# Pointer states and quantum Darwinism with 2-body interactions

Paul Duruisseau<EMAIL_ADDRESS>ENS Paris-Saclay, 91190,
Gif-sur-Yvette, France  Akram Touil<EMAIL_ADDRESS>Theoretical Division, Los
Alamos National Laboratory, Los Alamos, New Mexico 87545  Center for Nonlinear
Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
Sebastian Deffner<EMAIL_ADDRESS>Department of Physics, University of
Maryland, Baltimore County, Baltimore, MD 21250, USA

(August 28, 2024)

###### Abstract

Quantum Darwinism explains the emergence of classical objectivity within a
quantum universe. However, to date most research in quantum Darwinism has
focused on specific models and their stationary properties. To further our
understanding of the quantum-to-classical transition it appears desirable to
identify the general criteria a Hamiltonian has to fulfill to support
classical reality. To this end, we categorize all models with 2-body
interactions, and we show that only those with separable interaction of system
and environment can support a pointer basis. We further show that “perfect”
quantum Darwinism can only emerge if there are no intra-environmental
interactions. Our analysis is complemented by the solution of the ensuing
dynamics. We find that in systems that exhibit information scrambling, the
dynamical emergence of classical objectivity is in direct competition with the
non-local spread of quantum correlations. Our rigorous findings are
illustrated with the numerical analysis of four representative models.

##  I Introduction

We live in a quantum universe, yet our everyday reality is well-described by
classical physics. Hence, the obvious question to ask is where all the quantum
information and correlations hide. The quantum nature of our universe is
captured by its ability to be in a superposition of classically allowed
states. The transition from quantum to classical is a two-step process. The
first, necessary but not sufficient, step is the destruction of quantum
superpositions, i.e., destruction of all interference phenomena. The theory of
decoherence teaches us that it is the interaction between the quantum system
and its environment that is the cause of this phenomenon [1]. The destruction
of quantum superpositions presupposes a privileged and unique quantum basis.
The elements of this basis are called pointer states [2, 3, 4]. Any quantum
superposition written in this basis decomposes into a classical mixture under
the effect of environmental interaction. Thus, pointer states are precisely
the only quantum states that are stable under this interaction.

Quantum Darwinism [5, 6, 7, 8, 9, 10, 11, 12, 13] builds on decoherence theory
and goes a step further, approaching the problem from the point of view of
quantum information theory. An outside observer has no direct access to a
system of interest $\mathfrak{S}$, but rather the environment $\mathfrak{E}$
acts as a communication channel. Since any real environment is tremendously
large, “observing” $\mathfrak{S}$ actually means that an observer intercepts
only a small, possibly even tiny fragment $\mathfrak{F}$ of $\mathfrak{E}$,
and then reconstructs the state of $\mathfrak{S}$ from the information carried
by $\mathfrak{F}$.

If the constituents of the environment do not interact, such as, for instance,
photons [14, 15], then the information about $\mathfrak{S}$ is accessible by
_local_ measurements on $\mathfrak{E}$. However, reality is a little more
complicated, and in general, the constituents of $\mathfrak{E}$ do interact.
Such intra-environmental interactions lead to the build-up of non-local
correlations, which is the root-cause of information scrambling [16, 17, 18,
19, 20, 21]. Thus, an observer has to access a macroscopic fraction of
$\mathcal{E}$ to reconstruct unambiguous information about $\mathfrak{S}$.

Despite the significant attention scrambling dynamics has received in the
literature [16, 17, 18, 19, 20, 21, 22, 23], curiously little is known about
the quantum-to-classical transition in the presence of scrambling. Only rather
recently, several studies have started to unveil the interplay of decoherence
and scrambling [23, 24, 25, 26, 27, 28, 29, 30, 31].

More directly relevant to our present work is Ref. [32], which analyzed a
specific model where intra-environmental interactions scramble the information
encoded in different fragments $\mathfrak{F}$. This scenario, scrambling only
in $\mathfrak{E}$, but not in $\mathfrak{S}$, makes it easier to highlight the
competition between the local transfer of information from $\mathfrak{S}$ to
each degree of freedom of $\mathfrak{E}$, and the scrambling of information
between the different $\mathfrak{F}$ of $\mathfrak{E}$ due to their
interactions.

Current research in quantum Darwinism [13] is driven by the analysis of
increasingly complex model systems. However, the focus has remained on
particular qubit-models [33, 34, 35, 36, 37, 32], since their dynamics is
tractable. Despite, or rather because of continued progress in our
understanding it appears desirable to elucidate the general properties of
Hamiltonians that support the emergence of quantum Darwinism. More precisely,
it is instrumental to sort all possible interacting many-body Hamiltonians
into classes that support a pointer basis for $\mathfrak{S}$, and which sub-
classes of these will further exhibit the emergence of classical objectivity.
Such a classification will also unveil if and under what circumstances,
quantum Darwinism can emerge in the presence of scrambling dynamics.

In the present work, we consider a qubit of interest $\mathfrak{S}$, that
interacts with an environment $\mathfrak{E}$ also comprised of qubits. Hence,
scrambling of information may only occur in $\mathfrak{E}$, but not in
$\mathfrak{S}$. Further, for the sake of simplicity we restrict ourselves to
arbitrary two-body interactions.

(a) Arbitrary two-body interaction model.

(b) $\mathfrak{S}$-$\mathfrak{E}$ separable two-body interaction model.

Figure 1: Two-body interactions between $\mathfrak{S}$ (red) and
$\mathfrak{E}$ (blue). Lines depict interaction terms.

In a first part of our analysis, we show that the existence of a pointer basis
for $\mathfrak{S}$ imposes a specific structure for the total Hamiltonian
describing the evolution of the universe $\mathfrak{S}\otimes\mathfrak{E}$. In
fact, we will see that a pointer basis for $\mathfrak{S}$ exists for any
interactions within $\mathfrak{E}$, yet $\mathfrak{S}$ may only interact with
all fragments of $\mathfrak{E}$ identically, cf. Fig. 1. The second part of
the analysis is then focused on the dynamics induced by such Hamiltonians that
support a pointer basis. We find that the efficiency of the information
transfer between $\mathfrak{S}$ and $\mathfrak{E}$ is governed by the
statistics of the interaction terms. The average information transfer is
irreversible if and only if the support of the coupling coefficients is
continuous, and the “speed of communication” is determined by the shape of the
distribution of the interaction coefficients. Our general findings are
illustrated with four models that correspond to a variety of situations,
including scrambling or no scrambling, pointer basis or no pointer basis,
quantum Darwinism or no quantum Darwinism.

