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Adrian Kent has recently criticized Masanes, Galley and M\"uller's work on postulates for quantum mechanics. MGM claim to find two contradictions in Kent's criticism. I argue that neither is a true contradiction unless some other premise is added., Comment: 5 pages, hopefully some of them constructive

I review some recent technical developments in quantum information theory by rephrasing them in the form of exercises., Comment: 9 pages of homework, just in time for the holidays

L. E. Ballentine's remarks in Physics Today about the QBist interpretation of quantum mechanics are generally wide of the mark., Comment: 6 pages nearly exactly

In a recent article, Khrennikov claims that a particular theorem about agreement between quantum measurement results poses a problem for the interpretation of quantum mechanics known as QBism. Considering the basic setup of that theorem in light of the meaning that QBism gives to probability shows that the claim is unfounded., Comment: 5 pages

A recent phase transition in the relational interpretation of quantum mechanics (RQM) is situated in its historical context, and the novelty of the post-transition viewpoint is questioned., Comment: 4 pages

Masanes, Galley and M\"uller claim to have derived a unique rule for quantum state update consequent upon a measurement outcome. Upon closer examination, their proof implicitly assumes its first step, namely that the state-update rule is linear., Comment: 6 pages; v2: added reply to their reply to my comment, trying to avoid talking past one another

I gamely try to disentangle ideas that have been confused with one another., Comment: 4 pages

I provide the backstory on how a historical error in the Feynman Lectures on Physics was corrected., Comment: 4 pages; 1 sense of puzzle- or vague bemusement

Brukner and Pienaar have critiqued the Relational Quantum Mechanics of Rovelli, and together with Di Biagio, the latter has replied. I point out a few places where, in my view, that reply needs clarification., Comment: 3 and a bit pages. arXiv admin note: text overlap with arXiv:2110.03610

Why I'm not happy with how Relational Quantum Mechanics has addressed the reconstruction of quantum theory, and why you shouldn't be either., Comment: 13 pages, 0 figures; v3: ~1 paragraph rewritten to asymptotically approach clarity, bibliography updated

I comment briefly on derivations of the Born rule presented by Masanes et al. and by Hossenfelder., Comment: 6 pages, 1 SciRate comment all grown up

We remark on John Earman's paper ``Quantum Bayesianism Assessed'' [The Monist 102 (2019), 403--423], illustrating with a number of examples that the quantum ``interpretation'' Earman critiques and the interpretation known as QBism have almost nothing to do with each other., Comment: 12 pages

The subjective Bayesian interpretation of probability asserts that the rules of the probability calculus follow from the normative principle of Dutch-book coherence: A decision-making agent should not assign probabilities such that a series of monetary transactions based on those probabilities would lead them to expect a sure loss. Similarly, the subjective Bayesian interpretation of quantum mechanics (QBism) asserts that the Born rule is a normative rule in analogy to Dutch-book coherence, but with the addition of one or more empirically based assumptions -- i.e., the "only a little more" that connects quantum theory to the particular characteristics of the physical world. Here we make this link explicit for a conjectured representation of the Born rule which holds true if symmetric informationally complete POVMs (or SICs) exist for every finite dimensional Hilbert space. We prove that an agent who thinks they are gambling on the outcomes of measurements on a sufficiently quantum-like system, but refuses to use this form of the Born rule when placing their bets is vulnerable to a Dutch book. The key property for being sufficiently quantum-like is that the system admits a symmetric reference measurement, but that this measurement is not sampling any hidden variables., Comment: 13 pages, 35 references, 3 appendices, 1 miserable year finally over and done with; v2: title change to match journal version

I introduce the problem of finding maximal sets of equiangular lines, in both its real and complex versions, attempting to write the treatment that I would have wanted when I first encountered the subject. Equiangular lines intersect in the overlap region of quantum information theory, the octonions and Hilbert's twelfth problem. The question of how many equiangular lines can fit into a space of a given dimension is easy to pose -- a high-school student can grasp it -- yet it is hard to answer, being as yet unresolved. This contrast of ease and difficulty gives the problem a classic charm., Comment: 9 pages; a bit about the Fano plane reused from arXiv:1911.05809

The Bohr--Einstein debate is one of the more remarkable protracted intellectual exchanges in the history of physics. Its influence has been lasting: One of the few clear patterns in a 2013 survey about quantum foundations was that the physicists who believed Bohr to be correct were apt to say that Einstein had been wrong. The exchanges began when Bohr and Einstein first met in 1920, continued at the Solvay conferences of the following decade, reached a dramatic crescendo with the EPR paradox in 1935, and continued thereafter. Not every episode in this long story has been investigated equally. In particular, one late statement attributed to Bohr has received much more intense examination than Einstein's equally pithy reply., Comment: presented at the Virtual Symposium on Borges and the Quantum, 32 March 2020 (session chair, Prof. W. Mozart)

