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Kluwer Academic Publishers eBooks

Quantum Measurement Problem and State Dual Representations

Sergey Nikolaevich Mayburov

Published Feb 24, 2006• 0 citations
Publisher

Abstract

The information aspects of quantum measurement problem studied in the simple quantum model of information processing and memorizing device — observer O . It’s argued that to describe probablistic events appearencei, i.e., ‘State Collapse’ consistently O complete states manifold must be tensor product of standard Hilbert space and dual to it novel set describing information acquired by O in the individual events.

DOI
10.1007/0-306-47114-0_24

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The information aspects of quantum measurement problem studied in the simple quantum model of information processing and memorizing device — observer O . It’s argued that to describe probablistic events appearencei, i.e., ‘State Collapse’ consistently O complete states manifold must be tensor product of standard Hilbert space and dual to it novel set describing information acquired by O in the individual events.
The quantum measurement problem considered for measuring system (MS) consist of measured state S (particle), detector D and information processing device (observer) O. It's shown that O states selfreference structure results in principal nonobservability of MS interference terms which discriminate pure and mixed S states. Such observables restriction permit to construct for MS states subjective representation (SR) which describes probabilistic evolution for measurement events observed by OOO and his subjective information about S values. SR is dual and nonequivalent to MS Hilbert space HHH for external observer O′O'O′. Due to it SR evolution is compatible with Schrodinger linear MS evolution observed by O′O'O′. It's argued that SR evolution corresponds to S state collapse for individual events observed by OOO.
Quantum Measurements regarded in Systems Selfdescription framework for measuring system (MS) consist of measured state S environment E and observer OOO processing input S signal. OOO regarded as quantum object which interaction with S,E obeys to Schrodinger equation (SE) and from it and Breuer selfdescription formalism S information for OOO reconstructed. In particular S state collapse obtained if OOO selfdescription state has the dual structure LT=H⨂LVL_T=\cal H \bigotimes L_VLT​=H⨂LV​ where H\cal HH is Hilbert space of MS states ΨMSΨ_{MS}ΨMS​. LV\cal L_VLV​ is the set with elements VO=∣Oj><Oj∣V^O=|O_j> < O_j|VO=∣Oj​><Oj​∣ describing random 'pointer' outcomes OjO_jOj​ observed by OOO in the individual events. The 'preferred' basis ∣Oj>|O_j>∣Oj​> defined by OOO state decoherence via OOO - E interactions. Zurek's Existential Interpretation discussed in selfmeasurement framework.
Information-Theoretical restrictions on the systems self-descriptions are applied to Quantum Measurements Theory. For the quantum object S measurement by information system O such restrictions are described by the restricted states formalism by Breuer. The analogous restrictions obtained in Algebraic QM from the analysis of Segal algebra U of O observables; O restricted states set is defined as U dual space. From Segal theorem for the associative subalgebra it's shown that such states describe the random pointer outcomes observed by O in the individual events.
Quantum Measurement problem studied in Information Theory approach of systems selfdescription which exploits the information acquisition incompleteness for the arbitrary information system. The studied model of measuring system (MS) consist of measured state S environment E and observer OOO processing input S signal. OOO considered as the quantum object which interaction with S,E obeys to Schrodinger equation (SE). MS incomplete or restricted states for OOO derived by the algebraic QM formalism which exploits Segal and C∗C^*C∗-algebras. From Segal theorem for systems subalgebras it's shown that such restricted states VO=∣Oj><Oj∣V^O=|O_j> < O_j|VO=∣Oj​><Oj​∣ describes the classical random 'pointer' outcomes OjO_jOj​ observed by OOO in the individual events. The 'preferred' basis ∣Oj>|O_j>∣Oj​> defined by OOO state decoherence via OOO - E interactions.
Left on its own, a quantum state evolves deterministically under the Schrödinger equation, forming superpositions. Upon measurement, however, a stochastic process governed by the Born rule collapses it to a single outcome. This dual evolution of quantum states – the core of the Measurement Problem – has puzzled physicists and philosophers for nearly a century. Yet, amid the cacophony of competing interpretations, the problem today is not as impenetrable as it once seemed. This paper reviews the current status of the Measurement Problem, distinguishing between what is well understood and what remains unresolved. We examine key theoretical approaches, including decoherence, many-worlds interpretation, objective collapse theories, hidden-variable theories, dualistic approaches, deterministic models, and epistemic interpretations. To make these discussions accessible to a broader audience, we also reference curated online resources that provide high-quality introductions to central concepts.
This paper introduces Intention Field Theory (IFT), a theoretical framework extending the standard quantum formalism to address the Measurement Problem and the origin of non-local correlations. Postulating that the observer is not external to the laws of physics but resides in a Higher Hilbert Space, we define an ``Intention Vector'' that acts as a non-local boundary condition on the physical manifold. We propose that wavefunction collapse is governed by an Interaction Hamiltonian mediated by this vector, effectively transforming quantum probabilities into actualized history through an "Alignment Metric". By treating the agent's internal state as a control parameter originating outside spacetime, we derive a Lindblad-form Master Equation that recovers standard unitary evolution in the absence of observation while describing state reduction as a dynamic alignment process. Crucially, we demonstrate that this framework is consistent with, and provides a unifying interpretation for, major foundational experiments including the Quantum Zeno Effect, Delayed Choice Quantum Eraser, and “Before-Before” non-local coordination. The theory offers a rigorous mathematical model for “non-material coordination” in entanglement and proposes falsifiable experimental tests involving macroscopic superposition.