##  II Structure of the Hamiltonian

We start by defining the problem in mathematically rigorous terms. Consider a
set of $(N+1)$ qubits, where the $0$th qubit is the system $\mathfrak{S}$.
Hence, the environment $\mathfrak{E}$ is comprised of $N$ qubits. For the sake
of simplicity, we further restrict ourselves to 2-body interaction models. The
most general Hamiltonian corresponding to this scenario then reads

$H=\sum_{i,j}\sum_{\alpha,\beta}J^{\alpha\beta}_{ij}\sigma^{\alpha}_{i}\otimes\sigma^{\beta}_{j}+\sum_{i}\vec{B_{i}}\cdot\vec{\sigma_{i}},$
(1)

where $\alpha$ and $\beta$ take the values $x$, $y$ and $z$, corresponding to
the Pauli matrices $\sigma^{x}$, $\sigma^{y}$ and $\sigma^{z}$. Indices $i$
and $j$ count the qubits, with $i,j=0$ for $\mathfrak{S}$ and $i,j\geq 1$ for
$\mathfrak{E}$. Further, $J_{ij}$ and $\vec{B_{i}}$ are real coefficients,
which in the following we will choose to be random variables.

###  II.1 Existence of a pointer basis

The natural question now is, what conditions $J_{ij}$ and $\vec{B_{i}}$ have
to fulfill, such that $H$ in Eq. (1) supports a pointer basis for
$\mathfrak{S}$. Pointer states are _the_ particular states of $\mathfrak{S}$
that are stable under the interaction with $\mathfrak{E}$ [2, 3, 4]. Formally,
these states can be identified in the following way:
$\left|\psi_{\mathfrak{S}}\right\rangle\in\mathfrak{S}$ is a pointer state of
$\mathfrak{S}$ if and only if for any
$\left|\psi_{\mathfrak{E}}\right\rangle\in\mathfrak{E}$, an initial product
state
$\left|\psi_{\mathfrak{S}}\right\rangle\otimes\left|\psi_{\mathfrak{E}}\right\rangle$
evolves under $H$ (1) to remain within an epsilon ball around the product
state
$\left|\psi_{\mathfrak{S}}\right\rangle\otimes\left|\psi_{\mathfrak{E}}(t)\right\rangle$.
In other words, the reduced state $\left|\psi_{\mathfrak{S}}\right\rangle$
remains pure under the evolution of the total Hamiltonian.

It will prove convenient to separate the total Hamiltonian into terms
corresponding to $\mathfrak{S}$, $\mathfrak{E}$, and their interaction. Hence,
we write

$H=H_{\mathfrak{S}}\otimes\mathbb{I}_{\mathfrak{E}}+\mathbb{I}_{\mathfrak{S}}\otimes
H_{\mathfrak{E}}+H_{\mathfrak{SE}}\,.$ (2)

Comparing with Eq. (1) we identify the system Hamiltonian as

$H_{\mathfrak{S}}=\vec{B_{0}}\cdot\vec{\sigma_{0}}\,,$ (3)

whereas we have for the environment

$H_{\mathfrak{E}}=\sum_{1\leq i<j\leq
N}\sum_{\alpha,\beta}J^{\alpha\beta}_{ij}\sigma^{\alpha}_{i}\otimes\sigma^{\beta}_{j}+\sum_{i=1}^{N}\vec{B_{i}}\cdot\vec{\sigma_{i}}\,.$
(4)

Notice that the first term in Eq. (4) describes the intra-environmental
interactions. The interaction between $\mathfrak{S}$ and $\mathfrak{E}$ is
given by

$H_{\mathfrak{SE}}=\sum_{\alpha}\sigma^{\alpha}_{0}\otimes\sum_{j=1}^{N}\sum_{\beta}J^{\alpha\beta}_{0j}\sigma^{\beta}_{j}\,.$
(5)

From this separation of terms it becomes immediately obvious that the pointer
basis for $\mathfrak{S}$ can only exist if certain necessary and sufficient
conditions for the interaction term $H_{\mathfrak{SE}}$ are fulfilled.

These conditions become particularly intuitive by considering the original
motivation for _pointer_ states. These states are not only immune to the
dynamics induced by the interaction with the environment, but can be also
thought of as states that correspond to the _pointer_ of a measurement
apparatus. Mathematically, such an apparatus is described by the pointer
observable

$A\equiv A_{\mathfrak{S}}\otimes\mathbb{I}_{\mathfrak{E}}\,.$ (6)

By construction, the pointer observable $A$ commutes with the total
Hamiltonian (1), and hence $A$ and $H$ share an eigenbasis. Due to form of $A$
the corresponding eigenstates can be written in tensor-product form
$\left|S_{i}\right\rangle\otimes\left|E_{j}\right\rangle$ with
$\left|S_{i}\right\rangle\in\mathfrak{S}$ and
$\left|E_{j}\right\rangle\in\mathfrak{E}$.

Correspondingly, we can factorize the time-evolution operator as

$U=\sum_{i}|S_{i}\rangle\\!\langle
S_{i}|\otimes\exp{\left(-i/\hbar\,H_{i}\,t\right)}$ (7)

where the $H_{i}$ act only on $\mathfrak{E}$. Now, choosing any (reduced)
eigenstate of $A$ as initial state of $\mathfrak{S}$,
$\left|\psi_{\mathfrak{S}}\right\rangle=\left|S_{i}\right\rangle$, the product
state
$\left|\psi_{\mathfrak{S}}\right\rangle\otimes\left|\psi_{\mathfrak{E}}\right\rangle$
evolves by remaining in product form
$\left|\psi_{\mathfrak{S}}\right\rangle\otimes\left|\psi_{\mathfrak{E}}(t)\right\rangle$.

In Appendix A we show that by enforcing the commutation relation
$\left[A,\,H\right]=0$ we have that any Hamiltonian (4) supporting a pointer
basis for $\mathfrak{S}$ has to be of the form

$H=H_{\mathfrak{S}}\otimes\mathbb{I}_{\mathfrak{E}}+\mathbb{I}_{\mathfrak{S}}\otimes
H_{\mathfrak{E}}+H_{\mathfrak{S}}\otimes\sum_{i=1}^{N}h_{i},$ (8)

where $H_{\mathfrak{S}}$ is the system Hamiltonian (3), and the $h_{i}$ are
arbitrary traceless Hermitian operators acting on the $i$th qubit of
$\mathfrak{E}$.

In conclusion, we have shown that any model of interacting qubits that
supports a pointer basis for a system qubit $\mathfrak{S}$ may have at most a
separable interaction term $H_{\mathfrak{SE}}$. Moreover, this interaction
terms has to be factorizable into the system Hamiltonian $H_{\mathfrak{S}}$
and traceless terms acting on the environmental qubits $\mathfrak{E}$. It is
important to emphasize that no additional conditions are required pertaining
to, for instance, the intra-environmental interactions. Schematically, our
findings are illustrated in Fig. 1.

###  II.2 Further conditions for quantum Darwinism

It was shown in Ref. [11] that only a special structure of states is
compatible with the emergence of quantum Darwinism. These states are of the
_singly-branching form_ [38, 12], which are the only states to support epsilon
quantum correlations as measured by quantum discord [39].

