The teleportation scheme works as follows. Alice has the particle 1 in the | initial state Ψ 1 and the ancillary particle 2. Particle 2 is entangled with the other ancillary particle 3 in the hands of Bob. Although this establishes the possibility of nonclassical correlations between Alice and Bob, the entangled | pair at this stage contains no information about Ψ 1. Indeed the entire system, comprising Alice’s unknown particle 1 and the entangled pair is in | | − a pure product state, Ψ 1 Ψ 23, involving neither classical correlation nor quantum entanglement between the unknown particle and the entangled pair. Therefore no measurement on either member of the entangled pair, or both | together, can yield any information about Ψ 1.

The essential point to achieve teleportation is to perform a joint Bell-state measurement on particles 1 and 2 which projects them onto one of the four entangled states :

|Ψ± = √1 (|H |V ±|V |H ) 12 2 1 2 1 2 (3.3) |Φ± = √1 (|H |H ±|V |V ) . 12 2 1 2 1 2

Note that these four states are a complete orthonormal basis for particles 1 and 2. The complete state of the three particles before Alice’s measurement 38 CHAPTER 3. QUANTUM TELEPORTATION is

|Ψ = √α (|H |H |V −|H |V |H ) 123 2 1 2 3 1 2 3 (3.4) + √β (|V |H |V −|V |V |H ) . 2 1 2 3 1 2 3

| | In this equation, each direct product 1 2 canbeexpressedintermsofthe four Bell states, and one can thus rewrite Eq.(3.4)as

| 1 | − − | − | Ψ 123 = 2 [ Ψ 12 ( α H 3 β V 3) | + − | | + Ψ 12 ( α H 3 + β V 3) | − | | (3.5) + Φ 12 (α V 3 + β H 3) | + | − | + Φ 12 (α V 3 β H 3)] .

| It follows that, regardless of the unknown state Ψ 1, the four Bell-state measurement outcomes are equally likely, each occurring with probability 1/4. Quantum physics predicts that once particles 1 and 2 are projected into one of the four entangled states, particle 3 is instantaneously projected into one of the four pure states superposed in Eq.(3.5). Denoting | H by the 1 0 vector and | V by , they are thus, respectively, 0 1

−10 −|Ψ , |Ψ , 3 01 3 (3.6) 01 0 −1 |Ψ , |Ψ . 10 3 10 3

| | | where Ψ 3 = α H 3+β V 3. Each of these possible resultant states for Bob’s | EPR particle 3 is related in a simple way to the original state Ψ 1 which Alice sought to teleport. In the case of the ﬁrst (singlet) outcome, the state of particle 3 is the same as the initial state of particle 1 except for an irrelevant phase factor, so Bob need do nothing further to produce a replica of Alice’s unknown state. In the other three cases, Bob could accordingly apply one of the unitary transformations in Eq.(3.6) to convert the state of particle 3 into the original state of particle 1, after receiving via a classical communication channel the information which of the Bell-state measurement results was 3.2. QUANTUM TELEPORTATION–THE IDEA 39 obtained by Alice. (For the photon polarization state, one can use a suitable combination of half-wave plates to perform these unitary transformations.) After Bob’s unitary operation, the ﬁnal state of particle 3 is therefore

| | | Ψ 3 = α H 3 + β V 3 . (3.7)

Note that during the Bell-state measurement particle 1 loses its identity | because it becomes entangled with particle 2. Therefore the state Ψ 1 is destroyed on Alice’s side during teleportation.

The result in Eq.(6.2) deserves some further comments. The transfer of quantum information from particle 1 to particle 3 can happen over arbitrary distances, hence the name teleportation. Experimentally, quantum entanglement has been shown to survive over distances of the order of 10 km [Tittel et al., 1998a; 1998b]. We note that in the teleportation scheme it is not necessary for Alice to know where Bob is. Furthermore, the initial state of particle 1 can be completely unknown not only to Alice but to anyone. It could even be quantum mechanically completely undeﬁned at the time the Bell-state measurement takes place. This is the case when, as already re- marked by Bennett et al. [Bennett et al., 1993], particle 1 itself is a member of an entangled pair and therefore has no well-deﬁned properties on its own. This ultimately leads to entanglement swapping [Zukowski et al., 1993, Bose et al., 1998].

It is also important to notice that the Bell-state measurement does not reveal any information on the properties of any of the particles. This is the very reason why quantum teleportation using coherent two-particle superpositions works, while any measurement on one-particle superpositions would fail. The fact that no information whatsoever on either particle is gained is also the reason why quantum teleportation escapes the verdict of the no- cloning theorem [Wootters and Zurek, 1982]. After successful teleportation particle 1 is not available in its original state anymore, and therefore particle 3 is not a clone but really the result of teleportation. 40 CHAPTER 3. QUANTUM TELEPORTATION

3.3 Experimental teleportation

3.3.1 Experimental scheme

Teleportation necessitates both production and measurement of entangled states; these are the two most challenging tasks for any experimental realization. Thus far there are only a few experimental techniques by which one can prepare entangled states, and there exists no experimentally realized procedure to identify all four Bell-states for any kind of quantum system composed of two separate particles. However, as discussed in Chapter 2 entangled pairs of photons can readily be generated and they can be projected onto at least two of the four Bell states.

Using the technique in section 2.3, we were able to produce the entangled photons 2 and 3 by type II parametric down-conversion (Fig. 3.2). Inside a non-linear crystal, an incoming pump photon can decay spontaneously into two photons which are in the polarization entangled state given by equation (3.2).

For practical convenience, in the experiment we decided to analyze only | − the projection onto Ψ 12 . As discussed in section 2.2.1, this projection is realized by interfering the two photons, 1 and 2, at a 50 : 50 beam splitter and, detecting a coincidence between the two detectors at the diﬀerent output ports of the beam splitter. Here, such a coincident detection acts as a | − projection onto Ψ 12. Since originally the polarization state of photon 2 is completely undetermined, the combined state between photons 1 and 2 is in an equal superposition of the four Bell states. As a result, in one out of four cases on average a coincidence will be recorded by the two detectors behind | − the beam splitter; that is, a projection onto Ψ 12 takes place.

| − It is clear that once particles 1 and 2 are projected into Ψ 12, particle | 3 is instantaneously projected into Ψ 3. Yet we note, with emphasis, that | − even we choose to identify only one of the four Bell states, here Ψ 12, 3.3. EXPERIMENTAL TELEPORTATION 41

Figure 3.2: Experimental Setup. A pulse of ultraviolet light passing through a non-linear crystal creates the ancillary pair of entangled photons 2 and 3. After retroﬂection during its second passage through the crystal the ultraviolet pulse creates another pair of photons, one of which will be prepared in the initial state of photon 1 to be teleported, the other one serves as a trigger indicating that a photon to be teleported is under way. Alice then looks for coincidences after a beam splitter (BS) where the initial photon and one of the ancillaries are superposed. Bob, after receiving the classical information that Alice obtained a coincidence count in detectors f1 and f2 identifying the − |Ψ 12 Bell-state, knows that his photon 3 is in the initial state of photon 1 which he then can check using polarization analysis with the polarizing beam splitter (PBS) and the detectors d1 and d2. The detector P provides the information that photon 1 is under way. 42 CHAPTER 3. QUANTUM TELEPORTATION teleportation is successfully achieved, albeit only in a quarter of the cases. As mentioned already, if we further insert a two-channel polarizer into each of the outputs of the beam splitter BS, then a coincident detection between the two outputs of the polarizer on either side of the BS acts as a projection onto | + Ψ 12. Thus, a slight change of our scheme is actually capable of achieving teleportation in 50% of the time–in those occasions when Alice happened to | − | + detect state Ψ 12 or Ψ 12.

Note that in our teleportation scheme the Bell-state analysis relies on the interference of two independently created photons. One, therefore, has to guarantee good spatial and temporal overlap at the beam splitter and, above all, one has to erase all kinds of path information for photons 1 and 2. Especially the strong time and frequency correlations of the two photons 2 and 3 created by parametric down-conversion can give rise to Welcher-Weg information for the interfering photons [Herzog et al., 1995].

There are two possibilities for quantum erasure. In the ﬁrst one, Welcher- Weg information is erased by detecting photons 1 and 2 within time intervals much shorter than their coherence time [Zukowski et al., 1993]. Then, such ultracoincident registrations are too close in time to discriminate which of the detected photons shares the source with photon 3 or with photon 4, respectively. Yet, this method cannot be used in practice due to the poor time resolution of the existing single-photon detectors (typically 0.5ns for Si-avalanche photodiodes as compared to typical coherence times of about 500fs).

The second possibility involves increasing the coherence times of the interfering photons to become much longer than the time interval within which they are created [Zukowski et al., 1995]. Then again, one cannot infer anymore which of the detected photons 1 or 2 was created together with photon 3, or with photon 4, respectively. In our experiment UV pulses with a duration of 200fs are used to create the photon pairs. We then chose narrow bandwidth filters (∆λ =4.6nm) in front of the detectors registering photons 1 and 2. The resulting coherence time of about 500fs is sufficiently longer 3.3. EXPERIMENTAL TELEPORTATION 43 than the pump pulse duration. Furthermore, single-mode fiber couplers act- ing as spatial filters were used to guarantee good mode overlap of the detected photons.

Figure 3.2 is a schematic drawing of the experimental setup. UV pulses are produced by frequency doubling the pulses of a commercial mode locked

Ti:sapphire laser from 788 to 394nm using a nonlinear LBO crystal (LiB3O5). For a repetition rate of 76 MHz we obtained an averaged power of 500 mW.

Passing the UV pulses through a BBO crystal (β−BaB2O4) creates via type- | − II down-conversion a pair of photons, 2 and 3, in the entangled state Ψ 23. After reﬂection, the pump pulse passes the crystal again and produces the second pair of photons 1 and 4. Photon 1 is prepared in the initial state to be teleported, and its partner, photon 4, serves to indicate that it was emitted.

How can one experimentally prove that an arbitrary unknown quantum state can be teleported? First, one has to show that teleportation works for a (complete) basis, a set of known states into which any other state can be decomposed. A basis for polarization states has just two components, and in principle we could choose as the basis horizontal and vertical polarization as emitted by the source. Yet this would not demonstrate that teleportation works for any general superposition, because these two directions are preferred directions in our experiment. Therefore, in the ﬁrst demonstration we chose as the basis for teleportation the two states linearly polarised at -45◦ and +45◦ which are already superpositions of the horizontal and vertical polarizations. Second, one has to show that teleportation works for superpositions of these base states. Therefore we also demonstrate teleportation for circular polarisation. This covers all three mutually orthogonal axes of the Poincar´e sphere.

