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Introduction to Quantum Theory
Published in Andrei Khrennikov, Social Laser, 2020
This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [191], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Ltiders’s forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (LUders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac’s notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.
Rome teleportation experiment analysed in the Wigner representation: the role of the zeropoint fluctuations in complete one-photon polarization-momentum Bell-state analysis
Published in Journal of Modern Optics, 2018
A. Casado, S. Guerra, J. Plácido
Following the standard particle-like Hilbert space approach, the application of the projection postulate implies that the quantum information that remains in the electromagnetic field after the BSM is stored at Bob's station into a given one-photon polarization state, depending on the result obtained at Alice's station (see Equation (8)). This interpretation contrasts sharply with the one provided by the Wigner formalism: the autocorrelations of the light field corresponding to do not give any information about the preparation (see Equations (52) and (54)). Hence, the properties concerning Bob's photon after the BSM are supported by the cross-correlations associated to the field (see Equations (55) to (58)) where the spacetime condition given in Equation (80) must be fulfilled: The behaviour of Bob's photon is inherently linked to Alice's photon, so that there is no disconnection between them mediated through the collapse of the state vector at Alice's station. The WRHP approach adds a new point of view of the reconstruction of the prepared state, as it has been analysed in Section 4.4.1. As a matter of fact, the requirements given in Equations (66) and (67), along with an appropriate balance between the number of amplified ZPF sets of modes entering the analyser and the number of ZPF sets of modes entering the idle channels inside Alice's station (see Equation (81)), constitute an alternative way to understand teleportation which contrasts to the standard interpretation based on superpositions in the tensor product of Hilbert spaces (entanglement) and the projection postulate. In the Wigner formalism, teleportation arises from the correlation properties of the field mediated by its fundamental state, so that the vacuum inputs determine the quantity of information that can be generated and processed.