We explore the concepts of coherence and entanglement as they apply to both the classical and quantum natures of light. In the classical domain, we take inspiration from the tools and concepts developed in foundational quantum mechanics and quantum information science to gain a better understanding of classical coherence theory of light with multiple degrees of freedom (DoFs). First, we use polarization and spatial parity DoFs to demonstrate the notion of classical entanglement, and show that Bell's measure can serve as a useful tool in distinguishing between classical optical coherence theory. Second, we establish a methodical yet versatile approach called 'optical coherency matrix tomography' for reconstructing the coherency matrix of an electromagnetic beam with multiple DoFs. This technique exploits the analogy between this problem in classical optics and that of tomographically reconstructing the density matrix associated with multipartite quantum states in quantum information science. Third, we report the first experimental measurements of the 4 x 4 coherency matrix associated with an electromagnetic beam in which polarization and a spatial DoF are relevant, ranging from the traditional two-point Young's double slit to spatial parity and orbital angular momentum modes. In the quantum domain, we use the modal structure of classical fields to develop qubits and structure Hilbert spaces for use in quantum information processing. Advancing to three-qubit logic gates is an important step towards the success of optical schemes for quantum computing. We experimentally implement a variety of two- and three- qubit, linear and deterministic, single-photon, controlled, quantum logic gates using polarization and spatial parity qubits. Lastly, we demonstrate the implementation of two-qubit single-photon logic using polarization and orbital angular momentum qubits.
We explore the concepts of coherence and entanglement as they apply to both the classical and quantum natures of light. In the classical domain, we take inspiration from the tools and concepts developed in foundational quantum mechanics and quantum information science to gain a better understanding of classical coherence theory of light with multiple degrees of freedom (DoFs). First, we use polarization and spatial parity DoFs to demonstrate the notion of classical entanglement, and show that Bell's measure can serve as a useful tool in distinguishing between classical optical coherence theory. Second, we establish a methodical yet versatile approach called 'optical coherency matrix tomography' for reconstructing the coherency matrix of an electromagnetic beam with multiple DoFs. This technique exploits the analogy between this problem in classical optics and that of tomographically reconstructing the density matrix associated with multipartite quantum states in quantum information science. Third, we report the first experimental measurements of the 4 x 4 coherency matrix associated with an electromagnetic beam in which polarization and a spatial DoF are relevant, ranging from the traditional two-point Young's double slit to spatial parity and orbital angular momentum modes. In the quantum domain, we use the modal structure of classical fields to develop qubits and structure Hilbert spaces for use in quantum information processing. Advancing to three-qubit logic gates is an important step towards the success of optical schemes for quantum computing. We experimentally implement a variety of two- and three- qubit, linear and deterministic, single-photon, controlled, quantum logic gates using polarization and spatial parity qubits. Lastly, we demonstrate the implementation of two-qubit single-photon logic using polarization and orbital angular momentum qubits.
The coherence of an optical beam having multiple degrees of freedom (DoFs) is described by a coherency matrix G spanning these DoFs. This optical coherency matrix has not been measured in its entirety to date--even in the simplest case of two binary DoFs where G is a 4 × 4 matrix. We establish a methodical yet versatile approach--optical coherency matrix tomography--for reconstructing G that exploits the analogy between this problem in classical optics and that of tomographically reconstructing the density matrix associated with multipartite quantum states in quantum information science. Here G is reconstructed from a minimal set of linearly independent measurements, each a cascade of projective measurements for each DoF. We report the first experimental measurements of the 4 × 4 coherency matrix G associated with an electromagnetic beam in which polarization and a spatial DoF are relevant, ranging from the traditional two-point Young's double slit to spatial parity and orbital angular momentum modes.
