My research has three main foci, one which is of interest to the fields of quantum information and quantum optics is to characterize the effects of coupling between quantum systems. Advances in technology made since the conception of quantum mechanics spurred the development of quantum technologies like quantum optics and quantum computing. These applications require a good understanding of the quantum systems that compose them, and the clever ways in which the information in these systems is stored and manipulated. By understanding the ways which quantum systems interact with their environment, their usefulness can be further utilized. The second is to develop methods of calculation which simplify the calculational and computational side of quantum dynamics. Understanding how quantum systems evolve in time is very important, but traditional methods of solving for the dynamics of quantum systems struggle to deal with time dependent parameters. Using results from the theory of Lie algebras, I will show how quantum dynamics problems can be made less computationally intensive by avoiding the partial differential equations typically associated with such problems and reducing the problem to a system of coupled ordinary differential equations. The third focus is to develop methods of visualizing properties of quantum systems. I think this is important as an educational tool and as a way to develop intuition about how quantum systems evolve considering how few opportunities there are to visualize quantum effects on a macroscopic scale. This problem is particularly challenging because quantum mechanics is non-deterministic, meaning that the only nontrivial definitive statements that can be made regarding quantum systems deal with probabilities. This coupled with the fact that many of the systems of interest depend on upwards of six variables and also evolve in time make creating useful visual representations challenging. Despite the challenges, I feel that useful insights can be gained from the visualizations, and will illustrate a few examples.
In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at
2n×2n unitary matrices, which describe operations on a closed
n-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general
2n×2n unitary matrix to the product of no more than
2n−1(2n−1) single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element
Φ among a given set of quantum channels
S that will produce the optimal value of a scalar function
D(ρ1,Φ(ρ2)), on two fixed quantum states
ρ1 and
ρ2. Some of the functions we considered for
D(⋅,⋅) are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when
S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers
a1≥…≥a2n≥0, we want to determine if there exist a
2n×2n density matrix
ρ having eigenvalues
a1,…,a2n and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on
a1,…,a2n for
n≤6 and state some observations on general values of
n. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that
Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k) for given
ρ(1),…,ρ(k)∈Dn and
σ(1),…,σ(k)∈Dm, (b) constructing a multipartite state
ρ having a prescribed set of reduced states
ρ1,…,ρr on
r of its subsystems, (c) constructing a multipartite state
ρ having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix
A can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets
K1 and
K2 of $\IC$ given by
K1K2={ab:a∈K1,b∈K2}, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that
K1K2 is star-shaped when
K1 and
K2 are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set
K1K2 is star-shaped if
K1 is a line segment or a circular disk.
Quantum information theory has become a central area in the two fields of quantum mechanics and information science. The focus of this work is mathematical modelling of quantum information with the use of operator algebras, in a way that accentuates the modelling, and acting or manipulation of observables and quantum states. The fundamental aim is to utilise C*-algebras, von Neumann algebras, and other frameworks in operator algebra to formally describe the behaviour of quantum systems, understand the nature of quantum entanglement, and support quantum error correction. Operator algebra usage provides a firm mathematical foundation that guarantees accuracy and contributes to the incorporation of topological mechanisms and algebraic rules necessary for explaining noncommutative concepts in quantum mechanics. In this paper, we will present some original notions of operator algebra and explain their usefulness when it comes to modeling the key dynamics, features of quantum states, measurement, and evolution. We will use a method that combines algebraic techniques to chart qubit interactions, entropic inequalities, and characterize transformations of level in unitary evolution. Through the use of Hilbert spaces and bounded linear operators, we build models in multi-qubit systems and see what the specifications have to say about quantum computing and quantum communication protocols. The paper provide a comprehensive discussion, including algebraic formulations, simulation outputs, and a critical comparison with traditional Hilbert space methods. It is quite notable that the operator algebras can say more about the non-local correlations and decoherence processes. There are extensive literature reviews, useful modelling tools, and implementation outcomes in the paper when symbolic computation tools have been utilised. This combination gives both theoretical strength and modelling instruments to develop quantum technologies in the future.
