Redundant imprinting of information in nonideal environments: Objective reality via a noisy channel

Quantum Darwinism provides an information-theoretic framework for the emergence of the objective, classical world from the quantum substrate. The key to this emergence is the proliferation of redundant information throughout the environment where observers can then intercept it. We study this process for a purely decohering interaction when the environment, $\mathcal{E}$, is in a nonideal (e.g., mixed) initial state. In the case of good decoherence, that is, after the pointer states have been unambiguously selected, the mutual information between the system, $\mathcal{S}$, and an environment fragment, $\mathcal{F}$, is given solely by $\mathcal{F}$’s entropy increase. This demonstrates that the environment’s capacity for recording the state of $\mathcal{S}$ is directly related to its ability to increase its entropy. Environments that remain nearly invariant under the interaction with $\mathcal{S}$, either because they have a large initial entropy or a misaligned initial state, therefore have a diminished ability to acquire information. To elucidate the concept of good decoherence, we show that, when decoherence is not complete, the deviation of the mutual information from $\mathcal{F}$’s entropy change is quantified by the quantum discord, i.e., the excess mutual information between $\mathcal{S}$ and $\mathcal{F}$ is information regarding the initial coherence between pointer states of $\mathcal{S}$. In addition to illustrating these results with a single-qubit system interacting with a multiqubit environment, we find scaling relations for the redundancy of information acquired by the environment that display a universal behavior independent of the initial state of $\mathcal{S}$. Our results demonstrate that Quantum Darwinism is robust with respect to nonideal initial states of the environment: the environment almost always acquires redundant information about the system but its rate of acquisition can be reduced.

Figure 1

The behavior of the mutual information, $I(\mathcal{S}:\mathcal{F})$, between the system, $\mathcal{S}$, and a fragment, $\mathcal{F}$, of a symmetric environment, $\mathcal{E}$, as a function of the fraction of the environment intercepted, $f={}^{♯}\mathcal{F}/{}^{♯}\mathcal{E}$. The black solid line is for an initially pure $\mathcal{E}$ and the black dashed line is for an initially hazy $\mathcal{E}$. Here, ${}^{♯}\mathcal{F}$ and ${}^{♯}\mathcal{E}$ are the number of components in the fragment and the environment, respectively. The information $H_{\mathcal{S}}$ sets the limit of classical information about $\mathcal{S}$ that $\mathcal{E}$ can acquire, resulting in a plateau region at $H_{\mathcal{S}}$ (the classical plateau) that signifies the redundant proliferation of classical information into the environment. We define a fragment size, ${}^{♯}{\mathcal{F}}_{\delta }$, that gives the value of the mutual information $(1-\delta )H_{\mathcal{S}}$, i.e., to within the information deficit, $\delta$, of the classical plateau. The redundancy, $R_{\delta }$, at this information deficit is then given by Eq. (2). The initial haziness of $\mathcal{E}$ reduces the redundancy. The data for this figure is from an actual simulation like those performed in Sec. 5.

Figure 2

Bloch sphere representation of (a) haziness and (b) misalignment. Pure states in the $x$-$y$ plane (highlighted) have the maximum capacity to accept information about the system’s pointer states. Haziness of an environment qubit contracts its Bloch sphere, reducing the qubits ability to increase its entropy and therefore decreasing its capacity to store information. Misalignment rotates the state out of the $x$-$y$ plane, which also decreases the qubits capacity to store information. For both haziness and misalignment, the decrease in capacity is due to a reduction in the environment qubits’ ability to branch into two orthogonal states correlated with the two pointer states of the system.