In this thesis, I present experimental and theoretical work which assesses and develops a range of tools that are required for performing quantum information processing, particularly in photonic systems. I investigate the three degrees of freedom of a single photon—its polarisation, spatialmomentum and time-frequency distributions. For polarisation, I show that using wave plates to implement arbitrary, single-qubit rotations is more complicated than is commonly appreciated. In the spatial-momentum and time-frequency domains, I develop ways to perform tomographic analysis of quantum states, and I report the first demonstrations of these techniques. In the time-frequency domain, the tomography technique utilises entanglement in the photon polarisation as a resource to store and provide access to the time-frequency information. In my first two experiments, I use spontaneous parametric down-conversion to produce entanglement between pairs of single photons in all three degrees of freedom. I demonstrate the first characterisation of entanglement in spatial modes and the time-frequency domain, the first quantitative measurement of entangled qutrit states, and the highest quality entangled states yet measured in both polarisation and spatial modes. I also report the first realisation of complete hyperentanglement, and a full, black-box tomography of a 36-dimensional two-photon state—the largest system to be characterised in this way to date. In my final experiment, I model and implement a new architecture for a controlled-Z gate which is much simpler to align than previous implementations. This gate requires only one non-classical interference condition, the visibility of which is the main limitation to its performance. I show that the gate operates effectively as a means of both creating entanglement and discriminating between the four elements in a basis of maximally entangled, Bell-type states. Indeed, its observed performance as a Bell analyser would be sufficient to build a quantum state teleporter which would guarantee that the recipient would be left with a better copy of an unknown input state than any eavesdropper. In a series of numerical simulations, I investigate some of the practicalities that arise when using tomographic reconstruction techniques, including how to estimate errors, which measurements to make and what is actually the optimal reconstruction. In particular, I show that tomographies perform better when based on the results from over-complete sets of measurements. Finally, I discuss the important issue of how to compare two processes, particularly when trying to assess the quality of a measured process by comparing it to some expected ideal. Judging possible candidate measures against a set of experimentally and theoretically motivated
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.
Optical coherence is a well-established branch of physical optics in which the statistical properties of fluctuating optical fields are described in terms of correlation functions over continuous spatial and temporal degrees of freedom (DoFs). Nevertheless, in any practical setting, only discrete DoFs are ever accessible. In these scenarios, the modes are fixed, stable, and deterministic, and partial coherence arises from random relative complex amplitudes, a configuration we refer to as 'structured coherence'. Advances in structured coherence have recently unveiled new conceptual ground in optical communications and information processing in which partial coherence may be preferable to full coherence, which we call a 'coherence advantage'. We formulate structured coherence in terms of coherence matrices to investigate these recent theoretical and experimental breakthroughs. We first review optical fields characterized by a binary DoF via 2x2 Hermitian coherence matrices and introduce key concepts that take on new significance for larger-dimensional modal sets. Next, we examine the structured coherence of two binary DoFs, which can be described by 4x4 coherence matrices. We introduce the concept of coherence rank, entropy swapping, and optical cross-purity. In the perspective outlined here, coherence is viewed as a 'resource', which can be exchanged between DoFs, concentrated into a DoF or into particular modes, or spread over the DoFs. We then examine larger-dimensional modal sets, which allow for more versatile applications in optical information processing. The formulation presented here lends itself particularly to the manipulation of partial optical coherence in integrated photonic platforms, thereby opening myriad avenues for novel fundamental investigations of structured coherence and exploiting the coherence advantage in optical communications and information processing.
Introduction In this chapter, we will be concerned with entangled states of light. Historically, quantum states of light, i.e., photons, have played a dominant role in the experimental study of quantum entanglement. For instance, the first Bell tests and the first realizations of quantum teleportation were performed using two-dimensional polarization-entangled twophoton states [3]. From an experimental point of view these states are readily prepared and manipulated using conventional polarization optics. Recently, however, both theoretical and experimental efforts have shifted towards entanglement in more than two modes , which is often referred to as high-dimensional entanglement. This is much richer than two-dimensional entanglement, as is, for instance, reflected in stronger violations of various measures of nonlocality which reveal the complexity of high-dimensional Hilbert spaces [4–7]. Also, from an applications perspective, it promises a larger channel capacity [8, 9] and improved security for quantum communication [10, 11]. There are several ways to implement high-dimensional two-photon entanglement. One can exploit the temporal [12], frequency [13], or spatial [14–18] degrees of freedom of the optical field. A promising way to implement high-dimensional two-photon entanglement is to use the orbital-angular-momentum (OAM) degree of freedom of light. Light beams that carry OAM have a helical wavefront that spirals around the optical axis during propagation [19]. The winding number m , which characterizes the OAM eigenmodes, can adopt any integer value between −∞ and ∞; since all coaxial OAM modes are orthogonal they can thus span a very high-dimensional Hilbert space.