Photonic Multipartite Entanglement

Massimiliano Smania Photonic multipartite entanglement

Generation, measurement and applications

Photonic multipartite entanglement Massimiliano Smania

Massimiliano Smania received his BSc and MSc degrees in Physics from the University of Padua. His Master's thesis on quantum communication was carried out at Stockholm University, where he also completed his PhD in Physics.

ISBN 978-91-7911-030-7

Department of Physics

Doctoral Thesis in Physics at Stockholm University, Sweden 2020

Photonic multipartite entanglement Generation, measurement and applications Massimiliano Smania Academic dissertation for the Degree of Doctor of Philosophy in Physics at Stockholm University to be publicly defended on Thursday 10 September 2020 at 09.00 in FB41, AlbaNova universitetscentrum, Roslagstullsbacken 21, digitally via conference (Zoom), public link https:// stockholmuniversity.zoom.us/s/239996391.

Abstract We are currently witnessing a fundamental change in the field of quantum information, whereby protocols and experiments previously performed in university labs are now being implemented in real-world scenarios, and a strong commercial push for new and reliable applications is contributing significantly in advancing fundamental research. In this thesis and related included papers, I first look at a keystone of quantum science, Bell's theorem. In particular, I will expose an issue that we call apparent signalling, which affects many current and past experiments relying on Bell tests. A statistical test of the impact of apparent signalling is described, together with experimental approaches to successfully mitigate it. Next, I consider one of the most refined ideas that recently emerged in quantum information, device-independent certification. Device-independent quantum information aims at answering the question: "Assuming we trust quantum mechanics, what can we conclude about the quantum systems or the measurement operators in a given experiment, based solely on its results, while making minimal assumptions on the physical devices used?". In my work, the problem was successfully approached in two different scenarios, one based on entangled photons and the other on prepare-and-measure experiments with single photons, with the aim of certifying informationally-complete quantum measurements. Finally, I conclude by presenting an elegant and promising approach to the experimental generation of multi-photon entanglement, which is a fundamental prerequisite in most modern quantum information protocols.

Keywords: quantum information, entanglement, Bell tests, POVM, device-independent, self-testing, quantum optics, prepare-and-measure.

Stockholm 2020 http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-182523

ISBN 978-91-7911-030-7 ISBN 978-91-7911-031-4

Department of Physics

Stockholm University, 106 91 Stockholm

PHOTONIC MULTIPARTITE ENTANGLEMENT

Massimiliano Smania

Photonic multipartite entanglement

Generation, measurement and applications

Massimiliano Smania ©Massimiliano Smania, Stockholm University 2020

ISBN print 978-91-7911-030-7 ISBN PDF 978-91-7911-031-4

Thesis for the degree of Doctor of Philosophy in Physics Department of Physics, Stockholm University, Sweden.

©Paper II: 2020 Optical Society of America. ©Paper III: 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY- NC).

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Printed in Sweden by Universitetsservice US-AB, Stockholm 2020

Distributor: Department of Physics, Stockholm University To my parents. Ai miei genitori.

Abstract

We are currently witnessing a fundamental change in the field of quantum information, whereby protocols and experiments previously performed in university labs are now being implemented in real-world scenarios, and a strong commercial push for new and reliable applications is contributing significantly in advancing fundamental research. In this thesis and related included papers, I first look at a keystone of quantum science, Bell’s theorem. In particular, I will expose an issue that we call apparent signalling, which affects many current and past experiments relying on Bell tests. A statistical test of the impact of apparent signalling is described, together with experimental approaches to successfully mitigate it. Next, I consider one of the most refined ideas that recently emerged in quantum information, device-independent certification. Device-independent quantum information aims at answering the question: “Assuming we trust quantum mechanics, what can we conclude about the quantum systems or the measurement operators in a given experiment, based solely on its results, while making minimal assumptions on the physical devices used?”. In my work, the problem was successfully approached in two different scenarios, one based on entangled photons and the other on prepare-and-measure experiments with single photons, with the aim of certifying informationally-complete quantum measurements. Finally, I conclude by presenting an elegant and promising approach to the experimental generation of multi-photon entanglement, which is a fundamental prerequisite in most modern quantum information protocols.

v vi Acknowledgements

It is not a coincidence that most acknowledgement sections in dissertations begin with the doctoral supervisor. My PhD would not have been possible without the scientific, moral and financial support of Mohamed Bouren- nane. I especially thank you for all our never-ending discussions on the most diverse topics, which, besides keeping me from supper, have in time come to constitute the foundations of some of the skills I have developed outside of the scientific realm in the past few years. I also thank my co-supervisor Markus Hennrich, for always being knowl- edgeable and clear whenever explaining complex quantum optics concepts to me, and Per-Erik Tegner for his guidance and help as a doctoral mentor. Ingemar Bengtsson, your generosity and dedication in helping students are lights in the sometimes dark corridors of Academia, and have proven useful multiple times in my path. Five years are a rather long time in academic terms, which is why I had the chance to meet and collaborate with several researchers without whom this work would have never reached a satisfactory end. Among them, I would like to thank Ashraf, Sadiq, Hammad and Nawareg, who shared with me countless hours of work and discussion inside and outside the lab. I am also grateful to my co-authors Adán, Matthias, Piotr and Armin. There would be no papers in this thesis without you. My experience in research so far has also been invaluably enhanced by the strong sense of community I felt in my everyday work, especially among fellow students. I thank Marco, Alley, Walid, Natalie, Alban, Alexander, Gerard, Fabian, Irina, Pil, and each and every person in the KIKO group. My time in Albanova would have been miserable without all of you. Unlike my Masters thesis, these acknowledgements will be part of the final printed version, thus I have to resist switching to more emotional southern languages. Ornella and Franco, although I have never had the chance to try other parents, I am absolutely and scientifically certain that you are the best in the world. You and my brother are the ground I walk on and the sky I look up to. Thank you. Nicola, thank you also for the ever-too-brief times spent together in these few years. The distance that separates us is but a number, and if this goes to plan, I will soon achieve total control over numbers. This section of my acknowledgements would

vii not be complete without the latest addition to my family: Vani, the efforts and sacriﬁces you made so I could reach this goal are what made it worth in the ﬁrst place. Thank you for taking care of me as if your life depended on it, your attention is what kept me going. Karin and Paolo, thank you for being the best friends to live next to in a pandemic, and so much more. Actually, thank you Karin for the Sam- manfattning too, for giving me a place to sleep, for covering my rent when I could not pay, for sharing apartments with me throughout my PhD, and additional crucial help which would need a separate section just to list. Thank you Andrea for exchanging doubts on our aptitude as researchers, and for long and intricate discussions on all of the big themes in the world. Both made us stronger. I am also grateful to all the friends that have made my time outside Albanova so much fun, in particular the Desert Island team and the ever-expanding Italian club. Sincere thanks also to all the friends who, although not in Stockholm, played a fundamental role during the time of my PhD studies anyway. In particular, I would like to thank Davide, Luca, Alice, Ale, Ale, Degia, Joey, Danel and Tri. Your friendship has time and again proven stronger than the distance that divides us. Last but not least, the research work carried out in this thesis was funded by the Knut and Alice Wallenberg Foundation and the Swedish Research Council.

viii Contents

Abstract v

Acknowledgements vii

List of Papers xii

Author’s contribution xiii

Relevant papers not included in the thesis xiv

List of Figures xv

List of Tables xvii

Sammanfattning xviii

Preface xix

1 Bits of quantum information 1 1.1 The qubit ...... 1 1.2 Multi-qubit systems ...... 2 1.3 Mixed states ...... 4 1.4 Quantum measurements ...... 5 1.4.1 Projective measurements ...... 6 1.4.2 Generalised measurements ...... 7 1.4.3 SIC-POVM ...... 8 1.4.4 Trine-POVM ...... 9 1.5 Quantum state ﬁdelity ...... 10 1.6 Entanglement ...... 10 1.6.1 Entanglement witness ...... 11 1.7 Bell states ...... 12

ix 2 Bell tests 13 2.1 The EPR paradox ...... 13 2.2 Bell’s theorem and the CHSH inequality ...... 14 2.3 Loopholes ...... 16 2.4 Applications ...... 18

3 Experimental background 21 3.1 Single photon polarisation as a qubit ...... 21 3.2 The main actors in polarisation manipulation ...... 22 3.2.1 Wave-plates ...... 22 3.2.2 Polarisers ...... 23 3.2.3 (Polarising) beam-splitters ...... 24 3.2.4 Polarisation-dependent ﬁlter ...... 25 3.3 Polarisation qubit state preparation ...... 27 3.4 State analysis and detection ...... 27

4 Signalling 31 4.1 A deﬁnition ...... 32 4.1.1 A mathematical deﬁnition ...... 32 4.2 Some key motivations ...... 33 4.3 Experimental sources of signalling ...... 35 4.3.1 Solutions and results ...... 36 4.4 Concerning common assumptions ...... 37 4.5 Additional experimental work ...... 37 4.5.1 State characterisation ...... 38 4.5.2 Wave-plate motor hysteresis ...... 38 4.5.3 Laser power stability ...... 39

5 Certifying a generalised quantum measurement 41 5.1 Device independent quantum information ...... 42 5.2 Two ways to certify a generalised quantum measurement . 42 5.3 Entanglement-based certiﬁcation ...... 44 5.3.1 Summary of results ...... 46 5.3.2 Experimental assumptions ...... 47 5.4 Prepare-and-measure certiﬁcation ...... 47 5.4.1 Summary of results ...... 48 5.4.2 Experimental assumptions ...... 49

6 Multi-photon entanglement source 51 6.1 Spontaneous-parametric down-conversion: a qubit source . 52 6.1.1 Pump ...... 53 6.1.2 Polarisation-entangled pair source ...... 54 x 6.2 Scaling up to GHZ states ...... 55 6.3 General setup characterisation ...... 58 6.3.1 Source brightness ...... 60 6.3.2 Source ﬁdelity ...... 63 6.4 Characterisation of individual sources ...... 68 6.4.1 2-photon sources ...... 68 6.4.2 4-photon sources ...... 70 6.4.3 6-photon source – γ6 ...... 72 6.5 Comparison with literature ...... 74 6.5.1 Future outlook ...... 75

7 Summary and outlook 77

References 79

xi List of Papers

The following papers, referred to in the text by their Roman numerals, are included in this thesis.

PAPER I: Avoiding apparent signaling in Bell tests for applications Massimiliano Smania, Matthias Kleinmann, Adán Cabello, Mohamed Bourennane Arxiv, 1801.05739 (2018) – Not the latest version, see attached paper. DOI: https://arxiv.org/abs/1801.05739

PAPER II: Experimental certiﬁcation of an informationally complete quantum measurement in a device-independent protocol Massimiliano Smania, Piotr Mironowicz, Mohamed Nawareg, Marcin Pawłowski, Adán Cabello, Mohamed Bourennane Op- tica, 7, pp. 123-128 (2020). DOI: https://doi.org/10.1364/OPTICA.377959

PAPER III: Self-testing nonprojective quantum measurements in prepare-and-measure experiments Armin Tavakoli, Massimiliano Smania, Tamás Vértesi, Nico- las Brunner, Mohamed Bourennane Science Advances, vol. 6 no. 16 (2020). DOI: https://doi.org/10.1126/sciadv.aaw6664

PAPER IV: Experimental observation of photonic multipartite entanglement Hammad Anwer, Muhammad Sadiq, Massimiliano Smania, Mohamed Bourennane – In preparation.

Reprints were made with permission from the publishers.

xii Author’s contribution

PAPER I: I designed the experimental setup, built it, and performed the measurements. I carried out a relevant part of the data analysis and participated in theoretical discussions. I contributed to writing the paper.

PAPER II: I and M. Nawareg designed and built the experimental setup. I performed all the measurements independently, except for the full state tomography, which I did with M. Nawareg. I carried out the data analysis and participated in developing the theoretical background. I contributed to writing the paper.

PAPER III: I designed and built the experimental setup. I performed the measurements and their analysis. I participated in discussions of the theoretical ideas in the paper, and I contributed in writing the experimental sections.

PAPER IV: M. Sadiq, M. Nawareg and M. Bourennane designed the experimental setup. M. Sadiq, M. Nawareg, H. Anwer and I built the experimental setup. M. Sadiq, H. Anwer and I performed the experiments reported in Chapter 6. I carried out the data analysis for the results reported in the same chapter. All authors are contributing to writing the paper.

xiii Relevant papers not included in the thesis

Experimental quantum multiparty communication protocols Massimiliano Smania, Ashraf Mohamed El Hassan, Armin Tavakoli, Mohamed Bourennane npj Quantum Information, 2, 16010 (2016). DOI: https://doi.org/10.1038/npjqi.2016.10

xiv List of Figures

1.1 A vector on a Bloch sphere ...... 2 1.2 Eigenstates of Pauli matrices on the Bloch sphere . . . . .3 1.3 SIC-POVM and Trine-POVM projectors on the Bloch sphere.9

3.1 A beam-splitter...... 24 3.2 Polarisation-dependent ﬁlter...... 25 3.3 Single qubit state preparation...... 27 3.4 A measurement station for single photon polarisation qubits. 28

4.1 Simulated CHSH result for equal and very unequal overall efﬁciencies. Reported from Paper I...... 34 4.2 Example of experimental data showing high apparent signalling. Reported from Paper I...... 35 4.3 Test of wave-plate motor hysteresis...... 38 4.4 Test of power stability of pump laser...... 39

6.1 Simple scheme of pump light preparation...... 54 6.2 Depiction of the entangled photon source...... 54 6.3 Down-converted photons as photographed by a single-photon camera...... 55 6.4 Spectrum of down-converted photons, before and after spectral filtering...... 56 6.5 Gold-coated ten-facet mirror to split SPDC rings...... 57 6.6 Entanglement of independent photons...... 57 6.7 Multi-photon entanglement source - experimental setup. . 59 6.8 RP against pump power, with linear fit...... 62 6.9 Singles, 2- and 4-photon generation rates against pump power. 62 6.10 4-, 6- and 8-photon rates as function of 2-photon coincidences. 63 6.11 φ + visibility as function of pump power and collection efficiency.| i ...... 66 6.12 The Hong-Ou-Mandel effect in our source...... 68

xv 6.13 A measure of photon indistinguishability as function of pump power...... 69 6.14 Experimental quantum state tomography of 2-qubits. . . . 70 6.15 Experimental quantum state tomography of 4-qubits. . . . 71 6.16 Experimental results of measurement of operators in Eq. (6.11)...... 72 6.17 Experimental results of measurement of operators in Eq. (6.12)...... 73 6.18 Expected 8-photon rates as function of pump power and collection efﬁciency...... 76

xvi List of Tables

1.1 Pauli operators and their eigenstates, labelled with light polarisation directions...... 3

3.1 Wave-plate angle settings are reported for Pauli measurement operators, together with the PBS outputs associated to each projector...... 29

5.1 Comparison of the main features of the entanglement-based and prepare-and-measure certiﬁcation approaches. Pros and cons are coloured in green and red respectively...... 44

6.1 List of multi-photon entanglement sources found in the setup in Fig. 6.7. In the source name, the superscript represents the number of photons involved, while the subscript helps to distinguish between sources with the same number of photons. 58 6.2 Experimental visibility in σz and σx bases for the 2-photon sources in the setup. The different uncertainties are mainly due to different measurement duration...... 70 6.3 Comparison of the performance of the multi-photon entanglement source reported in this work with the state-of-the- art experiments found in literature, and a previous source built in our lab...... 74

xvii Sammanfattning

Vi bevittnar numera en grundläggande förändring inom kvantinformations- fältet: protokoll och experiment som tidigare utfördes i universitetslabora- torier genomförs nu i verkliga scenarier. Dessutom bidrar en stark kommer- siell satsning på nya och tillförlitliga tillämpningar avsevärt till att främja grundforskningen. I denna avhandling och tillhörande bifogade artiklar, betraktar jag först en grundpelare inom kvantvetenskapen: Bells sats. I synnerhet kommer jag att uppenbara ett problem som vi kallar “apparent signalling” - skenbar signalering. Detta problem påverkar många nu- varande och tidigare experiment som bygger på Bell-tester. Det beskrivs ett statistiskt test av skenbar signalering, samt experimentella metoder för att framgångsrikt förmildra dem. Därefter betraktar jag en av de mest inno- vativa idéer som nyligen framkommit inom kvantinformationen: “device- independent certification” - enhetsoberoende certifiering. Enhetsoberoende kvantinformation syftar till att besvara frågan: “Förutsatt att vi litar på kvantmekanik, vilka slutsatser kan vi dra gällande kvantsystemen eller mä- toperatörerna i ett visst experiment, baserat enbart på dess resultat, sam- tidigt som att det framställs minimala antaganden om de fysiska enheter som används?”. I mitt arbete är problemet framgångsrikt betraktat i två olika scenarier, den ena baserad på sammanflätade fotoner och den andra på förbered-och-mät enstaka fotoner, i syfte att försäkra informativt kom- pletta kvantmätningar. Slutligen avslutar jag med att presentera en elegant och lovande inställning till den experimentella alstring av multifotonsam- manflätning, vilket är en grundläggande förutsättning i de flesta moderna kvantinformationsprotokoll.

xviii Preface

One evening around halfway through my PhD studies, I found myself demonstrating a Bell test experiment to a group of rather enthusiastic students as part of a modern physics course. The course was included in a general physics program, which skimmed over theorems and equations in favour of the more popular aspects of the discipline. After going through the optics included in the experimental setup, I quickly performed the few measurements required for the estimation of the final result. Result which, I an- nounced in darkness, turned out to be safely beyond the boundary of classical physics, which I had previously explained on the whiteboard. A few hands popped up, and I swiftly dispensed equation-free, dogmatic answers to the apparent satisfaction of the class. That was, until one perplexed- looking student asked: “What is entanglement?”. Realising that the standard definition would require knowledge of at least quantum superposition and state separability, I scrambled for an answer which the students could cope with. I went with the all-time favourite: two identical objects are of two colours at the same time, and when we check the colour of one of them, they both immediately turn the same colour, no matter how far apart they are. The class was not satisfied. They pointed out there is nothing wrong with the objects being one of two colours from the beginning, and no weird instantaneous influence is required to explain why they always turn out of the same colour. That was correct of course, but I could not possibly say “that is the difference between a mixed quantum state and entanglement”, could I? Unable to find a better answer, I resorted to playing the card of dogmatic knowledge once again, and simply said that if the students’ remark were true, our little experiment would not have violated Bell’s inequality, and that more knowledge was of course required to really understand the definition of entanglement. Unsurprisingly perhaps, there really is no way to explain the concept of entanglement without invoking some of the features which are unique to quantum mechanics. Although the consequences of entanglement used to puzzle some of the best minds of the previous century [1], they are now accepted and mostly understood by the scientific community. In recent years, quantum information has been transitioning from books and labs to cata- logues and (online) shops, by means of multi-national public and private ef-

xix forts. While the industry shows a growing interest in investing in quantum information and its applications, researchers are sharpening their existing tools and exploring new paradigms in view of real-world implementations. In this context, my PhD work aimed at both refining current experimental approaches to quantum information applications, and probing some of the more recent ideas emerging in the field. After a brief introduction of some fundamental theoretical concepts in Chapters 1 and 2, I will discuss in the present thesis how such theoretical concepts were translated into experiment in Chapter 3. Chapter 4 will report on how, in Paper I, we ex- posed and characterised a potential issue common to some of the best Bell experiments found in literature. On the other hand, in Paper II and III, we explored the recent concept of device-independent quantum information in relation to a type of measurement rather unusual in quantum optics, that is, non-projective measurements. The different approaches to the problem of device-independent certification taken in the two papers are outlined in Chapter 5. However, none of the work in the mentioned papers could be carried out without the development and characterisation of a source of entangled photons. Our novel approach on multi-photon entanglement generation is what took most of my time and efforts during my PhD and, since not yet published, it is reported in details in Chapter 6. The results will constitute part of Paper IV, which is currently in preparation. The three introductory chapters are built upon my licentiate thesis, which was defended on December 20, 2018, half-way through my PhD studies. Among the papers discussed here, Paper I was included in that dissertation, although in the form of a previous version. As a consequence, Chapter 4 too is partially taken from my licentiate thesis. More in detail, the contri- butions from my previous dissertation are as follows: Chapter 1 : Sections 1.1, 1.2, 1.3, the first part of 1.6, and 1.7 were present in the licentiate thesis. They have been limitedly extended. Chapter 2 was present in the licentiate thesis. It has been reviewed and corrected where needed. Chapter 3 was present in the licentiate thesis, except for Section 3.2.4. Section 3.4 has been extended. Chapter 4 : Sections 4.1, 4.2, 4.3 and 4.5 were present in my licentiate thesis. They have been significantly reworked and extended. Chapter 6 : Section 6.1 was present in my licentiate thesis. It has been reviewed and limitedly extended. All the remaining parts are new. xx 1. Bits of quantum information

In this introductory chapter we will quickly go through the main ingredients necessary to comprehend the slightly more advanced topics that are the object of this work. Far more complete presentations of the basics of quantum information can be found in literature (for example in [2]), while the in- tention here is to limit the discussion to the necessary notions and intuition useful in giving the performed experiments context and meaning.