Quantum information theory is a branch of science at the frontier of physics, mathematics, and information science, and offers a variety of solutions that are impossible using classical theory. This book provides a detailed introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. The second edition contains new sections and entirely new chapters: the hot topic of multipartite entanglement; in-depth discussion of the discrete structures in finite dimensional Hilbert space, including unitary operator bases, mutually unbiased bases, symmetric informationally complete generalized measurements, discrete Wigner function, and unitary designs; the Gleason and Kochen–Specker theorems; the proof of the Lieb conjecture; the measure concentration phenomenon; and the Hastings' non-additivity theorem. This richly-illustrated book will be useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.
In recent decades, substantial theoretical and experimental progress was achieved in understanding the quantum nature of physical phenomena that serves as the foundation of present and future quantum technologies. Quantum correlations like the entanglement of the states of composite systems, the phenomenon of quantum discord, which captures other aspects of quantum correlations, quantum contextuality and, connected with these phenomena, uncertainty relations for conjugate variables and entropies, like Shannon and Rényi entropies, and the inequalities for spin states, like Bell inequalities, reflect the recently understood quantum properties of micro and macro systems. The mathematical methods needed to describe all quantum phenomena mentioned above were also the subject of intense studies in the end of the last, and beginning of the new, century. In this section of CAMOP 'Mathematical Methods of Studying Physical Phenomena' new results and new trends in the rapidly developing domain of quantum (and classical) physics are presented. Among the particular topics under discussion there are some reviews on the problems of dynamical invariants and their relations with symmetries of the physical systems. In fact, this is a very old problem of both classical and quantum systems, e.g. the systems of parametric oscillators with time-dependent parameters, like Ermakov systems, which have specific constants of motion depending linearly or quadratically on the oscillator positions and momenta. Such dynamical invariants play an important role in studying the dynamical Casimir effect, the essence of the effect being the creation of photons from the vacuum in a cavity with moving boundaries due to the presence of purely quantum fluctuations of the electromagnetic field in the vacuum. It is remarkable that this effect was recently observed experimentally. The other new direction in developing the mathematical approach in physics is quantum tomography that provides a new vision of quantum states. In the tomographic picture of quantum mechanics, the states are identified with fair conditional probability distributions, which contain the same information on the states as the wave function or the density matrix. The mathematical methods of the tomographic approach are based on studying the star-product (associative product) quantization scheme. The tomographic star-product technique provides an additional understanding of the associative product, which is connected with the existence of specific pairs of operators called quantizers and dequantizers. These operators code information on the kernels of all the star-product schemes, including the traditional phase-space Weyl–Wigner–Moyal picture describing the quantum-system evolution. The new equation to find quantizers, if the kernel of the star product of functions is given, is presented in this CAMOP section. For studying classical systems, the mathematical methods developed in quantum mechanics can also be used. The case of paraxial-radiation beams propagating in waveguides is a known example of describing a purely classical phenomenon by means of quantum-like equations. Thus, some quantum phenomenon like the entanglement can be mimicked by the properties of classical beams, for example, Gaussian modes. The mathematical structures and relations to the symplectic symmetry group are analogous for both classical and quantum phenomena. Such analogies of the mathematical classical and quantum methods used in research on quantum-like communication channels provide new tools for constructing a theoretical basis of the new information–transmission technologies. The conventional quantum mechanics and its relation to classical mechanics contain mathematical recipes of the correspondence principle and quantization rules. Attempts to find rules for deriving the quantum-mechanical formalism starting from the classical field theory, taking into account the influence of classical fluctuations of the field, is considered in these papers. The methods to solve quantum equations and formulate the boundary conditions in the problems with singular potentials are connected with the mathematical problems of self-adjointness of the Hamiltonians. The progress and some new results in this direction are reflected in this CAMOP section. The Gaussian states of the photons play an important role in quantum optics. The multimode electromagnetic field and quantum correlations in the Gaussian states are considered in this section. The new results in the statistical properties of the laser radiation discussed here are based on applications of mathematical methods in this traditional domain of physics. It is worth stressing that the universality of the mathematical procedures permitted to consider the physical phenomena in the ocean is on the same footing as the phenomena in the microworld. In this CAMOP section, there are also papers devoted to traditional problems of solving the Schrödinger equation for interesting quantum systems. Recently obtained results related to different domains of theoretical physics are united by applying mathematical methods and tools, that provide new possibilities to better understand the theoretical foundations needed to develop new quantum technologies like quantum computing and quantum communications. The papers are arranged alphabetically by the name of the first author. We are grateful to all authors who accepted our invitation to contribute to this CAMOP section.
