Photonic Multipartite Communication: Complexity, measurements and Bell inequalities

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Photonic Multipartite

Communication

Complexity, measurements and Bell inequalities

Hammad Anwer

Hammad Anwer Photonic Mul tipartite Comm unica tion

Doctoral Thesis in Physics at Stockholm University, Sweden 2021

Department of Physics

ISBN 978-91-7911-418-3

Hammad Anwer

received his Master degree in Nanotechnology from the Royal Institute of Technology (KTH) . He carried out his Master's thesis and PhD studies in the field of Quantum Information and Quantum Optics at the Department of Physics, Stockholm Univeristy.

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Photonic Multipartite Communication

Complexity, measurements and Bell inequalities

Hammad Anwer

Academic dissertation for the Degree of Doctor of Philosophy in Physics at Stockholm University to be publicly defended on Friday 5 March 2021 at 09.00 in sal C5:1007, AlbaNova universitetscentrum, Roslagstullsbacken 21, online via Zoom, public link https:// stockholmuniversity.zoom.us/j/64005105009.

Abstract

The interdisciplinary field of quantum communication and quantum information processing merges quantum mechanics, optics, photonics, information processing, and electronics to solve information and communication tasks that are impossible to solve efficiently with classical resources. Here in this thesis experimental demonstrations of some of such tasks are presented. In particular, using a single qubit system and quantum Zeno effect we investigated a class of communication complexity problems (CCP) for multi-parties. As solutions, three different quantum strategies are evaluated by proof of concept experimental demonstrations. Our results go beyond the classical limits. Furthermore, the same single qubit system is used to show that preparation contextuality can be shared among multiple observers through a quantum state ensemble while implementing sequential unsharp measurement. We showed that this is possible for any amount of white noise and presented experimental demonstration for three parties. In addition, characterization of unsharp measurements based on quantum random access code and quantifying the degree of incompatibility of sequential measurements in a wide range of sharpness parameters are also presented.

Finally, I present the experimental generation of multi-photon entanglement to meet the basic requirement of modern quantum information processing.Using this source we produced a state with high fidelity that can violate a tight Bell inequality maximally with maximally incompatible local measurements.

Stockholm 2021

http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-189572 ISBN 978-91-7911-418-3

ISBN 978-91-7911-419-0

Department of Physics

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PHOTONIC MULTIPARTITE COMMUNICATION

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Photonic Multipartite

Communication

Complexity, measurements and Bell inequalities

Hammad Anwer

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©Hammad Anwer, Stockholm University 2021 ISBN print 978-91-7911-418-3

ISBN PDF 978-91-7911-419-0

copyright papers American Physical Society.

Thesis for the degree of Doctor of Philosophy in Physics Department of Physics Stockholm University Sweden. Printed in Sweden by Universitetsservice US-AB, Stockholm 2021

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Abstract

The interdisciplinary field of quantum communication and quantum formation processing merges quantum mechanics, optics, photonics, in-formation processing, and electronics to solve inin-formation and commu-nication tasks that are impossible to solve efficiently with classical re-sources. Here in this thesis experimental demonstrations of some of such tasks are presented. In particular, using a single qubit system and quan-tum Zeno effect we investigated a class of communication complexity problems (CCP) for multi-parties. As solutions, three different quantum strategies are evaluated by proof of concept experimental demonstra-tions. Our results go beyond the classical limits. Furthermore, the same single qubit system is used to show that preparation contextuality can be shared among multiple observers through a quantum state ensem-ble while implementing sequential unsharp measurement. We showed that this is possible for any amount of white noise and presented exper-imental demonstration for three parties. In addition, characterization of unsharp measurements based on quantum random access code and quantifying the degree of incompatibility of sequential measurements in a wide range of sharpness parameters are also presented.

Finally, I present the experimental generation of multi-photon entan-glement to meet the basic requirement of modern quantum information processing. Using this source we produced a state with high fidelity that can violate a tight Bell inequality maximally with maximally incompat-ible local measurements.

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Acknowledgements

First of all, I would like to express my modest and sincere thanks to my supervisor, Prof. Mohamed Bourennane, for not only believing in me, but also for giving me the opportunity to perform state-of-the art experiments in the field of Quantum Information and Technologies. I am very grateful to him for his supervision during my doctoral stud-ies. I would also like to express my gratitude to my co-supervisor Prof. Markus Hennrich and my doctoral studies mentor Prof. Fawad Hassan.

A special thanks of Prof. Ingemar Bengtsson, Prof. Per-Erik Tegnér and Prof. Sten Hellman for useful discussions and excellent guidance. I am also grateful to my collaborators Armin Tavakoli, Muhammed Sadiq, Ralph Silva, Nikolai Miklin, Prof. Adan Cabello, Prof. Michał Horodecki, Prof. Karol Horodecki and Prof. Pawel Horodecki for their help and discussion. I would also like to thank all the members of the group Kvant Information and Kvant Optik (KIKO) group for creating a joyful, inspiring and pleasant working atmosphere in the laboratory and during Fika. A special thanks for my colleagues in the laboratory Mohamed Nawareg, Ashraf Abdelrazig, Alley Hameedi, Massimiliano Smania, Alexander Mortiz, Alban Seguinard and Guillermo Andler. I would like to express my thanks and gratitude to Prof. Emeritus Kjell Carlsson from the Department of Biomedical and X-ray Physics at KTH, it should be for his support during my studies in Albanova.

Finally, I would like to thank my family for their continuous and unending support. I would like to express my deepest thanks to my parents, my brother and a sister. And last but not least, l would like to thank my beloved wife, who has always supported and loved me. I thank you, my dear for your endless support and trust in me. Hussain Raza, Hadi Raza and Zainab I love you more than you will ever know!

Research work carried out in this thesis was funded by the Knut and Alice Wallenberg Foundation and the Swedish Research Council.

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Sammanfattning på svenska

Det tvärvetenskapliga området för kvantinformation och kommunika-tion kopplar samman kvantmekanik, optik, fotonik, materialvetenskap och elektronik med klassisk informationsteori för att lösa uppgifter inom information och kommunikation som är omöjliga med klassiska metoder. Kvantinformationsvetenskap har också återupplivat diskussionerna om kvantteoriens grundval. Fältet är för närvarande mycket aktivt och ger drivkraft för framsteg inom både fysik och informationsteknik.

I denna avhandling har vi använt enstaka fotoner för att demonstrera kvant-Zeno-effekten för ett enda kvanttillstånd på d-nivå (qudits) för att minska kommunikationskomplexitetsproblemet mellan de tre parterna. Våra resultat ligger utanför de klassiska gränserna. Dessutom används samma kvantsystem på en enda nivå för att implementera sekventiell kommunikation för tre parter i ett envägskommunikationsnätverk som består av förberedelse-, transformations- och mätanordningar för att utföra oskarp mätning i två olika scenarier i) för att studera inkom-patibilitet för sekventiell mätning i ett brett spektrum genom slump-mässig åtkomstkod ii) för att observera förberedd kontextualitet som delas mellan flera observatörer. Vi har undersökt och utfört experi-mentet för maximal kvantöverträdelse av en stram Bell-olikhet med max-imalt oförenliga lokala mätningar för treparts fotoniska qubitar. Slutli-gen avslutar jag med att presentera den experimentella Slutli-generationen av multifoton-snärjelse för att uppfylla den grundläggande förutsättningen för modern kvantinformationsbehandling QIP

Forskningsarbetet i denna avhandling finansierades av Knut och Alice Wallenberg Foundation och Vetenskapsrådet.

