  The interest in quantum-optical states confined in finite-dimensional Hilbert
spaces has recently been stimulated by the progress in quantum computing,
quantum-optical state preparation, and measurement techniques, in particular,
by the development of the discrete quantum-state tomography. In the first part
of our review we present two essentially different approaches to define
harmonic oscillator states in the finite-dimensional Hilbert spaces. One of
them is related to the truncation scheme of Pegg, Phillips and Barnett [Phys.
Rev. Lett. 81, 1604 (1998)] -- the so-called quantum scissors device. The
second method corresponds to the truncation scheme of Leo\'nski and Tana\'s
[Phys. Rev. A 49, R20 (1994)]. We propose some new definitions of the states
related to these truncation schemes and find their explicit forms in the Fock
representation. We discuss finite-dimensional generalizations of coherent
states, phase coherent states, displaced number states, Schr\"odinger cats, and
squeezed vacuum. We show some intriguing properties of the states with the help
of the discrete Wigner function.

Quantum Physics