1.1 The qubit

Just as bits represent the unit of information in classical information and computation theory, quantum information theory has its own unit, called qubit. The key difference between the two is that while a bit can only be 1 or 0, a qubit can be in any linear superposition of 1 and 0. This distinction stems from the fact that while bits describe states of classical systems, qubits refer to quantum systems, thus obeying the laws of quantum mechanics, among which we ﬁnd the superposition principle. Therefore, any bi-dimensional pure quantum state ψ can be written as | i ψ = α 0 + β 1 , (1.1) | i | i | i where 0 and 1 are the two orthonormal vectors, written in Dirac notation, that| i constitute| i the so-called computational basis of the bi-dimensional Hilbert space where the qubit lives, and α and β are complex numbers that satisfy α 2 + β 2 = 1 whenever ψ is normalised. A different, often more | | | | useful way of describing qubit states| i is with vectors. The state in Eq. (1.1) is equivalent to α ψ = . (1.2) | i β Disregarding any global phase factor in front of a qubit state, two real parameters are sufﬁcient to describe a pure state. These parameters can be two angles, and the state is then very intuitively visualised on a three- dimensional Bloch sphere (see Fig. 1.1). Such a vector is usually written as θ θ ψ = cos 0 + eiφ sin 1 , (1.3) | i 2 | i 2 | i 1 with 0 θ π and 0 φ 2π respectively being latitude and longitude on the sphere.≤ ≤ ≤ ≤

Figure 1.1: A vector on a Bloch sphere

In the same way that we can express states either as linear combinations of basis states or as vectors, these two representations can be used for operators, or observables. For example, Pauli’s σz operator is:

1 0 σz = 0 0 1 1 = . (1.4) | ih | − | ih | 0 1 − Pauli operators (reported in Tab. 1.1) take on a very special role in quantum information theory: their eigenvectors constitute the three mutually unbiased bases in the qubit space. Moreover, Pauli matrices, together with the identity matrix (sometimes referred to as σ0), span the 2 2 matrix space. For this reason, any operation on a qubit can be expressed× as a linear combination of these four matrices. As observables, Pauli operators fulfil Heisenberg’s uncertainty relation. They will be used throughout this work. When acting on a qubit, Pauli operators σx, σz and σy respectively perform a bit-flip, a phase-flip, and both of these operations, in the computational basis (see Tab. 1.1). The six states making up the three Pauli bases are depicted in Fig. 1.2.

1.2 Multi-qubit systems

Some of the signature features of quantum information, entanglement as a predominant example, require combining multiple qubits to make up a more

2 Operator Matrix Eigenstates Operation

0 1 0 1 σx 0 = 1 σx = | i±| i | i | i 1 0 |±i √2 σx 1 = 0 | i | i 0 i 0 i 1 0 +i 1 σy 0 = i 1 σy − R = | i− | i , L = | i | i | i | i i 0 | i √2 | i √2 σy 1 = i 0 | i − | i 1 0 σz 0 = 0 σz H = 0 , V = 1 | i | i 0 1 | i | i | i | i σz 1 = 1 − | i −| i

Table 1.1: Pauli operators and their eigenstates, labelled with light polarisation directions.

Figure 1.2: The six eigenstates of Pauli operators σx, σy and σz are drawn in, respectively, red, blue and green on the Bloch sphere.

complex quantum system. Such a system lives in a bigger Hilbert space which consists of the tensor product between each of the qubits’ Hilbert spaces. As the simplest example of a multi-qubit system, let us take a look at a two-qubit system. Labelling the single qubits A and B, the Hilbert space of 2 the composite system AB is HAB = HA HB, which has dimension 2 = 4. A natural basis for this new system is the⊗ tensor product of the single qubit space bases:

1 0 0 0 0 1 0 0 00 = 0 0 =  , 01 =  , 10 =  , 11 =  . (1.5) | i | i ⊗ | i 0 | i 0 | i 1 | i 0 0 0 0 1                 3 Similarly to the case of one qubit in Eq. (1.1), we can write the general pure state of two qubits as

ψ = α 00 + β 01 + γ 10 + δ 11 , (1.6) | i | i | i | i | i with the normalisation condition α 2 + β 2 + γ 2 + δ 2 = 1. In matrix | | | | | | | | notation, the equivalent state would be

α β ψ =  . (1.7) | i γ δ      As a ﬁnal note, even though qubits are often associated with a particular physical quantum system, for example photons, to the point where the two terms are used interchangeably (as will be the case further on in this thesis), it is important to notice that while a qubit is a mathematical entity, a photon or an atom are instead physical ones. Single photons or atoms can in principle store several - inﬁnite even - qubits [3].

1.3 Mixed states

In an ideal scenario, we would have access to the totality of a quantum state and relative information, and the quantum system would not decohere due to interaction with the surrounding environment: we would then describe such a system with a pure state of the form of Eq. (1.1). While this scenario might apply to a quantum system in extremely limited cases, for example an infinitesimally short time just after a projective measurement, the reality is usually more complex. Because a system always interacts with its environment, or whenever we do not have access to each part of a composite quantum system, a more accurate way of describing it are mixed states. These are statistical ensembles of pure states, where the distribution reflects the amount of information we have on the system state. Usually named ρ, or density operator, a mixed state is defined as

m ρ = ∑ pi ψi ψi (1.8) i=1 | ih | where each pi represents the probability (in the ensemble of m states) to ﬁnd the system in the state ψi . | i Pure states do not become mixed just in case of decoherence due to interaction with the surroundings. In fact, whenever a composite system

4 includes any entanglement among its sub-parts, each part alone will be described by a mixed state. The other way around is also true in a way: mixed states can always be described as a reduction of a pure state living in a bigger Hilbert space. Intuitively, a single concept can explain the mixing of a pure state due to two as seemingly different causes as decoherence and entanglement: a system (for simplicity, we will consider a qubit) in a pure state interacting with the environment exchanges information with it. This interaction gen- erates entanglement, and the initial information representing the pure state is thus spread on a far bigger system. Unfortunately, we are usually unable to measure the environment at the same time as our qubit, and therefore the information yielded to the former is lost. As a consequence of measuring just the qubit part of the greater entangled system (which also includes the environment), we necessarily obtain a mixed state. We will consider entanglement at greater length in the next section. As far as representation is concerned, the most convenient way to work with mixed states is within the density matrix formalism, which also works for pure states. Each matrix element is given by i ρ j , where i, j = 1,..N refer to basis vectors in the N-dimensional Hilberth | | spacei where the state{ is} deﬁned. The density matrix is therefore basis-dependent, although the computational basis is normally used. As an example, the state in Eq. (1.1) can be written as

αα αβ ρ = ψ ψ = ∗ ∗ . (1.9) | ih | α∗β ββ ∗

The eigenvalues of the density matrix are the probabilities pi of Eq. (1.8), thus ρ is Hermitian and positive semi-deﬁnite. Finally, a normalised state always has tr(ρ) = 1 and tr ρ2 1, where the inequality is saturated only for pure states, therefore establishing≤ a simple distinction between pure and mixed states. Moreover, a density matrix represents a pure state if and only if it is idempotent (i.e. ρ = ρ2), or if it has rank one.

1.4 Quantum measurements

Any measurable quantity in quantum mechanics (QM) is called observable, and is described by an Hermitian operator. For example, suppose the ob- Rank(A) servable A is deﬁned by the Hermitian matrix A; eigenvectors ai of {| i}i=1 the matrix are the observable’s eigenstates, associated to eigenvalues ai such that A ai = ai ai . (1.10) | i | i 5 Since the eigenvectors form an orthonormal basis, any pure state ψ can always be expressed as a function of them of the form: | i

ψ = ∑ ci ai (1.11) | i i=1 | i where complex coefﬁcients ci = ai ψ are probability amplitudes, with 2 h | i ∑ ci = 1. | | Translated into words, Eq. (1.10) means that if we measure the observable A on a system described by the state ai , we will obtain the (real) | i result ai, and the system will remain unchanged. However, if the system originally was not in an eigenstate of A, we can use Eq. (1.11) to predict 2 2 the transition probability P(ai) = ci = ai ψ of obtaining the ai eigen- | | |h | i| value, while the system after measurement will be in the corresponding ai eigenstate. | i If we are not dealing with a pure state, the density matrix formalism can be used to estimate the expectation value of observable A for state ρ as

A ρ = tr(Aρ) ∑(Aρ)ii . (1.12) h i ≡ i According to the (most widely accepted) Copenhagen interpretation of QM, if a system is not in an eigenstate of a measurement operator, the corresponding property of the system is indefinite before measurement. Upon measurement however, the system state collapses necessarily on one of the eigenstates of the measured observable. This apparently counter-intuitive feature of QM is at the core of one of the greatest scientific debates of the twentieth century, as we will see in the next chapter. Interestingly enough though, it all becomes much more eccentric once complex systems come into play. While the above gives an intuitive, operational definition of measurements in QM, I will now give a slightly more abstract description of the subject, which will let us distinguish between two experimentally very different types of measurements used in the works presented in this thesis.

1.4.1 Projective measurements A particular measurement in QM is in general associated with a set of measurement operators Ei . These operators are deﬁned in the Hilbert space of the system upon which{ } they act, and the set size is in general unconstrained, while each Ei corresponds to a measurement outcome. The only constraint on the set Ei descending from QM postulates is that of completeness: { } † 1 ∑Ei Ei = . (1.13) i

6 † In the special case in which the operators are also self-adjoint (Ei = Ei ) and orthogonal to each other, i.e. EiE j = δi jEi, the measurement is called projective, and the operators are orthogonal projectors. Therefore, an ob- servable1 E describing a projective measurement can be decomposed as

E = ∑eiEi, (1.14) i where Ei is a projector onto its corresponding eigenspace of E, with eigen- value ei. Since Ei’s are orthogonal to each other, the observable E deﬁned in a Hilbert space of dimension n can only have up to n eigenvectors, and therefore its measurement will require up to n outcomes. Hence a projective measurement on a qubit will have one or two outcomes. A simple formula describes the expectation value of a projective measurement E on a pure state ψ : | i E ψ = ψ E ψ , (1.15) h i| i h | | i while for mixed states Eq. (1.12) is in general used. Although being a very special case of quantum measurements, it should be noted that projective measurements are the most commonly implemented in quantum information experiments, by a wide margin.

1.4.2 Generalised measurements Starting from the more general definition of quantum measurement, with the only requirement of completeness in Eq. (1.13), we can define a more general class of measurements. These are usually called generalised quantum measurements, where one observable is now represented by a Positive Operator-Valued Measure (POVM), a set of positive semi-definite operators. From Eq. (1.13), we can rewrite:

† 1 ∑Fi Fi = ∑Ei = . (1.16) i i

† The set Ei = F Fi consists of positive operators that are in general not { i } orthogonal to each other. Each Ei is a POVM element, or effect, and the whole set is simply called a POVM. Upon measuring the POVM on a pure state, the probability of outcome i is:

† P(i) ψ = ψ Fi Fi ψ = ψ Ei ψ . (1.17) | i h | | i h | | i 1Starting from here, observables and their corresponding operators will be used interchangeably.

7 For mixed states, the probability is again estimated as

P(i)ρ = tr(ρEi). (1.18)

Two speciﬁc examples of POVMs that were realised for qubits in our labs will be introduced in the following: the SIC-POVM and the Trine-POVM.

1.4.3 SIC-POVM

In the quest for the best possible POVM, two properties are highly desirable: ﬁrst, the measurements should be able to reconstruct any quantum state; second, it should do so in the optimal way. A measurement with the ﬁrst property is called informationally complete (IC), since its statistics uniquely identify the state it acted upon. The property of optimality is instead associated with the POVM having the minimal amount of outcomes, and the outcomes being maximally independent. Such a POVM is called symmetric. Symmetric informationally-complete (SIC) POVMs are thus the ideal quantum measurements in terms of information extracted from the measuring process. The simplest way to describe a SIC-POVM in dimen- 2 sion d is with a set of d normalised states φi satisfying [4] {| i}

2 1 φ j φi = , j = i. (1.19) h | i 1 + d 6

Starting from these states, the POVM effects can be deﬁned as the sub- normalised projectors:

φi φi Ei | ih |, (1.20) ≡ d which are a complete set (see Eq. (1.16)). It is clear from Eq. (1.19) that effects Ei are not orthogonal to each other. The same equation also 2 expresses the symmetry condition EiE j = 1/ d (1 + d) , for i = j. On the other hand, starting from Eq. (1.19) one can also conclude that6 SIC-POVM effects are linearly independent, which implies that the set is IC [4]. The existence of SIC-POVMs in arbitrary dimensions is to date an open question, although algebraic or numerical solutions in many dimensions are known [5; 6]. In our work, we focused on the simple d = 2, where pen-and-

8 paper calculations lead to the four projectors

φ1 = 0 , | i | i 0 + √2 1 φ2 =| i | i, | i √3 0 + √2ei2π/3 1 (1.21) φ3 =| i | i, | i √3 0 + √2ei4π/3 1 φ4 =| i | i, | i √3 from which the SIC-POVM effects can be calculated with the help of Eq. (1.20). The four vectors are easily remembered thanks to their representation in the Bloch sphere, where they point to the corners of a regular tetrahedron, as depicted in Fig. 1.3 (left).

∣ϕ1⟩ ∣ψ1⟩

∣ϕ4⟩ ∣ϕ ⟩ ∣ϕ2⟩ 3 ∣ψ ⟩ ∣ψ2⟩ 3

Figure 1.3: The SIC-POVM (left) and Trine-POVM (right) projectors, depicted on the Bloch sphere, from Eqs. (1.19) and (1.22) respectively.

1.4.4 Trine-POVM In Paper III, we also considered a different non-projective measurement for qubits, called Trine-POVM. The name comes from the fact that this measurement has three outcomes, whose corresponding projectors on the Bloch sphere form an equilateral triangle as in Fig. 1.3 (right). Due to the normalisation condition in Eq. (1.16), the three vectors lie on the same plane in the Bloch sphere, and can for example be written as:

ψ1 = 0 | i | i 0 + √3 1 ψ2 =| i | i | i 2 (1.22) 0 √3 1 ψ3 =| i − | i, | i 2 9 for a Trine-POVM on the xz disk. Thanks to its symmetry and simplicity, the Trine-POVM made for a good candidate for demonstrating the self-testing methods explained in Paper III, and in particular the concept of robustness to experimental noise.

1.5 Quantum state ﬁdelity

At the beginning of Section 1.4, we have defined the quantity ψ φ 2 as |h | i| the probability of state φ “transitioning” into state ψ upon measurement. Such transition probability| i represents the overlap between| i the two states, and because it gives a measure of how similar they are to each other, it is also called quantum state fidelity. The transition probability used to determine fidelity of pure states ( φ , ψ ) can be generalised to the case of mixed | i | i states (σ,ρ) [7] with the formula:

2 F (σ,ρ) = tr √σρ√σ . (1.23) q In case of pure states, the ﬁdelity is equivalent to the transition probability. Moreover, it is symmetric (F (σ,ρ) = F (ρ,σ)) and bounded, i.e 0 F (σ,ρ) 1. The bounds are respectively met when σ and ρ are orthogonal≤ or identical.≤ Eq. (1.23) was used in all the experiments discussed in this thesis.

1.6 Entanglement

Perhaps the most striking and essential concept in QM, entanglement never fails to puzzle anyone who comes across it, from non-scientists to the greatest physicist of the last century. The idea that quantum objects can be bound by correlations that are stronger than any classical correlation one might conceive of, was so alien and counter-intuitive that, after Einstein, Podol- sky and Rosen pointed out some of its apparently unnatural consequences in 1935 [1], it took almost thirty more years to ﬁgure out a way to test these concepts in a lab. These tests are one of the main concerns of this work. From a mathematical perspective, a pure quantum state is not entangled if it can be expressed as the tensor product of its subsystems, while a mixed state is not entangled if it is equivalent to a mixture of non-entangled pure states. A general non-entangled mixture can be expressed as

1 2 n ρ = ∑ piρi ρi ... ρi , (1.24) i ⊗ ⊗ ⊗

10 where the probabilities pi sum to unity and superscripts refer to each part of the composite system. Because entanglement is a concept that inherently involves two parts of a system, it is fairly well studied and characterised in bipartite quantum systems. Up to six total dimensions - thus in case of qubit-qubit or qubit-qutrit systems - the Peres-Horodecki criterion gives a necessary and sufﬁcient condition for separability [8; 9], therefore clearly indicating whether a quantum system contains any entanglement. In higher dimensions though, the criterion only provides sufﬁciency for separability. It should be noted however, that in the special case of pure states, a simple method to determine if the system is entangled consists in calculating its partial trace. If any of its sub-partitions turns out to be mixed, then the original system necessarily contained entanglement. A system composed n 1 by n parts is entangled even if only one of its 2 − 1 possible partitions contains entanglement. If all such partitions are entangled− the system is said to possess genuine entanglement.

1.6.1 Entanglement witness The matter of identifying entanglement becomes even more problematic when dealing with systems composed by more than two parts. In these sit- uations, no general criterion applies, rather what is usually employed is a so-called entanglement witness [10]. These are functionals of the density matrix that distinguish between a specific type of entangled states and sep- arable states. They are usually built ad-hoc for a particular state and measurement scenario, and when linear, they are simply a set of observables. The most famous entanglement witness, known to be sub-optimal, is Bell’s (CHSH) inequality operator, which will be analysed in the next chapter. A simple and convenient entanglement witness devised for a class of multi-qubit entangled states that will be studied in this thesis, GHZN = | i ( 01...0N + 11...1N )/√2 states, was introduced by Gühne and colleagues | i | i in [11]. The witness operator is defined as WGHZ = 1/2 D. D is the following decomposition of the GHZN state: − | i D = GHZN GHZN | ih | N 1 1 N N 1 − k N (1.25) = ( 0 0 )⊗ + ( 1 1 )⊗ + ∑ ( 1) Mk⊗ , 2 | ih | | ih | 2N k 0 − h i = where Mk = cos(πk/N)σx +sin(πk/N)σy and k = 0,1,...,N 1. A negative − value of WGHZ indicates that a state is genuinely entangled. Furthermore, measuring F¯ = tr(ρD) returns an estimate of the fidelity of state ρ with GHZN . An experiment that determines F¯ requires exactly N +1 measurements,| asi opposed to a full state tomography which involves an amount of

11 measurements that grows exponentially with N. For this reason, the witness above is commonly used in multi-photon entanglement experiments [12– 15], and will be employed extensively in Chapter 6 when characterising the multi-photon entanglement source used in our work.