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List of papers

List of papers that are included in this PhD thesis.

PAPER I: Quantum Communication Complexity using the

Quantum Zeno Effect.

Armin Tavakoli, Hammad Anwer, Alley Hameedi, and Mohamed Bourennane. Phys. Rev. A, 92, 012303 (2015). DOI:10.1103/PhysRevA.92.012303

PAPER II: Experimental test of maximal tripartite

nonlocal-ity using an entangled state and local measure-ments that are maximally incompatible.

Hammad Anwer*, Mohamed Nawareg*, Adan Cabello,

and Mohamed Bourennane. Phys. Rev. A, 100, 022104 (2019).

DOI:10.1103/PhysRevA.100.022104

PAPER III: Noise-robust preparation contextuality shared

be-tween any number of observers via unsharp mea-surements.

Hammad Anwer, Natalie Wilson, Ralph Silva, Sadiq

Muhammad, Armin and Mohamed Bourennane.

arXiv:1904.09766 : Submitted to PRL

PAPER IV: Experimental characterisation of unsharp qubit

ob-servables and sequential measurement incompati-bility via quantum random access codes.

Hammad Anwer, Muhammad Sadiq, Walid Cherfi,

Niko-lai Miklin, Armin Tavakoli and Mohamed Bourennane.

Phys. Rev. Lett. 125, 080403.

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PAPER V: Practical No-Signalling proof Randomness

Ampli-fication using Hardy paradoxes and its experimen-tal implementation.

Ravishankar Ramanathan, Michal Horodecki, Hammad

Anwer, Stefano Pironio, Karol Horodecki, Marcus

Grün-feld, Sadiq Muhammad, Mohamed Bourennane and Pawel Horodecki. (Not included in this thesis).

PAPER VI: Experimental observation of photonic multipartite

entanglement.

Massimiliano Smania, Hammad Anwer, Muhammad Sadiq, Mohamed Bourennane. (In preparation...).

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Author’s contribution

PAPER I: I contributed in designing the experimental setup. I was the one who built the setup and performed all the mea-surements and error analysis. I was also actively involved and contributed to the writing of the manuscript.

PAPER II: I contributed in building the optical setup. I performed the measurements and analyzed the data together with M. Nawareg. I was also actively involved and contributed to the writing of the manuscript.

PAPER III: I performed all the experimental work, which included design and construction of the optical setup. I performed the measurements, as well as data analysis. I was also actively involved and contributed to the writing of the manuscript.

PAPER IV: I designed and built the optical setup for the experiment. I performed all measurements as well as analyzed the data including error calculation and contributed to the writing of the paper.

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List of Figures

2.1 A qubit state |ψi is described on the surface of Bloch sphere. A state with its parametrization variable θ and φ is illustrated. . . 8

3.1 The fast (no) and slow (ne) axes of a waveplate. . . 19

3.2 Beam splitter with two input and two output ports. . . . 21 3.3 Two inputs and two outputs ports of a polarization beam

splitter. . . 22 3.4 Polarization analysis station (measurement station). . . . 24 3.5 A schematic diagram of a frequency doubler box. Optical

setup consist of a nonlinear Bismuth Triborate (BiB3O6)

crystal to obtain SHG, two identical lenses are used to fo-cus and collimate the SHG beam, and broadband mirrors. 27 3.6 Spectra of up-converted 390 nm light. . . 28 3.7 Energy and momentum conversion of pump photons into

down converted signal and idler photon. . . 29 3.8 Full spectral image of type I down converted horizontal

polarized cone appeared as a ring shape of diameter ≈ 12 mm taken with a single photon CCD camera. . . 29 3.9 Image of type II down converted horizontal and vertical

polarized cones appeared as two intersecting rings [1]. . . 30 3.10 Two photons polarization entangled source is used to

gen-erate the Bell state φ−. . . 32 3.11 In image (a) real and (b) imaginary parts of the density

matrix ρ_Exp of the Bell state φ−. . . 33

4.1 Communication complexity scenario between Alice and Bob for computing the function f (x, y) by receiving n-bits string for x and y. . . 37

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xvi List of Figures

4.2 Communication problem scenario, ‘No jump’ means that x equal to y or adjacent to y and ‘Jump’ means that x is opposite to y or adjacent to opposite site of y. Note that not all dots are shown in his picture. . . 40 4.3 Preparation, transformation, and measurement

configu-ration between Alice, Bob, and Charlie. . . 41

5.1 Opitcal setup illustrate the quantum interference mecha-nism for orthogonally polarized photons (modes). . . 49 5.2 Alice receives two bits x₀, x1 and sends the qubit state

ρx0,x1 to Bob who receives an input y and produces a

classical output b and a quantum output ρy,b_x

0,x1 which is

measured by Charlie according to his input z, yielding an outcome c [2]. . . 56

6.1 Triangular shaped Sagnac interferometer setup used for generation of photons polarization entanglement by using type (I) SPDC from two nonlinear crystals. . . 60 6.2 In image (a) vertical (b) horizontal polarized ring.

Down-converted photons are distributed in a cone which appear as a ring in a single photon CCD camera. Two rings corresponds to two nonlinear BBO crystals that are used in the optical setup as illustrated in Fig. (6.1). . . 61 6.3 In image full and filtered wavelength spectra of

down-converted photons is measured with single photon spec-trometer [3] . . . 61 6.4 Complete schematic design and strategy of optical setup

for photon polarization entangled source. . . 63 6.5 Implemented design and strategy of one of our two-photons

entanglement source. . . 64 6.6 Star configuration approach is used for combining the two

independent two-photons source. . . 66 6.7 Two photons interference scheme with beam splitter BS.

(a) Photons from input 1 and 2, both are transmitted from BS. (b) Photon from input 1 is transmitted and photon from input 2 is reflected by BS. (c) Photons from input 1 and 2, both are reflected by BS. (d) Photons from input 1 is reflected and photon from input 2 is transmitted from BS. . . 67 6.8 Quantum interference (H-O-M dip) between photons pair

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List of Figures xvii

6.9 Optical setup illustrate the quantum interference mecha-nism for orthogonally polarized photons (modes). . . 68 6.10 Schematic diagram of a PDF box showing two outputs

of a Sagnac interferometer from a single input, which is divided into two optical paths. Each path contains a half-wave plate to adjust the phase between two paths and coefficient α and β weights for the desired state. . . . 70 6.11 Complete four qubits GHZ state tomography results

plot-ted as a density matrix elements. . . 73 6.12 Six qubits entanglement witness result obtained from seven

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List of Tables

3.1 Specifications of Chameleon Ultra and Chameleon XR laser systems used to produce entangled photon pairs. . . 26

4.1 Experimental results of quantum success probability for protocols P1 and P2 solving the family of CCPs. . . 42

5.1 Alice’s four states preparation angles for a polarizer and HWP_A. . . 56

6.1 List of all two photons entangled pairs in the setup and their visibility measurement in σz and σx basis. . . 65

6.2 Two four-photons sources consists of the two different two-photons sources to produce two maximal entangled |GHZ4_{i state.} _{. . . 65}

