Nonlocality under Computational Assumptions

Nonlocality and its connections to entanglement are fundamental features of quantum mechanics that have found numerous applications in quantum information science. A set of correlations is said to be nonlocal if it cannot be reproduced by spacelike-separated parties sharing randomness and performing local operations. An important practical consideration is that the runtime of the parties has to be shorter than the time it takes light to travel between them. One way to model this restriction is to assume that the parties are computationally bounded. We therefore initiate the study of nonlocality under computational assumptions and derive the following results: (a) We define the set $\mathsf{NeL}$ (not-efficiently-local) as consisting of all bipartite states whose correlations arising from local measurements cannot be reproduced with shared randomness and \emph{polynomial-time} local operations. (b) Under the assumption that the Learning With Errors problem cannot be solved in \emph{quantum} polynomial-time, we show that $\mathsf{NeL}=\mathsf{ENT}$, where $\mathsf{ENT}$ is the set of \emph{all} bipartite entangled states (pure and mixed). This is in contrast to the standard notion of nonlocality where it is known that some entangled states, e.g. Werner states, are local. In essence, we show that there exist (efficient) local measurements producing correlations that cannot be reproduced through shared randomness and quantum polynomial-time computation. (c) We prove that if $\mathsf{NeL}=\mathsf{ENT}$ unconditionally, then $\mathsf{BQP}\neq\mathsf{PP}$. In other words, the ability to certify all bipartite entangled states against computationally bounded adversaries gives a non-trivial separation of complexity classes. (d) Using (c), we show that a certain natural class of 1-round delegated quantum computation protocols that are sound against $\mathsf{PP}$ provers cannot exist.Comment: 65 page

Demonstrating and testing the Deutsch-Jozsa Quantum Algorithm Towards the Realization of Quantum Computing at BSU

The world is changing, and fast. Quantum computing and photonic engineering are revolutionary new technologies that could change the way humans interact with information; though the eld hasn\u27t always been that way. As with most new elds, proof of concept is needed to show that this new technology isn\u27t just hear to stay, but it\u27s hear to take the lead. In this, nothing is more important the the Deutsch-Jozsa Quantum algorithm; as it did just that . The majority of this research paper revolves around understanding the very essence of quantum computing. As the eld of quantum computing is in its extreme infancy, most of this paper focuses on the theoretical background needed to understand how quantum computers function with a main use of the Deutsch Jozsa algorithm to really drive forward an application the theoretical research done here. At its core, this paper uses many works from Quantum Mechanics: Theory and Experiment by Mark Beck, and Quantum Optics: An Introduction by Mark Fox. If at any point one wishes to delve deeper into the topics discussed here, please refer to these texts. Addendum addressing early 2020 COVID outbreak. Unfortunately; while this thesis presents many interesting ideas in the elds of quantum computing, quantum teleportation, and quantum algorithms, the breadth of the text isn\u27t as deep as initially hoped. Just before spring break of Spring 2020 Dr. Serna and myself had laid out some plans to do testing of some of the quantum properties outlined in this paper. We were to use several cutting edge devices that had recently been purchased via LEAP grants provided to the BSU Photonics program, but the eects of the COVID outbreak prohibited us from getting into the lab and running these useful parallel experiments

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue

Towards higher-dimensional structured light

Structured light refers to the arbitrarily tailoring of optical fields in all their degrees of freedom (DoFs), from spatial to temporal. Although orbital angular momentum (OAM) is perhaps the most topical example, and celebrating 30 years since its connection to the spatial structure of light, control over other DoFs is slowly gaining traction, promising access to higher-dimensional forms of structured light. Nevertheless, harnessing these new DoFs in quantum and classical states remains challenging, with the toolkit still in its infancy. In this perspective, we discuss methods, challenges, and opportunities for the creation, detection, and control of multiple DoFs for higher-dimensional structured light. We present a roadmap for future development trends, from fundamental research to applications, concentrating on the potential for larger-capacity, higher-security information processing and communication, and beyond

Cryptography from quantum uncertainty in the presence of quantum side information

The thesis starts with a high-level introduction into cryptography and quantum mechanics. Chapter 2 gives a theoretical foundation by introducing probability theory, information theory, functional analysis, quantum mechanics and quantum information theory. Chapter 3, 4 and 5 are editions of work published earlier. In Chapter 3, we present a quantum-information-theoretic tool to analyze random sampling in a quantum setting. In particular, we present two new rigorous security proofs that make use of our new sampling tool: one for BB84 quantum key distribution, and one for a quantum reduction from oblivious transfer (OT) to bit commitment. Chapter 4 studies the problem of message authentication from a weak key (which is a key that is not uniformly random, e.g., a password) in a new scenario. In this scenario, the weak key is a one-time session key that is derived from a public source of randomness with the help of a long-term key (e.g., a password). We propose a new four-round protocol for message authentication from a weak (session) key. In Chapter 5 we present a new entropic uncertainty relation and furthermore we consider the task of password-based identification. We devise a new quantum identification protocol that is secure in two security models simultaneously.LEI Universiteit LeidenNWO Vrije competitieAlgebra en meetkund

The Schmidt rank for the commuting operator framework

In quantum information theory, the Schmidt rank is a fundamental measure for the entanglement dimension of a pure bipartite state. Its natural definition uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which does not exist (or at least is not canonically given) if the observable algebras of the local systems are allowed to be general C*-algebras. In this work, we generalize the Schmidt rank to the commuting operator framework where the joint system is not necessarily described by the minimal tensor product but by a general bipartite algebra. We give algebraic and operational definitions for the Schmidt rank and show their equivalence. We analyze bipartite states and compute the Schmidt rank in several examples: The vacuum in quantum field theory, Araki-Woods-Powers states, as well as ground states and translation invariant states on spin chains which are viewed as bipartite systems for the left and right half chains. We conclude with a list of open problems for the commuting operator framework.Comment: 44 pages, 3 figure