UNIVERSITY OF THE WITWATERSRAND DOCTORAL THESIS Towards Teleporting Quantum Images Author: Bereneice SEPHTON Supervisor: Prof. Andrew FORBES A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Structured Light Laboratory School of Physics March 27, 2023 iii Declaration of Authorship I, Bereneice SEPHTON, declare that this thesis titled, “Towards Teleporting Quantum Images” and the work presented in it are my own. I confirm that: • This work was done wholly or mainly while in candidature for a research de- gree at this University. • Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly attributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. • I have acknowledged all main sources of help. • Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed my- self. Signed: Date: 27 March 2023 v “If you think you understand quantum mechanics, you don’t understand quantum mechan- ics.” Richard P. Feynman vii UNIVERSITY OF THE WITWATERSRAND Abstract Doctor of Philosophy Towards Teleporting Quantum Images by Bereneice SEPHTON Achieving higher dimensionality in quantum protocols is receiving increasing interest at the promise of a range of benefits, starting from increased information capacity to noise resilience. This also arises naturally in many quantum systems, from the multitude of photonic states in the temporal, frequency or spatial domains to many atomic levels in an atom. Generalising these to superpositions of states with unique amplitude and phases, we can call such encoded information quantum images. The ability to transfer, either the state of a system directly or information encoded as quantum images thus becomes a pressing frontier with teleportation an important building block. Bearing two distinct features, teleportation forms a fun- damentally different way of communicating such that the information does not pass directly between communicating parties and when used with quantum carriers, can form the basis for a variety of quantum networks, starting from quantum repeaters to a type of quantum computing. In this thesis, the over-arching goal involves physically implementing a high- dimensional system in an effort towards teleporting quantum images. To do so, Chapter 1 considers the basic language, protocol, characterising techniques and spa- tial states being used in this work. We then consider bringing in nonlinear optical strategies to the physical imple- mentation of both the generation and detection aspects in Chapter 2. Particular em- phasis is made on orbital angular momentum (OAM) states as they are used as a test-bed for our spatial degree of freedom. For this property, the naturally generated states in spontaneous parametric down-conversion are considered where the ampli- tude structure is neglected and an in-depth investigation done. Here, we look at the radial modal purities in both the generational and detection aspects of such phase- only structuring and emphasise the general nature with experiments spanning from quantum to classical. Mitigating measures and corrective steps are also introduced which show the effects of maintaining the naturally-preserved eigenmode structures of the wave-equation. Next, the time-reversed phenomenon, sum-frequency generation, was explored as a detection mechanism. We test this classically by interacting different modal structures and looking at the generated modes. In doing so, we showed that the phase-flattening nature of conjugate states in the interaction allows for modal de- tection that is not confined to the same wavelength. The measurement-conditioned process could then be modelled in quantum formalism as reverse SPDC and intro- duced as a projector for quantum teleportation. Chapter 3 explores the full theoretical formalism of a non-linear detection-based teleportation system for spatial modes, which is numerically modelled for OAM and implemented as a test-bed. Here, the physical system was characterised and control explored by changing the generation and detection mode sizes. In doing so, we demonstrated a 15-dimensional teleportation system in OAM that exceeds the viii classical limit and extended the implementation to include a variety of states across different bases, extending from polar to Cartesian coordinates. We achieved this by encoding the information on bright coherent laser light. Despite this protocol extending to quantum carriers, present inefficiencies in the non-linear interaction require many copies to to be present, stimulating the process and yielding what we call stimulated teleportation. While the ability to develop efficiencies is both beyond the scope of this work and an active area of research, with promising advances in metasurfaces, we look towards how this system may be further applied and enhanced in Chapter 4. No- tably, we numerically map out an optimisation space for pixel states and look at requirements for teleporting images of complex objects, directly. Here, we derive a technique for characterising such teleportation, based on single pixel quantum ghost imaging where signal at higher resolution is conserved, wavelength dependency is lessened and the ability to fully distinguish the complex spatial structure; the latter being is an active challenge in quantum imaging systems. Further applications are also discussed where pump engineering could lead to better fidelities, how a de- terministic system could be implemented and how hybrid entanglement channels could lead to the teleportation of hybrid entangled states. ix Acknowledgements What a crazy journey its been with twists and turns I’d never have imagined! I’d like to express my sincerest gratitude for everyone that has been part of this under- taking, be it for a smaller moment or ’along for the ride’. Each input, experience and encounter has made this a truly valuable undertaking. Specifically, I’d like to thank the following: Supervisors: First, I would like to thank my supervisor, Professor Andrew Forbes for his invaluable guidance, enthusiasm and expertise throughout this work. I have been afforded so many wonderful experiences and opportunities, both locally and abroad that I am thoroughly thankful for. Thank you for having time to discuss the strange things things I saw in the lab and always having more questions than I have answers. My gratitude also extends to Dr. Adam Vallés for being a strong voice of reason, great fun to both learn from and work with and everlasting patience as I ex- plored the ropes of my first quantum experiments. I really appreciate every effort, video chat, update meeting and word of support. Seniors: I’d like to thank the postdocs that have come and gone in my time here, who have come to feel like friends. Thank you, Valeria, for being an unbreakable source of strength, Najmeh, for the fun chats and coffee breaks, Wagner, for your in- teresting take on physics - your enthusiasm is inspiring. I’d like to thank Dr. Dudley for the quick chats and understanding when things were tough as well as the con- stant supply of recommendation letters. It has been a privilege learning from and working with all of you. Colleagues and friends: I’d like to thank Hend, BV, Isaac and Chané for all the help and emotional support throughout this PhD. Hend, despite the travels over- seas, you never felt too far off with the constant checkups, calls, cycles in the wild and ’backup’ in every tricky situation. BV, thank you for your calm mindset and shroud of sureness always made the mountains molehills. Isaac, your passion for science and relentlessness persuit for the answer. Chané, the morbid jokes, looks of understanding, jokes only we seem to get and moments of feeling girly have made you the best fellow desk mate. To all the ’new’ recruits: Ravin, Keshaan, Lehola, Pedro, Leerin, Cade and those too many to mention - thanks for the countless jokes, epic stories to be retold and fun moments that will be fond memories of the Struc- tured Light Lab as I leave. Family: Last, but not least, thank you to my family: Eline, Abigail and Kyle. Thank you for your understanding, fun holidays away from the lab and countless prayers, and support that has always been there. Without you all, I’d not have made it through. Thank you, Lord, for being my refuge when I’d not had the strength and joy in every moment, good and bad. xi Contents Declaration of Authorship iii Abstract vii Acknowledgements ix 1 Introduction 1 1.1 The language of quantum states . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Qubits and the Bloch sphere . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Density matrices and purity . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Qudits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.4 Multi-particle states and entanglement . . . . . . . . . . . . . . 7 1.1.5 Schmidt decomposition and dimensionality . . . . . . . . . . . 10 1.2 Projectors and measurement . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Traditional teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Transverse spatial modes for high-dimensional encoding . . . . . . . . 14 1.4.1 Laguerre-Gaussian modes . . . . . . . . . . . . . . . . . . . . . . 14 1.4.2 Hermite-Gaussian modes . . . . . . . . . . . . . . . . . . . . . . 16 1.4.3 Pixel basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Interrogating the state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5.1 Quantum state tomography . . . . . . . . . . . . . . . . . . . . . 18 1.5.2 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . 22 1.5.3 Sampling dimensionality and purity . . . . . . . . . . . . . . . . 24 2 Bringing in Non-linear Optics 27 2.1 Second-order nonlinear optics . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Spontaneous parametric down-conversion . . . . . . . . . . . . . . . . 29 2.2.1 Phase-matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.2 Wavefunction and efficiencies . . . . . . . . . . . . . . . . . . . . 31 2.2.3 Spatial entanglement in SPDC . . . . . . . . . . . . . . . . . . . 32 2.2.4 Dimensional extent of OAM in SPDC . . . . . . . . . . . . . . . 33 2.3 Taking care: the modal purity of orbital angular momentum photons . 35 2.3.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.2 Experimental implementation . . . . . . . . . . . . . . . . . . . 39 Dielectric metasurface . . . . . . . . . . . . . . . . . . . . . . . . 39 Enhanced generation scheme . . . . . . . . . . . . . . . . . . . . 40 Enhanced two photon detection scheme . . . . . . . . . . . . . . 41 2.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Enhanced generation of OAM modes . . . . . . . . . . . . . . . 41 Enhanced quantum detection of OAM modes . . . . . . . . . . 43 2.3.4 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . 45 2.4 Sum-frequency generation . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.1 Phase-matching and efficiencies . . . . . . . . . . . . . . . . . . 46 xii 2.4.2 Classical demonstration of upconversion for spatial mode de- tection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.4.3 A wavefunction description for a non-linear mode projector . . 53 3 Stimulated high-dimensional spatial teleportation 55 3.1 Theoretical and numerical framework . . . . . . . . . . . . . . . . . . . 56 3.1.1 Teleportation with SFG . . . . . . . . . . . . . . . . . . . . . . . 56 3.1.2 The teleportation channel: physical limitations . . . . . . . . . . 58 3.1.3 Numerically simulating teleportation for OAM . . . . . . . . . 60 3.2 Experimental implementation . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2.1 Working principle . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2.2 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.3 Probing the limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.4 Rotating states, reconstructing the channel and breaking the state of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.5 Changing the basis symmetry - a step towards pixel states . . . 71 3.2.6 Bringing it all together: A similarity comparison . . . . . . . . . 71 3.2.7 Challenges and errors . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.8 Looking forward: From stimulated to quantum teleportation . 