Negative probabilities
We explain, on the example of Wigner's quasiprobability distribution, how negative probabilities may be used in the foundations of probability.
quant-ph↗
arXiv subjects
Yuri Gurevich
Publications and source records attributed to Yuri Gurevich.
Circuits: An abstract viewpoint
Our primary purpose is to isolate the abstract, mathematical properties of circuits -- both classical Boolean circuits and quantum circuits -- that are essential for their computational interpretation. A secondary purpose is to clarify the similarities and differences between the classical and quantum situations. The general philosophy in this note is to include the mathematically essential aspects of circuits but to omit any of the additional structures that are usually included for convenience. We shall, however, retain the assumption that circuits are finite; this assumption does no harm to the applicability of our approach and is necessary for some of our work.
quant-ph↗
Means-fit effectivity
Historically, the notion of effective algorithm is closely related to the Church-Turing thesis. But effectivity imposes no restriction on computation time or any other resource; in that sense, it is incompatible with engineering or physics. We propose a natural generalization of it, means-fitting effectivity, which is effectivity relative to the (physical or abstract) underlying machinery of the algorithm. This machinery varies from one class of algorithms to another. Think for example of ruler-and-compass algorithms, arithmetical algorithms, and Blum-Shub-Smale algorithms. We believe that means-fitting effectivity is meaningful and useful independently of the Church-Turing thesis. Means-fitting effectivity is definable, at least in the theory of abstract state machines (ASMs). The definition elucidates original effectivity as well. Familiarity with the ASM theory is not assumed. We tried to make the paper self-contained.
cs.LO↗
Braided distributivity
In category-theoretic models for the anyon systems proposed for topological quantum computing, the essential ingredients are two monoidal structures, $\oplus$ and $\otimes$. The former is symmetric but the latter is only braided, and $\otimes$ is required to distribute over $\oplus$. What are the appropriate coherence conditions for the distributivity isomorphisms? We came to this question working on a simplification of the category-theoretical foundation of topological quantum computing, which is the intended application of the research reported here. This question was answered by Laplaza when both monoidal structures are symmetric, but topological quantum computation depends crucially on $\otimes$ being only braided, not symmetric. We propose coherence conditions for distributivity in this situation, and we prove that our conditions are (a) strong enough to imply Laplaza's when the latter are suitably formulated, and (b) weak enough to hold when --- as in the categories used to model anyons --- the additive structure is that of an abelian category and the braided $\otimes$ is additive. Working on these results, we found a new redundancy in Laplaza's conditions.
math.CT↗
Who needs category theory?
In mathematical applications, category theory remains a contentious issue, with enthusiastic fans and a skeptical majority. In a muted form this split applies to the authors of this note. When we learned that the only mathematically sound foundation of topological quantum computing in the literature is based on category theory, the skeptical author suggested to "decategorize" the foundation. But we discovered, to our surprise, that category theory (or something like it) is necessary for the purpose, for computational reasons. The goal of this note is to give a high-level explanation of that necessity, which avoids details and which suggests that the case of topological quantum computing is far from unique.
cs.LO↗
Circuit pedantry
Boolean and quantum circuits have commonalities and differences. To formalize the syntactical commonality we introduce syntactic circuits where the gates are black boxes. Syntactic circuits support various semantics. One semantics is provided by Boolean circuits, another by quantum circuits. Quantum semantics is a generalization of Boolean but, because of entanglement, the generalization is not straightforward. We consider only unitary quantum circuits here.
quant-ph↗
Unconstrained Church-Turing thesis cannot possibly be true
The Church-Turing thesis asserts that if a partial strings-to-strings function is effectively computable then it is computable by a Turing machine. In the 1930s, when Church and Turing worked on their versions of the thesis, there was a robust notion of algorithm. These traditional algorithms are known also as classical or sequential. In the original thesis, effectively computable meant computable by an effective classical algorithm. Based on an earlier axiomatization of classical algorithms, the original thesis was proven in 2008. Since the 1930s, the notion of algorithm has changed dramatically. New species of algorithms have been and are being introduced. We argue that the generalization of the original thesis, where effectively computable means computable by an effective algorithm of any species, cannot possibly be true.
