arXiv subjects

Blake C. Stacey

Publications and source records attributed to Blake C. Stacey.

46 records · Page 3Linked to original sources

arXiv2017

Introducing the Qplex: A Novel Arena for Quantum Theory

We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, "Unperformed experiments have no results." The tools of modern quantum information theory, and in particular the symmetric informationally complete (SIC) measurements, provide a concise expression of how exactly Peres's dictum holds true. That expression is a constraint on how the probability distributions for outcomes of different, mutually exclusive experiments mesh together, a type of constraint not foreseen in classical thinking. Taking this as our foundational principle, we show how to reconstruct the formalism of quantum theory in finite-dimensional Hilbert spaces. Along the way, we derive a condition for the existence of a d-dimensional SIC.

quant-ph↗

arXiv2016

Geometric and Information-Theoretic Properties of the Hoggar Lines

We take a tour of a set of equiangular lines in eight-dimensional Hilbert space. This structure defines an informationally complete measurement, that is, a way to represent all quantum states of three-qubit systems as probability distributions. Investigating the shape of this representation of state space yields a pattern of connections among a remarkable spread of mathematical constructions. In particular, studying the Shannon entropy of probabilistic representations of quantum states leads to an intriguing link between the questions of real and of complex equiangular lines. Furthermore, we will find relations between quantum information theory and mathematical topics like octonionic integers and the 28 bitangents to a quartic curve.

quant-ph↗

arXiv2016

SIC-POVMs and Compatibility among Quantum States

An unexpected connection exists between compatibility criteria for quantum states and symmetric informationally complete POVMs. Beginning with Caves, Fuchs and Schack's "Conditions for compatibility of quantum state assignments" [Phys. Rev. A 66 (2002), 062111], I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.

quant-ph↗

arXiv2016

Sporadic SICs and the Normed Division Algebras

Recently, Zhu classified all the SIC-POVMs whose symmetry groups act doubly transitively. Lattices of integers in the complex numbers, the quaternions and the octonions yield the key parts of these symmetry groups.

quant-ph↗

arXiv2016

Von Neumann Was Not a Quantum Bayesian

Wikipedia has claimed for over three years now that John von Neumann was the "first quantum Bayesian." In context, this reads as stating that von Neumann inaugurated QBism, the approach to quantum theory promoted by Fuchs, Mermin and Schack. This essay explores how such a claim is, historically speaking, unsupported.

physics.hist-ph↗

arXiv2015

Multiscale Structure in Eco-Evolutionary Dynamics

In a complex system, the individual components are neither so tightly coupled or correlated that they can all be treated as a single unit, nor so uncorrelated that they can be approximated as independent entities. Instead, patterns of interdependency lead to structure at multiple scales of organization. Evolution excels at producing such complex structures. In turn, the existence of these complex interrelationships within a biological system affects the evolutionary dynamics of that system. I present a mathematical formalism for multiscale structure, grounded in information theory, which makes these intuitions quantitative, and I show how dynamics defined in terms of population genetics or evolutionary game theory can lead to multiscale organization. For complex systems, "more is different," and I address this from several perspectives. Spatial host--consumer models demonstrate the importance of the structures which can arise due to dynamical pattern formation. Evolutionary game theory reveals the novel effects which can result from multiplayer games, nonlinear payoffs and ecological stochasticity. Replicator dynamics in an environment with mesoscale structure relates to generalized conditionalization rules in probability theory. The idea of natural selection "acting at multiple levels" has been mathematized in a variety of ways, not all of which are equivalent. We will face down the confusion, using the experience developed over the course of this thesis to clarify the situation.

q-bio.PE↗

arXiv2015

My Struggles with the Block Universe

This document is the second installment of three in the Cerro Grande Fire Series. Like its predecessor arXiv:quant-ph/0105039, "Notes on a Paulian Idea," it is a collection of letters written to various friends and colleagues, most of whom regularly circuit this archive. The unifying theme of all the letters is that each has something to do with the quantum. Particularly, the collection chronicles the emergence of Quantum Bayesianism as a robust view of quantum theory, eventually evolving into the still-more-radical "QBism" (with the B standing for no particular designation anymore), as it took its most distinctive turn away from various Copenhagen Interpretations. Included are many anecdotes from the history of quantum information theory: for instance, the story of the origin of the terms "qubit" and "quantum information" from their originator's own mouth, a copy of a rejection letter written by E. T. Jaynes for one of Rolf Landauer's original erasure-cost principle papers, and much more. Specialized indices are devoted to historical, technical, and philosophical matters. More roundly, the document is an attempt to provide an essential ingredient, unavailable anywhere else, for turning QBism into a live option within the vast spectrum of quantum foundational thought.