Singly branching states are pointer states of $\mathfrak{S}$ correlated with
the environment states in the special form,

$\left|\psi(t)\right\rangle=\alpha_{0}\left|0\right\rangle\bigotimes_{i\in\mathfrak{E}}\left|\mathfrak{0}_{i}(t)\right\rangle+\beta_{0}\left|1\right\rangle\bigotimes_{i\in\mathfrak{E}}\left|\mathfrak{1}_{i}(t)\right\rangle\,.$
(9)

It is easy to see that such a singly branching form can emerge if and only if
there are no intra-environmental interactions.

Thus, we conclude that quantum Darwinism can only be supported by Hamiltonians
with separable interaction between $\mathfrak{S}$ and $\mathfrak{E}$, cf. Eq.
(8), and no intra-environmental interactions, i.e., $J^{\alpha\beta}_{ij}=0$
in Eq. (4). The remaining question now is whether all such Hamiltonians
provide so-called good decoherence, which makes their corresponding
$\mathfrak{E}$ good channels for information transfer.

##  III Coefficients of the Hamiltonian

To analyze the _dynamical_ emergence of quantum Darwinism, we now solve for
the average dynamics under an arbitrary, random Hamiltonian for which the
system $\mathfrak{S}$ has a pointer basis of the single branching form (9). We
will find that the efficiency of information transfer within $\mathfrak{E}$ is
governed by the randomness of the interaction coefficients.

###  III.1 Solving the dynamics

To this end, consider an arbitrary Hamiltonian of the form (8), where we
further enforce vanishing intra-environmental interactions
$J^{\alpha\beta}_{ij}=0$. Note that in Eq. (8) the $h_{i}$ are hermitian, and
traceless. Hence, we can write equivalently (and without loss of generality)

$H=\sigma^{z}_{0}\otimes\sum_{i=1}^{N}B_{i}\,\sigma^{z}_{i},$ (10)

where the $B_{i}$ are real random variables.

We are now interested in the dynamics induced by Eq. (10), and we choose an
arbitrary separable initial condition. Therefore, we write

$\left|\psi_{0}\right\rangle=(\alpha_{0}\left|0\right\rangle+\beta_{0}\left|1\right\rangle)\,\bigotimes_{i=1}^{N}(\alpha_{i}\left|0_{i}\right\rangle+\beta_{i}\left|1_{i}\right\rangle),$
(11)

where the $\alpha_{i}$ and $\beta_{i}$ are arbitrary, complex coefficients.
Evolving this $\left|\psi_{0}\right\rangle$ under the corresponding
Schrödinger equation,
$i\partial_{t}\left|\psi\right\rangle=H\left|\psi\right\rangle$ we obtain the
time-dependent solution,

$\left|\psi(t)\right\rangle=\alpha_{0}\left|0\right\rangle\bigotimes_{i\in\mathfrak{E}}\left|\mathfrak{0}_{i}(t)\right\rangle+\beta_{0}\left|1\right\rangle\bigotimes_{i\in\mathfrak{E}}\left|\mathfrak{1}_{i}(t)\right\rangle,$
(12)

where we introduced

$\begin{split}\left|\mathfrak{0}_{i}(t)\right\rangle&=\alpha_{i}\,\exp{\left(iB_{i}t\right)}\left|0\right\rangle+\beta_{i}\,\exp{\left(-iB_{i}t\right)}\left|1\right\rangle\\\
\left|\mathfrak{1}_{i}(t)\right\rangle&=\alpha_{i}\,\exp{\left(-iB_{i}t\right)}\left|0\right\rangle+\beta_{i}\,\exp{\left(iB_{i}t\right)}\left|1\right\rangle\,.\end{split}$
(13)

As usual, the reduced density matrix of $\mathfrak{S}$ is given by tracing out
$\mathfrak{E}$,
$\rho_{\mathfrak{S}}(t)=\mathrm{tr_{\mathfrak{E}}}\left\\{\left|\psi(t)\right\rangle\left\langle\psi(t)\right|\right\\}$.
The corresponding decoherence factor [32] is given by the amplitude of the
off-diagonal coefficients of the reduced density matrix $\rho_{\mathfrak{S}}$
in the basis $\left\\{\left|0\right\rangle,\left|1\right\rangle\right\\}$. We
have

$\left\langle\left|\left\langle\mathfrak{1}_{i}(t)|\mathfrak{0}_{i}(t)\right\rangle\right|^{2}\right\rangle\equiv\left\langle\left|\Gamma_{i}(t)\right|^{2}\right\rangle\,,$
(14)

and, since the $B_{i}$ are stochastically independent, we can write

$\left\langle\left|\prod\Gamma_{i}(t)\right|^{2}\right\rangle=\prod\left\langle\left|\Gamma_{i}(t)\right|^{2}\right\rangle.$
(15)

It is easy to see that we have,

$\Gamma_{i}(t)=\left|\alpha_{i}\right|^{2}\,\exp{\left(-2iB_{i}t\right)}+\left|\beta_{i}\right|^{2}\,\exp{\left(2iB_{i}t\right)}\,.$
(16)

For random $B_{i}$ it is, however, more instructive to compute the decoherence
factors averaged over all possible values for $B_{i}$. Further denoting the
probability density function of $B_{i}$ as $P(B_{i}=x)=f_{i}(x)$, we show in
Appendix B that we obtain

$\begin{split}\left\langle\left|\Gamma_{i}(t)\right|^{2}\right\rangle&=\left|\alpha_{i}\right|^{4}+\left|\beta_{i}\right|^{4}\\\
&+2\left|\alpha_{i}\right|^{2}\left|\beta_{i}\right|^{2}\left|\widetilde{f_{i}}(4t)\right|\cos(\arg(\widetilde{f_{i}}(4t))),\end{split}$
(17)

where $\widetilde{f_{i}}$ is the characteristic function of $B_{i}$

$\widetilde{f_{i}}(k)=\int_{-\infty}^{+\infty}dx\,f_{i}(x)\,\exp{\left(ikx\right)}\,.$
(18)

In conclusion, we have derived an analytic expression for the average
decoherence function, which governs the rate with which information about
$\mathfrak{S}$ is communicated through $\mathfrak{E}$.

###  III.2 Rate and irreversibility of information transfer

Equation (17) demonstrates the relationship between the emergence of
classicality and the randomness of system-environment interactions. Indeed,
the probability distributions $f_{i}$ of the couplings $B_{i}$ between
$\mathfrak{S}$ and $\mathfrak{E}$ play a central role in the rate of
information transfer. Observe that the decoherence factors decrease rapidly if
$\widetilde{f_{i}}$ decrease rapidly. Since $\widetilde{f_{i}}$ is the Fourier
transform of the probability distribution $f_{i}$, the order of
differentiability of $f_{i}$ gives us the order of decay of
$\widetilde{f_{i}}$, while the smallest characteristic length in the
distribution $f_{i}$ gives us the inverse of the characteristic time of decay
of $\widetilde{f_{i}}$.

