Engineering robust photonic
quantum states for quantum

communication and information

Isaac Mphele Nape

A Thesis submitted to the Faculty of Science

in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

School of Physics

University of Witwatersrand

Supervisor: Andrew Forbes
28 February 2022



Declaration

I declare that this Dissertation is my own, unaided work. It is being submitted for

the Degree of Doctor of Philosophy at the University of the Witwatersrand, Johan-

nesburg. It has not been submitted before for any degree or examination at any other

University.

(Signature of candidate)

day of 20 in

i

isaac
Typewriter
28

isaac
Typewriter
February

isaac
Typewriter
22

isaac
Typewriter
Johannesburg



Abstract

Quantum communication and information processing with photons achieves the over-

arching goals of transferring, encrypting, and processing digital information using ma-

chinery provided by fundamental physics principles that are established in quantum

mechanics. In the last decade, the field has matured rapidly, from being the bedrock

of simple demonstrations of quantum key distribution protocols with typical polarisa-

tion qubits that have two dimensional (𝑑 = 2) alphabets to now overseeing accelerated

developments with high dimensional encoding using alternative photonic degrees of

freedom (DOFs) that span larger Hilbert spaces of dimensions 𝑑 > 2. Excitingly, the

transverse spatial DOF of light offers an infinite encoding alphabet. While spatial

modes may be transported over most propagation media, i.e. free-space, optical fiber

and underwater channels, they are easily perturbed by various noise mechanisms,

e.g., rapidly varying refractive index profiles, diffraction, mode dependent loss, in-

hibiting their performance in practical applications. Most potential approaches for

undoing these deleterious effects require full knowledge of the channel dynamics or

the state evolution. In relation to the latter we can highlight the following challenges

for transverse spatial mode encoding that are prevalent in the field: i) the internal

modal scattering due to the perturbations from a quantum channel for spatial modes

can be difficult to predict; ii) and when possible, accurate characterisation methods

are required before the effects of the channel can be undone; iii) in higher dimensions,

characterising quantum states become increasing difficult due to the quadratic scaling

of the number of measurements with respect to the dimensions.

In this Thesis we tackle these issues by engineering techniques for creating, con-

trolling and characterising photons that are subject to a diverse range of perturbative

ii



channels. For channels, that cause diffraction induced losses, we tailor non-diffracting

higher dimensional vectorial photon fields, that have coupled polarisation and az-

imuthal spatial components, modulated with self-healing radial profiles. We show

that these fields can be used to transmit secure quantum information in the presence

of disturbances. We overcome the scattering effects of optical media with spatially

varying refractive index, by invoking channel state duality and the invariance of non-

separable states to unitary channels, but in locally entangled vectorial photon fields.

This approach enables us to devise a procedure for undoing the effects of a channel

in order to preserve information encoded in spatial modes. This method advances

the use of so called classical entanglement in quantum and classical optics. Next,

we develop a technique that manipulates heterogeneous channels to deliver multiple

hybrid non-locally entangled states using a single mode fiber channel. The nonlocal

hybrid entanglement between the polarisation and high dimensional spatial modes

of two spatially separated photons is used as main resource. Lastly, we develop a

novel technique for characterising high dimensional quantum states that are affected

by white noise. The procedure involves the use of conditional measurements that re-

turn crucial information about the underlying states’ occupied dimensions and purity.

We demonstrate the feasibility and adaptability of our approach using photons that

have nonlocal entanglement between their transverse spatial modes of orbital angular

momentum, and separately using the pixel position basis.

iii



Acknowledgments

To my supervisor: Distinguished Prof. Andrew Forbes, thank you for all the

valuable advice, guidence and encouragement. You have been instrumental in my

training. You have taught me not to work for the bare minimum but rather to do far

better than is expected. I also want to thank you for giving me the opportunity to

travel around the world and learn from other experts in our field.

To my seniors: Dr. Adam Valles, Dr. Bienvenue Ndagano, Dr. Carmelo.

Rosales-guzman, Dr. Valeria Rodŕıguez-Fajardo, Dr. Najmeh TabeBordbar and Dr.

Wagner Tavares Buono: The work in this dissertation would not be successfully

conducted without your valuable nurturing;

To my colleges and team mates: you are all such wonderful people to work

with. I have learned the value of friendship and teamwork through all of you.

To our collaborators: Dr. Eileen Otte, Dr. Feng Zhu and Dr. Jun Liu, you

all made my visits in Germany, Scotland and China hospitable, respectively. I would

like to thank your group leaders Prof. Cornelia Denz, Prof. Jian Wang and Prof.

Jonathan Leach for allowing me to work in their labs and also giving me an excellent

experience outside South Africa.

Family and friends: thank you for your continuous support and encouragement.

It has really brought me far.

Funders: Lastly, I would like to acknowledge the University of the Witwatersrand,

the joint Council of Scientific and Industrial Research (CSIR) and Department of

Science and Technology (DST)-Interbursary Support (IBS), the National Laser center

(NLC) and the SPIE optics society for their financial assistance.

iv



Contents

Declaration i

Abstract ii

Acknowledgments iv

List of Figures ix

List of Tables xii

Publications xiii

1 Introduction 1

1.1 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Polarisation modes . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Qudits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 High dimensional states . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 High dimensional spatial modes with orbital angular momentum 9

1.2.3 High dimensional hybrid polarisation modes . . . . . . . . . . 13

1.3 Two photon states in high dimensions . . . . . . . . . . . . . . . . . . 15

1.3.1 Entangled states . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.2 Nonlocal hybrid entanglement and classical hybrid entanglement 19

1.4 Quantum channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

v



2 Self healing quantum communication through obstructions 26

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Self-healing Bessel modes . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.1 Non-diffracting information basis . . . . . . . . . . . . . . . . 29

2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Single photon heralding . . . . . . . . . . . . . . . . . . . . . 32

2.3.2 Tailoring the desired spatial profile . . . . . . . . . . . . . . . 34

2.3.3 Generation and detection . . . . . . . . . . . . . . . . . . . . 34

2.3.4 Scattering probability . . . . . . . . . . . . . . . . . . . . . . 36

2.4 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.1 Procedure and analysis . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 A single vector beam can be used to charactersise turbulence chan-

nels 46

3.0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1.1 Nonseparable vector modes . . . . . . . . . . . . . . . . . . . 48

3.1.2 Vector mode propagation through turbulence from the perspec-

tive of quantum mechanics . . . . . . . . . . . . . . . . . . . . 52

3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.2 Vector quality factor measurement . . . . . . . . . . . . . . . 56

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.1 Propagation of vector modes through turbulence . . . . . . . . 57

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

vi



4 Unraveling the invariance of vectorial photon fields in unitary chan-

nels 62

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 Vectorial light and unitary channels . . . . . . . . . . . . . . . 64

4.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3.1 Vector beam generation . . . . . . . . . . . . . . . . . . . . . 68

4.4 Non-separability measurements . . . . . . . . . . . . . . . . . . . . . 69

4.4.1 Adjusted basis measurement . . . . . . . . . . . . . . . . . . . 71

4.5 Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5.1 Experimental demonstration: the tilted lens. . . . . . . . . . . 72

4.5.2 The role of measurement . . . . . . . . . . . . . . . . . . . . . 75

4.5.3 Reversing turbulence distortions . . . . . . . . . . . . . . . . . 78

4.6 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Transporting multiple hybrid entangled states through optical fibers 82

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Concept and principle . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3.1 The experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3.2 Characterisation . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.4.1 A hybrid quantum eraser . . . . . . . . . . . . . . . . . . . . . 94

5.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 97

6 Quantifiying Dimensionality and Purity in High Dimensional En-

tanglement 99

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.2.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

vii



6.3.1 High dimensional state projections . . . . . . . . . . . . . . . 107

6.3.2 Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . 109

6.3.3 Optimal purity and dimensionality calculation . . . . . . . . . 109

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.4.1 Orbital angular momentum basis measurements . . . . . . . . 110

6.4.2 Pixel basis measurements. . . . . . . . . . . . . . . . . . . . . 113

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7 Conclusions 118

7.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

A Scattering probability of OAM in turbulence 124

B Synthesis of turbulence 126

C Supplementary material: invariant vectorial photon fields 129

C.1 Transmitting vector beams through unitary single sided channels . . . 129

C.2 Nonseparability of the transformed vector mode . . . . . . . . . . . . 131

C.3 Undoing the effects of the channel . . . . . . . . . . . . . . . . . . . . 132

C.4 Examples with a titled lens . . . . . . . . . . . . . . . . . . . . . . . 134

C.5 Titled lens mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

C.5.1 Wave optics description . . . . . . . . . . . . . . . . . . . . . 138

C.6 Basis dependent and basis independent non-separability measurements 139

C.6.1 Basis independent non-separability . . . . . . . . . . . . . . . 139

C.6.2 Basis dependent non-separability . . . . . . . . . . . . . . . . 141

C.7 Basis independent VQF propagation of uncertainty . . . . . . . . . . 143

C.8 Stokes Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

D Supporting data for multi-mode SMF fiber channel 147

D.0.1 Modal spectrum data after the two meter fiber . . . . . . . . . 147

D.0.2 Supporting density matrix reconstruction data . . . . . . . . 147

viii



D.0.3 Supporting quantum eraser data . . . . . . . . . . . . . . . . . 148

D.0.4 Tabulated concurrence and fidelity data . . . . . . . . . . . . 149

E 151

E.1 Dimensionality of Pure States . . . . . . . . . . . . . . . . . . . . . 151

E.2 High dimensional state projections . . . . . . . . . . . . . . . . . . . 153

E.3 Decomposition of Entangled Photons . . . . . . . . . . . . . . . . . . 155

E.4 Detection Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

E.5 Visibility for Different Spectra . . . . . . . . . . . . . . . . . . . . . . 160

E.6 Visibility of Mixed States . . . . . . . . . . . . . . . . . . . . . . . . . 161

E.7 Visibility of Separable States . . . . . . . . . . . . . . . . . . . . . . . 163

E.8 Verification of the Technique . . . . . . . . . . . . . . . . . . . . . . . 164

E.9 Simulations in the Pixel Basis . . . . . . . . . . . . . . . . . . . . . . 165

E.10 Quantum State Fidelity and Schmidt Rank . . . . . . . . . . . . . . . 166

E.11 Measurements in the Pixel Basis . . . . . . . . . . . . . . . . . . . . . 167

E.12 Comparison to State-of-the-art . . . . . . . . . . . . . . . . . . . . . . 168

ix



List of Figures

1-1 BlochSphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1-2 Spatial profiles of LG modes . . . . . . . . . . . . . . . . . . . . . . . 9

1-3 OAM Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1-4 Higher order Poincareśphere . . . . . . . . . . . . . . . . . . . . . . . 14

1-5 Effects of quantum channels on photons . . . . . . . . . . . . . . . . 20

2-1 Polarisation profiles of self-healing modes. . . . . . . . . . . . . . . . 30

2-2 QKD elements for self-healing vector modes. . . . . . . . . . . . . . . 32

2-3 Experimental setup for heralded self healing photons. . . . . . . . . . 37

2-4 Photon count rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2-5 Crosstalk measurements. . . . . . . . . . . . . . . . . . . . . . . . . . 41

2-6 Security analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3-1 Vector modes in turbulence . . . . . . . . . . . . . . . . . . . . . . . 49

3-2 Experimental setup for VQF measurements. . . . . . . . . . . . . . . 50

3-3 Modal spectrum in turbulence. . . . . . . . . . . . . . . . . . . . . . . 52

3-4 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . 58

4-1 Concept of vectorial fields undergoing a unitary transformation. . . . 65

4-2 Vectorial light through a tilted lens. . . . . . . . . . . . . . . . . . . . 66

4-3 Impact of scattering across multiple subspaces. . . . . . . . . . . . . . 67

4-4 The unitary channel mapping and its inversion. . . . . . . . . . . . . 73

4-5 The choice of measurement basis. . . . . . . . . . . . . . . . . . . . . 76

4-6 Unravelling turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . . 78

x



5-1 Concept of multi-dimensional entanglement . . . . . . . . . . . . . . . 84

5-2 Experimental setup schematic and modal spectrum . . . . . . . . . . 86

5-3 State reconstruction of multiple hybrid states . . . . . . . . . . . . . 88

5-4 Hybrid state Bell violations. . . . . . . . . . . . . . . . . . . . . . . . 94

5-5 Quantum eraser experiment on the channel . . . . . . . . . . . . . . . 95

6-1 Concept of dimensionality measurement . . . . . . . . . . . . . . . . 102

6-2 Visibility, dimensionality and purity extraction. . . . . . . . . . . . . 105

6-3 Experimental visibilities . . . . . . . . . . . . . . . . . . . . . . . . . 111

6-4 Pixels basis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

B-1 Turbulence characterisation. . . . . . . . . . . . . . . . . . . . . . . . 126

C-1 Basis dependent measurement for tilted lens with input mode of ℓ = 1. 142

C-2 Basis dependent measurement for tilted lens with input mode of ℓ = 4. 142

C-3 Stokes parameters through a tilted lens. . . . . . . . . . . . . . . . . 144

C-4 Stokes Parameters of corrected modes . . . . . . . . . . . . . . . . . . 146

C-5 Stokes parameters in turbulence . . . . . . . . . . . . . . . . . . . . . 146

D-1 Spectral decomposition of the hybrid channel. . . . . . . . . . . . . . 148

D-2 Experimental tomography measurements. . . . . . . . . . . . . . . . . 149

D-3 Bell-inequality and quantum eraser measurements. . . . . . . . . . . . 150

E-1 Modal spectrum shapes. . . . . . . . . . . . . . . . . . . . . . . . . . 152

E-2 Modal decomposition of the state projectors . . . . . . . . . . . . . . 156

E-3 Detection probability vs relative orientation and dimensionality. . . . 157

E-4 Visibility of entangled pure states with differing spectral shapes. . . . 158

E-5 Impact of purity and dimensionality on visibility . . . . . . . . . . . . 169

E-6 Simulations for the pixel basis . . . . . . . . . . . . . . . . . . . . . . 170

E-7 Dimensionality witness comparison . . . . . . . . . . . . . . . . . . . 171

E-8 Dimensionality and purity measurements in the pixels basis . . . . . . 172

xi



List of Tables

1.1 Eigenvalues and eigenvectors of the Pauli matrices. . . . . . . . . . . 6

2.1 Wave-plate orientation angles . . . . . . . . . . . . . . . . . . . . . . 36

2.2 Security parameters for the channel . . . . . . . . . . . . . . . . . . . 43

3.1 Analysis of vector modes in turbulence . . . . . . . . . . . . . . . . . 59

6.1 Purity and dimensionality measurements . . . . . . . . . . . . . . . . 113

C.1 VQF of selected subspace. . . . . . . . . . . . . . . . . . . . . . . . . 132

C.2 Decomposition of LG modes into HG modes. . . . . . . . . . . . . . 137

C.3 Modal decomposition of LG after a tilted lens transformation. . . . . 138

C.4 VQF Values and errors. . . . . . . . . . . . . . . . . . . . . . . . . . . 145

D.1 Fidelity and concurrence values. . . . . . . . . . . . . . . . . . . . . . 149

E.1 Dimensionality and purity measurements in the OAM basis. . . . . . 165

E.2 Measurement scaling with dimensions . . . . . . . . . . . . . . . . . . 168

xii



Publications

Contributed patents

A. Forbes, B. Ndagano, I. Nape, M. Cox, and C. Rosales-guzman, “Method and

system for hybrid classical-quantum communication,” Dec. 26 2019. US. Patent

App. 16/480,008.