Figure 3

(a and b) Mutual information, $I(\mathcal{S}:\mathcal{F})$, and (c and d) quantum discord, $\mathit{\delta }(\mathcal{S}:\mathcal{F}){}_{\{{\sigma }_{\mathcal{S}}^z\}}$, versus the fragment size, ${}^{♯}\mathcal{F}$, and time, $t$, with $s_{00}=1/2$, $r_{00}=1/2$, and ${}^{♯}\mathcal{E}=200$. (a and b) The mutual information for $H_{\mathcal{S}}(0)=0$ and $H_{\mathcal{S}}(0)=0.8$, respectively. Initially $\mathcal{E}$ and $\mathcal{S}$ are uncorrelated, but as time develops, $\mathcal{E}$ acquires information about $\mathcal{S}$. After an initial transient region, signified by a nonzero quantum discord in (c and d), a plateau develops in the mutual information. A sudden increase in the von Neumann mutual information occurs for large ${}^{♯}\mathcal{F}~{}^{♯}\mathcal{E}$ because complementary information about $\mathcal{S}$ is accessible via global measurements. For a system initially in a superposition, this jump is large and so is the discord in the transient region. The jump is reduced by any existing decoherence of $\mathcal{S}$ when it is placed in contact with $\mathcal{E}$. However, the level and size of the classical plateau is identical regardless of the initial entropy of $\mathcal{S}$. (c and d) The quantum discord with respect to the eigenstates of ${\Pi }_{\mathcal{S}}={\sigma }_{\mathcal{S}}^z$ for $H_{\mathcal{S}}(0)=0$ and $H_{\mathcal{S}}(0)=0.8$, respectively. The environment, $\mathcal{E}$, can be pure or mixed, and, in fact, the discord is equivalent to the mutual information between $\mathcal{S}$ and $\mathcal{F}$ for a diagonal initial state ${\rho }_r$. There is a transient region, just after $\mathcal{S}$ and $\mathcal{E}$ have come into contact, where nonzero discord exists. Its duration depends on the size of the environment. Except for this region, the discord is negligibly small since both $\mathcal{E}$ and $\mathcal{E}/\mathcal{F}$ are sufficient to decohere $\mathcal{S}$. The discord is reduced by the preexisting entropy of $\mathcal{S}$ before coming in contact with $\mathcal{E}$, as shown in (b). This is because the discord signifies complementary information about $\mathcal{S}$, i.e., information about the initial coherence between pointer states of $\mathcal{S}$.

Figure 4

Mutual information, $I(\mathcal{S}:\mathcal{F})$, versus the fragment size, ${}^{♯}\mathcal{F}$, and time, $t$, for an initially pure $\mathcal{S}$ and (a) an initially hazy $\mathcal{E}$ with $r_{00}=1/2$ and $h\approx 0.8$, and (b) an initially hazy and misaligned $\mathcal{E}$ with $\sigma =0.8$ and $h/h_m\approx 0.8$. The other parameters are $s_{00}=1/2$ and ${}^{♯}\mathcal{E}=200$. As with Fig. 3, $\mathcal{E}$ initially contains no information about $\mathcal{S}$, but correlations develop, transmitting information about $\mathcal{S}$ throughout $\mathcal{E}$, then going over to a classical regime signified by the formation of the classical plateau. Misalignment and haziness both reduce the information a given fragment $\mathcal{F}$ gains about $\mathcal{S}$. However, since the classical plateau forms, the environment still maintains the ability to redundantly encode classical information about $\mathcal{S}$.

Figure 5

(a and b) Mutual information, $I(\mathcal{S}:\mathcal{F})$, versus the fragment size, ${}^{♯}\mathcal{F}$, and haziness, $h$, of the environment qubits, and (c and d) the redundancy (limiting redundancy), $R_{\delta }$ ($\mathrm{R̄}$), versus $h$. The system is initially pure, and $s_{00}=1/2$, $r_{00}=1/2$, and ${}^{♯}\mathcal{E}=200$. (a and b) Mutual information at $t=\pi /2$ and $t=\pi /4$, respectively. The classical plateau forms in all but the haziest of conditions where the environment is ab initio in a perfect mixture. (c) The redundancy, $R_{\delta }$, at $t=\pi /2$ (and $t=\pi /4$ in the inset). The black line is the exact data. The redundancy initially drops fairly rapidly because of the symmetry of the model but then goes over to a region where it behaves as $1-h$ (shown as the red, dashed line). The blue, dotted curve shows the scaling behavior from Eq. (29), which already reasonably approximates the exact behavior even though ${}^{♯}{\mathcal{F}}_{\delta }$ is small. (d) The limiting redundancy, $\mathrm{R̄}$, at $t=\pi /2$. There are four different initial states of $\mathcal{S}$: $s_{00}=1/2$ (blue circles), $s_{00}=1/8$ (red crosses), $s_{00}=1/64$ (green squares), and $s_{00}=1/4096$ (magenta triangles). Equation (30) with $R_{\delta }$ from Eq. (29), is plotted (black line) with the exact data for the different initial states of $\mathcal{S}$.