The field of Quantum Information Theory provides the theoretical foundation for the pursuit of quantum computers. The ongoing questions of how quantum computers will be realized and what they will achieve, are both very uncertain. However, worldwide efforts are beginning to converge on some answers, and the future of quantum computers is looking brighter than ever. In contribution to the grand goal that is quantum computing, this thesis serves as a demonstration to the usefulness of quantum over classical computing. The central theme of my work, and my collaborators, is the exploration of using quantum systems as a tool for path finding and search algorithms. First, we explore a specific problem of sequential measurements in Quantum Information Theory, namely quantum retrodiction. In experiments involving quantum systems with numerous intermediate interactions, often times the only thing that matters to the experimenters is the final measurement. However, quantum retrodiction is the problem of extrapolating information about the past of a quantum system, by studying the present state. The case studied in this dissertation is the retrodiction of a series of two-outcome measurements, the double and triple qubit-interferometers. In both cases, we are presented with the problem of inferring the results of the two-outcome measurements, using only the final state of the system. Specifically, all of the possible outcomes from the measurements result in a different final state, and we are tasked with distinguishing which final state we have. One version of this problem can be seen as light traversing a series of half-silver mirrors, and we would like to know the path the light took. To distinguish which final state we are given, we employ the use of POVMs (Positive-Operator Valued Measurements) that reflect the symmetry of the system. The effectiveness of these POVMs in solving the retrodiction problem are then presented, along with a few comments as to other types of information that can be extracted. Second, we examine a different type of path finding via a quantum system. For a given graph G that can be expressed by connections and nodes, the same graph can be mapped onto a quantum system with a Hilbert space H. All of the connections in the graph can be represented as states in H, and the nodes act as sites for local unitary operations. The sum of all these local operations give rise to U, which is the unitary operator that acts on H, and describes the behavior of the system. Using this formalism, we explore a particular type of U called a scattering quantum random walk. The quantum random walk is the quantum mechanical analogy to the classical random walk, and serves as a powerful tool for searching on a graph. In this dissertation, we present the case of searching for a marked final node, on a graph that consists of a series of stars. The initial state of the system has an equal probability distribution among all states, reflecting no a priori knowledge about the location of the final node. But by applying the quantum random walk, the final state of the system reaches a superposition where nearly all of the probability is concentrated along the path leading to the marked final node. Lastly, we extend this idea of using a quantum random walk to reveal a path even further. We examine a specific type of graph G, called an “N-tree” graph, and apply our quantum random walk scheme. An N-tree graph can be described as a maze where the solver is presented with a junction consisting of N choices. Beyond each choice, is an identical junction of N choices, and so on. The depth of the maze is categorized by a parameter M, thus, to correctly solve the maze one must make M correct choices in succession. Labeling the marked final nodes as F, we present a quantum algorithm that uses a quantum random walk to locate F faster than any classical algorithm. Specifically, the quantum random walk results in a final state of the system with a high probability concentration on all the states leading to F. Two methods for searching for F are then presented. The first approach mimics the ideology of a Grover Search, relying on repetitive searches until F is found. The second approach presents a novel use of past measurements to dictates how the algorithm proceeds next. Specifically, we take advantage of the fact that states along the path leading to F hold most of the probability in the system. In the end, the second quantum algorithm approach provides an speedup over the classical algorithm of the order O().