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Studies in plausibility theory, with applications to physics
KTH, School of Information and Communication Technology (ICT), Microelectronics and Applied Physics, MAP.
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

The discipline usually called `probability theory' can be seen as the theory which describes and sets standard norms to the way we reason about plausibility. From this point of view, this `plausibility theory' is a province of logic, and the following informal proportion subsists:

plausibility theory is to the common notion of `plausibility', as deductive logic is to the common notion of `truth'.

Some studies in plausibility theory are here offered. An alternative view and mathematical formalism for the problem of induction (the prediction of uncertain events from similar, certain ones) is presented. It is also shown how from plausibility theory one can derive a mathematical framework, based on convex geometry, for the description of the predictive properties of physical theories. Within this framework, problems like state assignment - for any physical theory - find simple and clear algorithms, numerical examples of which are given for three-level quantum systems. Plausibility theory also gives insights on various fashionable theorems, like Bell’s theorem, and various fashionable `paradoxes', like Gibbs' paradox.

Place, publisher, year, edition, pages
Stockholm: KTH , 2007. , xv, 105 p.
Series
Trita-ICT/MAP, 2007:6
Keyword [en]
Bayesian probability theory, quantum theory, state assignment, state estimation
National Category
Physical Sciences
Identifiers
URN: urn:nbn:se:kth:diva-4421ISBN: 978-91-7178-688-3 (print)OAI: oai:DiVA.org:kth-4421DiVA: diva2:12252
Public defence
2007-06-12, D, Forum, Isafjordsgatan 39, Kista, 10:00
Opponent
Supervisors
Note
QC 20100816Available from: 2007-06-04 Created: 2007-06-04 Last updated: 2010-08-16Bibliographically approved
List of papers
1. Consistency of the Shannon entropy in quantum experiments
Open this publication in new window or tab >>Consistency of the Shannon entropy in quantum experiments
2004 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 69, no 6, 062108- p.Article in journal (Refereed) Published
Abstract [en]

The consistency of the Shannon entropy, when applied to outcomes of quantum experiments, is analyzed. It is shown that the Shannon entropy is fully consistent and its properties are never violated in quantum settings, but attention must be paid to logical and experimental contexts. This last remark is shown to apply regardless of the quantum or classical nature of the experiments.

Keyword
Composition; Entropy; Experiments; Mathematical operators; Matrix algebra; Probability distributions; Statistical methods; Thermodynamics; Conditional entropy; Cummutative property; Density matrix; Shannon entropy; Quantum theory
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-7279 (URN)10.1103/PhysRevA.69.062108 (DOI)000222471400015 ()2-s2.0-4043141550 (Scopus ID)
Note
QC 20100816Available from: 2007-06-04 Created: 2007-06-04 Last updated: 2011-10-26Bibliographically approved
2. Distinguishability of non-orthogonal density matrices does not imply violations of the second law
Open this publication in new window or tab >>Distinguishability of non-orthogonal density matrices does not imply violations of the second law
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The hypothetical possibility of distinguishing preparations described by non-orthogonal density matrices does not necessarily imply a violation of the second law of thermodynamics, as was instead stated by von Neumann. On the other hand, such a possibility would surely mean that the particular density-matrix space (and related Hilbert space) adopted would not be adequate to describe the hypothetical new experimental facts. These points are shown by making clear the distinction between physical preparations and the density matrices which represent them, and then comparing a "quantum" thermodynamic analysis given by Peres with a "classical" one given by Jaynes.

National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-7280 (URN)
Note
QC 20100816Available from: 2007-06-04 Created: 2007-06-04 Last updated: 2010-08-16Bibliographically approved
3. Schrödinger-cat states: size classification based on evolution or dissipation
Open this publication in new window or tab >>Schrödinger-cat states: size classification based on evolution or dissipation
2004 (English)In: Proceedings of SPIE - The International Society for Optical Engineering: Vol. 5468 / [ed] Heszler P; Abbott D; GeaBanacloche JR; Hemmer PR, 2004, 335-343 p.Conference paper, Published paper (Refereed)
Abstract [en]

The issue of estimating how "macroscopic" a superposition state is, can be addressed by analysing the rapidity of the state's evolution under a preferred observable, compared to that of the states forming the superposition. This fast evolution. which arises from the larger dispersion of the superposition state for the preferred operator, also represents a useful characteristic for interferometric applications. This approach can be compared to others in which a superposition's macroscopality is estimated in terms of the fragility to dissipation.

