Bottleneck Problems: Information and Estimation-Theoretic View

Information bottleneck (IB) and privacy funnel (PF) are two closely related optimization problems which have found applications in machine learning, design of privacy algorithms, capacity problems (e.g., Mrs. Gerber's Lemma), strong data processing inequalities, among others. In this work, we first investigate the functional properties of IB and PF through a unified theoretical framework. We then connect them to three information-theoretic coding problems, namely hypothesis testing against independence, noisy source coding and dependence dilution. Leveraging these connections, we prove a new cardinality bound for the auxiliary variable in IB, making its computation more tractable for discrete random variables. In the second part, we introduce a general family of optimization problems, termed as \textit{bottleneck problems}, by replacing mutual information in IB and PF with other notions of mutual information, namely $f$-information and Arimoto's mutual information. We then argue that, unlike IB and PF, these problems lead to easily interpretable guarantee in a variety of inference tasks with statistical constraints on accuracy and privacy. Although the underlying optimization problems are non-convex, we develop a technique to evaluate bottleneck problems in closed form by equivalently expressing them in terms of lower convex or upper concave envelope of certain functions. By applying this technique to binary case, we derive closed form expressions for several bottleneck problems

Semidefinite programming relaxations for quantum correlations

Semidefinite programs are convex optimisation problems involving a linear objective function and a domain of positive semidefinite matrices. Over the last two decades, they have become an indispensable tool in quantum information science. Many otherwise intractable fundamental and applied problems can be successfully approached by means of relaxation to a semidefinite program. Here, we review such methodology in the context of quantum correlations. We discuss how the core idea of semidefinite relaxations can be adapted for a variety of research topics in quantum correlations, including nonlocality, quantum communication, quantum networks, entanglement, and quantum cryptography.Comment: To be submitted to Reviews of Modern Physic

Improvements on Device Independent and Semi-Device Independent Protocols of Randomness Expansion

To generate genuine random numbers, random number generators based on quantum theory are essential. However, ensuring that the process used to produce randomness meets desired security standards can pose challenges for traditional quantum random number generators. This thesis delves into Device Independent (DI) and Semi-Device Independent (semi-DI) protocols of randomness expansion, based on a minimal set of experimentally verifiable security assumptions. The security in DI protocols relies on the violation of Bell inequalities, which certify the quantum behavior of devices. The semi-DI protocols discussed in this thesis require the characterization of only one device - a power meter. These protocols exploit the fact that quantum states can be prepared such that they cannot be distinguished with certainty, thereby creating a randomness resource. In this study, we introduce enhanced DI and semi-DI protocols that surpass existing ones in terms of output randomness rate, security, or in some instances, both. Our analysis employs the Entropy Accumulation Theorem (EAT) to determine the extractable randomness for finite rounds. A notable contribution is the introduction of randomness expansion protocols that recycle input randomness, significantly enhancing finite round randomness rates for DI protocols based on the CHSH inequality violation. In the final section of the thesis, we delve into Generalized Probability Theories (GPTs), with a focus on Boxworld, the largest GPT capable of producing correlations consistent with relativity. A tractable criterion for identifying a Boxworld channel is presented.Comment: This PhD thesis consists of 212 pages, with 16 figures and presents content that intersects with the author's previously published work R. Bhavsar, S. Ragy, and R. Colbeck. Improved device independent randomness expansion rates using two sided randomness. New Journal of Physics 25.9 (2023): 09303

A framework for quantum-secure device-independent randomness expansion

A device-independent randomness expansion protocol aims to take an initial random seed and generate a longer one without relying on details of how the devices operate for security. A large amount of work to date has focussed on a particular protocol based on spot-checking devices using the CHSH inequality. Here we show how to derive randomness expansion rates for a wide range of protocols, with security against a quantum adversary. Our technique uses semidefinite programming and a recent improvement of the entropy accumulation theorem. To support the work and facilitate its use, we provide code that can generate lower bounds on the amount of randomness that can be output based on the measured quantities in the protocol. As an application, we give a protocol that robustly generates up to two bits of randomness per entangled qubit pair, which is twice that established in existing analyses of the spot-checking CHSH protocol in the low noise regime.Comment: 26 (+9) pages, 6 (+1) figures. v2: New result included (Fig. 7) and several updates made based on referee comment

