Quantum Darwinism-encoding transitions on expanding trees

Quantum Darwinism (QD) proposes that classical objectivity emerges from the broadcast of information about a microscopic degree of freedom into multiple fractions of a many-body environment. Such a broadcast of information is in sharp contrast with its scrambling under strong interaction. It was recently shown that quantum dynamics interpolating between broadcasting and scrambling may display sharp phase transitions of information propagation, named QD-encoding transitions. Here we initiate their systematic study in generic, non-Clifford settings. First, in a general theoretical setup where the information propagation is modeled as an isometry, whose input qudit is entangled with a reference, we propose a probe of the transitions—the distribution of the density matrix of the reference after measuring an environment fraction. This probe measures the classical correlation between the fraction and the injected information. We then apply the framework to two similar models defined by a tensor network on an expanding tree, modeling a noisy apparatus that attempts to broadcast the $z$ component of a spin-half. We derive an exact recursion relation of the density matrix distribution, which we analyze analytically and numerically. As a result we find three phases: QD, intermediate, and encoding, and two continuous transitions. The encoding-intermediate transition describes the establishment of nonzero correlation between the reference and a small environment fraction, and can be probed by a “coarse-grained” measure of the total spin-$z$ of the fraction, which becomes non-Gaussian and symmetry breaking in the intermediate space. The QD-intermediate transition is about whether the correlation is perfect. It must be probed by fined-grained measures and corresponds to a subtler symmetry breaking in the replica space.

Figure 1

General theoretical setup for studying information propagation in an environment. $V$ is an isometry from $A$ (injected qudit) to $E$ (output environment). $A$ is initially entangled with a reference $R$. To probe the classical correlation between a fraction $F\subset E$ and $R$, we perform a measurement on $F$ and consider the postmeasurement density matrix of $R$. By definition, in the QD phase, ${\rho }_{R,m}$ is almost surely a pure state; in the encoding phase, ${\rho }_{R,m}$ is almost surely maximally mixed. (The density matrices are represented using the Bloch sphere.) In the tree models studied in this work, the two phases appear at small and large values of a “scrambling parameter” $J$ (that controls $V$). The two phases are separated by an intermediate phase and two transitions. See Figs. 2 and 3 below.

Figure 2

The matrix distribution ${\mathbb{P}}_{t,k}\left(\tilde{Q}\right)={\mathbb{P}}_{t,k}(u,v)$ where $\tilde{Q}=\mathbf{1}+u{\sigma }^z+v{\sigma }^x$, for three values of the parameter $J$ in the deterministic (a)–(c) and random (d)–(f) tree models. The distributions are obtained by numerically solving the recursion relations (81) and (82), by methods in Sec. 3c1. These distributions describe the injected information retrievable from a fraction of size $\left|F\right|=2^t$ in an environment of size $\left|E\right|=2^{t+k}$; here $t=7$ and $k=2$. (a), (d) When $J$ is close enough to 0, the distribution is supported on the unit circle $u^2+v^2=1$ (indicated in green); this indicates a QD phase where the injected information is entirely retrievable from $F$. (b), (e) For intermediate values of $J$, the distribution has a nontrivial shape supported throughout the unit disk. The injected information is partially accessible from $F$. (c), (f) When $J$ is close enough to 1, the distribution is peaked at $u=v=0$. This is the hallmark of an encoding phase where no information is accessible from $F$.

Figure 3

The purity $\langle r^2\rangle =\langle u^2+v^2\rangle$ of the density matrix distribution ${\mathbb{P}}_{t,k}$ as a function of the parameter $J$ for different values of $t$ and $k=3$ (the fraction size is $\left|F\right|=2^t$ and its relative size is $\left|F\right|/\left|E\right|=2^{-k}$), in the deterministic (a) and random (b) tree models. In both models, $\langle r^2\rangle \to 1$ rapidly as $t$ increases when $J$ is small enough, and $\langle r^2\rangle \to 0$ rapidly when $J$ is close enough to 1. At intermediate values, the convergence to the thermodynamics limit is slow. Further analysis (see below) will show that an intermediate phase where $0<{\langle r^2\rangle }_{t\to \infty }<1$ separates the QD phase and the encoding phase. The data are obtained by numerically solving the recursion relations (81) and (82); see Sec. 3c1 for methods.

Figure 4

Dependence of the purity on the relative fraction size $\left|F\right|/\left|E\right|=2^{-k},k=2,\cdots ,8$, for two values of $J$ and a few values $t$ ($\left|F\right|=2^t$). At small enough $J$, the purity converges rapidly to a Darwinism plateau $\langle r^2\rangle =1$ for all values of $k$. At intermediate $J$, the convergence is slower, yet $\langle r^2\rangle$ tends to a $k$-independent value between 0 and 1 in the thermodynamic limit.

Figure 5

The purity $\langle r^2\rangle$ as a function of $J$ near the encoding transition $J_c=1/2$. The numerical data (dots connected by lines, which are a guide to the eye) are obtained in the random model, up to $t=160$, using the efficient sampling method; see Sec. 3c1. To reduce statistical noise, we averaged over data in a small time interval $[0.9t,t]$ for each data point. The red dashed curve represents the analytical prediction (103) in the $t\to \infty$ limit.

Figure 6

Perfectly QD distribution ${\mathbb{P}}_{t,k=1}$ for a few values of $J$ and $t=9$ and $t=10$ obtained by solving the recursion relation (82) (random model) with initial condition (86) and $k=1$. These distributions are supported on the unit circle $u^2+v^2=1$ and symmetric with respect to $v\to -v$, so we plot them as a distribution of the angle $\alpha =arccos\left(u\right)$. We observe a fast convergence to the $t\to \infty$ limit and no abrupt dependence on $J$.

