Date of Award

Spring 2022

Document Type

Open Access Dissertation

Department

Mathematics

First Advisor

George Androulakis

Abstract

In this manuscript we study entanglement measures defined via the convex roof construction. In the first chapter we build the notion of an entanglement measure from the ground up and discuss various issues that arise when trying to measure the amount of entanglement present in an arbitrary density operator. Through this introduction we will motivate the use of the convex roof construction. In the second chapter we will show that the infimum in the convex roof construction is achieved for a specific set of entanglement measures and provide canonical examples of such measures. We also describe LOCC operations via a tree structure and show this tree structure’s utility in proving LOCC monotonicity for candidate entanglement measures. In the final chapter we supply a variational quantum algorithm which allows for the approximation of the convex roof construction and show that the algorithm experiences the problem of exponentially vanishing gradients for a functional we call an entanglement detector. We showcase some numerical experiments of the algorithm that illustrate convergence for a small number of qubits. The work presented herein is the merging of two works coauthored with Dr. George Androulakis.

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Mathematics Commons

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