Journal of Physics A Mathematical and General

Factorization in finite quantum systems

Apostol Vourdas

Published May 8, 2003• 23 citations

Abstract

Unitary transformations in an angular momentum Hilbert space H (2 j + 1), are considered. They are expressed as a finite sum of the displacement operators (which play the role of SU (2 j + 1) generators) with the Weyl function as coefficients. The Chinese remainder theorem is used to factorize large qudits in the Hilbert space H (2 j + 1) in terms of smaller qudits in Hilbert spaces H (2 j i + 1). All unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits.

DOI: 10.1088/0305-4470/36/20/319

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Journal of Physics A Mathematical and General

Factorization in finite quantum systems

Apostol Vourdas

Published May 8, 2003• 23 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/36/20/319

Related papers

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Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU (2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU (2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.

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Quantum symmetries and the Weyl Wigner product of group representations

Anthony J. Bracken, G Cassinelli, J G Wood

In the usual formulation of quantum mechanics, groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex Hilbert space, in accordance with Wigner's theorem. In the phase-space formulation, they have real, true unitary representations in the space of square-integrable functions on phase space. Each such phase-space representation is a Weyl–Wigner product of the corresponding Hilbert space representation with its contragredient, and these can be recovered by 'factorizing' the Weyl–Wigner product. However, not every real, unitary representation on phase space corresponds to a group of automorphisms, so not every such representation is in the form of a Weyl–Wigner product and can be factorized. The conditions under which this is possible are examined. Examples are presented.In the usual formulation of quantum mechanics, groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex Hilbert space, in accordance with Wigner's theorem. In the phase-space formulation, they have real, true unitary representations in the space of square-integrable functions on phase space. Each such phase-space representation is a Weyl–Wigner product of the corresponding Hilbert space representation with its contragredient, and these can be recovered by 'factorizing' the Weyl–Wigner product. However, not every real, unitary representation on phase space corresponds to a group of automorphisms, so not every such representation is in the form of a Weyl–Wigner product and can be factorized. The conditions under which this is possible are examined. Examples are presented.

arXiv:math-ph/0211001Journal of Physics A Mathematical and GeneralJan 15, 200313 citations

Abstract Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space

D=d^n

D = d n with d an odd integer, and is inspired by the Cooley-Tukey formalism. The ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (together with some other transforms) in quantum systems with d-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the n Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space

D=d_0...d_{n-1}

D = d 0 . . . d n - 1 with

d_0,...,d_{n-1}

d 0 , . . . , d n - 1 odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (that involve some constants related to the number theory that describes the formalism). The ‘small Fourier transforms’ can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from

\mathcal{O}(D^2)

O ( D 2 ) to

\mathcal{O}(D\log D)

O ( D log D ) . The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.