3.3.2 Results

In the ﬁrst experiment photon 1 is polarized at 45◦. Teleportation should − work as soon as photon 1 and 2 are detected in the |Ψ 12 state, which occurs 44 CHAPTER 3. QUANTUM TELEPORTATION

− in 25% of all possible cases. The |Ψ 12 state is identiﬁed by recording a coincidence between two detectors, f1 and f2, placed behind the beamsplitter (Fig. 3.2).

If we detect a f1f2 coincidence (between detectors f1 and f2), then photon 3 should also be polarized at 45◦. The polarization of photon 3 is analysed by passing it through a polarizing beamsplitter selecting +45◦ and −45◦ polarization. To demonstrate teleportation, only detector d2 at the +45◦ output of the polarizing beamsplitter should click (that is, register a detection) once detectors f1 and f2 click. Detector d1 at the −45◦ output of the polarizing beamsplitter should not detect a photon. Therefore, recording a three-fold coincidence d2f1f2 (+45◦ analysis) together with the absence of a three-fold coincidence d1f1f2 (−45◦ analysis) is a proof that the polarization of photon 1 has been teleported to photon 3.

To meet the condition of temporal overlap, we change in small steps the arrival time of photon 2 by changing the delay between the ﬁrst and second down conversion by translating the retroﬂection mirror (Fig. 3.2). In this way we scan into the region of temporal overlap at the beamsplitter so that teleportation should occur.

Outside the region of teleportation photon 1 and 2 each will go either to f1 or to f2 independent of one another. The probability to have a coincidence between f1 and f2 is therefore 50%, which is twice as high as inside the region of teleportation. Photon 3 should not have a well-deﬁned polarization because it is part of an entangled pair. Therefore, d1 and d2 have both a 50% chance of receiving photon 3. This simple argument yields a 25% probability both for the −45◦ analysis (d1f1f2 coincidences) and for the +45◦ analysis (d2f1f2 coincidences) outside the region of teleportation. Figure 3 summarizes the predictions as function of the delay. Successful teleportation of the +45◦ polarization state is then characterized by a decrease to zero in the −45◦ analysis (Fig. 3.3a), and by a constant value for the +45◦ analysis (Fig. 3.3b). 3.3. EXPERIMENTAL TELEPORTATION 45

Theory: +45° Teleportation

0,25

y 0,20

0,15

0,10

0,05 -45° (a) 0,00 0,25

0,20

0,15

0,10

3-fold coincidence probabilit 0,05 +45° (b) 0,00 -100 -50 0 50 100 Delay (µm)

Figure 3.3: Theoretical prediction for the three-fold coincidence probability between the two Bell-state detectors (f1, f2) and one of the detectors analysing the teleported state. The signature of teleportation of a photon polarization state at +45◦ is a dip to zero at zero delay in the three-fold coincidence rate with the detector analysing -45◦ (d1f1f2) (a) and a constant value for the detector analysis +45◦ (d2f1f2) (b). The shaded area indicates the region of teleportation. 46 CHAPTER 3. QUANTUM TELEPORTATION

The theoretical prediction of Fig. 3.3 may easily be understood by realizing that at zero delay there is a decrease to half in the coincidence rate for the two detectors of the Bell-state analyser, f1 and f2, as compared to outside the region of teleportation. Therefore, if the polarization of photon 3 were completely uncorrelated to the others the three-fold coincidence should also show this dip to half. That the right state is teleported is indicated by the fact that the dip goes to zero in Fig. 3.3a and it is ﬁlled to a ﬂat curve in Fig. 3.3b.

We note that about as likely as production of photons 1, 2, and 3 is emission of two pairs of down-converted photons by a single source. Although there is no photon coming from the second source (photon 1 is absent), there will still be a signiﬁcant contribution to the three-fold coincidence rates. These coincidences have nothing to do with teleportation and can be identi- ﬁed by blocking the path of photon 1.

The probability for this process to yield spurious two- and three-fold coincidences can be estimated by taking into account the experimental parameters. The experimentally determined value for the percentage of spurious three-fold coincidences is 68% ± 1%. In the experimental graphs of Fig. 3.4 we have subtracted the experimentally determined spurious coincidences.

The experimental results for teleportation of photons polarized under +45◦ is shown in the left column of Fig. 3.4;Fig.3.4a and b should be compared with the theoretical predictions as shown in Fig. 3.3. The strong decrease in the −45◦ analysis, and the constant signal for the +45◦ analysis, indicate that photon 3 is polarized along the direction of photon 1, conﬁrming teleportation.

The results for photon 1 polarized at −45◦ demonstrate that teleportation works for a complete basis for polarization states (right-hand column of Fig. 3.4). To rule out any classical explanation for the experimental results, we have produced further conﬁrmation that our procedure works by additional experiments. In these experiments we teleported photons linearly 3.3. EXPERIMENTAL TELEPORTATION 47

+45° Teleportation -45° Teleportation

600 600

500 500

400 400

300 300

200 200 -45° -45° 100 100 (a) (c) 0 0

500 500

400 400

300 300

200 200 +45° +45° 100 100 (b) (d)

3-fold coincidences per 2000 seconds 0 0 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 Delay (µm) Delay (µm)

Figure 3.4: Experimental results. Measured three-fold coincidence rates d1f1f2 (-45◦) and d2f1f2 (+45◦) in the case that the photon state to be teleported is polarized at +45◦ (a and b) or at -45◦ (c and d). The coincidence rates are plotted as function of the delay (in µm) between the arrival of photon 1 and 2 at Alice’s beamsplitter (see Fig.1.2). The three-fold coincidence rates are plotted after subtracting the spurious three-fold background contribution (see text). These data, compared with Fig.1.3, together with similar ones for other polarisations (Table 1) conﬁrm teleportation for an arbitrary state. 48 CHAPTER 3. QUANTUM TELEPORTATION polarized at 0◦ and at 90◦, and also teleported circularly polarized photons. The experimental results are summarized in Table 1, where we list the visibility of the dip in three-fold coincidences, which occurs for analysis orthogonal to the input polarization.

polarization visibility +45◦ 0.63 ± 0.02 −45◦ 0.64 ± 0.02 0◦ 0.66 ± 0.02 90◦ 0.61 ± 0.02 circular 0.57 ± 0.02

Table 1

As mentioned above, the values for the visibilities are obtained after subtracting the oﬀset caused by spurious three-fold coincidences. These can experimentally be excluded by conditioning the three-fold coincidences on the detection of photon 4, which eﬀectively projects photon 1 into a single- particle state. We have performed this four-fold coincidence measurement for the case of teleportation of the +45◦ and +90◦ polarization states, that is, for two non-orthogonal states. The experimental results are shown in Fig. 3.5. Visibilities of 70% ± 3% are obtained for the dips in the orthogonal polarization states! Here, these visibilities are directly the degree of polarization of the teleported photon in the right state without any background subtracted.

As can be seen from the measured visibilities, the teleportation ﬁdelity is rather high in our experiment. Typically, it is of the order of 0.85. This very clearly surpasses the limit of 2/3 [Massar and Popescu, 1995] which at best could have been obtained by Alice performing a polarisation measurement on the given photon, informing Bob about the measurement result via classical communication, and by Bob repreparing the photon state at his output. The measured high ﬁdelity proves that we demonstrated teleportation of the quantum state of a single photon. 3.3. EXPERIMENTAL TELEPORTATION 49

45° Teleportation 90° Teleportation

120 100 100 80 80 60 60 40 40 -45° 0° 20 (a) 20 (c) 0 0

100 120 100 80 80 60 60 40 +45° 40 +90° 20 (b)20 (d)

4-fold coincidences per 4000 seconds 0 0 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 Delay (µm) Delay (µm)

Figure 3.5: Four-fold coincidence rates (without background subtraction). Conditioning the three-fold coincidences as shown in Fig.1.4 on the registration of photon 4 (see Fig.1.2) eliminates the spurious three-fold background. Graphs a and b show the four-fold coincidence measurements for the case of teleportation of the +45◦ polarization state, and graphs c and d show the results for the +90◦ polarization state. The visibilities, and thus the polarizations of the teleported photons, obtained without any background subtraction are 70% ± 3%. These results for teleportation of two non-orthogonal states prove that we demonstrated teleportation of the quantum state of a single photon. 50 CHAPTER 3. QUANTUM TELEPORTATION

3.4 Discussion

In our experiment, we used pairs of polarization entangled photons as produced by pulsed down-conversion and two-photon interferometric methods to transfer the polarization state of one photon onto another one. But, teleportation is by no means restricted to this system. In addition to pairs of entangled photons or entangled atoms [Hagley 1997, Fry 1995], one could imagine entangling photons with atoms or photons with ions, and so on. Then teleportation would allow us to transfer the state of, for example, fast- decohering, short-lived particles onto some more stable systems. This opens the possibility of quantum memories, where the information of incoming photons is stored on trapped ions, carefully shielded from the environment.

Furthermore, entanglement puriﬁcation [Bennett et al., 1996a] is a scheme of improving the quality of entanglement if it was degraded by decoherence during storage or transmission of the particles over noisy channels. Then it becomes possible to send the quantum state of a particle to some place, even if the available quantum channels are of very poor quality and thus sending the particle itself would very probably destroy the fragile quantum state. The feasibility of preserving quantum states in a hostile environment will have great advantages in the realm of quantum computation. The teleportation scheme can be used to provide links between quantum computers.

Quantum teleportation is not only an important ingredient in quantum information tasks; it also allows new types of experiments and investigations on the foundations of quantum mechanics. As any arbitrary state can be teleported, so can the fully undetermined state of a particle which is member of an entangled pair. By doing so, one can transfer the entanglement between particles. This allows us not only to chain the transmission of quantum states over distances, where decoherence would have already destroyed the state completely, but it also enables us to perform a test of Bell’s theorem on particles which do not share any common past, a new step in the investigation of the features of quantum mechanics. Last but not least, novel experiments 3.4. DISCUSSION 51 disproving concepts of a the local realistic character of nature become possible if one uses features of the experiment presented here to generate entanglement between more than two spatially separated particles [Greenberger et al., 1990; Zeilinger et al., 1997]. A ﬁrst such experiment will be reported in Chapter 6.