Peer reviewed articles

1. I. Nape, V. Rodri´guez-Fajardo, F. Zhu, H.-C. Huang, J. Leach, and A. Forbes,

“Measuring dimensionality and purity of high-dimensional entangled states,”

Nature Communications, vol. 12, no. 5159, pp. 1–8, 2021.

2. I. Nape, K. Singh, A. Klug, W. Buono, C. Rosales-Guzm´an, S. Franke-Arnold,

A. Dudley, and A. Forbes, “Revealing the invariance of vectorial structured light

in perturbing media,” arXiv preprint arXiv:2108.13890, 2021.

3. A. Klug, I. Nape, and A. Forbes, “The orbital angular momentum of a tur-

bulent atmosphere and its impact on propagating structured light fields,” New

Journal of Physics, vol. 23, p. 093012, 2021.

4. I. Nape, N. Mashaba, N. Mphuthi, S. Jayakumar, S. Bhattacharya, and A.

Forbes, “Vector-mode decay in atmospheric turbulence: An analysis inspired

by quantum mechanics,” Physical Review Applied, vol. 15, no. 3, p. 034030,

2021.

xiii



5. B. Sephton, A. Vall´es, I. Nape, M. A. Cox, F. Steinlechner, T. Konrad, J.

P. Torres, F. S. Roux, and A. Forbes, “High-dimensional spatial teleportation

enabled by nonlinear optics,” arXiv preprint arXiv:2111.13624, 2021.

6. Y. Shen, I. Nape, X. Yang, X. Fu, M. Gong, D. Naidoo, and A. Forbes,

“Creation and control of high-dimensional multi-partite classically entangled

light,” Light: Science & Applications, vol. 10, no. 50, pp. 1–10, 2021.

7. J. Liu, I. Nape, Q. Wang, A. Vall´es, J. Wang, and A. Forbes, “Multidimen-

sional entanglement transport through single-mode fiber,” Sci Adv, vol. 6, no.

4, p. eaay0837, 2020.

8. I. Nape, B. Sephton, Y.-W. Huang, A. Vall´es, C.-W. Qiu, A. Ambrosio,

F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular

momentum photons,” APL Photonics, vol. 5, no. 7, p. 070802, 2020.

9. A. Forbes and I. Nape, “A scramble to preserve entanglement,” Nature Physics,

vol. 16, no. 11, pp. 1091–1092, 2020.

10. M. de Oliveira, I. Nape, J. Pinnell, N. TabeBordbar, and A. Forbes, “Ex-

perimental high-dimensional quantum secret sharing with spin-orbit-structured

photons,” Physical Review A, vol. 101, no. 4, p. 042303, 2020.

11. A. Manthalkar, I. Nape, N. T. Bordbar, C. Rosales-Guzm´an, S. Bhattacharya,

A. Forbes, and A. Dudley, “All-digital stokes polarimetry with a digital mi-

cromirror device,” Optics Letters, vol. 45, no. 8, pp. 2319–2322, 2020.

12. A. Forbes and I. Nape, “Quantum mechanics with patterns of light: Progress

in high dimensional and multidimensional entanglement with structured light,”

AVS Quantum Science, vol. 1, no. 1, p. 011701, 2019.

13. J. Pinnell, I. Nape, M. de Oliveira, N. TabeBordbar, and A. Forbes, “Ex-

perimental demonstration of 11-dimensional 10-party quantum secret sharing,”

Laser & Photonics Reviews, vol. 14, no. 9, p. 2000012, 2020.

xiv



14. M. A. Cox, N. Mphuthi, I. Nape, N. Mashaba, L. Cheng, and A. Forbes,

“Structured light in turbulence,” IEEE Journal of Selected Topics in Quantum

Electronics, vol. 27, no. 2, pp. 1–21, 2020.

15. E. Otte, I. Nape, C. Rosales-Guzm´an, C. Denz, A. Forbes, and B. Ndagano,

“High-dimensional cryptography with spatial modes of light: tutorial,” JOSA

B, vol. 37, no. 11, pp. A309–A323, 2020.

16. J. Pinnell, I. Nape, B. Sephton, M. A. Cox, V. Rodr´ıguez-Fajardo, and A.

Forbes, “Modal analysis of structured light with spatial light modulators: a

practical tutorial,” JOSA A, vol. 37, no. 11, pp. C146–C160, 2020.

17. E. Toninelli, B. Ndagano, A. Vall´es, B. Sephton, I. Nape, A. Ambrosio, F.

Capasso, M. J. Padgett, and A. Forbes, “Concepts in quantum state tomogra-

phy and classical implementation with intense light: a tutorial,” Advances in

Optics and Photonics, vol. 11, no. 1, pp. 67–134, 2019.

18. I. Nape, E. Otte, A. Vall´es, C. Rosales-Guzm´an, F. Cardano, C. Denz, and

A. Forbes, “Self-healing high-dimensional quantum key distribution using hy-

brid spin-orbit bessel states,” Optics express, vol. 26, no. 21, pp. 26946–26960,

2018.

19. E. Otte, I. Nape, C. Rosales-Guzm´an, A. Vall´es, C. Denz, and A. Forbes,

“Recovery of nonseparability in self-healing vector bessel beams,” Physical Re-

view A, vol. 98, no. 5, p. 053818, 2018.

20. B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Cre-

ation and detection of vector vortex modes for classical and quantum commu-

nication,” Journal of Lightwave Technology, vol. 36, no. 2, pp. 292–301, 2018.

Conference papers

1. I. Nape, J. Liu, Q. Wang, A. Valles, J. Wang, and A. Forbes, “Transmitting

multiple hybrid entangled states using a conventional single mode fiber,” in

xv



Laser Science, pp. JTh4A–31, Optical Society of America, 2020.

2. I. Nape, V. Rodr´ıguez-Fajardo, H.-C. Huang, and A. Forbes, “Measuring

high dimensional entanglement using fractional orbital angular momentum.,”

in Frontiers in Optics, pp. FW7C–8, Optical Society of America, 2020.

3. K. Singh, I. Nape, A. Manthalkar, N. Tabebordbar, C. Rosales-Guzm´an,

S. Bhattacharya, A. Forbes, and A. Dudley, “Polarization reconstruction with

a digital micro-mirror device,” in Laser Beam Shaping XX, vol. 11486, p.

1148609, International Society for Optics and Photonics, 2020.

4. A. Vall´es, I. Nape, J. Liu, Q. Wang, J. Wang, and A. Forbes, “Multidimen-

sional spatial entanglement transfer through our existing fiber optic network,”

in Optical Manipulation and Structured Materials Conference 2020, vol. 11522,

p. 1152218, International Society for Optics and Photonics, 2020.

5. M. de Oliveira, J. Pinnell, I. Nape, N. TabeBordbar, and A. Forbes, “Realising

high-dimensional quantum secret sharing with structured photons,” in Quan-

tum Communications and Quantum Imaging XVIII, vol. 11507, p. 1150709,

International Society for Optics and Photonics, 2020.

6. V. Rod´rıguez-Fajardo, S. Scholes, R. Kara, J. Pinnell, C. Rosales-Guzm´an,

N. Mashaba, I. Nape, and A. Forbes, “Controlling light with dmds,” in 2020

International Conference Laser Optics (ICLO), pp. 1–1, IEEE, 2020.

7. J. Pinnell, I. M. Nape, M. De Oliveira, N. Tabebordbar, and A. Forbes, “Quan-

tum secret sharing with twisted light,” in Complex Light and Optical Forces XV,

vol. 11701, p. 117010K, International Society for Optics and Photonics,2021.

8. Y. Shen, Z. Wang, X. Yang, I. Nape, D. Naidoo, X. Fu, and A. Forbes, “Clas-

sically entangled vectorial structured light towards multiple degrees of freedom

and higher dimensions,” in CLEO: Science and Innovations, pp. STh1B–1,

Optical Society of America, 2021.

xvi



Conference Presentations

1. I. Nape, V. Rodr´ıguez-Fajardo, F. Zhu, HC. Huang, J. Leach, A. Forbes, “A

method for characterising high dimensional entangled states”, Student Confer-

ence on Optics and Photonics, India, 2021.

2. I. Nape, V. Rodr´ıguez-Fajardo, F. Zhu, HC. Huang, J. Leach, A. Forbes,

“Quantitative measurements of the purity and dimensionality of high dimen-

sional entagled states”, Annual conference of the South African Institute of

Physics, South Africa, 2021.

3. I. Nape, N. Mashaba, N. Mphuthi, S. Jayakumar, S. Bhattacharya, A. Forbes,

“Characterising laser beams through turbulence using vector beams and a sim-

ple quantum trick”, Annual conference of the South African Institute of Physics,

South Africa, 2021.

4. I. Nape, V. Rodr´ıguez-Fajardo, F. Zhu, HC. Huang, J. Leach, A. Forbes,

“Measuring the dimensionality of twisted modes”, Photon 2020 by the Institute

of Physics (IOP), United Kingdom, 2020.

5. I. Nape, E. Otte, J. Liu., A. Vall´es, Q. Wang, J. Wang, C. Rosales-guzman,

C. Denz and A. Forbes, “Reaching high dimensions by spinning and twisting

photons”, Quantum Africa (5) Conference, Stellenbosch, South Africa, 2019.

6. I. Nape, E. Otte, J. Liu., A. Vall´es, Q. Wang, J. Wang, C. Rosales-guzman,

C. Denz and A. Forbes,“Hybrid entanglement for quantum information pro-

cessing”, at the Intentional conference on orbital angular momentum , Ottawa,

Canada, 2019.

7. I. Nape, E. Otte, J. Liu., A. Vall´es, Q. Wang, J. Wang, C. Rosales-guzman,

C. Denz and A. Forbes “Self-healing locally entangled modes for secure com-

munication”, at the 2nd International OSA Network of Students (IONS) South

Africa, South Africa, 2018.

xvii



Chapter 1

Introduction

Initial developments in quantum mechanics unveiled several intriguing features about

the fundamental nature of quantum systems that make them distinct from their clas-

sical counterparts. For example, the most non-classical manifestation of quantum

correlations, i.e. entanglement, was discovered indirectly when Einstein, Podolsky,

and Rosen (EPR) [1] attempted to expose the incompleteness of quantum mechanics

by demonstrating that two particle systems with space-like separation can remain

correlated, therefore violating local hidden variable theories. It took several decades

of theoretical studies (see Ref. [2] for a concise review) and convincing experimental

validation [3], to prove that quantum entanglement, though paradoxical, is a funda-

mental aspect of nature. Another, intriguing feature of quantum systems that has no

classical equivalent is captured in the no-cloning theorem [4], preventing the creation

of exact copies of quantum states. Today, these intriguing properties, among many

others, serve as constituent elements of rapidly advancing technologies in quantum

communication, computing and information science that offer a variety of solutions

for digital information transmission, storage and processing.

While quantum computation harnesses the superposition principle and entangle-

ment for executing computational algorithms and calculations, quantum communi-

cation on the other hand offers solutions for secure information transmission over

long distances. A topical sub-field of quantum communication is quantum cryptog-

raphy [5–7], where single photons are used to generate encryption keys in a provably

1



Chapter 1 Isaac Nape 2

secure manner instead of relying on computationally difficult algorithms that may

fail with advancing computing power. Reaching long transmission distances as well

as maintaining high information capacities and communication speeds is at the core

of quantum communication research. Photons are favoured in this area as opposed to

other elementary particles (protons, electrons, neutrons), because they couple weakly

with the environment and have the highest propagation speeds.

Initially, numerous newly developed quantum communication protocols, e.g., quan-

tum key distribution (QKD) [8], superdense coding [9], quantum teleportation [10],

entanglement swapping [11], quantum secret sharing (QSS) [12] and some fundamen-

tal test of quantum mechanics, e.g., early demonstrations of Bell inequality viola-

tions [13], quantum erasures [14], etc., were demonstrated with photon polarisation

qubits having a two digit alphabet with dimensions of 𝑑 = 2. For this reason, access-

ing higher dimensions (𝑑 > 2) using transverse spatial mode is topical since spatial

modes offer the benefit of increasing the information capacity of photonic quantum

communication protocols [15–17], overcoming the capacity limit imposed by polari-

sation encoding. Transverse spatial modes are receiving much attention, with some

demonstrations of quantum encryption (QKD and QSS) ranging from 𝑑 = 4 [18,19],

𝑑 = 7 [20, 21], 𝑑 = 8 [22] and 𝑑 = 11 [23] dimensions. Current research efforts are

focused on deploying spatial modes in practical scenarios. However, spatial modes

are sensitive to numerous decay mechanisms (or perturbations) in the environment,

therefore limiting their performance. The perturbations can include spatially depen-

dent phase variations [24, 25], optical diffraction [26] and environmental noise [27],

known to scramble information in quantum channels.