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Holographic Studies of Entanglement Measures

This thesis consists of four research papers and an introduction covering the most important concepts appearing in the papers. The papers deal with applications of gauge/gravity dualities in the study of various physical quantities and systems. Gauge/gravity dualities are equivalences between certain quantum field theories and classical theories of gravity. These dualities can be used as computational tools in a wide range of applications across different fields of physics, and as such they have garnered much attention in the last two decades. The great promise of these new tools is the ability to tackle difficult problems in strongly interacting quantum field theories by translating them to problems in classical gravity, where progress is much easier to make. Quantum information theory studies the information contained in quantum systems. Entanglement is the fundamental property of quantum mechanics that sets it apart from classical theories of physics. Entanglement is commonly quantified by entanglement entropy, a quantity which is difficult to compute in interacting quantum field theories. Gauge/gravity dualities provide a practical way for computing the entanglement entropy via the Ryu-Takayanagi formula. The primary focus of this thesis is to use this formula for computing various entanglement measures in strongly interacting quantum field theories via their gravity duals. The purpose of this thesis is to introduce quantum information theory concepts that have been important in our research. When applicable, quantities of interest are first introduced in the classical setting in order to build intuition about their behaviour. Quantum properties of entanglement measures are discussed in detail, along with their holographic counterparts, and remarks are made concerning their applications.Kvanttilomittuminen on ilmiö jossa fysikaalisen systeemin eri osat käyttäytyvät kollektiivisesti tavalla, jolle ei ole klassista vastinetta. Lomittumisen olemassaolo on tunnettu teoreettisesti jo kvanttimekaniikan aikaisista ajoista asti ja ilmiö on myös havaittu kokeellisesti. Kvanttilomittuminen on tärkeä ominaisuus joka erottaa klassisen fysiikan kvanttimekaniikasta ja siksi sen tutkimus on olennaista niin fysiikan ymmärryksen kannalta kuten myös sen roolin vuoksi esimerkiksi kvanttilaskennassa. Holografiset dualiteetit ovat ekvivalensseja tiettyjen kvanttikenttäteorioiden ja gravitaation välillä. Näitä dualiteetteja käyttämällä on mahdollista tutkia suureita vahvasti kytkeytyneissä kvanttikenttäteorioissa esittämällä ongelma gravitaation avulla. Näin on mahdollista analysoida monia tilanteita joissa perinteiset laskukeinot eivät ole riittäviä. Tärkeä esimerkki helposti holografialla tutkittavasta suureesta on lomittumisentropia, suure joka mittaa kvanttilomittumisen määrää systeemin osien välillä. Tässä väitöskirjassa käytetään holografisia dualiteetteja erityisesti lomittumisentropian ja sen sukulaissuureiden kuten keskinäisinformaation tutkimiseen erilaisissa fysikaalisissa systeemeissä

Extremal problems in Fourier analysis, Whitney's theorem, and the interpolation of data

This dissertation deals with three problems in interpolation theory. The first two, the Beurling-Selberg box minorant problem and Turán's extremal problem, are optimization problems involving constrained interpolation by bandlimited functions. The Beurling-Selberg box minorant problem is a higher dimensional version of Selberg's minorant problem for the interval. We study the problem of minorizing the indicator function of the unit cube Q [subscript d] = [-1, 1] [superscript d] by a function bandlimited to Q [subscript d]. We show that there exists a dimension d* ≤ 710 such that if d > d* then there do not exist d-dimensional minorants. We also construct the first non-trivial minorants for dimensions 2, 3, 4, and 5. Next, we show how to compute upper and lower bounds for the value of Turán's extremal problem by solving finite dimensional linear programs. The problem depends on a convex body K; our bounds have been used to compute the sharpest known upper bound in the case in which K is the 3 dimensional ℓ₁ ball. The third problem we study concerns the interpolation of data by C [superscript m] functions. We give a new proof of the Brudnyi-Shvartsman-Fefferman finiteness principle for C [superscript m-1,1] (R [superscript d]) functions. We hope that this proof will lead to practical algorithms for C[superscript m] interpolationMathematic