Since our first experimental realization, quantum teleportation has been one of the main focus in the field of quantum information physics. In the past two years, several groups in the world have achieved a number of experimental advances in quantum teleportation. It is worth comparing these experiments with our scheme and, briefly discussing the differences among them.

Though Popescu’s suggestion, as realized by De Martini’s group in Rome [Boschi et al., 1998], provides an elegant way to demonstrate some relevant Hilbert-space formalism in the absence of a two-photon Bell measurement, it is not quite proper to claim that the Rome experiment does constitute teleportation. What really constitutes teleportation? An essential criterion is to be able to teleport any independent quantum state coming from outside. This is obviously not possible in the Rome experiment, where the initial photon has to be entangled from the beginning with the ﬁnal one. One might argue that the scheme by Popescu is equivalent to the original teleportation scheme up to a local operation since, in principle, any unknown state of a particle from outside could be swapped onto the polarization degree of freedom of Alice’s EPR particle by a local unitary operation. However, such a local unitary operation would require a quantum C-NOT gate that does not exist yet. Further we note, with emphasis, that an experimental realization of quantum C-NOT gate itself leads a complete Bell measurement and thus a full realization of the original teleportation scheme, Popescu’s scheme is therefore not necessary anymore.

Generally speaking, the basic criterions to constitute a bona fide teleportation should be (1) the experimental scheme is capable of teleporting any state that is designed to teleport, (2) a fidelity better than 2/3 [Massar and Popescu, 1995], (3) for future applications, at least in principle, the scheme should be able to be extended to long-distance teleportation, that is, the real 52 CHAPTER 3. QUANTUM TELEPORTATION teleportation. Indeed, at the moment only the interference of independently created photons makes teleportation of any independent and even undefined (not just simply unknown) photon state possible. Our realization of entanglement swapping reported in Chapter 4 will underline the quantum nature of our teleportation procedures.

The two most recent teleportation experiments were reported by Kim- ble’s group [Furusawa et al., 1998] and Laflamme’s group [Nielson et al., 1998], respectively. An obvious advantage of these two schemes is that the input quantum state can be, in principle, teleported with an efficiency close to 100%. However, it should be pointed out that while a rather low fidelity (0.58) was observed in the experiment of Furusawa et al., their scheme is hardly to be extended to long-distance case because of the unavoidable dis- persion of squeezed-states during the distribution of entanglement, which consequently leads to a fast degrading of the quality of squeezed-state entanglement. Finally, it is also worthwhile noting that the NMR method used by Nielson et al. can never teleport a quantum state over macroscopic distance. Chapter 4

Entanglement Swapping

4.1 Introduction

The phenomenon of entanglement is a remarkable feature of quantum theory. It plays a crucial role in the discussions of the Einstein-Podolsky-Rosen paradox, of Bell’s inequalities, and of the non-locality of quantum mechanics. Thus far entanglement has either been realized by having the two entangled particles emerge from a common source [Freedman 1972, Rarity 1990], or from having two particles interact either directly or indirectly with each other [Lamehi-Rachti and Mittig, 1976; Hagley et al., 1997]. Yet, an alternative possibility to obtain entanglement is to make use of a projection of the state of two particles onto an entangled state. This projection measurement does not necessarily require a direct interaction between the two particles: When each of the particles is entangled with one other partner particle, an appropriate measurement, for example, a Bell-state measurement, of the partner particles will automatically collapse the state of the remaining two particles into an entangled state. This striking application of the projection postulate is referred to as entanglement swapping [Zukowski et al., 1993]. Here, we report the ﬁrst experimental realization for entanglement swapping [Pan 1998b]. In our experiment we take two pairs of polarization entangled photons and subject one photon from each pair to a Bell-state measurement.

53 54 CHAPTER 4. ENTANGLEMENT SWAPPING

This results in projecting the other two outgoing photons into an entangled state.

4.2 Theoretical scheme

Consider the arrangement of Fig. 4.1. There are two EPR sources, each one simultaneously emitting a pair of entangled particles. In anticipation to our experiments we assume that these are polarization entangled photons in the state

| 1 | | −| | Ψ 1234 = 2 ( H 1 V 2 V 1 H 2) × | | −| | . (4.1) ( H 3 V 4 V 3 H 4)

Here |H and |V indicate the states of a horizontally and a vertically polarized photon, respectively. The total state describes the fact that photons 1 and 2 are entangled in a singlet state in polarization and photons 3 and 4 are also entangled in the singlet state. Yet, the state of pair 1-2 is factorizable from the state of pair 3-4, that is, there is no entanglement of any of the photons 1 or 2 with any of the photons 3 or 4.

We now perform a joint Bell-state measurement on photons 2 and 3, that is, photons 2 and 3 are projected into one of the four Bell states which form a complete basis for the combined state of photons 2 and 3

|Ψ± = √1 (|H |V ±|V |H ) 23 2 2 3 2 3 (4.2) |Φ± = √1 (|H |H ±|V |V ) . 23 2 2 3 2 3

This measurement projects photons 1 and 4 also onto a Bell state, a diﬀerent one depending on the result of the Bell-state measurement for photons 2 and 3. To consider a speciﬁc example let us assume that the result of the Bell- state measurement of photons 2 and 3 is |Ψ− then it can be seen that the resulting state for photons 1 and 4 is also |Ψ−. In fact, close inspection shows that for the initial state given in Eq. (4.1) the emerging state of photons 1 4.2. THEORETICAL SCHEME 55

Figure 4.1: Principle of entanglement swapping. Two EPR sources produce two pairs of entangled photons, pair 1-2 and pair 3-4. One photon from each pair (photon 2 and 3) is subjected to a Bell-state measurement (BSM). This results in projecting the other two outgoing photons 1 and 4 into an entangled state. Change of the shading of the lines indicates a change in the set of possible predictions that can be made. and 4 will be identical to the one photon 2 and 3 collapsed into. This is a consequence of the fact that the state of Eq. (4.1) can be rewritten as

| 1 | + | + −| − | − Ψ 1234 = 2 ( Ψ 14 Ψ 23 Ψ 14 Ψ 23 −| + | + | − | − . (4.3) Φ 14 Φ 23 + Φ 14 Φ 23)

In all cases photons 1 and 4 emerge entangled despite the fact that they never interacted in the past. In Fig. 4.1 entangled particles are indicated by the same line darkness. Note that particles 1 and 4 become entangled after the Bell-state measurement (BSM) on particles 2 and 3.

The result above can also be interpreted as teleportation of the unknown state of, say, photon 2 onto photon 4 [Bennett et al., 1993]. In that case one could consider Alice performing the Bell-state measurement on photons 2 and 3, telling Bob, who is in possession of photon 4, the result of the Bell-state measurement. Then, by performing one of a fixed set of unitary operations on photon 4, photons 1 and 4 will be left in an singlet state, 56 CHAPTER 4. ENTANGLEMENT SWAPPING which is exactly the same as the state of photons 1 and 2 before the Bell- state measurement. It is conceptually most interesting to realize that in this case the teleported photon state does not have any well-defined polarization, because it is entangled with photon 1. It is fair to say that here we do not teleport some unknown state of a photon but rather an in principle undefined state. The state of photon 2, and therefore also of the teleported photon 4, is certainly undefined before any measurements are performed on photons 1 or 4.

It is worthwhile noting that the process of entanglement swapping also gives a means to define that an entangled pair of photons, 1 and 4, is available. As soon as Alice completes the Bell-state measurement on particles 2 and 3, we know that photons 1 and 4 are on their way ready for detection in an entangled state. An experimental realization of entanglement swapping will thus give, for the first time, the possibility to perform a test of Bell’s inequality using a pair of photons that never interacted. That is a big step towards the final realization of so called ”event-ready detections” of the entangled particles, a concept suggested by John Bell [Clauser and Shimony, 1978; Bell, 1980].

4.3 Experimental entanglement swapping

As in our teleportation experiment, here we also analyzed only the projection | − onto Ψ 23.Figure4.2 is a schematic drawing of the experimental setup. A UV laser pulse passing through a BBO crystal creates via type-II parametric down-conversion the ﬁrst pair of entangled photons, 1 and 2, in the state | − Ψ 12. After reﬂection the pump pulse passes the crystal again and creates | − the second pair of photons, 3 and 4, in the state Ψ 34. Note that the two pairs are created independently of one another although the same pulse and the same crystal are used twice. Photon 2 and 3 are subjected to a Bell- state measurement (BSM). In order to project the state of photons 2 and 3 | − onto the anti-symmetric Bell-state Ψ 23, the same technique developed for 4.3. EXPERIMENTAL ENTANGLEMENT SWAPPING 57 quantum teleportation has been used in the experiment.

According to the entanglement swapping scheme, upon projection of pho- | − ton 2 and 3 into the Ψ 23 state, photon 1 and 4 should be projected into | − the Ψ 14 state. To verify that this entangled state is obtained we have to analyze the polarization correlation between photons 1 and 4 conditioned on coincidences between the detectors of the Bell-state analyzer. If photon 1 | − and 4 are in the Ψ 14 state their polarizations should be orthogonal upon detection in any polarization basis. Using a λ/2 retardation plate at 22.5◦ and two detectors (D1+ and D1−) behind a polarizing beamsplitter we chose to analyze the polarization of photon 1 both along the +45◦ axis (D1+)and along the −45◦ axis (D1−). Photon 4 is analyzed by detector D4 at the variable polarization direction Θ.

If entanglement swapping happens, then both the two-fold coincidences + − | − between D1 and D4, and between D1 and D4, conditioned on the Ψ 23 detection, should show two sine curves as a function of Θ which are 90◦ out of phase. The D1+D4 curve should, in principle, go to zero for Θ = 45◦ whereas the D1−D4 curve should show a maximum at this position. Figure 4.3 shows the experimental results for the coincidences between D1+ and D4, and between D1− and D4, given that photons 2 and 3 have been registered by the two detectors in the Bell-state analyzer. Note that this method requires four-fold coincidences. The result clearly demonstrates the expected sine curves, complementary for the two detectors (D1+ and D1−) of photon 1 registering orthogonal polarizations. We veriﬁed by additional measurements that the sine curves are independent (up to the corresponding shift in Θ) on the detection basis of photon 1, that is, independent of the rotation angle of the λ/2 retardation plate. In other words, the observed sinusoidal behavior of the coincidence rates depends only on the relative angle between the polarizers in beams 1 and 4.