78 3.3 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4 Application, future work and conclusions 83 4.1 Towards teleporting phase objects . . . . . . . . . . . . . . . . . . . . . 83 4.1.1 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . 84 4.1.2 Retrieving phase images . . . . . . . . . . . . . . . . . . . . . . . 87 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.1.3 Outlook and considerations . . . . . . . . . . . . . . . . . . . . . 95 4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.1 Deterministic teleportation . . . . . . . . . . . . . . . . . . . . . 95 4.2.2 Hybrid entanglement state teleportation: Expanding degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.2.3 Tailoring the source for optimal teleportation . . . . . . . . . . . 97 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 A Supplementary information for modal purity of OAM photons 101 A.1 OAM in the Laguerre-Gaussian basis . . . . . . . . . . . . . . . . . . . . 101 A.2 Modal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 A.3 Holograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 B Suplementary information for stimulated high-dimensional spatial tele- portation 107 B.1 Suitable experimental conditions for teleportation with SFG . . . . . . 107 B.2 Fidelity of the teleportation channel . . . . . . . . . . . . . . . . . . . . 108 B.3 Procrustean filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 B.4 Background subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 xiii B.5 Qutrit teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.6 Unbalanced teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . 115 B.7 Raw experimental measurements with uncertainties . . . . . . . . . . . 115 B.8 Teleportation fidelity results . . . . . . . . . . . . . . . . . . . . . . . . . 117 C Supplementary information for detecting phase images 121 C.1 Revealing the embedded phase information . . . . . . . . . . . . . . . . 121 Bibliography 127 xv List of Figures 1.1 Bloch sphere. The Bloch sphere with key qubit states and mapping parameters indicated. Poles contain the basis states with equal super- positions positioned on the equator. . . . . . . . . . . . . . . . . . . . . 4 1.2 Pure and mixed quantum systems. Exemplary particles for a mixed and pure quantum system which will always give a 50% outcome of measuring a |0⟩ or a |1⟩. Determination of the nature of the system can be made by finding the purity. . . . . . . . . . . . . . . . . . . . . . 5 1.3 Traditional teleportation protocol. Initially, a pair of entangled pho- tons (purple) is shared between Alice (particle A) and Bob (particle B), establishing an entanglement channel. A third particle C (green) containing an unknown or prepared quantum state to be teleported is then indirectly interacted with Alice’s particle. Alice then performs a Bell state measurement on photons A and C. The Bell state projection collapses the entanglement between particles A and B, while simulta- neously entangling photons A and C as well as destroying the state of photon C. Alice then transmits classical information to Bob about the Bell measurement projection (j), allowing him to correct for any unitary rotation on the state being teleported (Uj). The state has then been teleported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Laguerre-Gaussian mode family. Intensity profiles for the spatial dis- tributions described by LG modes, at the waist plane (z = 0), with indices ℓ = [−1, 2] and p = [0, 2] with the phase profile given in the top-right inset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Hermite-Gaussian mode family. Intensity profiles for the spatial dis- tributions described by HG modes, at z = 0, with indices n = [0, 2] and m = [0, 2] with the phase profile given in the top-right inset. . . . . 16 1.6 Pixel basis states. An example of the discretised pixel basis states superimposed on an SPDC cone for (a) striaghtforward 1-to-1 images as well as a (b) tailored pixel basis for entanglement concentration. . . 17 1.7 Concept of tomography. Projective measurements (a) are made on an object which are then used to (b) ’work backwards’ so that a picture of the object can be rebuilt. . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.8 Tomographic MUB projection OAM states. Projective OAM states for exemplary d = 3 tomographic measurements for an ℓ = {−1, 0, 1} quantum system based on the (a) d-dimensional MUB approach and (b) restricted two-dimensional MUB approach. Sj refers to the basis state indicated in Table 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 21 xvi 2.1 Spontaneous parametric downconversion. Diagrams of (a) the inter- action geometry for spontaneous parametric downconversion with a χ(2) non-linear crystal and interaction geometries for (b) collinear as well as (c) non-collinear phase matching scenarios. Inset shows the energy level diagram for such a process. . . . . . . . . . . . . . . . . . . 29 2.2 Quasi-phase matching with periodic poling. Example of (a) periodic poling in a non-linear crystal comprising alternating domains to com- pensate for the (b) wavevector mismatch. . . . . . . . . . . . . . . . . . 31 2.3 SPDC generational OAM bandwidth. Example of an (a) SPDC OAM bandwidth marked in the (b) pump beam waist size wp and the non- linear crystal length L parameter space determining the OAM Schmidt number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4 SPDC measurable OAM bandwidth. Example of measurable band- width as a function of the pump and detection beamsizes for a (a) L = 5 mm crystal and (b) L = 1 mm crystal. Dotted line indicates the optimal ratio for maximum bandwidth between detection and signal beam waists. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.5 (a) A Gaussian beam modulated by an azimuthal phase, shown here as a metasurface, results in an OAM mode with many radial orders, shown in (b) as the sum of beams with several intensity rings. By com- bining dynamic and geometric phase control using holograms and metasurfaces, high purity OAM modes can be produced, as seen in (c). For detection of the modal composition of a beam, conjugate mod- ulation of the mode being evaluated is applied and the Fourier plane on-axis intensity measured, as shown in (d). . . . . . . . . . . . . . . . . 38 2.6 (a) Schematic of a J-plate design. The J-plate imprints two kinds of helical phase profiles for x- and y-incident polarization, resulting in output beams with orbital angular momentum mh̄ and nh̄, where m and n can be any independent integers. The metasurface elements are rectangular nano-posts made of amorphous TiO2 with a fixed height h = 600 nm. By changing the width along x- and y-direction (wx and wy), the nano-posts impart phase delays given by δx and δy. (b-d) Optical micrographs (b) and Scanning electron micrographs (SEMs) (c and d) of a representative J-plate with OAM number (m, n) = (10, 100). The SEMs show a top view (c) and angled view (d) of the device center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.7 Illustrations of the (a) classical and (b) quantum experimental setups. SLM: spatial light modulator functioning in reflective mode but drawn here as transmissive; F,L: lens; CCD: charge-coupled device; NC: non- linear crystal; BS: beam splitter; M: mirror; SMF: single mode fibre; APD: avalanche photodiode; BPF: band-pass filter; s: signal; i: idler; C.C.: coincidence counter. Overhead insets show representative phase maps for the respective beam manipulations carried out at each point. 41 xvii 2.8 Normalised detection probabilities with variation in the p-mode in- dex and α = wℓ/w0 for (a) ℓ = 10 and (b) ℓ = 1. Measured p- mode spectra (blue bars) are given for detection with (i) the unad- justed mode size (wℓ = w0) and (ii) the optimal encoded mode size (wℓ = wopt) shown with the theoretical predictions (black bars). The normalisation was performed with respect to the p-mode distribution. False colour map images of an ℓ = 100 generated beam with (c) phase- only modulation and (d) amplitude corrected phase modulation are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.9 The p-mode spectrum when full amplitude and phase control is ap- plied for (a) ℓ = 10 and (b) ℓ = 1. The bars represent the theory while the points represent the experiment. Insets show the processed trans- verse spatial distributions. The small intensity oscillations are due to the discrete nature of the metasurface nano-posts. . . . . . . . . . . . . 43 2.10 (a) Experimental and (b) simulated coincidence measurements based on the detection probabilities of OAM photons measured with the en- coded holograms. (c) The normalized coincidences as a function of α = wℓ/w0 for (c) ℓ = 1 (d) ℓ = 2 (e) ℓ = 5. The bars represent the experimental data while the points represent the simulation. (f) Spiral-bandwidth plots obtained from the diagonal of the inset mode projections for (left) w0 and (right) wopt. Insets are density plots rep- resenting measured coincidences for OAM projections on the SLMs when the encoded LG beam size is w0 (black squares) and the opti- mal mode sizes (wopt) (blue triangles). Each data point was measured over a 5 seconds time interval. . . . . . . . . . . . . . . . . . . . . . . . . 44 2.11 Sum-frequency generation. Diagram of the interaction geometry for sum-frequency generation with a χ(2) non-linear crystal. Top right inset shows the energy level diagram. . . . . . . . . . . . . . . . . . . . 46 2.12 (a) In a traditional quantum experiment, a Gaussian mode pumps a nonlinear crystal (NLC) mediating the generation of two entangled photons with OAM values of ℓ and −ℓ. (b) Example of flat and anti- correlated modal spiral spectrum of the paired down-converted pho- tons. (c) In the frequency up-conversion process, two incoming sig- nals are engineered to be in specific states. The up-converted signal is detected in the far field, so that there is a non-zero signal only when the phases are conjugate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.13 Schematic of the experimental up-conversion spatial mode detection setup. 806nm laser: high-power single frequency diode laser (Roith- ner); 1565 nm laser: single mode fiber pigtailed laser diode (Thor- labs); FL: Fourier lens; PH: pin hole; SLM: spatial light modulator (Holoeye); HWP: half-wave plate; DM: dichroic mirror; PPKTP: 5- mm-long nonlinear periodically-poled potassium titanyl phosphate crystal (Raicol); SPF: short-pass filter; CCD: CCD camera (Spiricon). . . 50 2.14 Cross-talk expressed in dB relative to the power in an input OAM mode with a particular helical charge for the 1565 nm (ℓA = [−3, 3]) and performing a modal decomposition by varying the helical charge of the input OAM mode for the 806 nm (ℓB = [−5, 5]). We have obviated the cross-talk values below -60 dB. The inset rightmost col- umn shows the captured intensities for the OAM input mode ℓA = 1, marked with a dotted white rectangle, to indicate where the central pixels are extracted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 xviii 2.15 HG modal cross-talk given in dB for two different input HGm,n modes. The 1565 nm laser beam is encoded with (a) HG0,0 and (c) HG1,1, per- forming a modal decomposition with the 806 nm input laser beam such as HGn,m, with the m, n indexes ranging within [0,3]. We show the captures obtained in the right indicating the weightings of the de- tected modes in the left, by extracting the center pixel value in the intensity plots for (b) HG0,0 and (d) HG1,1. Dotted cross-hairs indicate the central coordinate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.16 Up-converted images measured as MC with MA (1565 nm) as a Gaus- sian, shown in (a), and MB (806 nm) set to be (b) lambda, (c) yin yang and (d) jay symbols. The insets show the applied amplitude mod- ulated mask and the corresponding hologram to imprint the image onto mode MB. The emerging fringes are the Moiré pattern due to the finite resolution of the SLM screen. . . . . . . . . . . . . . . . . . . . . . 