Kraus representation of a damped harmonic oscillator and its application

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.

Figure 1

Difference of fidelities: $F_1-\eta$ (solid curves), given by Eqs. (28, 34), and $F_2-\eta$ (dashed curves), given by Eqs. (30, 34), versus scaled time $\gamma t$ for different values of amplitude $\alpha$.

Figure 2

Evolution of fidelities $f_0\left(\tau \right)$ (solid curves) and $f_1\left(\tau \right)$ (dashed curves) are given by Eq. (54) for different values of intensity ${\mid \alpha \mid }^2$. It is seen that the curves for $f_0\left(\tau \right)$ and $f_1\left(\tau \right)$ coincide if $\mid \alpha \mid {}^2⩾4$.

Figure 3

Evolution of fidelities $F^{\prime }\left(\tau \right)$ (dashed curve), given by Eq. (62), and $F\left(\tau \right)$ (solid curves), given by Eq. (58), for different values of intensity ${\mid \alpha \mid }^2$.