Probabilistic reversing operation with fidelity and purity gain for macroscopic quantum superposition

It is shown that a large class of weak disturbances on macroscopic quantum superpositions can be canceled by a probabilistic reversing operation on the system. We illustrate this for spin systems undergoing an Ising-type interaction with the environment and demonstrate that both the fidelity to the original state and the purity of the amended state can simultaneously be increased by the reversing operation. A possible experimental scheme to implement our scheme is discussed.

Figure 1

Fidelity $F^{\prime }$ and purity $P^{\prime }$ after an interaction with the environment as functions of spin $s$, which is equal to one half of the degrees of the system. The parameters used are $c_{\pm }=1∕\sqrt{2}$, $j=50$, $g=0.01$, and $\theta =\pi ∕6$. The fidelity $F^{\prime }$ oscillates near $s\simeq 1$ due to the effect of statistics (i.e., $s$ is an integer or a half integer) which becomes negligible as $s$ becomes large.

Figure 2

Probability $p_m$, fidelity $F_m^{″}$, and purity $P_m^{″}$ after a measurement as functions of outcome $m$ for $s=j=50$, $c_{\pm }=1∕\sqrt{2}$, $g=0.01$, and $\theta =\pi ∕6$. The mean fidelity and purity after the measurement are $0.945$ and $0.913$, respectively.