The experimentally obtained four-fold coincidences have been ﬁtted by a joint sine function with the same amplitudes for both curves. Note that the observed visibility of 0.65 clearly surpasses the 0.5 limit of a classical wave 58 CHAPTER 4. ENTANGLEMENT SWAPPING

Figure 4.2: Experimental setup. A UV-pulse passing through a non-linear crystal creates pair 1-2 of entangled photons. Photon 2 is directed to the beamsplitter (BS). After reflection, during its second passage through the crystal the UV-pulse creates a second pair 3-4 of entangled photons. Photon 3willalsobedirectedtothebeamsplittertoperformaBell-statemeasure- ment (BSM) of photons 2 and 3. When photons 2 and 3 yield a coincidence | − click at the two detectors behind the beamsplitter a projecting into the Ψ 23 state takes place. As a consequence of this Bell-state measurement the two remaining photons 1 and 4 will also be projected onto an entangled state. To analyse their entanglement we look at coincidences between detectors D1+ and D4, and between detectors D1− and D4, for different polarization angles Θ. By rotating the λ/2 plate in front of the polarizing beamsplitter (PBS) we can also analyze photon 1 in a different orthogonal polarization basis which is necessary to obtain statements for relative polarization angles between photons 1 and 4. Note that, since the detection of coincidences between detectors D1+ and D4, and D1− and D4 are conditioned on the detection of the Ψ− state, we are looking at 4-fold coincidences. Narrow bandwidth filters (F) are positioned in front of each detector. 4.3. EXPERIMENTAL ENTANGLEMENT SWAPPING 59 theory. A visibility of 0.72 ± 0.04 was observed in a few initial measurements for analysis along 45o. A future experiment for showing a significant violation of Bell’s inequalities requires a stable visibility better than 0.71.

In order to ﬁnd a practical way to improve the visibility of four-photon interference fringes, let us perform a semi-quantitative estimation of the visibility via our experimental parameters. If we simply assume that our EPR pairs are of perfect entanglement quality, i.e. a visibility of 100% for two entangled photons, then according to Zukowski et al. [Zukowski et al., 1995; Zeilinger et al., 1997] the visibility of four-photon fringes is given by

σP Videal = (4.4) 2 2 σP + σF

where σP and σF are the spectral width of the pump pulse and the bandwidth of the interference ﬁlters, respectively. Our narrow bandwidth ﬁlters σF ≈

4.6nm and the measured pulse spectral width σP ≈ 8nm yield Videal ≈ 87%.

However, as observed in the experiment the EPR pairs produced by pulsed pump have at best a visibility of VEPR ≈ 90% and cannot fully satisfy the requirement of perfect entanglement. Therefore, one must also consider such a degrading eﬀect on the experimental visibility of four-photon fringes. Because we observe interference fringes between two independent pairs, one possible × 2 way to estimate the visibility may be the product Videal VEPR.Thisconse- quently leads to an estimated visibility of 87% × 90% × 90% ≈ 72%, which compares favourably with our best experimental results.

Our estimation indicates two possible ways to improve the visibility beyond the 0.71 limit. In principle, one could use interference ﬁlters with more narrow bandwidth or try to produce EPR pairs with higher quality of entanglement, though both approaches are technically certainly challenging due to the very low coincidence rate. Therefore, we also expect to improve the very low four-fold coincidence rate, the main diﬃculty of the present experiment, by using a new laser system currently being installed in Vienna, leading to 60 CHAPTER 4. ENTANGLEMENT SWAPPING a better performance of the experiment.

Figure 4.3: Veriﬁcation of entanglement swapping via veriﬁcation of entanglement between the two photons 1 and 4 from separate pairs. Four-fold coincidences, resulting from two-fold coincidence D1+D4 (D1−D4) conditioned on the Bell-state measurement two-fold coincidences at detectors D2 and D3, as a function of the polarization angle Θ. The two complementary sine curves with a visibility of 0.65 ± 0.02 demonstrate that photons 1 and 4 are polarization entangled.

Again, we mention that, obviously, registration of a coincidence in the two detectors behind the beam splitter could also have been caused by two pairs created in either source. That possibility could clearly be ruled out by sophisticated detection procedures. It certainly does not have any implication on those events in our experiment where we indeed obtain four registration events. 4.4. GENERALIZATION AND APPLICATIONS 61

4.4 Generalization and applications

While experimental entanglement swapping itself is a further demonstration of teleportation, i.e. teleportation of quantum mechanically undeﬁned state, a test of Bell’s theorem using entanglement swapping could test nonlocality with a pair of particles that never interacted. Such a test would certainly help us to further understand nonlocality. We could even perform a delayed choice experiment for entanglement swapping as recently suggested by Peres [Peres, 1999], where one could delay the instant of time of the to perform Bell-state measurement on photons 2 and 3, and thus entanglement between photons 1and4isproduceda posteriori, after they have already been measured and may no longer exist.

Further, various generalizations of the present scheme are at hand [Bose 1997, Pan 1998a, 1999b]. One could have many different kinds of entanglement to begin with, perform various different measurements, and obtain novel kinds of entanglement for the emerging particles. A first clear possibility [Zukowski et al., 1995; Rarity and Tapster, 1995] is to project three particles, each from an entangled pair, into a GHZstate [Greenberger et al., 1990] whenceforth the other three emerging particles are also projected into aGHZstate.

Secondly, for example, one could use a polarizing beam splitter instead of the beam splitter in the set-up (Fig. 4.2). Then, using the same reasoning line as used in section 2.2.2 (or more directly refer to Fig. 2.7 and Eq.(2.33)) we could arrive the following conclusions: (1) The outgoing four-particle state will be in a conditional GHZstate immediately after photons 2 and 3 passing through the polarizing beam splitter; (2) as suggested in chapter 2, one could perform ±45◦ polarization analysis at one of the two output ports of the polarizing beam splitter, then conditioned on the detection of a single photon with 45◦ (or −45◦) polarization, the remaining three photons will be correspondingly projected into a 62 CHAPTER 4. ENTANGLEMENT SWAPPING freely propagating GHZ-state; (3) one could also perform ±45◦ polarization analysis at the two output ports in the meantime, that is, perform a Bell-state measurement on photons 2 and 3 by using our modiﬁed Bell-state analyzer (refer to Fig. 2.5), then by + − Eq. (4.3) photons 1 and 4 will be projected into state |Φ 14 (or |Φ 14)with respect to a projection of the state of photons 2 and 3 into the Bell-state + − |Φ 23 (or |Φ 23).

The results above exactly imply that one could interpret our experimental setup in diﬀerent ways, corresponding to what kind of speciﬁc measurement one intends to perform. It should be noted that such a scheme can be easily generalized to N particles case by just adding more EPR pairs and polarizing beam splitters. Again all these schemes above require pulsed pump technology, with pulses of even higher power than in the present experiment, yet in principle achievable with current technology.

We might also remark that the present results, taken together with our veriﬁcation of quantum teleportation in chapter 3, are easily understood in the framework of the Copenhagen interpretation of quantum mechanics [Nagel, 1989]. They cause no conceptual problems if one accepts that information about quantum systems is a more basic feature than any possible ”real” properties these systems might have [Zeilinger, 1998].

Finally, it is foreseen that entanglement swapping, besides its interest to fundamental physics, will have a number of important applications in future quantum communication schemes. First of all, as mentioned by Bose et al. [Bose et al., 1998], our entanglement swapping scheme opens up a way to speed up the distribution of entanglement for any particles possessing mass. The importance of the distribution of entanglement between distant parties is obvious as Bell pairs are essential for the implementation of many quantum communication schemes over large distance, such as secret-key distribution [Ekert, 1991], teleportation [Bennett et al., 1993] and dense coding [Bennett and Wiesner, 1992]. 4.4. GENERALIZATION AND APPLICATIONS 63

On the other hand, due to the unavoidable decoherence caused by coupling with the environment, the quality of entanglement will be degraded during the distribution and storage of entanglement. Therefore, entanglement purification [Bennett et al., 1996a; 1996b; Deutsch et al., 1996] is of great sig- nificance to achieve quantum communication with perfect fidelity. Because all purification schemes involve collective measurements on many photons at once, or need two-qubit logic gates for polarization-entangled photons, whose physical implementation is very difficult under the current technology, and remains to be realized in an experiment. Furthermore, for distances much larger than the coherence length of a corresponding noisy quantum channel, the fidelity of transmission is so low that the standard purification methods above are not applicable. However, while the most recent research [Bose et al., 1999] shows that a simple variant of our entanglement swapping scheme can be directly used to purify single pairs of polarization-entangled photons, it is possible to divide the quantum channel into shorter segments that are purified separately and then connected by entanglement swapping [Briegel et al, 1998; Duer et al., 1999]. All this underlines that entanglement swapping is one of the most important key procedures in quantum communication networks. Chapter 5

Three-photon GHZ entanglement

5.1 Introduction

Ever since the seminal work of Einstein, Podolsky and Rosen [Einstein 1935] there has been a quest for generating entanglement between quantum particles. Although two-particle entanglements have long been demonstrated experimentally [Wu and Shaknov, 1950; Freedman and Clauser, 1972; As- pect et al., 1982; Kwiat et al., 1995; Hagley et al., 1997], the preparation of entanglement between three or more particles remains an experimental challenge. Proposals have been made for experiments with photons [Green- berger et al., 1990; Zeilinger et al., 1997; Pan and Zeilinger, 1998a] and atoms [Cirac and Zoller, 1994; Haroche, 1995], and three nuclear spins within a single molecule have been prepared such that they locally exhibit three-particle correlations [Lloyd, 1998; Laﬂamme et al., 1998]. However, until now there has been no experiment which demonstrates the existence of entanglement of more than two spatially separated particles. Here we present the ﬁrst experimental observation of polarization entanglement of three spatially separated photons [Bouwmeester, Pan et al., 1999]. Such states, known as Greenberger- Horne-Zeilinger (GHZ) states, are interesting from both a fundamental and

64 5.2. EXPERIMENTAL SET-UP 65 a technological point of view.

