Oscillations of the purity in the repeated-measurement-based generation of quantum states

Repeated observations of a quantum system interacting with another one can drive the latter toward a particular quantum state, irrespective of its initial condition, because of an effective nonunitary evolution. If the target state is a pure one, the degree of purity of the system approaches unity, even when the initial condition of the system is a mixed state. In this paper we study the behavior of the purity from the initial value to the final one, that is, unity. Depending on the parameters, after a finite number of measurements, the purity exhibits oscillations that brings about a lower purity than that of the initial state, which is a point to be taken care of in concrete applications.

Figure 1

The purity as a function of the number of measurements, starting with $\rho \left(0\right)=0.1|\downarrow ⟩⟨\downarrow |+0.9|\uparrow ⟩⟨\uparrow |$, in the parameter region characterized by $ϵ\tau =7.82$, $\theta =2.25$, $ϵ/\Omega =0.1$.

Figure 2

The purity as a function of the number of measurements, starting with $\rho \left(0\right)=0.5|\downarrow ⟩⟨\downarrow |+0.5|\uparrow ⟩⟨\uparrow |$, in the parameter region characterized by $ϵ\tau =2.50$, $\theta =1.0$, $ϵ/\Omega =0.1$.

Figure 3

Contour plot of the quantity $\eta =\text{max}\mathbf{\left(}0,\frac{1}{2}\left\{1+\text{ln}\left[p_{\uparrow \uparrow }/\left(1-p_{\uparrow \uparrow }\right)\right]/\text{ln}\left|\text{cos}ϵ\tau \right|\right\}\mathbf{\right)}$ vs the population of $|\uparrow ⟩$, $p_{\uparrow \uparrow }$, and the dimensionless time between two measurements $2ϵ\tau /\pi$ both in the range [0.1, 0.9]. The bold line corresponds to $\eta =1$, which discriminates between monotonic behavior of the purity ($\eta <1$ on the right-hand side of the line) and oscillations ($\eta >1$ in the left-hand side). In the dark region (in the bottom of the left-hand side) the value of $\eta$ is very high, meaning that one needs a large number of measurements to guarantee monotonicity.