Reports on Progress in Physics

Quantum systems with finite Hilbert space

Apostol Vourdas

Published Feb 2, 2004• 321 citations

Abstract

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.

DOI: 10.1088/0034-4885/67/3/r03

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Reports on Progress in Physics

Quantum systems with finite Hilbert space

Apostol Vourdas

Published Feb 2, 2004• 321 citations

Publisher

Abstract

DOI: 10.1088/0034-4885/67/3/r03

Related papers

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Quantum systems with finite Hilbert space are considered. Position and mo- \nmentum states and their relation through a Fourier transform, displacement \nin the position-momentum phase-space, and symplectic transformations are \nintroduced and their properties are studied. Symplectic Sp(2l;Zp) trans- \nformations in l-partite finite system are explicit constructed. The general \nmethod is applied to bi-partite and tri-partite systems. The effect of these \ntransformations on the correlations is discussed. Entanglement calculations \nbetween the subsystems in a bi-partite system and a tri-partite system are \npresented. The effect of measurements is also studied.Quantum systems with finite Hilbert space are considered. Position and mo- \nmentum states and their relation through a Fourier transform, displacement \nin the position-momentum phase-space, and symplectic transformations are \nintroduced and their properties are studied. Symplectic Sp(2l;Zp) trans- \nformations in l-partite finite system are explicit constructed. The general \nmethod is applied to bi-partite and tri-partite systems. The effect of these \ntransformations on the correlations is discussed. Entanglement calculations \nbetween the subsystems in a bi-partite system and a tri-partite system are \npresented. The effect of measurements is also studied.

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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.