cs.LO↗
Evolving Algebras 1993: Lipari Guide
Computation models and specification methods seem to be worlds apart. The project on abstract state machines (in short ASMs, also known as evolving algebras) started as an attempt to bridge the gap by improving on Turing's thesis. We sought more versatile machines which would be able to step-for-step simulate arbitrary algorithms on their natural abstraction levels. The ASM thesis asserts that ASMs are such versatile machines. The guide provides the definitions of sequential, parallel and distributed ASMs.
cs.LO↗
Witness Algebra and Anyon Braiding
Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of the present framework necessary? The computations of associativity and braiding matrices can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.
quant-ph↗
Negative probabilities, II: What they are and what they are for
A signed probability distribution may extend a given traditional probability from observable events to all events. We formalize and illustrate this approach. We also illustrate its limitation. We argue that the right question is not what negative probabilities are but what they are for.
quant-ph↗
Common Denominator for Value and Expectation No-go Theorems: Extended Abstract
Hidden-variable (HV) theories allege that a quantum state describes an ensemble of systems distinguished by the values of hidden variables. No-go theorems assert that HV theories cannot match the predictions of quantum theory. The present work started with repairing flaws in the literature on no-go theorems asserting that HV theories cannot predict the expectation values of measurements. That literature gives one an impression that expectation no-go theorems subsume the time-honored no-go theorems asserting that HV theories cannot predict the possible values of measurements. But the two approaches speak about different kinds of measurement. This hinders comparing them to each other. Only projection measurements are common to both. Here, we sharpen the results of both approaches so that only projection measurements are used. This allows us to clarify the similarities and differences between the two approaches. Neither one dominates the other.
quant-ph↗
Common Denominator for Value and Expectation No-Go Theorems
Hidden-variable (HV) theories allege that a quantum state describes an ensemble of systems distinguished by the values of hidden variables. No-go theorems assert that HV theories cannot match the predictions of quantum theory. The present work started with repairing flaws in the literature on no-go theorems asserting that HV theories cannot predict the expectation values of measurements. That literature gives one an impression that expectation no-go theorems subsume the time-honored no-go theorems asserting that HV theories cannot predict the possible values of measurements. But the two approaches speak about different kinds of measurement. This hinders comparing them to each other. Only projection measurements are common to both. Here, we sharpen the results of both approaches so that only projection measurements are used. This allows us to clarify the similarities and differences between the two approaches. Neither one dominates the other.
quant-ph↗
Test statistics and p-values
We point out that the traditional notion of test statistic is too narrow, and we propose a natural generalization that is arguably maximal. The study is restricted to simple statistical hypotheses.
stat.ME↗
Fundamentals of p-values: Introduction
We explain the concept of p-values presupposing only rudimentary probability theory. We also use the occasion to introduce the notion of p-function, so that p-values are values of a p-function. The explanation is restricted to the discrete case with no outcomes of zero probability. We are going to address the general case elsewhere.
math.ST↗
Impugning Randomness, Convincingly
John organized a state lottery and his wife won the main prize. You may feel that the event of her winning wasn't particularly random, but how would you argue that in a fair court of law? Traditional probability theory does not even have the notion of random events. Algorithmic information theory does, but it is not applicable to real-world scenarios like the lottery one. We attempt to rectify that.
stat.OT↗
On quantum computation, anyons, and categories
We explain the use of category theory in describing certain sorts of anyons. Yoneda's lemma leads to a simplification of that description. For the particular case of Fibonacci anyons, we also exhibit some calculations that seem to be known to the experts but not explicit in the literature.
quant-ph↗
On Hidden Variables: Value and Expectation No-Go Theorems
No-go theorems assert that hidden-variable theories, subject to appropriate hypotheses, cannot reproduce the predictions of quantum theory. We examine two species of such theorems, value no-go theorems and expectation no-go theorems. The former assert that hidden-variables cannot match the predictions of quantum theory about the possible values resulting from measurements; the latter assert that hidden-variables cannot match the predictions of quantum theory about the expectation values of measurements. We sharpen the known results of both species, which allows us to clarify the similarities and differences between the two species. We also repair some flaws in existing definitions and proofs.
quant-ph↗
Inverse Privacy
An item of your personal information is inversely private if some party has access to it but you do not. We analyze the provenance of inversely private information and its rise to dominance over other kinds of personal information. In a nutshell, the inverse privacy problem is unjustified inaccessibility to you of your inversely private information. We argue that the inverse privacy problem has a market-based solution.
cs.CY↗