Furthermore, if the support of $B_{i}$ is discrete and finite, then the
characteristic function $\widetilde{f_{i}}(k)$ is a periodic (or quasi-
periodic) function and therefore does not converge to 0. Hence, having
continuous support for the $f_{i}$ is essential for the emergence of truly
classical behavior. In fact, if the $f_{i}$ distribution is continuous, then
the information is transferred _irreversibly_. In this case, the $f_{i}$ are
integrable

$\int_{-\infty}^{+\infty}dx\,\left|f_{i}(x)\right|=\int_{-\infty}^{+\infty}dx\,f_{i}(x)=1<\infty,$
(19)

and thus, by virtue of the Riemann-Lebesgue Lemma,

$\left|\widetilde{f_{i}}(k)\right|\xrightarrow[k\rightarrow\infty]{}0\quad\text{and
thus}\quad\left\langle\left|\Gamma_{i}(t)\right|^{2}\right\rangle\xrightarrow[t\rightarrow\infty]{}\epsilon_{i}<1.$
(20)

where $\epsilon_{i}$ depends on the initial state.

Finally, we note that a perfect record of the information about $\mathfrak{S}$
in the $i$th qubit corresponds to $\Gamma_{i}=0$. This is typically not the
case. However, as the
$\left\langle\left|\Gamma_{i}(t)\right|^{2}\right\rangle$ become strictly less
than one, Eq. 15 shows that
$\left|0\right\rangle\bigotimes_{i\in\mathfrak{F}}\left|\mathfrak{0}_{i}(t)\right\rangle$
and $\left|1\right\rangle\bigotimes_{i\in
F}\left|\mathfrak{1}_{i}\right\rangle$ become orthogonal on average for a
sufficiently large fragment $\mathfrak{F}$. Thus, a small set of qubits of the
environment is enough to obtain an almost complete record of the state of
$\mathfrak{S}$.

###  III.3 Quantum Darwinism – the classical plateau

The hallmark result of quantum Darwinism is the emergence of the “classical
plateau” [5, 6, 7, 8, 9, 10, 11, 12, 13], cf. Fig. 2. This classical plateau
is a consequence of redundant encoding of the same information in
$\mathfrak{E}$.

Figure 2: Asymptotic mutual information (27) and Holevo quantity (28) as
functions of the fragment size. The classical plateau corresponds to the value
$\mathcal{S}_{max}=1$.

To this end, consider again a fragment $\mathfrak{F}$ of $\mathfrak{E}$. If
any $\mathfrak{F}$ carries the same information about $\mathfrak{S}$, then any
two observers accessing different $\mathfrak{F}$ learn exactly the same
information about $\mathfrak{S}$. The amount of information that a fragment
$\mathfrak{F}$ of $\mathfrak{E}$ contains about the system $\mathfrak{S}$ can
be quantified with the mutual information $I(\mathfrak{S}:\mathfrak{F})$
defined as

$I(\mathfrak{S}:\mathfrak{F})=\mathcal{S}_{\mathfrak{S}}+\mathcal{S}_{\mathfrak{F}}-\mathcal{S}_{\mathfrak{SF}}\,,$
(21)

where $\mathcal{S}(\rho)=-\mathrm{tr}\left\\{\rho\log(\rho)\right\\}$. The
maximal classical information that can be accessed by any observer is upper-
bounded by the Holevo quantity [40, 41]

$\chi(\mathfrak{S}:\check{\mathfrak{F}})=\mathcal{S}_{\mathfrak{S}}-\mathcal{S}_{\mathfrak{S}|\check{\mathfrak{F}}}$
(22)

where $\mathcal{S}_{\mathfrak{S}|\check{\mathfrak{F}}}$ is the conditional von
Neumann entropy defined as the minimal von Neumann entropy of $\mathfrak{S}$
obtained after a measurement on $\mathfrak{F}$.

The difference of the mutual information, $I(\mathfrak{S}:\mathfrak{F})$, and
the Holevo quantity, $\chi(\mathfrak{S}:\check{\mathfrak{F}})$ has been called
quantum discord [39],

$D(\mathfrak{S}:\check{\mathfrak{F}})\equiv
I(\mathfrak{S}:\mathfrak{F})-\chi(\mathfrak{S}:\check{\mathfrak{F}})\geq 0\,.$
(23)

Quantum discord measures the genuinely quantum information encoded in
$\mathfrak{F}$.

For each $\mathfrak{F}$ and its complement,
$\overline{\mathfrak{F}}=\mathfrak{E}\setminus\mathfrak{F}$ we define the
corresponding decoherence factors

$\Gamma_{\mathfrak{F}}=\prod_{i\in\mathfrak{F}}\Gamma_{i}\quad\text{and}\quad\Gamma_{\overline{\mathfrak{F}}}=\prod_{i\notin\mathfrak{F}}\Gamma_{i}.$
(24)

With these definitions, one can then show that for small enough decoherence
factors [32] we have

$I(\mathfrak{S}:\mathfrak{F})\simeq\mathcal{S}_{max}-\frac{\xi(\left|\alpha_{0}\right|^{2})}{2}\left[\left|\Gamma\right|^{2}+\left|\Gamma_{\mathfrak{F}}\right|^{2}-\left|\Gamma_{\overline{\mathfrak{F}}}\right|^{2}\right],$
(25)

where
$\mathcal{S}_{max}=-\left|\alpha_{0}\right|^{2}\log(\left|\alpha_{0}\right|^{2})-(1-\left|\alpha_{0}\right|^{2})\log(1-\left|\alpha_{0}\right|^{2})$,
which is the maximal value that the von Neumann entropy of $\mathfrak{S}$.
Correspondingly, we obtain for the Holevo quantity (see Appendix C)

$\chi(\mathfrak{S}:\check{\mathfrak{F}})\simeq\mathcal{S}_{max}-\frac{\xi(\left|\alpha_{0}\right|^{2})}{2}\left|\Gamma_{\mathfrak{F}}\right|^{2}\,,$
(26)

which is fully consistent with earlier findings [42].

Finally, in the limit of long times, $t\gg 1$, and for smooth enough $f_{i}$
(20) we obtain the following asymptotic expression for the mutual information

$\begin{split}I(\mathfrak{S}:\mathfrak{F})\simeq\mathcal{S}_{max}-\frac{\xi(\left|\alpha_{0}\right|^{2})}{2}\left[\prod_{i\in\mathfrak{E}}\epsilon_{i}+\prod_{i\in\mathfrak{F}}\epsilon_{i}-\prod_{i\notin\mathfrak{F}}\epsilon_{i}\right]\,,\end{split}$
(27)

and the Holevo quantity

$\chi(\mathfrak{S}:\check{\mathfrak{F}})\simeq\mathcal{S}_{max}-\frac{\xi(\left|\alpha_{0}\right|^{2})}{2}\prod_{i\in\mathfrak{F}}\epsilon_{i}\,.$
(28)

Further, averaging over every possible separable initial states, we have

$I(\mathfrak{S}:n)_{\infty}\simeq\mathcal{S}_{max}-\frac{\xi(\left|\alpha_{0}\right|^{2})}{2}\,\left[\overline{\epsilon}^{N}+\overline{\epsilon}^{n}-\overline{\epsilon}^{N-n}\right]$
(29)

and

$\chi(\mathfrak{S}:n)_{\infty}\simeq\mathcal{S}_{max}-\frac{\xi(\left|\alpha_{0}\right|^{2})}{2}\,\overline{\epsilon}^{n}\,,$
(30)

where $n$ is the size of $\mathfrak{F}$ and $\overline{\epsilon}=2/3$ (see
Appendix C).