In the last five years of his life Itamar Pitowsky developed the idea that the formal structure of quantum theory should be thought of as a Bayesian probability theory adapted to the empirical situation that Nature's events just so happen to conform to a non-Boolean algebra. QBism too takes a Bayesian stance on the probabilities of quantum theory, but its probabilities are the personal degrees of belief a sufficiently-schooled agent holds for the consequences of her actions on the external world. Thus QBism has two levels of the personal where the Pitowskyan view has one. The differences go further. Most important for the technical side of both views is the quantum mechanical Born Rule, but in the Pitowskyan development it is a theorem, not a postulate, arising in the way of Gleason from the primary empirical assumption of a non-Boolean algebra. QBism on the other hand strives to develop a way to think of the Born Rule in a pre-algebraic setting, so that it itself may be taken as the primary empirical statement of the theory. In other words, the hope in QBism is that, suitably understood, the Born Rule is quantum theory's most fundamental postulate, with the Hilbert space formalism (along with its perceived connection to a non-Boolean event structure) arising only secondarily. This paper will avail of Pitowsky's program, along with its extensions in the work of Jeffrey Bub and William Demopoulos, to better explicate QBism's aims and goals., Comment: 32 pages; Appendix B contains extracts that were formerly buried deep within arXiv:1405.2390

Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, "probability" distributions that can go negative. Informationally complete POVMs, a much younger subject than phase space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born Rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonalizations of minimal informationally complete measurements (MICs). By not imposing a particular discrete phase space structure at the outset, we push Wigner functions to their limits in a suitably quantified sense, revealing a new way in which the symmetric informationally complete measurements (SICs) are significant. Finally, we speculate that astute choices of MICs from the orthogonalization preimages of Wigner bases may in general give quantum measurements conceptually underlying the associated quasiprobability representations., Comment: 13 pages, sequel to arXiv:1812.08762; v2: minor updates in response to feedback; v3: 16 pages, substantial revision and additions

The interpretation of quantum mechanics known as QBism developed out of efforts to understand the probabilities arising in quantum physics as Bayesian in character. But this development was neither easy nor without casualties. Many ideas voiced, and even committed to print, during earlier stages of Quantum Bayesianism turn out to be quite fallacious when seen from the vantage point of QBism., Comment: 21 pages, 8 footnotes, 1 demonstration that not everything from the '90s has held up as well as Daria and OK Computer

Sometimes, mathematical oddities crowd in upon one another, and the exceptions to one classification scheme reveal themselves as fellow-travelers with the exceptions to a quite different taxonomy., Comment: 20 pages, 1 figure; based on guest posts at the n-Category Caf\'e

The L\"uders rule provides a way to define a quantum channel given a quantum measurement. Using this construction, we establish an if-and-only-if condition for the existence of a $d$-dimensional Symmetric Informationally Complete quantum measurement (a SIC) in terms of a particular depolarizing channel. Moreover, the channel in question satisfies two entropic optimality criteria., Comment: 9 pages. v2: Title and other changes to match journal version. Important postscript pointing to arXiv:1805.04583 by Pandey et al. Added proof of special case of conjecture appearing in v1

I correct two misapprehensions, one historical and one conceptual, in the recent literature on extensions of the Wigner's Friend thought-experiment. Perhaps fittingly, both concern the accurate description of some quantum physicists' beliefs by others., Comment: 3 pages, 17 references; v2: date corrected, opportunity taken to adjust a few sentences, conclusions unchanged

I summarize a research program that aims to reconstruct quantum theory from a fundamental physical principle that, while a quantum system has no intrinsic hidden variables, it can be understood using a reference measurement. This program reduces the physical question of why the quantum formalism is empirically successful to the mathematical question of why complete sets of equiangular lines appear to exist in complex vector spaces when they do not exist in real ones. My primary goal is to clarify motivations, rather than to present a closed book of numbered theorems, and consequently the discussion is more in the manner of a colloquium than a PRL., Comment: 31 pages, 1 figure; the tone and topic of a Chris Fuchs samizdat, but 1/100th the size; v5: results of complex and quaternionic SIC searches brought up to date

Coherence, treated as a resource in quantum information theory, is a basis-dependent quantity. Looking for states that have constant coherence under canonical changes of basis yields highly symmetric structures in state space. For the case of a qubit, we find an easy construction of qubit SICs (Symmetric Informationally Complete POVMs). SICs in dimension 3 and 8 are also shown to be equicoherent., Comment: 7 pages

Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, number theory and probability. Among our results are some equivalent conditions for unbiased MICs, a characterization of rank-1 MICs through the Hadamard product, several ways in which immediate properties of MICs capture the abandonment of classical phase space intuitions, and a numerical study of MIC Gram matrix spectra. We also present, to our knowledge, the first example of an unbiased rank-1 MIC which is not group covariant. This work provides further context to the discovery that the symmetric informationally complete quantum measurements (SICs) are in many ways optimal among MICs. In a deep sense, the ideal measurements of quantum physics are not orthogonal bases., Comment: 23 pages, 4 figures, 4 open conjectures; v5: title change to journal version, minor updates

In 1996, N. David Mermin proposed a set of desiderata for an understanding of quantum mechanics, the "Ithaca Interpretation". In 2012, Mermin became a public advocate of QBism, an interpretation due to Christopher Fuchs and Ruediger Schack. Here, we evaluate QBism with respect to the Ithaca Interpretation's six desiderata, in the process also evaluating those desiderata themselves. This analysis reveals a genuine distinction between QBism and the IIQM, but also a natural progression from one to the other., Comment: 13 pages, no figures except those of speech

