Redundancy from Subsystem Thermalization

Abstract

In the theory of decoherence, redundancy is the correlation between a quantum system and fractions of the environment. It underlies the emergence of classical behavior. We show that redundancy can persist despite thermalizing dynamics in the environment. This follows an initial broadcasting interaction that changes the density of a conserved quantity. The mutual information between the system and a fraction of the environment is estimated using the large deviation principle governing subsystem thermalization.

Figures (5)

• Figure 1: Mutual information $I(\mathcal{F}, \mathcal{S})$ between the system and a fraction $\mathcal{F}$ of the environment, as a function of its size $|\mathcal{F}|$, after a broadcasting interaction \ref{['eq:H-int']} (with $t_0 = \pi / 4\lambda \times 0.75$), followed by intrinsic environment evolution under \ref{['eq:HE-ex']} for $t = 1, 2, 4, \dots, 32$ (light to dark). (a) The initial redundancy plateau crosses over to encoding behavior due to thermalization. (b) Upon replacing $Z_j$ by $Y_j$ in \ref{['eq:H-int']}, the redundancy plateau persists at late times and the approach to it does not depend on the environment size $N$.
• Figure 2: (a) Conserved energy density large deviation rate functions \ref{['eq:large-deviation']} in the example of Fig. \ref{['fig:example']}-(b). The intersection point gives $\alpha_*$ in \ref{['eq:alpha']}. (b) Approach the plateau of $I(\mathcal{F}, \mathcal{S})$ [same condition as Fig. \ref{['fig:example']}-(b), $N = 20$, larger $t$] compared to the bound \ref{['eq:plateaumain']}.
• Figure 3: System-fraction mutual information $I(\mathcal{F}, \mathcal{S})$ in an interpolation between Fig. \ref{['fig:example']}-(a) and (b). The parameters are the same as in Fig. \ref{['fig:example']}, except that $Z_j \to Z_j \cos(\lambda \pi / 2) + Y_j \sin(\lambda \pi / 2)$ in \ref{['eq:H-int']}. In the main plot, we show the data collapse as a function of $\lambda^2 |\mathcal{F}|$. In the inset, we show the raw data.
• Figure 4: Distribution of expectation value $\left< \psi | Z_{\mathcal{S}} | \psi \right>$ where $\psi$ is drawn from the projective ensemble obtained from measuring the environment in the computational basis in the final states in Fig. \ref{['fig:example']} with $N = 20$ and $N = 10$ (same other parameters). In the encoding case [Fig. \ref{['fig:example']}-(a)], the distribution is approximately uniform; in the redundancy case [Fig. \ref{['fig:example']}-(b)], it is peaked at $\pm 1$.
• Figure 5: System-fraction mutual information with partial energy degeneracy. At late times, a barely visible partial redundancy plateau at $s = \ln (3)/ 3 + 2 \ln (3/2) /3$ is followed by an encoding jump at $|\mathcal{F}| = |\mathcal{E}| / 2$, where $|\mathcal{E}| = 20.$

Theorems & Definitions (1)

• Proposition 1