The original motivation to prepare three-particle entanglements stems from the observation by Greenberger, Horne and Zeilinger that three-particle entanglement leads to a conﬂict with local realism for non-statistical predictions of quantum mechanics [Greenberger et al., 1989; 1990; Mermin, 1990a; 1990b]. This is in contrast to the case of Einstein-Podolsky-Rosen experiments with two entangled particles testing Bells inequalities, where the con- ﬂict only arises for the statistical predictions of quantum theory [Bell, 1964].

The incentive to produce GHZstates has been signiﬁcantly increased by the advance of the ﬁeld of quantum communication and quantum information processing. Entanglement between several particles is the most important feature of many such quantum communication and computation protocols [Bennett, 1995].

5.2 Experimental Set-up

The experiment described here is based on techniques that have been developed for our previous experiments on quantum teleportation [Bouwmeester, Pan et al., 1997] and entanglement swapping [Pan et al., 1998]. In fact, one of the main complications in the those experiments, namely, the creation of two pairs of photons by a single source, is here turned into a virtue.

The main idea, as was put forward in [Zeilinger et al., 1997; Pan and Zeilinger, 1998], is to transform two pairs of polarization entangled photons into a triplet of entangled photons and a fourth independent photon. Our experimental arrangement is such that we start with two pairs of entangled photons and register the photons in a way that any information as to which pair each photon belongs to is erased. Fig. 5.1 is a schematic drawing of our experimental setup. Pairs of polarization-entangled photons are generated by a short pulse of ultraviolet (UV) light (≈ 200 fs, λ = 788 nm from a frequency-doubled, mode-locked Ti-Sapphire laser), which passes through an 66 CHAPTER 5. THREE-PHOTON GHZ ENTANGLEMENT optically nonlinear crystal (Beta-Barium-Borate, BBO). The probability per pulse to create a single pair in the desired modes is rather low and of the order of a few 10−4. The pair creation is such that the following polarization entangled state is obtained [Kwiat et al., 1995]:

1 √ (| H | V −|V | H ) . (5.1) 2 a b a b

This state indicates that there is a superposition of the possibility that the photoninarma is horizontally polarized and the one in arm b vertically | | | | polarized ( H a V b), and the opposite possibility, i.e., V a H b.Themi- nus sign indicates that there is a ﬁxed phase diﬀerence of π between the two possibilities. For our GHZexperiment this phase factor is actually allowed to have any value, as long as it is the same for both pairs.

The setup is such that arm a continues towards a polarizing beam splitter, where V photons are reflected and H photons are transmitted towards detector T (behind an interference filter δλ =4.6 nm at 788 nm). Arm b continues towards a 50/50 polarization-independent beam splitter. From each beam splitter, one output is directed to a final polarizing beam splitter. In between the two polarizing beam splitters there is a λ/2 wave plate at an angle of 22.5◦ which rotates the vertical polarization of the photons reflected by the first polarizing beam splitter into a 45◦ polarization, i.e. a superposition of | H and | V with equal amplitudes. The remaining three output arms continue through interference filters (δλ =3.6 nm) and single-mode fibers towards the single-photon detectors D1, D2,andD3. Including filter losses, coupling into single-mode fibers, and the Si-avalanche detector efficiency, the total collection and detection probability of a photon is about 10%.

Consider now the case that two pairs are generated by a single UV-pulse, and that the four photons are all detected, one by each detector T, D1, D2, and D3. Our claim is that by the coincident detection of four photons and because of the brief duration of the UV pulse and the narrowness of the ﬁlters, one can conclude that a three-photon GHZstate has been recorded 5.2. EXPERIMENTAL SET-UP 67

Figure 5.1: Schematic drawing of the experimental setup for the demonstration of Greenberger-Horne-Zeilinger entanglement for three spatially separated photons. The UV pulse incident on the BBO crystal produces two pairs of entangled photons. Conditioned on the registration of one photon at the trigger detector T, the three photons registered at D1,D2,andD3 exhibit the desired GHZ correlations. 68 CHAPTER 5. THREE-PHOTON GHZ ENTANGLEMENT

by detectors D1,D2,andD3. The reasoning is as follows. When a four- fold coincidence recording is obtained, one photon in path a must have been horizontally polarized and detected by the trigger detector T. Its companion photon in path b must then be vertically polarized, and it has a 50% chance to be transmitted by the beam splitter (see Fig. 5.1) towards detector D3 and a 50% chance to be reflected by the beam splitter towards the final polarizing beam splitter where it will be reflected to D2. Consider the first possibility, i.e. the companion of the photon detected at T is detected by

D3 and necessarily carried polarization V . Then the counts at detectors D1 and D2 were due to a second pair, one photon traveling via path a and the other one via path b. The photon traveling via path a must necessarily be V polarized in order to be reflected by the polarizing beam splitter in path a; thus its companion, taking path b,mustbeH polarized and after reflection at the beam splitter in path b (with a 50% probability) it will be transmitted by the final polarizing beam splitter and arrive at detector D1. The photon detected by D2 therefore must be H polarized since it came via path a and had to transit the last polarizing beam splitter. Note that this latter photon was V polarized but after passing the λ/2 plate it became polarized at 45◦ which gave it a 50% chance to arrive as an H polarized photon at detector

D2. Thus we conclude that if the photon detected by D3 is the companion of the T photon, then the coincidence detection by D1,D2,andD3 corresponds to the detection of the state

| | | H 1 H 2 V 3 . (5.2)

By a similar argument one can show that if the photon detected by D2 is the companion of the T photon, the coincidence detection by D1,D2,andD3 corresponds to the detection of the state

| | | V 1 V 2 H 3 . (5.3)

In general, the two possible states (5.2)and(5.3) corresponding to a four- fold coincidence recording will not form a coherent superposition, i.e. a GHZ 5.2. EXPERIMENTAL SET-UP 69 state, because they could, in principle, be distinguishable. Besides possible lack of mode overlap at the detectors, the exact detection time of each photon can reveal which state is present. For example, state (5.2) is identified by noting that T and D3,orD1 and D2, fire nearly simultaneously. To erase this information it is necessary that the coherence time of the photons is substantially longer than the duration of the UV pulse (approximately 200 fs) [Zukowski et al., 1995]. We achieved this by detecting the photons behind narrow band-width filters which yield a coherence time of approximately 500 fs. Thus, the possibility to distinguish between states (5.2)and(5.3)isno longer present, and, by a basic rule of quantum mechanics, the state detected by a coincidence recording of D1,D2,andD3, conditioned on the trigger T, is the quantum superposition

1 √ (| H | H | V + | V | V | H ) , (5.4) 2 1 2 3 1 2 3 which is a GHZstate 1

The plus sign in Eq. (5.4) follows from the following more formal deriva- tion. Consider two down-conversions producing the product state

1 (| H | V −|V | H ) | H | V −|V | H . (5.5) 2 a b a b a b a b

| | Initially we assume that the components H a,b and V a,b created in one | down-conversion might be distinguishable from the components H a,b and | V a,b created in the other one. The evolution of the individual components of state (5.5) through the apparatus towards the detectors T, D1,D2,and

D3 is given by

| →| H a H T , (5.6) 1Rigorously speaking, this erasure technique is perfect, hence produces a pure GHZ state, only in the limit of infinitesimal pulse duration and infinitesimal filter bandwidth, but detailed calculations [Zukowski et al., 1995; Horne, 1998] reveal that our pulse and filter values are sufficient to create a clearly observable entanglement, as confirmed by our experimental data. 70 CHAPTER 5. THREE-PHOTON GHZ ENTANGLEMENT

1 | V → √ (| V + | H ) , (5.7) a 2 1 2 1 | H → √ (| H + | H ) , (5.8) b 2 1 3 1 | V → √ (| V + | V ) . (5.9) b 2 2 3

Identical expressions hold for the primed components. Inserting these expressions into state (5.5) and restricting ourselves to those terms where only one photon is found in each output we obtain, after normalization

1 | H | V | V | H + | H | H | V 2 T 1 2 3 1 2 3 . (5.10) | | | | | | | + H T V 1 V 2 H 3 + H 1 H 2 V 3

If now the experiment is performed such that the photon states from the two down-conversions are indistinguishable, we ﬁnally obtain the desired state (up to an overall minus sign)

1 √ | H (| H | H | V + | V | V | H ) . (5.11) 2 T 1 2 3 1 2 3

Note that, even conditioned upon trigger T detecting a single photon, the total state of the remaining three photons before detection still contains terms in which, for example, two photons enter the same detector. Thus, the GHZentanglement is observed only under the condition that both the trigger photon and the three entangled photons are actually detected, and in the following experiments the four-fold coincidence detection actually plays thedoubleroleofbothprojectingintothedesiredGHZstate(5.11)and performing a speciﬁc measurement on the state. This, we submit, in practice will not be a severe limitation because, on the one hand, in any realistic scheme one always has losses, and information is only obtained if the photons are actually observed, as, for instance, in third-man quantum cryptography. On the other hand, many applications explicitly use speciﬁc measurement results. For example, as we will show in chapter 6, the GHZargument 5.2. EXPERIMENTAL SET-UP 71 for testing local realism is based on detection events, and knowledge of the underlying quantum state is actually not even necessary.

The eﬃciency for one UV pump pulse to yield such a four-fold coincidence detection is very low (of the order of 10−10). Fortunately, 7.6×107 UV-pulses are generated per second, which yields about one double pair creation and detection per 150 seconds, which is just enough to perform our experiments 2. Triple pair creations can be completely neglected since they can give rise to a four-fold coincidence detection of only very few per day.

Comparing with the scheme suggested in Chapter 2 ( refer to Fig. 2.7 ), the scheme presented here has a significant advantage for the experimental alignment. That is that one can easily scan into the region of time overlap of the photons at the final polarizing beamsplitter simply by observing bunching effect of two correlated photons generated from a single pump pulse. In contrast, in the scheme of Fig. 2.7 one has to observe bunching effect of two photons created by independent sources to find the region of quantum superposition, this correspondingly decreases the two-fold coincidence rate by at least three orders of magnitude and thus one has to take much longer scanning time to see clear two-photon bunching effect.