In this chapter, we introduce the qubit (two-dimensional states) and transition

to qudits (high dimensional states). We draw our attention to the spatial degree

of freedom (DOF) of light together with polarisation modes and focus on how both

DOFs can be used to increase the encoding dimensions of photons in single and

two photon states. Subsequently, we discuss some of the challenges that come with

deploying spatial mode encoding in practical quantum channels and finally provide

an outline of each chapter in the thesis.

2



Chapter 1 Isaac Nape 3

1.1 Qubits

Discrete quantum states can be expressed as elements of the Hilbert space; an inner

product vector space ℋ, that can be spanned by an orthonormal computational basis,

ℬ𝑑, where the subscript 𝑑, is the number of unique elements corresponding to the

dimensions of the vector space. Moreover, the basis, ℬ𝑑, is complete because any

state on the Hilbert space can be expanded in terms of its elements. The simplest

example of a Hilbert space for discrete quantum states is the two dimensional qubit

space. For qubits, we can construct vectors, |0⟩ =

⎛⎝1

0

⎞⎠ and |1⟩ =

⎛⎝0

1

⎞⎠, which form

our basis. Note that ℬ𝑑 is not unique, e.g., from our computational basis, we can

create another basis, {|+⟩ = 1/
√

2 (|0⟩ + |1⟩) , |−⟩ = 1/
√

2 (|0⟩ − |1⟩)}, that can also

span the qubit vector space. Accordingly, any qubit state can be expanded in terms

of the basis vectors. For example, using our initial basis vectors, {|0⟩ , |1⟩}, i.e. the

computational basis, any state can be expressed as

|Ψ⟩ = 𝑎 |0⟩ + 𝑏 |1⟩ , (1.1)

=

⎛⎝𝑎
𝑏

⎞⎠ , (1.2)

where 𝑎 and 𝑏 are complex coefficients that determine the state |Ψ⟩. Since | ⟨Ψ|Ψ⟩ |2 =

1, the coefficients must satisfy |𝑎|2 + |𝑏|2 = 1. Physically, |𝑎|2 and |𝑏|2 are the prob-

abilities of obtaining the states |0⟩ and |1⟩, respectively. We can represent qubits on

a three dimensional sphere, called the Bloch sphere (shown in Fig. 1-1(a)), where

states are parameterised by angles 𝜃 and 𝜒. We can express Eq. (1.2) using these

parameters according to the mapping

|Ψ⟩ = cos(𝜃/2) |0⟩ + sin(𝜃/2)𝑒−𝑖𝜒 |1⟩ . (1.3)

Here 𝜃, the zenith angle controls the relative weighting between the basis states in

the computational basis while 𝜒 determines the relative phases. A photon with the

3



Chapter 1 Isaac Nape 4

Figure 1-1: (a) Bloch sphere, for qubit states. The poles contain the computational basis
states while the equator contains their equally weighted superpositions. (b) Poincareśphere
for polarisation modes.

state in Eq. (1.3) is pure. We can represent it using outer-products of the pure vector

state, |Ψ⟩, mapping it to a density matrix

𝜌Ψ = |Ψ⟩ ⟨Ψ| ,

=

⎛⎝𝑎
𝑏

⎞⎠⊗
(︁
𝑎* 𝑏*

)︁

=

⎛⎝|𝑎|2 𝑎𝑏*

𝑏𝑎* |𝑏|2

⎞⎠ . (1.4)

The matrix version is still normalised under the trace norm, so that Tr(𝜌Ψ) = 1,

where Tr denotes the trace operator. This can represent a process that produces

photons that are identical. In the case where the photons are not identical, or in

other words occupy states that are not necessarily the same, the system can be an

4



Chapter 1 Isaac Nape 5

ensemble (statistical mixture) of several pure states,

𝜌 =
∑︁
𝑖

𝑝𝑖𝜌𝑚,

=
∑︁
𝑖

𝑝𝑖 |Ψ𝑚⟩ ⟨Ψ𝑚| , (1.5)

where the states 𝜌𝑚 occur with a probability 𝑝𝑚 = Tr(𝜌†𝑚𝜌), and the states 𝜌𝑚 are not

necessarily orthogonal. If the density matrices are indeed othogonal and complete,

i.e., Tr(|Ψ𝑖⟩ ⟨Ψ𝑖|Ψ𝑗⟩ ⟨Ψ𝑗|) = 𝛿𝑖𝑗 and
∑︀

𝑖 |Ψ𝑖⟩ ⟨Ψ𝑖| = 1 then the state is said to be

maximally mixed and not pure. The purity of a density matrix can be computed

from Tr(𝜌2) ranging from zero for mixed states to one for pure states. Just as there

is a decomposition of pure states, there is also a compact decomposition with a basis

for two dimensional matrices given by,

{ 12 =

⎛⎝1 0

0 1

⎞⎠ , 𝜎𝑥 =

⎛⎝0 1

1 0

⎞⎠ , 𝜎𝑦 =

⎛⎝0 −𝑖

𝑖 0

⎞⎠ , 𝜎𝑧 =

⎛⎝1 0

0 −1

⎞⎠ },

which are the identity matrix, 12 and the three Pauli matrices satisfying Tr(𝜎𝑖) = 0.

Accordingly, any qubit density matrix can be decomposed as

𝜌 =
12

2
+
∑︁
𝑖

𝑏𝑖𝜎𝑖. (1.6)

Here 12 is the identity matrix, b = (𝑏1, 𝑏2, 𝑏3) is called the Bloch vector while 𝜎𝑖 are the

Pauli matrices. Since the density matrix must be semi-positive definite, |b| ≤ 1. The

eigenvalues and eigenvectors corresponding to each Pauli matrix are shown in Table

1.1. Crucially, the eigenvectors of 𝜎𝑧, corresponding to the computational basis, are

on the poles of the Bloch sphere and are aligned with the 𝑧-axis. The 𝜎𝑥 eigenvectors

map the states on the 𝑥-axis and correspond to the Hadamard basis states |±⟩ and

the 𝜎𝑦 eigenstates are on the poles of the 𝑦-axis. Therefore the Pauli matrices form

an over complete basis for the qubit states. Next, we explore an internal DOF of

photons that can be used to encode qubit states.

5



Chapter 1 Isaac Nape 6

Table 1.1: Eigenvalues and eigenvectors of the Pauli matrices.

Eigenvalues Eigenvector

𝜎𝑧 ±1 {|0⟩ ≡
(︂

1
0

)︂
, |1⟩ ≡

(︂
0
1

)︂
}

𝜎𝑦 ±1 { 1√
2

(︂
1
−𝑖

)︂
, 1√

2

(︂
1
𝑖

)︂
}

𝜎𝑥 ±1 {|+⟩ ≡ 1√
2

(︂
1
1

)︂
, |−⟩ ≡ 1√

2

(︂
1
−1

)︂
}

1.1.1 Polarisation modes

The polarisation of light is often associated with its spin angular momentum [28], a

property that enables light to rotate objects about their origin. This phenomenon is

observed when the light is circularly polarised. The direction of the rotating object

indicates the handedness of the circular polarisation field.

At the single photon level each photon carries exactly ±1ℏ per photon, where the

sign is associated with the handedness of the circular polarisation photon field and

hence can form a two level system using the circular polarisation basis. In optics, the

Bloch sphere equivilent for polarisation states is called the Poincare´ sphere, shown

in Fig. 1-1(b). Here the poles contain the right |𝑅⟩ ≡ |0⟩ and left |𝐿⟩ ≡ |1⟩ circular

polarisation modes.

Assuming the decomposition of qubits states in Eq. (1.3), we see that 𝜃 = 𝜋/2 can

be associated with states on the equator, corresponding to electric field oscillations

about the xy plane. Here well known linear polarisation states can be found to include

|𝐻⟩ = 1/
√

2 (|𝑅⟩ + |𝐿⟩) , (1.7)

|𝑉 ⟩ = 1/
√

2 (|𝑅⟩ − |𝐿⟩) , (1.8)

|𝐷⟩ = 1/
√

2 (|𝑅⟩ + 𝑖 |𝐿⟩) , (1.9)

|𝐴⟩ = 1/
√

2 (|𝑅⟩ − 𝑖 |𝐿⟩) , (1.10)

listed as the linear the horizontal (H) and vertical (V) polarisation states followed by

the rectilinear diagonal (D) and anti-diagonal (A) polarisation states, corresponding

6



Chapter 1 Isaac Nape 7

to phases 𝜒 = 0, 𝜋, 𝜋/2 and 3𝜋/2, respectively.

It should be trivial to see that the right and left circular polarisations are deter-

mined by 𝜃 = 0 and 𝜃 = 𝜋, respectively. For 𝜃 = (0, 𝜋), one obtains various elliptical

polarisation states. Therefore using the parameters (𝜃, 𝜒) it is possible to express any

polarisation qubit.

1.2 Qudits

Upon recognising that polarisation qubits imposed fundamental limits on quantum

communication protocols, i.e., admitting only one bit of information per photon,

numerous protocols were extended to multi-level encoding schemes [29], ushering in

high dimensional (𝑑 > 2) quantum information [17]. The migration from traditional

qubit to high dimensional qudits is motivated by the fact that higher dimensional

states offer increased information capacity and security [29–31], protection against

optimal quantum cloning machines [22, 32,33] and resilience to noise [34].

1.2.1 High dimensional states

In higher dimensions, 𝑑 > 2, we can express states on the Hilbert space by simply

increasing the size of our encoding basis. We achieve this by increasing the number

of elements in the computational basis. That is by constructing a basis with 𝑑 > 2

elements; ℬ𝑑 = {|𝑗⟩ , 𝑖 = 0, 1, ..𝑑 − 1} satisfying, | ⟨𝑖|𝑗⟩ | = 𝛿𝑖𝑗), for all |𝑖⟩ , |𝑗⟩ ∈ ℬ𝑑.

For example, in three dimensions we have the basis

ℬ3 = {|0⟩ =

⎛⎜⎜⎜⎝
1

0

0

⎞⎟⎟⎟⎠ , |1⟩ =

⎛⎜⎜⎜⎝
0

1

0

⎞⎟⎟⎟⎠ , |2⟩ =

⎛⎜⎜⎜⎝
0

0

1

⎞⎟⎟⎟⎠}, (1.11)

spanning the state-space for qutrits (𝑑 = 3). Accordingly, a qutrit pure state can be

written as

|Ψ⟩ = 𝑎1 |0⟩ + 𝑎2 |1⟩ + 𝑎3 |2⟩ , (1.12)

7



Chapter 1 Isaac Nape 8

having a corresponding density matrix,

𝜌𝜓 =

⎛⎜⎜⎜⎝
𝑎0

𝑎1

𝑎2

⎞⎟⎟⎟⎠⊗
(︁
𝑎*0 𝑎*2 𝑎*2

)︁
,

=

⎛⎜⎜⎜⎝
|𝑎0|2 𝑎0𝑎

*
1 𝑎0𝑎

*
2

𝑎1𝑎
*
0 |𝑎2|2 𝑎1𝑎

*
2

𝑎2𝑎
*
0 𝑎2𝑎

*
1 |𝑎3|2

⎞⎟⎟⎟⎠ . (1.13)

In general, any high dimensional purestate can be written as the superpositon state

|Ψ⟩ =
𝑑−1∑︁
𝑗=0

𝑎𝑗 |𝑗⟩ , (1.14)

where 𝑎𝑖 are complex coefficients that determine the state |Ψ⟩ up to a global phase.

Similarly to the qubit states, the normalisation condition requires that the coefficients

satisfy
∑︀

𝑗 |𝑎𝑗|2 = 1.

The density matrices can also be expressed using the computational basis as

𝜌 =

(∑︁
𝑚,𝑛=0

𝑑− 1)𝑐𝑚𝑛 |𝑚⟩ ⟨𝑛| . (1.15)

The components of the density matrix are given by 𝑐𝑚𝑛 and there are exactly 𝑑2 of

them. In addition to the decomposition in Eq. (1.15), there exists another decompo-

sition with 𝑑2 − 1 Gell-Mann matrices, 𝜏𝑘, following

𝜌 =
1𝑑

𝑑
+

𝑑2−1∑︁
𝑘=0

𝑡𝑘𝜏𝑘, (1.16)

where t = (𝑡0, 𝑡1, ..𝑡𝑑2−1) are coefficients of the Gell-Mann matrices. The Gell-mann

matrices are also trace-less, othorgonal and have 𝑑 eigenvectors. In order to determine

the coefficients 𝑡𝑘, a set of tomographically complete measurements are required and

these can be the eigenvectors of each matrix. This means that (𝑑2 − 1) observables

8



Chapter 1 Isaac Nape 9

Figure 1-2: High dimensional spatial modes of light. (a) Normalised intensity and (b) phase
profiles of LG modes for various discrete radial (𝑝) and OAM indices (ℓ).

are required to completely describe the state. Alternatively mutually unbiased bases

can be used [35], requiring 𝑑+ 1 observables, though there are restrictions on 𝑑 [36].

Now that we have established what qudits (or high dimensional states) are, the

goal is to use photonic DOFs that have the properties listed above.

1.2.2 High dimensional spatial modes with orbital angular

momentum

Photons have internal DOFs that span higher dimensional Hilbert spaces, e.g., time

[37–39], path [40,41] and transverse spatial modes [15,16,42]. In particular, high di-

mensional transverse spatial modes that carry quantised amounts of orbital angular

momentum (OAM) are slowly showing feasibility in freespace [20, 22, 43–46], optical

fiber [47] and underwater [48, 49] quantum communication channels. Here, we focus

on the transverse spatial mode DOF. Firstly, let us revisit the wave description light

where the transverse oscillations of the electric field are characterised by field func-

tions, 𝑈(r, 𝑧, 𝑡), satisfying the wave equation [50]. The parameters r = (𝑥, 𝑦), 𝑧 and

𝑡 are the transverse, longitudinal and temporal coordinates, respectively.

