arXiv subjects

Blake C. Stacey

Publications and source records attributed to Blake C. Stacey.

At least 37 records · Page 2Linked to original sources

arXiv2020

The Varieties of Minimal Tomographically Complete Measurements

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.

quant-ph↗

arXiv2020

Maximal Sets of Equiangular Lines

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.

quant-ph↗

arXiv2020

Discrete Wigner Functions from Informationally Complete Quantum Measurements

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.

quant-ph↗

arXiv2020

An Underappreciated Exchange in the Bohr--Einstein Debate

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.

quant-ph↗

arXiv2020

Lüders Channels and the Existence of Symmetric Informationally Complete Measurements

The Lüders 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.

quant-ph↗

arXiv2019

Are Non-Boolean Event Structures the Precedence or Consequence of Quantum Probability?

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.

quant-ph↗

arXiv2019

Ideas Abandoned en Route to QBism

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.

quant-ph↗

arXiv2019

Sporadic SICs and Exceptional Lie Algebras

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.

quant-ph↗

arXiv2019

QBism: Quantum Theory as a Hero's Handbook

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.

quant-ph↗

arXiv2019

On QBism and Assumption (Q)

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.

quant-ph↗

arXiv2019

Invariant Off-Diagonality: SICs as Equicoherent Quantum States

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.

quant-ph↗

arXiv2019

FAQBism

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.

quant-ph↗

arXiv2018

QBism and the Ithaca Desiderata

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.

quant-ph↗

arXiv2018

Some Negative Remarks on Operational Approaches to Quantum Theory

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.

quant-ph↗

arXiv2018

Is the SIC Outcome There When Nobody Looks?

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.

quant-ph↗

arXiv2018

The Stock Market Has Grown Unstable Since February 2018

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.

q-fin.ST↗

arXiv2017

The SIC Question: History and State of Play

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.

quant-ph↗

arXiv2017

Multiscale Structure of More-than-Binary Variables

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.

cond-mat.stat-mech↗