It is also worth noting that one could simply use a 50:50 beam splitter instead of the polarizing beam splitter in front of the detector T and take oﬀ the half wave plate λ/2 to observe conditional four-photon GHZentan- glement. Following the same reasoning line above, one can easily verify that the state detected by a four-fold coincidence recording of D1, D2, D3 and T is in the superposition

1 √ (| H | V | V | H + | V | H | H | V ) , 2 T 1 2 3 T 1 2 3 which is a four-photon GHZstate.

2 The singles detection rate at detectors D1,D2 and D3, is about 15,000 counts per second, and at the trigger detector T about 100,000 counts per second, due to the larger ﬁlter bandwith and mode acceptance. Four-fold coincidence is registered with logic AND circuitry with a coincidence time of 6 ns. 72 CHAPTER 5. THREE-PHOTON GHZ ENTANGLEMENT

5.3 Observation of three-photon entanglement

To experimentally demonstrate that a GHZstate has been obtained by the method described above, we first verified that, conditioned on a photon detection by the trigger T, both the H1H2V3 and the V1V2H3 components can be observed with the same intensity, but no others. This was done by comparing the count rates of the eight possible combinations of polarization measurements, H1H2H3,H1H2V3, ..., V1V2V3. The observed intensity ratio between the desired and the undesired states was 12:1. Existence of the two terms as just demonstrated is a necessary but not yet sufficient condition for demonstrating GHZentanglement. In fact, there could in principle be just a statistical mixture of those two states. Therefore, one has to prove that the two terms coherently superpose. This we did by a measurement of linear polarization of photon 1 along +45◦, bisecting the H and V direction. Such a measurement projects photon 1 into the superposition

1 | +45◦ = √ (| H + | V ) , (5.12) 1 2 1 1 what implies that the state (5.11) is projected into

1 √ | H | +45◦ (| H | V + | V | H ) . (5.13) 2 T 1 2 3 2 3

Thus photon 2 and 3 end up entangled as predicted under the notion of ”entangled entanglement” [Krenn and Zeilinger, 1996]. We conclude that demonstrating the entanglement between photon 2 and 3 conﬁrms the coherent superposition in state (5.11) and thus the existence of the GHZentangle- ment. In order to proceed to our experimental demonstration we represent the entangled state of photons 2 and 3 in a linear basis rotated by 45◦.The state then becomes

1 √ (| +45◦ | +45◦ −|−45◦ |−45◦ ) , (5.14) 2 2 3 2 3 5.3. OBSERVATION OF THREE-PHOTON ENTANGLEMENT 73 which implies that if photon 2 is found to be polarized along -45◦ (or +45◦), photon 3 is also polarized along the same direction. We test this predic- | ◦ |− ◦ tion in our experiment. The absence of the terms +45 2 45 3 and |− ◦ | ◦ 45 2 +45 3 is due to destructive interference and thus indicates the desired coherent superposition of the terms in the GHZstate ( 5.11). The experiment therefore consisted of measuring four-fold coincidences between the detector T, detector 1 behind a +45◦ polarizer, detector 2 behind a -45◦ polarizer, and measuring photon 3 behind either a +45◦ polarizer or a -45◦ polarizer. In the experiment, the difference of arrival time of the photons at the final polarizer, or more specifically, at the detectors D1 and D2, was varied.

The data points in Fig. 5.2(a) are the experimental results obtained for the polarization analysis of the photon at D3, conditioned on the trigger and the detection of two photons polarized at 45◦ and −45◦ by the two detectors

D1 and D2, respectively. The two curves show the four-fold coincidences for a ◦ ◦ polarizer oriented at −45 (squares) and +45 (circles) in front of detector D3 as function of the spatial delay in path a. From the two curves it follows that ◦ for zero delay the polarization of the photon at D3 is oriented along −45 , in accordance with the quantum-mechanical predictions for the GHZstate. For non-zero delay, the photons traveling via path a towards the second polarizing beam splitter and those traveling via path b become distinguishable. Therefore increasing the magnitude of delay gradually destroys the quantum superposition in the three-particle state.

Note that one can equally well conclude from the data that at zero delay, the photons at D1 and D3 have been projected onto a two-particle entangled ◦ state by the projection of the photon at D2 onto −45 . The two conclusions are only compatible for a genuine GHZstate. We note that the observed visibility was as high as 75%. 3

3The limited visibility is due mainly to the finite width of the interference filters, the finite pulse duration, and the limited quality of the polarization optics. detector noise or accidental coincidences do not play any role. 74 CHAPTER 5. THREE-PHOTON GHZ ENTANGLEMENT

Figure 5.2: Experimental confirmation of GHZ entanglement. Graph (a) shows the results obtained for polarization analysis of the photon at D3,conditioned on detection of the trigger photon and detection of one photon at ◦ ◦ D1 polarized at 45 and one photon at detector D2 polarized −45 .Thetwo curves show the four-fold coincidence rates for a polarizer oriented at −45◦ ◦ and 45 respectively in front of detector D3 as a function of the spatial delay in path a. The difference between the two curves at zero delay confirms the GHZ entanglement. By comparison (graph (b)) no such intensity difference ◦ occurs if the polarizer in front of detector D1 issetat0. Error bars are given by the square root of the coincidence counts. 5.4. DISCUSSION AND CONCLUSION 75

For an additional conﬁrmation of state (5.11) we performed measurements ◦ conditioned on the detection of the photon√ at D1 under 0 polarization (i.e. V polarization). For the GHZstate (1/ 2)(H1H2V3 +V1V2H3) this implies that the remaining two photons should be in the state V2H3 which cannot give rise to any correlation between these two photons in the 45◦ detection basis. The experimental results of these measurement are presented in Fig. 5.2(b). The data clearly indicate the absence of two-photon correlations and thereby conﬁrm our claim of the observation of GHZentanglement between three spatially separated photons.

5.4 Discussion and conclusion

Although the extension from two to three entangled particles might seem to be only a modest step forward, the implications are rather profound. First of all, GHZentanglements allow for novel tests of quantum mechanics versus local realistic models [Greenberger et al., 1989; 1990; Mermin, 1990a; 1990b; Zukowski, 1998; Pan et al., 1999a]. Secondly, three-particle GHZ states might ﬁnd a direct application, for example, in third-man quantum cryptography. And thirdly, the method developed to obtain three-particle entanglement from a source of pairs of entangled particles can be extended to obtain entanglement between many more particles [Bose et al., 1998], which is the basis of many quantum communication and computation protocols. Finally, we note that our experiment, together with our earlier realization of quantum teleportation [Bouwmeester, Pan et al., 1997] and entanglement swapping [Pan et al., 1998b] provides the necessary tools to implement a number of novel entanglement distribution and network ideas as recently proposed [Grover, 1997; Duer et al., 1999]. Chapter 6

Experimental tests of the GHZ theorem

6.1 Introduction

Ever since its introduction by Schrödinger [Schrödinger, 1935] entanglement has commanded a central position in the discussions of the interpretation of quantum mechanics. Originally that discussion has focused on the proposal by Einstein, Podolsky and Rosen (EPR) of measurements performed on two spatially separated entangled particles [Einstein et al., 1935]. Most signifi- cantly, John Bell then showed that there is a conflict between any attempt to explain the correlations observed in such systems by a local realistic model and the predictions made by quantum mechanics [Bell, 1964]. In the deriva- tion of Bell’s inequalities one makes the seemingly innocuous assumption that perfect correlations can be understood using such a local realistic model and the conflict then arises for the statistical predictions of quantum theory.

An increasing number of experiments on entangled particle pairs having conﬁrmed the statistical predictions of quantum mechanics [Freedman et al., 1972; Aspect et al., 1982; Weihs 1998] have thus provided increasing evidence against local realistic theories. Yet, one might ﬁnd some comfort in the fact

76 6.2. THE CONFLICT WITH LOCAL REALISM 77 that such a realistic and thus classical picture can explain perfect correlations and is only in conﬂict with statistical predictions of the theory. After all, quantum mechanics is statistical in its core structure. In other words, for entangled particle pairs the cases where the result of a measurement on one particle can deﬁnitely be predicted on the basis of a measurement result on the other particle can be explained by a local realistic model. It is only that subset of statistical correlations where the measurement results on one particle can only be predicted with a certain probability which cannot be explained by such a model.

Yet in 1989 it was shown by Greenberger, Horne and Zeilinger (GHZ) that for certain three- and four-particle states [Greenberger et al., 1989; 1990] a conflict with local realism arises even for perfect correlations. That is, even for those cases where, based on measurement on N − 1 of the particles, the result of the measurement on particle N can be predicted with certainty. Local realism and quantum mechanics here both make definite but completely opposite predictions. A particularly elegant demonstration of that conflict is due to Mermin [Mermin, 1990a].

Utilizing our recently developed source for three-photon GHZ-entanglements it is the purpose of this chapter to present a ﬁrst realization of such a three- particle test against local realism [Pan et al., 1999a].

6.2 The conﬂict with local realism

6.2.1 GHZ theorem

How are the quantum predictions of a three-photon GHZ-state in stronger conﬂict with local realism than the conﬂict for two-photon states as implied 78 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM by Bell’s inequalities 1? To answer this, consider the state

1 √ (| H | H | H + | V | V | V ) , (6.1) 2 1 2 3 1 2 3 where H and V denote horizontal and vertical linear polarizations. This state indicates that the three photons are in a quantum superposition of | | | the state H 1 H 2 H 3 (the three photons are horizontally polarized) and | | | V 1 V 2 V 3 (the three photons are vertically polarized). We choose this speciﬁc state because it is symmetric with respect to the interchange of all photons which simpliﬁes the arguments below. The same line of reasoning holds, however, for any three-particle entangled state.

Consider now some speciﬁc predictions following from state (6.1) for measurements of linear polarization along directions rotated by 45◦ with respect to the original H-V directions, denoted by H-V , or of circular polarization denoted by L-R (left-handed, right-handed). These new polarizations can be expressed in terms of the original ones as

1 1 | H = √ (| H + | V ), | V = √ (| H−|V )(6.2), 2 2 1 1 | R = √ (| H + i | V ), | L = √ (| H−i | V )(6.3). 2 2

For convenience we will refer to a measurement of H-V linear polarization as a l measurement and of L-R circular polarization as a c measurement.