52 3.1 Teleportation channel scheme with optimisation parameters. A pump photon with a waist size of wp impinges on a nonlinear crystal, gen- erating two photons, photon B and photon C. Photon C is transmitted to a second crystal for SFG where it is absorbed with another indepen- dent photon encoded with the mode |ΘA⟩, corresponding to a mode field with a waist size of w0. We will call this independent photon, photon A. To recover the spatial information of photon A, we scan the spatial mode of photon B with spatial projections mapping onto the state |ΦB⟩ with a corresponding mode field that also has a waist size of wD. ℓA and ℓB refer to the encoded and projected vortex states displayed on the spatial light modulators (SLMs). . . . . . . . . . . . . 60 3.2 Simulated spiral bandwidths and dimensionalities from measure- ments of photon B relating to encoded states of photon A. Example renormalised teleported spiral bandwidths for vortex modes when (a) α = wp/w0 = 1 and (b) α = 5, using a fixed β = wp/wD = 1. These are marked as dashed lines in a wider (c) density plot of the spiral spectrum diagonal (ℓA = ℓB) as a function of α and (d) the calculated dimensionality (K), measured from the Schmidt number as a function of α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3 Generalised OAM teleportation channel capacity analysis. Contour plot of the dimensionality (K) as a function of β and α. For higher dimensionality we need a small β < 1 and large α. The blue single dash line corresponds to the minimum α = α0 for teleporting a spatial mode through the setup. The red double dashed line corresponds to the minimum α = α5 for teleporting OAM modes with ℓ = [−2, 2] giving access to no more than K = 5 dimensions. . . . . . . . . . . . . . 61 xix 3.4 Realising a teleportation channel based on nonlinear optical detec- tion. (a) A pump photon (λp = 532 nm) undergoes spontaneous para- metric downconversion (SPDC) in a nonlinear crystal (NLC1), pro- ducing a pair of entangled photons (signal B and idler C), at wave- lengths of λB = 1565 nm and λC = 808 nm, respectively. Photon B is directed to a spatial mode detector comprising a spatial light modu- lator (SLMB) and a single mode fibre coupled avalanche photo-diode detector (APD). The state to be teleported is prepared as photon A us- ing SLMA (λA = 1565 nm), and is overlapped in a second nonlinear crystal (NLC2) with photon C, resulting in an upconverted photon D which is sent to a single mode fibre coupled APD. Photons B and D are measured in coincidence to find the joint probability of the pre- pared and measured states using the two SLMs. (b) The teleportation channel’s theoretical modal bandwidth (K) as a function of the pump (wp) and detected photons’ (w0 and wD) radii, with experimental con- firmation shown in (c) through (e) corresponding to parameter posi- tions C, D and E in (b). Kth and Kex are the theoretical and experimen- tal teleportation channel capacities, respectively. The cross-talk plots are shown as orbital angular momentum (OAM) modes prepared and teleported. The raw data is reported with no noise suppression or background subtraction, and considering the same pump power con- ditions in all three configurations. . . . . . . . . . . . . . . . . . . . . . . 65 3.5 Detailed experimental setup description for high-dimensional spatial teleportation without ancillary photons. Ap: Aperture; BPF: Band- pass filter; DM: Dichroic mirror; f: Lens focal length; LPF: Lowpass filter; HWP: Half-waveplate; NLC: χ(2) Non-linear crystal; SLM: Spa- tial light modulator (phase-only). . . . . . . . . . . . . . . . . . . . . . . 67 3.6 Quality of the teleportation process. Experimental fidelities (points) for teleportation channel dimensions up to the maximum achievable channel capacity of K = 15 ± 1, all well above the classical limit (dashed line). The solid line forms a maximum fidelity for the mea- sured teleported state. The inset shows the measured OAM modal spectrum of the optimised teleportation channel with maximum co- incidences of 320 per second for a 5 minute integration time. The raw data is reported with no noise suppression or background subtrac- tion. Error bars are produced from error propagation of Poissonian statistics and uncertainties the analysis method. . . . . . . . . . . . . . 68 xx 3.7 Visibilities and quantum state tomography. (a) Measured coinci- dences (points) and fitted curve (solid) as a function of the phase angle (θ) of the corresponding detection analyser for the state |ϕ⟩ = |ℓ⟩ + exp{(iθ)} |−ℓ⟩, for three OAM subspaces of ℓ = ±1, ±2, and ±3 (further details in Appendix B.4). (b) The real (Re[ρ]) and imagi- nary (Im[ρ]) parts of the density matrix (ρ) for the qutrit state |Ψ⟩ = |−1⟩+ |0⟩+ |1⟩ as reconstructed by quantum state tomography. The inset shows the raw coincidences with maximum coincidences of 220 detected per second from the tomographic projections (full details in the Appendix B.3 and B.7). (c) Measurements for the teleportation of a 4-dimensional state, constructed from the states ℓ = {±1,±3}, with the inset graph showing the detection (solid bars) of all the prepared (transparent bars) OAM states comprising one of the MUB states. The raw data is reported with no noise suppression or background sub- traction and error bars produced from multiple measurements. . . . . 70 3.8 Teleportation in the Hermite-Gaussian basis. Coincidence measure- ments for teleportation of a (a) 3-dimensional and (b) 9-dimensional HGn,m state, constructed from the states (n, m) = {(0, 1), (1, 0), (1, 1)} and (n, m) = {(0, 0), (2, 0), (0, 2), (2, 2), (2, 4), (4, 2), (4, 4)}, respectively. The teleported state (solid bars) is in good agreement with the pre- pared state (transparent bars). The raw data is reported with no noise suppression or background subtraction. . . . . . . . . . . . . . . . . . . 71 3.9 Summary of teleported states. Similarities for teleportation of a 2,3,4 and 9-dimensional superposition states in the OAM (represented as |φ⟩) and HG (represented as |γ⟩) bases shown and labelled to the left. Raw data are reported without noise suppression or background subtraction Teleported states are |φ1⟩ = |0⟩+ |1⟩, |φ2⟩ = |−1⟩+ |1⟩, |φ3⟩ = |0⟩+ |−1⟩, |φ4⟩ = |−2⟩+ |0⟩+ |2⟩, |γ1⟩ = |HG1,0⟩+ |HG1,1⟩+ |HG0,1⟩, |φ5⟩ = |−3⟩ − i |−1⟩+ |1⟩+ i |3⟩, |γ2⟩ = |HG0,0⟩+ |HG1,0⟩+ |HG1,1⟩+ |HG0,1⟩ and |γ3⟩ = |HG0,0⟩+ |HG2,0⟩+ |HG0,2⟩+ |HG2,2⟩+ |HG4,0⟩+ |HG0,4⟩+ |HG4,2⟩+ |HG2,4⟩+ |HG4,4⟩. Error bars given are from multiple measurements . . . . . . . . . . . . . . . . . . . . . . . . 72 3.10 Modal detection efficiencies with experimental parameters. Nu- merical simulation of the change in detection efficiency for ℓ = 0 as the experimental parameters are optimised for higher dimensionality. Points C-E indicated correspond to experimental parameters tested in Fig. 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.11 Mode dependent detected noise. Spiral bandwidth plot spanning ℓ = [−8, 8] showing the measured background noise indicating higher noise for lower order modes. . . . . . . . . . . . . . . . . . . . . . . . . 74 3.12 Visibilities with detection parameters. Experimental visibility curves obtained by rotating θ = [0, 2π] in the detection mode (|−1⟩+ eiθ |1⟩) for teleported state |ψ⟩ = |−1⟩+ |1⟩ for (a) 1 µs deadtime and 1.5 ns coincidence window, (b) 5 µs deadtime and 1.5 ns coincidence win- dow as well as (c) 1 µs deadtime and 0.5 ns coincidence window. Error bars given are from multiple measurements. . . . . . . . . . . . . . . . 75 xxi 3.13 Channel SPDC characterisation. Experimental (a) spiral bandwidth of the channel SPDC and (b) visibility curve obtained by rotating phase angle, θ = [0, 2π], in the detection mode (|−1⟩+ eiθ |1⟩) in Bob’s arm for the state |ψ⟩ = |−1⟩ + |1⟩ in the arm used for teleportation. No noise subtraction or error correction was performed on the data. Error bars given are from multiple measurements . . . . . . . . . . . . . . . 77 3.14 Nonlinear efficiency vs quantumness of the teleportation. Concep- tual plot showing the convergence of the number of copies carrying the d-dimensional teleportee state as the SFG is increased. Red dots help to identify the different examples of commercial nonlinear crys- tals (ppKTP type-0 being our case), and the all-dielectric metasurface of Ref. [298] with a notably increased nonlinear coefficient. The plot in the inset shows an example of the amount of extra ancillary photons required to teleport the same d-dimensional teleportee state using lin- ear optics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1 Simulated pixel bandwidths and dimensionalities from measure- ments of photon B relating to encoded states of photon A. Example renormalised teleported pixel bandwidths for a 5 × 5 discretised po- sition space when (a) α = wp/w0 = 1 and (b) α = 10, using a fixed β = wp/wD = 1. These are marked as dashed lines in a wider (c) den- sity plot of the spiral spectrum diagonal (jA = jB) as a function of α and (d) the calculated dimensionality (K), measured from the Schmidt number as a function of α. The pixel order in the space is indicated in the top-right inset of (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 Generalised pixel teleportation channel capacity analysis. Contour plot of the dimensionality (K) as a function of β and α. For higher di- mensionality we need a larger β > 5 and large α. The white dash line corresponds to the general value of β = 5, which indicates parameter values in which close to maximal dimensionality (K = 25) is achievable. 86 4.3 A schematic diagram of the implemented quantum ghost imaging setup. (a) Entangled bi-photons are produced via spontaneous para- metric down-conversion (SPDC) at the NLC. The entangled photons are spatially separated and each imaged onto a SLM. Required holo- grams are displayed on each SLM. Coincidence measurements are done between both paths and the coincidences are used to reconstruct an image of the object. For each object two measurements are taken, as numerically simulated in (b). These two images are then combined such that the argument reveals the total phase. (c) Shows the digital phase-only objects used in the experiment, while the simulated image reconstructions are shown in (d) showing how the digital objects are expected to perform. L1 = 50 mm, L2 = 750 mm, L3 = 750 mm, L4 = 2 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4 Reconstructed amplitude-only images using the Walsh-Hadamard masks for (a) an intensity slit, and (b) an intensity ring. The outer area of the dashed white circle indicates the region in which noise was sup- pressed due to lack of SPDC signal. The amplitude-only objects are shown as insets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 xxii 4.5 Numerical simulations (Sim) showing excellent agreement with the Experimental reconstructions (Exp) for the pi-phase slit and the az- imuthal gradient ring using (a) Walsh-Hadamard and (b) random masks. Experimental images were denoised with image processing tools and contrast adjustment after reconstruction. The outer area of the dashed white circle indicates the region in which noise was suppressed due to lack of SPDC signal. Insets show the corresponding cosine and sine components of the experimental reconstructions. . . . . . . . . . . . . . 92 4.6 Phase image cross-sections showing the phase value per pixel for the numerically simulated reconstruction given by the grey dotted line and experimental reconstructions for both the Walsh-Hadamard (blue diamonds) and random masks (red dots) for (a) the π-phase slit in the horizontal direction, and (b) phase ring in the azimuth direction for a radius set inside the ring. Error bars shown are from error propaga- tion of Poissonian statistics. . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.7 Experimental ghost image of a (a) pure phase and (b) complex ampli- tude gradient spiral flower using Walsh-Hadamard masks. The full phase profile was denoised after reconstruction by image processing tools and contrast adjustment. The cosine (middle) and sine (right) components are shown for completeness in (a). The complex aplitude object (left) was reconstructed to give phase (middle) and threshold amplitude (right) in (b). Three-dimensional renders of the raw recon- struction data are shown below. . . . . . . . . . . . . . . . . . . . . . . . 