Under the paraxial approximation (the limit of small beam divergence in the

traverse plane) it can be assumed that transverse components of spatial modes are a

slowly varying function of 𝑧, i.e. satisfying the paraxial inequality [28]

| 𝜕
2

𝜕2𝑧
𝑈(r, 𝑧)| << 𝑘| 𝜕

𝜕𝑧
𝑈(r, 𝑧)|. (1.17)

9



Chapter 1 Isaac Nape 10

It follows that the differential equation that governs the propagation of these fields

can be written as [28]

(∇2
⊥ + 𝑖𝑘2

𝜕

𝜕𝑧
)𝑈(r, 𝑧) = 0, (1.18)

where ∇2
⊥ is the transverse component of the Laplacian differential operator. Equa-

tion (1.18) is called the paraxial approximation of the Helmholtz equation. In the

cylindrical coordinates, general solutions to Eq. (1.18) have the form

𝑈(𝑟, 𝜑, 𝑧) = 𝑢(𝑟, 𝑧)𝑒𝑖ℓ𝜑. (1.19)

Here 𝜑 and 𝑟 are the azimuthal and radial coordinates, respectively, while ℓ ∈ Z

is an integer, and 𝑢(𝑟, 𝑧) is the radial profile of the beam. These solutions have a

characteristic complex profile, 𝑒𝑖ℓ𝜑, indicative of light beams that carry orbital angu-

lar momentum OAM. The integer, ℓ, has a physical significance: ℓ is the helicity or

topological charge of the vortex present in the field and each photon has an OAM of

ℓℏ. Moreover, the phase profiles have |ℓ| characteristic dislocations that correspond

to complete cycles of 2𝜋. It was Allen et. al. [51] who first generated OAM light

beams and since then they are used throughout the photonics community. Current

methods for generating and detecting photons carrying OAM include the use of phase

elements that utalise dynamic phase control on spatial light modulators (SLM) [52],

geometric phase control using birefringent liquid crystals [53–55] and recently emerg-

ing metasurface technology [56,57].

An example of a photon field that carries OAM is a Gaussian mode that is im-

printed with the characteristic azumthal phase profile of OAM modes, i.e. exp
(︁
− 𝑟2

𝑤2
0

)︁
×

exp(𝑖ℓ𝜑). However, such a field is instead a Hypergoemetric-Gaussian mode [58]and

has a radial profile that changes with propagation though maintaining its azimuthal

profile. On the contrary, a well known mode family that maintains the radially depen-

dent amplitude and phase profiles in the paraxial regime is the of Laguerre-Gaussian

10



Chapter 1 Isaac Nape 11

(LG) mode family, expressed here as

𝑈𝐿𝐺
ℓ𝑝 (𝑟, 𝜑, 𝑧) = 𝐶ℓ𝑝

[︂√
2

𝑟

𝑤(𝑧)

]︂ℓ
𝐿𝑝

ℓ

(︂
2𝑟2

𝑤(𝑧)2

)︂
𝑤0

𝑤(𝑧)
exp (−𝑖𝜓𝑝ℓ(𝑧))×

exp

(︂
−𝑖 𝑘

2𝑞(𝑧)
𝑟2
)︂

exp

(︂
− 𝑟2

𝑤(𝑧)2

)︂
exp(𝑖ℓ𝜑), (1.20)

where the function 𝐿ℓ𝑝(·) is the associated Laguerre polynomial, 𝑤(𝑧) = 𝑤0

√︀
1 + (𝑧/𝑧𝑅)

is the beam size of the Gaussian envelope as the field propagates in the 𝑧 direction

while 𝑤0 is its corresponding radius at the 𝑧 = 0 plane, with 𝑧𝑅 = 𝜋𝑤0
2/𝜆 represent-

ing the Gaussian mode Rayleigh range. The term depending on 𝑞(𝑧) = 𝑧− 𝑖𝑧𝑅 is the

complex beam parameter and 𝜓𝑝ℓ(𝑧) = (2𝑝+ |ℓ| + 1) tan−1(𝑧/𝑧𝑅) is the Gouy phase.

The constant factor 𝐶ℓ𝑝 is a normalisation constant so that
∫︀
|𝑈𝐿𝐺

ℓ𝑝 (𝑟, 𝜑, 𝑧)|2𝑑2𝑟 = 1.

The second moment radius of LG beams increases in size with |ℓ| and 𝑝 according to

𝑤ℓ,𝑝 =
√︀

2𝑝+ |ℓ| + 1 × 𝑤0.

In Fig. 1-2(a) and Fig. 1-2(b) various LG intensity, (|𝑈𝐿𝐺
ℓ𝑝 (𝑟, 𝜑, 𝑧)ℓ𝑝|2), and phase

profile, mod
[︀
arg
(︀
𝑈𝐿𝐺
ℓ𝑝 (𝑟, 𝜑, 𝑧 = 0)

)︀
, 2𝜋
]︀
, are shown. The intensity profile has a region

of null intensity, centered at the origin and a radius that increases with |ℓ|. Further-

more, the field has 𝑝+ 1 or 𝑝 concentric rings for |ℓ| > 0 or |ℓ| = 0, respectively.

Interestingly, OAM basis modes that have LG profiles have been measured at

the single photon level using interferometers [59] and refractive optical elements [60].

Moreover, they have been used in high dimensional encoding schemes for single pho-

tons [21], entangled two [61] and three photon GHZ states [62]. In such applications,

the field profiles in Eq. (1.20) are used to describe basis states of the transverse DOF

of photons following the expansion

|ℓ, 𝑝⟩ =

∫︁∫︁
𝑈𝐿𝐺
ℓ𝑝 (𝑥, 𝑦) |𝑥⟩ |𝑦⟩ 𝑑𝑥𝑑𝑦, (1.21)

where |𝑥⟩ |𝑦⟩ are continuous position state vectors in Cartesian coordinates, such

that ⟨𝑥′|𝑥⟩ = 𝛿(𝑥−𝑥′) and ⟨𝑦′|𝑦⟩ = 𝛿(𝑦− 𝑦′), are inner products in the position basis

resulting in Dirac delta functions [63]. It follows that ⟨𝑦′| ⟨𝑥′|𝑥⟩ |𝑦⟩ = ⟨𝑥′|𝑥⟩ ⟨𝑦′|𝑦⟩ =

𝛿(𝑥−𝑥′)𝛿(𝑦−𝑦′). Accordingly, the overlap between any two states |ℓ1, 𝑝1⟩ and |ℓ2, 𝑝2⟩

11



Chapter 1 Isaac Nape 12

Figure 1-3: Any two LG basis modes can be used to construct a Bloch (or equivalently

Poincare)́ sphere for spatial qubits. We show an example for OAM modes in the ℓ = ±1
subspace where 𝑝 = 0.

is

⟨𝑝1, ℓ1|ℓ2, 𝑝2⟩ =

∫︁∫︁
𝑈𝐿𝐺*
ℓ1,𝑝1

(𝑥1, 𝑦1)𝑑𝑥1𝑑𝑦1

∫︁∫︁
𝑈𝐿𝐺
ℓ2,𝑝2

(𝑥2, 𝑦2) ⟨𝑥1|𝑥2⟩ ⟨𝑦1|𝑦2⟩ 𝑑𝑥2𝑑𝑦2,

=

∫︁∫︁
𝑈𝐿𝐺*
ℓ1,𝑝1

(𝑥1, 𝑦1)𝑑𝑥1𝑑𝑦1

∫︁∫︁
𝑈𝐿𝐺
ℓ2,𝑝2

(𝑥2, 𝑦2)𝛿(𝑥2 − 𝑥1)𝛿(𝑦2 − 𝑦1)𝑑𝑥2𝑑𝑦2,

=

∫︁∫︁
𝑈𝐿𝐺*
ℓ1,𝑝1

(𝑥, 𝑦)𝑈𝐿𝐺
ℓ2,𝑝2

(𝑥, 𝑦)𝑑𝑥𝑑𝑦,

= 𝛿ℓ1,ℓ2𝛿𝑝1,𝑝2 , (1.22)

owing to the orthonormality of the LG basis modes. This means that any two distinct

LG modes are orthogonal. Therefore a qubit state-space analogous to the polarisation

Poincaré sphere, can be constructed for any two independent OAM modes [64]. In

Fig. 1-3, we show an example of a qubit space for 𝑝 = 0 and ℓ = ±1. Here, the north

and south poles are the basis states |ℓ⟩ ≡ |ℓ, 𝑝 = 0⟩ and |−ℓ⟩ ≡ |−ℓ, 𝑝 = 0⟩. The

superposition states are located on the equator. Any qubit state on this subspace can

12



Chapter 1 Isaac Nape 13

be written as

|𝜓ℓ𝜃𝜒⟩ = cos (𝜃/2) 𝑒𝑖𝜒 |ℓ⟩ + sin (𝜃/2) 𝑒−𝑖𝜒 |−ℓ⟩ , (1.23)

parameterised by 𝜃 and 𝜒, similar to qubits on the Bloch sphere. Since ℓ ∈ Z and 𝑝 ∈

Z
+, there are infinitely many such qubit subspaces. In general, the basis states |ℓ, 𝑝⟩

span the high dimensional Hilbert space and is overcomplete over the azimuthal and

radial coordinate. Accordingly, the collection of 𝑑 modes, i.e. {|ℓ𝑗, 𝑝𝑗⟩ , 𝑗 = 0, 1..𝑑−1}

spans a 𝑑 dimensional Hilbert space, ℋ𝑑. This means that we can map the spatial

mode ket states to state vectors with 𝑑 entries. Using the vector notation we can

proceed in discussing photons states with coupled DOF and two photon states in

higher dimensions.

1.2.3 High dimensional hybrid polarisation modes

Another avenue for increasing the dimensions of photon is through the coupling of

independent DOFs to create hybrid states. In particular, the polarisation (ℋ𝑆𝑃𝐼𝑁
2 )

and OAM (ℋ𝑂𝐴𝑀
𝑑 ) (see Fig. 1-4(a)) Hilbert spaces can be combined to construct a

higher dimensional subspace [65], called the higher order Poincaré sphere (see Fig.

1-4(b) and (c)). If the spatial DOF also spans the two dimensional Hilbert space,

then the subspace is a tensor product of the polarisation qubit and spatial qubit

state spaces, respectively. The resulting Hilbert space is ℋ𝑆𝑃𝐼𝑁
2 ⊗ ℋ𝑂𝐴𝑀

2 . Each

qubit, subspace is a span of the scalar mode basis {|𝑅⟩ |ℓ⟩ , |𝐿⟩ |−ℓ⟩} (Fig. 1-4(a))

and {|𝑅⟩ |−ℓ⟩ , |𝐿⟩ |ℓ⟩} (Fig. 1-4(c)). Poles contain the scalar basis modes where

the polarisation and OAM are just scalar separable products. The equator contains

nonseparable superpositions states that cannot be written as independent as scalar

separable products. We can construct a complete basis from modes on the equator

13



Chapter 1 Isaac Nape 14

Figure 1-4: The Higher order Poincaré (HOP) sphere is a 4 dimensional state space formed
from the (a) tensor product between the polarisation and two dimensional (±ℓ) state-space.
(b)-(c) The resulting subspaces contain spatial modes ranging from scalar fields to vectorial
fields. Orthogonal vector modes namely the radially (|Ψ1⟩), azimuthally (|Ψ2⟩) polarised
and hybrid electric |Ψ3,4⟩ modes (from left to right).

of the spheres:

⃒⃒
Ψℓ

1

⟩︀
=

1√
2

(|𝑅⟩ |ℓ⟩ + |𝐿⟩ |−ℓ⟩), (1.24)⃒⃒
Ψℓ

2

⟩︀
=

1√
2

(|𝑅⟩ |ℓ⟩ − |𝐿⟩ |−ℓ⟩), (1.25)⃒⃒
Ψℓ

3

⟩︀
=

1√
2

(|𝑅⟩ |−ℓ⟩ + |𝐿⟩ |ℓ⟩), (1.26)⃒⃒
Ψℓ

4

⟩︀
=

1√
2

(|𝑅⟩ |−ℓ⟩ − |𝐿⟩ |ℓ⟩), (1.27)

resulting in a high dimensional (𝑑 = 4) encoding basis that can be used for quantum

key distribution [66]. We show the polarisation profiles for each mode in Fig. 1-

4(d). The polarisation of the fields is nonuniform across the transverse plane of

the fields. These light fields have been a topic of great interest in the classical and

14



Chapter 1 Isaac Nape 15

quantum optics community [67–69], especially for possessing correlations that are

ubiquitous to nonlocal quantum entangled states. Before introducing this topic, we

first establish the description of the two photon Hilbert space and then subsequently

introduce entanglement.

1.3 Two photon states in high dimensions

Two independent photons, A and B, defined on Hilbert spaces ℋ𝐴 and ℋ𝐵, respec-

tively, have a joint state that is an element of the tensor product space ℋ𝐴𝐵 =

ℋ𝐴⊗ℋ𝐵. For example, if each photon is a qutrit state |0⟩𝐴 =

⎛⎜⎜⎜⎝
1

0

0

⎞⎟⎟⎟⎠ and |2⟩𝐵 =

⎛⎜⎜⎜⎝
0

0

1

⎞⎟⎟⎟⎠
then the combined state is given by

|0⟩𝐴 ⊗ |2⟩𝐵 =

⎛⎜⎜⎜⎝
1

0

0

⎞⎟⎟⎟⎠⊗

⎛⎜⎜⎜⎝
0

0

1

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1

⎛⎜⎜⎜⎝
0

0

1

⎞⎟⎟⎟⎠

0

⎛⎜⎜⎜⎝
0

0

1

⎞⎟⎟⎟⎠

0

⎛⎜⎜⎜⎝
0

0

1

⎞⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

𝑛1 = 0

𝑛2 = 0

𝑛3 = 1

.

