Influence of dissipation on the extraction of quantum states via repeated measurements

A quantum system put in interaction with another one that is repeatedly measured is subject to a nonunitary dynamics, through which it is possible to extract subspaces. This key idea has been exploited to propose schemes aimed at the generation of pure quantum states (purification). All such schemes have so far been considered in the ideal situations of isolated systems. In this paper, we analyze the influence of non-negligible interactions with environment during the extraction process, with the scope of investigating the possibility of purifying the state of a system in spite of the sources of dissipation. A general framework is presented and a paradigmatic example consisting of two interacting spins immersed in a bosonic bath is studied. The effectiveness of the purification scheme is discussed in terms of purity for different values of the relevant parameters and in connection with the bath temperature.

Figure 1

(Color online) The purity of the extracted state vs the parameters $ϵ\tau$ and $\theta ∕\pi$ ($\chi =0$ for all cases), in different situations: (a) in the ideal case, i.e., in the absence of interaction with the bath, and in the presence of interaction with the bath with (b) $k_{\text{B}}T∕\hslash \Omega =0.01$, (c) $k_{\text{B}}T∕\hslash \Omega =1$, and (d) $k_{\text{B}}T∕\hslash \Omega =10$. In all cases, ratios between salient physical quantities are fixed as $\Omega ∕ϵ=10$, ${\gamma }_2∕ϵ=0.1$, ${\gamma }_1∕{\gamma }_2=0.95$.

Figure 2

(Color online) The purity of the extracted state as a function of the temperature $T$ (in units of $\hslash \Omega ∕k_{\text{B}}$) and $ϵ\tau$. The other parameters are $\theta =3\pi ∕4$, $\chi =0$, $\Omega ∕ϵ=10$, ${\gamma }_2∕ϵ=0.1$, ${\gamma }_1∕{\gamma }_2=0.95$.

Figure 3

(Color online) The quantity $\text{ln}\left|{\lambda }_0∕{\lambda }_1\right|$ vs the parameters $ϵ\tau$ and $\theta ∕\pi$ ($\chi =0$ for all cases), in different situations: (a) in the ideal case, i.e., in the absence of interaction with the bath, and in the presence of interaction with the bath with (b) $k_{\text{B}}T∕\hslash \Omega =0.01$, (c) $k_{\text{B}}T∕\hslash \Omega =1$, and (d) $k_{\text{B}}T∕\hslash \Omega =10$. In all cases, ratios between salient physical quantities are fixed as $\Omega ∕ϵ=10$, ${\gamma }_2∕ϵ=0.1$, ${\gamma }_1∕{\gamma }_2=0.95$.

Figure 4

(Color online) The purity of the state of system $S$ as a function of the number of measurements on $X$. The initial state is $\rho \left(0\right)=0.5{|\uparrow ⟩}_S⟨\uparrow |+0.5{|\downarrow ⟩}_S⟨\downarrow |$ and its purity is 0.5. Observe that the purity reaches the maximum value at $N=2$ and then decays to an asymptotic value. Here, the parameters are $\chi =0$, $\theta =2.25$, and $ϵ\tau =7.82$, and we have set $k_{\text{B}}T∕\hslash \Omega ={10}^{-3}$, $\Omega ∕ϵ=10$, ${\gamma }_2∕ϵ=0.1$, ${\gamma }_1∕{\gamma }_2=0.95$.