waves are parametrically coupled and can give rise to mirrorless **optical** parametric oscillation (OPO)[19, 20, 21, 22, 23, 24, 25]. The reason for the similarity is that, in the scheme presented here, even though the signal and the idler copropagate with the pump pulse in the laboratory frame, they counterpropagate in the frame of the moving pump pulse, because one is faster than the pump and one is slower. Hence the moving pump pulse provides both an effective cavity and parametric gain, leading to oscillation. In reality, however, the interaction among the pulses should be ultimately limited by the finite device length. It is shown in Section 7.4, with a Laplace analysis, that the parametric gain should abruptly increase above the threshold, where infinite gain is predicted by the Fourier analysis, but a finite medium length will always limit the gain to a finite value. Still, as previous proposals of TWM mirrorless OPO have never been experimentally achieved due to the requirement of a continuous-wave (CW) pump and the difficulty in phase matching counterpropagating waves, the presented analysis suggests the exciting possibility that mirrorless OPO can be realized with an ultrashort pump pulse and a practical poling period for phase matching of copropagating modes, if a long enough medium can be fabricated and parasitic effects can be controlled. By analyzing the scheme in the Heisenberg picture in Section 7.5, a high spontaneous parametric down conversion rate is also predicted, in excellent agreement with the experimental result reported in Ref. [17]. The result should be useful for many **quantum** **information** **processing** applications, such as **quantum**-enhanced synchronization [26] and multiphoton entanglement for **quantum** cryptography [27]. Finally, numerical results are presented in Section 7.6, which confirm the theoretical predictions.

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Before we can dive into the use and application of an Atomic Force Microscope (AFM), we must first see the bigger picture. With the apparent trend of Moore’s Law – which today states that the data density of integrated circuits doubles about every 18 months – holding fast, it is inevitable that as circuits get smaller we will reach a limit. Eventually integrated circuits will get so small they will cross over from being appropriately governed by **classical** mechanics to **quantum** mechanics. Not only is this inevitable, but if Moore’s Law holds up, it will happen in the next couple of decades.

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Integrated **optical** **quantum** circuits based on photonic waveguiding structures are increasingly gaining attention as a possible solution for scalable **quantum** tech- nologies with important applications to **quantum** simulations. **Quantum** commu- nication provides secure **information** transmission, but the distance over which **quantum** states of light can be distributed without significant disturbance is lim- ited due to inescapable losses and noise in **optical** elements. Loss is the greatest challenge facing the implementation of integrated photonic technologies, and it is inescapable in experimental reality. In recent years there is a rise of interest in structures with spatially inhomogeneous losses. Light propagation in waveg- uiding structures with spatially distributed sections of loss can be used for im- plementation of **quantum** plasmonic circuits, which are able to strongly confine light to sub-wavelength dimensions, as well as for parity-time (PT) symmetric structures, with phase transition associated with PT-symmetry breaking, which opens new possibilities for light manipulation. The PhD thesis contains research on the controllable **classical** and **quantum** dynamics of **optical** frequency con- version processes in quadratically **nonlinear** photonic integrated circuites in the presence of losses. Namely, I discuss spontaneous parametric down-conversion (SPDC), sum-frequency generation (SFG) and **optical** parametric amplification (OPA) in **nonlinear** structures governed by non-Hermition Hamiltonians. I ex- plore the fundamental features of multi-photon generation in integrated nonlin- ear waveguides. I have been shown that arrays of coupled **nonlinear** waveguides can serve as a robust integrated platform for the generation of entangled pho- ton states with nonclassical spatial correlations through spontaneous parametric down-conversion (SPDC), and that the operation of such **quantum** circuit is tol- erant even to relatively high losses. Furthermore, I have studied the bi-photon multimode **quantum** emission tomography in waveguide structures with spatially inhomegeneous losses. The PhD thesis also covers the research on the effect of these losses in waveguide couplers possessing parity-time (PT) symmetry. I have identified an anti-PT spectral symmetry of a parametric amplifier based on those

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comparatively large distribution of transition frequencies within an ensemble of ions (∼GHz). These two facts allow many different groups of ions to be ad- dressed individually, based on their resonant frequencies. The ratio between the inhomogeneous and homogeneous broadening can be as large as 10 7 . The nuclear splittings are generally O(10 MHz) in zero field. This is much bigger than the homogeneous linewidths of the transitions, which means the nuclear states of the ions can be manipulated optically. The fact that the hyperfine structure is resolvable optically means **quantum** **information** encoded in the nuclear transitions can be transfered to the **optical** transitions and vice versa. Rare earth ions in non-centro-symmetric hosts also have large (as big as GHz) interactions involving their **optical** transitions and these provide the ion-ion interactions needed for multi-qubit gates. The advantage of these interactions is that **quantum** **information** is only sensitive to their action when it is encoded in the **optical** states. Putting the **quantum** **information** into the ground state hyperfine structure would free one from having to rephase the interaction when it is not wanted. This isolation was investigated in the recent work of Alexander [100].

