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arXiv:quant-ph/0412213 · International Journal of Quantum Information

PARTICLE STATISTICS IN QUANTUM INFORMATION PROCESSING

Yasser Omar

Published Dec 17, 2004• 10 citations
arXivHTMLPDF

Abstract

Particle statistics is a fundamental part of quantum physics, and yet its role and use in the context of quantum information have been poorly explored so far. After briefly introducing particle statistics and the Symmetrization Postulate, we argue that this fundamental aspect of nature can be seen as a resource for quantum information processing and present examples showing how it is possible to do useful and efficient quantum information processing using only the effects of particle statistics.

DOI
10.1142/s021974990500075x

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Particle statistics is a fundamental part of quantum physics, and yet its role and use in the context of quantum information have been poorly explored so far. After briefly introducing particle statistics and the Symmetrization Postulate, we argue that this fundamental aspect of nature can be seen as a resource for quantum information processing and present examples showing how it is possible to do useful and efficient quantum information processing using only the effects of particle statistics.
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Identical quantum particles exhibit only two types of statistics: bosonic and fermionic. Theoretically, this restriction is commonly established through the symmetrization postulate or (anti)commutation constraints imposed on the algebra of creation and annihilation operators. The physical motivation for these axioms remains poorly understood, leading to various generalizations by modifying the mathematical formalism in somewhat arbitrary ways. In this work, we take an opposing route and classify quantum particle statistics based on operationally well-motivated assumptions. Specifically, we consider that a) the standard (complex) unitary dynamics defines the set of single-particle transformations, and b) phase transformations act locally in the space of multi-particle systems. We develop a complete characterization, which includes bosons and fermions as basic statistics with minimal symmetry. Interestingly, we have discovered whole families of novel statistics (dubbed transtatistics) accompanied by hidden symmetries, generic degeneracy of ground states, and spontaneous symmetry breaking – effects that are (typically) absent in ordinary statistics.
A review of the interpolative quantum statistics is presented. It is shown, by imposing a positivity condition on the distribution function, that this statistics subsumes Gentile's intermediate statistics and is far richer than Gentile's statistics, apart from being a genuine quantum statistics. It is also capable of accounting for the possible very small violations of statistics in physical processes. It is finally pointed out that this statistics offers an elegant and useful parametrization of Haldane's 'statistical interaction', while retaining all the interesting features of the latter. Some of the statistical mechanics that these distribution functions lead to are also examined.
The quantum reconstruction program seeks to uncover the deeper physical meaning of the quantum formalism by distilling its content into a set of physically graspable postulates. Once reconstructed, one can further elucidate the theory by reflecting philosophically upon the postulates rather than the inscrutable formalism in which the theory is couched. Quantum reconstruction thereby opens up a two-step strategy for developing a quantum conceptual framework that reflects the full physical content of quantum theory and can be used to establish precisely what aspects of the classical conceptual framework are retained, modified, or abandoned. The chapter is organized as follows. Section 11.2 describes the tripartite structure of classical physics, whose coherency I argue underlies its intelligibility. Section 11.3 discusses the lack of intelligibility of quantum physics, analyses the limitations of the main traditional elucidative approaches, and describes how reconstruction can remove the fundamental interpretative bottleneck. Section 11.4 discusses the methodology of reconstruction per se , showing that it is part of the natural life-cycle of physical theories, and summarizes its use in the elucidation of the theories of classical physics. Finally, circling back to quantum theory, Section 11.5 summarizes the reconstruction of quantum theory, and Section 11.6 describes some of the interpretative insights obtained thus far through interpretation of reconstructions of the abstract quantum formalism and the quantum symmetrization postulate.
As a result of the capabilities of quantum information, the science of quantum information processing is now a prospering, interdisciplinary field focused on better understanding the possibilities and limitations of the underlying theory, on developing new applications of quantum information and on physically realizing controllable quantum devices. The purpose of this primer is to provide an elementary introduction to quantum information processing, and then to briefly explain how we hope to exploit the advantages of quantum information. These two sections can be read independently. For reference, we have included a glossary of the main terms of quantum information.