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arXiv:1206.6292

Observability and Computability in Physical Systems

Subhash C. Kak

Published Jun 26, 2012• 0 citations
arXivHTMLPDF

Abstract

This paper considers the relevance of the concepts of observability and computability in physical theory. Observability is related to verifiability which is essential for effective computing and as physical systems are computational systems it is important even where explicit computation is not the goal. Specifically, we examine two problems: observability and computability for quantum computing, and remote measurement of time and frequency.

DOI
10.48550/arxiv.1206.6292

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This paper considers the relevance of the concepts of observability and computability in physical theory. Observability is related to verifiability which is essential for effective computing and as physical systems are computational systems it is important even where explicit computation is not the goal. Specifically, we examine two problems: observability and computability for quantum computing, and remote measurement of time and frequency.
This paper explores the problem of quantum measurement complexity. In computability theory, the complexity of a problem is determined by how long it takes an effective algorithm to solve it. This complexity may be compared to the difficulty for a hypothetical oracle machine, the output of which may be verified by a computable function but cannot be simulated on a physical machine. We define a quantum oracle machine for measurements as one that can determine the state by examining a single copy. The complexity of measurement for a realizable machine will then be respect to the number of copies of the state that needs to be examined. A quantum oracle cannot perform simultaneous exact measurement of conjugate variables, although approximate measurement may be performed as circumscribed by the Heisenberg uncertainty relations. When considering the measurement of a variable, there might be residual uncertainty if the number of copies of the variable is limited. Specifically, we examine the quantum measurement complexity of linear polarization of photons that is used in several quantum cryptography schemes and we present a relation using information theoretic arguments. The idea of quantum measurement complexity is likely to find uses in measurements in biological systems.
We explore in the framework of quantum computation the notion of computability, which holds a central position in mathematics and theoretical computer science. A quantum algorithm that exploits the quantum adiabatic processes is considered for Hilbert's tenth problem, which is equivalent to the Turing halting problem and known to be mathematically non-computable. Generalized quantum algorithms are also considered for some other mathematical non-computables in the same and in different non-computability classes. The key element of all these algorithms is the measurability of both the values of physical observables and the quantum-mechanical probability distributions for these values. It is argued that computability, and thus the limits of mathematics, ought to be determined not solely by mathematics itself but also by physical principles.
This article develops an expanded critical-propositional analysis of D. Hucul, I. V. Inlek, G. Vittorini, C. Crocker, S. Debnath, S. M. Clark, and C. Monroe’s experiment, “Modular Entanglement of Atomic Qubits Using Photons and Phonons,” published in Nature Physics 11, 37–42, in systematic dialogue with the Theory of Objectivity (TO) formulated by Vidamor Cabannas and Denivaldo Silva. Hucul et al. demonstrated a modular quantum architecture in which three trapped 171Yb+ atomic-ion qubits distributed between two physically separated modules were connected through two complementary mediating systems. Remote entanglement between modules was generated probabilistically through spontaneously emitted photons, optical interference, Bell-state-sensitive coincidence detection, and heralding, whereas deterministic entanglement between ions within the same module was established through Coulomb-coupled phonon modes. The experiment achieved an average remote Bell-state fidelity of Fremote = 0.78 ± 0.03, a remote entanglement generation rate of approximately R = 4.5 s−1 , a coherence time of approximately τ = 1.12 s, and intramodular entanglement fidelity of approximately Flocal = 0.85 ± 0.01. It further integrated photonic and phononic buses to produce experimentally observable three-particle correlations. The analysis examines possible correspondences between these results and the seven Absolute Truths of TO, while maintaining a rigorous distinction among modal necessity, empirical compatibility, indirect corroboration, and theoretical identity. Par- ticular attention is given to VA2, concerning the individualized field associated with each element; VA4, concerning boundary and interface; VA5, according to which an element must be observed or situated within the frequency relation of at least two other elements for logical existence; VA6, concerning composition; and VA7, associated in the contemporary formulation of TO with transcendent knowledge or information generated in atomic relations and considered equivalent to atomic radiations. The Hucul experiment is especially significant for the latter proposition because spontaneously emitted radiation physically carries quantum correlations dependent on atomic states and can mediate the constitution of new atom-atom relations across spatially separated modules. The article also analyzes possible relations with TO’s phenomenic elements, Inducing Effects, cosmogonic theorem, and cosmological Eras. It argues that Hucul et al. provide substantial empirical material for constructing operational bridges concerning radiation-information, relational composition, fields, frequencies, interfaces, and non-anthropocentric physical observation. Nevertheless, the experiment does not establish the modal necessity of TO’s axioms, does not empirically demonstrate logical Nothingness, and does not directly verify its cosmogenesis or cosmological Eras. Its strongest contribution to TO is therefore not a confirmation of the theory as a whole but the empirical realization of a class of physical processes that can be translated into explicit and measurable bridge propositions. On this basis, the article assigns Hucul et al. a dialogue score of 8.0/10 relative to TO. Keywords: Theory of Objectivity; modular quantum computing; quantum entangle- ment; atomic qubits; photons; phonons; radiation-information; modal necessity; atomic relations; non-anthropocentric observation; Inducing Effects; phenomenic elements; Vidamor Cabannas; Denivaldo Silva; Hucul; quantum information.
This paper explores the problem of quantum measurement complexity. In computability theory, the complexity of a problem is determined by how long it takes an effective algorithm to solve it. This complexity may be compared to the difficulty for a hypothetical oracle machine, the output of which may be verified by a computable function but cannot be simulated on a physical machine. We define a quantum oracle machine for measurements as one that can determine the state by examining a single copy. The complexity of measurement for a realizable machine will then be respect to the number of copies of the state that needs to be examined. A quantum oracle cannot perform simultaneous exact measurement of conjugate variables, although approximate measurement may be performed as circumscribed by the Heisenberg uncertainty relations. When considering the measurement of a variable, there might be residual uncertainty if the number of copies of the variable is limited. Specifically, we examine the quantum measurement complexity of linear polarization of photons that is used in several quantum cryptography schemes and we present a relation using information theoretic arguments. The idea of quantum measurement complexity is likely to find uses in measurements in biological systems.
We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically noncomputable. Generalised quantum algorithms are also considered for some other mathematical noncomputables in the same and of different noncomputability classes. The key element of all these algorithms is the measurability of both the values of physical observables and of the quantum-mechanical probability distributions for these values. It is argued that computability, and thus the limits of Mathematics, ought to be determined not