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Various physical implementations of NLA have been proposed and experimentally demonstrated, including the **quantum** scissor setup [69, 70, 71, 72], the photon-addition and -subtraction [73, 74], and noise addition [75] schemes. In all these approaches, a large truncation is often imposed on the unbounded amplification operator in the photon-number basis. The high-fidelity operating region of the amplification is conse- quently restricted to small input amplitude and small gains [89, 91]. The current realisa- tions require non-**classical** light sources and non-Gaussian operations like photon count- ing, thereby rendering their application to many systems and protocols very challeng- ing. Intriguingly, as recently proposed [79, 92] and experimentally demonstrated [81], the benefits of noise-reduced amplification can be retained via **classical** post-**processing**, provided that the NLA precedes a dual homodyne measurement directly. Although the simplicity of this measurement-based noiseless linear amplifier (MB-NLA) is appeal- ing, its post-selective nature confines it to point-to-point applications such as **quantum** key distribution. To overcome this drawback, the concatenation of an NLA and a de- terministic linear amplifier (DLA) that uses MB-NLA and yet outputs a **quantum** state was proposed recently and studied in the context of **quantum** cloning [94], where the production of clones with fidelity surpassing the deterministic no-cloning bound was demonstrated.

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The issues we will address in this section are the same as those that dogged early researchers in **quantum** computing in the 80s, that of what constitutes a **quantum** model for computation, if such a thing were possible, and whether there are qualitative differences between such a model and the preceeding **classical** models for computaton. We will summarize below some of the key conceptual discoveries that resulted from this research. A landmark paper on this subject, which considered some of these issues as well as the existence of a **quantum** analog to the Turing machine, was published in 1985 by Deutsch [7]. The two important issues to address when it comes to building a computer model is that of storage and **processing**. That is, what will be the memory units of our **quantum** computer, and how the **information** contained in them will be processed to do the computation. This will give us an idea of what constraints are placed on the kind of computations that can take place in this system.

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As we have mentioned in the introduction, some frac- tional transforms arise under consideration of di ﬀ erent problems: description of paraxial diﬀraction in free space and in a quadratic refractive index medium, resolution of the nonstationary Schr¨odinger equation in **quantum** mechanics, phase retrieval, and so forth. Other fractional transforms can be constructed for their own sake, even if their direct ap- plication may not be obvious yet. In particular, in Section 9 we consider a general algorithm for the fractionalization of a given linear cyclic integral transform. The application of a particular fractional transform for **optical** **information** pro- cessing then depends on its properties and on the possibility of its experimental realization in optics.

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It is natural to consider methods from the study of device-independent **quantum** **information** **processing** or self-testing **quantum** devices (e. g., [49, 67, 54, 50]). For example, such ideas have successful applications in achieving the **classical** command of **quantum** systems as shown in [63]. A key ingredient behind such device-independent methods is the rigidity of nonlocal games such as the CHSH game. By the definition of rigidity, however, the players will essentially share a specific entangled state, such as the EPR state or the GHZ state (| 000 i + | 111 i)/ √ 2, and perform prescribed measurements on the state. This seems contradictory to what we need here—the ability to store the **quantum** witness state distributed among the players. The **quantum** witness state is usually an entangled state with complex structures that are far away from what EPR or GHZ states can represent.

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Part I (Chapters 1–4) explores what is possible if the two parties may use only **classical** communication. A well-known result by M. Nielsen says that this is inti- mately connected to the majorization relation: if x is the vector of eigenvalues of the initial state, then y can be the vector of eigenvalues of the final state if and only if x is majorized by y. It was recently observed that it is possible for x ⊗ z to be majorized by y ⊗ z, even if x is not majorized by y; physically, this means that the presence of a state with eigenvalues z is a catalyst that allows a certain transformation to occur. If such a z exists, then x is said to be trumped by y. Part I is mainly a study of the structure of this trumping relation, an extension of the majorization relation. Notably, we show that for almost all probability vectors y ∈ R d where d ≥ 4, there is no finite dimension n such that the set of vectors trumped by y can be determined by restricting attention to catalysts of dimension n. We also study some concrete examples to illustrate various aspects of the trumping relation.

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The main research lines described here are related ei- ther to the field of **Optical** Solitons or **Quantum** In- formation and Computation. In Section 2 we address one of the most mature research lines of the group: that on spatial solitons as solutions of the Nonlin- ear Helmholtz (NLH) Equation. The activities in the field of temporal **optical** solitons in dispersion man- aged **optical** fiber links are summarised in Section 3. Research on **quantum** **information** based on photonic schemes has focused on the multiplexing and rout- ing of **quantum** data and an overview is presented in Section 4. As regards Quatum Computation, several schemes for **optical** gates have been proposed and are reviewed in Section 5. Back to **nonlinear** optics and **optical** solitons, we have recently started a research line for the development of new materials with high **nonlinear** **optical** response. This is briefly reported in Section 6. Transversally to most of the research ac- tivities we find a strong computational background. Some specific works in the field of Computational Photonics are described in Section 7.

