Quantum Darwinism in a Mixed Environment

Quantum Darwinism recognizes that we—the observers—acquire our information about the “systems of interest” indirectly from their imprints on the environment. Here, we show that information about a system can be acquired from a mixed-state, or hazy, environment, but the storage capacity of an environment fragment is suppressed by its initial entropy. In the case of good decoherence, the mutual information between the system and the fragment is given solely by the fragment’s entropy increase. For fairly mixed environments, this means a reduction by a factor $1-h$, where $h$ is the haziness of the environment, i.e., the initial entropy of an environment qubit. Thus, even such hazy environments eventually reveal the state of the system, although now the intercepted environment fragment must be larger by $\sim (1-h{)}^{-1}$ to gain the same information about the system.

Figure 1

Mutual information between $\mathcal{S}$ and a fragment $\mathcal{F}$ for ${}^{♯}\mathcal{E}=100$, ${\rho }_{\mathcal{S}}(0)=|+⟩⟨+|$, and $r_{00}=1/2$, where $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$. An initially (a) pure $\mathcal{E}$, $h=0$, and (b) hazy $\mathcal{E}$, $h\approx 0.8$. At $t=0$, $\mathcal{S}$ and $\mathcal{E}$ are not correlated but, with time, correlations develop. As the system approaches good decoherence, a long plateau region forms where information is redundantly recorded in $\mathcal{E}$. This level occurs at the value of the entropy of $\mathcal{S}$ when decohered, which here is at $H_{\mathcal{S}}=1$. For sufficiently large ${}^{♯}\mathcal{E}$, the mutual information will eventually reach $H_S$ regardless of the initial haziness, except for a completely mixed initial $\mathcal{E}$, $h=1$. However, the plateau is attained more slowly and only for larger fragments as $\mathcal{E}$ gets more mixed. When all of $\mathcal{E}$ is captured, the mutual information jumps to its maximum value of $2H_S$ (signifying complete quantum correlation of $\mathcal{E}$ with $\mathcal{S}$) so long as a well-defined plateau exists.

Figure 2

(a) Mutual information at $t=\pi /2$ versus ${}^{♯}\mathcal{F}$ and $h$ for the same conditions as in Fig. 1. (b) Redundancy versus $h$ for the information deficit $\delta =0.1$ and at $t=\pi /2$ (the inset shows $t=\pi /3$). The black line (squares) is the exact data. The redundancy can only take on rational values with ${}^{♯}\mathcal{E}$ in the numerator because of the discrete nature of the spin environment, which is particularly visible at high redundancy. The blue dashed line is that obtained by the scaling $1-h$, which is a good approximation when $h$ is near one. Thus, even initially mixed $\mathcal{E}$ can store information about $\mathcal{S}$ in many copies. However, it takes larger ${}^{♯}\mathcal{F}$ to acquire the same information about $\mathcal{S}$.

Figure 3

(a) Bimodal probability distribution for the state of $\mathcal{F}$ to be in a subspace $n$, where $n=0,\dots ,{}^{♯}\mathcal{F}$ indexes the number of identical records. The left (blue) peak, $P_L$, is correlated with $|0⟩$ in $\mathcal{S}$ and the right (red), $P_R$, with $|1⟩$. When $\mathcal{E}$ is initially nearly pure, $h\sim 0$, the two peaks are well separated. Upon increasing $h$, the two peaks start to overlap. (b) The information deficit $\delta$ versus the overlap $O=\sum _n{P}_L(n)P_R(n)$ for ${}^{♯}\mathcal{F}=50$ and different $h$. A larger overlap is associated with a decreased capacity for information about $\mathcal{S}$, i.e., an increased information deficit. The parameters are the same as Fig. 1.