These results are depicted in Fig. 2 for $N=50$ environmental qubits. Both,
the mutual information and the Holevo quantity exhibit a steep initial rise
with increasing fragment size $n$, as larger fragments provide more data about
$\mathfrak{S}$. This initial rise is followed by the classical plateau.

##  IV Representative examples

Table 1: Models classification | Pointer Basis | Continuous Support | No Scrambling
---|---|---|---
CPDI (31) | ✓ | ✓ | ✓
DPDI (32) | ✓ | $\times$ | ✓
CODI (33) | $\times$ | ✓ | ✓
CPDI-S (34) | ✓ | ✓ | $\times$

We conclude the analysis with the numerical solution of four representative
examples. To support quantum Darwinism a Hamiltonian must obey the following
conditions: existence of a pointer basis, continuous support, and no intra-
environment interactions. Our first example exhibits these three conditions,
and for each following example we successively remove one of these conditions,
cf. Tab. 1.

(a)

(b)

(c)

(d)

Figure 3: Mutual information $I(\mathfrak{S}:\mathfrak{F})$ as a function of
time and fragment size for an arbitrary separable initial state of 9 qubits
($N=8$), divided by the von Neumann entropy of $\mathfrak{S}$ (averaged over
$10^{2}$ realizations). (a) CPDI (31): irreversible information transfer and
classical objectivity. (b) DPDI (32): non-irreversible (periodic) information
transfer. (c) CODI (33): no local information transfer or redundancy. (d)
CPDI-S (34): competition of emergence of classical objectivity and scrambling
in $\mathfrak{E}$.

###  IV.1 Continuous Parallel Decoherent Interaction

The first model has a pointer basis, random coupling coefficients with a
continuous spectrum, and does not exhibit scrambling in $\mathfrak{E}$. The
corresponding Hamiltonian reads

$H_{\mathrm{CPDI}}=\sigma^{z}_{0}\otimes\sum_{i=1}^{N}B_{i}\sigma^{z}_{i}$
(31)

where $B_{i}$ are independent random variables, drawn uniformly from
$B_{i}\in\left[-1,1\right]$. For specificity, we call this model _Continuous
Parallel Decoherent Interaction (CPDI)_.

In Fig. 3(a) we plot the resulting mutual information (21), as a function of
the fragment size, which rapidly converges towards the asymptotic expression
(27). Note the distinct classical plateau indicative of quantum Darwinism.
Moreover, we observe relaxation of $\mathfrak{S}$ into its stationary pointer
states over a typical time $\tau\simeq 1/4$, at which point the information
transfer becomes irreversible.

###  IV.2 Discrete Parallel Decoherent Interaction

Our second example is called, _Discrete Parallel Decoherent Interaction
(DPDI)_. The corresponding Hamiltonian is

$H_{\mathrm{DPDI}}=\sigma^{z}_{0}\otimes\sum_{i=1}^{N}B_{i}\sigma^{z}_{i}$
(32)

where $B_{i}$ are again independent random variables. However, in contrast to
the continuous case in Eq. (31), the $B_{i}$ are now drawn uniformly from the
_discrete_ set $B_{i}\in\left\\{-1,-0.5,0.5,1\right\\}$.

In Fig. 3(b) we depict the resulting mutual information. As expected, we
observe that the classical plateau appears and disappears periodically, and
hence the information transfer is no longer irreversible. At instants at which
the plateau completely vanishes, $I(\mathfrak{S}:\mathfrak{F})$ is linear in
the fragment size. This indicates that the information about $\mathfrak{S}$
encoded in $\mathfrak{E}$ is distributed throughout the entire environment (no
redunancy).

###  IV.3 Continuous Orthogonal Decoherent Interaction

As a next example, we consider _Continuous Orthogonal Decoherent Interaction
(CODI)_ , which refers to our first model (31) but with an added external
field. This additional term is chosen to be not parallel to the interaction
between $\mathfrak{S}$ and $\mathfrak{E}$, and hence CODI does not support a
pointer basis. The Hamlitonian reads

$H_{\mathrm{CODI}}=\sigma^{y}_{0}\otimes\mathbb{I}_{E}+\sigma^{z}_{0}\otimes\sum_{i=1}^{N}B_{i}\sigma^{z}_{i}.$
(33)

where, as before, $B_{i}$ are independent random variables, drawn uniformly
from $B_{i}\in\left[-1,1\right]$.

In this model, information about the observable $\sigma^{z}_{0}$ can be
registered in $\mathfrak{E}$, but the eigenstates of this observable are not
stable and hence not classically objective. This observation is further
supported by Fig. 3(c), which does not exhibit any form of classical plateau.
Moreover, at all instants $I(\mathfrak{S}:\mathfrak{F})$ is a linear function
of the size of $\mathfrak{F}$, which is a consequence of the complete absence
of any redundancy.

###  IV.4 Continuous Parallel Decoherent Interaction with Scrambling

As a final example, we again consider Eq. (31) but now design $\mathfrak{E}$
to exhibit scrambling dynamics. Accordingly, this model is called Continuous
Parallel Decoherent Interaction with Scrambling (CPDI-S), and the Hamiltonian
becomes

$H_{\mathrm{CPDI-S}}=\sigma^{z}_{0}\otimes\sum_{i=1}^{N}B_{i}\sigma^{z}_{i}+\sum_{1\leq
i<j\leq N}J_{ij}\sigma^{z}_{i}\otimes\sigma^{z}_{j}$ (34)

where again $B_{i}$ are independent random variables, drawn uniformly from
$B_{i}\in\left[-1,1\right]$, and $J_{ij}$ are independent random variables,
drawn uniformly from $J_{ij}\in\left[-0.03,0.03\right]$.

As Fig. 3(d) shows, a classical plateau rapidly emerges over a time scale of
$\tau_{\mathfrak{SE}}\simeq 1/4$. However, this plateau quickly “disperses” as
the quantum information becomes non-local due to scrambling in $\mathfrak{E}$.
We refer to Ref. [32] for a more detailed analysis of this particular model.
Furthermore, it is interesting to note that Eq. (34) is another example that
demonstrates the competition of decoherence and scrambling as a “sink for
quantum information” as analyzed by (some of) us in Ref. [25].

##  V Concluding remarks

In the present work we determined the set of qubit models, which support the
emergence of classicality. In particular, we established a classification of
2-body interaction models based on the structure of the Hamiltonian and on the
nature of its coefficients.