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In t his Chapter it is shown t hat classical light in 2D linear waveguide arrays can be used to simulate nonlinearity and quantum statistics of photon-pair generation and quantum walks i[r]

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and he assumes that the **classical** **information** is encoded in to pure states. As we will show, our maps are expansive, and we do not have access to the entire state space, so we will modify Martin’s approach and show how to maximize the **information** transfer. With our model, we show that even with an arbitrarily noisy preparation procedure, **information** can be sent perfectly through the process by choosing an appropriate communication basis. The advantage to calculating the scope of a **quantum** process is that it not only gives a procedure for maximizing the capacity, but it does so without any additional error cor- recting codes (Martin et al. 2015). In some of the forth- coming examples, one of the communication bases appears not to suffer the effects of relativistic noise. So the naive approach would be to use that basis to encode **information**. However, as we will show, that is not always the optimal communication basis. Additionally, we will provide specific examples of relativistic noise induced by kine- matics and gravitational fields. While both kinematic and gravitational noise produce non-trivial Wigner rotations, we will show that in our example, the largest effects are induced by the gravitational field of a black hole.

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I was born the 3rd of January of 1977 in Buenos Aires (Argentina). I started my under- graduate Physics studies in March 1996, at the University of Buenos Aires (UBA). In Sep- tember 2002, I completed my ‘Licenciado’ Thesis (equivalent to a Master’s Thesis) at UBA, working on the theory of **quantum** algorithms and **quantum** maps under the supervision of Professors M. Saraceno and J. P. Paz. This work allowed me to ﬁnish my Master’s studies with a honors degree. After this, I received a six month grant to join the Imaging **Processing** Laboratory at UBA, where I investigated experimental **optical** implementations of **quantum** algorithms. On April 2003, I was employed by the ‘Stichting voor Fundamenteel Onder- zoek der Materie’ (FOM) to do my PhD research work at the **Quantum** Optics and **Quantum** **Information** Group directed by Professor J. P. Woerdman, in Leiden University (The Nether- lands). For the ﬁrst year of my graduate studies, I worked on the theory of chaotic **optical** cavities, and for the last three years, I worked on light scattering experiments with entan- gled photons. During my PhD, I attended three international summer schools: ‘**Quantum** Logic and Communications’-France (2004), ‘**Quantum** Computers, Algorithms and Chaos’- Italy (2005) and ‘ICAP06’-Austria (2006). I also presented talks at: Veldhoven and Lunteren (The Netherlands), Imperial College and Oxford University (UK), University of Innsbruck (Austria) and ICFO-Barcelona (Spain). Additionally, during my PhD years, I worked as a teaching assistant in an undergraduate course on **Quantum** Mechanics and in a Master’s level course on **Quantum** Theory, at Leiden University.

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In the realm of **quantum** **information**, a **quantum** cir- cuit model of **quantum** computation is one wherein a computation is a sequence of **quantum** gates. These **quantum** gates are reversible transformations on a quan- tum register, a system comprising multiple qubits. The key paradigm shift, going from **classical** computation to **quantum** computation is the presence of reversible (quan- tum) logic gates. These mappings preserve the Hermitian inner product and a general n-qubit (reversible) quan- tum gate is a unitary mapping U from the Hilbert space of n-qubits onto itself. The pertinent point to be ad- dressed here is regarding the number of **quantum** gates and resources required that can optimally approximate any **quantum** computation.

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Superconducting circuit **quantum** electrodynamics (circuit QED) [] is increasingly being used to study systems in the **quantum** regime. This experimental context sees a supercon- ducting coplanar waveguide act as a microwave cavity, in contrast to the **optical** frequency cavities of traditional cavity **quantum** electrodynamics (cavity QED). The microwave res- onator is made from aluminium on a silicon substrate, and Josephson junctions are created by allowing the aluminium to oxidise before adding more aluminium. Such devices are placed in a dilution refrigerator, and experiments take place at cryogenic temperatures. Such low temperatures, close to the **quantum** ground state, allow **quantum** mechanical phenomena to become manifest. Recent engineering progress means that fabrication of these devices is possible [].

