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arXiv:1906.12099

Maximal Entanglement: Applications in Quantum Information and Particle Physics

Alba Cervera-Lierta

Published Jun 28, 2019• 1 citation
arXivHTMLPDF

Abstract

In this PhD thesis, several aspects regarding maximal entanglement are analyzed. In the first chapter, Bell Inequalities are analyzed from an operational perspective as well as novel Bell inequalities are obtained together with their optimal settings for a maximal violation. Multipartite figures of merit, in particular, the hyperdeterminant, are the subject of the second chapter. They are applied to detect quantum phase transitions in several spin models. The third chapter focuses on the simulation of the XY model in a quantum computer. The quantum circuit obtained is tested in three current quantum devices. Quantum computers must be able to generate and hold highly entangled states in order to show a quantum advantage. This statement is analyzed in chapter four, where quantum circuits for maximally entangled states are presented. Finally, chapter five analyzes how maximal entanglement is generated at its fundamental level. Maximal entanglement constraints the QED vertex interaction and predicts a value for the weak mixing angle close to the experimental value.

DOI
10.48550/arxiv.1906.12099

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In this PhD thesis, several aspects regarding maximal entanglement are analyzed. In the first chapter, Bell Inequalities are analyzed from an operational perspective as well as novel Bell inequalities are obtained together with their optimal settings for a maximal violation. Multipartite figures of merit, in particular, the hyperdeterminant, are the subject of the second chapter. They are applied to detect quantum phase transitions in several spin models. The third chapter focuses on the simulation of the XY model in a quantum computer. The quantum circuit obtained is tested in three current quantum devices. Quantum computers must be able to generate and hold highly entangled states in order to show a quantum advantage. This statement is analyzed in chapter four, where quantum circuits for maximally entangled states are presented. Finally, chapter five analyzes how maximal entanglement is generated at its fundamental level. Maximal entanglement constraints the QED vertex interaction and predicts a value for the weak mixing angle close to the experimental value.
Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, Gisin's Theorem ), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio R = 1/Tr(ρ 2 ) or their maximum eigenvalue λ max . This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so-called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four-qubit systems, where nonlocality for generalized Greenberger–Horne–Zeilinger states of arbitrary number of parties is computed.
Quantum information science has grown from being a very small subfield in the 70s until being one of the most dynamic fields in physics, both in fundamentals and applications. In the theoretical section, perhaps the feature that has attracted most interest is the notion of entanglement, the ghostly relation between particles that dazzled Einstein and has provided fabulous challenges to build a coherent interpretation of quantum mechanics. While not completely solved, we have today learned enough to feel less uneasy with this fundamental problem, and the focus has shifted towards its potential powerful applications. Entanglement is now being studied from different perspectives as a resource for performing information processing tasks.\n\n\t\t\t\t With bipartite entanglement being largely understood nowadays, many questions remain unanswered in the multipartite case. The first part of this thesis deals with multipartite entanglement in different contexts. In the first chapters it is studied within the whole corresponding Hilbert space, and we investigate several entanglement measures searching for states that maximize them, including violations of Bell inequalities. Later, focus is shifted towards hamiltonians that have entangled ground states, and we investigate entanglement as a way to establish a distance between theories and we study frustration and methods to efficiently solve hamiltonians that exhibit it.\n\n\t\t\t\t In the practical section, the most promised upcoming technological advance is the advent of quantum computers. In the 90s some quantum algorithms improving the performance of all known classical algorithms for certain problems started to appear, while in the 2000s the first universal computers of few atoms began to be built, allowing implementation of those algorithms in small scales. The D-Wave machine already performs quantum annealing in thousands of qubits, although some controversy over the true quantumness of its internal workings surrounds it. Many countries in the planet are devoting large amounts of money to this field, with the recent European flagship and the involvement of the largest US technological companies giving reasons for optimism.\n\n\t\t\t\t The second part of this thesis deals with some aspects of quantum computation, starting with the creation of the field of cloud quantum computation with the appearance of the first computer available to the general public through internet, which we have used and analysed extensively. Also small incursions in quantum adiabatic computation and quantum thermodynamics are present in this second part.
We explore the set of unitary matrices characterized by a given structure in the context of their applications in the field of Quantum Information. In the first part of the Thesis we focus on classification of special classes of unitary matrices and possibility of introducing certain internal parameterizations. Several new results and conjectures are discussed. Second part of the Thesis is devoted to the concept of multipartite quantum entanglement. We present a solution to the problem of absolutely maximally entangled states of four subsystems with six levels each. Finally, we analyze the excess of a matrix and the corresponding Bell inequalities. This combination allows us to draw new conclusions related to quantum nonlocality.
This dissertation describes the application of symmetry to multipartite quantum systems from the viewpoint of quantum information theory. It is written in the framework of abstract quantum information theory, i.e. without distinguishing the physical nature of the d-level systems considered. The results are mathematically rigorous and aim at investigating entanglement properties as well as operational properties of states of multipartite systems. The first chapter gathers the mathematical tools and concepts needed in the following chapters and can be omitted by the experienced reader. In chapter two we apply the concept of symmetry to multipartite systems and show how to construct families of symmetric states that can be described by few parameters only regardless of the size of single systems. Chapter three is devoted to the states of tripartite systems having Werner symmetry. We characterize their separability properties and analyze the strength of the known separability criteria. Furthermore we investigate their entanglement properties in terms of the relative entropy of entanglement, the entanglement monotone induced by the trace norm distance and the maximal violation of Bell inequalities. As a third aspect we analyze the possibility of embedding bipartite Werner states and explore the inner geometry of the manifold given by the statistical distance of two neighbouring tripartite Werner states. In the fourth chapter we introduce multipartite quantum data hiding as an application of multipartite Werner states. We prove the security of this protocol for any coalition and give various examples. The last chapter is concerned with the shared fidelity, its frustration and how it can be maximized in certain multipartite systems.