94 A.1 Diagram of the holograms and field structure for holographic anal- ysis of the meta-surface generated modes. Grating, amplitude and phase components are combined to form the holograms shown for the 3 cases evaluated in this paper. (a) Detection of the modes in the azimuthal and radial bases with varying encoded beam waists. (b) Correction of the modal amplitude to generate OAM modes with pure radial profiles. We believe that the residual rings observed in the farfield are due to pixelation from the nano-posts and possibly due to no aperturing before the CCD. (c) Traditional phase-only OAM gen- eration with Gaussian input onto the meta-surface. Far-field images shown in (b) and (c) were post-processed to remove a central inten- sity spot, resulting from the presence of the unmodulated Gaussian remaining after the meta-surface. This was achieved by setting the central values to 0 and thus allows one to see only the amplitude cor- rection and modulation effects of the meta-surface as well as demon- strating what one would see in detection for a 100% efficient device. . 104 B.1 Procrustean filtering of the OAM modes. (a) Unflattened and (b) flat- tened spiral bandwidths by decreasing the grating depth for lower ℓ- values. Here (i) gives the density plot, (ii) shows the diagonal of (i) and (iii) renders the data in 3D where the diagonal values are high- lighted. Error bars are calculated from multiple measurements. . . . . 110 B.2 Illustration of background measurement. Histogram showing the arm delays with the coincidence windows for a 3s integration time, demonstrating the measured background values for noise correction. . 111 xxiii B.3 Effects of applying noise correction to the results. Plots showing the (a) raw measured coincidences and (b) coincidences corrected by subtracting the background measured in an uncorrelated time-bin for the (i) spiral bandwidth with a (ii) 3D rendering and (iii) the visibility measurable for rotating the projected state for the ℓ = ± 1 teleported state. Error bars are calculated from multiple measurements. . . . . . . 112 B.4 Teleported MUB states. Experimental fidelities measured for state to- mography performed on all 12 teleported MUB states for a d = 3 space comprised OAM modes ℓ = {-1,0,1}. Dashed (solid) lines indicate the classical (genuine qutrit) limit and phase insets along the x-axis show the MUB state phase profiles. The data is raw and not background subtracted. Error bars are calculated from multiple measurements. . . 114 B.5 Teleportation of unevenly weighted states. Experimental measure- ments (filled bars) of uneven encoded superposition states (bar out- lines) for (a) |ψ⟩ = 2 |−1⟩+ 3 |0⟩+ |1⟩, (b) |ψ⟩ = 2 |−2⟩+ 3 |0⟩+ |2⟩, (c) |ψ⟩ = |−2⟩+ 2 |0⟩+ |2⟩ and (d) |ψ⟩ = 2 |−3⟩+ |−1⟩+ |1⟩+ 2 |4⟩. Uncertainties are calculated from multiple measurements. . . . . . . . 116 B.6 Channel tomography measurements with uncertainties. Experimen- tal measurement plot shown with the detected coincidences given by the false colormap and the associated uncertainties, calculated from multiple measurements, printed in each measurement block. θ = 0, π 2 , π and 3π 2 as indicated for each of the bracketed superposition subspaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 B.7 Four-dimensional MUB measurements with uncertainties. Experi- mental measurement plot shown with the detected coincidences given by the false colormap and the associated uncertainties, calculated from multiple measurements, printed in each measurement block. . . . . . . 117 B.8 Teleported MUB states tomography. Experimental state tomography measurements measured with an integration with of 120s performed on the 12 MUB states for a d = 3 space comprised OAM modes ℓ = {-1,0,1}. Numbers printed on the measurement blocks indicate the as- sociated standard deviations measured from multiple measurements. . 118 B.9 Spiral bandwidth measured for high dimensionally tuned setup. Experimental measurements with standard deviations, calculated from multiple rounds, printed on the average coincidences shown by the false colormap for the spiral bandwidth of an optimally tuned system where K ≈ 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 B.10 Teleportation Fidelities. The measured teleportation fidelities for the channel (Fchannel) for states that bob receives (FBob). The classical (Fclassic) and cloning bounds (Fclone) are also shown. Our system shows the possibility of teleporting up to d = 4 dimensions. Moreover, since our spectrum for the OAM basis was not flat (maximal correlations), we show how a flat spectrum would improve the teleportation fidelities (Fmax) under the same experimental conditions. . . . . . . . . . . . . . . 119 B.11 Size-matched flattened modes. Example of the experimentally mea- sured teleported spectrum diagonal for the lowest three modes used to comprise the twelve d = 3 MUB teleported states. Error bars are calculated from multiple measurements. . . . . . . . . . . . . . . . . . . 119 xxiv C.1 Examples of the computed terms in the ghost imaging protocol. Here the labels t1,2,3 correspond to the first, second and third term in GIcos. The inset in the firs column corresponds to the ghost image obtained from the algorithm in Eq C.1. The insets in the fourth column show the DC component to be subtracted. . . . . . . . . . . . . . . . . . . . . 122 C.2 Example theoretical phase reconstruction from the ghost images. . . . 124 C.3 Uncropped experimentally measured ghost images. . . . . . . . . . . . 125 xxv List of Tables 1.1 Mutually unbiased basis states. Mutually unbiased basis state sets for d = 2 and d = 3, calculated using Hadamard constructions. Sj refers to the basis state in each indicated MUB and ω = ei 2π 3 . . . . . . . 20 2.1 Phase matching types. Summary of various phase-matching regimes for second order non-liner interactions which differ for input and out- put field polarisation with respect to parallel orientation to the crystal ordinary (o) or extraordinary (e) axes. . . . . . . . . . . . . . . . . . . . 30 3.1 Experimentally tested parameters. Parameters values used experi- mentally to test the numerically simulated dimensionality trends. . . . 66 3.2 Visibilities for different parameters. Comparison of the visibility and maximum detected coincidences with different detector dead- times and coincidence windows for results obtained by rotating θ = [0, 2π] in the detection mode (|−1⟩+ eiθ |1⟩) for teleported state |ψ⟩ = |−1⟩+ |1⟩. Uncertainties are calculated from multiple measurements . 76 B.1 Results summary of background subtracted and raw data. Experi- mental visibilities, fidelities and similarities calculated for the telepor- tation channel and teleported states comparing raw and background subtracted (B. Sub.) outcomes. Abbreviated states are: |φ1⟩ = |0⟩+ |−1⟩, |φ2⟩ = |−1⟩ + |1⟩, |φ3⟩ = |0⟩ − |1⟩, |φ4⟩ = |−2⟩ + |0⟩ + |2⟩, |γ1⟩ = |HG1,0⟩+ |HG1,1⟩+ |HG0,1⟩, |φ5⟩ = |−3⟩ − i |−1⟩+ |1⟩+ i |3⟩, |γ2⟩ = |HG0,0⟩ + |HG1,0⟩ + |HG1,1⟩ + |HG0,1⟩ and |γ3⟩ = |HG0,0⟩ + |HG2,0⟩+ |HG0,2⟩+ |HG2,2⟩+ |HG4,0⟩+ |HG0,4⟩+ |HG4,2⟩+ |HG2,4⟩+ |HG4,4⟩, as given in Section 3.9. Slightly better similarities may be noted in some cases for the raw values of the superposition on states as the Procrustean filtering applied was optimised for the raw data. Uncertainties are calculated from multiple measurements. . . . . . . . 113 B.2 Comparison of d = 3 teleported MUB states. Comparison of the teleported fidelities calculated from raw (not background subtracted) data. for all 12 MUB states with those in reported in existing literature for high-dimensional teleportation. Values estimated from the graphs presented in the respective publications are labelled (app.). . . . . . . . 115 xxvii List of Publications 1. B. Sephton, Nape, I., A. Vallés, M.A. Cox, F. Steinlechner, T. Konrad, J.P. Torres, F.S. Roux and A. Forbes, "Stimulated teleportation of high-dimensional infor- mation with a nonlinear spatial mode detector", arXiv:2111.13624v2 [quant-ph], (under review at Nature Photonics, 2022) 2. B. Sephton, C. Moodley, J. Francis and A. Forbes, "Revealing the embedded phase in single pixel quantum ghost imaging", Optica, 10 2(2023), 286-291 3. I. Nape, B. Sephton, Y-W. Huang, A. Ambrosio, C-W. Qui, A. Vallés, F. Capasso and A. Forbes, "Enhancing the modal purity of orbital angular momentum photons", APL Photonics 5 7(2020), 070802 4. B. Sephton, A. Vallés, F. Steinlechner, T. Konrad, J. P. Torres, F. S. Roux and A. Forbes, "Spatial mode detection by frequency upconversion", Optics Letters 44 3(2019), 586-589 xxix Dedicated to my family. Eline, Kyle and Abigail: thank you for always being there. Lucius: I miss you, always. 1 Chapter 1 Introduction With the reference "Beam me up Scotty", the idea of being able to move an object from one position in space to another without having to physically traverse the dis- tance has become commonplace in science fiction. This, after first being proposed by H. Forte [1] in the 1970s and then made popular by Star Trek. In this sense, the classical format may be considered as moving all the particles comprising the object and so moving the ’baking ingredients’ between two positions. Quantum mechani- cally, however, Vaidman [2] discussed an interesting view of teleportation where an object may be rather considered as the quantum state of the elementary particles of which it is made and thus rather the ’recipe’ that combines the ingredients into mak- ing the object. Accordingly, reconstruction of this state on other such elementary particles at another location can be thought of as actually teleporting a quantum me- chanical object [2]. More formally, however, we can describe quantum teleportation as a way to broadcast unknown information between two distant locations without using a physical carrier in between. A key feature here is that the state forms on the other side without needing to know what it is and does not travel directly be- tween the two positions as in traditional methods of communication, be it classical or quantum. This formed the basis of the protocol proposed by Bennett et al. [3] some 20 years later in 1993 which coined the term quantum teleportation. In doing so, the state need not be intercepted as it does not exist between the communicating parties and does not necessarily need to suffer the distortion effects of the commu- nication channel as it uses a pre-shared resource. As there is no direct observation of the state being sent, the superposition as well as all the correlations are maintained, preserving the quantum nature. In order to achieve this, the protocol relies on the quantum mechanical property of entanglement to first form a channel between two distant parties. Each party then possesses one of a pair of entangled entities and, as such, shares the quantum me- chanical correlations existing between them. Information may then be transferred to within a unitary rotation between them by manipulating the state of one of the en- tangled particles such that it becomes entangled with a secondary entity that carries an unknown quantum state that one wishes to be sent. It thus follows that infor- mation is sent between parties without that actual information being transmitted physically between them i.e. the particle with the unknown state is never sent di- rectly to the other party, but stays with the one sending it. A measurement is then required to determine the unitary rotation that needs to be applied in order to re- trieve the state on the other side. This is communicated classically and results in excluding the protocol from communication faster than the speed of light and thus remains in the bounds of physics. Quantum teleportation thus forms a unique method for sending states which utilises quantum entanglement at its core and can be exploited as a quantum infor- mation protocol where inherent advantages can be exploited due to its properties. 