Representing state (6.1) in the new states using Eqs. (6.2)and(6.3) one obtains predictions for measurements of these new polarizations. For example, for the case of measurement of circular polarization on, say, both photon 1 and 2, and measurement of linear polarization H-V on photon 3,

1For two-photon states Hardy [Hardy, 1993] has found situations where quantum mechanics predicts a speciﬁc result to occur sometimes and local realism predicts the same result never to occur [Boschi et al., 1997] 6.2. THE CONFLICT WITH LOCAL REALISM 79 denoted as a ccl experiment, the state may be expressed as

This expression has a number of signiﬁcant implications. Firstly, we note that any speciﬁc result obtained in any individual or in any two-photon joint measurement is maximally random. For example, photon 1 will exhibit polarization R or L with the same probability of 50%, or photons 1 and 2 will exhibit polarizations RL, LR, RR,orLL with the same probability of 25%.

Yet secondly we realize that, given any two results of measurements on any two photons, we can predict with certainty what the result of the corresponding measurement performed on the third photon will be. For example, suppose photons 1 and 2 are both found to exhibit right-handed (R) circular polarization. Then by the third term in the expression above, photon 3 will deﬁnitely be V polarized.

By cyclic permutation, we can obtain analogous expressions for any case of any experiment measuring circular polarization on two photons and H-V linear polarization on the remaining one. Thus, in any one of the three ccl, clc and lcc experiments any individual measurement result both for circular polarization and for linear H-V polarization can be predicted with certainty for any one of the three photons given the corresponding measurement results of the other two.

Now we will analyze the implications of these predictions from the point of view of local realism. First note that the predictions are independent of the spatial separation of the photons and independent of the relative time order of the measurements. Let us thus consider the experiment to be performed such that the three measurements are performed simultaneously in a given reference frame, say, for conceptual simplicity, in the reference frame of the source. Thus we can employ the notion of Einstein locality which implies that no information can travel faster than the speed of light. Hence the 80 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM specific measurement result obtained for any photon must not depend on which specific measurement is performed simultaneously on the other two nor on the outcome of these measurements. The only way then to explain from a local realist point of view the perfect correlations discussed above is to assume that each photon carries elements of reality for both l and c measurements considered and that these elements of reality determine the specific individual measurement result [Greenberger et al., 1989; 1990; Mermin, 1990a].

We now consider a fourth experiment measuring linear H-V polarization on all three photons, i.e. a lll experiment. What possible outcomes will a local realist predict here based on the elements of reality introduced to explain the earlier ccl, clc and lcc experiments? The state of Eqn. (6.4) and its permutations imply that whenever we obtain the result H [V ]for any one photon, the other two photons must carry elements of reality imply- ing opposite [identical] circular polarizations. Suppose that for one speciﬁc run of the lll experiment we ﬁnd, say, the result V both for photon 2 and for photon 3. Because photon 3 is a V , both photon 1 and 2 must carry identical circular polarization elements of reality; and because photon 2 is a V , both photons 1 and 3 must carry identical circular polarization elements of reality. This means that all three photons must carry identical circular polarization elements of reality. Thus, since photons 2 and 3 carry identical circular polarization elements of reality, photon 1 must necessarily exhibit linear polarization V , it cannot be H polarized. Hence the existence of elements of reality leads to the conclusion that the result V1 V2 V3 is one possible outcome and H1V2 V3 is impossible. By parallel constructions, one can verify that H1H2V3 , H1V2 H3,andV1 H2H3 are the only other possible outcomes from a local realistic viewpoint if we elect to measure H-V polarizations of all three particles, i.e. if we perform a lll measurement.

How do these predictions of local realism compare with those of quantum physics? If we express the state given in Eq. (6.1)intermsofH-V 6.2. THE CONFLICT WITH LOCAL REALISM 81 polarization using Eq. (6.2)weobtain

Here the local realistic model predicts none of the terms occurring in the quantum prediction. This implies that whenever local realism predicts a specific result definitely to occur for a measurement on one of the photons based on the results for the other two, quantum physics definitely predicts the opposite result. For example, if two photons are both found to be H polarized, local realism predicts the third photon to carry polarization V while the quantum state predicts H.

Thus, while in the case of Bell’s inequalities for two photons the difference between local realism and quantum physics happens for statistical predictions of the theory, for three entangled particles the difference occurs already for the definite predictions, statistics is now only due to inevitable measurement errors occurring in any and every experiment, even in classical physics.

6.2.2 Generalization to conditional GHZ state

The experiment reported here is based on the observation of three-photon GHZentanglement that was achieved in Chapter 5. Conditioned upon that detector T observes a single photon, the total photon state out of our setup (Fig. 5.1) actually reads2

|H1|V 1|V 2 + |H1|V 1|H3+ |H |V |H + |H |V |V + 2 2 1 2 2 3 (6.6) |H3|V 3|V 1 + |H3|V 3|H2+ |H1|H2|H3 + |V 1|V 2|V 3 2For simplicity of argumentation we have assumed here that for photon 3 H and V are defined at right angles compared to photons 1 and 2, and also we already exclude the case in which only one EPR pair is produced by a single pulse or two photons enter detector T. Yet we note that such a simplification does not change the physical conclusion. 82 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM which implies that the total photon state produced by our setup, i.e., the state before detection, also contains terms in which two photons enter the same detection station. For example, the first two terms of Eq. 6.6 imply there are two photons in mode 1, and so on. Therefore, the four-fold coincidence detection here acts as a projection measurement onto the desired GHZstate (6.1) and filters out those undesirable terms. This might raise doubts about whether such a source can be used to test local realism.

Actually, the same doubts had been raised earlier for certain Bell-type experiments involving photon pairs [Ou et al., 1988; Shih and Alley, 1988]. Al- though these experiments have successfully produced the expected quantum- mechanical correlations, in the past it was often believed [Kwiat et al., 1994, Caro and Garuccio, 1994] that they could never, not even in their idealized versions, be consider as genuine tests of local realism. To avoid this problem, a possible solution is that one could directly prepare unconditional three- photon GHZstate using the scheme suggested in Chapter 2 (refer to Fig. 2.7).

However, fortunately, Popescu, Hardy and Zukowski [Popescu et al., 1997] showed that this general belief is wrong and that the above experiments indeed constitute (modulo the usual detection loopholes) true tests of local realism. Following the same reasoning line, Zukowski has recently shown that our GHZentanglement source enables one to perform a three-particle test of local realism [Zukowski 1999]. Here we brieﬂy discuss this analysis.

For clarity of the argumentation, let’s ﬁrst deﬁne a local hidden-variable model for our GHZentanglement source. In such a model the local events (i.e., events at one of the observation stations) are determined by the value of the hidden variables describing the experiment, usually denoted by the symbol λ, and the local macroscopic controllable parameter set by the local observer; here the settings of the polarizers in front of detectors D1,D2, and D3, respectively denoted by x1,x2,andx3. As mentioned already, in the GHZargument xi(i =1, 2, 3) could independently indicate settings for linear H − V polarization or for circular R − V polarization, i.e. l or c 6.2. THE CONFLICT WITH LOCAL REALISM 83 measurements respectively.

Let Λ be the space of the parameter λ for an ensemble comprised of a very large number of the states produced by our GHZentanglement source. From Eq. 6.6, it is easy to see that such an ensemble can be partitioned into two disjoint sub-ensembles Λa and Λb, i.e. corresponding to those for which (a) one photon each is detected in each of the outgoing beams 1, 2 and 3, (b) two photons are observed in one of the three outgoing beams, and one photon is observed in one of the remaining two beams. Denoting by ρ(λ) the distribution of the hidden variable λ, then according to local realism, the distribution ρ(λ) of the union of the two subsets is clearly independent of the settings of the polarizers in front of detectors D1,D2,andD3. However, it is not evident whether the mode of partitioning is also independent of the settings of the polarizers. If one wants to construct the GHZargument within the sub-ensemble Λa, one must thus demonstrate that the distribution ρa(λ) of the sub-ensemble Λa is also independent of the polarizer settings. For a thorough discussion of the essence of that argument, refer to [Clauser and Shimony, 1978] and references therein.

To do so, imagine that detectors D1 and D2 each detects a single photon

(under two speciﬁed polarizer settings, x1, x2 respectively), then there is deﬁnitely a single photon to be detected by D3, no matter what the polarizer setting is there3. This implies that, (1) the occurrence of such a joint event, i.e. whether a joint event belongs to the subset Λa or not, is irrespective of the local polarizer setting of D3; (2) the local occurrence of a single photon detection by D3, which belongs to the subset Λa, is also independent of the polarizer setting of D3.

By parallel reasoning, one can ﬁnally conclude that the occurrence of a joint event belonging to the sub-ensemble Λa depends only on the hidden variable λ and not upon the local settings of the polarizers. Furthermore, in

3 Because the four photons were produced by a single pulse, then conditioned upon that detectors D1,D2 and T each detects a single photon, detector D3 must also correspondingly detect a single photon. Here, we always suppose the detectors have perfect 100% detection eﬃciency. 84 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM the subset the occurrence of a single photon detection at any local observation station also does not depend on the polarizer settings at D1,D2,andD3.

These exactly conﬁrm that the distribution ρa(λ) of the sub-ensemble Λa is independent of the polarizer orientations, by which we could deﬁne the probability measures only on the subset Λa and take ρa(λ)tohavenormone:

dρa =1 (6.7) Λa

Thus, by limiting ourselves only to the sub-ensemble Λa we can predict the possible outcomes in a lll experiment via local realism, then compare with the quantum mechanical prediction. Therefore, we ﬁnally arrive at the following conclusion:

In essence, the GHZ argument for testing local realism is based on detection events, and knowledge of the underlying quantum state is not even necessary. It is indeed enough to consider only those four-fold coincidences discussed above and ignore totally the contributions by the other terms.

6.3 Experimental results

As explained in section 6.2.1 demonstration of the conﬂict between local realism and quantum mechanics for GHZentanglement consists of four experiments each with three spatially separated polarization measurements. First, one performs ccl, clc,andlcc experiments. If the results obtained are in agreement with the predictions for a GHZstate then the predictions for an lll experiment are exactly opposite for a local realist theory as to that of quantum mechanics.

For each experiment we have 8 possible outcomes of which ideally 4 should never occur. Obviously, no experiment neither in classical physics nor in quantum mechanics can ever be perfect and therefore, due to principally unavoidable experimental errors, even the outcomes which should not occur will 6.3. EXPERIMENTAL RESULTS 85 occur with some small probability in any realistic experiment. The question is how to deal with this problem in view of the fact that the GHZargument is based on perfect correlations.

