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Mathematical statistical structure for quantum systems: There is no measurement problem

Mohamed Mostafa

Published Mar 24, 2023• 0 citations
Publisher

Abstract

Abstract In this work, we prove that the behavior of quantum systems during various measurement processes is fully understood through the statistical structure in quantum mechanics represented by the uncertainty principle. We also prove that any measurement process is never different from any interaction between quantum particles, and this is evidenced only by the existence of a minimum uncertainty in position characteristic of each measurement system imposed by the statistical-mathematical structure of quantum mechanics.

DOI
10.21203/rs.3.rs-2715742/v1

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Abstract In this work, we prove that the behavior of quantum systems during various measurement processes is fully understood through the statistical structure in quantum mechanics represented by the uncertainty principle. We also prove that any measurement process is never different from any interaction between quantum particles, and this is evidenced only by the existence of a minimum uncertainty in position characteristic of each measurement system imposed by the statistical-mathematical structure of quantum mechanics.
Abstract The measurement problem in quantum mechanics still appears to be an unresolved issue. Here we present a new quantum theory of measurement that overcomes many of the difficulties previously found. It is based on a consistent use of the linear superposition principle and distinguishes two aspects: recording and observation. A recording elicits the full interaction of the object quantum system with the quantum measuring apparatus. No wavefunction collapse is introduced. Statistics may appear at the observation of the recording only and depends on filtering processes. The theory presented here uses the existing mathematical structure of quantum mechanics but requires no ad hoc measurement postulates. Well‐known paradoxical aspects in standard quantum mechanics, for instance, wave–particle duality, Schrödinger's cat, and Zeno effects do not appear in the current formulation. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
Heisenberg's uncertainty principle, which imposes intrinsic restrictions on our ability to predict the outcomes of incompatible quantum measurements to arbitrary precision, demonstrates one of the key differences between classical and quantum mechanics.The physical systems considered in the uncertainty principle are static in nature and described mathematically with a quantum state in a Hilbert space.However, many physical systems are dynamic in nature and described with the formalism of a quantum channel.In this paper, we show that the uncertainty principle can be reformulated to include process measurements that are performed on quantum channels.Since both the preparation of quantum states and the implementation of quantum measurements are themselves special cases of quantum channels, our formalism encapsulates the uncertainty principle in its utmost generality.More specifically, we obtain expressions that generalize the Maassen-Uffink uncertainty relation and the universal uncertainty relations from quantum states to quantum channels.
We investigate here the concept of generalized complementarity introduced by Bohr. It is shown that it is not possible to ascertain the microscopic state of a system without, for the purpose of measurement, imposing upon it physical conditions so stringent that they preclude the application of the statistical description which was appropriate before the measurement. One may conclude that the statistical averages which appear in the formation of an ensemble should in quantum statistics be given an interpretation somewhat different from that of classical statistics. These results lead naturally to a critical survey of the foundations of quantum statistical mechanics. We discuss the physical interpretation of statistical matrices which are the quantum equivalent of the classical ensemble. By a straightforward generalization of classical methods one obtains a statistical representation of a system if a set of measurements are given. Let $\ensuremath{\rho}$ designate the statistical matrix which represents the system; the method consists in making the diagonal sum of $\ensuremath{\rho} log \ensuremath{\rho}$ a minimum, while the results of the performed measurements appear as conditions of the minimum. As an application we treat the composition of two separate systems and derive from this the general proof of the HHH theorem which was first given by Klein.
In this paper, we will see that we can reformulate the purely classical probability theory, using similar language to the one used in quantum mechanics. This leads us to reformulate quantum mechanics itself using this different way of understanding probability theory, which in turn will yield a new interpretation of quantum mechanics. In this reformulation, we still prove the existence of none classical phenomena of quantum mechanics, such as quantum superposition, quantum entanglement, the uncertainty principle and the collapse of the wave packet. But, here, we provide a different interpretation of these phenomena of how it is used to be understood in quantum physics. The advantages of this formulation and interpretation are that it induces the same experimental results, solves the measurement problem, and reduces the number of axioms in quantum mechanics. Besides, it suggests that we can use new types of Q-bits which are more easily to manipulate.
It is argued that the so-called minimal statistical interpretation of quantum mechanics does not completely resolve the measurement problem in that this view is unable to show that quantum mechanics can dispense with classical physics when it comes to a treatment of the measuring interaction. It is suggested that the view that quantum mechanics applies to individual systems should not be too hastily abandoned, in that this view gives perhaps the best hope of leading to a version of quantum mechanics which does provide a complete solution to the measurement problem.
This paper presents a conceptual reinterpretation of the Heisenberg Uncertainty Principle within the relational framework developed in the preceding papers of this series. In conventional quantum mechanics, the uncertainty relation between position and momentum is typically understood as a limitation on the simultaneous precision with which these quantities can be measured. In this work, we explore an alternative interpretation in which the uncertainty principle reflects a structural constraint on the types of physical states that can exist within spacetime. Within the relational lens framework, physical systems exist between two fundamental limiting conditions: gravitational singularities, representing perfect spatial localization, and the speed of light, representing maximal propagation through spacetime. From this perspective, position and momentum can be interpreted as expressions of two complementary tendencies—localization and propagation—that define the allowable configurations of physical systems. Under this interpretation, the uncertainty relation expresses the minimum balance that must exist between these tendencies for a physical system to remain within spacetime. Planck’s constant (ħ) therefore represents the smallest relational scale separating these two boundary conditions. Rather than reflecting a limitation of measurement alone, the uncertainty principle may be understood as a structural feature of relational spacetime itself. This paper is conceptual in nature and does not introduce new empirical claims. Instead, it proposes a relational interpretation of the uncertainty principle that links quantum behavior with the broader boundary structure explored in earlier papers in this series.