2 Chapter 1. Introduction As such, it is an active component in the development of quantum information sci- ence [4–7]. Here the conceptual scheme forms a fundamental step in many formal quantum information theories and its physical process forms a basic building block towards the development of many quantum technologies, allowing information ex- change protected against eavesdropping [8]. For instance, quantum technologies such as quantum repeaters [9], measurement-based quantum computing [10] as well as quantum gate teleportation [11] all derive from the quantum teleportation proto- col and extend to the idea of a quantum network [12]. Additionally, the scheme has been shown useful for exploring ’extreme’ scenarios such as closed timelike curves [13]. It follows that many avenues are being actively explored in order to develop physical systems that can easily and accurately implement quantum teleportation with systems ranging from nuclear magnetic resonance [14], atomic ensembles [15– 18], solid state systems [19–22] and trapped atoms [23–27] to optical modes [28–36] and discrete photonic states. Here, optical mode teleportation occurs in a continu- ous variable regime [37, 38] adapted from the originally proposed discrete variable case by Bennett et al. In this case, the information is carried in the electric field that is described by phase and amplitude or position and momentum-like quadrature op- erators [6, 7]. For discrete photonic states, the many degrees of freedom possessed by the particles have been utilised to form qubits, including the spin or polarisation states [39–45], time-binning [46–48], single- [49, 50] and dual- [51, 52] rail states and as well as spatial structures [53]. Much focus has been given to extending the distance and fidelities [54] at which the protocols may be executed across both fibre [40, 46, 48, 55, 56] and free-space [44, 45, 57] with efforts extending to a low orbit satellite [58]. These advances bring the practicality of implementing large quantum networks which can then form a quantum internet [12, 59, 60]. All of these, however, remain at the fundamental two-dimensional or qubit state space, which limits the bandwidth of the communication link to one bit of infor- mation per photon and thus limiting the communication link to how fast the signal can be sent, measured and corrected. Quantum systems at the core are also gen- erally multi-dimensional and so, either full reconstruction of one or extending the bandwidth of the information that can be sent at a given time would be a limiting factor on the practicality of implementing and using such a technique in quantum networks. Extension of this limitation then lies in increasing the dimensions that can be teleported by the physical system from qubits (two-dimensions) to qudits (d- dimensional). This has already gained attention in the broad quantum information field where advantages of using higher dimensional systems extend past increasing information capacity to offer protection against optimal quantum cloning machines [61–63], higher information security [64–66] and possible noise resilience [67]. In lieu of this, recent research has also begun to focus on experimentally extend- ing the degrees of freedom [53] and dimensions [68, 69] of the states being sent, bringing the current state of the art to three-dimensions or qutrit teleportation. The- oretical protocols for qudit teleportation using linear optics have already been put forward [70, 71], however, much of the limitation lies in the physical implementa- tion of a system that can support and entangle the higher dimensional states in a scalable manner. For instance, in the qutrit implementation schemes [68, 69], the protocols can be extended past the demonstrated dimensions, however, each addi- tional dimension requires another (ancillary) pair of entangled photons which in- volves additional optics and and nonlinear crystals that makes implementation for higher dimensions increasingly difficult. Here the scheme relies on a linear optical 1.1. The language of quantum states 3 approach which has a fundamental limitation in the entanglement step that restricts the ability to discriminate all of the teleportation variations. A method to side-step this involves a nonlinear approach [72–74] where the en- tanglement step involves up-conversion of the channel and prepared state photon, rather than an interaction with a beamsplitter. Studies show that the quantum prop- erties such as squeezing are conserved in up-conversion [75], allowing the state up- converted with the interacting channel photon to be transferred to the entangled partner. Here the approach has been used to demonstrate a complete Bell-state mea- surement for discrete-variables [43] as well as multiplexed teleportation of orbital angular momentum spatial states with continuous variables [76], however, neither of the demonstrations extend beyond qubits. With this nonlinear approach, it is possible to exploit the transverse spatial modes in structured light [77] such as orbital angular momentum (OAM) or pixel states as high-dimensional encoding bases. Higher dimensional spatial modes have been a topic of interest as a promising approach for increasing information capacity with experiments with other quantum information protocols demonstrating their poten- tial in freespace [63, 78–84], optical fibres [85] and underwater [86] quantum commu- nication channels. Using the pixel basis, one may achieve teleportation of quantum images, allowing compact, high bandwidth information to be securely teleported to another party and, with other bases like OAM, can introduce diversity for imple- menting versatile quantum networks. In the sections that follow, we introduce how states can be represented and then extend the formalism to cover the basic concept of teleportation before exploring transverse spatial modes as a convenient encoding basis that scales easily. We then look at how choice measurements can be used to reconstruct the states of interest. These form the basics which will be taken further in order to develop, implement and characterise high-dimensional quantum teleportation with non-linear optics in a way that scales conveniently and can be extended towards the teleportation of complex images. 1.1 The language of quantum states Due to the probabilistic nature of quantum entities, the outcome of a single-shot measurement is not directly predictable, but rather a picture of such is gained by the statistical outcomes of many choice single-shot measurements repeated on iden- tically prepared systems. The collective outcomes then allows one to determine a probability distribution which can be used to describe the particles. Mathematically, quantum states are used to describe the behaviour of quantum entities and as such represent key building blocks of quantum processes. Here we cover the basic math- ematical formalism used to describe and handle the quantum systems used to both understand the teleportation protocol as well as implement it in higher dimensions later on. 1.1.1 Qubits and the Bloch sphere Wavefunctions such as ψ can be used to describe a quantum state. With Dirac for- malism, this is can be represented as a vector, |ψ⟩ in an inner product vector space, known as Hilbert space, H. In the discrete case, where particles have quantised outcomes such as spin up or spin down, those form elements in a discrete Hilbert space, Hd where d represents the dimension of the space and as such is the number 4 Chapter 1. Introduction of unique elements that span it for an orthonormal computational basis. Outside of FIGURE 1.1: Bloch sphere. The Bloch sphere with key qubit states and mapping parameters indicated. Poles contain the basis states with equal superpositions positioned on the equator. the trivial one-dimensional case, two-dimensions represent the simplest example of discrete quantum states spanning a Hilbert space. Here the basis vectors |0⟩ = [ 1 0 ] ; |1⟩ = [ 0 1 ] (1.1) span the state space. A particle described by this space is called a qubit. It then follows that a general description of a pure qubit state |ψ⟩ is made from a linear superposition of the basis vectors such that, |ψ⟩ = α |0⟩+ β |1⟩ = [ α β ] (1.2) where α and β are complex coefficients weighting the contribution and phase be- tween the basis vectors. |α|2 and |β|2 thus give the probability of finding the qubit in the basis state |0⟩ or |1⟩, respectively. Additionally, |α|2 + |β|2 = 1 as the proba- bilities should sum to unity i.e. | ⟨ψ| ψ⟩ |2 = 1. Traditionally, qubits are graphically visualised on a Bloch sphere [4], shown in Fig. 1.1 for an arbitrary state. Here the poles (z-axis) represent the basis states with the hemisphere (x- and y-axes) then giv- ing equal superpositions of both with different relative phases. The pure qubit (|ψ⟩) is mapped to the surface of the sphere with the relation, |ψ⟩ = cos ( χ 2 ) |0⟩+ eiΦ sin ( χ 2 ) |1⟩ , (1.3) where 0 ≤ χ ≤ π and 0 ≤ Φ < 2π and the radius of the sphere is 1. The parameter χ gives the polar angle and thus dictates the contribution of the basis state,laterally positioning the qubit. For instance, at χ = π 2 an equal superposition of |1⟩ and 1.1. The language of quantum states 5 |0⟩ is specified. The second parameter Φ specifies the azimuthal angle, yielding the relative phase between the basis states and as such generates a rotation of the qubit vector about the z-axis. For example, Φ = π 2 then gives results in the state 1√ 2 (|0⟩+ |1⟩). In this case, the common phase value between the complex α and β in Eq. 1.2 is factored out as a global value that can be ignored, leaving only the relative phase between them. 1.1.2 Density matrices and purity When describing a quantum system, however, using a wavefunction or state vector such as given in Eq. 1.2 is not always sufficient. For instance, what if the particles in the system are not all identically prepared, but rather different particles have dif- ferent states with some accompanying probability? This would be different from a particle that is in a superposition of different basis states as in Eq. 1.2. Here the parti- cle can be seen to be in both states at the same time and simply collapses into one of these states based on the measurement or observation being made. Rather the par- ticle is in a specific state, but then the next particle being observed is in a different state, making the system a statistical ensemble of varying states. This latter case is known as a mixed state. This concept is illustrated in Fig. 1.2 for the two cases where three particles in such as ensemble are considered. Here measurements of the par- ticles in either system will always give a 50% outcome of measuring a |0⟩ or a |1⟩. In the mixed case, the quantum system can no longer only be described by a single state vector that has a superposition of these outcomes as in the pure case. It fol- lows that the state vector presented previously is no longer a viable way to describe the system being looked at. In order to extend the way we describe the quantum system, density matrices may be introduced along with another factor which is the purity. Purity in this case describes how ’identical’ the system is that is being ob- served. Additionally, the question may arise as to how a system can be described where the states of different particles are correlated to each other. Density matrices are also integral to dealing with such entangled particles. This is explored further on in Section 1.1.4. FIGURE 1.2: Pure and mixed quantum systems. Exemplary particles for a mixed and pure quantum system which will always give a 50% outcome of measuring a |0⟩ or a |1⟩. Deter- mination of the nature of the system can be made by finding the purity. More specifically, for a pure state, the density matrix can be obtained by taking the outer product of the state with itself. In terms of Eq. 1.2, the density matrix (ρ) is 6 Chapter 1. Introduction then ρ = |ψ⟩ ⟨ψ| (1.4) = [ α β ] [ α∗ β∗] = |0⟩ |1⟩ ⟨0| ⟨1|[ |α|2 αβ∗ α∗β |β|2 ] with ∗ indicating the complex conjugate. On the last line, the basis states are indi- cated outside the matrix to indicate the significance of each element. For instance. the first term (|α|2) yields the coefficient for the outer product, |0⟩ ⟨0|, in the pure state. It then follows that the density matrix of a mixed state is given by the weighted sum of pure matrices, ρ = ∑ j Pj ∣∣ψj 〉 〈 ψj ∣∣ = ∑ j Pjρj. (1.5) Here the Pj is the probability of finding a particle in the state ∣∣ψj 〉 with ∑ Pj = 1. Due to its significance, the density matrix has several properties to note. Firstly, when taking the trace of the of the quantum state ρ with Tr(ρ) = ∑d j=0 ⟨j| ρ |j⟩ for a d-dimensional state space with the basis |0⟩ , |1⟩ , ..., |d⟩, it should equal to one. This is as the sum of all the possible measurements in a particular basis must equate to one. Secondly, it should be Hermitian and so equal to the complex conjugate, ρ = ρ∗. Lastly, any projective measurements should have a non-negative outcome, ⟨ψ| ρ |ψ⟩ ≤ 0 ∀ |ψ⟩. In other words, the density matrix should be positive semi- definite which also corresponds to all the eigenvalues of ρ being positive values. Based on these restrictions, it is also possible to write a more generalised form of the density matrix with entries which satisfies these properties, but is not necessarily pure. In two-dimensions such a matrix takes on the form, ρ = [ a b + ic b − ic d ] , (1.6) where d = 1 − a as Tr(ρ) = 1 and a, b, c, d are real parameters. As this form of the quantum state allows one to describe both pure and mixed states, it follows that one can harness the properties of such to extract a measure that allows one to determine the degree of ’pureness’ or the purity of the system. This is achieved by taking the trace of ρ2, or Tr(ρ2). The outcome of which ranges from 0 indicating maximally mixed to 1 for a completely pure state. 