  We develop a new method of representation of quantum states in terms of the
displaced number states. We call it representation, where is an amplitude of
the base displaced states. In particular, representation was obtained for set
of the displaced number states with different amplitude of the displacement,
two-mode squeezed vacuum (TMSV) and superposition of vacuum and single photon.
The treatment is employed for building of elementary linear operators, in
particular, two-qubit control-sign gate, Hadamard matrix between two
two-dimensional Hilbert spaces. Main idea underlying the method is extraction
of the displaced number state from initial state. Coherent qubit with large
amplitude taken simultaneously two modes shifts two-mode target qubit on phase
plane and constructive interference appears after measurement of single photon.
The results are exact, any approximations are not exploited. We show
implementation of the gates with almost unity fidelity of the output states is
possible in realistic scenario.