.

.

𝑛8 = 0

𝑛9 = 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (1.28)

resulting in an a state vector with nine, 𝑛𝑖 (𝑖 = 1..9), entries. This is because the two

photon subspace has dimensions 𝑑2, resulting from the product of the dimensions of

the individual subspaces. We can further compute the two photon qutrit basis set

15



Chapter 1 Isaac Nape 16

from the tensor product between the bases of each photon resulting in

ℬ𝐴𝐵 = ℬ𝐴 ⊗ ℬ𝐵,

= {|0⟩𝐴 , |1⟩𝐴 , |2⟩𝐴} ⊗ {|0⟩𝐵 , |1⟩𝐵 , |2⟩𝐵},

= {|0⟩𝐴 ⊗ |0⟩𝐵 , |0⟩𝐴 ⊗ |1⟩𝐵 , .., |2⟩𝐴 ⊗ |1⟩𝐵 , |2⟩𝐴 ⊗ |2⟩𝐵},

(1.29)

having nine basis vectors. In general, for any dimensions 𝑑𝐴 and 𝑑𝐵, for each photon,

the basis can be expressed as

ℬ𝐴𝐵 = {|𝑚⟩𝐴 ⊗ |𝑛⟩𝐵 , ∀𝑚 = 0, 1, ..𝑑𝐴 − 1 and 𝑛 = 0, 1, ..𝑑𝐵 − 1}, (1.30)

where each photon is spanned by the bases {|𝑚⟩𝐴} and {|𝑛⟩𝐵}, respectively. As such,

an arbitrary two photon pure state can be written as a superstition

|Ψ⟩𝐴𝐵 =

𝑑𝐴−1∑︁
𝑚=0

𝑑𝐵−1∑︁
𝑛=0

𝑎𝑚𝑛 |𝑚⟩𝐴 ⊗ |𝑛⟩𝐵 , (1.31)

We drop the tensor product symbol (⊗) and use the short hand notation |·⟩𝐴 |·⟩𝐵.

Furthermore, the density matrix for an arbitrary quantum states has the form

𝜌𝐴𝐵 =
∑︁
𝑚,𝑛,𝑘,𝑙

𝑐𝑚𝑛𝑘𝑙 |𝑚⟩𝐴 |𝑛⟩𝐵 ⟨𝑙|𝐵 ⟨𝑘|𝐴 . (1.32)

with components |𝑚⟩𝐴 |𝑛⟩𝐵 ⟨𝑙|𝐵 ⟨𝑘|𝐴 having coefficients 𝑐𝑚𝑛𝑘𝑙.

1.3.1 Entangled states

There is a special class of quantum states that have correlations that have no classical

equivalent. Einstein, Padolsky and Rosen (EPR) introduced these states in their

seminal paper [1] with the intention of using them as examples of states that expose

the incompleteness of quantum mechanics. Paradoxically, the correlations produced

by these states violate local relativistic causality and realism in classical mechanics,

16



Chapter 1 Isaac Nape 17

allowing for measurements on one subsystem to effect the outcomes of measurements

of their other counterparts even when spatially separated. This phenomenon was later

called ’Verschränkung’ in German, by Schrödinger, and translates as entanglement in

English.

To clearly define entanglement, it is instructive to first introduce separable states.

Firstly, let us consider the decomposition in Eq. (1.31), which describes a system of

two spatially separated photons, A and B. If the photons are not entangled then the

final state can be written as the separable product state

|Ψ⟩ = |𝜓⟩𝐴 ⊗ |𝜙⟩𝐵 , (1.33)

where |𝜓⟩𝐴 =
∑︀

𝑖=0 𝑏𝑖 |𝑖⟩𝐴 and |𝜙⟩𝐵 =
∑︀

𝑗=0 𝑐𝑗 |𝑗⟩𝐵. Conversely, the state is entangled

if it cannot be written as a separable product of the individual subsystems, i.e.,

|Ψ⟩𝐴𝐵 ̸= |𝜓⟩𝐴 ⊗ |𝜙⟩𝐵, implying that the state is nonseparable. John Bell, devised

a statistical test [70], with an inequality that can be violated by states that are

nonseparable, and in particular the states

|Φ⟩±𝐴𝐵 =
1√
2

(|0⟩𝐴 |0⟩𝐵 + |1⟩𝐴 |1⟩𝐵),

|Φ⟩±𝐴𝐵 =
1√
2

(|0⟩𝐴 |0⟩𝐵 − |1⟩𝐴 |1⟩𝐵),

|Ψ⟩±𝐴𝐵 =
1√
2

(|0⟩𝐴 |1⟩𝐵 + |1⟩𝐴 |0⟩𝐵),

|Ψ⟩±𝐴𝐵 =
1√
2

(|0⟩𝐴 |1⟩𝐵 − |1⟩𝐴 |0⟩𝐵), (1.34)

are known as maximally entangled Bell-states. From these equations it is clear that

a projection onto of the eigenstates of subsystem A determines the outcome of sub-

system B. For example, in Eq. (1.34), a measurement of the state |0⟩𝐴 in photon A,

results in |1⟩𝐵 for photon B.

The Bell states shown in Eq. (1.34) are only two dimensional. In high dimensions,

17



Chapter 1 Isaac Nape 18

an entangled state can be written as

|Ψ⟩ =
𝑑−1∑︁
𝑗=0

𝜆𝑗 |𝑗⟩𝐵 |𝑗⟩𝐵 , (1.35)

where |𝜆𝑗|2 is the probability of detecting the biphoton state |𝑗⟩𝐴 |𝑗⟩𝐵 spanning a 𝑑

dimensional basis. Here, |𝑗⟩𝐴 |𝑗⟩𝐵 is called a Schmidt basis. For 𝜆𝑗 = 1
√
𝑑, the state

is maximally entangled and the number 𝑑 is called the Schmidt number [71]. For

states where 0 < 𝜆𝑗 ≤ 1 the Schmidt number can be estimated as [72],

𝐾 =

(︁∑︀
𝑗 |𝜆𝑗|2

)︁2∑︀
𝑗 |𝜆𝑗|4

, (1.36)

which evaluates as 𝐾 = 1 for completely separable states and is 𝐾 ≥ 2 for states

that are nonseperable. It is possible to generate entanglement with the spatial DOF.

Recent demonstrations have included different mode families such as the LG [73],

Ince-Gaussian [74] and Bessel Gaussian [75] basis states.

Spontaneous parametric down-conversion (SPDC) is one way of generating en-

tangled photons. In this process, a high frequency pump photon is absorbed by the

crystal and converted into two lower frequency (down-converted) photons which are

highly correlated in polarisation [76–79], momenta [41] and temporal DOF [38,80,81].

The conservation in momentum also results in the conservation of OAM and therefore

the entanglement thereof [75, 82–84]. In the LG basis for OAM modes, the Schmidt

decomposition for SPDC is given by [85,86]

|Ψ⟩ =
∑︁
ℓ1,ℓ2,𝑝

𝜆ℓ1ℓ2𝑝 |ℓ1, 𝑝⟩𝐴 |ℓ2, 𝑝⟩𝐵 , (1.37)

where |𝜆ℓ1ℓ2𝑝|2 is the probability of finding photon 𝐴 and 𝐵 in the state |±ℓ, 𝑝⟩. The

dimensionality of the state is limited by the distribution of |𝜆ℓ1ℓ2𝑝|2. Due to OAM con-

servation in the SPDC process, the OAM of the pump photon sets the restriction that,

ℓ𝑝𝑢𝑚𝑝 = ℓ1 + ℓ2, ensuring that OAM is always conserved. There are several methods

for increasing the number of accessible modes which include tuning phase-matching

18



Chapter 1 Isaac Nape 19

conditions (momentum conservation conditions) at the crystal [61], adjusting the de-

tection mode sizes or by shaping the modes [86] and even selecting different mode

families [87] thus maximising the maximum number of accessible modes. To this end,

valid tests of quantum entanglement using the Bell inequality violations in two [84]

and up to twelve dimensions [88], quantum state tomography [35, 89], entanglement

witnesses [42] have all been demonstrated with SPDC photons using the transverse

spatial DOF of photons.

1.3.2 Nonlocal hybrid entanglement and classical hybrid en-

tanglement

While the states in Eq. (1.37) are entangled in single DOFs, there exists a class of

quantum entangled states where the internal DOFs of each photon are completely

independent. Such states are said to be hybrid entangled and have previously been

prepared between polarisation and path [14, 90], polarisation and OAM [91] and po-

larisation and time-bin [92]. The states can be prepared as maximally entangled

qubits, for example, in the polarisation and OAM basis they can be expressed as

⃒⃒
Ψℓ
⟩︀
𝐴𝐵

=
1√
2

(︀
|𝑅⟩𝐴 |ℓ⟩𝐵 + |𝐿⟩𝐴 |−ℓ⟩𝐵

)︀
, (1.38)

where photon A and B are defined in the circular polarisation (spin) and OAM bases,

respectively. When photon A is projected onto the state |𝑅⟩, photon B will collapse

onto the state |ℓ⟩, similarly, a projection of photon A onto the state |𝐿⟩, will result in

the collapse of photon B into the state |−ℓ⟩. This means that the measurements are

strongly correlated even though the individual photons are defined in independent

DOFs. Interestingly these exotic states have been used for fundamental tests of

quantum mechanics such as the complementary principle [93] through quantum eraser

experiments [14,94,95].

Interestingly, vector modes also demonstrate entanglement like correlations [96].

First notice the resemblance between a vector mode, e.g., in Eq. (1.24), and the

hybrid entangled state in Eq. (1.38). Both these states represent systems that have

19



Chapter 1 Isaac Nape 20

Figure 1-5: (a) A vectorial field propagates through a channel that distorts the spatial
amplitude, phase and polarisation field. The input field is mapped to the output field
having evolved as a result of the channel perturbation. (b) An entanglement source (ES)
creates two strongly correlated photons where one goes through free-space while the other
is perturbed. At the end of the channel, Alice and Bob perform correlation measurements
on the two bi-photon state.

nonseparable correlations with the discrepancy being in the nature of the correlations:

for hybrid entanglement, spatial separation is involved and therefore nonlocality plays

a crucial role in the nonseparable correlations, while for vector modes the nonsepa-

rable correlations are between the internal DOFs, locally, within the photon field.

Therefore, for vector modes, the labels A and B mark the internal DOFs of the same

photons and are therefore classically entangled [97, 98]. Today classical entangle-

ment though controversial, is finding applications to quantum walks [99,100], process

tomography of entangled channels [101] and metrology [102].

1.4 Quantum channels

In Fig. 1-5(a), we show a vector mode before and after it propagates through a

noisy channel. After the channel, the entire field is completely distorted. A cur-

rent challenge has been to mitigate such deleterious effects so that the quality of

photons signals is preserved after a quantum channel. Turbid [103–106] and turbu-

lent media [107–114] can distort the transverse spatial amplitude and phase profiles of

photons due to rapid variations in the refractive index profile. Similarly, birefringence

20



Chapter 1 Isaac Nape 21

(polarisation dependent refractive index) and imperfections in optical fibers [115,116]

are main pertubations that can rotate polarsation fields or cause inter-modal cou-

pling. Moreover, partially obstructed photons can also result in information loss due

to diffraction [26]. On the other hand, high dark count rates in detectors, stray

light and inefficient photon sources can reduce the purity of higher dimensional quan-

tum states [27] since these mechanism can introduce some degree of mixture to the

state [117].

To see the impact of quantum channels on quantum states, we adopt the opera-

tional definition for channel operators, ℰ (·), acting on an initial state 𝜌 [118]

ℰ (𝜌) =
∑︁
𝑖

𝐴𝑖𝜌𝐴
†
𝑖 , (1.39)

that we assume is completely positive and trace preserving (CPTP), meaning that the

operators 𝐴𝑖 satisfy,
∑︀

𝑖𝐴𝑖𝐴
†
𝑖 = 1. Here operators 𝐴𝑖 are called Krauss operators,

and can be further decomposed using a matrix basis {�̃�𝑘, Tr
(︁
�̃�𝑘�̃�𝑙

)︁
= 𝑑𝛿𝑘𝑙} as

𝐴𝑖 =
∑︀

𝑘 𝑎𝑖𝑘�̃�𝑘. Subsequently, the channel mapping can be rewritten as [119]

ℰ (𝜌) =
𝑑2−1∑︁
𝑘𝑙

�̃�𝑘𝜌�̃�
†
𝑙 𝜒𝑘𝑙. (1.40)

The matrix components, 𝜒𝑘𝑙 =
∑︀

𝑝𝑞 𝑎𝑘𝑝𝑎
*
𝑙𝑞, are entries of the positive Hermitian ma-

trix, 𝜒, that determines the channel in the {�̃�𝑘} basis.

