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Numerical Bayesian state assignment for a three-level quantum system: I. Absolute-frequency data; constant and Gaussian-like priors
KTH, School of Information and Communication Technology (ICT), Microelectronics and Applied Physics, MAP.
KTH, School of Information and Communication Technology (ICT), Microelectronics and Applied Physics, MAP.
KTH, School of Information and Communication Technology (ICT), Microelectronics and Applied Physics, MAP.ORCID iD: 0000-0002-2082-9583
(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 [en]
Quantum Physics (quant-ph)
National Category
Telecommunications
Identifiers
URN: urn:nbn:se:kth:diva-7287OAI: oai:DiVA.org:kth-7287DiVA: diva2:12249
Note
QC 20100810Available from: 2007-06-04 Created: 2007-06-04 Last updated: 2010-08-16Bibliographically approved
In thesis
1. Studies in plausibility theory, with applications to physics
Open this publication in new window or tab >>Studies in plausibility theory, with applications to physics
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
Bayesian probability theory, quantum theory, state assignment, state estimation
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-4421 (URN)978-91-7178-688-3 (ISBN)
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
2. Quantum State Analysis: Probability theory as logic in Quantum mechanics
Open this publication in new window or tab >>Quantum State Analysis: Probability theory as logic in Quantum mechanics
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.

Place, publisher, year, edition, pages
Stockholm: KTH, 2007. xii, 65 p.
Series
Trita-ICT/MAP, 2007:5
Keyword
quantum, quantum mechanics, state, state analysis, probability, probability theory, logic
National Category
Telecommunications
Identifiers
urn:nbn:se:kth:diva-4417 (URN)978-91-7178-638-8 (ISBN)
Public defence
2007-06-07, Sal D, Forum, IT-universitetet, Isafjordsgatan 39, Kista, 10:00
Opponent
Supervisors
Note
QC 20100810Available from: 2007-06-01 Created: 2007-06-01 Last updated: 2010-08-10Bibliographically approved

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