6.3 Comparison of different multiqubits entanglement sources available in literature with our entanglement source . . . . 76

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1. Introduction

1.1 Overview

The classical picture of the world started to crumple in the last decade of the 19th century, when Max Planck laid the foundation of quan-tum mechanics, by proposing that radiation is emitted from the body in discrete energy packets called quanta. The energy E of a quanta is associated with frequency ν, defined as E = hν 1. Later in 1905, A. Einstein with photoelectric effect broadened the hypothesis of Planck’s by proving that light composed of corpuscles or photons and energy of these particles, is defined as Planck’s suggested. New and different kinds of phenomena observed at subatomic level which were impossible to de-scribe using classical mechanics. The behaviour of these microscopic objects demands new concepts and descriptions to understand them. Quantum theory differs from its classical counterpart as it takes into account the strange behaviour that we can observe at atomic scale such as electrons, photons and atoms etc., whereas classical physics fails to explain these traits. Quantum mechanics was developed almost a cen-tury ago, but still fascinates today due to the remarkable attributes of a quantum system e.g, superposition (a system coexists in two or more different states simultaneously) and entanglement, which are generally referred to as quantum resources for information processing. Basically, quantum theory provides tools not only to predict the outcomes, but also for justifying physical phenomena. It has applications in different disciplines such as in particle physics, magnetic resonance imaging and solid state devices etc.

An essential principle of quantum mechanics is the inseparability of some physical systems, often referred to as entanglement. This means that different parts of a system can be connected and influence each other no matter how far they are. These influences lead to correla-tions that have no classical analogue and hence can in principle be used to achieve some tasks that are considered classically impossible, or to

1_{Planck’s constant: h = 6,62607 x 10}−34 _joule.second

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2 Chapter 1. Introduction

perform some tasks more efficiently than it is possible using classical resources. A new theory is developed from the understanding and re-alization of quantum correlations known as quantum information, that has application in computation, communication and metrology etc.

Quantum correlations allow us to reduce communication complexity problems (CCP) defined as the minimum amount of communication

re-quired between parties (such as Alice and Bob) to solve a function whose inputs are distributed among them. We will see in this thesis that certain

CCPs can be solved efficiently not only by using quantum entanglement but also replacing classical communication by quantum communication among the parties. Nevertheless, entanglement remains a key resources in quantum information, as well as in research and investigation concern-ing foundation of quantum mechanics. In this thesis we have presented a new setup that is capable to produce multi-partite entangled states in somewhat simpler manner.

To extract the information from a quantum system or to learn its state one has to carry out measurements. Often these measurements are projective, where outcomes are mutually exclusive. Here, we de-scribe more general quantum measurements called POVM. Especially, we considered a variant of POVM known as unsharp or weak measure-ments. Using such measurement one can extract information from the state without completely destroying its coherence. Here in this thesis we characterize such measurements for a qubit case and use it to share contextuality among many observers.

1.2 Outline

The aim of this thesis is to provide theoretical and experimental back-ground of my work presented in research articles. To do this, the thesis is divided into two parts. The first covers the introduction and back-ground of the research field which includes the basic notion of quantum information and quantum mechanics. The second part is based on my experiments and research articles.

The first part of the thesis comprises chapters 2 and 3, and pro-vides an overview of theoretical concepts and relevant experimental background respectively. The second part contains chapters 4, 5 and 6. The chapters 4 and 5 cover topics that are directly related to quan-tum communication, which are carried out with polarization state of

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1.2. Outline 3

a single qubit system between multi-observers. In this sequential com-munication approach, reduction of comcom-munication complexity problem is achieved by implementing two different quantum protocols. One of these uses quantum "Zeno effect" QZE [4]. Moreover, using this simple quantum system we have also demonstrated that preparation contextu-ality can be shared between three observers, when sequential unsharp measurements are performed for a fixed sharpness parameter η before the final observer perform projective measurements. Besides this, char-acterization of the sharpness of a measurement using quantum random access code (QRACs) is also described in chapter 5, where we also shows that how this can be used to quantify the degree of incompatibility of two sequential measurements.

Chapter 6 covers the design, preparation, and generation of multi-photon polarization entangled multi-photon source. Entanglement quality and generation rate of the correlated photons pairs mentioned as is the char-acterization of entangled sources from two to six qubits. Additionally, we also present an experiment where we demonstrate the violation of a tight Bell inequality for the first time, using maximally incompatible local measurements for a three-qubit state.

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2. Introduction to Quantum

Information

In this introductory chapter, a brief introduction to the basic element of quantum information and processing known as a qubit is provided. The qubit represents the simplest unit of a quantum information and it is shown how a qubit state is visualized on the Bloch sphere. Additionally, I will give review on entanglement properties and a Bell inequality for a system of two qubits.

2.1 Qubit

In classical information and computation theory, a bit represents the smallest and elementary unit of classical information. One usually tie this information to states of a system. A system which has only one possible state is trivial and uninteresting as it can never be in any other state. Therefore, we start with systems that can have at least two pos-sible states i.e physical systems that can be found in two mutually ex-clusive states. Now a classical bit is amount of information that encodes and distinguishes between two possible mutually exclusive states. These two states are usually represented by binary digits (0 and 1). The ba-sic rules for operations and computation for these bits are outlined in

Boolean algebra. In particular there are two distinctive operations that

can be performed on a single classical bit named as identity and Not gate operations [5].

Identity : (0 → 0, 1 → 1)

N OT : (0 → 1, 1 → 0)

In contrast, a quantum bit or a qubit encodes or differentiates be-tween the states of a quantum system that can occupy two mutually exclusive states. In other words, qubit represent two dimensional quan-tum system. Moreover, a qubit can also exist in superposition of these two states, therefore the states of quantum systems are conveniently represented by vectors in Hilbert spaceH . Usually, we represent these 5

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6 Chapter 2. Introduction to Quantum Information

possible orthogonal states by |0i and |1i. A vertical bar and right bracket illustrates that the states are written in Dirac notation, and it is known as ket and its dual vector is known as bra. A single qubit state |ψi is represented by linear superposition of two orthogonal states as,

|ψi = α|0i + β|1i (2.1)

where the coefficients α and β represent the amplitudes of state-vectors |0i and |1i respectively, and α, β ∈ C are arbitrary complex numbers that obey the normalization condition

|α|2+ |β|2= 1. (2.2)

Here |α|2 and |β|2 represent the probability to find the state |ψi in |0i and |1i respectively. In matrix form the state vectors |0i and |1i are defined as |0i ≡ 1 0 ! , |1i ≡ 0 1 ! .

Furthermore, using the normalization condition, the state |ψi can also be parametrized as the following two dimensional vector representation with relative phase φ.

|ψi = α 1 0 ! + β 0 1 ! = α β ! = cos(θ/2) e(iφ)_sin(θ/2) !

where we have neglected global phase as |ψi ≡ eiξ_{|ψi, ∀ξ ∈ R. In classical} systems, an individual operation is required for both input states of a bit. In contrast, it is the beauty of quantum information that we can provide the superposition of both inputs instead of putting them individually for any operation. This is specified as quantum parallelism. Quantum information has advantage over its classical counterpart, which is often referred to as quantum supremacy [5].

