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A number of issues of quantum measurement theory

Author: LiuWeiHua
Tutor: WuJunDe
School: Zhejiang University
Course: Basic mathematics
Keywords: Quantum measurement quantum operation sequential effect algebra fixed point set bounded quantum obserbles infimum
CLC: O413
Type: Master's thesis
Year: 2010
Downloads: 57
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Abstract

Quantum measurement theory is one of the key problems in quantum theory, it contains a great many of mathematical problems and philosoph-ical problems. Also it has applications in quantum information theory and quantum correction theory. The essential difference between quantum measurement and classical measurement is that the quantum measurement would make the system collapsed. It has the follows four characteristics:(1). Randomness:It is unpredictable and uncontrollable.(2). Irreversibility:In general, measurement is entropy-increasing procedure.(3). Decoherence:Eliminate all the coherence of the original state.(4). Nonlocality:The collapse of the wave function is nonlocal.Thus, people would concern the following problems:(a). Why would quantum measurement incur the irreversible changes of the measured system?(b). Is there any efficient way to avoid or control the quantum decoherence so that people could get some meaningful information of the quantum states?(c). Does quantum, measurement mean that People involved the subjective view on the microscopic world?(d). Is there an appropriate logical theory to describe the intrinsic structure of quantum structure? In history, Heinsenberg, von Neumann, Birkhoff published some im-portant far-reaching fundamental works.In 1994, Foulis and Bennet defined and did some researches on a math-ematical structure-effect algebra to study the unprecise measurement in quantum theory. Boolean algebra, fuzzy set, projective operator lattice on Hilbert space are contained in this category. In 2002, in order to de-scribe the order relation be two quantum measurement, professor Gudder constructed the sequential effect theory base on effect algebra. To develop this theory, professor Gudder raised 25 open problems in this area, the 2th problem of them is:Is the operation B o C= B1/2CB1/2 the unique sequential product on the operator effect algebraε(H)?In Chapter 1, we give a negative answer to this question.As we know, a quantum measurement can be described as a quantum operation which is a completely positive map on the bounded operators set B(H) of a Hilbert space H. One of the key problems of quantum measure-ment is:which element in B(H) is not disturbed by the quantum operation, namely, to determine the fixed point set of the quantum operation.In chapter 2, we will study two problems in this aspect:(I). Solving Gudder’s 25th problem:Give a quantum opera-tion, determined by 3 elements, such that the fixed point set of the operation is not the commutant of the 3 elements.In 2002, Arias, Gheondea, Gudder conjectured:If any two elements of the quantum measurement are commutative, then the fixed point set of the quantum operation which is decided by the quantum measurement is exactly the commutant of the elements of the quantum measurement. (II). By the spectral theory of self-adjoint operators, we will not only prove the conjecture, but also give a stronger conclusion.In 2006, professor Gudder defined a partial order≤on bounded quan-tum observalbe set S(H). It is meaningful in physics so people called the order logical order. Based on partial order≤, professor Gudder asked the following question:Does the infimum of any two elements in S(H) always exists?In 2007, Pulmannova and Vincekova proved that for arbitrary set D of S(H), the infimum of D always exists under the partial order≤, but their proof is abstract and has no information for the infimum. In my undergraduate thesis, I gave a representation theorem of the infimum. In 2009, Xu, Du, Fang described a condition of the existence of the supre-mun under partial order≤. Nevertheless, their condition is difficult to be checked since the condition depend on an operator W, but W is not easy to get.In chapter 3, we get a sufficient and necessary condition for the existence of the supremum A V B under partial order≤and we also provide a nice representation theorem for A V B. By the way, our conclusion has a clear physical interpretation.

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