We answer several questions that have been Frequently Asked about QBism. These remarks (many of them lighthearted) should be considered supplements to more systematic treatments by the authors and others., Comment: 40 pages; 24 (Q,A) ordered pairs

Notwithstanding its great influence in modern physics, the EPR thought-experiment has been explained incorrectly a surprising number of times., Comment: 23 pages; an unknown number of bridges burned

Informationally complete measurements are a dramatic discovery of quantum information science, and the symmetric IC measurements, known as SICs, are in many ways optimal among them. Close study of three of the "sporadic SICs" reveals an illuminating relation between different ways of quantifying the extent to which quantum theory deviates from classical expectations., Comment: 19 pages, 0 figures, more footnotes than originally intended

On the fifth of February, 2018, the Dow Jones Industrial Average dropped 1,175.21 points, the largest single-day fall in history in raw point terms. This followed a 666-point loss on the second, and another drop of over a thousand points occurred three days later. It is natural to ask whether these events indicate a transition to a new regime of market behavior, particularly given the dramatic fluctuations --- both gains and losses --- in the weeks since. To illuminate this matter, we can apply a model grounded in the science of complex systems, a model that demonstrated considerable success at unraveling the stock-market dynamics from the 1980s through the 2000s. By using large-scale comovement of stock prices as an early indicator of unhealthy market dynamics, this work found that abrupt drops in a certain parameter $U$ provide an early warning of single-day panics and economic crises. Decreases in $U$ indicate regimes of "high co-movement", a market behavior that is not the same as volatility, though market volatility can be a component of co-movement. Applying the same analysis to stock-price data from the beginning of 2016 until now, we find that the $U$ value for the period since 5 February is significantly lower than for the period before. This decrease entered the "danger zone" in the last week of May, 2018., Comment: 3 pages, 1 figure

We describe a general procedure for associating a minimal informationally-complete quantum measurement (or MIC) and a set of linearly independent post-measurement quantum states with a purely probabilistic representation of the Born Rule. Such representations are motivated by QBism, where the Born Rule is understood as a consistency condition between probabilities assigned to the outcomes of one experiment in terms of the probabilities assigned to the outcomes of other experiments. In this setting, the difference between quantum and classical physics is the way their physical assumptions augment bare probability theory: Classical physics corresponds to a trivial augmentation -- one just applies the Law of Total Probability (LTP) between the scenarios -- while quantum theory makes use of the Born Rule expressed in one or another of the forms of our general procedure. To mark the irreducible difference between quantum and classical, one should seek the representations that minimize the disparity between the expressions. We prove that the representation of the Born Rule obtained from a symmetric informationally-complete measurement (or SIC) minimizes this distinction in at least two senses -- the first to do with unitarily invariant distance measures between the rules, and the second to do with available volume in a reference probability simplex (roughly speaking a new kind of uncertainty principle). Both of these arise from a significant majorization result. This work complements recent studies in quantum computation where the deviation of the Born Rule from the LTP is measured in terms of negativity of Wigner functions., Comment: 10 pages, 2 figures; v4: corrected error in the last step of Theorem 1, strangely enough making the theorem stronger than we had believed

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., Comment: 8 pages, 0 figures, 1 term from quantum information theory reappropriated for classical ends; v2: expansions and edits in response to feedback

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., Comment: 16 pages, 1 figure, many references; v3: updating bibliography, dimension eight hundred forty four

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., Comment: 39 pages, 8 figures (1 of a superhero); lectures based on arXiv:1003.5209, arXiv:1401.7254 and arXiv:1405.2390 plus new content; v3: bibliography glitch-fixes

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., Comment: 30 pages, 4 figures, 3 appendices; v2: typos and notational glitches corrected

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., Comment: 15 pages, 3 figures, 2 tables

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., Comment: 2 pages, at the expense of deleting the definition of a Moufang loop

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., Comment: PhD thesis, 274 pages. Includes and updates material from arXiv:1110.3845

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., Comment: 25 pages, written in response to 1 sentence on Wikipedia. v3: journal version, plus deleted scenes

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., Comment: 60 pages, 10 figures

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., Comment: CAUTION, do not unthinkingly print from a printer: 2,349 pages, 4 indices, 6 figures, with extensive hyperlinking. Foreword by M. Schlosshauer, edited by B. C. Stacey. v2: more footnotes, fewer typos

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., Comment: 15 pages, 4 MUBs, 2 errata for CFS (2002), 1 graph with chromatic number 4. v4: journal version

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 comes to mind of a brain-teaser puzzle, all sliding and interlocking pieces. 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., Comment: Edited transcript of a talk given by one of us (CAF) at the Conceptual Foundations and Foils for Quantum Information Processing workshop, Perimeter Institute, 10 May 2011. v2: affiliations updated, broken URL removed

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., Comment: report for the Chief of Naval Operations Strategic Studies Group, June 2008; made public 28 February 2013