2.2 Density Operator

In a laboratory environment, it is impossible to generate the noise free or perfectly pure quantum state, especially when the information is trans-ported from one point to another through a noisy channel. Additionally, imperfections in optical equipment influence the preparation and mea-surement outcomes of an isolated system and we usually end up having mixtures of different pure states. In quantum mechanics, any type of state, either mixed or pure, can be described by a density matrix. It is

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2.3. Bloch Sphere 7

an important tool for testing when a state is pure or mixed. The density operator of a mixture of pure states can be written a weighted sum of these states, where the weights describes the probability of finding that state. Mathematically, ρ = m X i=1 pi|ψiihψi| (2.3)

where |ψ_ii are denote pure states and p_i are real numbers representing probabilities of these states such that they fulfill the following conditions

X

i

pi= 1, pi≥ 0

From Eq. (2.3), it is clear that decomposition is not unique, which means that one can get the same density operator by mixing the different pure states. Also, note that the dimension of relevant Hilbert space H depends upon the orthonormality condition of vectors |ψii. A pure state

can provide the maximal information of a system hence in such case all

pi will be zero except the one that comes with the relevant state. In

this case, the density operator is also known as a projection operator (which is essentially a product of state vector with its transpose) as it can project a given state onto its 1-D subspace.

The density operator ρ is Hermitian (ρ = ρ†) and positive semi-definite with unit trace. It means that the eigenvalues λ_i of density operator are real and λi≥ 0. It has following significant properties:

1) It should obey the normalization condition Tr(ρ) = 1. 2) For pure states ρ2_{= ρ.}

3) Also Tr(ρ2) ≤ 1, equality holds for only pure states.

4) The expectation value of an operator O is defined as hb Oi = T r(b Oρ).b

Here, trace (Tr) is define as the sum of all the diagonal elements of a density matrix. A mixed state can also be obtained as a reduction of a pure state when it interacts with the surrounding that leads to decoherence.

2.3 Bloch Sphere

A qubit state either pure |ψi or mixed state ρ can be visualized on the Bloch sphere in three dimensions. A pure state is represented as a single point on the surface of the Bloch sphere and a mixed state is

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Figure 2.1: A qubit state |ψi is described on the surface of Bloch sphere.

A state with its parametrization variable θ and φ is illustrated.

represented as a point inside the Bloch sphere. The space of a Bloch sphere is embedded in three-dimensional space with Cartesian coordi-nates (x = cos φ sin θ, y = sin φ sin θ, z = cos θ) which fulfills the normal-ization condition x2_{+ y}2_{+ z}2_{= 1. Angles φ and θ are used to define the}

Bloch vector in it as shown in Fig. (2.1). A pure state in Eq. (2.1) can also be parametrized as following with relative phase φ,

|ψi = cos(θ

2)|0i + e

iφ_sin(θ

2)|1i. (2.4)

By ignoring the global phase in Eq. (2.1) of a qubit state, then only two real parameters known as angle θ and φ are used in Eq. (2.4) to represent a state on the Bloch sphere. In order to envision the pure state on the Bloch sphere, one should identify the Bloch vector with the help of angles θ and φ. Angle φ has range 0 ≤ φ < 2π and θ has range 0 ≤ θ ≤ π to define a position of Bloch vector corresponding to their longitude and latitude respectively. A qubit can be represented by two orthogonal states such as |0i and |1i written in the computational basis {|0i, |1i}. These states are also known as the ground and excited states. Besides computational basis, there are other bases of qubit that play a significant role in quantum computation and communication protocols. But we can distinguish two unbiased bases known as diagonal and anti-diagonal, left and right basis and can be represented as (D/A) and (L/R)

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2.4. Higher Dimensional Quantum System 9

respectively. Hadamard basis can also be define in computational basis:

|Di →|+i ≡|0i + |1i√

2 (2.5)

|Ai →|−i ≡ |0i − |1i√

2 (2.6)

The density operator of a state written in Eq. (2.4) has the following matrix representation. ρ = cos 2₍θ 2) e −iϕ_sin(θ 2) cos( θ 2)

eiϕsin(₂θ) cos(θ₂) sin2(θ₂)

!

(2.7)

= 1 2

1 + cos(θ) sin(θ)(cos(ϕ) − i sin(ϕ) sin(θ)(cos(ϕ) + i sin(ϕ)) 1 − cos(θ)

!

(2.8)

which can be parametrized as,

=1 2 1 + z x − iy x + iy 1 − z ! (2.9)

Single qubit measurement

In quantum theory, complete and precise characterization of an unknown single qubit is impossible as it can not be fully identified by a single or sequence of experiments [6]. For example, if the qubit state |ψi ex-ists in either basis states |0i or |1i then the measurement statistics can easily reveal the state of the qubit instantly. However, this identifica-tion is not possible when the state of the qubit is in superposiidentifica-tion as given in Eq. (2.1). In characterization of a state we need to deter-mine the value of complex variables α and β. We get answers either in 0 or 1 if measurement σ_z is performed and we randomly get one of these answer with probability |α2| and |β2_{| respectively. But if the σ}

x

measurement is performed then the probability with which the system project to |Di = √1

2(|0i + |1i) or |Ai = 1 √

2(|0i − |1i) will be (

α+β_√ 2 ) 2 _or (α−β√ 2 ) 2 _{respectively [6; 7].}

2.4 Higher Dimensional Quantum System

Until now we have discussed only 2-dimensional quantum systems or qubits but in general a quantum mechanical system can consist of one

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or more multidimensional subsystems. Here in this thesis, we will cover single qubit d-level quantum systems and the generation of entangled systems with several qubits, e.g entangled system of four and six qubits. A d-level system is commonly known as a qudit system and the general mathematical expression for this is illustrated as follows

|φi =

d−1

X

i=0

ci|ii

where i = (0,1, 2, 3, .... d-1) represents the number of qudits and c_i is probability amplitude of the respective terms. The state space of the cor-respondingH has a finite dimension d ∈ R. The inner product between two states ψ and φ is considered as hψ, φi ∈ C. Also multi d-level quan-tum system can be created by adding the two or more systems. Qutrit (α|0i + β|1i + γ|2i) and ququad (ququart) (α|0i + β|1i + γ|2i + δ|3i) are examples of such kind of systems.

Similarly, adding the number of qubits in the system that converts a single qubit into a multi-qubit system may lead to a new opening of more complex and interesting tasks for multi-qubit quantum communication and computation. The state of an n-qubit system is described by the product of the individual subspaces. Mathematically,

H = H1⊗H2⊗H3.... ⊗Hn (2.10)

where H₁,H₂, H₃ .... H_n represents the individual Hilbert subspaces of systems. The dimensionality of Hilbert space increases exponentially with the number of qubits. In the case of n number of qubits the di-mensionality of space is 2n. In the case of two qubits we can define four vectors (|00i, |01i, |10i and |11i) as a tensor product of two qubits vectors |0i and |1i as following

|0i₁⊗ |0i₂= |00i =

     1 0 0 0     

, |0i₁⊗ |1i₂= |01i =

     0 1 0 0     

|1i1⊗ |0i2= |10i =      0 0 1 0     

, |1i1⊗ |1i2= |11i =      0 0 0 1     

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2.5. EPR Paradox 11

where, subscripts refer to subsystems. Similarly, for three qubits we can define eight basis vectors e.g (|000i, |001i, |010i, |011i, |100i, |101i, |110i, |111i) and for four qubits there will be sixteen basis eigenvectors.