In the present chapter we follow two independent possible strategies. In the first strategy we simply compare our experimental results with the predictions both of quantum mechanics and of a local realist theory for GHZ correlations assuming that the particles carry the hidden variables necessary to explain the perfect quantum ccl, clc and lcc correlations. The spurious events are then just due to experimental imperfection not correlated to the hidden variables a photon carries. A local realist might argue against that approach and suggest that the non-perfect detection events indicate that the GHZargumentation cannot succeed. In our second strategy we therefore give maximum leeway to local realist theories assuming that the non-perfect events in the first three experiments indicate a set of hidden variables (elements of reality) which are in extreme conflict with quantum mechanics. We then compare the local realist prediction for the lll experiment obtained under that assumption with the experimental results.

In the experiment, all measurements are conditioned upon the detection of a photon by the trigger detector T and we will only refer to the remaining three photons which are detected by detectors D1,D2,andD3. To meet the condition of time overlap of the photons at the final polarizing beamsplitter PBS (Fig. 5.1) we change in small steps the time difference between the photons from arm a and b by translating the position of the beamsplitter BS. In this way, we can scan into the region of quantum superposition. Polar- izers and λ/4 plates have been used to perform polarization analysis. More specifically, we insert a polarizer oriented at 45◦ or -45◦ in front of a certain detector to perform a H or V polarization measurement respectively, and further insert a λ/4 plate in front of the polarizer to perform a R or L circular polarization measurement.

The observed results for two possible outcomes in a ccl experiment are shown in Fig. 6.1(a). The remaining possible outcomes of a ccl experiment 86 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM have also been measured. At large delay, i.e. outside the region of coherent superposition, it was observed that within the experimental accuracy the eight possible outcomes have the same coincidence rate, whose mean value was chosen as a normalization standard. After normalizing we determined the fractions for all eight possible outcomes simply by dividing the normal- ized four-fold coincidences of a speciﬁc outcome by the sum of all possible outcomes in a ccl experiment. For instance the two black bars in Fig. 6.1(b) indicate the fractions of the RRV and RRH contributions at zero delay respectively.

All individual fractions which were obtained in our ccl, clc and lcc experiments are shown in Figs. 6.2(a), (b) and (c), respectively. From the data we conclude that we observe the GHZterms of Eq. 6.4 predicted by quantum mechanics in 85% of all cases and in 15% we observe spurious events.

Adopting our first strategy we assume the spurious events are just due to experimental errors and thus conclude within the experimental accuracy that for each photon 1, 2 and 3, quantities corresponding to both c and l measurements are elements of reality. Consequently a local realist if he accepts that reasoning would thus predict that for a lll experiment, the combinations V V V ,HHV ,HV H,andV HH will only be observable (Fig. 6.3(b)). However referring back to our original discussion we see that quantum mechanics predicts the exact opposite terms should be observed (Fig. 6.3(a)). To settle this conflict we then perform the actual lll experiment. Our results, shown in Fig. 6.3(c), disagree with the local realism predictions and are con- sistent with the quantum mechanical predictions. The individual fractions in Fig. 6.3(c) clearly show within our experimental uncertainty that only those triple coincidences predicted by quantum mechanics occur and not those predicted by local realism. In this sense, we claim that we experimentally realized the first three-particle test of local realism following the GHZ argument.

We have already seen that the observed results for a lll experiment con- ﬁrm the quantum mechanical predictions when we assume that deviations 6.3. EXPERIMENTAL RESULTS 87

Figure 6.1: A typical experimental result used in the GHZ argument, in this case four-fold coincidences (top) between the trigger detector T, detectors D1 and D2 both set to measure a right-handed polarized photon, and detector D3 set to measure a linearly polarized H (lower) and V (upper curve) photon as a function of the delay between photon 1 and 2 at the ﬁnal polarizing beamsplitter. At zero delay maximal GHZ entanglement results and the bottom graph shows the experimentally determined fractions of RRV and RRH triples (out of the eight possible outcomes in the ccl experiment) as deduced for the zero delay measurements 88 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM

Figure 6.2: Fractions of the various outcomes observed in the ccl, clc,and lcc experiments. The experimental data show that we observe the GHZ terms predicted by quantum physics in (85 ± 4)% of all cases and in (15 ± 2)% the 6.3. EXPERIMENTAL RESULTS 89

Figure 6.3: The conﬂicting predictions of quantum physics (a) and local realism (b) of the fractions of the various outcomes in a lll experiment for perfect correlations. The experimental results (c) are in agreement with quantum physics within experimental errors and in disagreement with local realism. 90 CHAPTER 6. EXPERIMENTAL TESTS OF THE GHZ THEOREM from perfect correlations in our experiment, and in any experiment for that matter, are just due to unavoidable experimental errors. However what are the predictions of local realism for the lll experiment when the correlations are not perfect, as is the case for our experiment where not all events observed in the ccl, clc and lcc experiments agree with the quantum GHZpredictions? Is it possible that by using a local realistic theory, these non-GHZterms can explain all our experimental results?

To answer this we adopt our second strategy and consider the best prediction that a local realistic theory could obtain using these spurious terms. How could, for example, a local realist obtain the quantum prediction HHH? One possibility is to assume that triple events producing HHH would be described by a speciﬁc set of local hidden variables such that they would give events in agreement with quantum theory both in a lcc and clc experiment, for example the results HLR and LHR, but a spurious event for a ccl experiment, namely LLH. In this way any local realistic prediction for an event predicted by quantum theory in our lll experiment will use at least one spurious event in the earlier measurements together with two correct ones. Therefore the fraction of quantum predictions in the lll experiment can at most be equal to the sum of the fractions of all spurious events in the ccl, clc,andlcc experiments, that is 0.45. However, we experimentally observed such terms with a fraction of 0.87 ± 0.04 (Fig. 6.3(c)), in clear contradiction to the hidden variable prediction.

6.4 Discussion and Prospects

Since the first tests of quantum mechanics versus local realism there have been strong debates as to what extent these experiments fully refute the notion of local realism. In this chapter we presented the first experimental test of quantum nonlocality in three-particle entanglement where the theories make definite but opposite predictions. Our experiment fully confirms the predictions of quantum mechanics and is in conflict with local hidden variable 6.4. DISCUSSION AND PROSPECTS 91 theories. We would like to remark that our second analysis presented above, succeeds because our average visibility of (71 ± 4)% clearly surpasses the minimum of 50% necessary for a violation of local realism [Mermin, 1990b; Roy and Singh, 1991; Zukowski and Kaszlikowski, 1997; Ryff, 1997].

However, we have by no means the illusion that our new test will once and for all convince the disbelievers of quantum mechanics. Our experiment shares with all existing two-particle tests of local realism the property that the detection eﬃciencies are rather low. Therefore we had to invoke the fair sampling hypothesis [Pearle, 1970; Clauser and Shimony, 1978] where it is assumed that the registered events are a faithful representative of the whole ensemble.

It will be interesting to further study GHZcorrelations over large distances with space-like separated randomly switched measurements [Weihs et al., 1998], to extend the techniques used here to the observation of multiphoton entanglement [Bose et al., 1998], to observe GHZ-correlations in massive objects like atoms [Hagley et al., 1997], and to investigate possible applications in quantum computation and quantum communication protocols [Briegel et al., 1998; Cleve and Buhrman, 1997]. Chapter 7

Conclusions and outlook

In this work, we have used pairs of polarization-entangled photons as produced by pulsed parametric down-conversion to experimentally explore interference phenomena of multiparticle quantum systems. Our research has been mainly concentrated on the experimental demonstration of quantum teleportation, on the experimental realization of entanglement swapping, on the production of three-particle GHZentanglement, and on the experimental realization of a three-particle test of local reality versus quantum mechanics. For the ﬁrst time, these experiments open the door to study various novel phenomena for quantum systems of three or more particles. It is foreseen that the techniques developed in our experiments, besides their interest to the foundations of quantum physics, will have many important applications in future quantum communication schemes, such as third-man quantum cryptography, entanglement puriﬁcation, and entanglement distribution.

Although teleportation has been realized using polarization-entangled photons, we are still on the way to long-distance quantum teleportation, i.e. transmission of quantum states over large distance. Utilizing the tools that were recently developed for those long-distance Bell-type experiments [Weihs et al., 1998; Tittel et al., 1998a; 1998b], a slight modiﬁcation of our teleportation scheme will allow us to realize long-distance teleportation with an eﬃciency of 50%. As discussed in Chapter 3, a stable visibility better than

92 93

71% is necessary to violate a Bell inequality. It is of great interest to investigate the possibility to improve the visibility of our entanglement swapping experiment, this will ultimately leads to an experimental test of nonlocality both with a pair of photons that never interacted and truly independent ob- servers. It is proposed that one can produce four-photon GHZstate [Pan and Zeilinger, 1999b]. The long-distance GHZ experiment will constitute a test for local realism under strict Einstein locality condition. It is also suggested to perform long-distance quantum cryptography based on multiparticle (GHZ) entanglement, which enables a more advanced cryptography system.

We have seen in the thesis that our interferometric Bell-state analyzer does not give the full capacity of the new quantum communication schemes. Three instead of four messages can be encoded in one photon and the teleportation of polarization states of photon can be performed only with a maximum of 50% eﬃciency. To perform all possible unitary transformations strong coupling between quantum systems is necessary. We propose to continue the development of photon-photon coupling in optical systems. As suggested by Roch et al. [Roch et al., 1992], the experiments recently performed by Kimble’s group at Caltech [Turchette et al., 1995, Hood et al., 1998] show a way to increase the coupling between weak light beams by atoms in high ﬁnesse cavities. However, the bandwidth requirements of these devices are too high to combine this approach with down-conversion experiments.

It will be a real future challenge to study similar systems with slightly relaxed bandwidth requirements, in order to adapt the technique for our experiments. This would ultimately lead to the realization of complete long- distance teleportation of quantum states of photon and atom. Moreover, this would also open a wholly new ﬁeld of experimental investigations, since such photon-photon-coupling devices are needed for quantum nondemolition measurements, quantum logic gates, and in various experiments on the foundations of quantum mechanics. Bibliography

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