1.7 Bell states

When dealing with simpler 2-qubit states and entanglement, a set of four pure states are usually employed as reference or starting point. These are called Bell states [16], and are traditionally identiﬁed as 1 ψ± = ( 01 10 ), | i √2 | i ± | i (1.26) 1 φ ± = ( 00 11 ). | i √2 | i ± | i Each of the Bell states is maximally entangled: if two qubits in the state ψ− , for example, are spatially separated and measured in the same measurement| i basis, perfect anti-correlated outcomes will result, independently of the particular chosen basis. This correlation goes much further than any classical physics theory could explain, and also implies that if an experimenter only knows the measurement result of their local qubit, they can infer no conclusion whatsoever on the other qubit unless they are willing to collaborate with the experimenter on the other side. Mathematically, this operation is also known as “tracing out” one of the qubits. For example, if Alice holds one qubit and ignores Bob’s, the state of her subsystem will be: 1 ρA = TrB( ψ− ψ− ) = ( 0 0 + 1 1 ) (1.27) | ih | 2 | ih | | ih | which is the totally mixed state, devoid of any correlation with Bob’s qubit. For this reason Bell states are key candidates in secure communication and cryptographic protocols [17], where collaboration and detectable eaves- dropping are absolute requirements. The four states also constitute an orthonormal basis for the Hilbert space of two qubits, and can be converted into each other by means of local operations based on Pauli operators (see Tab. 1.1).

12 2. Bell tests

The terminology “Bell tests” encompasses a rather large and diverse group of experiments that since the ’70s have aimed at shedding light on some of the most fundamental questions about reality: is Nature “real” and independent of our knowledge and perception of it? Is it local, that is, only inﬂuenced by its surroundings? In this chapter we will go deeper into the notion of local realism and exactly how quantum mechanics seems to give a way-out of the contradictions that Einstein, among others, ran into.

2.1 The EPR paradox

In their memorable 1935 paper, Einstein, Podolsky and Rosen (EPR) proposed a thought experiment that, with what they considered reasonable assumptions, lead them to conclude that “either the description of reality given by the wave function in quantum mechanics is not complete” or two non- commuting observables “cannot have simultaneous reality” [1]. The paradox, as per EPR’s reasoning, goes as follows: imagine two particles, A and B, that are separated while in a singlet (entangled) state of position and momentum. Following Heisenberg’s uncertainty principle, if we measure the position of particle A, its wave function will collapse in a state of indeterminate momentum. Because the system is entangled though, the momentum of particle B will also be indeterminate after the measurement on A, even though in principle no disturbance has occurred to B. Conversely, B’s position will be perfectly deﬁned and opposite to A. How can B “know” to have well deﬁned position and indeterminate momentum, instantaneously after a measurement is performed on A, no matter how far away it is from A? EPR’s solution is that the system included a variable, hidden from the wave function, that dictated to B how to behave in case of such a measurement performed on A. In their words, we are therefore “forced to conclude that the quantum mechanical description of physical reality given by the wave function is not complete”. The explicit assumption that EPR make is that a complete physical theory must contain all elements of the physical reality, that is, it should be able to predict with certainty the values of physical quantities, without in-

13 fluencing them, prior to them being measured. Such a theory is said to be local realist, because it satisfies the principle of realism, according to which reality is independent of our description of or interaction with it, and its physical properties are totally defined at any point in space and time. Quantum mechanics, on the other hand, not only cannot predict values of incompatible observables with certainty, but goes as far as suggesting that these values are not defined before measurement, all of it while seemingly admitting simultaneous non-local interactions. The idea that elements of reality could be influenced instantaneously by far-away events was too much to accept for the inventor of special relativity, who suggested that a theory that does not explicitly comply with locality cannot be correct. As we know now, quantum mechanics is indeed not local realist, but it does not violate causality [18, p. 426-428], therefore complying - at least in principle - with Einstein’s idea of locality. While most interpretations of the theory, including the ever-favourite Copenhagen Interpretation, reject realism, therefore admitting that observables are not defined prior to measurement, other interpretations - most fa- mously De Broglie-Bohm’s - do not discard determinism as easily, prefer- ring instead to explicitly break the locality principle. To date, no definitive argument has closed the discussion on which principle to follow. We thus concluded that quantum theory cannot fulfil both locality and realism at the same time. A careful reader might think that we are missing the important question though: what can be inferred on Nature and local realism from real-world experiments?

2.2 Bell’s theorem and the CHSH inequality

EPR’s solution to their paradox, namely that quantum mechanics is incom- plete, was of course a much easier pill to swallow at the time than giving up local realism. After all, the idea that there were hidden variables that a fairly young theory had not considered almost seemed natural. This situation of uncertainty remained until 1964, when John Bell proved a breakthrough theorem [16] that could potentially settle the question once and for all: an inequality that any local realist theory has to satisfy, regardless of known or hidden variables. Since a theory that violates such inequality cannot comply with local realism, and quantum mechanics does indeed predict this violation, the theorem transforms EPR’s conclusion into: either Nature is not local realist, or quantum mechanics is plainly wrong. While Bell’s proof is a remarkably simple and elegant one, we will consider here a slightly modiﬁed version of his inequality, which proved more useful and manageable for experimenters: the Clauser-Horne-Shimony-Holt

14 inequality [19], as reported by Bell himself in [20, p. 36-37]. Suppose we have two qubits that are sent to two stations which perform measurements along directions ~a and~b respectively. The initial state of the qubits is described entirely by (possibly hidden) variables λ, and measurement outcomes can be A(~a,λ) = 1 and B(~b,λ) = 1. The fact that A and B do not depend on~b and ~a respectively,± is the locality± assumption. The expectation value of the experiment will be

E ~a,~b = A ~a,λ B ~b,λ ρ(λ)dλ (2.1) Z where ρ(λ) is the probability distribution of λ and A and B are averages which satisfy A 1 and B 1. Let a, a0, b and b0 be four particular measurement settings,| | ≤ then| | ≤

E a,b E a,b0 = A a,λ B b,λ A a,λ B b0,λ ρ(λ)dλ − − Z = A a,λ B b,λ 1 A a0,λ B b0,λ ρ(λ)dλ ± Z A a,λ B b0,λ 1 A a0,λ B b,λ ρ(λ)dλ . − ± Z Using the constraints mentioned above on the outcomes and the triangle inequality, we get

E a,b E a,b0 1 A a0,λ B b0,λ ρ(λ)dλ | − | ≤ ± Z + 1 A a0,λ B b,λ ρ(λ)dλ ± Z = 2 E a0,b0 + E a0,b , ± which can ﬁnally be rewritten as

S E a,b E a,b0 + E a0,b + E a0,b0 2. (2.2) ≡ | − | | | ≤ Bell’s theorem proves that if a theory fails to satisfy Eq. (2.2), it is necessarily not local realist. Checking for quantum mechanics is now a basic exercise: suppose the two qubits are in the maximally entangled state ψ− (see Eq. (1.26)). If we choose the four measurement projectors to be on| thei xz plane of the Bloch sphere, we can express them as a function of the angle θ they make with the north pole (or state 0 ). The expectation value of a measurement then becomes | i

E aˆ,bˆ = cos θaˆ θˆ . (2.3) − − b 15 Choosing suitable settings, Inequality (2.2) is comfortably violated:

E 0◦,45◦ E 0◦,135◦ + E 90◦,45◦ + E 90◦,135◦ = 2√2 > 2. | − | | | (2.4) The value 2√2 is also the maximum reachable by quantum physics, and known as Tsirelson’s bound [21]. We have therefore concluded that quantum mechanics cannot be local realist. What about Nature? Since the 70s, a great amount of Bell experiments have been performed, mainly with photons as qubits [22–25], but also ions [26] and other systems as atom condensates [27] or nitrogen-vacancies in diamonds [28]. Needless to say, all of these experiments, and countless others, have overwhelmingly shown that Nature is indeed not compatible with local realism. It is important to notice that all of these experimental tests are not directed at proving that quantum mechanics is correct or complete, but rather at demonstrating that Nature cannot be described by any local realist theory. The fact that some of the results come really close to Tsirelson’s bound, without overcoming it, is at best an additional indication that quantum mechanics is the best theory of its kind that we have at the moment. Moreover, the degree of assurance these experiments offer changes from case to case, mainly because of assumptions that the authors had to make in order to violate the inequality. These assumptions are often called loopholes, and we shall take a look at them in the next section.

2.3 Loopholes

During the years, as new and more advanced Bell experiments were proposed, researchers came up with weaknesses in the setups that left room for (local) hidden variable models to invalidate results that would have otherwise ruled out local realism. Some of these so-called loopholes are setup- speciﬁc, but the most important and experimentally demanding ones are found in most experimental realisations until very recently. A selection of them includes: Locality loophole Originally identiﬁed by Bell himself [20], this loophole corresponds to the breaking of the main assumption in Bell’s work, i.e. that the outcome of a measurement on one of the qubits has to be independent from the measurement setting on the other qubit. This is usually solved by involving relativistic causality: if measurement settings on each qubit are chosen after the two qubits are far apart, enough that no signal can travel between the two before the measurement is completed, then there can be no causal correlation between

16 settings and outcomes. This loophole was already closed in a famous experiment by Aspect and others in 1982 [23], when fast polarisers were used for measurements of photon polarisation.

Detection loophole Because of imperfections in experimental setups - attenuation, scattering, low detection efficiency - only a small fraction of measurements actually output any result at all. A very simple hidden variable model could contain information on whether the qubit should be detected or not, depending on the measurement setting. The lower bound on overall detection efficiency in order to close this loophole is 2/3, which was fairly prohibitive, especially in photonics setups, until recently. The usual adjustment to this problem is the fair sampling assumption, according to which the fraction of detected qubits is a fair representation of all the qubits originally prepared. This loophole was originally closed in 2001 using trapped ions, which allow for near-unity detection efficiency [26]. Photonics experiments were able to conquer the issue for the first time in 2013 [29], thanks to a new type of highly efficient single photon detector.

Free-will loophole We have already seen that Bell’s key assumption is that outcomes A and B cannot depend on settings b and a respectively. Suppose we go back in time to when the measurement devices and the source of qubits were stored together in the same room, or even much further to the Big Bang if necessary: if all future events have been determined by an interaction that happened before the experiment, then there can be no independence between the measurement devices at all! This metaphysical concept is also known as super- determinism. To-date, physicist have not yet ﬁgured out a solution to this matter, and it is likely that one is not even possible. Whether this is within the realm of Physics is obviously debatable, but considering the astounding philosophical implications of Bell’s theorem, his reply upon being accused of doing metaphysics, for considering this matter in the ﬁrst place, is easily shared: “Disgrace indeed, to be caught in a metaphysical position! But it seems to me that in this matter I am just pursuing my profession of theoretical physics.” [20].

For the ﬁrst time in 2015, in a span of a few weeks, three research groups managed to independently close both the locality and detection loopholes at the same time. Two of them used entangled photons [24; 25], and one combined photons and electronic spin in nitrogen-vacancies [28]. These experiments have shown, covering all known and addressable loopholes until proven otherwise, that Nature is not local realist.

17 2.4 Applications

As we have seen in the previous sections, Bell tests were originally con- ceived as a way of determining whether Nature could be compatible with any local realist theory of hidden variables, or if non-local theories, as quantum mechanics, are the right path to a better understanding of the universe. Besides the fundamental scope of Bell’s work, and its profound philosophical implications, researchers have, throughout the years, identiﬁed several other ﬁelds that can take advantage of his theorem. Some interesting examples are reported here.

Cryptography One of the ﬁrst, and most fascinating still, applications of Bell tests was formulated by Arthur Ekert in 1991 [17]: suppose Al- ice and Bob want to share a secure key for encryption and decryption of secret messages. Ekert’s idea consists in exploiting the correlations contained in 2-qubit entangled states. If, upon testing for the CHSH inequality, the two users obtain a maximal violation (that is, the value 2√2), they can be unconditionally sure that no eavesdrop- per has tampered± with their qubits, the measurement devices, or even the qubit source. The protocol has been successfully implemented (although assuming fair-sampling and no communication) in a very suggestive experiment in the Canary Islands in 2006 [30].

Communication complexity The advantage of quantum over classical protocols in the complexity scenario, where Alice and Bob (and potentially more users) try to solve a problem with the least communication possible, has been shown in several experiments that used entanglement and non-locality as resources [31; 32]. Even more importantly, in 2004 Brukner and colleagues proved that in communication complexity scenarios, the “violation of Bell’s inequalities is the necessary and sufﬁcient condition for quantum protocols to beat the classical ones” [33].

Randomness certiﬁcation Similarly to how Alice and Bob can securely share a secret key, they can also obtain private bits that are certiﬁably random through a violation of a Bell-type inequality [34].

Entanglement measure In the ever-growing number of scenarios where entanglement is used as a resource, Bell-type inequalities can be effective in characterising the experimental setup by quantifying the amount of entanglement in the prepared states [35; 36]. The possibility of measuring entanglement, especially in high-dimensional systems, is of course interesting also from a fundamental perspective.

18 All of the above-mentioned applications require either loophole-free Bell’s inequality violations, or additional assumptions to meet the setting/outcome independence requirement. In particular, a very common assumption consists in expecting there to be no signalling (or no communication) between Alice’s and Bob’s measurement stations. As we shall see in Chapter 4, this assumption is often taken for granted and not suitably tested.

19 20 3. Experimental background

So far, we have discussed quantum systems, notably qubits, without referring to a particular physical implementation of such mathematical entities. In this chapter I will introduce single photon polarisation as a qubit realisa- tion. This was the system of choice throughout the experiments reported in this thesis, and is also one of, if not the most common qubit implementation found in literature. We will consider the key parts of the experimental setups used to achieve the results presented in the following chapters, including some polarisation manipulation components, and measurement and detection devices.

3.1 Single photon polarisation as a qubit

In general, any physical system with two well-defined, isolated states can be visualised as a qubit. Around the world, an extremely diverse range of possibilities has unfolded throughout the years: electronic or nuclear spin, (trapped) ions or neutral atoms, quantum dots, superconducting Josephson junctions or, more recently, topological systems. All of these besides, of course, photons. Photonic qubits are so convenient and practical that different degrees of freedom have been taken advantage of to realise them, or even higher-dimensional quantum systems, in a single photon. Photons have the advantage of interacting extremely weakly with the environment, thus preventing decoherence. They are also fairly easy to create, move around (at the highest speed!), and detect. Single photon polarisation was the two-level system of choice for this work, but photon number, time of arrival, or orbital angular momentum are also commonly used. Our choice has several benefits: polarised photons are easily prepared with extremely high precision, in superposition or entangled states. Two variations of the same optical components, the wave-plate, are sufficient to have access to the whole two-dimensional Hilbert space. Finally, distinguishing between orthogonal states at detection can be done with almost unitary efficiency. In this work, horizontal ( H ) and vertical ( V ) linear polarisations are used as the 0 and 1 computational| i states from| i which all other relevant | i | i 21 states are derived, as reported in Tab. 1.1. Once a photon is prepared in a given polarisation, it will maintain it as long as it propagates through air or other non-birefringent media, while optical fibres and glass will in general modify the polarisation state.

3.2 The main actors in polarisation manipulation

A significant advantage of polarisation qubits consists in the ease of manip- ulating this degree of freedom. Very few, passive optical components, are sufficient for most tasks. We will now take a look at the most significant ones.

3.2.1 Wave-plates Birefringent materials can present two different refractive indexes to light propagating through them. In particular, in our work we used wave-plates made of quartz, which is a positive (ne > no) uni-axial crystal. Light polarisation components along the ordinary and extraordinary axes will experience a relative phase retardance φ which depends on wavelength λ, crystal thickness d and refractive index difference in the following way:

2πd (ne no) φ = − . (3.1) λ In matrix notation, this phase shift corresponds to 1 0 W (φ) = . (3.2) 0 0 eiφ There are two particular values of the phase φ that are of interest: when π φ = π, the wave-plate is called half wave-plate (HWP), while φ = 2 corresponds to a quarter wave-plate (QWP). These components usually differ in thickness d. By changing the angle θ between the optic axis and the incident light polarisation, we can vary the relative phase between the H and V components. | i | i In the lab, we conventionally set θ = 0 when the vertical polarisation is aligned to the optic axis. In this case, the effect of a wave-plate is expressed by Eq. (3.2), for given d and λ. To describe the more interesting scenario of θ = 0, one can rotate the frame of reference to the one where θ = 0, apply 6 matrix W 0 (φ), and then rotate back to the lab frame. Using the rotation matrix cosθ sinθ R(θ) = , (3.3) sinθ cosθ − 22 we can deﬁne the generic wave-plate matrix as

T W (θ,φ) = R (θ)W 0 (φ)R(θ) cos2 θ + eiφ sin2 θ sinθ cosθ 1 eiφ (3.4) = − . sinθ cosθ 1 eiφ sin2 θ + eiφ cos2 θ − π Thus, for the two interesting cases of φ = π and φ = 2 , we obtain respec- tively1 cos(2θ) sin(2θ) HWP(θ) = (3.5) sin(2θ) cos(2θ) − and cos2 θ + isin2 θ (1 i)sinθ cosθ QWP(θ) = − . (3.6) (1 i)sinθ cosθ sin2 θ + icos2 θ − To get some intuition, we can refer once again to the Bloch sphere: if we start with a horizontal polarisation state, a HWP will rotate it around the y-axis, while a QWP will do the same around the x-axis. Combinations of HWPs and QWPs can realise any unitary single-qubit gate; for example, HWP(22.5◦) corresponds to a Hadamard gate.

3.2.2 Polarisers

While a wave-plate is essential in performing (ideally) unitary operations on polarisation qubits, polarisers are very useful whenever a speciﬁc state is required, or if we want to go from a mixed to a pure state. These are in practice polarisation ﬁlters that only let through one particular polarisation, depending on the orientation angle at which they are set. The polarisers we use in the lab consist of thin parallel nano-wires that absorb the polarisation component orthogonal to the wire orientation. By rotating the component around the propagation axis, any linear polarisation on the xz equator can be prepared. The action of a polariser, at an angle θ with the vertical polarisation, can be found by applying again the same trick as used for wave-plates, of changing frame of reference:

0 0 sin2 θ sinθ cosθ Pol(θ) = RT (θ) R(θ) = − . (3.7) 0 1 sinθ cosθ cos2 θ − It is worth noting that the operation of such a polariser is not unitary: the absorbed component is “lost”.

1Overall phases in front of operator matrices are purposefully ignored.

23 3.2.3 (Polarising) beam-splitters Perhaps the most crucial optical component in quantum optics experiments, a beam-splitter (BS) is a semi-transparent planar - or cubic - piece of glass with (at least) two input and two output directions. It can be used to split, combine, or interfere light beams or single photons. The BSs we used are cubic glass devices consisting of two triangular halves, adhering to each other’s hypotenuse facet through a dielectric ﬁlm. If we consider the BS in Fig. 3.1, we can describe its operation within a second-quantisation-type of formalism [37, Section 6.2], using annihilation operators as ﬁeld amplitudes:

aˆ t r aˆ 2 = 0 0 (3.8) aˆ r t aˆ 3 0 1 where (r,r0) and (t,t0) are complex reﬂectance and transmittance as indi- cated in Fig. 3.1, which are constrained by energy conservation.

Figure 3.1: A beam-splitter with two input ports (a0, a1) and two output ports (a2, a3).

For a dielectric BS, the phase difference between transmitted and reflected beams is i. If we assign that phase to the reflected beam, and further simplify to± the case of a 50:50 lossless BS, we can rewrite Eq. (3.8) as aˆ 1 1 i aˆ 2 = 0 . (3.9) aˆ3 √2 i 1 aˆ1 Polarisation-insensitive BSs as the one described are used for path-encoded qubits and photon counting implementations, or wherever interferometry plays a role. In our case of polarisation qubits, a version of BS aptly called polarising beam-splitter (PBS) was used. This device effectively combines the polarisation degree of freedom with path, usually reflecting vertical polarisation and transmitting its horizontal component. In mathematical terms,

24 following the same notation as above

aˆ0,H aˆ2,H −→ aˆ0,V iaˆ3,V −→ (3.10) aˆ1,H aˆ3,H −→ aˆ1,V iaˆ2,V −→ with the H and V subscripts indicating photon polarisation. Naturally, PBSs in the lab are never ideal. A well-aligned, high quality PBS will normally have a so called “polarisation leakage” of around 0.1%, meaning that for every one thousand horizontally-polarised photons, one will be wrongly reﬂected. This imperfection directly limits measurement quality.

