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Quantum State Analysis: Probability theory as logic in Quantum mechanics
KTH, School of Information and Communication Technology (ICT), Microelectronics and Applied Physics, MAP.
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 [en]
quantum, quantum mechanics, state, state analysis, probability, probability theory, logic
National Category
Telecommunications
Identifiers
URN: urn:nbn:se:kth:diva-4417ISBN: 978-91-7178-638-8 (print)OAI: oai:DiVA.org:kth-4417DiVA: diva2:12219
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
List of papers
1. Quantum mechanics as "space-time statistical mechanics"?
Open this publication in new window or tab >>Quantum mechanics as "space-time statistical mechanics"?
(English)Manuscript (Other academic)
Abstract [en]

In this paper we discuss and analyse the idea of trying to see (non-relativistic) quantum mechanics as a ``space-time statistical mechanics'', by using the classical statistical mechanical method on objective microscopic space-time configurations. It is argued that this could perhaps be accomplished by giving up the assumption that the objective ``state'' of a system is independent of a future measurement performed on the system. This idea is then applied in an example of quantum state estimation on a qubit system.

National Category
Telecommunications
Identifiers
urn:nbn:se:kth:diva-7256 (URN)
Note
QC 20100810Available from: 2007-06-01 Created: 2007-06-01 Last updated: 2010-08-10Bibliographically approved
2. 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
3. '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
4. 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
5. 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
6. 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

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