Figure 6

(a and b) Mutual information, $I(\mathcal{S}:\mathcal{F})$, versus the fragment size, ${}^{♯}\mathcal{F}$, and misalignment, $\sigma$, of the environment qubits, and (c and d) the redundancy, $R_{\delta }$, versus $\sigma$. The system and environment are initially pure, and $s_{00}=1/2$ and ${}^{♯}\mathcal{E}=200$. (a and b) The mutual information at $t=\pi /2$ and $t=\pi /4$, respectively. The classical plateau is formed and is quite large for all but very misaligned states ($\sigma$ near 1). (c and d) The redundancy versus the misalignment at $t=\pi /2$ and $t=\pi /4$, respectively. The black lines are the exact data obtained numerically and the blue dotted line is the scaling given by Eq. (33), which already fits quite well with the numerical results. The red dashed line is the redundancy given by $R_{\delta }\propto \ln |{\Lambda }_k(t)|{}^2$ with the constant of proportionality found by retaining all the factors when using Eq. (30). The inserts are the limiting redundancy, $\mathrm{R̄}$, given by Eq. (27) (using both the exact data shown as squares and data from Eq. (33) shown as a dashed blue line), which demonstrates that the scaling result is obtained when discrete effects are not present (i.e., in the limit of vanishing information deficit, $\delta \to 0$).

Figure 7

(a and b) Mutual information, $I(\mathcal{S}:\mathcal{F})$, versus the fragment size, ${}^{♯}\mathcal{F}$, and normalized haziness, $h/h_m$, of the environment qubits, and (c) the redundancy, $R_{\delta }$, versus $h/h_m$. The system is initially pure, and $t=\pi /2$, $s_{00}=1/2$, and ${}^{♯}\mathcal{E}=200$. (a and b) Mutual information for the misalignments $\sigma =0.4$ ($h_m\approx 0.88$) and $\sigma =0.8$ ($h_m\approx 0.47$), respectively. The plateau region still forms at the same level, $H_{\mathcal{S}}$, essentially regardless of how hazy or misaligned the environment is initially. (c) Redundancy for $\sigma =0.4$ (the inset is for $\sigma =0.8$). The black line is the exact data and the red dashed line is the linear approximation. As we see, there is still a linear region of the redundancy for fairly hazy environments.

Figure 8

Schematic diagram of the rotating technique used to diagonalize the density matrix ${\rho }_{\mathcal{F}}(t)$. The density matrix is split into two parts, $s_{00}{\mathrm{\rho ̃}}_r(t){}^{\otimes {}^{♯}\mathcal{F}}$ and $+s_{11}{\mathrm{\rho ̃}}_r(-t){}^{\otimes {}^{♯}\mathcal{F}}$, which are each separately rotated into the basis $|j,m\rangle {}_z$ by first going through the basis in which the state ${\mathrm{\rho ̃}}_r(t)$ is diagonal.

Figure 9

(a) Deviation of the mutual information from its plateau value, $\Delta I(\mathcal{S}:\mathcal{F})=H_{\mathcal{S}}-I(\mathcal{S}:\mathcal{F})$, versus the fragment size ${}^{♯}\mathcal{F}$. The exact deviation is plotted for $h=0.5$, $s_{00}=1/2$ (black squares); $h=0.5$, $s_{00}=1/16$ (red triangles); $h=0.9$, $s_{00}=1/2$ (blue diamonds); and $h=0.9$, $s_{00}=1/16$ (green inverse triangles), with the approximate data plotted as a line of the same color as its corresponding exact data. For all but the smallest ${}^{♯}\mathcal{F}$, the approximation gives the correct decay of the mutual information to its plateau value. Further, changing the value of $H_{\mathcal{S}}$ (by shifting $s_{00}$) does not change the decay behavior to the plateau. (b) Relative error of the asymptotic approximation versus ${}^{♯}\mathcal{F}$. The errors are for $h=0.5$, $s_{00}=1/2$ (black line); $h=0.5$, $s_{00}=1/16$ (red dashed line); $h=0.9$, $s_{00}=1/2$ (blue dotted line); and $h=0.9$, $s_{00}=1/16$ (green dash-dotted line). The errors decay initially as the approximation of the binomial coefficient by a constant becomes better, but the approximation will contain a finite relative error as ${}^{♯}\mathcal{F}\to \infty$ due to the approximation of the sum by an integral.