1.1.3 Qudits Real particles are not only confined to two-dimensional states, however. Consider- ing photons for instance, contrary to spin or polarisation, properties such as the tem- poral [87], frequency [88] and spatial [89–91] domains all have many modes and thus the photon can exist in a vast array of different states. Atomic levels offer another example with many different possible excited states in which an atom can exist [92]. Here we accommodate this by simply extending the dimension of the state space and 1.1. The language of quantum states 7 thus the number of possible basis states from {|0⟩, |1⟩} spanning H2 in 2-dimensions to {|0⟩, |1⟩ , ..., |d − 1⟩} spanning Hd in d-dimensions. Here, the higher-dimensional particles are qudits and the mathematically, the pure states are described by the vec- tor |ψ⟩ = ∑d−1 j=0 cj |j⟩. As before, cj are complex coefficients giving the probability, |cj|2, of the particle being measured in the basis state |j⟩ and ∑j cj = 1. Unpacking this using three dimensions (d=3) as an example, the basis states are now written as |0⟩ = 1 0 0  ; |1⟩ = 0 1 0  ; |2⟩ = 0 0 1  (1.7) with any pure state described as |ψ⟩ = c0 |0⟩+ c1 |1⟩+ c2 |2⟩ . (1.8) The density matrix describing Eq. 1.8 is then computed as follows: ρ = |ψ⟩ ⟨ψ| (1.9) = c0 c1 c2  [c∗0 c∗1 c∗2 ] = |0⟩ |1⟩ |2⟩ ⟨0| ⟨1| ⟨2||c0|2 c0c∗1 c0c∗2 c1c∗0 |c1|2 c1c∗2 c2c∗0 c2c∗1 |c2|2 . (1.10) By expanding on this example, it can thus be seen that the for a qudit (d-dimensional), the denisty matrix can also be expressed as ρ = d−1 ∑ j,k cjk |j⟩ ⟨k| , (1.11) where {j, k} ∈ {0, 1, ..., d − 1} are the row and column indices of the matrix and there are d2 elements. For mixed states, the density matrix can then be described by d x d pure matrices as in Eq. 1.5. Furthermore, the corresponding generalised form of the density matrix in higher dimensions can thus be also be extended as show in the three-dimensional example, ρ =  a1 b1 + ic1 b2 + ic2 b1 − ic1 a2 b3 + ic3 b2 − ic2 b3 − ic3 a3  , (1.12) where a1 + a2 + a3 = 1 and {aj, bj, cj} are real parameters for j ∈ 1, 2, 3. 1.1.4 Multi-particle states and entanglement So far, only the states of single particles or one degree of freedom (DoF) have been described. A quantum system may entail more than a single particle, however, or we may way to look at more than one property (DoF). Furthermore, the state of a single particle may depend on that of another, resulting in a particular property known as entanglement, as will be seen further on. In order to incorporate this, we describe 8 Chapter 1. Introduction how to write the state of more than one particle in this section and what it means in cases where there is non-separability present. When considering the state of more than one particle such as two photons or even two DoFs such as the spin and path in a single photon, tensor products of each individual state is used. For instance, for a system comprised two d-dimensional pure particles or pure qudits with the individual states, |ψ1⟩ =  a0 a1 ... ad−1  ; |ψ2⟩ =  c0 c1 ... cd−1  , (1.13) the quantum state of the system is given as |ψ1⟩ ⊗ |ψ2⟩ =  a0 a1 ... ad−1 ⊗  c0 c1 ... cd−1  =  a0  c0 c1 ... cd−1  a1  c0 c1 ... cd−1  ... ad−1  c0 c1 ... cd−1   =  a0c0 a0c1 ... a0cd−1 a1c0 a1c1 ... a1cd−1 ad−1c0 ad−1c1 ... ad−1cd−1  . (1.14) Furthermore, the joint state (|ψ1⟩ ⊗ |ψ2⟩) forms an element in the corresponding Hilbert space that is the tensor product of the individual Hilbert spaces: Hd2 1,2 = Hd 1 ⊗Hd 2. The dimensions then scale as product of the two composite Hilbert spaces. In this example, both dimensions were d, making the dimensions of the system then d2. Following the convention in Eq. 1.2, the basis states of the system then be- comes the tensor product of all the individual basis states with each other. This can be seen from Eq. 1.14 where the coefficients for each basis states for particle 1 (a0, a1, . . . , ad−1) is multiplied with each of the coefficients of the basis states for par- ticle 2 (c0, c1, . . . , cd−1). The new set of basis states then becomes {|0⟩1 ⊗ |0⟩2 , |0⟩1 ⊗ |1⟩2 , . . . , |0⟩1 ⊗ |d − 1⟩2 , |1⟩1 ⊗ |0⟩2 , |1⟩1 ⊗ |1⟩2 , . . . , |1⟩1 ⊗ |d − 1⟩2 , |d − 1⟩1 ⊗ |0⟩2 , |d − 1⟩1 ⊗ |1⟩2 , . . . , |d − 1⟩1 ⊗ |d − 1⟩2}. More generally, the composite basis, M1,2, resulting from the tensor product of two other bases, i.e. M1 spanning the Hilbert space H1 of dimension d1 and M2 spanning the Hilbert space H2 of dimen- sion d2, is given by M1,2 = M1 ⊗M2 = {|j⟩1 ⊗ |k⟩2}j,k. (1.15) 1.1. The language of quantum states 9 Here {}j,k denotes the set where j ∈ {0, 1, . . . , d1 − 1} and k ∈ {0, 1, . . . , d2 − 1}. The dimension of the new basis is then d1 x d2. Accordingly, the arbitrary pure state of two particles can thus be written as |ψ⟩1,2 = d1−1 ∑ j=0 d2−1 ∑ k=0 bj,k |j, k⟩1,2 (1.16) with |j⟩ ⊗ |k⟩ is rewritten in shorthand notation as |j, k⟩1,2 and the position of the element in the combined ket indicates the composite basis (particle) being described. Following on from Eq. 1.11, the density matrix then takes on the form ρ1,2 = ∑ j,k,l,n=0 bj,k,l,n |j, k⟩1,2 ⟨l, n|2,1 (1.17) where indices j, n correspond to basis states spanning the basis of particle 1 and k, l refer to those spanning the basis of particle 2. b is then the complex coefficient weighting each of the elements in the matrix. An important aspect, integral to many quantum protocols being developed to- day, makes use of a specific type of two-particle state. Here, the two particle state cannot be written as a tensor product of the composing single-particle states, but in- stead must be described using the basis states in the combined Hilbert space (H1,2). For instance, we can consider a system comprised two qubit particles such as de- scribed in Eq. 1.2. The basis in which we can describe the particles is readily formed from the tensor product of the individual bases: M1,2 = M1 ⊗M2 = {|00⟩1,2 ; |10⟩1,2 ; |01⟩1,2 ; |11⟩1,2}. (1.18) If each individual particle was in a separate arbitrary superposition as given in Eq. 1.2, where the coefficients are non-zero, the state of the composite system is |ψ⟩1,2 = α1α2 |00⟩1,2 + β1α2 |10⟩1,2 + α1β2 |01⟩1,2 + β1β2 |11⟩1,2 = (α1 |0⟩1 + β1 |1⟩1)⊗ (α2 |0⟩2 + β2 |1⟩1) = |ψ⟩1 ⊗ |ψ⟩2 . (1.19) As shown in Eq. 1.19, the composite state can be factorised and thus written as a product or tensor state of the individual particles. The state of each particle can thus be produced individually and does not show any specific interaction between them. Interesting properties arise when this is no longer true. For example, we can consider the state |ψ⟩1,2 = b0 |00⟩1,2 + b1 |11⟩1,2 , (1.20) where the coefficients are non-zero. Here, it is clearly not possible to factor the state of either particle and so is said to be non-separable. The implications of this is that one can no longer describe either particle indi- vidually (in each basis alone), but rather requires the both at the same time. Such a system holds correlations where the measurement of one particle results in the de- termination of the state of the other particle. This is due to the act of measurement collapsing the state. Accordingly, a common (Copenhagen) interpretation of this is that, if a |0⟩ was measured on particle 1, particle 2 would then also be in the state |0⟩, even though before measurement it was in a superposition of both. Similarly, a measurement of |1⟩ ensures the other particle will also be in that state, despite any arbitrary distance of separation. This prompted Einstein questioning the reality [93], 10 Chapter 1. Introduction calling this "spooky action at a distance" and has since been termed quantum entan- glement [94]. It follows that this property is a direct consequence of the inability to factorise the state into its subsystems. As such, composite states for which one is not able to write as a product state i.e. |ψ⟩1,2 ̸= |ψ⟩1 ⊗ |ψ⟩2 can be called entangled. 1.1.5 Schmidt decomposition and dimensionality It may now be noted that there is a particular type of decomposition on pure state composite systems such as described in Section 1.1.4 which assist in the character- isation thereof. More specifically, it can be shown [4] that there exists a particular decomposition of a pure composite state, such as given in Eq. 1.15, where the sum of the basis state combinations no longer run over all possible indices. Instead, a particular basis can be found, known as the Schmidt basis, such that b = 0 for all i ̸= j with |ψ⟩S 1,2 = d−1 ∑ j=0 bj |j, j⟩1,2 . (1.21) Here bj now becomes the Schmidt coefficients where b ≥ 0. They are also the square roots of the eigenvalues for partial traces with respect systems 1 and 2 across the composite density matrix in this basis, ρS 1 = Tr2 ( ρS 1,2 ) = d−1 ∑ j=0 |b|2j |j⟩1 ⟨j|1 ρS 2 = Tr1 ( ρS 1,2 ) = d−1 ∑ j=0 |b|2j |j⟩2 ⟨j|2 . (1.22) These partial traces are thus diagonal matrices with the entries given by the squared Schmidt coefficients. It follows that several characteristics can be determined from the coefficients. If there is only one non-zero coefficient, the system is separable, if there is more than one, the system is entangled and if all the coefficients are non-zero and equal (bj = 1√ d ), the system is said to be maximally entangled. Furthermore, the value d in this case is the Schmidt number [95]. This is has a significant implication in high-dimensional entangled systems as this quantified the ’amount’ of entanglement in these systems. For example, in a three-dimensional maximally entangled bipartite system, d = 3 and so the Schmidt number, K, is three which is indicative of maximal entanglement across thee states. Generalised to sys- tems that are not necessarily maximally entangled, this number can instead be ap- proximated by [96, 97] K = 1 ∑j ∣∣bj ∣∣4 , (1.23) where 0 < bj ≤ 1 so that K = 1 dictates a separable system and K > 1 an entangled one. K thus provides an effective number of the contributing modes to the entangled state and thus refers to the dimensionality of the system. 