There is a one-to-one correspondence between CPTP maps and specific density

matrices that the channels can map onto, i.e., ℰ → 𝜌ℰ , owing to the Choi–Jamio lkowski

isomorphism [120,121] or commonly referred to as channel-state duality [122]. To elu-

cidate this concept, let us consider a two photon state that is maximally entangled,

|Φ+⟩ = 1/
√
𝑑
∑︀𝑑−1

𝑗=0 |𝑗⟩ |𝑗⟩ and subsequently interacts with a quantum channel that

transforms it as

𝜌ℰ = 1⊗ ℰ
⃒⃒
Φ+
⟩︀ ⟨︀

Φ+
⃒⃒
. (1.41)

The operator 1⊗ℰ represents a single sided channel acting on a maximally entangled

state, where nothing happens to one photon (hence the identity, 1) while its twin

21



Chapter 1 Isaac Nape 22

interacts with the channel ℰ . The situation is illustrated in Fig. 1-5(b). Suppose the

channel can be represented by a unitary operator that is trace preserving, therefore

mapping the initial state onto another pure state, i.e. |𝑗⟩ →
∑︀

𝑖 𝑡𝑖𝑗 |𝑖⟩. For transverse

spatial modes, channels of this nature can be, for example, represented by atmospheric

turbulence [24], optical aberrations in underwater channels [123] or refractive index

imperfections in short optical fibers [124]. Accordingly, if we represented the channel

transmission matrix as 𝑇 =
∑︀

𝑖𝑗 𝑡𝑖𝑗 |𝑖⟩ ⟨𝑗|, the transformed two photon state is [125]

(1⊗ ℰ) |Φ⟩ =
1√
𝑑

𝑑−1∑︁
𝑖,𝑗=0

𝑡𝑖𝑗 |𝑗⟩ |𝑖⟩ , (1.42)

where the coefficients, relating to the channel are imprinted onto the final state,

and can be used to undo the effects of the channel [125]. Konrad et al. [126] also

revealed an intriguing aspect of entangled states that are subjected to single sided

channels. The authors, showed that the input and output degree of entanglement of

photons, concurrence [127], after a single sided channel only depends on the evolution

of maximally entangled states as depicted in the relation

𝐶(1⊗ ℰ |𝜓𝑖𝑛)⟩ = 𝐶 (𝜌ℰ) × 𝐶 (|𝜓𝑖𝑛⟩) , (1.43)

where 𝐶 (|𝜓𝑖𝑛⟩), is the concurrence of the input state |𝜓𝑖𝑛⟩ before traversing the

channel while 𝐶 (𝜌ℰ) is the concurrence of the maximally entangled state after the

channel ℰ . This means that the degree of entanglement of any input state |𝜓𝑖𝑛⟩ decays

proportional to a maximally entangled state. This finding has been confirmed through

numerical [128] and experimental [101] studies focusing on spatial mode entanglement

decay and characterisation through turbulence.

Another class of quantum channels that are encountered in practical settings, are

a type of depolarisation channel, that models the influence of external noise entering

the system and reducing the coherence of quantum states [27]. The growing interest

in these classes of states is due to the the potential resilience of high dimensional

quantum states to noise [34]. The state decomposition after such channels is given

22



Chapter 1 Isaac Nape 23

by

𝜌𝑝 = 𝑝 |Ψ𝑑⟩ ⟨Ψ𝑑| +
1 − 𝑝

𝑑2
1𝑑2 , (1.44)

where |Ψ𝑑⟩ is the transmitted high dimensional entangled state, and 1𝑑2 is the identity

operator containing the noise contribution to the state. Here 𝑝 can be associated

with the purity of the state, ranging from a maximally mixed (𝑝 = 0) to a pure state

(𝑝 = 1). Interestingly, the isotropic state is separable for 𝑝 ≤ 1/(𝑑+ 1) and entangled

otherwise. It is important to have knowledge of 𝑝 and 𝑑 since the quality of the

generated state depends on them. For example, in generalised high dimensional Bell

inequality tests, non-local correlations can only be confirmed when 𝑝 > 2/𝑆𝑑 where

𝑆𝑑 is the Bell parameter [129].

Standard procedures for overcoming some of these perturbations includes quantum

error correction methods [125,130,131], that require full knowledge of how the states

evolve through the channel, often requiring a process tomography (reconstruction of

the channel operator) to determine all 𝑑4 components of the channel operator [132],

e.g. in Eq. 1.40.

23



Chapter 1 Isaac Nape 24

1.5 Outline

This thesis will explore several approaches for engineering robust photon states that

span higher dimensional Hilbert spaces by either manipulating their intrinsic prop-

erties to make them immune to malignant effects of perturbative channels, or by

tailoring efficient characterisation methods that yield important parameters about

the state after the perturbation. We draw attention to the spatial (and sometimes

combined polarisation) DOF both locally and nonlocaly with entangled states. Our

overarching goals are executed in five subsequent chapters as follows:

In Chapter 2, we introduce our first perturbation, solid obstructions, that are

diffractive, mimicking dust particles in the air, and show that we can tailor the

radial profile so that the generated photons can be used for quantum key distribu-

tion (QKD) in the presence of obstructions. The photon states that we tailor are

self-reconstructing vector modes that are modulated with a non-diffracting Bessel-

Gaussian (BG) envelope. Using a prepare-measure (BB84) protocol, we demonstrate

that these mode fields offer higher information capacity after solid diffractive objects

in comparison to Laguerre-Gaussian modes.

In Chapter 3, we introduce our second perturbation, optical turbulence. Here, we

will show that a single maximally nonseparable vector mode is sufficient to predict

the behaviour of arbitrary vector OAM states through a unitary channel, showing

interesting features about the performance of various modes through the channel

such as the decay dynamics for different mode orders. We will invoke channel state

duality in our demonstration, a quantum tool, to devise our approach. This method

illustrates the benefit of applying quantum tools to the study of coherent laser light

and single photon states with nonseparable (equivalently entangled) internal DOFs.

In Chapter 4, we build onto the work from chapter three by considering a hidden

feature of vector modes, namely their invariance to unitary channels, thanks to their

entanglement like properties. Here, we introduce a third family of pertubations,

i.e. optical aberrations, that can emanate from, element misalignment, or dynamic

processes during propagation in transparent media with a spatially varying refractive

24



Chapter 1 Isaac Nape 25

index profile and are unitary in the spatial DOF. We will show that the unitary nature

of the channel makes vector fields immune to the perturbations and that the channel

operation can easily be undone. Moreover, the work will address prior contradictions

in the community on the robustness of vectorial photon fields in both quantum and

classical channels.

In Chapter 5, we demonstrate a novel approach for transmitting multiple entangled

states through a heterogeneous free-space and fiber quantum channel. We engineer

hybrid entanglement non-locally, between two photons that are spatially separated.

Each photon will be described in an independent DOF; the polarisation (spin) state

of one photon will be entangled to the OAM state of its twin photon. Enabling for

the polarised photon to be transmitted through a long distance fiber channel while

the high dimensional OAM photon will be transmitted through free-space. Since the

photon that carries the spatial mode can take on any OAM state, we will demonstrate

that the channel can transmit multiple hybrid entangled states subspaces through the

same conventional single mode fibre.

Lastly, in Chapter 6, we will encounter our last channel, a noisy quantum entangled

two photon channel that is encroached in white noise, known to affect the purity and

dimensionlity of quantum entangled states. The focus will be on devising a technique

that can characterise the effective dimensions and purity of entangled states defined

on large Hilbert spaces. A set of conditional measurements that return a visibility

that scales monotonically with state dimensionality and purity will be constructed to

demonstrate the versatility of the approach, we will showcase it with two bases, the

OAM and pixel bases, showing that it works over a wide range of noise levels.

25



Chapter 2

Self healing quantum

communication through

obstructions

The work in this chapter was published in:

� Nape, I., Otte, E., Vallés, A., Rosales-Guzmán, C., Cardano, F., Denz, C.

and Forbes, A., 2018. “Self-healing high-dimensional quantum key distribution

using hybrid spin-orbit Bessel states.”, Optics express, 26(21), pp.26946-26960.

Nape, I., lead the experiments with the second author, analysed the data and con-

tributed to writing the manuscript under the guidance of the contributing coauthors.

26



Chapter 2 Isaac Nape 27

2.1 Introduction

Quantum key distribution (QKD) enables two parties to securely exchange informa-

tion detecting the presence of eavesdropping [133]. Unlike conventional cryptogra-

phy, with unproven computational assumptions, the security of QKD relies on the

fundamental laws of quantum mechanics [134], prohibiting the cloning of quantum

information encoded in single photons [4]. Although current state of the art imple-

mentations have successfully transfered quantum states in free-space [135], optical

fibers [136], and between satellites [137], efficient high capacity key generation and

robust security are still highly sought-after.

Spatial modes of light hold significant promise in addressing these issues. The

channel capacity can be exponentially increased by encoding information in the spa-

tial degree of freedom (DoF) of photons and has been demonstrated with classical

light in free-space and fibres [138]. Implementing QKD with high-dimensional (HD)

states (𝑑 > 2) has also been demonstrated [20, 139], by exploiting the ability of each

photon to carry up to log2(𝑑) bits per photon while simultaneously increasing the

threshold of the quantum bit error rate (QBER). This makes HD QKD protocols

more robust [29–31], even when considering extreme perturbing conditions, i.e., un-

derwater submarine communication links [123]. While most studies to date have used

spatial modes of light carrying orbital angular momentum (OAM) [140], reaching up

to 𝑑 = 7 [21], higher dimensions are achievable with coupled spatial and polarization

structures, e.g. vector modes. These states have received recent attention in classical

communication [141–144], in the quantum realm as a means of implementing QKD

without a reference frame [145, 146], but only recently have both DoFs been used to

increase dimensionality in QKD [147,148].

To date, there has been only limited work on the impact of perturbations on HD

entanglement and QKD with spatial modes [148–152]. In turbulence, for example,

the key rates are known to decrease [153], with the latter to be compensated for large

OAM states in the superposition. There has been no study on HD QKD through

physical obstacles.

27



Chapter 2 Isaac Nape 28

In this chapter, we take advantage of the self-healing properties in non-diffracting

vector beams to show that the bit rate of a QKD channel, affected by partial obstruc-

tions, can be ameliorated by encoding information onto diffraction-free single photons.

To this end, we generate a non-diffracting (self-reconstructing) set of mutually unbi-

ased bases (MUB), formed by hybrid scalar and vector modes with a Bessel-Gaussian

(BG) transverse profile. We herald a single photon with a BG radial profile by means

of spontaneous parametric down-conversion (SPDC), generating paired photons and

coupling OAM and polarization using a 𝑞-plate [53]. We characterize the quantum link

by measuring the scattering probabilities, mutual information and secret key rates in

a prepare-measure protocol for BG and Laguerre-Gaussian (LG) photons, comparing

the two for various obstacle sizes. We find that the BG modes outperform LG modes

for larger obstructions by more than 3×, highlighting the importance of radial mode

control of single photons for quantum information processing and communication.

2.2 Self-healing Bessel modes

Since Bessel modes cannot be realized experimentally, a valid approximation, the

Bessel-Gaussian (BG) mode, is commonly used [154]. This approximation inherits

from the Bessel modes the ability to self-reconstruct in amplitude, phase [155, 156],

and polarization [157–159], even when considering entangled photon pairs [152] or

non-separable vector modes [160,161]. Mathematically, they are described by

𝒥ℓ,𝑘𝑟(𝑟, 𝜙, 𝑧) =

√︂
2

𝜋
𝐽ℓ

(︂
𝑧𝑅𝑘𝑟𝑟

𝑧𝑅 − i𝑧

)︂
exp (iℓ𝜙− i𝑘𝑧𝑧)

· exp

(︂
i𝑘2𝑟𝑧𝑤0 − 2𝑘𝑟2

4(𝑧𝑅 − i𝑧)

)︂
, (2.1)

where (𝑟, 𝜙, 𝑧) represents the position vector in the cylindrical coordinates, ℓ is the

azimuthal index (topological charge). Furthermore, 𝐽ℓ(·) defines a Bessel function

of the first kind , 𝑘𝑟 and 𝑘𝑧 are the radial and longitudinal components of the wave

number 𝑘 =
√︀
𝑘2𝑟 + 𝑘2𝑧 = 2𝜋/𝜆. The last factor describes the Gaussian envelope with

beam waist 𝑤0 and Rayleigh range 𝑧𝑅 = 𝜋𝑤2
0/𝜆 for a certain wavelength 𝜆.

28



Chapter 2 Isaac Nape 29

The propagation distance over which the BG modes approximate a non-diffracting

mode is given by 𝑧max = 2𝜋𝑤0/𝜆𝑘𝑟. In the presence of an obstruction of radius 𝑅 in-

serted within the non-diffracting distance, a shadow region of length 𝑧min ≈ 2𝜋𝑅/𝑘𝑟𝜆

is formed [162]. The distance 𝑧min determines the minimum distance required for

the beam to recover its original form, whereby full reconstruction is achieved at

2𝑧min [155,156].

We exploit this property with single photons that have non-separable polarization

and OAM DoFs. By carefully selecting a 𝑘𝑟 value, we show that the information of

hybrid entangled single photon encoded with a Bessel radial profile can be recovered

after the shadow region of an obstruction. Traditionally hybrid modes, while still new

in the communication context, have not been controlled in radial profile. Indeed, the

traditional generation approaches often result in very complex radial structures [163].

To control and exploit all spatial and the polarization DoFs for QKD we introduce

a high-dimensional self-healing information basis constructed from non-orthogonal

vector and scalar OAM BG spatial modes.

2.2.1 Non-diffracting information basis

In order to demonstrate the concept we will use the well-known BB84 protocol, but

stress that this may be replaced with more modern and advantageous protocols with

little change to the core idea as outlined here. In the standard BB84 protocol, Alice

and Bob unanimously agree on two information basis. The first basis can be arbitrar-

ily chosen in 𝑑 dimensions as {|Ψ𝑖⟩ , 𝑖 = 1..𝑑}. However, the second basis must fulfill

the condition

| ⟨Ψ𝑖|Φ𝑗⟩ |2 =
1

𝑑
, (2.2)

making |Ψ⟩ and |Φ⟩ mutually unbiased. Various QKD protocols were first imple-

mented using polarization states, spanned by the canonical right |𝑅⟩ and left |𝐿⟩

circular polarization states constituting a two-dimensional Hilbert space, i.e., ℋ𝜎 =

span{|𝐿⟩ , |𝑅⟩}. More dimensions where later realized with the spatial DoF of pho-

29



Chapter 2 Isaac Nape 30

(c) (d)

|Φ⟩00

(a) (b)

0

2𝜋
0

1

|Φ⟩01

|Φ⟩10

|Φ⟩11

|Ψ⟩00

|Ψ⟩01

|Ψ⟩10

|Ψ⟩11

Figure 2-1: Intensity and polarization mappings of vector (first row) and scalar (second
row) MUB modes with (a) BG and (b) LG radial profiles for ℓ = ±1. The polarization
projections on the (c) vector |Ψ⟩ and (d) scalar |Φ⟩ basis BG modes. The vector modes
have spatially varying polarizations which consequently render the polarization and spatial
DoF as non-separable. This is easily seen in the variation of the transverse spatial profile
when polarization projections are performed (orientation indicated by white arrow) on the
|Ψ⟩ modes. In contrast, the scalar modes have separable polarization and spatial DoF hence
polarization projections only cause fluctuations in the intensity of the transverse profile for
the |Φ⟩ modes.

tons [20,21], using the OAM DoF spanning the infinite dimensional space, i.e. ℋ∞ =⨁︀
ℋℓ, such that ℋℓ = {|ℓ⟩ , |−ℓ⟩} is qubit space characterized by a topological charge

ℓ ∈ Z.