The existence of an “exact” pointer basis for the qubit $\mathfrak{S}$
requires the interaction Hamiltonian to be separable between $\mathfrak{S}$
and its environment $\mathfrak{E}$ such that the part acting on $\mathfrak{S}$
is proportionnal to the self Hamiltonian of $\mathfrak{S}$. We call that type
of structure a _parallel decoherent interaction_. Furthermore, without any
intra-environment interactions, this Hamiltonian structure leads to a
branching state structure, the only one compatible with quantum Darwinism
[11].

Furthermore, intra-environment interactions can lead to information scrambling
in $\mathfrak{E}$, which deteriorates the branching structure. In such
situations, the state of $\mathfrak{S}$ is still a classical mixture, but this
classicality is hidden to an outside observer who must take a measurement on a
non-local part of $\mathfrak{E}$ in order to recover almost all the
information about $\mathfrak{S}$. This indicates a clear competition of the
emergence of classical objectivity and scrambling dynamics.

The conceptual notions and the gained insight of our work may open the door
for further inquiry, such as the study of $k-$body interactions. In
particular, there is every reason to believe that this type of interaction
leads to a non-local information transfer. Indeed, for such interactions, the
information about $\mathfrak{S}$ is directly encoded in $\mathfrak{E}$ by the
entanglement of $\mathfrak{S}$ with fragments $\mathfrak{F}$ of size $k$. This
results in a lower redundancy of information. However, the analytical analysis
of the dynamics is much more involved than the present 2-interaction case,
which is why we leave $k-$body interactions for future work.

###### Acknowledgements.

This work was carried out during a 15 week internship at the University of
Maryland, Baltimore County (P.D.). We gratefully acknowledge several
discussions with Joshua Chiel, who has provided us with perceptive comments.
A.T. acknowledges support from the Center for Nonlinear Studies and the U.S
DOE under the LDRD program at Los Alamos. S.D. acknowledges support from the
John Templeton Foundation under Grant No. 62422.

##  Appendix A  Hamiltonians with a pointer basis

In this appendix we provide further details that lead to the Hamiltonian
structure (8). We start by decomposing the interaction term (5) into

$H_{\mathfrak{SE}}=\sigma^{x}_{0}\otimes\sum_{i}\vec{J^{x}_{i}}\cdot\vec{\sigma_{i}}+\sigma^{y}_{0}\otimes\sum_{i}\vec{J^{y}_{i}}\cdot\vec{\sigma_{i}}+\sigma^{z}_{0}\otimes\sum_{i}\vec{J^{z}_{i}}\cdot\vec{\sigma_{i}}.$
(35)

Further, we can expand for any pointer observable
$A=A_{\mathfrak{S}}\otimes\mathbb{I}_{\mathfrak{E}}$ the commutator

$\begin{split}\left[H,A\right]&=\left[H_{\mathfrak{S}},A_{\mathfrak{S}}\right]\otimes\mathbb{I}_{\mathfrak{E}}+\left[\sigma^{x}_{0},A_{\mathfrak{S}}\right]\otimes
H^{x}_{\mathfrak{SE}}\\\
&\quad+\left[\sigma^{y}_{0},A_{\mathfrak{S}}\right]\otimes
H^{y}_{\mathfrak{SE}}+\left[\sigma^{z}_{0},A_{\mathfrak{S}}\right]\otimes
H^{z}_{\mathfrak{SE}}\,,\end{split}$ (36)

where $H^{x}_{\mathfrak{SE}}$, $H^{y}_{\mathfrak{SE}}$, and
$H^{z}_{\mathfrak{SE}}$ are linear combinations of $\sigma^{x}_{i}$,
$\sigma^{y}_{i}$ and $\sigma^{z}_{i}$, which are orthonormal to
$\mathbb{I}_{\mathfrak{E}}$. Thus, a vanishing commutator,
$\left[H,A\right]=0$ immediately yields

$\left[\sigma^{x}_{0},A_{\mathfrak{S}}\right]\otimes
H^{x}_{\mathfrak{SE}}+\left[\sigma^{y}_{0},A_{\mathfrak{S}}\right]\otimes
H^{y}_{\mathfrak{SE}}+\left[\sigma^{z}_{0},A_{\mathfrak{S}}\right]\otimes
H^{z}_{\mathfrak{SE}}=0$ (37)

as well as
$\left[H_{\mathfrak{S}},A_{\mathfrak{S}}\right]\otimes\mathbb{I}_{\mathfrak{E}}=0$
. Therefore, for any non-trivial $A_{\mathfrak{S}}$ we must have,

$H^{x}_{\mathfrak{SE}}\propto H^{y}_{\mathfrak{SE}}\propto
H^{z}_{\mathfrak{SE}}\,,$ (38)

and the Hamiltonian (35) becomes

$H_{\mathfrak{SE}}=\vec{J_{0}}\cdot\vec{\sigma_{0}}\otimes\sum_{i=1}^{N}\vec{J_{i}}\cdot\vec{\sigma_{i}}\,.$
(39)

Thus, we have $\left[\vec{B_{0}}\cdot\vec{\sigma_{0}},A_{S}\right]=0$ and
$\left[\vec{J_{0}}\cdot\vec{\sigma_{0}},A_{S}\right]=0$, which is equivalent
to

$\vec{B_{0}}\wedge\vec{J_{0}}=\vec{0}.$ (40)

It means that $\vec{J_{0}}$ and $\vec{B_{0}}$ are parallel. We finally obtain
the desired result Eq. (8).

##  Appendix B Average decoherence factors

Equation (17) can be obtained by direct derivation. Consider

$\left\langle\left|\Gamma_{i}(t)\right|^{2}\right\rangle=\left|\alpha_{i}\right|^{4}+\left|\beta_{i}\right|^{4}+2\left|\alpha_{i}\right|^{2}\left|\beta_{i}\right|^{2}\left\langle\cos(4B_{i}t)\right\rangle$
(41)

where the average is given by

$\left\langle\cos(4B_{i}t)\right\rangle=\int dx\,f_{i}(x)\cos(4xt)\,.$ (42)

As above, $f_{i}(x)$ denotes the probability density of the magnetic field
$B_{i}$. Employing its Fourier transform $\widetilde{f_{i}}(k)$, we write

$\left\langle\cos(4B_{i}t)\right\rangle=\int dx\int
dk\,\frac{\widetilde{f_{i}}(k)}{2\pi}\exp{\left(-ikx\right)}\,\cos(4xt).$ (43)

Now using [43]

$\int
dx\,\exp{\left(-ikx\right)}\cos(ax)=\pi\,\left(\delta(k+a)+\delta(k-a)\right),$
(44)

we have

$\left\langle\cos(4B_{i}t)\right\rangle=\frac{1}{2}\,\left(\widetilde{f_{i}}(4t)+\widetilde{f_{i}}(-4t)\right)\,$
(45)

and hence

$\left\langle\cos(4B_{i}t)\right\rangle=\left|\widetilde{f_{i}}(4t)\right|\,\cos(\arg(\widetilde{f_{i}}(4t)))\,.$
(46)

