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Journal of Optics B Quantum and Semiclassical Optics

Entangling transformations in composite finite quantum systems

Apostol Vourdas

Published Oct 16, 2003• 5 citations
Publisher

Abstract

Phase space methods are applied in the context of finite quantum systems. 'Galois quantum systems' (with a dimension which is a power of a prime number) are considered, and symplectic transformations are studied. Composite systems comprising two finite quantum systems are also considered. Symplectic transformations are classified into local and entangling ones and the necessary matrices which perform such transformations are calculated numerically.

DOI
10.1088/1464-4266/5/6/006

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Phase space methods are applied in the context of finite quantum systems. 'Galois quantum systems' (with a dimension which is a power of a prime number) are considered, and symplectic transformations are studied. Composite systems comprising two finite quantum systems are also considered. Symplectic transformations are classified into local and entangling ones and the necessary matrices which perform such transformations are calculated numerically.
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Quantum systems in which the position and momentum take values in the ring calZd{\\cal Z}_dcalZd​ and which are described with ddd-dimensional Hilbert space, are considered. When ddd is the power of a prime, the position and momentum take values in the Galois field $GF(p^ \\ell)$, the position-momentum phase space is a finite geometry and the corresponding `Galois quantum systems' have stronger properties. The study of these systems uses ideas from the subject of field extension in the context of quantum mechanics. The Frobenius automorphism in Galois fields leads to Frobenius subspaces and Frobenius transformations in Galois quantum systems. Links between the Frobenius formalism and Riemann surfaces, are discussed.
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A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in , i = 1, ..., k , is considered. Its Hilbert space is given by a k -fold tensor product of Hilbert spaces of dimensions n 1 , ..., n k . The symmetry group of the respective finite Heisenberg group is given by the quotient group of certain normalizer. This paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type. It provides detailed description of the normalizers and the corresponding symmetry groups. The new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings. As an application, a new proof of existence of the maximal set of mutually unbiased bases in Hilbert spaces of prime power dimensions is provided.