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Axioms
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Quantum Statistical Inference
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Quantum Statistical Inference

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (31 March 2015) | Viewed by 27819

Special Issue Editor

Dr. James D. Malley

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Guest Editor
Research Mathematical Statistician Building 12A, Room 2052 Mathematical and Statistical Computing Laboratory Division of Computational Bioscience Center for Information Technology National Institutes of Health Bethesda, MD 20892 USA

Special Issue Information

Dear Colleagues,

As quantum information theory advances, as new experimental quantum protocols are devised, and as the reality of practical quantum computing approaches, it is important to match these developments with appropriate statistical and data analysis methods. For this Special Issue the topic is understood to cover: advances in the probabilistic foundations of quantum mechanics; new developments for statistical methods and data analysis of quantum outcomes; quantum information perspectives on the analysis quantum outcomes and experiment; examination of classical statistical methods that have possible application to quantum outcomes and decisions, such as quantum based statistical learning machines, and Bayesian quantum state estimation. The emphasis in this Special Issue will be on experiment and applications of theory that advance our understanding of what quantum data can mean and how it might be used. More speculative revisions of the foundations of quantum mechanics are not obviously included in this discussion, unless grounded in current experiment or near term realizations of the proposed ideas and methods.

Dr. James D. Malley
Guest Editor

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Published Papers (5 papers)

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Research

429 KiB  
Article
A Model for the Universe that Begins to Resemble a Quantum Computer
by Stan Gudder
Axioms 2015, 4(1), 102-119; https://doi.org/10.3390/axioms4010102 - 9 Mar 2015
Viewed by 4469
Abstract
This article presents a sequential growth model for the Universe that acts like a quantum computer. The basic constituents of the model are a special type of causal set (causet) called a c-causet. A c-causet is defined to be a causet [...] Read more.
This article presents a sequential growth model for the Universe that acts like a quantum computer. The basic constituents of the model are a special type of causal set (causet) called a c-causet. A c-causet is defined to be a causet that has a unique labeling. We characterize c-causets as those causets that form a multipartite graph or equivalently those causets whose elements are comparable whenever their heights are different. We show that a c-causet has precisely two c-causet offspring. It follows that there are 2n c-causets of cardinality n + 1. This enables us to classify c-causets of cardinality n + 1 in terms of n-bits. We then quantize the model by introducing a quantum sequential growth process. This is accomplished by replacing the n-bits by n-qubits and defining transition amplitudes for the growth transitions. We mainly consider two types of processes, called stationary and completely stationary. We show that for stationary processes, the probability operators are tensor products of positive rank-one qubit operators. Moreover, the converse of this result holds. Simplifications occur for completely stationary processes. We close with examples of precluded events. Full article
(This article belongs to the Special Issue Quantum Statistical Inference)
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1385 KiB  
Article
Positive-Operator Valued Measure (POVM) Quantization
by Jean Pierre Gazeau and Barbara Heller
Axioms 2015, 4(1), 1-29; https://doi.org/10.3390/axioms4010001 - 25 Dec 2014
Cited by 17 | Viewed by 8846
Abstract
We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We [...] Read more.
We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on various probabilistic aspects of these constructions. Simple ormore elaborate examples illustrate the procedure: circle, two-sphere, plane and half-plane. Links with Positive-Operator Valued Measure (POVM) quantum measurement and quantum statistical inference are sketched. Full article
(This article belongs to the Special Issue Quantum Statistical Inference)
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227 KiB  
Article
Classical Probability and Quantum Outcomes
by James D. Malley
Axioms 2014, 3(2), 244-259; https://doi.org/10.3390/axioms3020244 - 26 May 2014
Viewed by 5279
Abstract
There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict [...] Read more.
There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs. Full article
(This article belongs to the Special Issue Quantum Statistical Inference)
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164 KiB  
Communication
Joint Distributions and Quantum Nonlocal Models
by James D. Malley and Anthony Fletcher
Axioms 2014, 3(2), 166-176; https://doi.org/10.3390/axioms3020166 - 15 Apr 2014
Cited by 3 | Viewed by 4433
Abstract
A standard result in quantum mechanics is this: if two observables are commuting then they have a classical joint distribution in every state. A converse is demonstrated here: If a classical joint distribution for the pair agrees with standard quantum facts, then the [...] Read more.
A standard result in quantum mechanics is this: if two observables are commuting then they have a classical joint distribution in every state. A converse is demonstrated here: If a classical joint distribution for the pair agrees with standard quantum facts, then the observables must commute. This has consequences for some historical and recent quantum nonlocal models: they are analytically disallowed without the need for experiment, as they imply that all local observables must commute among themselves. Full article
(This article belongs to the Special Issue Quantum Statistical Inference)
178 KiB  
Article
Orthogonality and Dimensionality
by Olivier Brunet
Axioms 2013, 2(4), 477-489; https://doi.org/10.3390/axioms2040477 - 13 Dec 2013
Cited by 7 | Viewed by 4202
Abstract
In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as [...] Read more.
In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the cardinality of a maximal collection of mutually orthogonal elements (which, for instance, can be seen as spatial directions). Following this idea, we develop a formalism based on two basic ingredients, namely an orthogonality relation and matroids which are a very generic algebraic structure permitting to define a notion of dimension. Having obtained what we call orthomatroids, we then show that, in high enough dimension, the basic constituants of orthomatroids (more precisely the simple and irreducible ones) are isomorphic to generalized Hilbert lattices, so that their presence is a direct consequence of an orthogonality-based characterization of dimension. Full article
(This article belongs to the Special Issue Quantum Statistical Inference)
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