Series
Proceedings of SPIE, ISSN 0277-786X
Keyword
Disconnectivity; Interference; Macroscopic superpositions; Schrödinger-cat states; Dispersion (waves); Interferometry; Mathematical models; Microwaves; Photons; Probability distributions; Thermodynamics; Quantum theory
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-7281 (URN)10.1117/12.547048 (DOI)000223225300035 ()2-s2.0-4344689602 (Scopus ID)0-8194-5391-9 (ISBN)
Conference
Fluctuations and Noise in Photonics and Quantum Optics II; Maspalomas; 26-28 May 2004, Maspalomas, Spain
Note
QC 20100816Available from: 2007-06-04 Created: 2007-06-04 Last updated: 2011-11-02Bibliographically approved
4. Why can states and measurement outcomes be represented as vectors?
Open this publication in new window or tab >>Why can states and measurement outcomes be represented as vectors?
(English)Manuscript (Other academic)
Abstract [en]

It is shown how, given a "probability data table" for a quantum or classical system, the representation of states and measurement outcomes as vectors in a real vector space follows in a natural way. Some properties of the resulting sets of these vectors are discussed, as well as some connexions with the quantum-mechanical formalism.

National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-7282 (URN)
Note
QC 20100816Available from: 2007-06-04 Created: 2007-06-04 Last updated: 2010-08-16Bibliographically approved
5. Probability tables
Open this publication in new window or tab >>Probability tables
2004 (English)In: Proceedings of International Conference Quantum Theory: Reconsideration of Foundations, Växjö (Småland), Sweden, 1-7 June 2003 / [ed] Khrennikov, Andrei, Växjö: Växjö Univ. Press , 2004, 387-401 p.Conference paper, Published paper (Other academic)
Abstract [en]

The idea of writing a table of probabilistic data for a quantum or classical system, and of decomposing this table in a compact way, leads to a shortcut for Hardy's formalism, and gives new perspectives on foundational issues.

Place, publisher, year, edition, pages
Växjö: Växjö Univ. Press, 2004
Series
Mathematical modelling in physics, engineering and cognitive sciences, ISSN 1651-0267 ; 10
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-7283 (URN)91-7636-409-7 (ISBN)
Conference
Quantum Theory: Reconsideration of Foundations 2, 2003, Växjö
Note
QC 20100816Available from: 2007-06-04 Created: 2007-06-04 Last updated: 2010-08-16Bibliographically approved
6. On distinguishability, orthogonality, and violations of the second law: contradictory assumptions, contrasting pieces of knowledge
Open this publication in new window or tab >>On distinguishability, orthogonality, and violations of the second law: contradictory assumptions, contrasting pieces of knowledge
(English)Manuscript (Other academic)
Abstract [en]

Two statements by von Neumann and a thought-experiment by Peres prompts a discussion on the notions of one-shot distinguishability, orthogonality, semi-permeable diaphragm, and their thermodynamic implications. In the first part of the paper, these concepts are defined and discussed, and it is explained that one-shot distinguishability and orthogonality are contradictory assumptions, from which one cannot rigorously draw any conclusion, concerning e.g. violations of the second law of thermodynamics. In the second part, we analyse what happens when these contradictory assumptions comes, instead, from _two_ different observers, having different pieces of knowledge about a given physical situation, and using incompatible density matrices to describe it.

National Category
Telecommunications
Identifiers
urn:nbn:se:kth:diva-7257 (URN)
Note
QC 20100810Available from: 2007-06-01 Created: 2007-06-01 Last updated: 2010-08-16Bibliographically approved
7. 'Plausibilities of plausibilities': an approach through circumstances
Open this publication in new window or tab >>'Plausibilities of plausibilities': an approach through circumstances
(English)Manuscript (Other academic)
Abstract [en]

Probability-like parameters appearing in some statistical models, and their prior distributions, are reinterpreted through the notion of `circumstance', a term which stands for any piece of knowledge that is useful in assigning a probability and that satisfies some additional logical properties. The idea, which can be traced to Laplace and Jaynes, is that the usual inferential reasonings about the probability-like parameters of a statistical model can be conceived as reasonings about equivalence classes of `circumstances' - viz., real or hypothetical pieces of knowledge, like e.g. physical hypotheses, that are useful in assigning a probability and satisfy some additional logical properties - that are uniquely indexed by the probability distributions they lead to.

Keyword
Quantum Physics (quant-ph); Artificial Intelligence (cs.AI)
National Category
Telecommunications
Identifiers
urn:nbn:se:kth:diva-7258 (URN)
Note
QC 20100810Available from: 2007-06-01 Created: 2007-06-01 Last updated: 2010-08-16Bibliographically approved
8. The Laplace-Jaynes approach to induction
Open this publication in new window or tab >>The Laplace-Jaynes approach to induction
(English)Manuscript (Other academic)
Abstract [en]

An approach to induction is presented, based on the idea of analysing the context of a given problem into `circumstances'. This approach, fully Bayesian in form and meaning, provides a complement or in some cases an alternative to that based on de Finetti's representation theorem and on the notion of infinite exchangeability. In particular, it gives an alternative interpretation of those formulae that apparently involve `unknown probabilities' or `propensities'. Various advantages and applications of the presented approach are discussed, especially in comparison to that based on exchangeability. Generalisations are also discussed.

