# Fast-Scrambling and Operator Confinement Using an Auxiliary Qubit

Joseph C. Szabo  Department of Physics, The Ohio State University, Columbus,
Ohio 43210, USA  Nandini Trivedi  Department of Physics, The Ohio State
University, Columbus, Ohio 43210, USA

###### Abstract

We introduce a minimal model for realizing a fast-to-slow scrambling
transition mediated by an auxiliary central qubit (c-qubit). The c-qubit is
coupled to a spin-$1/2$ Ising model with local Ising interactions and tunable
c-qubit-spin coupling. Each spin becomes next-nearest neighbor to all others
through the c-qubit, which mediates effective all-to-all interactions. As the
interaction with the c-spin increases, we find a surprising transition from
super-ballistic scrambling and information growth to continuously restricted
sub-ballistic entanglement and operator growth. This slow growth occurs on
intermediate timescales that extend exponentially with increasing coupling and
system size, indicative of logarithmic entanglement growth. We find that in
the slow-scrambling regime, the c-qubit Ising interaction allows commuting
operators to grow support on all sites rapidly, while operators orthogonal to
the interaction become echoed out. This projects local operators to lie in a
restricted subspace and prevents extensive operator entanglement growth. We
provide exact dynamics of small systems working with non-equilibrium,
effective infinite temperature states, and additionally contribute analytic
early-time expansions that support the observed rapid scrambling to quantum
Zeno-like crossover. Tracing out the central qubit provides a unique
translation from the full, closed unitary dynamics to a simple open system
construction consisting of a typical spin-chain with hidden qubit degree of
freedom.

##  I Introduction

Operator scrambling and entanglement entropy spreading are unambiguous
discriminators of purely quantum mechanical nonequilibrium dynamics;
fascinating properties underlying quantum thermalization, dynamical phase
transitions, and topological order [1, 2, 3, 4, 5, 6, 7]. In the Heisenberg
picture, quantum operator scrambling details how initially localized operators
propagate over spatiotemporal degrees of freedom due to noncommutative many-
body interactions. In the complementary Schrodinger picture, entanglement
entropy captures growing information complexity: from initially classical
states to those with nontrivial entangled structure. Quantum information
dynamics bridge both theoretical and experimental communities as primary
measures for quantum complexity and expressivity [8, 9, 10, 11, 12]. These
concepts combined with quantum simulation/circuit devices have coalesced into
many enriching, recent experiments [13, 10, 14, 15]. The accelerating pace of
results and drive to continually advance the corresponding theory extend these
successes to further research at the intersection of quantum chaos,
thermalization, and computability, extending from qubits to black-holes and
quantum gravity [16, 17, 18, 19, 20, 21, 22, 23, 21, 24].

The primary research thrusts among the quantum information dynamics community
fall along the lines of uncovering the minimal mechanisms behind myriad
information dynamical phases and understanding the fate of quantum to
classical thermalization. In studying generalized quantum information
dynamics, there are typically two disparate perspectives: closed and open
quantum systems. Closed quantum systems exhibit rich scrambling physics
ranging from frozen [25] to fast [18] dynamics, with the typical questions
relating to how well-preserved is such physics under driving and dissipation
contributions from an external environment [26, 27, 28, 29]. The environment
is oftentimes reduced to a memoryless, effective Markovian description, which
hinges on assumptions including weak-coupling and a separation in the
timescales associated with system and environment [30, 31]. Though solving for
the exact dynamics for a full complex environment is beyond the capabilities
of current devices, taking into account the structure and interaction with the
environment poses interesting research questions: what is the fate of
entangled information within the system, how can a structured environments
drive effective interactions and information dynamics, and how does the
environment serve as a probe in an information theoretic/entropic capacity?

A simple avenue for exploring the impact of a structured quantum environment
is by considering composite, unitary models. Focusing on a particular
subsystem of a full closed quantum system and tracing over the additional
degrees of freedom (DoFs), then termed the environment, captures the
subsystem’s effective dynamics/interactions. This is a popular focus of study
as it provides an open quantum system perspective for the subsystem and allows
full consideration of the environment’s structure and interaction topology.
This construction allows us to specifically evaluate how variable structured
environments impact the overall quantum information dynamical phase as
expressed by the underlying subsystem. Previous work considered the validity
of Markovian assumptions provided variable system-environment coupling, and
here we are looking to add an information scrambling perspective. Considering
a system-environment construction in this manner directly applies to those
codes/models investigating the information physics of auxiliary bits or those
systems with inherent auxiliary DoFs such as mechanical or optical modes [32,
33, 34, 35].