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To test this experimentally the reflected infrared field was examined with a scanning diffraction grating whilst the cavity was scanned repeatedly through the TEM0o mode 1. Scanning the monolith ensures that the cavity experiences a wide range of dispersions, and thus does not limit the possible resonances of the signal and idler. The output of the scanning grating is monitored with a silicon photodiode (EG & G FND-100). Fig. 7.2 shows the scanning grating output for the most nondegenerate case, A = 31nm. This is the broadest nondegeneracy reported to date for this system. The large peak in the centre of the plot is the fundamental, at 1064 nm. The peak at the far left of the plot is the signal, that at the far right is the idler. The signal intensity is 4.3% of the 1064nm peak; the idler is much weaker. This may be intrinsic, i.e. idler production is weaker than signal, however it is more probable that this is due to the very steep roll-off in **quantum** efficiency in silicon photodetectors between 1000 & 1100 nm. The signal wavelength of 1033 nm agrees well w ith the calculated maximum of **nonlinear** gain centred at 1031 nm. The idler appears displaced slightly (2.5 nm) from its expected position of 1096 nm. This is an unavoidable artifact caused by hysteresis in the scanning motor of the scanning diffraction grating.

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We investigated the **information** geometry of non-equilibrium processes involved in **classical** and **quantum** systems. Specifically, we introduced τ ( t ) as a dynamical time scale quantifying **information** change and calculated L( t ) by measuring the total elapsed time t in units of τ. As a unique representation of the total number of statistically different states that a PDF evolves through in reaching a final PDF, L ∞ was demonstrated to be a novel diagnostic for mapping out an attractor structure. In particular, L ∞ preserves a linear geometry of a linear process while manifesting **nonlinear** geometry in a cubic (**nonlinear**) process; it takes its minimum value at the stable equilibrium point. In the case of a chaotic attractor, L ∞ exhibits a sensitive dependence on initial conditions like a Lyapunov exponent. Thus, L ∞ is a useful diagnostic for mapping out an attractor structure. To illustrate that L can be applied to any data as long as time-dependent PDFs can be constructed from the data, we presented the analysis of different **classical** music (e.g. see [16]). Finally, the width of PDFs was shown to play a dual role in **information** length in **quantum** systems. It cannot be over-emphasized that L is path-specific and is a dynamical measure of the metric, capturing the actual statistical change that occurs during time evolution. This path-specificity would be crucial when it is desirable to control certain quantities according to the state of the system (e.g. time-dependent PDFs) at any given time (e.g. see [18]). Due to the generality of our methodology, we envision a large scope for further applications to different phenomena.

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channels, and **quantum** measurements, etc. **Quantum** measurement is an inevitable part of all **quantum** informa- tion **processing** tasks. The purpose of discriminating **quantum** measurements is to find out the identity of an un- known device secretly chosen from two known **quantum** measurements. Therefore, studying the distinguishabil- ity of **quantum** measurements is an interesting and important task. Ji et al. [2] have shown that perfect discrimi- nation between two different projective **quantum** measurements is possible with finite number of uses of the measuring apparatus. Since then, various kinds of **quantum** measurement discrimination (QMD) schemes [2]-[5] have been put forward. It is easy found that, up to now, employing the local operations and **classical** communi- cation is necessary in all the previous works (e.g. [2]-[5]) for discrimination of **quantum** measurements. For two spacelike separated parties, however, it has been shown theoretically [6]-[9] that since the operators at distance commute, the averages of the observable at distant site remain the same and do not depend on the operations employed by the other distant party, i.e., the no-signaling constraint [10] holds that entanglement cannot be used for transmission of **information** without help of **classical** communication. This constraint says that, consider two observers, Alice and Bob, who perform measurements in separated locations, the marginal probabilities for Alice’s observations of results for her measurements are independent of Bob’s choice of measurement setting. For example, assume that Alice and Bob have a bipartite **quantum** system in a known state ρ , they perform lo- cal **quantum** projection operators, with elements †

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For the physical implementation of the qubit, a **quantum** system which is sufficiently isolated from their surround- ings and can be individually manipulated is required. Individual manipulation means qubits are initializable, controllable and measureable. A single atomic ion confined by a physical platform which is called “ion trap” satisfies the requirements [16-19]. Thus the ion trap has become one of the leading technologies among the various qubit platforms including superconducting circuit [20-22], **optical** lattice [23,24], nuclear magnetic resonance (NMR) [25,26], and **quantum** dot [27,28]. The ion trap was initially developed by Wolfgang Paul and Hans Georg Dehmelt who are the co-winners of the Nobel Prize in Physics in 1989. Since Cirac and Zoller have proposed using trapped ions as a physical implementation of qubit [16], the feasibility of ion qubits has been verified through many experiments [19,29,30]. Recently, in 2012, Serge Haroche and David Wineland received the Nobel Prize in Physics owing to the measurement and manipulation of individual **quantum** systems, using cavity **quantum** electrodynamics (QED) and ion traps, respectively. There has been several review articles on the subject of **quantum** **information** **processing** [18,31-34].

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