An argument is presented that the consistent description of the evolution of a system is possible only with the statistical operator and not with the wave function. A reconstruction of the logic of quantum mechanics is proposed with the statistical operator as the basic quantity to describe the state in one-to-one correspondence. The problem of measurement is reconsidered in the light of this new logic. It is shown for the first time how the irreversible change in the measuring apparatus leads to the truncation of von Neumann's chain between the object and the subject.
An argument is presented that the consistent description of the evolution of a system is possible only with the statistical operator and not with the wave function. A reconstruction of the logic of quantum mechanics is proposed with the statistical operator as the basic quantity to describe the state in one-to-one correspondence. The problem of measurement is reconsidered in the light of this new logic. It is shown for the first time how the irreversible change in the measuring apparatus leads to the truncation of von Neumann's chain between the object and the subject.
Theory of measurement is presented which is based on the statistical interpretation of quantum mechanics. The reduction of wave packet is ascribed to undue comparison of the state of an ensemble before measurement and the state of an individual system after measurement. By studying the evolution of statistical operators for subsystems and composite systems comprising an object, a measuring apparatus and an observer, one can definitely locate where the measurement is done, thus negating the principle of psycho-physical parallelism due to von Neumann. Time needed for measurement is also discussed. The Einstein-Podolsky-Rosen state is described as an eigenstate of correlation. The argument lends support to the standpoint that the physical state of a system is to be represented by the statistical operator, not by the wave function, in the conceptual foundation of quantum mechanics.
Theory of measurement is presented which is based on the statistical interpretation of quantum mechanics. The reduction of wave packet is ascribed to undue comparison of the state of an ensemble before measurement and the state of an individual system after measurement. By studying the evolution of statistical operators for subsystems and composite systems comprising an object, a measuring apparatus and an observer, one can definitely locate where the measurement is done, thus negating the principle of psycho-physical parallelism due to von Neumann. Time needed for measurement is also discussed. The Einstein-Podolsky-Rosen state is described as an eigenstate of correlation. The argument lends support to the standpoint that the physical state of a system is to be represented by the statistical operator, not by the wave function, in the conceptual foundation of quantum mechanics.
Discussing quantum theory foundations, von Neumann noted that the measurement process should not be regarded in terms of a temporal evolution. A reason for his claim is insurmountability of the gap between reversible and irreversible processes. The time operator formalism that goes beyond such a gap is adequate framework for elaboration of the measurement problem. It considers signals to be stochastic processes, whether they correspond to variables or distribution densities. Signal processing like that utilizes statistical properties to perform its tasks, which is the definition of statistical signal processing. A hierarchy of the measurement process is indicated by crossing between states and devices, which implies an evolution in the temporal domain. The concept has generalized to an open system by the use of duality in frame theory.
It is shown that the quantum-mechanical state vector correctly describes not only the probabilities for the outcomes of measurements, but also the correlations between the outcomes of successive measurements. In particular, von Neumann's axiom M is shown to be redundant. Consequently, no extra---quantum-mechanical "reduction" of the joint object-apparatus state vector is required for a full statistical description of a sequence of measurements. It is also shown that any attempt to determine experimentally whether or not a reduction of the joint state vector has taken place during a measurement is incompatible with the preservation of the outcome of that measurement.
The quantum measurement problem has to do with the compatibility of two different prescriptions for the end state of a quantum measurement process: (1) a superposition representing an entangled state of object system and apparatus; (2) a mixture of correlated states of object system and apparatus. The first prescription is a consequence of the linearity of time evolution in quantum mechanics; the second one (corresponding to what is usually called the 'collapse of the wave function') is a basic feature of the axiomatic structure of the theory. This paper proves that the two prescriptions are exactly equivalent whenever information about the results of the measurement persists, i.e. whenever there is some kind of record or trace of the results (all the measurements we are ordinarily concerned with fulfill this requirement). Experimental evidence is cited suggesting that indeed we should expect the equivalence of prescriptions (1) and (2) to hold when and only when information about measurement results persists. Finally the notion of 'element of reality' introduced by Einstein, Podolsky and Rosen is discussed in the context of this description of the measurement process, and a brief remark is offered about the 'classical' nature of the world we experience.
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.