2.4.1 State Fidelity

The concept of quantum state fidelity is an important ingredient for preparation and transmission of a quantum state through a quantum channel. The fidelity between two quantum states is described as a measure of how identical they are or the distance between two quantum states vectors in the system state space [8]. Assume ρ₁ and ρ₂ are two quantum states in a finite dimensional in Hilbert space H then the fidelity between ρ1 and ρ2 is measured as a transition probability. It

is the probability outcome of state ρ₁ transitioning into state ρ₂ upon measurement. It shows how much these two states overlap, are similar and close to each other. The general notion of Uhlmann’s fidelity formula [9] for these states is defined as

F (ρ1, ρ2) = T r( q√

ρ1ρ2

√

ρ1) (2.11)

Although it is not obvious, the fidelity is in fact symmetric in its argument i.e. F (ρ1, ρ2) = F (ρ2, ρ1), and invariant under unitary

trans-formation, and bounded as 0 ≤ F (ρ1, ρ2) ≤ 1. Here, the upper bound

1 means that two states have maximum resemblance "ρ₁ = ρ₂" and the lower bound 0 means that both states are orthogonal to each other. If one of the states is considered pure state then the fidelity of state will be

F (ρ1, ρ2) = q

T r(ρ2|ψihψ|) (2.12)

2.5 EPR Paradox

Quantum mechanics was developed during the 1920s–1930s and pro-posed an extra-ordinary point of view that appeared to be different from the classical view of nature. According to standard interpreta-tion of quantum mechanics, unobserved physical properties of a system does not exist unless observed. Such physical properties of a system occurred as a result of a measurement performed on it. Many physicists did not agree with this new view of nature [10]. In 1935 Albert Ein-stein, together with Podolsky and Rosen, questioned and criticized the quantum mechanics concepts about the nature and proposed a thought

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experiment which apparently showed that quantum mechanics is an in-complete theory [11]. This thought experiment is know as EPR paradox. In this paper, they argued that either the quantum mechanical descrip-tion of reality, given by the wave funcdescrip-tion, is not complete or that two observables corresponding to non-commuting operators cannot have a simultaneous reality.

The EPR paradox is easier to comprehend if one considers that a system consist of two particles called P_A and P_B. They are separate and exist in the singlet state of position and momentum. According to Heisenberg’s uncertainty relation, if the position of the particle PA

is measured with certainty, then the wave function would collapse in a state of unknown momentum. The momentum of the particle P_B remains unclear after the measurement on PA. Surprisingly, how can PB

recognize that it has a position and an unknown momentum immediately after the measurement on P_A. If one does not allow non-local effects this will not be possible. The EPR expect that a theory giving a complete description of reality (in their sense) is possible. In a sense, this means that there might exist variables, called hidden variables (HV), that have not been discover yet but may be discover in the future, which can explain how this is possible. When one of the measurements is taken on a particle, these variables control how the system should react to a particular measurement.

2.6 Bell’s Theorem and Famous CHSH Inequality

Correlation predicted by quantum mechanics in situations like EPR ex-periment are term as entanglement and concedes stronger correlations in a system based on two or more subsystems, as compare to the corre-lations in classical physics. It means that both systems can affect each other without any physical connection between them. This caught the attention of Einstein in his correspondence with Bohr, where he broadly criticized the entanglement phenomena as "spooky action at a distance". However, it remains unclear until 1965, when John Bell found that by sticking to the assumptions in the EPR paper it is possible to derive an inequality which is violated by quantum mechanics. This goes beyond EPR, who thought that QM is correct but incomplete (in their sense). However, Bell showed that a complete theory (in the EPR sense) must disagree with QM, or else it must be non-local (again in the EPR sense).

Bell inequality provides an upper bound for classical correlations permitted by the HV presumptions [12]. Bell inequality is important

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2.6. Bell’s Theorem and Famous CHSH Inequality 13

for testing the foundation of quantum mechanics. Different variants of Bell theorem have been derived and experimentally confirmed. In 1969, an equality was derived by Clauser, Horne, Shimony, and Holt which known as CHSH inequality. It shows that the quantum systems cannot be described by an HV model and again confirmed the existence of en-tanglement in quantum state.

Proof of Clauser, Horne, Shimony, and Holt Inequality

The Clauser Horne Shimony and Holt (CHSH) inequality for two party is one of the most famous Bell inequalities [13–16] and we can derive it as follows.

Assume Alice and Bob are located in two different laboratories. In both of these laboratories, there is a device with two indicating lights on it, one indicator is for outcome value +1 and other is for outcome value -1. There is a switch on this device that can be set in two different positions [17]. Alice’s lab, this adjustable setting is named as " a " and in Bob lab is named as " b ". The probability of outcome for the setting " a " on Alice’s side is P (l|a) for l ∈ (−1, +1). Similarly, Bob can cal-culate the probability P (m|b) for " b " setting from two measurements outcome m ∈ (−1, +1). After finishing the experiment, they both can calculate the joint probability distribution P (l, m|a, b) from their records by acquiring the outcome " l " with setting " a " for Alice side and for Bob side the outcome " m " with setting " b ". In experiment, we impose two significant assumptions of reality and locality. Before deriving the Bell inequality, the formal definition of realism and locality is as follows:

Realism

The idea of realism is expressed as, "A physical objects and its prop-erties pre-exist without the effect of an observer". Therefore, it implies that physical quantities have pre-defined values which are autonomous of the measurement proceedings. Let’s assume, a state of system |ψi is an eigenstate of operator B, then

B|ψi = b|ψi

thus the eigenvalue b is an element of physical reality for an observable

B [5].

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According to the proposition of locality, a physical event happening at one place cannot have any effect on another event happening at a place outside its light cone. This locality rule is established by following the special theory of relativity, which does not allow that information can transfer (or travel) faster than the speed of light. These two events are not correlated if

(∆x)2> c2(∆t)2

where (∆x) and (∆t) represents the space and time separation of two events respectively and c is the speed of light.

Both parties have a space like separation between them and they have agreed upon the same definition of a coincidence for every mea-surement setting. According to the first assumption, which allows us to assume Alice’s probabilities are only dependent upon her choice of settings with some parameter λ and the same argument holds true for Bob’s. Here, we are not considering that Alice can get information about the outcome deterministically if λ is known to her. However, we know that one can always extend HV of a non-deterministic model into a de-terministic model where the probabilities P (l|a, λ) results are either 1 or 0. Now from the second assumption, we considered that both systems are linked with classical parameter λ, but otherwise they are completely distinguished from one another.

The classical correlations built up between Alice and Bob from joint probabilities P (l, m|a, b) = Σ_λP (l|a, λ)P (m|b, λ)P (λ) are described by

lo-cal realistic models.