3.2.4 Polarisation-dependent ﬁlter By combining the components described above in a two-path Sagnac-type interferometer, a polarisation-dependent ﬁlter (PDF) can be obtained as depicted in Fig. 3.2. As the name suggests, the PDF can arbitrarily attenuate

Output 2

Input 1

PBS

Output 1

Input 2

Figure 3.2: A two-path Sagnac-type interferometer with two HWPs can be used to independently attenuate H and V components of an input state, or to enable operations in a higher-dimensional| i | i space. the H and V components of an input state, independently. In order to describe| i how| thei PDF works, we need to work in a Hilbert space bigger than a qubit’s, since besides the usual polarisation directions, there are two spatial modes (or paths) at play. A convenient deﬁnition of a basis in this 4-dimensional space is H,a , V,a , H,b , V,b , where a represents {| i | i | i | i} | i 25 the path corresponding to Input 1 clock-wise rotation Output 1, while b stands for Input 2 counter-clock-wise→ rotation →Output 2. In this notation,| i we can describe→ the PBS as → 1 0 0 0 0 0 0 i U = , (3.11) PBS 0 0 1 0 0 i 0 0     while the two HWPs act jointly as

U = HWP(θH1) HWP(θH2) = θH1θH2 ⊕ cos2θH1 sin2θH1 0 0 sin2θH1 cos2θH1 0 0 (3.12)  − . 0 0 cos2θH2 sin2θH2  0 0 sin2θH2 cos2θH2  −    The PDF operator is then found as UPDF(θH1,θH2) = UPBSUθH1θH2 UPBS. In Dirac notation, the action of the PDF on the state ψ = α H + β V sent through Input 1 is | i | i | i

UPDF(θH1,θH2)(α H,a + β V,a ) = | i | i = α cos2θH1 H,a + β cos2θH2 V,a (3.13) | i | i + iβ sin2θH2 H,b + iα sin2θH1 V,b . | i | i If we were to disregard Output 2 completely, we see that the amplitudes of the horizontal and vertical components at Output 1 can indeed be tuned arbitrarily by simply setting the desired angles on the two wave plates, hence the name PDF. In our work reported in Papers II and III, we used both outputs of the PDF in order to implement non-projective measurements. This was achieved by adding a polarisation analysis setup as in Fig. 3.4 to every output, and setting all the required angles and phases on the interferometer and the measurement wave-plates. As an example, in the case of a 4-outcome measurement we can express the Kraus operators for each outcome as

F1 = a UPBSQWP(θQa)HWP(θHa)W 0(φa) a a UPDF a , h | | ih | | i F2 = a UPBSQWP(θQb)HWP(θHb)W 0(φb) a b UPDF a , h | | ih | | i (3.14) F3 = b UPBSQWP(θQb)HWP(θHb)W 0(φb) a b UPDF a , h | | ih | | i F4 = b UPBSQWP(θQa)HWP(θHa)W 0(φa) a a UPDF a , h | | ih | | i as functions of the PDF HWPs angles and the HWP, QWP and phase-plate (W 0) angles in outcomes 1 (θHa, θQa, φa) and 2 (θHb, θQb, φb). From these, † POVM effects (see Section 1.4.2) are found as Ei = Fi Fi.

26 3.3 Polarisation qubit state preparation

The simplest and cleanest way to prepare a single qubit in a desired pure state is by using a combination of polariser and HWP plus QWP, as in Fig. 3.3.

Figure 3.3: A generic, possibly mixed state ρ is sent through a polariser, which transforms it into pure state ψ0 . A combination of HWP and QWP can then rotate this to any desired pure| i state ψ . | i

Suppose we have a qubit described by a generic – possibly mixed – state ρ. We can select a well deﬁned linear polarisation with a polariser, after which the state will be Pol(θP)ρPol(θP). Now we can apply any rotation to this pure state and prepare any other pure state on the Bloch sphere. The result will be

† † ψ = QWP (θQ)HWP (θH )Pol(θP)ρPol(θP)HWP(θH )QWP(θQ). | i (3.15) As an example, we can prepare the state L = ( 0 + i 1 )/√2 by setting | i | i | i polariser and HWP to 0◦, and QWP to 45◦. The resulting qubit will be in state ρ22 L , which is typically not normalised since the polariser is not | i a unitary element. In general, given a state ψ that we want to prepare, we can solve Eq. (3.15) to ﬁnd the angles| ati which we need to set the components in the lab. In the simpler case where we start from a pure state instead of a mixed one, the polariser is of course not necessary. Similarly, if the state we need is in the xz equator, just a polariser is sufﬁcient.

3.4 State analysis and detection

In our case of single photon polarisation qubits, performing a (projective) measurement means projecting the qubit state onto a basis of the bidimen- sional polarisation space (see Section 1.4 for more details). As explained above, a PBS can separate H and V polarisation components, by map- ping them to different output| pathsi | i

PBS ψ = α H + β V α H,a2 + iβ V,a3 . (3.16) | i | i | i −−→ | i | i 27 Positioning detectors at the outputs, we can count photons and derive probabilities. For example, referring to Fig. 3.4 – and ignoring wave-plates for now – we can experimentally determine the modulus squared of the coefﬁ- cients in the state above:

2 2 N0 α = H,a2 (α H,a2 + iβ V,a3 ) = | | |h | | i | i i| N + N 0 1 (3.17) 2 2 N1 β = V,a3 (α H,a2 + iβ V,a3 ) = | | |h | | i | i i| N0 + N1 where N0 and N1 are the number of “clicks” in detectors D0 and D1 during the measurement.

Figure 3.4: A measurement station to analyse single photon polarisation.

Such a measurement, with just a PBS and no wave-plates, corresponds to a projection on the σz basis (see Tab. 1.1). Looking at Eq. (3.16), we notice that the output mode notation is redundant, and it will therefore be dropped in the following. The expectation value of σz (see Eq. (1.15)) will be

σz = ψ σz ψ h iψ h | | i = (α∗ H iβ ∗ V ) ( H H V V ) (α H + iβ V ) h h | − h | | | ih | − | ih | | | i | i i N0 N1 = α 2 β 2 = − | | − | | N0 + N1 (3.18) and can be calculated directly from experimental frequencies. We can now appreciate why a low-quality (or badly aligned) PBS will limit the measurement accuracy: detectors cannot distinguish between photon polarisations, therefore a wrongly transmitted or reﬂected photon will distort the counts N0 and N1, resulting in incorrect probabilities and expectation values. A PBS projects a state on the σz basis because it naturally distinguishes between the two eigenvectors of that basis, H and V . If we want to perform a measurement in a different basis, we| cani add| ai HWP and a QWP in front of the PBS. The wave-plates effectively rotate the frame of reference

28 so that the PBS will separate different (orthogonal) polarisations. For projective measurements in general, given the operator σ of an observable that we would like to measure, we can ﬁnd the wave plate angles experimentally required by solving the following equation:

† QWP(θQ)HWP(θH )σ [QWP(θQ)HWP(θH )] = σz . (3.19)

Wave-plate settings for the most common projective measurements are reported in Tab. 3.1.

Pauli operator HWP QWP Projector PBS Output

+ + a2 σx 22.5◦ 0◦ | ih | a3 |−ih−| L L a2 σy 0◦ 45◦ | ih | R R a3 | ih | H H a2 σz 0◦ 0◦ | ih | V V a3 | ih |

Table 3.1: Wave-plate angle settings are reported for Pauli measurement operators, together with the PBS outputs associated to each projector.

Regarding the actual detection of a photon, we used fast avalanche photo-diode (APD) single-photon detectors. These are diodes that get re- verse-biased above breakdown voltage, so that a single photon creating one electron-hole pair can be sufficient for starting an avalanche discharge, and therefore a detection signal. The detection efficiency of such devices was around 50% at 780 nm, while the dead-time, necessary after a detection to recover from the avalanche, was approximately 50 ns. Because of the high bias voltage, thermal fluctuations can sometimes start an unwanted discharge. Such a “false” detection event is also called a dark count. The devices we used had dark count rates of around 400 to 600 per second.

29 30 4. Signalling

As we have seen in Section 2.4, besides the fundamental progress on the matter of local realism, a parallel branch of applications involving Bell tests has flourished throughout the years. Teleportation and super-dense coding are just some of the early ones, while Bell-type inequalities constitute the backbone of the modern fields of quantum cryptography [17], communication [32] and randomness certification [34], among others. These applications usually require near-optimal violations of Bell inequalities in order to be effective, and for this reason additional assumptions are made in the process. In particular, fair-sampling is often assumed, where the detected qubits are postulated to be a fair representation of all qubits emitted by the source (see Section 2.3 for more details). In these cases, the crucial assumption in Bell’s theorem that Alice’s outcome has to be independent of Bob’s setting, and vice-versa, can apparently be breached. This dependence is also called signalling, as it seems as if the two measurement stations were communicating. Because this event appears highly unlikely, testing and disproving it is usually not done in literature. Unfortunately, as we reported in Paper I, an actual test of apparent signalling on even the highest violation of the CHSH inequality [38], shows an extremely significant breach of the no-signalling assumption. In the work carried out in Paper I, we formulated a simple test that can estimate the significance of apparent signalling from the data of a CHSH test, and we experimentally identified the main explanations to such issue. Measurements that confirmed our conjectures were performed, eventually reaching a result that was reasonably free of such troubles. As we will see in Section 4.2, our tests are not only useful from the fundamental point of view of reliably disproving local realism, but also have more practical uses. Indeed, as we discussed in Paper I, failing to meet the non-signalling requirements may lead to a distortion and invalidation of any experimentally derived result. We shall see in this chapter how physicists usually deal with this problem, and what the consequences are in conjunction with some of the most common assumptions on Bell setups. While our work mainly addresses experiments where fair-sampling is assumed, I will discuss briefly the relevance of our results in absence of the detection loophole.

31 4.1 A deﬁnition

Bell’s requirement that measurement outcomes do not depend on far-away measurement settings imposes that no communication happens between Al- ice and Bob prior to or during measurement. Such communication, or trans- fer of information, is also called signalling. We can therefore say that a successful violation of a Bell inequality must be free from signalling. On the other hand, as we have seen in Section 2.3, Bell tests have rather strict requirements on overall experimental efﬁciency. For practical or techno- logical reasons though, the greatest majority of experiments and applications to date do not meet this requirement, choosing instead to assume fair- sampling, and taking for granted that no signalling took place in the particular scenario. After all, the idea that no communication happens between two distinct and separate measurement devices does not seem that much of a stretch. Still, in all of these cases it is important to rule out any dependence between Alice’s settings and Bob’s outcomes, and vice-versa.

4.1.1 A mathematical deﬁnition

In the usual CHSH scenario (see Section 2.2 for more details), we deﬁne the quantity P(a,b x,y) as the joint probability of Alice and Bob respectively obtaining outcomes| a and b upon measuring settings x and y. We can then express the non-signalling conditions in mathematical form in the following way: A αa,x P(a,_ x,0) P(a,_ x,1) = 0 x,a ≡ | − | ∀ (4.1) αB P(_,b 0,y) P(_,b 1,y) = 0 y,b b,y ≡ | − | ∀ where superscripts A and B refer to either party, and marginal probabilities P(a,_ x,y) and P(_,b x,y) correspond to ∑b P(a,_ x,y) and ∑a P(_,b x,y) respectively.| | | | In Alice’s case for example, these conditions can be understood as follows: since how often she obtains outcome a cannot depend on Bob’s setting y, such probability, after summing over (or tracing out) Bob’s outcomes, has to be the same whether he measures setting 1 or 0, thus the difference between the two cases be zero. Moreover, the eight conditions in Eq. (4.1) are really four independent ones upon considering normalisation of probabilities, P(_,_ x,y) = 1 for any x,y. | In a real experiment though, the four α’s are never expected to be exactly zero, no matter how ideal the scenario is. Because of statistical noise, the differences in Eqs. (4.1) will ﬂuctuate within one or two standard deviations from zero. For this reason, it is useful to apply a normalisation on

32 the experimental uncertainty, that is for example:

A α+1,0 P(+1,_ 0,0) P(+1,_ 0,1) A = | −A | , (4.2) σ+1,0 σ+1,0 which should ideally be below 2. A direct check of the experimental data can be done by inserting measured frequencies and uncertainties in the four Eqs. (4.2). In Paper I, we also reported a more advanced method based on likelihood for a joint test of all four conditions.

4.2 Some key motivations

There are various reasons for analysing whether conditions (4.1) are satis- ﬁed. In the following, some relevant ones are reported, ranging from fundamental to practical in scope:

No-signalling assumption As we discussed in Section 2.2, the independence of Alice and Bob in the test is a crucial starting point of Bell’s theorem. Violating the conditions in Eq. (4.1) is a clear, quantiﬁable sign of breaching such assumption. This renders, in principle at least, any conclusion of the Bell test invalid. This is the most fundamental motivation for making sure that the above conditions are met.

Result certification If any signalling in the form expressed above is present in an experiment, it may be the case that the final result for the parameter S (see Eq. (2.2)) is distorted as compared to the same experiment in a signalling-free scenario. As an example, in Paper I we identified the different overall efficiency between outcomes in the same measurement station as being a potential source of apparent signalling. We then performed a CHSH experiment with a fixed, previously set difference between Alice’s and Bob’s respective outcomes. The experiment violates, to an extreme degree, the non-signalling conditions, as verified with our own test, and yielded a rather striking S = 2.947 0.003, which is far above the quantum bound of 2√2 2.828.± Using the same efficiencies, we simulated the expec- ≈ + + tation value of S for the Werner state ρW = p φ φ +(1 p)1/4, as a function of mixing parameter p, reported| inih Fig. 4.1| (taken− from Paper I). While such a result would certainly raise concerns in the experimenter, the same effect can happen in a controlled, malicious man- ner: Eve could intercept all communication between the source and

33 3.0

2.5

2.0

1.5

S parameter 1.0

0.5

0.0 0.0 0.2 0.4 0.6 0.8 1.0 p

Figure 4.1: Expectation value of the S parameter as a function of the mixing + + parameter p for Werner state ρW = p φ φ + (1 p)1/4. The red line corresponds to the two outcomes in each| ih measurement| − station having equal efﬁciencies, and the blue line to ηA = 0.48ηA+, and ηB = 0.2ηB+, where − − ηA is the efﬁciency of Alice’s 1 outcome. In the latter case, the values of −p in between the dotted black− lines incorrectly establish CHSH violation. Reported from Paper I.

Alice’s and Bob’s stations, and send them classically correlated systems instead. If she has access to Alice and Bob’s detectors, she could then tamper with the efﬁciencies so that they get S > 2, and they will not know about her unless they perform tests that include additional measurements. Alternatively, the two parties can run the statistical test of the non-signalling conditions that we presented in Paper I, on the same data used to estimate S, which would show the failing of those conditions, thus prompting them to discard the data. This type of issue is particularly relevant for all applications where the CHSH result is used as a means of certiﬁcation, which could therefore be rendered invalid.

Systematic error discovery Looking at Eq. (4.2), it is clear that a systematic, constant violation of the non-signalling conditions (in the αA A nominator) will cause the ratio +1,0/σ+1,0 to increase as √N, with N indicating sample size. Therefore a signalling analysis that shows such a dependence can reveal the presence of otherwise unknown systematic effects in the experiment. An example of such a problem can be seen in Fig. 4.2 taken from Paper I. Not only is this helpful in case of setup optimisation, but even if the error cannot be ﬁxed, it can at least be accounted for, therefore giving a fairer estimation of S

34 40

35 ) σ 30

10 signalling signiﬁcance (

0 0 200 400 600 800 1000 time (s)

Figure 4.2: Example of apparent signalling showing dependence as square root of sample size (proportional to time in the experiment), due to constant systematic sources. Reported from Paper I.

in case its value is used for any sort of certiﬁcation.

For all these reasons, testing conditions (4.1) is not only a matter of fulﬁlling Bell’s original assumptions in order to obtain a true violation, but it is also crucial in order to have reliable qualitative and quantitative results to use for applications.

4.3 Experimental sources of signalling

In Paper I, we identiﬁed four important systematic issues that will cause the failing of conditions in Eq. (4.1):

1. Power drift of pump laser: if only one detector per station is used, the two outcomes of each measurement have to be checked in sequence. If the total rate of photon changes between different settings, in particular with a systematic drift, the fair-sampling assumption is breached.

2. Polarisation-dependent collection efficiency: rotating wave-plates or polarisers to apply different settings can slightly misplace the photon path. This can be a problem if single-mode fibres are used to collect photons at the outputs, since these fibres have core diameters of very few micrometers. On the other hand, using only multi-mode fibres would collect different spatial modes, significantly decreasing the visibility.

35 3. Measurement setting reproducibility: a low precision in setting the angles of wave-plates or polarisers can lead to apparent signalling, as it is equivalent, in a way, to Alice changing her measurement depending on Bob’s setting. This is particularly problematic if only one round of settings is performed.

4. Asymmetric collection efficiency: if two detectors per station are employed, having different efficiency at the two outputs of the same station may violate the fair sampling assumption. The reason why this issue causes signalling has to do with the coincidence filter that is applied on detection events in cases where fair sampling is assumed: since Alice’s marginal probabilities are calculated from events con- ditioned on Bob’s outcomes, they are not truly local, but will instead depend on Bob’s measurement, and in turn on the efficiency of his outcomes. An example with a short mathematical derivation is provided in Paper I.

While this list is of course not complete, it includes the most relevant errors which are induced by standard procedures in CHSH experiments, especially if one is not aware of the issue.

4.3.1 Solutions and results In approaching the mentioned problems, the first step consisted in trying to design a setup that would naturally be insensitive to as many of them as possible. Problem 1 above is shared among experiments that use only one detector on each side. A simple solution is providing another detector each to Alice and Bob, so that a complete measurement takes place every time a photon on every side is detected, and the normalisation is the same for all outcomes. A clean solution to problem 2 meanwhile, consists in replacing single- mode fibres with multi-mode ones, while moving the single-mode fibres “upstream” in the setup, before any rotating optics. This way, unwanted spatial modes are still filtered out, while the much larger multi-mode cores are effectively insensitive to small beam displacements. Experiments that suffer from these two limitations includes the works with the current highest CHSH violations [34; 38]. Including the improvements mentioned above, we proceeded to test the effects of the two remaining signalling sources on measured data, in a series of experiments. After showing the result of a typical experiment where visibility and collection efficiency were optimised (reported in Fig. 4.2), we

36 showed that repeating the four CHSH measurement settings several times (a rigorous Bell test would also include setting randomisation), and using more precise motors to rotate wave-plates, could significantly reduce the effects of problem 3, assuming that the motor precision is symmetric around a central position (more details about this are given in the next section). Regarding problem 4, we described a simple method to verify the difference in collection efficiency, and used variable attenuators to equalise them. Finally, after taking care of all the above-mentioned issues, we ran a CHSH experiment which yielded S = 2.812 0.003, with a signalling significance of 1.3 standard deviations. This is± the highest violation that we know of, free of apparent signalling. The reader is referred to Paper I for more details on the different experiments and their results.

4.4 Concerning common assumptions

As mentioned above, our work concerns experiments that assume fair sampling, and therefore only use coincident detections for data analysis. All works that do not suffer from the detection loophole, including in particular loophole-free ones [24; 25; 28], calculate marginal probabilities from single events alone, and will therefore not introduce any non-local effect in the data processing stage. As a consequence, the test presented in Paper I will not reveal any apparent signalling. While some of the experimental issues introduced in Section 4.3 will of course impact the ﬁnal result, for example asymmetric efﬁciency, these will in general only deteriorate the Bell parameter, or should be accounted for as systematic uncertainties. Regarding other common experimental assumptions, as locality or freedom-of-choice, they are effectively irrelevant for the work presented above, since apparent signalling is introduced after the measurements, due to the “non local” data processing.