Figure 7

The (approximate) stability eigenvalue, ${\lambda }_d=2\langle 1-u^2\rangle /\langle 1-{u^{\prime }}^2\rangle$ (108), evaluated on the perfectly QD distribution ${\mathbb{P}}_{t,k=1}$ as a function of $J$ for $t=9,10$ (the result is practically independent of $t$). See Fig. 6 for plots of such distributions. We estimate the QD-intermediate transition $J_d\approx 0.375\left(5\right)$ as the value where ${\lambda }_d=1$.

Figure 8

Purity $\langle r^2\rangle$ in the random model up to $t=160$, obtained by numerical solution to the recursion relation (82); see Sec. 3c1 for numerical methods. (a) $\langle r^2\rangle$ as a function of $J$, for $t=10,20,\cdots ,160$. To reduce statistical noise, we averaged over data in a small time interval $[0.9t,t]$ for each data point. The vertical dashed line represents the estimate of the QD-intermediate critical point (109). (b) $1-{\langle r^2\rangle }_t$ as a function of $t$, for different values of $J$ (see color bar). The green line indicates a power law decay $\propto t^{-1}$. The data curves above (below) the green line have $J>J_d$ ($J<J_d$), respectively.

Figure 9

Numerical study of the QD-intermediate transition in the deterministic model. (a) The approximate stability eigenvalue ${\lambda }_d=2\langle 1-u^2\rangle /\langle 1-{u^{\prime }}^2\rangle$ (108) as a function of $J$, evaluated on the perfect QD distribution ${\mathbb{P}}_{t,k=1}$ with $t=9$ and 10, at which value ${\mathbb{P}}_{t,k=1}$ has well converged to the $t=\infty$ limit. The QD-intermediate critical point $J_d\approx 0.35\left(1\right)$ is estimated as the value where $\lambda$ exceeds 1. (b) Main: Testing the scaling Ansatz (112) with $J_d=0.35$ (not adjusted) using numerical data from $t=7,\cdots ,10$ ($k=2$). Inset: raw data.

Figure 10

Outcome distribution of $\mathcal{M}={\sum }_{i\in F}{m}_i$ (total spin of the fraction) and the postmeasurement reference state, for $J=0.1$ (a), 0.3 (b), and 0.6 (c) in the deterministic model with fraction size $\left|F\right|=2^{12}$ and relative size $\left|F\right|/\left|E\right|=1/4$. (See also Fig. 12 for a closer look at very small $J$.) We plot the probability density of the total spin re-scaled by its standard deviation, $m=\mathcal{M}/{\sigma }_{\mathcal{M}}, P\left(m\right)=p_{\mathcal{M}}{\sigma }_{\mathcal{M}}$ as the filled area. $r_{\mathcal{M}}^{2}P\left(z\right)$ is plotted as a colored curve, where the color indicates the angle of $(u_{\mathcal{M}},v_{\mathcal{M}})$, indicating the polarization direction the postmeasurement reference state. The filled area below and above the colored curve represents the information revealed by and hidden from the measurement respectively.

Figure 11

Main: the averaged purity with different measurement schemes in the deterministic model with fraction size $\left|F\right|=2^{12}$ and relative size $\left|F\right|/\left|E\right|=2^{-2}$. $\tau =0$: only the total spin of $F$ is measured. $\tau =1$: the total spins of the left and right subtrees are measured separately. $\tau =2$: the total spins of the four “grandchildren” subtrees of the root are measured separately. $\tau =t$: the microscopic measurement. See also Fig. 13. Inset: log-log plot of the $\tau =1$ and $\tau =2$ data near $J=0$, compared to power laws. The prefactor of the $J^4$ power law is an exact prediction (133).

Figure 12

Outcome distribution of the total spin measurement with $J=0.02, \left|F\right|=2^{11}, \left|F\right|/\left|E\right|=2^{-2}$. See Fig. 10 for further description. Here, in addition, we plot $p_{\mathcal{M}}(1-r_{\mathcal{M}}^2)$ in black. At small $J$, the distribution has a “multifractal” structure of peaks; the dominant ones are $\mathcal{M}\approx \pm \left|F\right|$. The information retrieval is nearly perfect: $r_{\mathcal{M}}^2$ is close to 1 for all $\mathcal{M}$. The imperfections are concentrated at secondary peaks at $\mathcal{M}=0,\pm \left|F\right|/4$.

Figure 13

Illustration of coarse-grained measurements. The green squares indicate the spins in the fraction $F$, with size $\left|F\right|=2^t=8$ and relative size $\left|F\right|/\left|E\right|=2^{-k}=1/4$. $\mathcal{M}$, the total spin of $F$, is divided into that of the subtrees. The coarse-grained measurement with $\tau =2$ will measure ${\mathcal{M}}_{\ell \ell },\cdots ,{\mathcal{M}}_{rr}$. The corresponding density matrices can be obtained in two stages. First we obtain the total-spin ($\tau =0$) matrices at $t-\tau =1$ using the recursion (129); then we iterate (134) twice ($\tau =2$).

Figure 14

The covariance ${\langle mu\rangle }_{*}$ (150), skewness ${\langle m^3\rangle }_{*}$ (148), and negative excessive kurtosis ${\langle 3-m^4\rangle }_{*}$ (149) of the fixed point outcome distribution $m$, normalized so that $\langle m^2\rangle =1$.

Figure 15

The flow generated by the recursion relation (153) of the Clifford model, at $J=0.45<J_c$ (a) and $J=0.55>J_c$ (b). The asymptotic limit undergoes a discontinuous transition from a QD fixed point to the encoding fixed point.