Quantum information can be processed in two fundamentally different ways, using either discrete or continuous variable implementations. Each one of them provides different practical advantages and drawbacks. In this thesis we study theoretical means allowing to implement quantum information protocols originally formulated for discrete quantum systems in physical objects characterized by continuous variables. At the heart of our considerations is the use of modular variables as helpful technique to reveal discrete structure in continuous-variable states, operations and observables. The present work is strongly guided by the experimental applicability of our ideas in quantum optics experiments, with a particular focus on the transverse degrees of freedom of single photons.One of the main themes of this thesis is the formulation of a framework for quantum information processing in phase-space based on the use of modular variables. The term modular variables refers to a specific class of observables that are periodic with respect to some pair of conjugate variables. In our framework we use these periodic observables in order to encode discrete quantum information in Hilbert spaces of infinite dimension. In particular, we consider protocols that involve measurements of judiciously chosen logical observables enabling the readout of the encoded discrete quantum information from the corresponding logical states. Using this framework we show how to perform tests of fundamental properties of quantum mechanics, such as entanglement, Bell nonlocality and contextuality, in Hilbert spaces of various dimensions. Particularly, we generalize known tests of each of these properties, that were originally formulated for measurements with discrete outcomes, to more general measurement contexts comprising the case of bounded continuous outcomes. Concerning experimental implementations of the presented theoretical elaborations we discuss the transverse degrees of freedom of single photons as a natural platform to manipulate and measure modular variables. In particular, we demonstrate how to process discrete quantum information encoded in the spatial distribution of single photons via the optical Talbot effect - a near-field interference effect. Finally, we show how to produce d-dimensional entangled Talbot photon pairs without post-selection, using spontaneous parametric down-conversion and linear optical elements only. As last topic we explore the nonlocal properties of a specific class of hybrid entanglement between particle-like and wave-like optical qubits. Using a hybrid measurement scheme of Pauli and displaced parity measurements we show that, even after including realistic experimental losses, a violation of local-realism is theoretically possible.
Quantum information can be processed in two fundamentally different ways, using either discrete or continuous variable implementations. Each one of them provides different practical advantages and drawbacks. In this thesis we study theoretical means allowing to implement quantum information protocols originally formulated for discrete quantum systems in physical objects characterized by continuous variables. At the heart of our considerations is the use of modular variables as helpful technique to reveal discrete structure in continuous-variable states, operations and observables. The present work is strongly guided by the experimental applicability of our ideas in quantum optics experiments, with a particular focus on the transverse degrees of freedom of single photons.One of the main themes of this thesis is the formulation of a framework for quantum information processing in phase-space based on the use of modular variables. The term modular variables refers to a specific class of observables that are periodic with respect to some pair of conjugate variables. In our framework we use these periodic observables in order to encode discrete quantum information in Hilbert spaces of infinite dimension. In particular, we consider protocols that involve measurements of judiciously chosen logical observables enabling the readout of the encoded discrete quantum information from the corresponding logical states. Using this framework we show how to perform tests of fundamental properties of quantum mechanics, such as entanglement, Bell nonlocality and contextuality, in Hilbert spaces of various dimensions. Particularly, we generalize known tests of each of these properties, that were originally formulated for measurements with discrete outcomes, to more general measurement contexts comprising the case of bounded continuous outcomes. Concerning experimental implementations of the presented theoretical elaborations we discuss the transverse degrees of freedom of single photons as a natural platform to manipulate and measure modular variables. In particular, we demonstrate how to process discrete quantum information encoded in the spatial distribution of single photons via the optical Talbot effect - a near-field interference effect. Finally, we show how to produce d-dimensional entangled Talbot photon pairs without post-selection, using spontaneous parametric down-conversion and linear optical elements only. As last topic we explore the nonlocal properties of a specific class of hybrid entanglement between particle-like and wave-like optical qubits. Using a hybrid measurement scheme of Pauli and displaced parity measurements we show that, even after including realistic experimental losses, a violation of local-realism is theoretically possible.
Binary quantum information can be fault-tolerantly encoded in states defined in infinite-dimensional Hilbert spaces. Such states define a computational basis, and permit a perfect equivalence between continuous and discrete universal operations. The drawback of this encoding is that the corresponding logical states are unphysical, meaning infinitely localized in phase space. We use the modular variables formalism to show that, in a number of protocols relevant for quantum information and for the realization of fundamental tests of quantum mechanics, it is possible to loosen the requirements on the logical subspace without jeopardizing their usefulness or their successful implementation. Such protocols involve measurements of appropriately chosen modular variables that permit the readout of the encoded discrete quantum information from the corresponding logical states. Finally, we demonstrate the experimental feasibility of our approach by applying it to the transverse degrees of freedom of single photons.