4.5 Additional experimental work

While the core results are detailed in Paper I, additional experimental work which might give a more complete overview of what has been done in the lab is reported here.

37 4.5.1 State characterisation In order to saturate the CHSH inequality, a maximally entangled state is required. Any of the Bell states in Eq. (1.26) would be equally good, and we chose to prepare φ + = ( 00 + 11 )/√2. Naturally, the actual prepared | i | i | i state is not the ideal φ + , and in order to characterise it we measured its visibility in the horizontal/vertical| i and diagonal bases. The results were:

visibility = 0.994 0.001, σx ± (4.3) visibility = 0.998 0.001. σz ± 4.5.2 Wave-plate motor hysteresis One of the main sources of apparent signalling that was identiﬁed in Paper I is related to how precise the motorised mounts used to rotate wave-plates are. A potential solution, besides the obvious substitution with better motors, was found in repeating the measurement settings several times. The underlying assumption was that the motors have little to no hysteresis, so that position errors would not be carried from one setting to the next, and increase with the repetitions. A test to conﬁrm this, in a given motor, consisted in the following: a series of 200 identical measurement settings, each followed by another setting, was executed, and the expectation value of each of the former calculated. The deviation from the average is plotted in the histogram in Fig. 4.3.

8 Frequency 6

0 0.06 0.04 0.02 0.00 0.02 0.04 − − − < E > E − Figure 4.3: Histogram of deviation of expectation values from the average for 200 identical measurements, each interrupted by a rotation to a different position. The data was ﬁtted by a Gaussian curve (p-value 0.97).

As we supposed, the motor mostly went back to the same initial position, and deviations from it are distributed normally, showing a well-

38 behaved (unbiased) Gaussian behaviour.

4.5.3 Laser power stability Pump power drifts can also introduce apparent signalling if the different measurement settings are performed sequentially, with only one detector per user. This was not the case in the experiments in Paper I, but we made sure this was not an issue in our setup nonetheless. Each of the measurements reported lasted less than an hour. As part of characterising the stability of our experimental setup, the pump laser power, as measured by its internal power-meter, has been monitored for around 7 hours. A histogram of the data can be found in Fig. 4.4.

0.16

0.14

0.12

0.10

0.08 Frequency 0.06

0.04

0.02

0.00 3272.5 3275.0 3277.5 3280.0 3282.5 3285.0 3287.5 Power (mW)

Figure 4.4: Histogram of pump laser power output as measured every 5 seconds, for 7 hours. The data was ﬁtted by a Gaussian curve that revealed an emitted power of (3280 2) mW. ± An extremely good power variation of less than 0.1% was calculated, and the Gaussian shape of the histogram indicates that no power drift happened.

39 40 5. Certifying a generalised quantum measurement

In Chapter 2, we analysed Bell’s theorem, and explained how it was originally devised as a way to test the principle of local realism in Nature. While that is the only explicit concern of the theorem, physicists have luckily not resisted the urge to push its consequences far beyond overcoming classical theories. Throughout the years, Bell tests have found use in applications that combine their rejection of classicality with the most peculiar features of quantum mechanics, in a remarkably powerful mix that finds relevance in an ever-growing range of fields. Among the most elegant of such combinations is the currently developing subject of device-independent (DI) quantum information. In this chapter, we will analyse two different approaches to the DI certification of generalised quantum measurements, one leveraging entanglement and quantum non-locality, and the other based on the so-called prepare-and-measure scenario, realised respectively in Papers II and III. As we have seen in Section 1.4.2, generalised measurements are, as the name suggests, the most general version of measures that can be defined in quantum mechanics. For this reason, they have been called the “standard” measurements and play a key role in information-theoretic reconstruction of the theory of quantum physics [39]. They have also been proven to be optimal for quantum state tomography [40], and for many of the applications mentioned in Section 2.4, including quantum key distribution (QKD) [41], randomness certification [42] and entanglement verification [43]. Because of their broad range of applications, generalised measurements have been realised before, both outside the context of device-independent certification [44–46], and inside it for a three-outcome measurement on a qubit [47]. In our work however, we aimed for combining the minimal assumptions of DI certification with the best kind of realisable quantum measurement, the informationally-complete measurement introduced in Section 1.4.3. After a quick comparison of the two approaches we followed, I will describe them in more details as they were applied to the works in the two papers mentioned.

41 5.1 Device independent quantum information

While Ekert’s original proposal for entanglement based QKD [17] already contained the seeds of the device-independence paradigm, it took a few years longer until Mayers and Yao explicitly proposed that observable experimental frequencies could uniquely determine the quantum state and measurements used to obtain them [48]. Their idea was formalised in 2007, when Acín and collegues proved that quantum non-locality, as certified by a Bell inequality, could be used to enforce cryptographic security in presence of untrusted, black-box devices [49]. The security proof of DI-QKD lead the way to subsequent application of the DI paradigm to other fields. Most notably, Pironio and collaborators presented in 2010 a proof of DI randomness expansion, along with an experimental demonstration [50]. Since those two initial proofs, the key advancements in the field can be associated with a process of reconciling the abstract, ideal definitions of devices in DI theory, with the very real and imperfect equipment used in laboratories. Indeed, while DI protocols often make no theoretical a priori assumptions on the specific functioning of devices in the lab, some basic characterisation is usually still required, for example the isolation of these black-boxes from each other. This reality-check type of process leads to the analysis of scenarios where some assumptions are relaxed, and part of the equipment is considered to be trusted and characterised, in order to lower the extremely demanding experimental requirements imposed by fully DI protocols. Two important scenarios that have been considered are represented by the concepts of semi-DI and measurement-DI, where the dimension of the quantum system is upper-bounded [51], or the source is trusted [52], respectively. On the other hand, the idea of robust self-testing sprung up, whereupon results yielded by imperfect devices lead to certifiable properties in spite of their sub-optimality, thanks to DI proofs that are tolerant to experimental noise [53]. Papers II and III present two of these approaches, with the common aim of certifying generalised quantum measurements.

5.2 Two ways to certify a generalised quantum measurement

As explained in the previous section, the original idea of DI certiﬁcation relies on the proof of non-locality made possible by a Bell inequality. This was also the starting point of the work in Paper II, where we realised a proof-of-principle experimental certiﬁcation of a genuine 4-outcome POVM

42 on a qubit by taking advantage of a particular Bell inequality. Such an approach is characterised by the following steps:

1. A source sends entangled states to Alice’s and Bob’s measurement stations.

2. Alice and Bob perform simultaneous measurements that correspond to a modiﬁed Bell operator. The outcomes of such measurements are used to estimate the relevant correlations.

3. The upper bound in the context of such Bell operator is calculated, in case only measurements with two or three outcomes are performed.

4. If the experimental correlations overcome both bounds, the performed measurement is certiﬁed to be a genuine 4-outcome one.

While a similar procedure works very well when certifying a shared secure key or random bit sequence, it runs into a delicate problem when certifying non-projective measurements: because of Neumark’s dilation theorem, a non-projective measurement on a given Hilbert space can in general be re-cast as a projective measurement in a bigger space. As a consequence, the third step in the procedure above relies on the assumption that the measurement is performed on a qubit. As we will see further on, in Paper II we arguably found a way around this problem, and at least partially removed the qubit assumption. Nevertheless, this issue hints at a different way of certifying non-projectivity. This second method is based on the Prepare- and-measure paradigm, and it was realised in Paper III. Its most important steps are:

1. Alice has a source of heralded photons, which are explicitly assumed to be described by qubit states.

2. Alice acts on the qubit with a unitary operator, thus preparing some quantum state; she then transmits the qubit to Bob, who performs a measurement on it. Again, relevant correlations between preparation and measurement outcome are estimated.

3. The correlations are used to calculate the expectation value of a Wit- ness, the bounds of which are also estimated, in case only projective or 3-outcome measurements are performed.

4. If the experimental result turns out higher than those bounds, Bob’s relevant measurement certiﬁably has four outcomes.

43 The assumption on the dimension of the quantum systems used, though limiting, is arguably fairly natural in the scenario of generalised measurement certification. The concept of non-projectivity in fact hints at a Hilbert space of fixed dimension to begin with. On the other hand, the much simpler prepare-and-measure scenario comes with several significant practical advantages compared to entanglement-based experiments. From the experimental point of view, neither space-like separation between observers nor highly entangled states are required. On the theoretical/computational side, the much smaller Hilbert space allows for a remarkable simplification of the analysis, in turn enabling more advanced and versatile certification methods to be developed. This is at least in part why we were able to carry the certification methods much further in Paper III compared to Paper II. As a quick overview, a comparison of some of the main features of the two approaches used in the two papers is reported in Tab. 5.1.

Entanglement-based Prepare-and-measure Quantum states Entangled Single qubits Source Untrusted Partially trusted (dim bound) Alice & Bob Space-like separated Sequential Measurement Coincident Bob only DI Full Partial

Table 5.1: Comparison of the main features of the entanglement-based and prepare-and-measure certiﬁcation approaches. Pros and cons are coloured in green and red respectively.

In conclusion, for all applications where semi-device independence is sufﬁcient (and in particular the dimensionality of the quantum systems can be trusted), the prepare-and-measure approach is by far the better choice, thanks to its simplicity. Whenever full device-independent certiﬁcation is required though, the entanglement-based approach is to date the only viable option. I will now outline in more details the procedures followed in Papers II and III in order to certify non-projective measurements, together with principal respective results.

5.3 Entanglement-based certiﬁcation

The Bell-type inequality upon which our entanglement-based certiﬁcation relied is called the Elegant Bell Inequality (EBI). The EBI was ﬁrst introduced in an appendix of a preprint manuscript by Gisin, in 2007 [54]. The

44 reason behind its name will become clear by the end of this section. Simi- larly to its more famous CHSH sibling (or ancestor!), the EBI scenario involves two parties, Alice and Bob, sharing a 2-qubit system. Alice performs three binary measurements, while Bob four, out of which they each may get outcomes +1 or 1. From the combined measurements, they calculate joint − probabilities P(a,b x,y), where (x,y) are Alice’s and Bob’s measurement | settings, and (a,b) their respective outcomes. Using these frequencies, they may estimate the value of the Bell parameter

βel =E11 + E12 E13 E14 + E21 E22 − − − (5.1) + E23 E24 + E31 E32 E33 + E34, − − − where Exy = ∑ abP(a,b x,y). a,b | The EBI reads βel 6, since 6 is the upper bound for local hidden variable theories. From a quantum≤ perspective, the inequality is saturated by a maximally entangled state and measurements 1 A1 = σx, B1 = (σx σy + σz), √3 − 1 A2 = σy, B2 = (σx + σy σz), √3 − (5.2) 1 A3 = σz, B3 = ( σx σy σz), √3 − − − 1 B4 = ( σx + σy + σz), √3 − for which the Tsirelson-type bound of βel = 4√3 6.93 is found. While Al- ice’s measurements are along the three orthogonal≈ axes of the Bloch sphere, Bob’s vectors point at the corners of a tetrahedron, exactly like the SIC- POVM projectors introduced in Eq. (1.21), from which they differ only by a unitary rotation. The tetrahedron shape seems to be part of the reason behind the elegant attribute to the inequality, which became part of its signature name. In order to combine the non-locality certiﬁcation from the EBI with the SIC-POVM property of information completeness, we started from a modiﬁed version of βel introduced in Ref. [42]:

4 m βel = βel k ∑ P(a = i,b = +1 x = 4,y = i) 6. (5.3) − i=1 | ≤ Here, Alice’s additional measurement setting (x = 4) consists in a 4-outcome m non-projective measurement. Most importantly, βel has the same quantum

45 bound of 4√3 as βel, and the bound is reached when the four vectors of Alice’s A4 measurement are anti-aligned to Bob’s projective measurements, thus forming again the SIC-POVM tetrahedron:

1 α β(1 + i) A4,1 = − , 2 β( 1 + i) 1 α − − 1 1 α β( 1 + i) A4,2 = − − , 2 β(1 + i) α − (5.4) 1 1 α β(1 i) A4,3 = − − , 2 β(1 + i) α 1 α β(1 + i) A = , 4,4 2 β(1 i) 1 α − −

3 √3 √3 with α = −6 and β = 6 . Unfortunately, certifying the symmetric POVM with the above inequality is experimentally not feasible, since it would require to hit the optimal quantum bound exactly. What can deﬁnitely be done though, is certifying the arguably most interesting property of the SIC-POVM, that is, information completeness. An IC-POVM with the minimal amount of outcomes, that is, d2, is aptly called minimal informationally complete (MIC) POVM. As we have shown in the supplementary material to Paper II, proving IC for qubits is equivalent to showing that the measurement has four outcomes (three independent plus normalisation). In order to check that, we have computed the maximum value that can be obtained for a generalised variant of Eq. (5.3), in case Alice’s A4 is a 3-outcome measurement. The calculations were carried out with Navascués, Pironio, and Acín’s semi-deﬁnite programming method [55], also known as NPA.

5.3.1 Summary of results

First, the twelve EBI measurements produced βel = 6.909 0.007 (see Eqs. (5.1) and (5.2)). This result let us estimate a lower bound± on the fidelity of our quantum state to the maximally entangled φ + Bell state. Using the (DI) SWAP method [56], we calculated the fidelity| i to be at least 0.947. This step of the protocol is particularly delicate, since the method described above to certify a non-projective measurement relies on the assumption that Alice’s system is a qubit. That said, we can now look at the results involving m the 4-outcome measurement. After optimising βel to our data, we obtained a final expression for Eq. (5.3) which we labelled βIC, the explicit and lengthy form of which can be found in Paper II. With NPA’s methods, we

46 calculated the bounds on βIC for 3- and 4-outcome measurements to be:

3-outcome 4-outcome βIC 6.8782 6.9883. (5.5) ≤ ≤ Since four outcomes are IC for qubits, the higher bound also sets the limit for quantum correlations. Taking into account statistics from Alice’s A4 measurements, we obtained the experimental result:

β exp = 6.960 0.007. (5.6) IC ± The result lies more than 11 standard deviations above the 3-outcome bound, therefore certifying that Alice’s A4 was a genuine 4-outcome, MIC measurement.

5.3.2 Experimental assumptions

The certification protocol presented in Paper II is DI, in the sense that its conclusions are drawn based only on experimentally produced statistics, and do not require additional assumptions on the devices used, besides what is dictated by the scenario and quantum theory. On the other hand, our particular experimental results were obtained under the rather common assumptions of no-communication, fair-sampling and freedom of choice. In order to remove the no-communication assumption, and close the locality loophole (see Section 2.3), much faster polarisation rotation devices and wider separation between measurement devices are required. Fair-sampling can be avoided with very high overall system efficiency. As we demon- strated in the supplementary material to Paper II, a minimum efficiency of 94% is required in this experiment. Freedom of choice is of course a whole different matter, and its resolution goes beyond the scope of this thesis.

5.4 Prepare-and-measure certiﬁcation

Similarly to the entanglement-based case, the prepare-and-measure scenario in Paper III involves two parties, Alice and Bob. In this case though, Alice has a source of heralded single qubits. On each qubit, she applies one of four unitary transformations (corresponding to her input x 1,2,3,4 ) in order to prepare a speciﬁc qubit state, which she then sends∈ to{ Bob. He} in turn performs one of three binary-outcome (b = 1) measurements, according to y 1,2,3 ). From the measurement outcomes,± they estimate ∈ { } 47 the experimental probabilities P(b x,y), in order to calculate the value of witness | 1 ASIC0 = ∑P(b = Sx,y x,y), (5.7) 12 x,y | with 0 0 0 0 1 1 S = . (5.8) x,y 1 0 1 1 1 0     As in the entanglement-based protocol, the witness is maximised when Al- ice’s four prepared states form a regular tetrahedron on the Bloch sphere (as in Fig. 1.3 (left) for example), while Bob performs measurements σx, σy and σz. In this case, the theoretical maximum of ASIC0 = 1/2 1 + 1/√3 0.7887 is achieved. The next step consists in providing Bob with an addi-≈ tional 4-outcome measurement (y = 4), while modifying the witness above as 1 4 ASIC = ∑P(b = Sx,y x,y) k ∑ P(b = x x,y = 4). (5.9) 12 x,y | − x=1 |

The ideal result of ASIC = 1/2(1+1/√3) would then certify that Bob’s last measurement was the SIC-POVM. Due to the impossibility of eliminating all experimental noise though, what can be concluded when a lower-than- optimal result is obtained?

5.4.1 Summary of results

One way to answer the question above consists of course in following a procedure akin to the one in Paper II: numerically calculate upper bounds of ASIC in case Bob’s fourth measurement has two or three outcomes. This is what we did for the 3-outcome bound. For the projective measurement bound instead, the relative simplicity of the prepare-and-measure scenario allowed us to derive a tight analytical limit for the witness. After optimising ASIC over k to have the best resilience to noise, we obtained the bounds

2-outcome 3-outcome 4-outcome ASIC 0.7738 0.7836 0.7887 (5.10) ≤ ≤ ≤ for k = 1/5. Experimentally, the 16 combinations of preparation and measurement yielded the result

exp 4 A = 0.7851 1 10− , (5.11) SIC ± × 48 which is many standard deviations above both projective and 3-outcome bounds, therefore certifying that Bob’s fourth measurement is genuinely a 4-outcome one. Besides the qualitative conclusion just presented, the prepare-and-measure approach granted the possibility of a deeper, quantitative analysis of the measurement. In Paper III, we introduced a method to pursuit robust self-testing of a target non-projective measurement. Following that method, we targeted the SIC-POVM, in order to get a quantitative estimate of the fidelity of Bob’s fourth measurement to it. We therefore proceeded to numerically approximate the worst-case fidelity of around 3 105 random 3- and 4-outcome measurements, under the constraint of our experimental× correlations. The analysis returned a worst-case fidelity of approximately 0.98 to the qubit SIC-POVM. It is important to point out that this result is not tight, but represents a numerical approximation made feasible by the bound on the Hilbert space in the prepare-and-measure scenario. A tight lower bound can in many instances be obtained with the SWAP method, although for rather technical reasons this could not be pursued in the case of the SIC- POVM. The reader is referred to Paper III for more details on why that was the case, and for an example of a different non-projective measurement (the Trine-POVM introduced in Section 1.4.4) where tight results were achieved for both non-projectivity and fidelity to the target measurement.

5.4.2 Experimental assumptions In addition to the key upper bound on the dimension of Alice’s prepared quantum systems, fair-sampling was also assumed with regards to the measured events. In contrast with the work in Paper II though, studies on minimum requirements for overall system efﬁciencies were not carried out, since the work was done in the context of exploring methods for robust self-testing of non-projective measurements, rather than applications with adversarial or malicious players.

49 50 6. Multi-photon entanglement source

Entanglement is not only, in Schrödinger’s own words, “the characteris- tic trait of quantum mechanics” [57], but also a key resource in quantum information theory. In Section 2.4, we have examined some of the most common applications of Bell tests that make use of such resource. In the past few decades, a greatly diverse range of pairs of systems have been entangled, spanning across several orders of magnitude in size. Neverthe- less, the first system used to investigate entanglement, and still one of, if not the most prominent and accessible, is photons. Their role is even more important in application-oriented experiments: thanks to their availability, mobility, and quantum-limited detection efficiency, photons are obviously the way to go in all quantum communication schemes, and in general in all applications that require transferring information between distant parties. This is true both in the classical and in the quantum realms. In the recently staggering efforts toward developing real-world quantum networks, entangled photons are posed to be the carriers of quantum information between network nodes, for example in distributed quantum computation or a more ambitious quantum internet [58]. The ability to generate states of more than two entangled photons has opened the doors to more extensive fundamental and applied research in quantum mechanics, for example in relation to Greenberger-Horne-Zeilinger [59] or cluster [60] states and one-way quantum computing [61]. However, the number of entangled photons that can be feasibly generated is limited by the overall detection efficiency and the high requirements in terms of state fidelity. Current state of the art experiments have shown probabilistic generation of up to eight [12; 13] or ten [14; 15] entangled photons. All of these works use spontaneous-parametric down-conversion of blue light to produce red entangled photons. This was also the method of choice for our source, and I will therefore compare our results with the papers mentioned above whenever possible. I will start from a brief introduction of the optical down- conversion process, and then move on to the actual setup characterisation, which will include abundant details on our results concerning multipartite entanglement.

