  By definition, the Kraus representation of a harmonic oscillator suffering
from the environment effect, modeled as the amplitude damping or the phase
damping, is directly given by a simple operator algebra solution. As examples
and applications, we first give a Kraus representation of a single qubit whose
computational basis states are defined as bosonic vacuum and single particle
number states. We further discuss the environment effect on qubits whose
computational basis states are defined as the bosonic odd and even coherent
states. The environment effects on entangled qubits defined by two different
kinds of computational basis are compared with the use of fidelity.