quant-ph↗

arXiv2014

An Information-Theoretic Formalism for Multiscale Structure in Complex Systems

We develop a general formalism for representing and understanding structure in complex systems. In our view, structure is the totality of relationships among a system's components, and these relationships can be quantified using information theory. In the interest of flexibility we allow information to be quantified using any function, including Shannon entropy and Kolmogorov complexity, that satisfies certain fundamental axioms. Using these axioms, we formalize the notion of a dependency among components, and show how a system's structure is revealed in the amount of information assigned to each dependency. We explore quantitative indices that summarize system structure, providing a new formal basis for the complexity profile and introducing a new index, the "marginal utility of information". Using simple examples, we show how these indices capture intuitive ideas about structure in a quantitative way. Our formalism also sheds light on a longstanding mystery: that the mutual information of three or more variables can be negative. We discuss applications to complex networks, gene regulation, the kinetic theory of fluids and multiscale cybernetic thermodynamics.

cond-mat.stat-mech↗

arXiv2013

Eco-Evolutionary Feedback in Host--Pathogen Spatial Dynamics

Spatial extent is a complicating factor in mathematical biology. The possibility that an action at point A cannot immediately affect what happens at point B creates the opportunity for spatial nonuniformity. This nonuniformity must change our understanding of evolutionary dynamics, as the same organism in different places can have different expected evolutionary outcomes. Since organism origins and fates are both determined locally, we must consider heterogeneity explicitly to determine its effects. We use simulations of spatially extended host--pathogen and predator--prey ecosystems to reveal the limitations of standard mathematical treatments of spatial heterogeneity. Our model ecosystem generates heterogeneity dynamically; an adaptive network of hosts on which pathogens are transmitted arises as an emergent phenomenon. The structure and dynamics of this network differ in significant ways from those of related models studied in the adaptive-network field. We use a new technique, organism swapping, to test the efficacy of both simple approximations and more elaborate moment-closure methods, and a new measure to reveal the timescale dependence of invasive-strain behavior. Our results demonstrate the failure not only of the most straightforward ("mean field") approximation, which smooths over heterogeneity entirely, but also of the standard correction ("pair approximation") to the mean field treatment. In spatial contexts, invasive pathogen varieties can prosper initially but perish in the medium term, implying that the concepts of reproductive fitness and the Evolutionary Stable Strategy have to be modified for such systems.

nlin.CG↗

arXiv2013

Principles of Security: Human, Cyber, and Biological

Cybersecurity attacks are a major and increasing burden to economic and social systems globally. Here we analyze the principles of security in different domains and demonstrate an architectural flaw in current cybersecurity. Cybersecurity is inherently weak because it is missing the ability to defend the overall system instead of individual computers. The current architecture enables all nodes in the computer network to communicate transparently with one another, so security would require protecting every computer in the network from all possible attacks. In contrast, other systems depend on system-wide protections. In providing conventional security, police patrol neighborhoods and the military secures borders, rather than defending each individual household. Likewise, in biology, the immune system provides security against viruses and bacteria using primarily action at the skin, membranes, and blood, rather than requiring each cell to defend itself. We propose applying these same principles to address the cybersecurity challenge. This will require: (a) Enabling pervasive distribution of self-propagating securityware and creating a developer community for such securityware, and (b) Modifying the protocols of internet routers to accommodate adaptive security software that would regulate internet traffic. The analysis of the immune system architecture provides many other principles that should be applied to cybersecurity. Among these principles is a careful interplay of detection and action that includes evolutionary improvement. However, achieving significant security gains by applying these principles depends strongly on remedying the underlying architectural limitations.

cs.CR↗