##  Appendix C Mutual information and asymptotics

In this final appendix, we summarize the derivation leading to the asymptotic
expressions of the mutual information (27) and the Holevo quantity (28). We
start by considering the reduced density operator of fragment $\mathfrak{F}$,
which is given by
$\rho_{\mathfrak{F}}=\mathrm{tr_{\mathfrak{S}\overline{\mathfrak{F}}}}\left\\{\rho\right\\}$,
and we have

$\rho_{\mathfrak{F}}(t)=\left|\alpha_{0}\right|^{2}\left|F_{0}(t)\right\rangle\left\langle
F_{0}(t)\right|+\left|\beta_{0}\right|^{2}\left|F_{1}(t)\right\rangle\left\langle
F_{1}(t)\right|\,.$ (47)

Explicitly, the states $\left|F_{0}(t)\right\rangle$ and
$\left|F_{1}(t)\right\rangle$ read

$\begin{split}\left|F_{0}(t)\right\rangle&=\bigotimes_{j\in\mathfrak{F}}(\alpha_{j}\,\exp{\left(iB_{j}t\right)}\left|0\right\rangle+\beta_{j}\,\exp{\left(-iB_{j}t\right)}\left|1\right\rangle)\\\
\left|F_{1}(t)\right\rangle&=\bigotimes_{j\in\mathfrak{F}}(\alpha_{j}\,\exp{\left(-iB_{j}t\right)}\left|0\right\rangle+\beta_{j}\,\exp{\left(iB_{j}t\right)}\left|1\right\rangle).\end{split}$
(48)

The corresponding decoherence factor is simply given by $\left\langle
F_{1}(t)|F_{0}(t)\right\rangle=\Gamma_{\mathfrak{F}}(t)$.

Since we are working with qubits, it is then a simple exercise to show that

$\mathcal{S}_{\mathfrak{S}}=h\left[\frac{1}{2}(1+\sqrt{1-4\left|\alpha_{0}\right|^{2}\left|\beta_{0}\right|^{2}(1-\left|\Gamma_{\mathfrak{S}}\right|^{2})})\right]$
(49)

and

$\mathcal{S}_{\mathfrak{F}}=h\left[\frac{1}{2}(1+\sqrt{1-4\left|\alpha_{0}\right|^{2}\left|\beta_{0}\right|^{2}(1-\left|\Gamma_{\mathfrak{F}}\right|^{2})})\right]$
(50)

where

$h\left[x\right]=-x\log(x)-(1-x)\log(1-x)\,.$ (51)

These expressions can be further simplified, by expanding for small
decoherence factors. We have in leading order

$\mathcal{S}_{\mathfrak{S}}\simeq\mathcal{S}_{max}-\frac{\xi(\left|\alpha_{0}\right|^{2})}{2}\left|\Gamma\right|^{2}$
(52)

and

$\mathcal{S}_{\mathfrak{F}}\simeq\mathcal{S}_{max}-\frac{\xi(\left|\alpha_{0}\right|^{2})}{2}\left|\Gamma_{\mathfrak{F}}\right|^{2}\,,$
(53)

where we introduced the notation

$\xi(\left|\alpha_{0}\right|^{2})=\frac{4\left|\alpha_{0}\right|^{2}(1-\left|\alpha_{0}\right|^{2})\operatorname{arctanh}(1-2\left|\alpha_{0}\right|^{2})}{1-2\left|\alpha_{0}\right|^{2}}\,.$
(54)

Note that we obtain an equivalent expression for the complement
$\overline{\mathfrak{F}}$. Thus, the mutual information (21) becomes

$I(\mathfrak{S}:\mathfrak{F})\simeq\mathcal{S}_{max}-\frac{\xi(\left|\alpha_{0}\right|^{2})}{2}\left[\left|\Gamma\right|^{2}+\left|\Gamma_{\mathfrak{F}}\right|^{2}-\left|\Gamma_{\overline{\mathfrak{F}}}\right|^{2}\right]\,.$
(55)

Following similar steps as detailed in Ref. [42] the corresponding Holevo
quantity can be written as

$\begin{split}\chi(\mathfrak{S}:\check{\mathfrak{F}})&=h\left[\frac{1}{2}(1+\sqrt{1-4\left|\alpha_{0}\right|^{2}\left|\beta_{0}\right|^{2}(1-\left|\Gamma\right|^{2})})\right]\\\
&-h\left[\frac{1}{2}(1+\sqrt{1-4\left|\alpha_{0}\right|^{2}\left|\beta_{0}\right|^{2}(\left|\Gamma_{\mathfrak{F}}\right|^{2}-\left|\Gamma\right|^{2})})\right]\end{split}$
(56)

which for weak decoherence in leading order simply is

$\chi(\mathfrak{S}:\check{\mathfrak{F}})\simeq\mathcal{S}_{max}-\frac{\xi(\left|\alpha_{0}\right|^{2})}{2}\left|\Gamma_{\mathfrak{F}}\right|^{2}\,.$
(57)

Finally, we note that for continuous distributions and $t\gg 1$ we have (with
Eq. (20))

$I(\mathfrak{S}:\mathfrak{F})\simeq\mathcal{S}_{max}-\frac{\xi(\left|\alpha_{0}\right|^{2})}{2}\left[\prod_{i\in\mathfrak{E}}\epsilon_{i}+\prod_{i\in\mathfrak{F}}\epsilon_{i}-\prod_{i\notin\mathfrak{F}}\epsilon_{i}\right]\,,$
(58)

where the $\epsilon_{i}$ are given by

$\epsilon_{i}=\left|\alpha_{i}\right|^{4}+\left|\beta_{i}\right|^{4}\,.$ (59)

Averaging $\epsilon_{i}$ over all possible values $\alpha_{i}$ and $\beta_{i}$
we obtain

$\overline{\epsilon}=\int_{0}^{1}dx\,(x^{2}+(1-x)^{2})=\frac{2}{3}.$ (60)