Here, we exploit an even larger encoding state space by combining polarization

and OAM, ℋ∞ =
⨁︀

ℋ𝜎 ⊗ℋℓ where ℋ4 = ℋ𝜎 ⊗ℋℓ, is a qu-quart space spanned by

the states {|𝐿⟩ |ℓ⟩ , |𝑅⟩ |ℓ⟩ , |𝐿⟩ |−ℓ⟩ , |𝑅⟩ |−ℓ⟩}, described by the so-called higher-order

Poincaré spheres (HOPSs) [164, 165]. These modes feature a coupling between the

polarization and OAM DoFs, shown in Fig. 2-1. The HOPS concept neglects the

radial structure of the modes, considering only the angular momentum content, spin

30



Chapter 2 Isaac Nape 31

and orbital. Yet all modes have radial structure, shown in Fig. 2-1 (a) for BG and

(b) for LG profiles. We wish to create a basis of orthogonal non-separable vector BG

modes together with their MUBs for our single photon states.

Without loss of generality, we choose a mode basis on the ℋ4 subspace with ℓ = ±1

as our example. Our encoding basis is constructed as follows: we define the radial

profile 𝒥ℓ,𝑘𝑟(𝑟) representing the radial component of the BG mode in Eq. (2.1). Our

first mode set is comprised of a self-healing vector BG mode basis, mapped as

|Ψ⟩00 =
1√
2
𝒥ℓ,𝑘𝑟(𝑟)

(︀
|𝑅⟩ |ℓ⟩ + |𝐿⟩ |−ℓ⟩

)︀
, (2.3)

|Ψ⟩01 =
1√
2
𝒥ℓ,𝑘𝑟(𝑟)

(︀
|𝑅⟩ |ℓ⟩ − |𝐿⟩ |−ℓ⟩

)︀
, (2.4)

|Ψ⟩10 =
1√
2
𝒥ℓ,𝑘𝑟(𝑟)

(︀
|𝐿⟩ |ℓ⟩ + |𝑅⟩ |−ℓ⟩

)︀
, (2.5)

|Ψ⟩11 =
1√
2
𝒥ℓ,𝑘𝑟(𝑟)

(︀
|𝐿⟩ |ℓ⟩ − |𝑅⟩ |−ℓ⟩

)︀
, (2.6)

with some example polarization projections shown in Fig. 2-1 (c). The set of MUB

modes is given by

|Φ⟩00 = 𝒥ℓ,𝑘𝑟(𝑟) |𝐷⟩ |−ℓ⟩ , (2.7)

|Φ⟩01 = 𝒥ℓ,𝑘𝑟(𝑟) |𝐷⟩ |ℓ⟩ , (2.8)

|Φ⟩10 = 𝒥ℓ,𝑘𝑟(𝑟) |𝐴⟩ |−ℓ⟩ , (2.9)

|Φ⟩11 = 𝒥ℓ,𝑘𝑟(𝑟) |𝐴⟩ |ℓ⟩ , (2.10)

where 𝐷 and 𝐴 are the diagonal and anti-diagonal polarization states (see Fig. 2-1

(d) for polarization projections). The set |Ψ⟩𝑖𝑗 and |Φ⟩𝑖𝑗 are mutually unbiased and,

therefore, form a reputable information basis for QKD in high dimensions.

As a point of comparison to the self-healing properties of the non-diffracting

modes, we make use also of a similar alphabet but projecting the heralding photon

onto a Gaussian mode, obtaining a helical mode in the other photon after traversing

31



Chapter 2 Isaac Nape 32

a spin-to-orbital angular momentum converter [53]. We will refer to this as a LG

mode in the remainder of the manuscript.

2.3 Methods

2.3.1 Single photon heralding

Hologram

i
SLM

SPDC

SLM
s

0
2𝜋

O
P M

C.C

𝐽𝜆
2
(𝛼1) �̂� 𝐽𝜆

2
(𝛼2) 𝐽𝜆

4
(𝛽1) �̂� 𝐽𝜆

4
(𝛽2)

V
ec

to
r

Sc
al

ar

Vector
Scalar

P
re
p
ar
e

Measure

0

1

Vector Scalar

P
ro

b
al

il
it

y

(a) (b)

(c)

𝐿

Figure 2-2: (a) Conceptual drawing of the QKD with self-healing BG modes. The SLMs
post-select the self-healing BG radial profile from the SPDC source. The prepare (P)
and measure (M) optics modulate and demodulate the OAM and polarization DoF of the
heralded photon. The physical obstruction (O) is placed at a distance 𝐿 from the right-
most SLM, which decodes the radial information of Bob’s photon. The optics are within
𝑧max = 54 cm distance of the BG modes depicted as the rhombus shape. The propagation
of the post-selected BG mode can be determined via back-projection. (b) Numerical scat-
tering probability matrix for the vector and scalar modes sets in free-space. The channels
correspond to the probabilities |𝐶𝑖𝑗 |2 calculated from Eq. (2.22). (c) Optical elements re-
quired by Alice and Bob to prepare and measure the spin-coupled states of the heralded
photons (cf. Table 2.1).

Heralded photon sources have been used as a means of producing single photons

in QKD [166]. In this process, the heralded photon conditions the existence of its

correlated twin. Moreover, the statistics of the heralded photon have low multi-photon

probabilities which can be further remedied by using decoy states [167].

Here, we herald a single photon via SPDC where a high frequency photon (𝜆 =

32



Chapter 2 Isaac Nape 33

405 nm) was absorbed by a nonlinear crystal, generating a signal (𝑠) and idler (𝑖)

correlated paired photons at 𝜆 = 810 nm. In the case of a collinear emission of 𝑠 and

𝑖, the probability amplitude of detecting mode functions |𝑚⟩𝑠 and |𝑚⟩𝑖, respectively,

is given by [168]

𝑐𝑠,𝑖 =

∫︁ ∫︁
𝑚*
𝑠(x)𝑚*

𝑖 (x)𝑚𝑝(x)𝑑2𝑥, (2.11)

where 𝑚𝑝(x) is the field profile of the pump (𝑝) beam which best approximates the

phase-matching condition in the thin crystal limit; the Rayleigh range of the pump

beam is much larger than the crystal length. The probabilities amplitudes 𝑐𝑠,𝑖 can be

calculated using the Bessel basis,

𝑚𝑠,𝑖(𝑟, 𝜙) = 𝒥ℓ𝑠,𝑖,𝑘𝑟(𝑟) exp(iℓ𝑠,𝑖𝜙), (2.12)

where exp(iℓ𝜙) corresponds to the characteristic azimuthal phase mapping onto the

state vector |ℓ⟩. Taking into account a SPDC type-I process and a Gaussian pump

beam, the quantum state used to encode and decode the shared key can be written

in the Bessel basis as

|Ψ⟩𝐴𝐵 =
∑︁

𝑐ℓ,𝑘𝑟,1,𝑘𝑟,2 |ℓ, 𝑘𝑟,1⟩𝑠 |−ℓ, 𝑘𝑟,2⟩𝑖 |𝐻⟩𝑠 |𝐻⟩𝑖 , (2.13)

being |ℓ, 𝑘𝑟⟩𝑠 ∼ 𝐽ℓ,𝑘𝑟(𝑟) |ℓ⟩ and 𝐻 the horizontal polarization state. The probability

amplitudes 𝑐ℓ,𝑘𝑟,1,𝑘𝑟,2 can be calculated using the overlap integral in Eq. (2.11). Ex-

perimentally |𝑐ℓ,𝑘𝑟,1,𝑘𝑟,2|2 is proportional to the probability of detecting a coincidence

when the state |ℓ, 𝑘𝑟,1⟩𝑠 |−ℓ, 𝑘𝑟,2⟩𝑖 is selected. Coincidences are optimal when |𝑘𝑟,1|

and |𝑘𝑟,2| are equivalent.

In this experiment, the idler photon (𝑖) was projected into the state |0, 𝑘𝑟⟩𝑖, herald-

ing only the signal photons (𝑠) with the same spatial state |0, 𝑘𝑟⟩𝑠, as can be seen

in the sketch of Fig. 2-2(a). Therefore, a prepare-measure protocol can be carried

out by using the same 𝑠 photon. In other-words, Alice remotely prepared her single

photon with a desired radial profile from the SPDC before encoding the polarization

33



Chapter 2 Isaac Nape 34

and OAM information.

2.3.2 Tailoring the desired spatial profile

Spatial light modulators (SLMs) are a ubiquitous tool for generating and detecting

spatial modes [52, 169]. We exploit their on-demand dynamic modulation via com-

puter generated holograms to post-select the spatial profiles of our desired modes (see

hologram inset in Fig. 2-2(a)). For the detection of BG modes, we choose a binary

Bessel function as phase-only hologram, defined by the transmission function

𝑇 (𝑟, 𝜙) = sign{𝐽ℓ(𝑘𝑟𝑟)} exp(iℓ𝜙), (2.14)

with the sign function sign{·} [170,171]. Classically, this approach has the advantage

of generating a BG beam immediately after the SLM and, reciprocally, detects the

mode efficiently [152]. Importantly, a blazed grating is used to encode the hologram,

with the desired mode being detected in the first diffraction order [172] and spatial

filtered with a single mode fiber (SMF).

Here, we set 𝑘𝑟 = 18 rad/mm and ℓ = 0 for the fundamental Bessel mode and,

conversely, 𝑘𝑟 = 0 to eliminate the multi-ringed Bessel structure.

2.3.3 Generation and detection

Liquid crystals 𝑞-plates represent a convenient and versatile way to engineer several

types of vector beams [67]. In our setup, vector and scalar modes, described in Fig.

2-1, are either generated or detected, at Alice and Bob’s prepare (P) and measure

(M) stations in Fig. 2-2 (a), by letting signal photons pass through a combination

of these devices and standard wave plates (see Fig. 2-2 (c)). A 𝑞-plate consists of a

thin layer of liquid crystals (sandwiched between glass plates) whose optic axes are

arranged so that they form a singular pattern with topological charge 𝑞. By adjusting

the voltage applied to the plate it is possible to tune its retardation to the optimal

value 𝛿 = 𝜋 [173]. In such a configuration indeed the plate behaves like a standard

half-wave plate (with an inhomogeneous orientation of its fast axis) and can be used

34



Chapter 2 Isaac Nape 35

to change the OAM of circularly polarized light by ±2𝑞, depending on the associated

handedness being left or right, respectively. As such, the 𝑞-plate is used to achieve

spin orbit coupling. In the Jones matrix formalism, the 𝑞-plate is represented by the

operator

�̂� =

⎛⎝cos(2𝑞𝜙) sin(2𝑞𝜙)

sin(2𝑞𝜙) −cos(2𝑞𝜙)

⎞⎠ , (2.15)

where 𝜙 is the azimuthal coordinate. The matrix is then written in the following

linear basis {|𝐻⟩ =

⎛⎝1

0

⎞⎠ , |𝑉 ⟩ =

⎛⎝0

1

⎞⎠}. In our experiment we use 𝑞-plates with

𝑞 = 1/2, and half-wave (𝜆
2
) as well as quarter-wave (𝜆

4
) plates for polarization control,

represented by the Jones matrices

𝐽𝜆
2
(𝜃) =

⎛⎝cos(2𝜃) sin(2𝜃)

sin(2𝜃) −cos(2𝜃)

⎞⎠ , (2.16)

and

𝐽𝜆
4
(𝜃) =

⎛⎝ cos2(𝜃) + isin2(𝜃) (1 − i) sin(𝜃)cos(𝜃)

(1 − i) sin(𝜃)cos(𝜃) sin2(𝜃) + icos2(𝜃)

⎞⎠ . (2.17)

Here, 𝜃 represents the rotation angle of the wave plates fast axis with respect to the

horizontal polarization. The operator associated with the generation of the vector

mode is

𝑉 (𝛼1, 𝛼2) = 𝐽𝜆
2
(𝛼2)�̂�𝐽𝜆

2
(𝛼1)𝑃𝐻 , (2.18)

where 𝛼1 and 𝛼2 are the rotation angles for the half-wave plates and 𝑃𝐻 =

⎛⎝1 0

0 0

⎞⎠
represents the operator for a horizontal linear polarizer. Similarly, the operator for

35



Chapter 2 Isaac Nape 36

the scalar modes is

𝑆(𝛽1, 𝛽2) = 𝐽𝜆
4
(𝛽2)�̂�𝐽𝜆

4
(𝛽1)𝑃𝐻 , (2.19)

where 𝛽1 and 𝛽2 are the rotation angles for the quarter-wave plates.

Let the set ℳ1 = {𝑉𝑖 |𝑉𝑖 → |Ψ𝑖⟩ , 𝑖 = 1..4} be associated with the generation of

vector modes from 𝑉 (𝛼1, 𝛼2), and ℳ2 = {𝑆𝑗 |𝑆𝑗 → |Φ𝑗⟩ , 𝑗 = 1..4} for the scalar

modes from 𝑆(𝛽1, 𝛽2). The orientation of the angles required to obtain them is given

in Table 2.1 for the vector and scalar modes (see also schematics of wave plates

arrangement in Fig. 2-2 (c)).