51 The experimental work outlined in this chapter has been carried out in collaboration with Muhammad Sadiq, Hammad Anwer and Mohamed Nawareg.

6.1 Spontaneous-parametric down-conversion: a qubit source

Among the several ways of generating pairs of photons entangled in polarisation, we chose an extremely common – and reasonably simple – non- linear effect called spontaneous-parametric down-conversion (SPDC). This process relies on the fact that upon application of a strong electromagnetic ﬁeld, the polarisation of a dielectric material stops being linearly proportional to the ﬁeld, and higher order terms appear1:

(1) (2) 2 (3) 3 P(t) = ε0 χ E(t) + χ E (t) + χ E (t) + ... (6.1) h i (i) where ε0 is the free-space permittivity, χ are the i-th order dielectric sus- ceptibilities, and for simplicity we took both the polarisation P(t) and the electric field E(t) to be scalar quantities. In case of a material with large enough second order susceptibility, there is a chance that a photon propagating through it will down-convert into two photons of lower energy. The process has to preserve energy and momentum, that is, respectively: ωp = ωs + ωi (6.2) k~p = ~ks +~ki , where omegas refer to frequencies of, in order, the incident field (or pump photon), signal and idler photons. The second equation, involving mo- menta, is also called phase-matching condition. Only the case of ωs = ωi, known as degenerate down-conversion, concerns this work. Because of Eqs. (6.2), signal and idler photons will be naturally created with very strong correlations, which pave the way for entanglement. In our experiments we used Beta-Barium Borate (BBO), an uni-axial birefringent non-linear material. The polarisation perpendicular to the plane defined by ~k and the optic axis is called ordinary polarisation, and while propagating, it will experience the ordinary refractive index no. Conversely, a light field polarised in that plane is described as extraordinarily polarised, and will witness the extraordinary refractive index ne. A BBO crystal can

1A much more detailed description of non-linear processes can be found for example in Chapters 1 and 2 of [62].

52 satisfy the phase-matching conditions in two ways, depending on the angle of incidence of light with respect to the optic axis orientation:

Type-I down-conversion In this case the incident light has extraordinary polarisation, while both down-converted photons have ordinary polarisation. Schematically:

ωe ωo + ωo . (6.3) p → s i This process therefore yields two photons of identical polarisation.

Type-II down-conversion Again, incident light is extraordinarily polarised. However, one of the outgoing photons will be ordinarily polarised while the other will have extraordinary polarisation:

ωe ωe + ωo or ωe ωo + ωe . (6.4) p → s i p → s i

Because of the two possibilities given in Eq. (6.4), type-II down-conversion naturally offers polarisation-entangled photon pairs directly out of a single crystal. However, thanks to the identical polarisation, photons produced by type-I processes will exit the crystal in phase and are in general more indistinguishable, thus prone to quantum interference, compared to type-II.

6.1.1 Pump

Pump photons must have the exact spectral, temporal and polarisation properties needed for optimal down-conversion and entanglement yield. To this end, we used a mode-locked Ti:Sapphire pulsed laser emitting 140 fs long pulses at 780 nm, with a repetition rate frep of 80 MHz. The spectral width of the emitted light was approximately 8 nm, while the average power was 3 W. To achieve the required degenerate down-conversion that yields 780 nm photons, the pump light underwent a non-linear process called second- harmonic generation, which can be intuitively imagined as the opposite of down-conversion. A 1 mm thick Bismuth triborate (BiBO) non-linear crystal was employed in this case, producing a 390 nm – 1.1 nm wide – deep-violet beam of around 1 W of average power after focusing the pump on the crystal. The out-coming laser light, collimated again, contained a significant amount of residual 780 nm pump, which was filtered out with the help of multi-reflection on several dichroic mirrors (see Fig. 6.1). The resulting deep-violet pulses were then sent to the down-conversion source.

53 Figure 6.1: 780 nm red light pulses from the femto-second laser are focused on a BiBO crystal for second harmonic generation (SHG). The up-converted violet light at 390 nm is then collimated and sent to the down-conversion setup, while residual red light is ﬁltered out with the help of several dichroic mirrors (DM).

H D

Pump polarisation

SPDC polarisation

Figure 6.2: Femto-second pulses at 390 nm undergo SPDC in two non-linear crystals (BBO) inside a Sagnac interferometer, with a polarising beam-splitter (PBS) at the input port. A lambda-half wave-plate (HWP) sets the desired pump polarisation, while a dichroic mirror (DM) lets the pump light through and reﬂects down-converted photons on their way out.

6.1.2 Polarisation-entangled pair source

The source of polarisation-entangled photons is depicted in Fig. 6.2. In order to achieve relatively high down-conversion efﬁciencies, the pump pulses are focused into a triangular-shaped Sagnac interferometer through a polarising beam-splitter. This results in the horizontal and vertical components of the beam propagating in opposite directions along the interferometer. The lambda-half wave-plate before the DM can rotate the pump polarisation to cos(2θHWP)[H] + sin(2θHWP)[V], for any arbitrary wave-plate angle θHWP, thus allowing for free tuning of the relative weights between H and V components in the subsequent entangled state. | i | i At focal distance from the lens, 2 mm thick BBO crystals are placed on each side of the beam-splitter outputs. The two crystals are identical and with their axes oriented 90◦ to each other, so that each of the counter-

54 propagating polarisations is left unaffected by the ﬁrst crystal it encounters, while undergoing type-I down-conversion inside the second. The down- converted photon pairs of orthogonal polarisations, emitted in a cone whose aperture depends on the phase-matching condition, are then recombined at the beam-splitter, and reﬂected by a dichroic mirror onto a collimating lens. Because of the very short duration of the femto-second pump pulses, the down-converted light can be thought of as a ring. The rings coming from the two crystals are shown as photographed by a single-photon CCD camera in Fig. 6.3. After the collimating lens, their diameter is approximately 15 mm.

Figure 6.3: Down-converted photons distributed in a ring, as captured by a single-photon CCD camera. The two rings correspond to the two BBO crystals (see Fig. 6.2).

Keeping in mind the efﬁciency of the down-conversion process of ap- 12 proximately 10− , the pump power can be adjusted to have the required amount of pairs produced per pulse, on average. An investigation of the effect of changing the pump power on the produced entangled states will be presented in Section 6.3. When the down-conversion process is successful in only one crystal – and strictly not both – the generally entangled state iφ ψ = cos(2θHWP) HH + e sin(2θHWP) VV is prepared, provided that the| i events of down-conversion| i from either| crystali are completely indistinguishable. The two entangled qubits are found at diametrically opposite points of the down-conversion ring. Both θ and φ can be adjusted freely in our setup. The pronounced spectral width of the entangled photons can be appre- ciated in Fig. 6.4.

6.2 Scaling up to GHZ states

Thanks to the HWP in front of the SPDC Sagnac interferometer (see Fig. 6.2), a wide class of quantum states (entangled or not) can be produced in our setup. However, the source was built with the explicit goal of generating genuinely polarisation-entangled multi-photon states. For this reason, our

55 No ﬁlter 3 nm ﬁlter 4000

3000

2000 Intensity (AU)

1000

0 740 750 760 770 780 790 800 wave-length (nm)

Figure 6.4: Wavelength spectrum of down-converted photons (blue) as measured with a single photon spectrometer. To improve interference and indistinguishability, the photons are passed through a 3 nm high-transmittivity bandpass ﬁlter (yellow). work mainly focused on the production and characterisation of multi-qubit GHZ states, and the following sections will as a consequence concentrate on that. As we can see in Fig. 6.3, pairs of entangled photons are created across the entire 2π angle of the down-conversion ring. Since phase-matching Eqs. (6.2) do not set any preferential direction, the result is that the process happens across random opposite locations on the rings. If the probability of a pair being produced per pump pulse in a given direction is RP, then with n probability RP, n pairs will be produced in the same pulse, and they can be used to obtain states of more than two entangled qubits. A common technique to generate multiple down-conversion pairs found in literature consists in sending the pump beam through several sequential down-conversion sources, and collecting one pair from each (as in Refs. [12–15]). This method has the inherent disadvantage of pump power decreasing across subsequent crystals. Moreover, aligning, focusing and collimating the pump beam needs to be carried out for each crystal, and can result in different spectral properties for different pairs, besides requiring additional work and components. In our setup, we devised a novel way to carry out multi-pair generation that uses a single SPDC source, instead of one source for each pair. We achieve this (for ﬁve pairs) by splitting the down-conversion rings on a ten-facet pyramidal mirror, as depicted in Fig. 6.5, and collecting the various pairs from different sections of the same rings. Since the pump beam is identical for all pairs, our method is remarkably simpler compared to a multi-crystal scheme. It requires fewer

56 Figure 6.5: Depiction of the gold-coated ten-facet pyramidal mirror used to split the SDPC rings into ten different modes, from the side (left) and from the top (right). Each mode corresponds to a single qubit in the prepared GHZ state. The mode numbering is consistent with Fig. 6.7. components and can be more compact. It is also scalable to more photon pairs, which would simply involve a pyramidal mirror with accordingly higher number of facets. In order to entangle photons from independent pairs, we use a technique similar to entanglement swapping [63], illustrated in Fig. 6.6. From two

PBS

BBO BBO

390 390 Figure 6.6: Independent Bell pairs are entangled with each other by interfering one photon from each pair on a PBS. Upon post-selecting the events when one photon is obtained in each of the four output modes, the maximally entangled GHZ4 state is obtained. In our setup, different pairs correspond to different directions| i around the down-conversion cone of a single BBO crystal. pairs of entangled photons, we take the two signal photons and overlap them on a PBS. The resulting interaction, known as Hong-Ou-Mandel (HOM) interference, effectively entangles the four photons. Finally, to obtain a GHZ state, we post-select the events when each PBS interaction results in a photon at each outcome mode. Provided that the interacting photons are indistinguishable, all which-path information prior to the PBS is erased, though we can be certain that they will be both in H or V states. The resulting state after the PBSs (and post-selection) is| thei GHZ| i state of N

57 qubits N 1 GHZ = ( H1...HN + V1...VN ). (6.5) | i √2 | i | i The above experimental configuration is also known as star configuration, and is at the core of many recent realisations of multi-photon entangled states [12–15]. In our labs, we realised the star configuration for up to five photon pairs, although because of limits set by pair-generation rates, only four such pairs can be entangled, and the remaining one is used as an independent high- quality Bell state source. The setup configuration is depicted in Fig. 6.7. Each mode and its respective prime constitute a source of entangled pairs, and considering the entangling interference at the PBSs, we can identify the nine entanglement sources reported in Tab. 6.1. In the remaining of this

Source # photons Modes (in Fig. 6.7) 2 γ1 2 1 + 10 2 γ2 2 2 + 20 2 γ3 2 3 + 30 2 γ4 2 4 + 40 2 γ5 2 5 + 50 4 γ1 4 1 + 10 + 2 + 20 4 γ2 4 3 + 30 + 4 + 40 6 γ 6 1 + 10 + 2 + 20 + 4 + 40 8 γ 8 1 + 10 + 2 + 20 + 3 + 30 + 4 + 40

Table 6.1: List of multi-photon entanglement sources found in the setup in Fig. 6.7. In the source name, the superscript represents the number of photons involved, while the subscript helps to distinguish between sources with the same number of photons. chapter, I will discuss some of the key properties of the setup in general, and then show relevant experimental characterisation of the sources from Tab. 6.1.

6.3 General setup characterisation

As with any experimental optical setup, when designing and characterising a multi-photon entanglement source of the type described above, two aspects are of fundamental importance:

58 Pump

5 5' 1' 4'

2' 3'

4 3 2 1

Filter

HWP

QWP

PBS

SMF coupler

Figure 6.7: Multi-photon entanglement source - experimental setup. - Experimental setup used to generate and characterise multi-photon entangled states used throughout this work. More details on each part can be found in the text.

59 Quantitative yield Commonly referred to as source brightness, it • will in general depend on the photon collection efﬁciency ξ, and the total pair generation probability (per pump pulse) RP. The resulting N-fold coincidence probability for GHZN states will be very close to | i N/2 N N/2 1 (RP ξ )/2 − , (6.6) where the denominator is due to post-selection.

Qualitative yield Usually associated with quantum state ﬁdelity or • visibility, it gives an estimation of how close the produced states are to the ideal (possibly GHZN ) state. Among the many factors that impact the state quality| in thei case of multi-photon states, key ones are higher-order down-conversion events and photon-photon distinguishability.

While it is interesting to analyse the qualitative and quantitative properties of a source independently, the two aspects are clearly interdependent. As an obvious example, increasing the pump laser power trivially improves the pair generation probability RP, which in turn also increases the probability th n of n -order down-conversion RP , therefore decreasing the state ﬁdelity (due to so called accidental events). Nevertheless, we will now consider the two aspects separately, starting from quantitative considerations.

6.3.1 Source brightness As was mentioned before, the source brightness can be described as the product of the total (single photon) collection efﬁciency ξ and the pair production probability RP. For a given non-linear crystal, ξ depends on a few factors:

Optical attenuation in the beam path, due to scattering, reﬂection • and absorption.

Coupling of the photon spatial mode into the optical ﬁbre before the • detectors.

Detection efficiency. • The attenuation can be reduced by having the minimum amount of optical elements in the photon path, and having anti-reflection coated mirrors and lenses, while coupling into optical fibres is optimised through optimal mode-matching. As an example, the path corresponding to mode 1 in our

60 setup is attenuated by a factor of approximately 3/4 from the SPDC crystals to the fibre coupler, resulting in one every four photons not reaching the fibre. The main culprits are the golden decagon mirror, the Sagnac-loop PBS and the uncoated down-conversion crystals. Regarding detection efficiency, the avalanche photo-diodes (APDs) used in the setup have a quantum efficiency ranging from 0.5 to 0.6. As a side note, ξ is also inversely proportional to the spatial walk-off in the non-linear crystal, due to optical birefringence, which increases for longer crystals. While this is more of an issue in type-II sources because of the opposite polarisations of signal and idler photons, it still affects coupling in type-I SPDC, since the spatial modes exiting the crystals will be elliptical, and the longer the crystal the higher the ellipticity. The overall collection efficiency ξi for mode i can be experimentally C i estimated from the rate of coincident events i,i0 between and its twin i S C /S mode 0, and the rate of twin single events i0 , as ξi = i,i0 i0 . In our setup, ξ ranged from 0.16 to 0.21, depending on the particular mode considered. The pair production probability RP is instead determined by: SPDC efficiency. • SPDC crystal length. • Pump power. • While the non-linear process efficiency is related to the joint spectral am- plitude (JSA) and will not be discussed here, RP is roughly inversely proportional to the crystal length [15]. A longer crystal increases the down- conversion probability, though it also increases the distinguishability of independent pairs, since they might be created at locations far away along the crystal, farther than the coherence length of the wave-packets set by the spectral filters. For the above reasons, crystals of 2 mm thickness were cho- 1 sen for our source . Finally, RP is proportional to the pump power, which could be easily controlled in our setup thanks to a HWP positioned before the up-conversion crystal. The average power of the up-converted 390 nm beam in our source ranged from close to zero up to around 860 mW. For mode i in our setup, RP,i can be estimated as:

Si S Si R = i0 = . (6.7) P,i f C f rep i,i0 repξi

In Fig. 6.8, the experimental dependence of RP as function of pump power is reported, for mode 1. 1The choice of optimal crystal type and thickness was carried out before the author joined in the work.

61 0.10

0.08

P 0.06 R

0.04

0.02

100 200 300 400 500 600 700 800 Pump power [mW]

Figure 6.8: The experimental pair production probability for mode 1, as determined from Eq. (6.7), is plotted against pump power. The relation is linear for small RP, as shown by the linear ﬁt.

Characterising RP and ξ is of fundamental importance in order to be able to estimate the expected generation rates for multi-photon GHZ states, as per Eq. (6.6). For our source, we have measured such rates in the case of 1-, 2- and 4-photon generation, as function of pump power. The results are reported in Fig. 6.9.

500 2500 1500 400 2000 300 1000 1500 200 Singles [kHz] 1000 500 100

500 4-photon coincidences [Hz] 2-photon coincidences [kHz] 0 250 500 750 250 500 750 250 500 750 Power [mW] Power [mW] Power [mW]

Figure 6.9: Experimental rates for singles in mode 1, (2-photon) coincidences in modes 1 and 10, and 4-photon coincidences in modes 1, 10, 2, 20, all plotted against pump power. Interpolation curves show that the relation is linear for singles and coincidences, while it is quadratic for 4-photon coincidences, as expected from Eq. (6.6).

Most crucially, using Eq. (6.6), we can predict rates of N-photon coincidences as function of pump power and collection efﬁciency ξ. A simple way to visualise this dual-variable relation is by removing the dependence on ξ (since this is not an adjustable experimental parameter anyway). This simpliﬁcation can be carried out by plotting estimated rates as function of

62 measured 2-photon coincidences, which also depend on ξ. Such a plot is reported in Fig. 6.10 for 4-, 6- and 8-photon rates.

101

100

1 4-photons [kHz] Event rate 10− 6-photons [Hz] 8-photons [mHz] 2 10− 4-photon (data) [kHz] 6-photon (data) [Hz]

100 200 300 400 500 600 2-photon coincidences [kHz]

Figure 6.10: 4-, 6- and 8-photon rates as function of 2-photon coincidences, as expected from Eq. (6.6). Experimental data for 4- and 6-photon coincidences is also reported.

6.3.2 Source fidelity If brightness is one side of the “source coin”, the other is certainly the quality of the produced states. Depending on the task at hand, there might be a lower bound on the fidelity of a quantum state which needs to be overcome in order to succeed in the task, as is the case in Paper II and III in this thesis, or simply the highest possible fidelity might be desirable, in order to improve the absolute quality of an experimental result, as for example in Paper I. Either way, the higher the fidelity to the target quantum state, the better a source will perform. To understand the factors that impact the generated state fidelity, we need to start from a key clarification: in our setup, we consider photons as polarisation qubits; However, a photon’s wave function consists of several other degrees of freedom, as time, wavelength, orbital angular momentum, etc. In order for us to limit the discussion to polarisation, all other quantities have to be identical across wave functions of different photons, so that they can be effectively ignored in the quantum state description. In a real experiment though, different photons will of course never be completely identical, or indistinguishable. Their differences, together with other minor experimental imperfections, constitute the limits that prevent achieving unitary fidelity, although as we will see one can get remarkably close in certain cases.

63 In this section, we will go through the most relevant of such imperfections, and describe some of the methods we followed to mitigate their impact. In particular, we consider:

Spatial, temporal and spectral overlap of H and V photons’ wave • functions (in the same photon pair). | i | i

PBSs’ extinction ratios, i.e. their ability to transmit (reﬂect) all, • and only, horizontally (vertically) polarised photons. Also known as crosstalk or polarisation leakage.

Higher-order down-conversion events, which give rise to so called • accidental coincidences.

Distinguishability of independent photons, which depends on the • spatial, temporal and spectral overlap of wave functions of photons coming from different pairs.

Spatial overlap of orthogonally polarised photons coming from the two BBO crystals can be obtained first by superimposing the down-conversion rings far away from the source. We periodically did this with a single- photon camera at a distance of over three meters (a picture of the rings obtained during this process is reported in Fig. 6.3). Next, a far more precise overlap is achieved by optimising photon collection at several fibre couplers for one crystal, and then slightly adjusting the direction of the beams coming from the other crystal by fine-tuning the relevant mirror in the source Sagnac loop. Temporal overlap is adjusted by translating one of the crystals in the propagation direction. If more precise adjustments are required for some of the beams, thick birefringent quartz crystals can be used to retard one of the polarisations with respect to the other. This was not the case in the experiments discussed in this thesis. Spectral overlap is instead obtained by simply using the same (3 nm FWHM) spectral filters for both polariza- tions. The overall overlap directly affects the experimental visibility of the produced 2-qubit Bell states in the σx measurement basis. Visibilities of up to 0.997 were obtained in our setup for this basis, which is in line with some of the highest values reported in literature [34; 64]. The extinction ratio of each PBS is optimised at the time of the initial building of the setup. In our source, PBSs were selected and aligned so that a ratio of at least 300:1 correctly to wrongly polarised photons were found in every output mode. The non-zero crosstalk af a PBS can in general be accounted for as a systematic error in final results, although this was not necessary in the work carried out in this thesis, as it was not among the main sources of error.

