Many practical systems are non-deterministic, in the sense that available information about the initial states and control values does not uniquely determine the future states. For some such systems, it is important to take quantum effects into account. For that, we need to develop non-deterministic versions of quantum physics. In this paper, we show that for non-deterministic versions of quantum physics, we cannot require superposition principle -- one of the main fundamental principles of modern quantum mechanics. Specifically, while we can consider superpositions of states corresponding to the same version of the future dynamics, it is not consistently possible to consider superpositions of states corresponding to different versions of the future.
Many practical systems are non-deterministic, in the sense that available information about the initial states and control values does not uniquely determine the future states. For some such systems, it is important to take quantum effects into account. For that, we need to develop non-deterministic versions of quantum physics. In this paper, we show that for non-deterministic versions of quantum physics, we cannot require superposition principle -- one of the main fundamental principles of modern quantum mechanics. Specifically, while we can consider superpositions of states corresponding to the same version of the future dynamics, it is not consistently possible to consider superpositions of states corresponding to different versions of the future.
Modern physical theories are deterministic in the sense that once we know the current state of the world, we can, in principle, predict all the future states. This was true for classical (pre-quantum) theories, this is true for modern quantum physics. On the other hand, we all know that we can make decision that change the state of the world -- even if, for most of us, a little bit. This intuitive idea of free will permeates all our life, all our activities -- and it seems to contradict the determinism of modern physics. It is therefore desirable to incorporate the idea of free will into physical theories. In this paper, we show that in quantum physics, free will leads to nonconservation of energy. This nonconservation is a microscopic purely quantum effect, but it needs to be taken into account in future free-will quantum theories.
When we are presented with apparently bizarre properties of microsystems governed by quantum mechanics (as in our discussion of the double-slit experiment in the previous chapter), we naturally ask whether it might be possible to find another theory, or at least another story to go with the equations, that would make more sense to us. We might seek a theory and a view of fundamental physical processes that would accord more with our classically based, everyday common sense. This typically takes the form of the question of the possibility of the existence of a more detailed description of atomic processes than that afforded by the wave function (that is, by the Copenhagen interpretation of the previous chapter). In this chapter we discuss some of the central interpretive problems of the Copenhagen version of quantum mechanics. Then, in the next two chapters, we look at the severe, general restrictions that are placed on any such completion of quantum mechanics and at one particular extension of quantum mechanics. First, though, we begin here with the struggle that ensued over the status of causality in quantum theory. THE COMPLETENESS OF QUANTUM MECHANICS Although the Heisenberg uncertainty principle (Section 20.4) and the lack of absolute predictive power are an inherent feature of quantum mechanics, there have been attempts to preserve an in principle completely deterministic structure for physics. Even classically, in a macroscopic sample of gas there are so many molecules that it would be hopeless, as a practical matter, to predict the future locations of all these molecules from Newton's laws of motion.
Abstract Quantum states in superposition cannot be observed, suggesting that they merely embody information on possible measurement results. Yet, we think of them as describing physical systems that evolve in time according to given mathematical equations. Furthermore, evolution takes place in physical space-time, and local causality is imposed on space-time by Special Relativity (a sequence of cause and effect that constitutes a fundamental principle by which we think about and do scientific work). It is therefore strange that there is non–locality and violation of causality embedded in Quantum Mechanics. We try to give here a first step to see whether a deterministic and local interpretation is possible. We will show that Bell’s inequality is not conclusive about the non–local nature of Quantum Mechanics, in that there exists an interpretation of reality (which we put forward) which is local and in which Bell’s inequality cannot be derived. Assuming determinism and locality a new Bell-like inequality is derived, which is always satisfied by Quantum Mechanics and which allows for local hidden variables [1].
One of the fundamental conceptual difficulties in quantum mechanics arises from the fact that two of the axioms of the theory concerning the evolution of a microscopic system are contradictory. The first axiom assumes that the state of a quantum system evolving undisturbed by any observer obeys a linear deterministic law expressed by the SchrSdinger equation. The second one (wave-packet reduction principle) states that, when the quantum system interacts with an apparatus devised to measure some of its dynamical variables, its state changes abruptly and stochastically, jumping into a normalized eigenstate of the measured variable. This evolution is not deterministic nor linear and cannot be accounted for by the SchrSdinger equation. As a consequence the internal coherence of the theory becomes questionable. On one side the influence of the apparatus on the quantum system cannot be described as an interaction of the quantum constituents of the measuring macroscopic apparatus with the measured quantum system. On the other side, the impossibility of considering a macroscopic object as made by quantum constituents, i .e . the difficulty of quantum mechanics in describing the behaviour of macroscopic systems, is even more dramatic if we consider the evolution of the closed system consisting of the macroscopic apparatus and of the microscopic system. In such a case the appearance of superpositions of macr0scopically distinguishable states related to the apparatus cannot be avoided. These superpositions contradict both our classical experience and wave-packet reduction principle, which provides for statistical mixtures.
The study of quantum systems represents one of the most important directions in modern Quantum Physics. Based on the principles of quantum mechanics, it is possible to conduct an in-depth investigation of microscopic processes, particle interactions, and quantum states. The detailed analysis of quantum systems relies on the application of fundamental concepts such as the Hamiltonian, wave function, fidelity function, and quantum evolution operators. These methods make it possible to determine the energy states, dynamics, and temporal evolution of a system. In the detailed analysis of quantum systems, quality control is of particular importance because quantum systems are highly sensitive to external influences and noise. Quality control includes the evaluation of the accuracy, stability, and fidelity of quantum states, which is essential for ensuring the efficiency of quantum operations and the reliable transmission of information. This paper discusses the main theoretical and practical aspects of quantum system analysis, the role of the Hamiltonian and fidelity function in the evaluation of quantum states, as well as quality control mechanisms. The paper also examines the significance of detailed quantum system analysis and quality control in modern quantum technologies. Special attention is paid to quantum testing and quantum-level scanning devices used for monitoring qubit states, determining energy levels, and optimizing quantum systems. Fundamental principles of quantum physics — superposition, quantum entanglement, and energy discreteness — are also discussed. The paper presents the application of the density matrix, fidelity function, and Hamiltonian in the analysis of quantum states. In addition, spectroscopic methods, sensors based on Josephson Junction technology, and the control problems of multi-qubit systems are examined. Particular attention is devoted to reducing decoherence, noise, and temperature instability. The study demonstrates that quantum testing and scanning systems play a significant role in the development of quantum computers, quantum communication technologies, and advanced sensing systems.
The principle of superposition of states requires that the states of a dynamical system form a linear vector space. This hypothesis of linearity has usually been regarded as a fundamental postulate in quantum mechanics, of a kind that cannot be explained by classical concepts. Indeed, Dirac [2, p. 14] comments that "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory, as is shown by the fact that the quantum superposition principle demands indeterminacy in the results of observations in order to be capable of a sensible physical interpretation."