This thesis explores the use of entangled states in quantum computation and quantum information science. Entanglement, a quantum phenomenon with no classical counterpart, has been identified as an important and quantifiable resource in many areas of theoretical quantum information science, including quantum error correction, quantum cryptography, and quantum algorithms. We first investigate the equivalence classes of a particular class of entangled states (known as graph states due to their association with mathematical graphs) under local operations. We prove that for graph states corresponding to graphs with neither cycles of length 3 nor 4, the equivalence classes can be characterized in a very simple way. We also present software for analyzing and manipulating graph states. We then study quantum error-correcting codes whose codewords are highly entangled states. An important area of investigation concerning QECCs is to determine which resources are necessary in order to carry out any computation on the code to an arbitrary degree of accuracy, while simultaneously maintaining a high degree of resistance to noise. We prove that transversal gates, which are designed to prevent the propagation of errors through a system, are insufficient to achieve universal computation on almost all QECCs. Finally, we study the problem of creating efficient quantum circuits for creating entangling measurements. Entangling measurements can be used to harness the apparent extra computing power of quantum systems by allowing us to extract information about the global, collective properties of a quantum state using local measurements. We construct explicit quantum circuits that create entangling measurements, and show that these circuits scale polynomially in the input parameters.
This thesis explores the use of entangled states in quantum computation and quantum information science. Entanglement, a quantum phenomenon with no classical counterpart, has been identified as an important and quantifiable resource in many areas of theoretical quantum information science, including quantum error correction, quantum cryptography, and quantum algorithms. We first investigate the equivalence classes of a particular class of entangled states (known as graph states due to their association with mathematical graphs) under local operations. We prove that for graph states corresponding to graphs with neither cycles of length 3 nor 4, the equivalence classes can be characterized in a very simple way. We also present software for analyzing and manipulating graph states. We then study quantum error-correcting codes whose codewords are highly entangled states. An important area of investigation concerning QECCs is to determine which resources are necessary in order to carry out any computation on the code to an arbitrary degree of accuracy, while simultaneously maintaining a high degree of resistance to noise. We prove that transversal gates, which are designed to prevent the propagation of errors through a system, are insufficient to achieve universal computation on almost all QECCs. Finally, we study the problem of creating efficient quantum circuits for creating entangling measurements. Entangling measurements can be used to harness the apparent extra computing power of quantum systems by allowing us to extract information about the global, collective properties of a quantum state using local measurements. We construct explicit quantum circuits that create entangling measurements, and show that these circuits scale polynomially in the input parameters.
Graph states form a rich class of entangled states that exhibit important aspects of multi-partite entanglement. At the same time, they can be described by a number of parameters that grows only moderately with the system size. They have a variety of applications in quantum information theory, most prominently as algorithmic resources in the context of the one-way quantum computer, but also in other fields such as quantum error correction and multi-partite quantum communication, as well as in the study of foundational issues such as non-locality and decoherence. In this review, we give a tutorial introduction into the theory of graph states. We introduce various equivalent ways how to define graph states, and discuss the basic notions and properties of these states. The focus of this review is on their entanglement properties. These include aspects of non-locality, bi-partite and multi-partite entanglement and its classification in terms of the Schmidt measure, the distillability properties of mixed entangled states close to a pure graph state, as well as the robustness of their entanglement under decoherence. We review some of the known applications of graph states, as well as proposals for their experimental implementation.
Graph states form a rich class of entangled states that exhibit important aspects of multi-partite entanglement. At the same time, they can be described by a number of parameters that grows only moderately with the system size. They have a variety of applications in quantum information theory, most prominently as algorithmic resources in the context of the one-way quantum computer, but also in other fields such as quantum error correction and multi-partite quantum communication, as well as in the study of foundational issues such as non-locality and decoherence. In this review, we give a tutorial introduction into the theory of graph states. We introduce various equivalent ways how to define graph states, and discuss the basic notions and properties of these states. The focus of this review is on their entanglement properties. These include aspects of non-locality, bi-partite and multi-partite entanglement and its classification in terms of the Schmidt measure, the distillability properties of mixed entangled states close to a pure graph state, as well as the robustness of their entanglement under decoherence. We review some of the known applications of graph states, as well as proposals for their experimental implementation.
Quantum computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. \n \nQuantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. \n \nThis thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. \n \nThree techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.