## References

  * Schlosshauer [2019] M. Schlosshauer, Quantum decoherence, Phys. Rep. 831, 1 (2019).
  * Zurek [1981] W. H. Zurek, Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?, Phys. Rev. D 24, 1516 (1981).
  * Zurek [1982] W. H. Zurek, Environment-induced superselection rules, Phys. Rev. D 26, 1862 (1982).
  * Brasil and de Castro [2015] C. A. Brasil and L. A. de Castro, Understanding the pointer states, Euro. J. Phys. 36, 065024 (2015).
  * Zurek [2000] W. Zurek, Einselection and decoherence from an information theory perspective, Ann. Phys. 512, 855 (2000).
  * Ollivier  _et al._ [2005] H. Ollivier, D. Poulin, and W. H. Zurek, Environment as a witness: Selective proliferation of information and emergence of objectivity in a quantum universe, Phys. Rev. A 72, 042113 (2005).
  * Ollivier  _et al._ [2004] H. Ollivier, D. Poulin, and W. H. Zurek, Objective properties from subjective quantum states: Environment as a witness, Phys. Rev. Lett. 93, 220401 (2004).
  * Zurek [2003] W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75, 715 (2003).
  * W.H.Zurek [2009] W.H.Zurek, Quantum darwinism, Nature Phys 5, 181 (2009).
  * Girolami  _et al._ [2022] D. Girolami, A. Touil, B. Yan, S. Deffner, and W. H. Zurek, Redundantly amplified information suppresses quantum correlations in many-body systems, Phys. Rev. Lett. 129, 010401 (2022).
  * Touil  _et al._ [2022a] A. Touil, F. Anza, S. Deffner, and J. P. Crutchfield, Branching states as the emergent structure of a quantum universe (2022a), arXiv:2208.05497 [quant-ph] .
  * Blume-Kohout and Zurek [2006] R. Blume-Kohout and W. H. Zurek, Quantum darwinism: Entanglement, branches, and the emergent classicality of redundantly stored quantum information, Phys. Rev. A 73, 062310 (2006).
  * Deffner  _et al._ [2023] S. Deffner, R. Laflamme, J. P. Paz, and M. Zwolak, eds.,  _Quantum Darwinism and Friends_ (MDPI books, 2023).
  * Riedel and Zurek [2010] C. J. Riedel and W. H. Zurek, Quantum darwinism in an everyday environment: Huge redundancy in scattered photons, Phys. Rev. Lett. 105, 020404 (2010).
  * Riedel and Zurek [2011] C. J. Riedel and W. H. Zurek, Redundant information from thermal illumination: quantum darwinism in scattered photons, New J. Phys. 13, 073038 (2011).
  * Hayden and Preskill [2007] P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, J. High Energ. Phys 2007, 120 (2007).
  * Swingle [2018] B. Swingle, Unscrambling the physics of out-of-time-order correlators, Nat. Phys. 14, 988 (2018).
  * Touil and Deffner [2020] A. Touil and S. Deffner, Quantum scrambling and the growth of mutual information, Quantum Sci. Technol. 5, 035005 (2020).
  * Bao and Kikuchi [2021] N. Bao and Y. Kikuchi, Hayden-preskill decoding from noisy hawking radiation, J. High Energ. Phys. 2021 (2), 17.
  * Chen  _et al._ [2022] B. Chen, B. Czech, and Z.-Z. Wang, Quantum information in holographic duality, Rep. Prog. Phys. 85, 046001 (2022).
  * Fisher  _et al._ [2023] M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Ann. Rev. Cond. Mat. Phys. 14, 335 (2023).
  * Swingle  _et al._ [2016] B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, Measuring the scrambling of quantum information, Phys. Rev. A 94, 040302 (2016).
  * Yoshida and Yao [2019] B. Yoshida and N. Y. Yao, Disentangling scrambling and decoherence via quantum teleportation, Phys. Rev. X 9, 011006 (2019).
  * Xu  _et al._ [2019] Z. Xu, L. P. García-Pintos, A. Chenu, and A. del Campo, Extreme decoherence and quantum chaos, Phys. Rev. Lett. 122, 014103 (2019).
  * Touil and Deffner [2021] A. Touil and S. Deffner, Information scrambling versus decoherence—two competing sinks for entropy, PRX Quantum 2, 010306 (2021).
  * Zanardi and Anand [2021] P. Zanardi and N. Anand, Information scrambling and chaos in open quantum systems, Phys. Rev. A 103, 062214 (2021).
  * Xu  _et al._ [2021] Z. Xu, A. Chenu, T. c. v. Prosen, and A. del Campo, Thermofield dynamics: Quantum chaos versus decoherence, Phys. Rev. B 103, 064309 (2021).
  * Domínguez  _et al._ [2021] F. D. Domínguez, M. C. Rodríguez, R. Kaiser, D. Suter, and G. A. Álvarez, Decoherence scaling transition in the dynamics of quantum information scrambling, Phys. Rev. A 104, 012402 (2021).
  * Cornelius  _et al._ [2022] J. Cornelius, Z. Xu, A. Saxena, A. Chenu, and A. del Campo, Spectral filtering induced by non-hermitian evolution with balanced gain and loss: Enhancing quantum chaos, Phys. Rev. Lett. 128, 190402 (2022).
  * Han  _et al._ [2022] L.-P. Han, J. Zou, H. Li, and B. Shao, Quantum information scrambling in non-markovian open quantum system, Entropy 24, 10.3390/e24111532 (2022).
  * Andreadakis  _et al._ [2023] F. Andreadakis, N. Anand, and P. Zanardi, Scrambling of algebras in open quantum systems, Phys. Rev. A 107, 042217 (2023).
  * Riedel  _et al._ [2012] C. J. Riedel, W. H. Zurek, and M. Zwolak, The rise and fall of redundancy in decoherence and quantum darwinism, New J. Phys. 14, 083010 (2012).
  * Giorgi  _et al._ [2015] G. L. Giorgi, F. Galve, and R. Zambrini, Quantum darwinism and non-markovian dissipative dynamics from quantum phases of the spin-1/2 $xx$ model, Phys. Rev. A 92, 022105 (2015).
  * Ryan  _et al._ [2021] E. Ryan, M. Paternostro, and S. Campbell, Quantum darwinism in a structured spin environment, Phys. Lett. A 416, 127675 (2021).
  * Zwolak  _et al._ [2009] M. Zwolak, H. T. Quan, and W. H. Zurek, Quantum darwinism in a mixed environment, Phys. Rev. Lett. 103, 110402 (2009).
  * Campbell  _et al._ [2019] S. Campbell, B. Çakmak, O. E. Müstecaplıoğlu, M. Paternostro, and B. Vacchini, Collisional unfolding of quantum darwinism, Phys. Rev. A 99, 042103 (2019).
  * Blume-Kohout and Zurek [2008] R. Blume-Kohout and W. H. Zurek, Quantum darwinism in quantum brownian motion, Phys. Rev. Lett. 101, 240405 (2008).
  * Blume-Kohout and Zurek [2005] Blume-Kohout and W. H. Zurek, A simple example of “quantum darwinism”: Redundant information storage in many-spin environments, Found. Phys. , 1857 (2005).
  * Ollivier and Zurek [2001] H. Ollivier and W. H. Zurek, Quantum discord: A measure of the quantumness of correlations, Phys. Rev. Lett. 88, 017901 (2001).
  * Holevo [1973] A. S. Holevo, Bounds for the quantity of information transmitted by a quantum communication channel, Peredachi Informatsii 9, 3 (1973).
  * Nielsen  _et al._ [2002] M. A. Nielsen, I. Chuang, and L. K. Grover, Quantum Computation and Quantum Information, Am. J. Phys. 70, 558 (2002).
  * Touil  _et al._ [2022b] A. Touil, B. Yan, D. Girolami, S. Deffner, and W. H. Zurek, Eavesdropping on the decohering environment: Quantum darwinism, amplification, and the origin of objective classical reality, Phys. Rev. Lett. 128, 010401 (2022b).
  * Abramowitz and Stegun [1948] M. Abramowitz and I. A. Stegun,  _Handbook of mathematical functions with formulas, graphs, and mathematical tables_ , Vol. 55 (US Government printing office, 1948).