64 Higher-order SPDC events

Down-conversion of multiple pump photons in a single pulse, into the same modes, is without a doubt the most limiting factor on the final state fidelity among those listed above. Because most photons are lost on the way to detection, these additional photons can create spurious, uncorrelated coincidences which result in errors in the derived results. As we have seen in the previous section, the probability of generating n photon pairs for each pump n pulse is RP, while RP is directly proportional to the pump power (see Fig. 6.8). As a consequence, at low power the higher order terms can usually be neglected. However, as shown in Fig. 6.18, high pump power is required for generating many-photon states, and the issue quickly takes centre stage for 6- and 8-photon states. In case of 2-photon experiments, accidental coincidences can be estimated from experimental data by counting coincidences between the two outcomes in a given mode. These can then be subtracted from the relevant counts when processing data. While this is generally done when calculating state fidelity, it is not standard procedure in certification protocols as the ones presented in Papers II and III. Another way to estimate the rate of accidental coincidences aci j between outcomes i and j is with the help of the following argument. Given a rate of single events Si, the probability of outcome i being triggered is Si/ frep. The rate of coincidences between such events and a detection in mode j is then

Si S j SiS j aci j = frep = . (6.8) frep frep · frep

This method can be used for states of any number N of qubits, by substituting Si with the sum of all possible N 1 coincidences. From the equation above, we can predict the effect that− accidental coincidences will have on + the visibility of the two qubit φ state in σz basis. If we label Alice’s (Bob’s) H,V outcomes as 1, 2| (3, 4),i we can write the visibility explicitly in terms of accidental coincidences, as:

C13 + ac13 +C24 + ac24 C14 ac14 C23 ac23 Vσz = − − − − . (6.9) C13 + ac13 +C24 + ac24 +C14 + ac14 +C23 + ac23

To simplify, we can assume the state obtained if all higher-order events were neglected is the ideal φ + Bell state (as we will see in the following, this | i assumption is fairly close to reality). In this case, C13 = C24 = C, C14 = C23 = 0, S1 = S2 = S3 = S4 = S, and ac13 = ac14 = ac23 = ac24 = ac, thus we can rewrite: 2C 1 Vσz = = . (6.10) 2C + 4ac 1 + 2S/ξ frep

65 Since we know the relation between the rate of singles and pump power in our setup from the linear ﬁt in Fig. 6.9, we can plot Vσz as function of pump power and ξ. The resulting graph is reported in Fig. 6.11.

+ Figure 6.11: σz visibility of φ state is affected by accidental coincidences due to higher-order SPDC events,| i and is plotted here as function of pump power and collection efﬁciency ξ as predicted by Eq. (6.10).

In conclusion, the issue of accidental coincidences can be mitigated by improving collection efficiency or by decreasing pump power at the cost of longer measurement times. While we have a fairly simple way to roughly predict how accidental coincidences affect the visibility of 2-qubit φ + states, it is far less trivial to do so in the case of N-qubit states, with N|> 2.i This is mostly due to the fact that the visibility of such states depends on the HOM interference between independent photons. Modelling HOM interference in the case of squeezed states of multiple photons (as opposed to single-photon Fock states) is a difficult problem, and is still the subject of current research in the field [65].

Distinguishability of independent photons The possibility of entangling photons belonging to different, independent down-conversion pairs is at the core of our multi-photon source. Without HOM interference at the PBSs, and post-selection, only entangled states of two photons could be achieved in our source, and their tensor products would make up bigger states, with no genuine entanglement added. Just like the wave functions of H and V photons in a pair need to be identical in all degrees of freedom (except polarisation!) in order to produce maximally entangled Bell states, the wave functions of photons interfering at a PBS need

66 to be absolutely identical, or indistinguishable, in order for the interference process to give rise to maximally entangled states between different pairs. In addition to the measures described above, some further conditions have to be met for HOM interference to be optimal. First of all, the two photons coming from different pairs have to be spatially and temporally overlapped. Spatial overlap is achieved by optimising collection efficiency at the fibre couplers for one photons, and then using the mirrors along the path of the other photon to optimise collection efficiency of the latter (without moving the couplers!). As a result, all photons entering the SMFs and reaching the detectors will have followed the same path from the interference PBS on- wards. Temporal overlap depends instead on relative path length, and we adjusted it by adding two retro-reflectors on each path, one of which was mounted on a micro-metric translation stage, longitudinal to photon propagation. In addition, the spectra of independent photons also need to overlap perfectly. This is achieved with the help of the narrow-band spectral filters located before each measurement station (see Fig. 6.7). Most crucially though, the interference process will result in a maximally entangled state only in case Fock states of single photons enter the PBS inputs. Since the down-conversion process actually produces sub- Poisson distributed squeezed states, the interference quality is disturbed by the higher-order down-conversion events. In order to quantify this effect, we first have to define a “quality” parameter: the HOM visibility. Given two down-converted pairs of photons, of defined, identical polarisations, we can send the signal photons into the same input of a PBS. If the two photons are indistinguishable, they will always come out bunched together at one output. This is precisely the HOM effect [66]. The four-fold coincidence rate between the two trigger photons and the two detectors at the outputs of the PBS should therefore be zero. Every time the two photons are instead distinguishable, they will independently pick an output, and there will be a 50/50 chance that they end up in different detectors, thus giving rise to a four-fold coincident event. A relatively simple way to make the photons completely distinguishable is by delaying them by an amount of time greater than their wave-packet coherence. This is possible in our setup thanks to the retro- reflectors on translation stages. We can then define the HOM visibility as the difference between the rate of four-fold coincidences when the photons are out of coherence and when they are perfectly overlapped, divided by the former. An experimental example for the interference of modes 1 and 2 is shown in Fig. 6.12. In contrast with the case of Bell state visibility, there is no simple way to predict the HOM visibility as function of pump power (as in Fig. 6.11). Nevertheless, we can of course measure it, as shown in Fig. 6.13. From

67 100

40 4-photon rate [Hz]

0 -600 -400 -200 0 200 400 600 Relative delay [µm]

Figure 6.12: HOM interference as measured through photon-photon bunch- ing at a PBS, for modes 1 and 2 in setup from Fig. 6.7. The visibility is deﬁned as the ratio of the difference between maximum and minimum of the curve, divided by the maximum. The shape of the curve is representative of the physical shape of the photons’ wave functions. In the data shown, the Gaussian ﬁt returned a visibility of approximately 80%, and a FWHM of 209 µm. ≈ the measurements, we can identify the upper limit of photon indistinguishability for our source to be around 0.91, in line with other works reported in literature [14].

6.4 Characterisation of individual sources

After discussing theoretical predictions and experimental characterisation concerning the whole setup in general, we will now go into more details about the speciﬁc multi-photon sources listed in Tab. 6.1. We will start from the smaller 2-photon sources and move up to the bigger ones progressively.

6.4.1 2-photon sources The SPDC source discussed in this thesis naturally produces pairs of photons entangled in the polarisation degree of freedom. As we have seen above, any two diametrically opposite locations on the down-conversion ring will with some probability contain two entangled photons. In our setup, we decided to select ﬁve such pairs, therefore producing ten pair-wise en- 2 tangled photons in ten separate paths. Of these ﬁve 2-photon sources, γ5 is detached from the other ones, and was for this reason used in the experiments performed in Papers I, II and III. We have therefore characterised 2 γ5 more in depth compared to the other 2-photon sources. Nevertheless,

68 0.90

0.85

0.80 HOM visibility 0.75

0.70

0 200 400 600 800 Pump power [mW]

Figure 6.13: HOM interference visibility for modes 1 and 2 at several different pump powers. The exponential ﬁt (of p-value 1.0) returned an upper bound for the visibility of approximately 91%. since the source has in principle no preferential direction along the down- conversion ring, and neither has the rest of the setup, the other sources can 2 perform very similarly to γ5 , as we will see.

2 Source γ5

Since Bell states form a basis for 2-qubit states, the ability to prepare perfect Bell states effectively means that any pure 2-qubit state can be produced. In the source, a simple lambda-half wave plate (HWP) and a phase-plate (PP) in one of the two modes are enough to convert among any of the Bell states. Considering this, we focused our characterisation on the fidelity of our experimental state to the maximally entangled φ + = ( 00 + 11 )/√2 state. In order to characterise it, we performed a full| statei tomography| i | i with the maximum likelihood method [67]. In addition, we employed motorised wave-plate rotation stages so that we could repeat the measurements several times. Each setting was measured for 30 seconds, and the nine tomography settings were repeated 35 times. The results are reported in Fig. 6.14. The prepared state showed a median fidelity to φ + of (99.7 0.1)%, which is at least as good as some of the best results| i found in± literature [34; 64]. The uncertainty is the standard deviation of the 35 measurement repetitions. While these results were corrected for accidental coincidences, the measurements were performed at extremely low pump power (of the order of 1 mW), so that the correction does not affect the final result on state fidelity,∼ up to the uncertainty.

69 0.5 0.5

0.4 0.4 0.4

0.3 0.3 0.2

0.2 0.2 0.0 0.1 0.1 0.2 − 0.0 0.0 0.4 − VV VV HH VH| i HH VH| i | i HV | i | i HV | i | i VH HV | i VH HV | i VV HH| i | i VV HH| i | i | i | i | i

Figure 6.14: Full quantum state tomography of the maximally entangled 2 states produced by source γ5 , as reconstructed with the maximum likelihood method. The real (left) and imaginary (right) parts of the density matrix are plotted.

Other 2-photon sources

The visibility in H/V (σz) and +/ (σx) bases for the remaining four − 2 sources of photon pairs, together with the discussed results for source γ5 as a comparison, is reported in Tab. 6.2.

Source Vσz (%) Vσx (%) γ2 99.65 0.01 98.86 0.01 1 ± ± γ2 99.83 0.01 99.28 0.01 2 ± ± γ2 99.9 0.1 99.4 0.1 3 ± ± γ2 99.66 0.02 99.66 0.02 4 ± ± γ2 99.8 0.1 99.6 0.1 5 ± ±

Table 6.2: Experimental visibility in σz and σx bases for the 2-photon sources in the setup. The different uncertainties are mainly due to different measurement duration.

6.4.2 4-photon sources

By taking two pairs of entangled photons and interfering the two signal photons at a PBS, we can create maximally entangled GHZ4 states with the help of post-selection. In particular, in our setup in| Fig. 6.7,i we have 4 two 4-photon sources available by overlapping modes 1 and 2 (γ1 in Tab. 4 6.1), or modes 3 and 4 (γ2 ). In order to characterise these sources, we set out 4 to perform two different experiments: a full state tomography for γ2 , and

70 the estimation of ﬁdelity witness D introduced in Section 1.6.1 for source 4 γ1 .

4 Source γ2

To characterise this 4-photon source, we prepared the state GHZ4 = | i ( HVHV + VHVH )/√2. Each of the 81 measurement settings required for| a fulli state| tomographyi was measured for 600 seconds, at a rate of approximately 15 4-photon coincidences (post-selected) per second. Af- ter correcting for accidental events, a maximum likelihood ﬁdelity of +0.11 +0.02 90.65 0.14 0.01 % was estimated. Because of the lower rate of coincidences− compared− to the 2-photon sources discussed above, uncertainties were estimated with Monte Carlo simulations that included two relevant experimental errors: poissonian distribution of counts, and limited motor repeatability when setting wave-plate angles. Each uncertainty is reported in the result above in respective order. The resulting density matrix is depicted in Fig. 6.15.

Figure 6.15: Full quantum state tomography of the GHZ4 states produced 4 | i by source γ2 . The real (left) and imaginary (right) parts of the density matrix are plotted.

Since the ﬁdelity is higher than 50%, we can conﬁrm that the source produces genuinely entangled GHZ4 states. | i

4 Source γ1

For this experiment we have prepared the standard state GHZ4 = ( HHHH | i | i + VVVV )/√2. The operator D introduced in Eq. (1.25) requires 5 mea- | i 71 surements in case of 4 qubits. The measurement operators are:

4 4 4 Z⊗ = H H ⊗ + V V ⊗ , | ih | | ih | 4 4 0 0 ⊗ 4 Me ⊗ = cos π σx + sin π σy = σ ⊗ , 0 4 4 x 4 4 4 1 1 ⊗ σx + σy ⊗ M⊗ = cos π σx + sin π σy = , 1 4 4 √2 (6.11) 4 2 2 ⊗ M 4 = cos π σ + sin π σ = σ 4, 2⊗ 4 x 4 y y⊗ 4 4 4 3 3 ⊗ σx + σy ⊗ M⊗ = cos π σx + sin π σy = − . 3 4 4 √2 Measuring each setting for 600 seconds at a rate of approximately 15 4- photon coincidences per second, we estimated the values of the operators above as in Fig. 6.16. Through Eq. (1.25)), we derived a ﬁdelity of our 4 +0.7 +0.1 state with GHZ of 90.1 1.1 0.2 %, which again includes a correction for accidental| events.i − −

0.5

-0.5 Experimental result

-1 4 4 4 4 4 ⊗ M0⊗ M1⊗ M2⊗ M3⊗ Z e Figure 6.16: Experimental values for measurement operators in Eq. (6.11), 4 as measured for source γ1 . The bar outlines indicate theoretical predictions.

4 The result overlaps with the corresponding one reported for source γ2 , 4 within one standard deviation. As was the case for γ2 , uncertainties have been estimated through Monte Carlo simulations and correspond, respectively, to poissonian and wave-plate angle errors.

6.4.3 6-photon source – γ6 In the setup depicted in Fig. 6.7, the green PBS where modes 3 and 4 meet is mounted on a vertical translation stage. When the PBS is lifted out of the beam path, we can block mode 3, and mode 4 thus interferes with

72 modes 1 and 2 further down the setup, at the following PBS. This process enables the generation of GHZ6 quantum states. In the case of six qubits, | i a full state tomography would require 36 = 729 measurements. To obtain an uncertainty of the order of 1% in the fidelity, each measurement would need to last on the order of an hour, which clearly makes the full state tomography approach unfeasible. Once again then, we resort to operator D and Eq. (1.25) to estimate the state fidelity. This approach requires measuring the following seven operators: 6 6 6 Z⊗ = H H ⊗ + V V ⊗ , | ih | | ih | 6 √ ⊗ e 6 6 6 3σx + σy M0⊗ = σx⊗ , M1⊗ = , 2 ! 6 ⊗ (6.12) 6 σx + √3σy 6 6 M2⊗ = , M3⊗ = σy⊗ , 2 ! 6 6 ⊗ ⊗ 6 σx + √3σy 6 √3σx + σy M4⊗ = − , M5⊗ = − . 2 ! 2 ! We have measured each setting for two hours at a rate of approximately 1.2 6-photon coincidences per minute. The resulting fidelity with state ( HHHHHH + VVVVVV )/√2, corrected for accidental events, was es- | i +| 1.7 +0.2 i timated at 76.1 1.8 0.2 %, and results of all seven operators are reported in Fig. 6.17. − −

0.5

-0.5 Experimental result

-1 6 6 6 6 6 6 6 ⊗ M0⊗ M1⊗ M2⊗ M3⊗ M4⊗ M5⊗ Z e Figure 6.17: Experimental values, with uncertainties, for measurement operators in Eq. (6.12), as measured for source γ6. The bar outlines indicate theoretical predictions.

As for the 4-photon sources, uncertainties were estimated from Monte Carlo simulations, and correspond to statistical (poissonian) and systematic (wave-plate angle) errors respectively.

73 6.5 Comparison with literature

In the interest of comparing our multi-photon entanglement source with other works in literature, I will consider the already mentioned references [12; 14; 15], in addition to a previous source realised in our research group and reported in references [68; 69]. While all six experiments aimed at generating entangled states, reference [68] did not target the maximally entangled GHZ state, but a different rotationally invariant, non-maximally entangled state. Moreover, rather than working with ﬁrst-order SPDC, the setup took advantage of third-order events to generate six entangled photons. Tab. 6.3 reports a streamlined comparison of the different setups.

This work [12] [14] [15] [68] Crystals 2xBBO 4xBBO 10xBBO 5xBiBO 1xBBO SPDC type I II I II II Pairs/W 9.3 106 1.2 107 1.2 107 4 106 1.6 106 (RP frep) · · · · · · Collection 0.18 0.25 0.42 0.38 0.17 efﬁciency Max HOM 0.91 0.83 0.91 NA – visibility Fid: GHZ4 0.907(1) NA 0.833(4) NA 0.919(2)a at| rate i 15 Hz NA 6 kHz NA 0.8 Hz Fid: GHZ6 0.76(2) NA 0.71(2) NA 0.88(3)a at| rate i 0.02 Hz NA 39 Hz NA 1 mHz aRef. [68] did not target GHZN states. | i NA: result is not publicly available.

Table 6.3: Comparison of the performance of the multi-photon entanglement source reported in this work with the state-of-the-art experiments found in literature, and a previous source built in our lab.

First of all, it should be noted that our design employs only two non- linear crystals, which is a fraction of those used by the other setups for GHZ state generation [12; 14; 15]. This is an enormous experimental advantage| i when initially setting up the experiment. It also makes the setup more compact and arguably more stable in time. Moving on along Tab. 6.3, our source shows a remarkably lower overall collection efﬁciency com-

74 pared with references [14; 15]. The extremely high efficiency reported in reference [14] is very likely due to the SPDC crystals emitting Gaussian- distributed, collinear beams, which have much better mode-matching into SMFs. According to the authors of reference [15] instead, they were able to dramatically improve previous results thanks to the reduced walk-off of BiBO crystals compared to BBO, and by using shorter crystals. In terms of rate of pair production however, our source is roughly on par with, or above, the others reported. Both brightness and collection efficiency were improved compared to previous efforts in our lab (in reference [68]). In- distinguishability of independent photons, as measured through HOM interference, turned out to be as high as the best of the works considered in the comparison. With regards to fidelity of state preparation, comparing different results is a delicate matter, since higher fidelity may be achieved by decreasing pump power, at the cost however of an exponentially decreased generation rate. With that in mind, the currently available results referring to GHZ4 and GHZ6 show that fidelities higher than reported in other works| can bei achieved| ini our setup, although at lower generation rates. More data needs to be gathered in order to compare bigger states and fidelity at different rates.