The subject of quantum computing brings together ideas from classical information theory, computer science, and quantum physics. This review aims to summarize not just quantum computing, but the whole subject of quantum information theory. Information can be identified as the most general thing which must propagate from a cause to an effect. It therefore has a fundamentally important role in the science of physics. However, the mathematical treatment of information, especially information processing, is quite recent, dating from the mid-20th century. This has meant that the full significance of information as a basic concept in physics is only now being discovered. This is especially true in quantum mechanics. The theory of quantum information and computing puts this significance on a firm footing, and has led to some profound and exciting new insights into the natural world. Among these are the use of quantum states to permit the secure transmission of classical information (quantum cryptography), the use of quantum entanglement to permit reliable transmission of quantum states (teleportation), the possibility of preserving quantum coherence in the presence of irreversible noise processes (quantum error correction), and the use of controlled quantum evolution for efficient computation (quantum computation). The common theme of all these insights is the use of quantum entanglement as a computational resource. It turns out that information theory and quantum mechanics fit together very well. In order to explain their relationship, this review begins with an introduction to classical information theory and computer science, including Shannon's theorem, error correcting codes, Turing machines and computational complexity. The principles of quantum mechanics are then outlined, and the Einstein, Podolsky and Rosen (EPR) experiment described. The EPR-Bell correlations, and quantum entanglement in general, form the essential new ingredient which distinguishes quantum from classical information theory and, arguably, quantum from classical physics. Basic quantum information ideas are next outlined, including qubits and data compression, quantum gates, the `no cloning' property and teleportation. Quantum cryptography is briefly sketched. The universal quantum computer (QC) is described, based on the Church-Turing principle and a network model of computation. Algorithms for such a computer are discussed, especially those for finding the period of a function, and searching a random list. Such algorithms prove that a QC of sufficiently precise construction is not only fundamentally different from any computer which can only manipulate classical information, but can compute a small class of functions with greater efficiency. This implies that some important computational tasks are impossible for any device apart from a QC. To build a universal QC is well beyond the abilities of current technology. However, the principles of quantum information physics can be tested on smaller devices. The current experimental situation is reviewed, with emphasis on the linear ion trap, high- Q optical cavities, and nuclear magnetic resonance methods. These allow coherent control in a Hilbert space of eight dimensions (three qubits) and should be extendable up to a thousand or more dimensions (10 qubits). Among other things, these systems will allow the feasibility of quantum computing to be assessed. In fact such experiments are so difficult that it seemed likely until recently that a practically useful QC (requiring, say, 1000 qubits) was actually ruled out by considerations of experimental imprecision and the unavoidable coupling between any system and its environment. However, a further fundamental part of quantum information physics provides a solution to this impasse. This is quantum error correction (QEC). An introduction to QEC is provided. The evolution of the QC is restricted to a carefully chosen subspace of its Hilbert space. Errors are almost certain to cause a departure from this subspace. QEC provides a means to detect and undo such departures without upsetting the quantum computation. This achieves the apparently impossible, since the computation preserves quantum coherence even though during its course all the qubits in the computer will have relaxed spontaneously many times. The review concludes with an outline of the main features of quantum information physics and avenues for future research.
Entanglement is the resource that most sharply distinguishes quantum information processing from its classical counterpart, and this thesis examines it from two complementary directions: how it can be detected and quantified, and how it, together with the broader machinery of quantum correlations, interference, and measurement, can be put to use. We first study two simple, measurable conditions that detect bipartite entanglement, and show that the extent to which either is violated yields a lower bound on the negativity of a state. The bound is established for two qubits, extended to systems of arbitrary finite dimension, and sharpened when partial knowledge of the Schmidt vectors is available. We then turn to discrete-outcome quantum sensor networks, in which one seeks only to determine which detectors, or which group or spatial pattern of detectors, has responded to a signal. Working in the weak-interaction regime, where the output states are nonorthogonal, we use minimum-error and unambiguous state discrimination to show that entangled initial states and collective measurements confer an advantage when detectors are grouped, and we develop a shift-operator formalism that gives closed-form success probabilities for the detection of shifted patterns in one- and two-dimensional arrays. Next we consider the problem of learning two unknown pure qubit states, together with their probabilities of occurrence, from a scrambled stream; we show that receiving the systems in identically prepared pairs supplies exactly the extra quantum information needed, since the resulting two-qubit state in the symmetric subspace has a unique decomposition into pure product states, which we recover by tomography. Finally, we show how interference and measurement can convert information encoded in the phases of quantum amplitudes into information encoded in their magnitudes, amplifying the amplitudes that carry useful information and suppressing those that do not, and we apply this technique to states produced by the quantum approximate optimization algorithm for the MaxCut and minimum-vertex-cover problems. Throughout, a single theme recurs: the correlations and superpositions permitted by quantum mechanics are at once something to be measured and something to be exploited.
Quantum communication and quantum computation are two important parts of quantum information science. Quantum algorithm usually uses real equally weighted states, among them are graph states, whose entanglement is well studied. Quantum communication is inevitably connected with quantum error correcting codes (QECC). The most important and frequently used QECCs are quantum stabilizer codes, which can be seen as the combination of graph states with classical error correcting codes. We argue that the complexity of quantum coding is closely related to the entanglement of the code. We prove that entanglement measured by the geometric measure, the robustness and the relative entropy of entanglement are equal for a stabilizer quantum codeword. The entanglement upper and lower bounds are determined from the generators of a quantum code. CSS codes are quantum codes derived from classical codes. We prove that the entanglement of CSS code equals its number of classical generators when the CSS code is dual-containing. We give the entanglement of codewords for Gottesman codes and related codes. An iterative algorithm is developed to calculate the entanglement numerically.