In earlier work, my colleagues and I developed a formalism for using information theory to understand scales of organization and structure in multi-component systems. One prominent theme of that work was that the structure of a system cannot always be decomposed into pairwise relationships. In this brief communication, I refine that formalism to address recent examples which bring out that theme in a novel and subtle way. After summarizing key points of earlier papers, I introduce the crucial new concept of an ancilla component, and I apply this refinement of our formalism to illustrative examples. The goals of this brief communication are, first, to show how a simple scheme for constructing ancillae can be useful in bringing out subtleties of structure, and second, to compare this scheme with another recent proposal in the same genre.
Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott's code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.
This paper represents an elaboration of the lectures delivered by one of us (CAF) during "Course 197 -- Foundations of Quantum Physics" at the International School of Physics "Enrico Fermi" in Varenna, Italy, July 2016. Much of the material for it is drawn from arXiv:1003.5209, arXiv:1401.7254, and arXiv:1405.2390. However there are substantial additions of original material in Sections 4, 7, 8 and 9, along with clarifications and expansions of the older content throughout. Topics include the meaning of subjective probability; no-cloning, teleportation, and quantum tomography from the subjectivist Bayesian perspective; the message QBism receives from Bell inequality violations (namely, that nature is creative); the import of symmetric informationally complete (SIC) quantum measurements for the technical side of QBism; quantum cosmology QBist-style; and a potential meaning for the holographic principle within QBism.
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.
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.
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.
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.
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.
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.
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.
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.
Over the last 10 years there has been an explosion of "operational reconstructions" of quantum theory. This is great stuff: For, through it, we come to see the myriad ways in which the quantum formalism can be chopped into primitives and, through clever toil, brought back together to form a smooth whole. An image of an IQ-Block puzzle comes to mind, http://www.prismenfernglas.de/iqblock_e.htm. There is no doubt that this is invaluable work, particularly for our understanding of the intricate connections between so many quantum information protocols. But to me, it seems to miss the mark for an ultimate understanding of quantum theory; I am left hungry. I still want to know what strange property of matter forces this formalism upon our information accounting. To play on something Einstein once wrote to Max Born, "The quantum reconstructions are certainly imposing. But an inner voice tells me that they are not yet the real thing. The reconstructions say a lot, but do not really bring us any closer to the secret of the `old one'." In this talk, I hope to expand on these points and convey some sense of why I am fascinated with the problem of the symmetric informationally complete POVMs to an extent greater than axiomatic reconstructions.
We study the mean field approximation to a simple spatial host-pathogen model that has been shown to display interesting evolutionary properties. We show that previous derivations of the mean field equations for this model are actually only low-density approximations to the true mean field limit. We derive the correct equations and the corresponding equations including pair-correlations. The process of invasion by a mutant type of pathogen is also discussed. [This article was published as Physical Review E 67, 047102 (2003). Errata for the published version are corrected here and explicitly listed at the end of this document.]
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.
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.
A substantial number of studies have extended the work on universal properties in physical systems to complex networks in social, biological, and technological systems. In this paper, we present a complex networks perspective on interfirm organizational networks by mapping, analyzing and modeling the spatial structure of a large interfirm competition network across a variety of sectors and industries within the United States. We propose two micro-dynamic models that are able to reproduce empirically observed characteristics of competition networks as a natural outcome of a minimal set of general mechanisms governing the formation of competition networks. Both models, which utilize different approaches yet apply common principles to network formation give comparable results. There is an asymmetry between companies that are considered competitors, and companies that consider others as their competitors. All companies only consider a small number of other companies as competitors; however, there are a few companies that are considered as competitors by many others. Geographically, the density of corporate headquarters strongly correlates with local population density, and the probability two firms are competitors declines with geographic distance. We construct these properties by growing a corporate network with competitive links using random incorporations modulated by population density and geographic distance. Our new analysis, methodology and empirical results are relevant to various phenomena of social and market behavior, and have implications to research fields such as economic geography, economic sociology, and regional economic development.
The dynamic network of relationships among corporations underlies cascading economic failures including the current economic crisis, and can be inferred from correlations in market value fluctuations. We analyze the time dependence of the network of correlations to reveal the changing relationships among the financial, technology, and basic materials sectors with rising and falling markets and resource constraints. The financial sector links otherwise weakly coupled economic sectors, particularly during economic declines. Such links increase economic risk and the extent of cascading failures. Our results suggest that firewalls between financial services for different sectors would reduce systemic risk without hampering economic growth.