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arXiv:quant-ph/0406154 · International Journal of Quantum Information

QUBITS AND QUANTUM SPACES

Paola Zizzi

Published Dec 16, 2004• 26 citations
arXivHTMLPDF

Abstract

We consider the quantum computational process as viewed by an insider observer: this is equivalent to an isomorphism between the quantum computer and a quantum space, namely the fuzzy sphere. The result is the formulation of a reversible quantum measurement scheme, with no hidden information.
DOI
10.1142/s021974990500089x

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We give the logical description of a new kind of quantum measurement that is a reversible operation performed by an hypothetical insider observer, or, which is the same, a quantum measurement made in a quantum space background, like the fuzzy sphere. The result is that the non-contradiction and the excluded middle principles are both invalidated, leading to a paraconsistent, symmetric logic. Our conjecture is that, in this setting, one can develop the adequate logic of quantum computing. The role of standard quantum logic is then confined to describe the projective measurement scheme.
Fuzzy model theory is an alternative approach to formalizing uncertainty and/or incompleteness of knowledge to the formalization using fuzzy sets proposed by Lotfi Zadeh. This approach to information processing is also applied in quantum computing. All quantum computations are conducted under the assumption of incomplete knowledge and only at the last stage the measurement (i.e. digitization) of the obtained results is performed. The paper describes a model-theoretic formalization of a pure quantum system in the language of fuzzy model theory. It is shown that the fuzzy model can be considered as a generalization of the concept of quantum system. One of the central concepts of quantum computing is quantum entanglement. A quantum system is considered to be non-entangled or separable if the results of measurements of some elements of the system do not depend on measurements of other elements. The paper introduces the concept of entangled and partially entangled fuzzy system, formulates and proves the criteria of entanglement of a fuzzy system.
Starting from the idea that physical measurements may have residual imprecisions, the possibility of replacing the nonrelativistic, three-dimensional configuration space by a so-called fuzzy configuration space, having an isomorphic Borel structure, is discussed. A quantization procedure with respect to such a space is developed, and the invariance of nonrelativistic quantum mechanics under such Borel isomorphisms is exploited to prove the equivalence of this quantization procedure to the usual quantization procedure on a fuzzy-free configuration space. Further, any Galilean invariant dynamics is shown to be insensitive to such imprecisions in the measurements of position and momentum.