1.2 Projectors and measurement The crux of quantum information protocols and technology relies on the ability to perform various operations in order to manipulate quantum states and measure 1.3. Traditional teleportation 11 them. To do so, we may consider the role of operators. More specifically, we con- sider the role of projector operators which forms a ubiquitous type of measurement in quantum mechanics. Once applied to a system, they project it onto a specific state. As such, these operators are take on the form Pn = |n⟩⟨n| which results in the sys- tem being projected onto the state |n⟩. Here it can be noted that the density matrix for a pure state is actually a projection operator of the pure state itself, acting on the Hilbert space of the quantum system. Mixed states are then simply formed from the sum of such pure state projectors. The effect of such a projector on a quantum system, ρ can, thus be seen in the following [4] ρ′ = PnρP† n Tr(PnρP† n) = |n⟩⟨n| ρ |n⟩⟨n| Tr(P† nPnρ) = |n⟩Tr(Pnρ) ⟨n| Tr(Pnρ) = |n⟩⟨n| (1.24) where ρ′ is the resulting density matrix after performing the projection operation and is trace-preserved due to division by the trace of the operation. This is as a single operator is not always trace-preserving. It thus follows that the act of measurement on ρ′ causes the quantum system to be projected onto the pure state, |n⟩ as dictated by the projection operator. It can then be seen that the probability of obtaining the outcome, n, when doing such a measurement on a system with the state ρ is Pn = Tr(Pnρ) = ⟨n| ρ |n⟩ , (1.25) which is known as Born’s rule. When the density matrix is for a pure state ψ (ρ = |ψ⟩⟨ψ|), the probability reduces to Pn = ⟨n|ψ⟩ ⟨ψ|n⟩ = |⟨n|ψ⟩|2. (1.26) Using a variety of these projections and subsequent measurements, information on the quantum state can be gathered or, in the case of entangled systems, certain states can be heralded. 1.3 Traditional teleportation Using the concepts covered so far, it is possible to consider how teleportation works following on from the seminal paper by Bennett et al. in 1993 where they considered the spin- 1 2 states of particles [3]. Here, instead of spin states, we use the generic qubits states introduced previously. In the traditional protocol, Alice wishes to send information in the form of an unknown qubit quantum state |Φ⟩C = α |0⟩C + β |1⟩C to a second party, Bob, where |α|2 + |β|2 = 1 are complex coefficients. Here |0⟩ and |1⟩ represent specific states such as the horizontal and vertical polarisations of a photon or spin-up and spin- down states of an electron. Figure 1.3 shows the overall protocol with each step broken down. In order to do so, a maximally entangled state between particles A and B such as |Ψ00⟩AB = 1√ 2 (|00⟩AB + |11⟩AB) is generated. The particles are then 12 Chapter 1. Introduction FIGURE 1.3: Traditional teleportation protocol. Initially, a pair of entangled photons (pur- ple) is shared between Alice (particle A) and Bob (particle B), establishing an entanglement channel. A third particle C (green) containing an unknown or prepared quantum state to be teleported is then indirectly interacted with Alice’s particle. Alice then performs a Bell state measurement on photons A and C. The Bell state projection collapses the entanglement between particles A and B, while simultaneously entangling photons A and C as well as destroying the state of photon C. Alice then transmits classical information to Bob about the Bell measurement projection (j), allowing him to correct for any unitary rotation on the state being teleported (Uj). The state has then been teleported. shared between Alice and Bob to form a quantum channel such that particle A goes to Alice and particle B goes to Bob (Step 1). When describing this system where a third particle, C, possesses the unknown state Alice wishes to send, the initial total state of the system can be written as |Φ⟩C ⊗ |Ψ00⟩AB = [α |0⟩C + β |1⟩C]⊗ [ 1√ 2 (|00⟩AB + |11⟩AB)], (1.27) = 1√ 2 (α |000⟩CAB + α |011⟩CAB + β |100⟩CAB + β |111⟩CAB) Here the initially shared maximally entangled state forms one of the four varia- tions known as Bell states or EPR pairs [4] which are listed below: |Ψ00⟩ = 1√ 2 (|00⟩AB + |11⟩AB), (1.28) |Ψ01⟩ = 1√ 2 (|00⟩AB − |11⟩AB), (1.29) |Ψ10⟩ = 1√ 2 (|01⟩AB + |10⟩AB), (1.30) |Ψ11⟩ = 1√ 2 (|01⟩AB − |10⟩AB). (1.31) It can be noted that choice of the initial maximally entangled state could be any of the Bell states as long as it is known between Alice and Bob. For simplicity the first state was chosen in this example. In order to transfer the state carried by particle C to Bob, particle C must be indi- rectly interacted with her channel particle (A), such that the state of either particle is 1.3. Traditional teleportation 13 not directly measured (Step 2). This is due to direct measurement of either particle’s state results in collapse to a specific observable and, with this, destruction of the un- known state or loss of the entanglement channel. To see how one may achieve this, the state of the particles we wish to interact can be rewritten in the Bell basis such that their states remain undefined using the following relations: |Ψ00⟩ = 1√ 2 (|00⟩CA + |11⟩CA), (1.32) |Ψ01⟩ = 1√ 2 (|00⟩CA − |11⟩CA), (1.33) |Ψ10⟩ = 1√ 2 (|01⟩CA + |10⟩CA), (1.34) |Ψ11⟩ = 1√ 2 (|01⟩CA − |10⟩CA). (1.35) Expanding Equation 1.28 and then substituting in the indirect Bell state relations between particles A and C gives |Φ⟩C ⊗ |Ψ00⟩AB = 1 2 [ |Ψ00⟩CA ⊗ (α |0⟩B + β |1⟩B) + |Ψ01⟩CA ⊗ (α |0⟩B − β |1⟩B) (1.36) + |Ψ10⟩CA ⊗ (α |1⟩B + β |0⟩B) + |Ψ11⟩CA ⊗ (α |1⟩B − β |0⟩B)]. The total state here is still the same and thus there is not yet any teleportation as no operations have been done. It can clearly be seen, however, that should particles A and C be projected into one of the Bell states (Step 3), such as |Ψ01⟩, the relation collapses to one of the terms e.g. |Ψ01⟩CA ⊗ (α |0⟩B − β |1⟩B). Particle B then assumes the state that particle C carried to within some unitary rotation, say Uj, that can be described in terms of the Pauli matrices, σ [4], σx = [ 0 1 1 0 ] ; σy = [ 0 −i i 0 ] ; σz = [ 1 0 0 −1 ] (1.37) and the identity matrix, σI = [ 1 0 0 1 ] , (1.38) in the trivial case. For instance, Bell projection |Ψ01⟩CA results in particle B having the unitary rotation Uj = σz. It is upon this projection that a variation of the state transfer has occurred where the projection causes particles C and A to become entangled (the Bell state is a maximally entangled state), while the entanglement between particles A and B is broken such that B assumes a variation of the prepared state. Here the unitary variation of the prepared state that Bob’s particle is in can be de- termined from the Bell state that Alice projected onto. To complete the teleportation, recovery of the state requires Alice to then classically communicate to Bob which Bell state she projected onto. Bob then applies the corresponding unitary operation that undoes the variation and restores the original prepared state (Step 4). This classical communication limits the speed at which the information is transferred, causing the protocol to remain within the laws of relativity. Additionally, with the Bell state pro- jection, the state of particle C becomes undefined as it is entangled with particle A which implies that the state is not cloned, adhering to the no-cloning theorem [98]. 14 Chapter 1. Introduction 1.4 Transverse spatial modes for high-dimensional encoding The transverse spatial distribution of photons constitute a high-dimensional state space contained in its internal degrees of freedom (DoF) [91, 99, 100] that spans higher dimensional Hilbert spaces. By structuring this DoF, it is then possible to en- code in a qudit space that comprises a theoretically infinite number of orthonormal modes. It follows that a theoretically infinite-dimensional space in which to repre- sent information is available, limited only by the physical constraints of the system being constructed. Here we briefly review the properties and structures of three such bases which can be used for high-dimensional state teleportation. 1.4.1 Laguerre-Gaussian modes By considering the field functions, A(r, z, t), describing the transverse oscillations of the electric field which satisfies the wave equation [101], a basis set of eigenmodes can be derived. Here r, z and t respectively represent the transverse, longitudinal and temporal coordinates. A paraxial approximation is taken where it is assumed that the transverse spatial components vary slowly with propagation (z) and as such satisfy the inequality [102] | ∂2 ∂2z A(r, z)| << k| ∂ ∂z A(r, z)|. (1.39) A differential equation, known as the paraxial Helmholtz equation, then governs the propagation of these fields and is written as [102] (∇2 ⊥ + ik2 ∂ ∂z )A(r, z) = 0, (1.40) where ∇2 ⊥ is the transverse component of the Laplacian differential operator and k = 2π/λ is the wavenumber corresponding to wavelength λ . By considering different coordinate spaces, such as the cylindrical and Cartesian, varying families of solutions can be found that correspond to different orthonormal mode sets or bases. In cylindrical coordinates, solving for the paraxial Helmholtz equation results in the Laguerre-Gaussian (LG) family of modes which has the complex amplitude [102] A(r, ϕ, z) = LGℓ,p = √ 2p! w2π(p + |l|)! ( r √ 2 w )|l|L|l| p ( 2r2 w2 )e −r2 w2 e−ilϕe −ikr2 2R e−iϑp,l (1.41) where r retains the previous definition, ϕ is the azimuthal angle and L|l| p (x) = (x−|l| ex p! )( d dx ) p(x|l|+pe−x) is the generalized Laguerre polynomial of orders p ≥ 0 ∈ Z, l ∈ Z. The parameters zR, w and R govern how the field changes with propa- gation. Here zR = πw2 0/λ determines the propagation distance at which the beam width is √ 2 larger than the fundamental, Gaussian beam waist at the z = 0 plane, w0. w = w0 √ 1 + (z/zR) is the size of the transverse mode as it propagates and ra- dius of curvature, R = z[1 + (zR/z)2], represents the ’bending’ of the wavefront and ϑ = (2p + |l|+ 1) tan−1(z/zR) is the Gouy phase [102]. Figure 1.4 shows the inten- sity (|LGℓ,p|2) and phase (mod[arg(LGℓ,p), 2π]) spatial distributions for the first few LG modes (ℓ = [−1, 2] and p = [0, 2]) at the waist plane (z=0). Here the significance of the ℓ and p indices can be seen. For instance, p dictates the radial profile where the field results in p concentric radial nulls in the intensity as well as a singularity in the phase profile such that a π-phase jump occurs between the ring phase profiles. 1.4. Transverse spatial modes for high-dimensional encoding 15 FIGURE 1.4: Laguerre-Gaussian mode family. Intensity profiles for the spatial distributions described by LG modes, at the waist plane (z = 0), with indices ℓ = [−1, 2] and p = [0, 2] with the phase profile given in the top-right inset. The index ℓ results in an azimuthal phase variation with a phase singularity at the centre of the beam and a central intensity null. This results in the phase-front of the beam ’twisting’ about the axis of propagation with the number of intertwined twists per wavelength equal to the value of |ℓ|. Additionally, the sign of ℓ indicates the handedness of the twist. This rotation of the wavefront corresponds to a sig- nificant physical property which can be attributed directly to the eiℓϕ term. More specifically, light with this term carries orbital angular momentum (OAM) where each photon carries ℓh̄ OAM [102, 103]. It follows that light carrying OAM can be generated with beams carrying the gen- eral form A(r, ϕ, z) = a(r, z)eiℓϕ which are known as vortex beams. Here, a(r, z) need not be the remaining terms in Eq. 1.41. In fact, simplistic methods for generation of these OAM or vortex beams often do not involve modulation of the amplitude, but rather only a phase modulation of the fundamental Gaussian mode (LG00). Such methods utilise phase control ranging from dynamic phase with spatial light mod- ulators (SLM) [104] to geometric phase using birefringent liquid crystals [105–107] and metasurfaces [108]. The trade-off of this approach results the radial profile of the beam amplitude changing upon propagation as the distribution is no longer a solution to the paraxial Helmholtz equation. A shift to higher p-indices related to the OAM charge is seen as a result when decomposed into the LG basis [109]. It has been shown that these beams are accurately modelled as Hypergeometric-Gaussian modes [110]. 