Entanglement and contextuality are among the most intriguing and fundamental features of quantum mechanics. Entanglement concerns quantum states:it is a phenomenon that happens when a system can not be described by the knowledge of its subparts. Contextuality concerns quantum observables: it is the capacity they have to give rise to different experimental results depending on the observables with which they are measured. Both have received tremendous attention during the last decades, following the second quantum revolution, that saw the attitude towards quantum features switch from "why is that so?" to "how can we use this?".This thesis conceives protocols to detect and characterise entanglement and contextuality in specific setups, seeking to fill the gaps between the theoretical studies and the experimental implementations. Entanglement is studied in the case of optical hybrid variables, a recent approach that tries to join the advantages of the discrete and continuous variables encoding of information for quantum communication and computation protocols. In a first chapter, we study how the formalism of matrices of moments can be adapted to the hybrid case and apply it to several hybrid entangled states. We then explore various possible improvements and refinements using the theory of positive maps, in order to gain sensitivity with regard to our detection thresholds. In a second chapter, we focus on a specific setup, define an entanglement witness and explain how it can be implemented on available experiments. It is robust to noise currently observed in quantum optical setups and is based on measurements of genuinely hybrid observables. Contextuality is studied in a last part. A review is made of the theory, from its emergence through philosophical questions to the first mathematical proofs that quantum mechanics is contextual. The main structural theories that yield a global approach to this concept are presented. They are summoned to understand and discuss the loopholes that can affect experiments of contextuality. A protocol is given for a much-studied experiment, that of the Peres-Mermin square.
We introduce a novel strategy, based on the use of modular variables, to encode and deterministically process quantum information using states described by continuous variables. Our formalism leads to a general recipe to adapt existing quantum information protocols, originally formulated for finite dimensional quantum systems, to infinite dimensional systems described by continuous variables. This is achieved by using non unitary and non-gaussian operators, obtained from the superposition of gaussian gates, together with adaptative manipulations in qubit systems defined in infinite dimensional Hilbert spaces. We describe in details the realization of single and two qubit gates and briefly discuss their implementation in a quantum optical set-up.
There are two traditionally-separated approaches to optical quantum information: the continuous- and discrete-variable strategies respectively linked to the wave-like and particle-like nature of light. This thesis work is focused on the novel hybrid approach aiming to join the capabilities of both strategies into single systems. Using hybrid methods, we report on the use of optical parametric oscillators and superconducting-nanowire single-photon detectors to generate highly non-classical non-Gaussian states such as high-purity single-photons, Schr\"odinger cat states and finally hybrid entanglement of light between continuous- and discrete-variable encoding. The potential of this resource is studied in a number of protocols. We first consider non-locality tests with hybrid entanglement such as Bell inequality violation and experimentally demonstrate the violation of Einstein-Podolsky-Rosen steering inequalities. We finally report on the progress made towards the implementation of a new setup for the demonstration of hybrid quantum teleportation between continuous and discrete-variable encodings. Our demonstrations prove the versatility of this hybrid resource and open the possibility of implementing scalable quantum networks linking systems of dissimilar nature.
Light beams offer many degrees of freedom to be explored in discrete and continuous domains. In addition to the possibility of entangling photons in these many degrees of freedom, it makes light a very useful and versatile tool for quantum information and quantum foundation purposes. In this thesis, we endorse its importance and versatility by presenting novel contributions that further explore both discrete and continuous degrees of freedom. It begins with two experiments that use classical light and explore its analogous behavior to quantum systems. The first one is a classical optics simulation of the dynamics of a relativistic quantum particle. The second work is related to the theory of mutually unbiased measurements that are effectively discrete but constructed from continuous variables systems. In the second part of the thesis, three works are presented that use the polarization and path discrete degrees of freedom. The first one is a redefinition of the quantum nonlocal correlation called steering in the multipartite scenario, based on an inconsistency in the previous definition, namely the creation of this correlation from scratch using operations that supposedly would not be able to do so. We call this exposure of quantum steering. Steering exposure is observed with entangled photons. The other two works are related to the experimental implementation of quantum channels of qubits, one of them is a particular channel for which we test for non-Markovianity using a operational measure called conditional past-future (CPF) correlation. The thesis finishes with a proposal for an experimental realization of any quantum channel of a single qubit, where the qubit is realized by the polarization of single photons. The functioning of all optical devices used is didactically explained.
Today, the potential power of quantum information processing comes as no surprise to physicist or science-fiction writer alike. However, the grand promises of this field remain unrealized, despite significant strides forward, due to the inherent difficulties of manipulating quantum systems. Simply put, it turns out that it is incredibly difficult to interact, in a controllable way, with the quantum realm when we seem to live our day to day lives in a classical world. In an effort to solve this challenge, people are exploring a variety of different physical platforms, each with their strengths and weaknesses, in hopes of developing new experimental methods that one day might allow us to control a quantum system. One path forward rests in combining different quantum systems in novel ways to exploit the benefits of different systems while circumventing their respective weaknesses. In particular, quantum systems come in two different flavours: either discrete-variable systems or continuous-variable ones. The field of hybrid quantum information seeks to combine these systems, in clever ways, to help overcome the challenges blocking the path between what is theoretically possible and what is achievable in a laboratory. In this thesis we explore four topics in the context of hybrid methods in quantum information, in an effort to contribute to the resolution of existing challenges and to stimulate new avenues of research. First, we explore the manipulation of a continuous-variable quantum system consisting of phonons in a linear chain of trapped ions where we use the discretized internal levels to mediate interactions. Using our proposed interaction we are able to implement, for example, the acoustic equivalent of a beam splitter with modest experimental resources. Next we propose an experimentally feasible implementation of the cubic phase gate, a primitive non-Gaussian gate required for universal continuous-variable quantum computation, based off sequential photon subtraction. We then discuss the notion of embedding a finite dimensional state into a continuous-variable system, and propose a method of performing quantum computations on encrypted continuous-variable states. This protocol allows for a client, of limited quantum ability, to outsource a computation while hiding their information. Next, we discuss the possibility of performing universal quantum computation on discrete-variable logical states encoded in mixed continuous-variable quantum states. Finally, we present an account of open problems related to our results, and possible future avenues of research.