The measurement outcome of Alice and Bob with setting " a " and " b " for expected average E(a, b) value is define as,

E(a, b) = Σl,ml.m.P (l, m|a, b)

= Σ_l,mΣ_λl.m.P (l|a, λ)P (m|b, λ)P (λ)

= ΣλE(a, λ)E(b, λ)P (λ) (2.13)

Here, we notice that E(a, b), E(a, λ) and E(b, λ) functions have range (-1,1) and E(a, λ) or E(b, λ) also known as local expectation values.

E(a) = ΣlΣλl.P (l|a, λ)P (λ) = ΣλE(a, λ)P (λ)

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2.6. Bell’s Theorem and Famous CHSH Inequality 15

The measurements sequences for different settings of Alice and Bob are given by following

|E(a, b) + E(a, b0)| + |E(a0, b) + E(a0, b0)|

≤ Σλ(|E(a, λ)||E(b, λ) + E(b0, λ)| + |E(a0, λ)||E(b, λ) − E(b0, λ)|)P (λ)

≤ Σ_λ(|E(b, λ) + E(b0, λ)| + |E(b, λ) − E(b0, λ)|)P (λ) ≤ 2

where we employed that sum of absolute values are greater than the sum of real values and |E(a, λ)| ≤ 1. Thus for inequality we also used the lemma |x + y| + |x − y| ≤ 2 and x, y ∈ [−1, 1] and we got

|E(a, b) + E(a, b0)| + |E(a0, b) − E(a0, b0)| ≤ 2 (2.14)

In quantum mechanics, the expected average value is define as

E(a, b) = T r(A(a) ⊗ B(b).ρ) (2.15)

where ρ is the bipartite state shared between the parties produced by a source, A(a) and B(b) are quantum operators. Experimentally, we usu-ally calculate the quantity E(a, b) define as E(a, b) = Σl,m.l.m.P (l.m|a, b).

The quantum mechanical violation for Eq. (2.14) can be obtained for the operators

A(x)a=x= σx A(z)a0_=z= σ_z (2.16)

B(+)b=+= (σ_x+ σ_z) √ 2 B(−)b 0₌₋= (σ_x− σ_z) √ 2 (2.17)

The sum of the right-hand side of Eq. (2.14) gives us | − 2√_{2| 2,} which is much higher than what local realistic hidden variable theory predicts. This value shows us the maximum allowed violation by quan-tum mechanics, because the first three terms are equal to (√−1

2) and the

last term in equation is equal to (√1

2). Thus the value of violation

can-not exceed from (2√2) because this is the maximum value offered by a bipartite quantum mechanical system regardless of the way of mea-surements particularly defined for Tsirelson bound [18]. The violation of Bell inequality occurred in quantum mechanics but not in classical physics [19]. Tsirelson showed that this value is in fact the maximum violation allowed by quantum mechanic.

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2.6.1 Bell States

Bell states are a special kind of bipartite entangled quantum states that shows maximum violation of Bell inequality of a quantum system. This special scenario constitutes upon two parties such as Alice and Bob, two measurement settings for each party and two outcomes for each setting. A bipartite system consists of two qubits states whose Hilbert space H is 4-D dimensional. Four Bell states can be distinguished from one another by appropriate quantum measurements because they form an orthonormal basis. They are defined as,

|Ψ±i =√1

2(|0iA⊗ |1iB± |1iA⊗ |0iB) = 1 √

2(|01i ± |10i) |Φ±i =√1

2(|0iA⊗ |0iB± |1iA⊗ |1iB) = 1 √

2(|00i ± |11i)

(2.18)

where subscripts A and B clarify that two qubits are represented with two different subsystems (or in two different modes). These states pos-sess the property to predict the measurement outcomes of one qubit with certainty by performing the measurement on the second qubit. Ap-plications of Bell states are found within superdense coding, quantum networks and teleportation, quantum communication, and cryptography etc.

2.7 Greenberger Horne Zeilinger (GHZ) state

Greenberger Horne Zeilinger (GHZ) state is an entangled state, where all qubits on Alice’s side and Bob’s side exist either in ground state |0i or excited state |1i with relative phase ϕ.

|GHZi = √1 2(|0i

⊗n_{+ e}iϕ_|1i⊗n₎ _(2.19)

where n ≥ 3 is the minimum number of qubits (subsystems). In

pa-per II the four qubits GHZ state is produced to observe the optimal

non-locality from maximum incompatible local measurements. The four qubits GHZ state is a maximally entangled state which is a not bi-separable state.

|GHZi_n=4=√1

2(|0000i + e

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3. Experimental background

In this chapter, experimental realization of a qubit in terms of polariza-tion degree of freedom of light is discussed. A short introducpolariza-tion to the optical components, which are used for the generation, manipulation, and measurement of polarization states is also provided. Later, the ba-sic idea of how entangled states of light can be generated is also explored in detail.

3.1 Physical Realization of Qubit

Physical realization of a qubit has a very diverse range of two dimen-sional quantum systems such as nuclear or electronic spin, trapped ions, quantum dots, superconducting Josephson junction and topological sys-tems. In our experiments, apart from these, we have used the polariza-tion of the photon to realize a qubit. This implementapolariza-tion is practically simple and favourable, because polarization of photons is easy to gen-erate, measure, and manipulate. Photons are charge-less and massless particles travelling from one point to another at highest speed without any interaction with the environment, thus preventing the decoherence of the quantum system.

Polarization of a single photon can be treated as well isolated two-level quantum system with others degree of freedom e.g orbital angular momentum, arrival time and number of photon. In laboratory, horizon-tal |Hi and vertical |V i linear polarizations of photon are encoded into |0i and |1i computational states respectively. Both these vectors are define our standard or calculational basis for quantum information and computation.

3.2 Optical Components

This section contains a brief introduction of optical elements and polar-ization components which are used to manipulate the polarpolar-ization state of a photon. These optical components are polarizer (Pol), half and

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18 Chapter 3. Experimental background

quarter wave plates (HWP and QWP), beam splitters (BS) and polar-ized beam splitters (PBS), optical fibers, and single photon detectors.

3.2.1 Polarizer

A polarizer is an important and key optical component in an optical setup which is used to select a specific polarization from incoming light. There are many types of polarizers available in different designs for dif-ferent applications. In our lab, a linear absorptive polarizer is used for selecting the specific polarization. Absorptive polarizer allows only the light (polarization) that is parallel to its transmission axis to pass through and absorbs all others polarization. We choose zero degree of the polarizer such that it passes through vertical polarized light parallel to the transmission axis. Therefore, a polarizer at 90◦ will let through the horizontal polarization. One can easily prepare a pure-state from a mixed state by placing a polarizer in the path of the beam. The den-sity matrices of a state passing through the polarizer at 0◦ and 90◦ are represented as,

θ = 0◦→ Vertically polarized light = 0 0 0 1

!

θ = 90◦→ Horizontolly polarized light = 1 0 0 0

! (3.1)

Hence, the matrix representation of a polarizer at 0◦ will be Pol(0) = 0 0

0 1

!