16 Chapter 1. Introduction 1.4.2 Hermite-Gaussian modes By considering a solution to Eq. 1.40 in the Cartesian coordinate system, the Hermite- Gaussian (HG) family of basis modes as found with the distribution given by [102] A(x, y, z) = HGn,m = A0 w0 w Hn( √ 2x w )Hm( √ 2y w )e− x2+y2 w2 e −ik(x2+y2) 2R e−iϑn,m . (1.42) Here w0, w and R retain their definitions given for the LG modes with ϑn,m = (n + m + 1) tan−1(z/zR) similarly giving the Gouy phase. A0 is a normalisation factor and Hj(x) = (−1)jex2 dj dxj e−x2 is the Hermite polynomial of order j. It follows that the family of HG modes are characterised by two indices, n ≥ 0 and m ≥ 0. Figure 1.5 shows the intensity and phase distributions at the waist plane for the first few modes where n = [0, 2] and m = [0, 2]. FIGURE 1.5: Hermite-Gaussian mode family. Intensity profiles for the spatial distributions described by HG modes, at z = 0, with indices n = [0, 2] and m = [0, 2] with the phase profile given in the top-right inset. It is straightforward to see that due to the Hermite polynomials, indices n and m reflect a linear intensity null in the x and y coordinates, respectively. This cor- responds to a π-phase jump in the phase. Here these modes reflect a rectangular symmetry and is another structurally sound basis which may be used as qudits for teleportation. 1.4.3 Pixel basis Another mode set that may be exploited for encoding and teleporting information is the pixel basis. Conversely, however, this set of modes does not result from gen- erating a solution to the paraxial Helmholtz wave equation as was the case for the 1.5. Interrogating the state 17 LG and HG modes. Subsequently, they do not form eigenmodes of free-space and change structure with propagation, akin to the phase-only vortex modes. They do however form orthogonal states that span the spatial distribution that they comprise. These states, arise out of the transverse position-momentum entanglement gener- ated in the spontaneous parametric downconversion (SPDC) process [111]. Here the position plane in divided into a discretised grid of positions, akin to the Cartesian plane. This is illustrated in Fig 1.6(a). FIGURE 1.6: Pixel basis states. An example of the discretised pixel basis states superim- posed on an SPDC cone for (a) striaghtforward 1-to-1 images as well as a (b) tailored pixel basis for entanglement concentration. Here each position state (|j⟩) has an associated phase and amplitude that consti- tute a complex coefficient, cj, |Ψ⟩ = N d−1 ∑ j=0 cj |j⟩ , (1.43) where N is the normalisation term. Similarly to the other basis, a theoretically infi- nite number of modes are possible. Realistically, however, the pixel size is limited by the resolution of the system in the same way the LG and HG modes are limited by the system apertures and angular resolution. If a 1-to-1 image-to-pixel ratio is used, it follows that complex optical images can be generated and teleported where each pixel holds an tailored amplitude and relative phase. An advantage of using this basis is that the size and shape of each state can be tailored to the system such that they achieve optimal detection [91]. This idea is illustrated in Fig. 1.6(b). Using this method, reports indicate fidelities of over 98% for dimensions exceeding 50 [91]. 1.5 Interrogating the state When using quantum states, it is imperative to be able to both read out the informa- tion that has been encoded in the states as well as determine if the system is working correctly. In order to do so, the quantum states must be interrogated. This has been, and remains, a topic of interest for over 80 years with questions on how one can obtain such an estimate tracing at least back to Pauli in 1933 [112]. Studies have become more intense, however, since the 1990s as the applications in quantum in- formation and communications have become clearer due to the inherently quantum nature that is being exploited. One significant challenge is the estimation of systems 18 Chapter 1. Introduction with higher dimensions as a result of increasing complexity. As such, there is an expanding toolbox with a growing collection of techniques to date such as [113–130] that can be characterised as quantum tomographic methods and, more recently, a technique which serves to obtain the bounds of a system [131]. In this section, the methods used in this thesis will be highlighted which allows one to determine the quantum state and dimensionality of the system of interest, given judiciously chosen measurements. 1.5.1 Quantum state tomography The concept of tomography relies on the idea of projecting a quantum state onto observable basis states and measuring the relative probability that the particle is in the state. One then works backwards in order to determine what state would result in the outcomes measured. Tutorial references covering this topic may be found in Refs [132, 133] with a more brief overview tailored to this work detailed here. The concept is illustrated in Fig. 1.7. Here the projective measurements equate to using a light source to project the shadow of the object of interest onto a plane such as the x-, y- and z-axes shown in Fig. 1.7 (a). The shapes and dimensions of the projected profiles then allow one to reconstruct what the object was as indicated in Fig. 1.7 (b). It can be noted that here only three projective measurements were used which assumes a fairly simple and symmetric object. For more complex objects or measurements that introduce uncertainties, a larger number of projections onto different sets of planes can allow for a more accurate reconstruction. FIGURE 1.7: Concept of tomography. Projective measurements (a) are made on an object which are then used to (b) ’work backwards’ so that a picture of the object can be rebuilt. Quantum mechanically, we can carry out this reconstructive concept by perform- ing various operations in order to manipulate the state and thus characterise it. To do so, we apply a variety of projections on many copies of the quantum system and with the subsequent measurements, information on the quantum state being inter- rogated is built up, analogous to the projections of the object shown in Fig. 1.7 (a). 1.5. Interrogating the state 19 The question now arises: what are the optimal projections necessary in order to ac- curately determine the quantum state? Several types of approaches to such tomographic measurements have been put forward which allows one to extract the information needed to reconstruct the state. Here these range from generalised Bell tests [115–117] to using mutually unbiased bases [119–121] or incorporating self-guided approaches [129, 130], with each ap- proach having certain merits. One may also ask how many measurements are enough to accurately reconstruct the state in methods chosen? The answer to this lies in uncertainties in the measurements being made. Initially one may consider the pro- jections in the concept figure. Here, only three projective measurements are actually necessary in order to find the object. These would then form a tomographically com- plete set. If there is any uncertainty or ’blurriness’ in the projection, the actual size and perhaps the fine structural details of the outlines may be in question. To im- prove this, one may consider making additional projections on the object where the planes are rotated to some degree. While these projections may not give additional information, they will allow the reconstructed object to be checked against this and adjusted as necessary. This results in the ’blurriness’ or uncertainty being reduced. This then forms an overcomplete set of measurements, but allows a greater degree of accuracy in the reconstruction [134, 135]. In this work we made use of mutually unbiased bases (MUBs) and as such will explore it in more detail. Here, MUBs [113, 136] refer to basis states that are char- acterised by a minimal overlap with all other respective bases in the state space i.e. the overlap of any basis states between different MUBs gives a value of 1/d. That is to say, given two different bases: M1 with orthonormal bases {|j1⟩ , ..., |jd⟩} and M2 with orthonormal bases {|k1⟩ , ..., |kd⟩} in Hilbert space Hd, they are mutually unbiased if | ⟨jl |kn⟩ |2 = 1 d (1.44) for all l, n ∈ {1, ..., d}. This implies that they form an optimal set in which to retrieve information as they are not biased to any one of the bases and so serve to maximise the amount of information that can be extracted for each measurement and minimise redundancy. Despite their usefulness, some uncertainties are associated with the formulation and existence of higher-dimensional MUBs and as such are noted here. Firstly, it is only known that there are exactly d+1 MUBs for dimensions which are prime or powers of prime [113, 137]. For arbitrary d-dimensions, however, such as d = 6, it can only be shown that the number of MUBs do not exceed d+1 [113, 138] with the exact number unknown [139–142]. Secondly, construction of these higher-dimensional MUBs is a problem of increasing difficulty as one needs to be able to define them [137]. Methods for constructing the known bases include using Weyl groups or Hadamard matricies [139], with Ref. [143] outlining the Hadamard constructions for d = 2 to 5 in detail. Table 1.1 gives such basis states for all the MUBs (M) where d = 2 and d = 3 as an example. An initial approach uses the full MUB construction using all d-modes as con- stituents for interrogating a d-dimensional system [144, 145]. This naturally is lim- ited by the constraints of the construction noted above and thus would only be valid for a certain set of dimensional systems. Furthermore, if the quantum system is multi-particle, the dimensions to be measured over are dN for an N-particle sys- tem which implies an increasingly large-dimensional MUB which needs to be con- structed and measured as well as some measurements needed for entangled observ- ables. A way around this is to instead restrict the measurements to the localised space 20 Chapter 1. Introduction MUB M1 M2 M3 M4 d = 2 S1 |0⟩ 1√ 2 (|0⟩+ |1⟩) 1√ 2 (|0⟩+ i |1⟩) - S2 |1⟩ 1√ 2 (|0⟩ − |1⟩) 1√ 2 (|0⟩ − i |1⟩) - d = 3 S1 |0⟩ 1√ 3 (|0⟩+ |1⟩+ |2⟩) 1√ 3 (ω |0⟩+ |1⟩+ |2⟩) 1√ 3 (ω2 |0⟩+ |1⟩+ |2⟩) S2 |1⟩ 1√ 3 (|0⟩+ ω |1⟩+ ω2 |2⟩) 1√ 3 (|0⟩+ ω |1⟩+ |2⟩) 1√ 3 (|0⟩+ ω2 |1⟩+ |2⟩) S3 |2⟩ 1√ 3 (|0⟩+ ω2 |1⟩+ ω |2⟩) 1√ 3 (|0⟩+ |1⟩+ ω |2⟩) 1√ 3 (|0⟩+ |1⟩+ ω2 |2⟩) TABLE 1.1: Mutually unbiased basis states. Mutually unbiased basis state sets for d = 2 and d = 3, calculated using Hadamard constructions. Sj refers to the basis state in each indicated MUB and ω = ei 2π 3 . of each particle. Despite the restriction, interrogation of this reduced subspace still adheres to the conditions for a complete set of tomographic measurements [120]. In doing this, the measurements are derived from MUBs defined in a Hilbert space of d 1 N , which reduces to a (d(d + 1)2) number of measurements for a bipartite or two-particle system as not all of the dN space of the quantum system is probed, but rather each particle’s d-subspace. The dimension of the basis states comprising the MUBs then also reduce to d as well. It follows that the scaling of this approach, both with respect to the complexity of MUB construction as well as number and type of measurements required, is more favourable as higher dimensional systems are con- sidered. It can be noted that while the number of measurements are greatly reduced, it is still susceptible to the shortcomings outlined for using MUBs, just at a slower rate and without the possibility of needing to consider entangled observables. As a result, there still exists a restriction on the dimensionality of the quantum system one wants to interrogate. An alternate method, bypassing this, is to keep the dimensionality of the MUBs projections restricted to two [119] as the construction of such a basis is straightfor- ward and simple to implement. The localised measurements for each particle is then spread across combinations of these two-dimensional subspaces within the d- dimensional system. This naturally results in the many more measurements as each two-dimensional subspace combination in the d-dimensions of each particle needs to be measured along with all the different MUBs therein. As such th