6.5.1 Future outlook

From the comparison above, it is clear that the parameter which needs improving the most in our source is the photon collection efficiency ξ. Such an improvement would dramatically increase the generation rate, for a given state fidelity. As an example, we can see in Fig. 6.18 the 8-photon rate as predicted by Eq. (6.6), as function of both ξ and pump power. To have statistically significant results in acceptable measurement times of several hours, a rate on the order of 10 8-photon coincidences per hour is required. Such a rate falls in the grey areas on the top-right corner of Fig. 6.18. ξ greater than 0.2 is necessary to access that area, a requirement that has not yet been met by our setup at the time of writing this thesis, and which pre- vented us from characterising source γ8. A quick (and very expensive) way to achieve this would be by using more efficient single-photon detectors. As an example, superconducting nanowire single-photon detectors (also known as SNSPDs) are currently available with efficiency that exceeds 90% [70]. In our setup, such an up- grade would increase ξ to about 0.3. Another approach would consist in exploring the relation between ξ and SPDC crystal thickness, possibly with both BBO and BiBO crystals. A similar analysis lead to dramatic improve-

75 Figure 6.18: 8-photon rates as function of both pump power and collection efficiency ξ, as estimated from Eq. (6.6). The dependence on RP is substituted with that on pump power using the linear fit of experimental data in Fig. 6.8. ments in reference [15]. The combination of higher efficiency detectors and improved collection efficiency would also enable the use of all five photon pairs to generate GHZ10 states. | i

76 7. Summary and outlook

In this thesis, I have outlined the most important results obtained in Papers I-III. These works underline two crucial aspects of the current landscape in quantum information: the need for reliable and reproducible results for applications, and the possibility of a new level of security and certiﬁcation thanks to physical constraints and minimal assumptions. In discussing Paper I, we have seen how ever-present Bell tests can run into troubles when experimenters assume fair-sampling. The effects of this issue, that we named apparent signalling, include distorting results and invalidating conclusions. We have therefore experimentally investigated the causes behind apparent signalling, and proposed effective ways of testing and solving them. In Papers II and III, we have explored the recent notions of device- independence and self-testing. These outstanding concepts aim at drawing conclusions from experimental data with minimal assumptions on the devices used to collect it. Because they rely on trusting the laws of quantum mechanics, they are arguably the next step in quantum information’s path to maturity. That is, after coming to terms with the rejection of classical physics by means of Bell’s theorem. In the two papers, we employed the techniques of device-independence and self-testing to successfully certify non-projective and informationally complete generalised quantum measurements. As we have seen in the introduction to Chapter 5, these measurements are optimal in several different scenarios, and in higher dimension are still the object of current fundamental research. In order to certify the various properties of our experimental measures, we have taken two quite different approaches, one relying on entangled qubits, and the other on heralded single qubits. These two options have different advantages and disad- vantages which were discussed in Chapter 5. Throughout the reﬁnement of existing experiments and the development of new ideas, photonic entanglement has remained an omnipresent resource in quantum information protocols. The need for better and more complex multipartite entangled states, for both fundamental studies and applications as quantum computation, has pushed us towards the development of a source of multi-photon entanglement based on spontaneous- parametric down-conversion. The original design solutions we employed

77 were described in Chapter 6, together with thorough experimental characterisation, and they will be at the core of the results in Paper IV, which is still in the works at the time of writing. An extensive comparison with other state-of-the-art multi-photon sources found in literature indicates our source as a valid alternative, while being experimentally far less complex. Finally, while more work is still needed to fully characterise the entanglement source, potential improvements have already been identiﬁed and will be investigated in the near future.

78 References

[1]A.E INSTEIN,B.PODOLSKY, AND N.ROSEN. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47:777–780, May 1935. xix, 10, 13

[2]M.A.N IELSENAND I.L.CHUANG. Quantum Computation and Quantum Information, 53. 2000. 1

[3]X.-L.W ANG, Y.-H. LUO,H.-L.HUANG,M.-C.CHEN,Z.-E.SU,C.LIU,C.CHEN, W. LI, Y.-Q. FANG,X.JIANG,J.ZHANG,L.LI,N.-L.LIU, C.-Y. LU, AND J.-W. PAN. 18-Qubit Entanglement with Six Photons’ Three Degrees of Freedom. Phys. Rev. Lett., 120:260502, Jun 2018. 4

[4]J.M.R ENES,R.BLUME-KOHOUT,A.J.SCOTT, AND C.M.CAVES. Symmetric informationally complete quantum measurements. Journal of Mathematical Physics, 45(6):2171–2180, 2004. 8

[5]A.J.S COTT. SICs: Extending the list of solutions. arXiv:1703.03993, (2):1–15, Mar 2017. 8

[6]M.A PPLEBY, T.-Y. CHIEN,S.FLAMMIA, AND S.WALDRON. Constructing exact symmetric informationally complete measurements from numerical solutions. Journal of Physics A: Mathematical and Theoretical, 51(16):165302, Apr 2018. 8

[7]R.J OZSA. Fidelity for Mixed Quantum States. Journal of Modern Optics, 41(12):2315–2323, Dec 1994. 10

[8]A.P ERES. Separability Criterion for Density Matrices. Phys. Rev. Lett., 77:1413–1415, Aug 1996. 11

[9]M.H ORODECKI, P. HORODECKI, AND R.HORODECKI. Separability of mixed states: necessary and sufﬁcient conditions. Physics Letters A, 223(1-2):1–8, 1996. 11

[10]B.M.T ERHAL. Bell inequalities and the separability criterion. Physics Letters, Section A: General, Atomic and Solid State Physics, 271(5-6):319–326, 2000. 11

[11]O.G ÜHNE, C.-Y. LU, W.-B. GAO, AND J.-W. PAN. Toolbox for entanglement detection and ﬁdelity estimation. Physical Review A, 76(3):030305, Jun 2007. 11

[12] Y.-F. HUANG,B.-H.LIU,L.PENG, Y.-H. LI,L.LI, C.-F. LI, AND G.-C.GUO. Experimental generation of an eight-photon Greenberger–Horne–Zeilinger state. Nature Communications, 2(1):546, Sep 2011. 12, 51, 56, 58, 74

[13]X.-C.Y AO, T.-X. WANG, P. XU,H.LU,G.-S.PAN,X.-H.BAO,C.-Z.PENG, C.-Y. LU, Y.-A. CHEN, AND J.-W. PAN. Observation of eight-photon entanglement. Nature Photonics, 6(4):225–228, Apr 2012. 51

79 [14]X.-L.W ANG,L.-K.CHEN, W. LI,H.-L.HUANG,C.LIU,C.CHEN, Y.-H. LUO,Z.-E.SU, D.WU,Z.-D.LI,H.LU, Y. HU,X.JIANG,C.-Z.PENG,L.LI,N.-L.LIU, Y.-A. CHEN,C.- Y. LU, AND J.-W. PAN. Experimental Ten-Photon Entanglement. Physical Review Letters, 117(21):210502, Nov 2016. 51, 68, 74, 75

[15]L.-K.C HEN,Z.-D.LI,X.-C.YAO,M.HUANG, W. LI,H.LU,X.YUAN, Y.-B. ZHANG, X.JIANG,C.-Z.PENG,L.LI,N.-L.LIU,X.MA, C.-Y. LU, Y.-A. CHEN, AND J.-W. PAN. Observation of ten-photon entanglement using thin BiB_3O_6 crystals. Optica, 4(1):77, Jan 2017. 12, 51, 56, 58, 61, 74, 75, 76

[16]J.S.B ELL. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1(3):195–200, 1964. 12, 14

[17]A.K.E KERT. Quantum cryptography based on Bell’s theorem. Physical Review Letters, 67(6):661–663, 1991. 12, 18, 31, 42

[18]D.J.G RIFFITH. Introduction to Quantum Mechanics. Prentice Hall, 2004. 14

[19] J.F.CLAUSER,M.A.HORNE,A.SHIMONY, AND R.A.HOLT. Proposed Experiment to Test Local Hidden-Variable Theories. Phys. Rev. Lett., 23:880–884, Oct 1969. 15

[20]J.S.B ELL. Speakable and Unspeakable in Quantum Mechanics, 1989. 15, 16, 17

[21]B.S.C IREL’SON. Quantum generalizations of Bell’s inequality. Letters in Mathematical Physics, 4(2):93–100, 1980. 16

[22]S.J.F REEDMANAND J. F. CLAUSER. Experimental Test of Local Hidden-Variable Theories. Phys. Rev. Lett., 28:938–941, Apr 1972. 16

[23]A.A SPECT,J.DALIBARD, AND G.ROGER. Experimental test of Bell’s inequalities using time- varying analyzers. Physical Review Letters, 49(25):1804–1807, 1982. 17

[24]M.G IUSTINA,M.A.M.VERSTEEGH,S.WENGEROWSKY,J.HANDSTEINER, A.HOCHRAINER,K.PHELAN, F. STEINLECHNER,J.KOFLER,J.Å.LARSSON,C.ABEL- LÁN, W. AMAYA, V. PRUNERI, M. W. MITCHELL,J.BEYER, T. GERRITS,A.E.LITA, L.K.SHALM, S. W. NAM, T. SCHEIDL,R.URSIN,B.WITTMANN, AND A.ZEILINGER. Signiﬁcant-Loophole-Free Test of Bell’s Theorem with Entangled Photons. Physical Review Letters, 115(25):1–7, 2015. 17, 37

[25]L.K.S HALM,E.MEYER-SCOTT,B.G.CHRISTENSEN, P. BIERHORST,M.A.WAYNE, M.J.STEVENS, T. GERRITS,S.GLANCY,D.R.HAMEL,M.S.ALLMAN,K.J.COAK- LEY,S.D.DYER,C.HODGE,A.E.LITA, V. B. VERMA,C.LAMBROCCO,E.TORTORICI, A.L.MIGDALL, Y. ZHANG,D.R.KUMOR, W. H. FARR, F. MARSILI,M.D.SHAW,J.A. STERN,C.ABELLÁN, W. AMAYA, V. PRUNERI, T. JENNEWEIN, M. W. MITCHELL, P. G. KWIAT,J.C.BIENFANG, R. P. MIRIN,E.KNILL, AND S. W. NAM. Strong Loophole-Free Test of Local Realism. Physical Review Letters, 115(25):1–10, 2015. 16, 17, 37

[26]M.A.R OWE,D.KIELPINSKI, V. MEYER,C.A.SACKETT, W. M. ITANO,C.MONROE, AND D.J.WINELAND. Experimental violation of a Bell’s inequality with efﬁcient detection. Nature, 409(6822):791–794, 2001. 16, 17

[27]R.S CHMIED,J.D.BANCAL,B.ALLARD,M.FADEL, V. SCARANI, P. TREUTLEIN, AND N.SANGOUARD. Bell correlations in a Bose-Einstein condensate. Science, 352(6284):441– 444, 2016. 16

[28]B.H ENSEN,H.BERNIEN,A.E.DREAÚ,A.REISERER,N.KALB,M.S.BLOK,J.RUITEN- BERG, R. F.L. VERMEULEN,R.N.SCHOUTEN,C.ABELLÁN, W. AMAYA, V. PRUNERI, M. W. MITCHELL,M.MARKHAM,D.J.TWITCHEN,D.ELKOUSS,S.WEHNER, T. H. TAMINIAU, AND R.HANSON. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526(7575):682–686, 2015. 16, 17, 37

80 [29]M.G IUSTINA,A.MECH,S.RAMELOW,B.WITTMANN,J.KOFLER,J.BEYER,A.LITA, B.CALKINS, T. GERRITS, S. W. NAM,R.URSIN, AND A.ZEILINGER. Bell violation using entangled photons without the fair-sampling assumption. Nature, 497(7448):227–230, 2013. 17

[30]R.U RSIN, F. TIEFENBACHER, T. SCHMITT-MANDERBACH,H.WEIER, T. SCHEIDL,M.LIN- DENTHAL,B.BLAUENSTEINER, T. JENNEWEIN,J.PERDIGUES, P. TROJEK,B.ÖMER, M.FÜRST,M.MEYENBURG,J.RARITY,Z.SODNIK,C.BARBIERI,H.WEINFURTER, AND A.ZEILINGER. Entanglement-based quantum communication over 144km. Nature Physics, 3(7):481–486, 2007. 18

[31]R.C LEVEAND H.BUHRMAN. Substituting quantum entanglement for communication. Physical Review A - Atomic, Molecular, and Optical Physics, 56(2):1201–1204, 1997. 18

[32]H.B UHRMAN,R.CLEVE,S.MASSAR, AND R.DE WOLF. Nonlocality and communication complexity. Reviews of Modern Physics, 82(1):665–698, 2010. 18, 31

[33] C.Bˇ RUKNER,M.ZUKOWSKI, J. W. PAN, AND A.ZEILINGER. Bell’s inequalities and quantum communication complexity. Physical Review Letters, 92(12):1–4, 2004. 18

[34]B.G.C HRISTENSEN, K. T. MCCUSKER,J.B.ALTEPETER,B.CALKINS, T. GERRITS,A.E. LITA,A.MILLER,L.K.SHALM, Y. ZHANG, S. W. NAM,N.BRUNNER, C. C.W. LIM, N.GISIN, AND P. G. KWIAT. Detection-loophole-free test of quantum nonlocality, and applications. Physical Review Letters, 111(13):1–5, 2013. 18, 31, 36, 64, 69

[35] T. MORODER,J.D.BANCAL, Y. C. LIANG,M.HOFMANN, AND O.GÜHNE. Device- independent entanglement quantiﬁcation and related applications. Physical Review Letters, 111(3):1–5, 2013. 18

[36]O.G ÜHNEAND G.TÓTH. Entanglement detection. Physics Reports, 474(1-6):1–75, 2009. 18

[37]C.G ERRY AND P. KNIGHT. Introductory Quantum Optics. Cambridge University Press, 2005. 24

[38]H.S.P OH,S.K.JOSHI,A.CERÈ,A.CABELLO, AND C.KURTSIEFER. Approaching Tsirelson’s Bound in a Photon Pair Experiment. Physical Review Letters, 115(18):30–33, 2015. 31, 36

[39]C.A.F UCHS. Quantum Mechanics as Quantum Information (and only a little more). pages 1–59, May 2002. 41

[40]J. Rˇ EHÁCEKˇ ,B.-G.ENGLERT, AND D.KASZLIKOWSKI. Minimal qubit tomography. Physi- cal Review A, 70(5):052321, Nov 2004. 41

[41]C.A.F UCHSAND M.SASAKI. Squeezing Quantum Information through a Classical Chan- nel: Measuring the “Quantumness” of a Set of Quantum States. Quantum Info. Comput., 3(5):377–404, Sep 2003. 41

[42]A.A CÍN,S.PIRONIO, T. VÉRTESI, AND P. WITTEK. Optimal randomness certiﬁcation from one entangled bit. Physical Review A, 93(4):040102, Apr 2016. 41, 45

[43]J.S HANG,A.ASADIAN,H.ZHU, AND O.GÜHNE. Enhanced entanglement criterion via symmetric informationally complete measurements. Physical Review A, 98(2):022309, Aug 2018. 41

[44] T. DURT,C.KURTSIEFER,A.LAMAS-LINARES, AND A.LING. Wigner tomography of two- qubit states and quantum cryptography. Physical Review A, 78(4):042338, Oct 2008. 41

81 [45]Z.E.D.M EDENDORP, F. A. TORRES-RUIZ,L.K.SHALM,G.N.M.TABIA,C.A.FUCHS, AND A.M.STEINBERG. Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements. Physical Review A, 83(5):051801, May 2011.

[46] W.M.PIMENTA,B.MARQUES, T. O. MACIEL,R.O.VIANNA,A.DELGADO,C.SAAVEDRA, AND S.PÁDUA. Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure. Physical Review A, 88(1):012112, Jul 2013. 41

[47]E.S.G ÓMEZ,S.GÓMEZ, P. GONZÁLEZ,G.CAÑAS, J. F. BARRA,A.DELGADO, G.B.XAVIER,A.CABELLO,M.KLEINMANN, T. VÉRTESI, AND G.LIMA. Device- Independent Certiﬁcation of a Nonprojective Qubit Measurement. Physical Review Letters, 117(26):260401, Dec 2016. 41

[48]D.M AYERS AND A.YAO. Quantum cryptography with imperfect apparatus. In Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280), pages 503– 509, Nov 1998. 42

[49]A.A CÍN,N.BRUNNER,N.GISIN,S.MASSAR,S.PIRONIO, AND V. SCARANI. Device- independent security of quantum cryptography against collective attacks. Physical Review Letters, 98(23):1–4, 2007. 42

[50]S.P IRONIO,A.ACÍN,S.MASSAR,A.BOYER DE LA GIRODAY,D.N.MATSUKEVICH, P. MAUNZ,S.OLMSCHENK,D.HAYES,L.LUO, T. A. MANNING, AND C.MONROE. Ran- dom numbers certiﬁed by Bell’s theorem. Nature, 464(7291):1021–1024, 2010. 42

[51]M.P AWŁOWSKI AND N.BRUNNER. Semi-device-independent security of one-way quantum key distribution. Physical Review A - Atomic, Molecular, and Optical Physics, 84(1):1–4, 2011. 42

[52]H.K.L O,M.CURTY, AND B.QI. Measurement-device-independent quantum key distribution. Physical Review Letters, 108(13):1–5, 2012. 42

[53]M.M CKAGUE, T. H. YANG, AND V. SCARANI. Robust self-testing of the singlet. Journal of Physics A: Mathematical and Theoretical, 45(45), 2012. 42

[54]N.G ISIN. Bell Inequalities: Many Questions, a Few Answers. ArXiv:quant-ph/0702021, pages 125–138, 2009. 44

[55]M.N AVASCUÉS,S.PIRONIO, AND A.ACÍN. A convergent hierarchy of semideﬁnite pro- grams characterizing the set of quantum correlations. New Journal of Physics, 10, 2008. 46

[56] T.H.YANG, T. VÉRTESI,J.D.BANCAL, V. SCARANI, AND M.NAVASCUÉS. Robust and versatile black-box certiﬁcation of quantum devices. Physical Review Letters, 113(4):1–5, 2014. 46

[57]E.S CHRÖDINGER. Discussion of Probability Relations between Separated Systems. Mathe- matical Proceedings of the Cambridge Philosophical Society, 31(4):555–563, Oct 1935. 51

[58]H.J.K IMBLE. The quantum internet. Nature, 453(7198):1023–1030, Jun 2008. 51

[59]D.B OUWMEESTER, J.-W. PAN,M.DANIELL,H.WEINFURTER, AND A.ZEILINGER. Ob- servation of Three-Photon Greenberger-Horne-Zeilinger Entanglement. Physical Review Letters, 82(7):1345–1349, Feb 1999. 51

[60]N.K IESEL,C.SCHMID,U.WEBER,G.TÓTH,O.GÜHNE,R.URSIN, AND H.WEIN- FURTER. Experimental Analysis of a Four-Qubit Photon Cluster State. Physical Review Letters, 95(21):210502, Nov 2005. 51

82 [61] P. WALTHER,K.J.RESCH, T. RUDOLPH,E.SCHENCK,H.WEINFURTER, V. VEDRAL, M.ASPELMEYER, AND A.ZEILINGER. Experimental one-way quantum computing. Na- ture, 434(7030):169–176, Mar 2005. 51

[62] R.W.BOYD. Nonlinear Optics. Academic Press, 2008. 52

[63] J.-W. PAN,D.BOUWMEESTER,H.WEINFURTER, AND A.ZEILINGER. Experimental En- tanglement Swapping: Entangling Photons That Never Interacted. Physical Review Letters, 80(18):3891–3894, May 1998. 57

[64]B.G.C HRISTENSEN, Y.-C. LIANG,N.BRUNNER,N.GISIN, AND P. G. KWIAT. Exploring the limits of quantum nonlocality with entangled photons. Physical Review X, 5(4):33–36, Jun 2015. 64, 69

[65]A.P RADANAAND L. Y. CHEW. Quantum interference of multi-photon at beam splitter with application in measurement-device-independent quantum key distribution. New Journal of Physics, 21(5):053027, May 2019. 66

[66]C.K.H ONG, Z. Y. OU, AND L.MANDEL. Measurement of subpicosecond time intervals between two photons by interference. Physical Review Letters, 59(18):2044–2046, Nov 1987. 67

[67]J.B.A LTEPETER,E.R.JEFFREY, AND P. G. KWIAT. Photonic State Tomography. Elsevier, 2005. 69

[68]M.R ÅDMARK. Photonic quantum information and experimental tests of foundations of quantum mechanics. PhD thesis, Stockholm University, Department of Physics, 2010. 74, 75

[69]M.R ÅDMARK,M. Z˙ UKOWSKI, AND M.BOURENNANE. Experimental Test of Fidelity Limits in Six-Photon Interferometry and of Rotational Invariance Properties of the Photonic Six- Qubit Entanglement Singlet State. Physical Review Letters, 103(15):150501, Oct 2009. 74

[70] F. MARSILI, V. B. VERMA,J.A.STERN,S.HARRINGTON,A.E.LITA, T. GERRITS, I.VAYSHENKER,B.BAEK,M.D.SHAW, R. P. MIRIN, AND S. W. NAM. Detecting single infrared photons with 93% system efﬁciency. Nature Photonics, 7(3):210–214, Mar 2013. 75