. The action of a linear polarizer at an arbitrary angle θ is to prepare photons in a pure state on the x-z plane of the Bloch sphere depending on this angle. The matrix representation of its action on the incident light can be obtained by rotating the lab frame into a frame in which the polarizer is at 0◦ and then rotate back to the lab frame that is

P ol(θ) = RT(θ) P ol(θ)R (θ) (3.2) where R (θ) is a rotation matrix in anti-clockwise direction i.e.

R (θ) = cos θ sin θ − sin θ cos θ ! (3.3) P ol(θ) = sin 2_θ _{− sin θ cos θ}

− sin θ cos θ cos2θ !

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3.2. Optical Components 19

Mostly, absorptive linear polarizers are commonly made of some thin crystal known as Tourmaline, Herapathite or a glass substrate embed-ded with elongated metallic nano-particles in it, which shows strong absorption of parallelly polarized light. The matrix representation of a polarizer for vertical polarized light is obtained by inserting the Eq. (3.1), and Eq. (3.3) in Eq. (3.2).

Figure 3.1: The fast (no) and slow (ne) axes of a waveplate.

3.2.2 Waveplates (WP)

A WP is an essential optical device to manipulate the polarization of transmitted photons with very little attenuation. A WP is made of birefringent material that shows relative phase retardance between two orthogonal polarization components. It is made of a thin, only a few micrometer thick, single crystal such as Mica, Calcite or Quartz. The principal indices of refraction vary slowly in visible spectrum range for these crystals [20]. Here, the word birefringence declares that the crys-tal has two refractive indices named as no and ne for two orthogonal

polarizations. In the laboratory, we used WPs made of quartz, which is positive uni-axial crystal (n_e> no). In this case refractive indices of

fast axis and slow axis are defined as ordinary refractive index (no) and

extra-ordinary (ne) respectively as shown in Fig. (3.1). WPs are

de-signed to work perfectly at specific wavelengths, because the indices of refraction of WP are wavelength dependent. The relative phase retar-dance φ between two orthogonal polarization components depends upon the crystal thickness t, wavelength λ, and difference between refractive index (∆n = ne− no) in the following way:

φ =2πt(∆n)

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At an angle θ◦= 0, WP produces a phase shift between two orthogonal polarizations and its action is expressed using the following matrix

W P(θ=0,φ)=

1 0 0 eiφ

!

. (3.6)

The two most commonly used wave plates to manipulate the polarization of photons are known as the half wave plate (HWP) and the quarter wave plate (QWP). Matrix of WP for an arbitrary angle θ is defined as

W P(θ,φ)=

cos2θ + eiφ_sin2_θ _{cos θ sin θ(1 − e}iφ₎

cos θ sin θ(1 − eiφ) sin2θ + eiφcos2θ !

(3.7)

Half waveplate (HWP)

A WP acts as a half waveplate, if it introduces a π phase retardance between two orthogonal linearly polarized light components. A HWP can be used to rotate horizontal polarization to any other polarization on the real or x-z plane of the Bloch sphere. The unitary transformation matrix of a HWP is represented as HW P(θ)= cos(2θ) sin(2θ) sin(2θ) − cos(2θ) ! (3.8)

Note that, if the HWP is rotated by angle θ then the polarization plane of linearly polarized light is rotated by an angle 2θ after transmitting from HWP.

Quarter waveplate (QWP)

A WP acts as a quarter waveplate, if it gives (π₂) phase retardance for an angle φ between two orthogonal polarization. The rotation matrix of QWP at an angle θ is defined as following

QW P_(θ)=(1 + i) 2

1 − i cos(2θ) −i sin(2θ) −i sin(2θ) 1 + i cos(2θ)

!

(3.9)

If the axis of incident polarized light makes an angle of 45◦ with optical axes of WP then the light becomes circularly polarized. We can distinguish between two circularly polarized light components by right hand rule known as "from the point of view of receiver" convention. If the vector of light field is rotating in the direction of the anti-clockwise and the thumb is pointing in the direction of propagation then the polar-ization of light is known as right-circular polarized light and vice versa.

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Figure 3.2: Beam splitter with two input and two output ports.

3.2.3 Beam Splitter

A beam splitter (BS) is an optical device that splits an incident field of light into two or more fields of light at the output. Apart from splitting, it can also combine two beams of light. A standard BS cube is made of a pair of right angled glass prisms glued together diagonally. The hy-potenuse surface of one right angled prism is coated with a thin layer of dielectric material (partially reflective material coating). The thickness and material of this transparent thin film determines the splitting ratio for transmittance T and reflectance R = 1 − T of incident light intensity at a specific wavelength λ. A black dot is marked on the top of one right angled prism to distinguish which prism surface is coated with thin film. Most commonly used non-polarizing beam splitter (NPBS) cube in our optical setup has 50/50 splitting ratios, to split the light in equal inten-sities. Many types of BSs with different splitting ratios of an incident light are available for different applications, such as interferometry se-tups, imaging sese-tups, projectors and in auto-correlators.

In quantum mechanics, a lossless beam splitter can transform the amplitudes of input spatial modes individually from a and b to output modes a0 and b0 with transformation given as [1; 21],

a b ! = B00 B01 B10 B11 ! a0 b0 ! (3.10)

where Bij are the elements of beam splitter transformation matrix B

which is complex and introduces the phase shifts for the beam splitters that defines as B_ij = |B_ij|eiφij_{. The commutation relations for each}

mode corresponds to operators which satisfy the condition of superpo-sition at the output ports of BS because of unitary transformation of

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commutators [21]. Hence, the general form of unitary matrix B is fol-lowing with global phase eiφ0 _{that can be ignored without affecting the}

generality.

B = eiφ0 cos(θ)e

iφ1 _sin(θ)eiφ2

− sin(θ)e−iφ2 _cos(θ)e−iφ1

!

(3.11)

We used 50/50 or symmetric beam splitters in our optical setup that has the following transformation

a →√1 2(a 0_{+ ib}0₎ b → √1 2(b 0 + ia0)

Polarizing Beam Splitter

A polarization beam splitter (PBS) looks identical to a non-polarization beam splitter, but a PBS can split randomly polarized light into two specific linearly polarizations (known as horizontal and vertical). The reflectivity and transmissivity of PBS is polarization dependent. A PBS not only splits but combines the two light beams which are polarized in specific directions. Light that transmits through PBS is horizontally polarized and the light that reflects from an intermediate hypotenuse surface of PBS cube is vertically polarized. The position of incident beam on PBS hypotenuse and the placement of PBS (the facet of PBS normal to the input beam) in the optical setup is very important in or-der to get a good extinction ratio between two orthogonal polarizations.

In our experiments, we used cube shaped PBSs to combine two pho-tons to observe quantum interference similar to Hong-Ou-Mandel [22]

Figure 3.3: Two inputs and two outputs ports of a polarization beam

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effect from two input ports as seen in Fig. (3.3). A PBS is also an important optical component to convert polarization encoding into path encoding. A PBS transform input polarization modes to the outputs as follows. aH bH ! → e iφ1 ₀ 0 e−iφ1 ! a0_H b0_H ! (3.12) aV bV ! → 0 e iφ2 −e−iφ2 ₀ ! b0_V a0_V ! (3.13) 3.2.4 Optical Fiber

An optical fiber is a very thin cylindrical shape silica-based dielectric waveguide used to transmit light (photons) from one point to another (e.g from one laboratory to another). Typically, each fiber has two different regions in the middle, known as core and cladding, which have slightly different refractive indices. A fiber core has a higher refractive index than the cladding (n_core > n_cladding) [21; 23]. Light travels in the fiber mainly through the core by a phenomenon known as total internal reflection (TIR). The light collection efficiency and TIR depend on the angle of incidence of the light beam in the core of the fiber. If, the angle of incidence is smaller than the critical angle θc of the fiber facet then

TIR has no loss.