Table 2.1: Generation of vector and scalar modes from a horizontally polarized BG mode
(ℓ = 0) at the input. The angles 𝛼1,2 and 𝛽1,2 are defined with respect to the horizontal
polarization. For each 𝑉𝑖 and 𝑆𝑖 we present the angles needed to perform the mapping of
ℳ1 → {|Ψ𝑖⟩} and ℳ2 → {|Φ𝑖⟩} with a one-to-one correspondence.

Vector, 𝑉 (𝛼1, 𝛼2) Scalar, 𝑆(𝛽1, 𝛽2)

Operator 𝐽𝜆
2
(𝛼1) 𝐽𝜆

2
(𝛼2) Operator 𝐽𝜆

4
(𝛽1) 𝐽𝜆

4
(𝛽2)

𝑉1 0 – 𝑆1 −𝜋/4 0

𝑉2 𝜋/4 – 𝑆2 𝜋/4 𝜋/2

𝑉3 0 0 𝑆3 −𝜋/4 𝜋/2

𝑉4 𝜋/4 0 𝑆4 𝜋/4 0

2.3.4 Scattering probability

Let 𝐴𝑖, �̂�𝑗 ∈ ℳ1 ∪ℳ2 represent operators selected by Alice and Bob, respectively.

Alice first obtains a heralded pshoton from the SPDC with the input state |𝜓in⟩ =

𝒥0,𝑘𝑟 |𝐻⟩. Then, Alice prepares the photon in a desired state from the MUB with

|𝑎𝑖⟩ = 𝐴𝑖𝒥0,𝑘𝑟(𝑟) |𝐻⟩ , (2.20)

and Bob similarly measures the state

|𝑏𝑗⟩ = �̂�𝑗𝒥0,𝑘𝑟(𝑟) |𝐻⟩ . (2.21)

36



Chapter 2 Isaac Nape 37

BS f2
f1

f3

f3

f4
f4

D1

P
M

PumpPPKTP

C.C.D2O SLM

SMF

BPF

Figure 2-3: Experimental setup for the self-healing QKD. Pump: 𝜆 = 405 nm (Cobalt, MLD
laser diode); f: Fourier lenses of focal length f1,2,3&4 = 100 mm, 750 mm, 500 mm, 2 mm,
respectively; PPKTP: periodically poled potassium titanyl phosphate (nonlinear crystal);
BS: 50:50 beam splitter; s and i: signal and idler photon paths; P: preparation of the state
(Alice); O: variable sized obstacle; M: measurement of the state (Bob); SLM: spatial light
modulator (Pluto, Holoeye); BPF: band-pass filter; SMF: single mode fiber; D1&2: single
photon detectors (Perkin Elmer); C.C.: coincidence electronics.

The probability amplitude of Bob’s detection is

𝐶𝑖𝑗 = ⟨𝑏𝑗|𝑎𝑖⟩ =

∫︁ 2𝜋

0

∫︁ ∞

0

⟨𝐻| 𝒥 *
0,𝑘𝑟(𝑟)�̂�

†
𝑗𝐴𝑖𝒥0,𝑘𝑟(𝑟) |𝐻⟩ 𝑟𝑑𝑟𝑑𝜑, (2.22)

while the corresponding detection probabilities, |𝐶𝑖𝑗|2, are presented in Fig. 2-2 (b).

2.4 Experimental set-up

Figure 2-3 is a schematic representation of our experimental setup. The continuous-

wave pump laser (Cobalt MLD diode laser, 𝜆 = 405 nm) was spatially filtered to

deliver 40 mW of average power in a Gaussian beam of 𝑤0 ≈ 170 𝜇m at the crystal

(2-mm-long PPKTP nonlinear crystal), generating two lower-frequency photons by

means of a type-I spontaneous parametric down-conversion (SPDC) process. By

virtue of this, the signal and idler photons had the same wavelength (𝜆 = 810 nm)

and polarization (horizontal).

The two correlated photons, signal and idler, were spatially separated by a 50:50

beam splitter (BS), with the idler photon projected into a Bessel state of 0 OAM,

thus heralding a zero-order Bessel photon in the signal arm for the prepare-measure

BB84 protocol. The signal photon traversed the preparation stage (P) where Alice

37



Chapter 2 Isaac Nape 38

could prepare a vector or scalar state from the MUB alphabet using elements detailed

in Fig. 2-2 (c). The signal photon was then propagated in free-space with an obstacle

of variable size placed within the non-diffracting distance. This mimics a line-of-sight

quantum channel. In our experiment we used the spatial light modulators (SLMs)

to post-select a wave number of 𝑘𝑟 = 18 rad/mm, thus realising a non-diffracting

distance of 𝑧max = 54 cm. These values where verified by classical back-projection

through the system [87]. The state measurement (M) was implemented after the

obstacle by Bob. The SLM acted both as a horizontal polarization filter and as a

post-selecting filter for the radial wave number. To conclude the heralding experi-

ment, both photons were spectrally filtered by band-pass filters (10 nm bandwidth

at full-width at half-maximum) and coupled with single mode fibers to single photon

detectors (D1&2; Perkin-Elmer), with the output pulses synchronized with a coinci-

dence counter (C.C.), discarding also the cases where the two photons exit the same

output port from the BS.

2.4.1 Procedure and analysis

We measured the scattering matrix for the BG and, for comparison reasons, the

LG profiles under three conditions: (FS) in free-space; (R1) with a 600 𝜇m radius

obstruction placed strategically such that the complete decoding is performed after

𝐿 > 𝑧min (𝐿: distance between obstruction and decoding SLM); and (R2) with a

800 𝜇m radius obstruction, placed at the same position. In the (R2) the shadow

region overlaps the detection system (𝐿 < 𝑧min) so that the mode is not able to self-

reconstruct completely before being detected. We measure the quantum bit error rate

(QBER) in each of these cases and computed the mutual information between Alice

and Bob in 𝑑 = 4 dimensions by [29]

𝐼𝐴𝐵 = log2(𝑑) + (1 − 𝑒) log2(1 − 𝑒) + (𝑒) log2

(︂
𝑒

𝑑− 1

)︂
. (2.23)

Here, 𝑒 denotes the QBER. Lastly, we measured the practical secure key rate per

signal state emitted by Alice, using the Gottesman-Lo-Lütkenhaus-Preskill (GLLP)

38



Chapter 2 Isaac Nape 39

Figure 2-4: (a) Measured photon count rates and (b) average photon number (𝜇) per-gating
window of 25 ns in free-space (FS) and the two obstructions (R1 = 600 𝜇m and R2 = 800
𝜇m) for the radially polarized mode |𝜓⟩00. (c) and (d) show coincidence rates with the same
obstructions for the BG and LG radial profiles, respectively. The BG count rate is lower
for smaller obstructions due to the high 𝑘𝑟 hologram on the SLM [87].

method [174,175] for practical implementations with BB84 states, given by

𝑅Δ = 𝑄𝜇

(︂
(1 − ∆)

(︂
1 −𝐻𝑑

(︂
𝑒

1 − ∆

)︂)︂
− 𝑓EC𝐻𝑑(𝑒)

)︂
, (2.24)

where 𝐻𝑑(·) is the high-dimensional Shannon entropy and 𝑓EC is a factor that accounts

for error correction and is nominally 𝑓EC = 1.2 for error correction systems that are

currently in practice.

The photon gain is defined as 𝑄𝜇 =
∑︀

𝑛 𝑌𝑛𝑃𝑛(𝜇) (in the orders of 10−4 for our

experiment), where 𝑌𝑛 is the 𝑛-th photon yield while 𝑃𝑛 is the probability distribution

over 𝑛 with respect to the average photon number 𝜇, following sub-Poisson statistics

for heralded photons produced from a SPDC source [175]. 𝑌𝑛 can be calculated from

the background rate, 𝑝𝐷 = 2.5×10−6 photons per gating window (25 ns), and 𝑛-signal

detection efficiency 𝜂𝑛:

𝑌𝑛 = 𝜂𝑛 + 𝑝𝐷(1 − 𝜂𝑛), (2.25)

39



Chapter 2 Isaac Nape 40

where the 𝑛-signal detection efficiency 𝜂𝑛 is given by

𝜂𝑛 = 1 − (1 − 𝜂)𝑛. (2.26)

Here 𝜂 = 𝜂𝑑𝑡𝐵 is the transmission probability of each photon state with 𝜂 = 0.45×0.8

for Bob’s detection (when accounting for the SLM grating). Furthermore, ∆ is the

multi-photon rate computed as (1 − 𝑃0 − 𝑃1)/𝑄𝜇 [175] where 𝑃0,1 are the vacuum

and single photon emission probabilities, respectively. The term (1−∆) accounts for

photon splitting attacks [175]. In our experiment, we measured the photon intensities

for every obstruction from the photon detection rates of the obstructed photon and

deduced 𝑃1 and 𝑃0 assuming a thermal statistics of the heralded photon. We point

out that it may be necessary to implement decoy states with a heralded source to

ensure security against multi photon states owing to the thermal nature of the reduced

photon state of SPDC correlated pairs [166,175].

2.5 Results and Discussion

We performed the aforementioned experiment in four dimensions using heralded single

photons with either a heralded LG mode or BG mode for the radial spatial profile,

and compare their performance under the influence of varying sized obstructions.

2.5.1 Experimental results

The photon count-rates, mean-photon counts (per gating window) and coincidence-

rates are presented in Fig. 2-4 (a) and (b), for the |Ψ⟩00 input state. As shown, the

photon count rates decay for both the BG and LG radial profiles, however, more so

for LG profile under the R2 obstruction. The coincidences rates are recovered for

the BG mode (Fig. 2-4 (c)) under the R1 obstruction since L> 𝑧min (detection is

performed outside the shadow region of the obstruction). Further, the BG mode still

demonstrates less decay for R2 obstructed even when the mode has not reconstructed

(since 𝐿 < 𝑧min), as compared to LG (Fig. 2-4 (d)), where the coincidence rate is

40



Chapter 2 Isaac Nape 41

Figure 2-5: Crosstalk (scattering) matrix for vector and scalar modes in (a) (I) free-space
having post-selected in a BG radial profile. The vector and scalar measured probabilities
with the first obstruction (II) having a radius R1 = 600 𝜇m (𝐿 > 𝑧min) when taking into
account (b) BG and (c) LG radial profiles. Measured probabilities with (III) an obstruction
of R2 = 800 𝜇m (𝐿 < 𝑧min) when taking into account (d) BG and (e) LG radially profiled
single photons.

seen to completely decay.

Next, we present the measured detection probability matrices for three tested

cases in Fig. 2-5 using our high-dimensional information basis. In the free-space case,

we measure QBERs of 𝑒 = 0.04±0.004 for the BG and LG spatial profiles (see Fig. 2-5

(a) and Table 2.2). We compute a mutual information of I𝐴𝐵 = 1.69 bits/photon and

a secure key rate of 𝑅Δ/𝑄𝜇 = 1.32 bits/s per photon for both radial profiles.

Under the perturbation of the R1 = 600 𝜇m obstruction (0.53× the beam waist of

the down converted photon), we measure a QBER of 𝑒 = 0.05 for both spatial profiles,

indicative of information retention, i.e. high fidelity. The intensity fields from the

back-projected classical beam (see insets of Fig. 2-5 (b) and (c)), show self-healing of

the BG mode at the SLM plane (see Fig. 2-5 (b)), although the LG is not completely

reconstructed (see Fig. 2-5 (c)). The photons encoded with the LG profile may have

a large component of the input mode which is undisturbed in polarization and phase.

Furthermore, the coincidence counts decreases to 49% for the LG profile relative to

the counts in free-space, as highlighted in Fig. 2-6 (a). In comparison, the BG modes

show resilience thanks to the multiple concentric rings [176].

41



Chapter 2 Isaac Nape 42

N
C

co
u

n
t-

ra
te 1

0

0.5

BG

LG
FS

R1

R2

(a)

BG LG
0

0.5

1

BG LG
0

1

2

BG LG
0

1

2

(b)

Q
B

E
R

I 𝐴
𝐵

𝑅
Δ
/𝑄

𝜇

FS R1 R2

Figure 2-6: (a) Experimental normalized coincidence (NC) count-rate for the BG and LG
MUB for free-space (FS) and the two obstructions (R1 = 600 𝜇m and R2 = 800 𝜇m) on
the radially polarized mode |𝜓⟩00. (b) The QBER, mutual information (𝐼𝐴𝐵) and key
rate (𝑅Δ/𝑄𝜇) for the BG and LG modes with no perturbation and under the two tested
obstructions are shown.

Lastly, we investigate the security when the R2 = 800 𝜇m (0.71× the beam waist

of the down converted photon) obstruction is used. Remarkably, as illustrated in

Fig. 2-6 (a), the signal decreased by almost four orders of magnitude, remaining only

the 0.07% of the signal for the LG set, but up to 71% for the BG self-healing mode

set, owing to an earlier reconstruction of the BG radial profile in comparison to the

LG radial profile. Based on the measurement results shown in Fig. 2-5 (d) and (e),

we determine a QBER of 𝑒 = 0.15 ± 0.01 and 𝑒 = 0.51 ± 0.00 for the BG and LG

modes, respectively. The mutual information (I𝐴𝐵) and secure key rates are higher

for the BG basis than the LG, even though the BG MUB has not fully reconstructed

(see Fig. 2-6 (b)). Table 2.2 shows a summary of the measured security parameters

for the BG and LG mode sets.

2.5.2 Discussion

We have presented a proof-of-concept experiment highlighting the importance of struc-

turing photons in the complete spatial mode state. Here we have demonstrated the

advantage when doing so with BG spatial modes for obstacle-tolerant QKD. Fur-

ther, we have employed hybrid spin-orbital states to access high dimensions, with the

spin-orbit states used to encode the information and the radial mode used to ame-

42



Chapter 2 Isaac Nape 43

Table 2.2: Measured security parameters for the self-healing BG (LG) modes. NC represents
the normalized coincidence counts. The normalization was performed with respect to the
counts obtained from the free-space measurements.

BG (LG) modes

Free-space R1 = 600 µm R2 = 800 µm
QBER 0.04 ± 0.01 (0.04 ± 0.01) 0.05 ± 0.