Keyword
Data Analysis, Statistics and Probability (physics.data-an); Artificial Intelligence (cs.AI); Quantum Physics (quant-ph)
National Category
Telecommunications
Identifiers
urn:nbn:se:kth:diva-7261 (URN)
Note
QC 20100810Available from: 2007-06-01 Created: 2007-06-01 Last updated: 2010-08-16Bibliographically approved
9. Numerical Bayesian state assignment for a three-level quantum system: I. Absolute-frequency data; constant and Gaussian-like priors
Open this publication in new window or tab >>Numerical Bayesian state assignment for a three-level quantum system: I. Absolute-frequency data; constant and Gaussian-like priors
(English)Manuscript (Other academic)
Abstract [en]

This paper offers examples of concrete numerical applications of Bayesian quantum-state-assignment methods to a three-level quantum system. The statistical operator assigned on the evidence of various measurement data and kinds of prior knowledge is computed partly analytically, partly through numerical integration (in eight dimensions) on a computer. The measurement data consist in absolute frequencies of the outcomes of N identical von Neumann projective measurements performed on N identically prepared three-level systems. Various small values of N as well as the large-N limit are considered. Two kinds of prior knowledge are used: one represented by a plausibility distribution constant in respect of the convex structure of the set of statistical operators; the other represented by a Gaussian-like distribution centred on a pure statistical operator, and thus reflecting a situation in which one has useful prior knowledge about the likely preparation of the system. In a companion paper the case of measurement data consisting in average values, and an additional prior studied by Slater, are considered.

Keyword
Quantum Physics (quant-ph)
National Category
Telecommunications
Identifiers
urn:nbn:se:kth:diva-7287 (URN)
Note
QC 20100810Available from: 2007-06-04 Created: 2007-06-04 Last updated: 2010-08-16Bibliographically approved
10. Numerical Bayesian quantum-state assignment for a three-level quantum system: II. Average-value data with a constant, a Gaussian-like, and a Slater prior
Open this publication in new window or tab >>Numerical Bayesian quantum-state assignment for a three-level quantum system: II. Average-value data with a constant, a Gaussian-like, and a Slater prior
(English)Manuscript (Other academic)
Abstract [en]

This paper offers examples of concrete numerical applications of Bayesian quantum-state assignment methods to a three-level quantum system. The statistical operator assigned on the evidence of various measurement data and kinds of prior knowledge is computed partly analytically, partly through numerical integration (in eight dimensions) on a computer. The measurement data consist in the average of outcome values of N identical von Neumann projective measurements performed on N identically prepared three-level systems. In particular the large-N limit will be considered. Three kinds of prior knowledge are used: one represented by a plausibility distribution constant in respect of the convex structure of the set of statistical operators; another one represented by a prior studied by Slater, which has been proposed as the natural measure on the set of statistical operators; the last prior is represented by a Gaussian-like distribution centred on a pure statistical operator, and thus reflecting a situation in which one has useful prior knowledge about the likely preparation of the system. The assigned statistical operators obtained with the first two kinds of priors are compared with the one obtained by Jaynes' maximum entropy method for the same measurement situation. In the companion paper the case of measurement data consisting in absolute frequencies is considered.

Keyword
Quantum Physics (quant-ph)
National Category
Telecommunications
Identifiers
urn:nbn:se:kth:diva-7260 (URN)
Note
QC 20100810Available from: 2007-06-01 Created: 2007-06-01 Last updated: 2010-08-16Bibliographically approved
11. A size criterion for macroscopic superposition states
Open this publication in new window or tab >>A size criterion for macroscopic superposition states
2004 (English)In: Journal of Optics B-Quantum and Semiclassical Optics, ISSN 1464-4266, E-ISSN 1741-3575, Vol. 6, no 11, 429-436 p.Article in journal (Refereed) Published
Abstract [en]

An operational measure to quantify the sizes of some 'macroscopic quantum superpositions', realized in recent experiments, is proposed. The measure is based on the fact that a superposition presents greater sensitivity in interferometric applications than its superposed constituent states. This enhanced sensitivity, or 'interference utility', may then be used as a size criterion among superpositions.

Keyword
Disconnectivity; Interference; Macroscopic superpositions; Schrödinger cat states; Correlation methods; Eigenvalues and eigenfunctions; Harmonic analysis; Interferometry; Light interference; Oscillators (electronic); Sensitivity analysis; Quantum theory
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-7289 (URN)10.1088/1464-4266/6/11/001 (DOI)000225702000001 ()2-s2.0-9744264046 (Scopus ID)
Note
QC 20100816. QC 20110915. Uppdaterad från Manuskript till Artikel i tidskrift 20100816.Available from: 2007-06-04 Created: 2007-06-04 Last updated: 2011-09-15Bibliographically approved

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