θc= cos−1(ncladding/ncore) (3.14)

Occasionally, optical fibers exhibit low birefringence due to stress and bends in the fiber. This causes the polarization of the propagating light in the fiber to change or become depolarized. There are two types of optical fibers that we have used in an experimental setup; single mode fiber (SMF) and multi-mode fiber (MMF).

Single mode fiber

A single mode fiber has a sufficiently small core diameter, in the or-der of a few microns usually in range of 2-15 µm for a given polarization and wavelength λ. It allows only a single mode of incident light to propagate through it. We have used a single mode fiber to purify one single mode from many spatial light modes, so the fiber acts as a mode filter. Coupling of light into a SMF is slightly harder and the coupling efficiency depends on the Gaussianity of incident light beam and on the precise placement of the coupling lens in the beam path.

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Figure 3.4: Polarization analysis station (measurement station).

Multi-mode fiber

The multi-mode fiber allows us to couple several spatial light modes into it, and all these modes can propagate at different speeds, taking into account the internal reflection process. The light modes can enter the fiber at many different angles, so that the modes that are parallel to the core axis or propagate in a straight line are slower than the ser-pentine modes (which propagate in a zigzag pattern) due to the high refractive index [24]. Therefore the discrepancy between the different light modes is reduced and they all arrive almost simultaneously. The core diameter of the multi-mode fiber is larger than the SMF and is in the range of 50 µm to 100 µm. MMF has a wide acceptance angle, which makes it easy to couple the incident light into the fiber, with high cou-pling efficiency approximately ≈ 90% and above. We have used MMF to couple light (photons) from optical setup of polarization analysis station to the single photon detectors as shown in Fig (3.4).

3.2.5 Avalanche Photodiodes

Photons are detected with a single photon silicon based avalanche photo-diodes (APD) [25]. These APDs are operated with a reverse bias voltage to avoid high current flow in Geiger mode, where the photons are con-tinuously detected above the breakdown level without external trigger pulses. When a photon is detected and merged in an active region of the detector (area of the active region 200 µm2), it generates an electron-hole pair that develops an avalanche of secondary electron-hole pairs when they diffuse in the silicon lattice due to high electric field. Therefore, it brings about an abrupt increase in electron-hole pairs, which develop a high avalanche current that is easily detected. Then, the high bias voltage decreases due to the detection of the high current which further reduces the electron-hole pairs. The quantum efficiency of all our single photon detectors is in the range of (55 - 60) % at 780 nm wavelength,

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3.3. Entangled Photons Generation 25

with their average dark counts in the range of 300 Hz- 600 Hz. The dead time1 of the detector is ≈ 50 ns and the TTL (transistor-transistor logic ) output signal of APDs has a pulse width of 20 ns with an amplitude of 5 volts.

3.2.6 Multiple Coincidences Counting Unit

The optical pumping of nonlinear crystals produces several photon pairs during each pump pulse (see section 3.3 below). These down-converted photons are detected by APDs with different detection efficiencies. When a down-converted photon is detected, a single photon detector APD sends a TTL signal to the multiple coincidence counting unit. The task of the coincidence counting unit is to track and record the detected pho-tons.

The coincidence counting unit is able to count and record all coinci-dences between 16 channels. The time during which two events can be as regarded as simultaneous is called the coincidence time window. The upper limit of the time window is defined by the repetition rate of the laser. In our case, the repetition rate of the pump laser is 90 MHz, which means that the coincidence time window cannot taken greater than 11.1 ns. The limit of the lower time window is defined by the detector time jitters. A shorter coincidence window is preferred, since it has lower probability of getting dark counts and false events. A suitable coinci-dence window to receive signals from different detectors is larger than the time jitters. In our coincidence counting unit, we have used a time window of 1.7 ns to obtain the signals and coincidences from different detectors.

3.3 Entangled Photons Generation

To generate entangled particles (photons) in a laboratory, many different techniques and schemes have been studied and developed. Currently, the parametric down conversion process (PDC) using nonlinear crystals is considered to be an efficient way to generate entangled photons. In the laboratory, I have used the PDC process to generate entangled and heralded photons for my experiments. In this process, a laser is used to pump the nonlinear crystals.

1_{Dead time: Also known as pulse dead time of the diode. It is time between} two successive detection events during which a detector is insensitive to detect any incoming photon after the first photon has been detected.

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Table 3.1: Specifications of Chameleon Ultra and Chameleon XR laser

systems used to produce entangled photon pairs.

3.3.1 Pump Laser

Laser (Light Amplification by Stimulated Emission of Radiation) is a coherent light source. In 1960, Theodore Harold Maiman presented the very first experimental demonstration of an optically pumped solid state laser [26]. It is a process of energy transformation (such as chemi-cal energy, electrichemi-cal energy and optichemi-cal energy) from incoherent energy to coherent optical emission. Here, the word coherence means that the light emitted by this process has both a spatial2and spectral coherence3. Basically, laser light is a different light source compared to conventional illumination sources.

In all our experiments, we used the Chameleon UltraT M laser system which is a tunable mode locked high power Ti: Sapphire femtosecond laser from Coherent that provides an average output power ∼ 4.1 watts at 780 nm wavelength. It is a pulse laser with a pulse width ∼ 140 fs and 80 MHZ repetition rate. Table (3.1) shows the specifications of our two pump lasers that we used in our laboratory. The wavelength of this tunable laser is set to 780 nm with a spectral width of about 8 nm full width at half maximum (FWHM). The output beam of the laser is

2_{Spatial coherence: is a measure of the correlation among the phases of a light} wave at different positions transverse to the direction of propagation. It tells how uniform the phase of the wave front is.

3_{Spectral coherence: It is defined as phase relation between different spectral} components.

Photonic Multipartite Communication: Complexity, measurements and Bell inequalities

Photonic Multipartite

Communication

Complexity, measurements and Bell inequalities

Hammad Anwer

Department of Physics

Photonic Multipartite Communication

Complexity, measurements and Bell inequalities

Hammad Anwer

Photonic Multipartite

Communication

Hammad Anwer

Abstract

Acknowledgements

Sammanfattning på svenska

List of papers

Author’s contribution

Contents

List of Figures

List of Tables

1. Introduction

1.1

Overview

1.2

Outline

2.

Introduction to Quantum

Information

2.1

Qubit

2.2

Density Operator

2.3

Bloch Sphere

2.4

Higher Dimensional Quantum System

2.5

EPR Paradox

2.6

Bell’s Theorem and Famous CHSH Inequality

2.7

Greenberger Horne Zeilinger (GHZ) state

3. Experimental background

3.1

Physical Realization of Qubit

3.2

Optical Components

3.3

Entangled Photons Generation