Studying composite systems in this manner provides an interesting framework
extending current quantum information dynamic research. Significant recent
results focus on the range of interactions, the speed and nature of
information propagation, the role of inherent symmetries, and the effect of
emergent symmetries in the cases of many-body localization, Floquet periodic
driving, etc. The same phenomena can be similarly cast as an environment
mediated effect. The tunability of the system-environment network topology and
the inherent environment structure and interaction symmetries then allows for
systematic investigation into the particular contribution on the overall
dynamics.

In this paper we explore these aforementioned questions by considering the
simplest environment extension; an auxiliary central qubit (c-qubit) coupled
to 1-d chain system of interacting spin-$1/2$ (qubit) objects. Tracing out the
c-qubit and considering the dynamics of the 1-d system provides a translation
from a full, unitary model, to an effective long-range, non-Hermitian spin
chain with a hidden qubit degree of freedom. This provides a single long-range
quantum channel for transmitting information but at the same time imposes a
shared two-fold DoF across all spins. Though only a small addition to well-
understood nearest-neighbor qubit model, we observe an abundance of exciting
repercussions.

The system-environment coupling expresses various regimes: in the weakly
coupled regime, the c-qubit provides little feedback and acts as a free
channel for information to pass unimpeded; while when strongly coupled to the
low dimensional qubit environment, the c-qubit acts as a strong drive and
imposes an effective hidden symmetry on the underlying spin system and
generates disorder-free localization. We liken the physics observed here to
that seen in systems undergoing quantum measurement or strong Floquet driving.
Considering the c-qubit as a hidden degree of freedom provides unique insight
into how the quantum scrambling dynamics of the underlying spin chain maps to
an extended unitary model provided one additional qubit. Central qubit or a
higher dimensional qudit/cavity/register are popular theoretical and
experimental tools for providing non-invasive many-body measurements [36, 37,
38, 39], evaluating Hermitian and non-Hermitian response [40, 38], generating
effective interactions [41], and studying the fundamentals of decoherence and
information transport [42, 43].

Here we particularly focus on the dual effect of a tunable central qubit by
investigating the operator and entanglement growth in a nonintegrable, ring-
star Ising model. The model includes homogeneous spin-spin interactions in a
mixed magnetic field. We find an extremely surprising fast-to-slow quantum
information spreading transition that occurs due to the nonlocal and coherent
nature of the c-qubit. We summarize this result in Fig.1(d), where in the
weakly coupled regime (regime I), the central qubit mediates rapid scrambling
with a timescale that decreases with system size (green, upper curve in). In
the strong coupling regime (regime II), the scrambling time increases
exponentially with system size (red, lower curve). The mechanism behind this
transition is the interplay between the noncommuting, extensive c-qubit Ising
interactions and transverse field $h_{c}$. The metric here presented for
scrambling is $e^{S_{vN}(t)}/2^{L+1}$, which provides a measure of the span of
the quantum wavefunction throughout the full Hilbert space. As we detail in
what follows, in the strong coupling regime the central qubit rapidly
saturates its entanglement with the surrounding spin-chain environment and
becomes strongly driven by this extensive interaction. This strong interaction
rapidly aliases operators orthogonal to the central qubit Ising interaction on
the central qubit and even more surprisingly within the spin chain. The long
lifetime of states and operators that commute with the central Ising
interaction leads to slow multi-particle entanglement growth and operator
complexity.

Our work agrees with previous research that finds an extensively scaling,
nonlocal interaction leads to rapid scrambling, where the rate increases with
system size [44, 45, 46]. At the same time we find a surprising limit where
the purely quantum nature of the c-qubit imparts a coherent effect that slows
operator decoherence/entanglement and subsequent spreading. This phenomena
mirrors what is seen in strongly driven Floquet systems, where periodic
driving can impart an effective symmetry in all eigenstates and leads to
prethermalization and correspondingly slow entanglement spreading. We liken
the projective action of the central qubit in this time independent
Hamiltonian to the quantum Zeno effect where quantum measurement leads to a
ballistic to sub-ballistic entanglement growth transition. Here the c-qubit
imparts a highly nonlocal effect on operator projection in contrast to a local
purification/disorder network that redefines local spreading dynamics. We
illuminate this c-qubit physics by examining the growth of out-of-time-order
correlators (OTOCs) and the von Neumann entanglement entropy for sufficiently
high-energy initial product states.

##  II Scrambling Metrics

Many recent works have made significant progress on establishing the family of
scrambling dynamics that occur in various lattice models and geometric random
circuit designs, as characterized by the growth of OTOCs. The OTOC generically
given as

$C_{VW}=\langle[\hat{W}(j,t),\hat{V}(i,0)]^{\dagger}[\hat{W}(j,t),\hat{V}(i,0)]\rangle,$
(1)

examines how an initially prepared unitary operator $\hat{V}$ on site$-i$
commutes after Heisenberg evolution with operator $\hat{W}$ after time $t$
(here assumed a local operator on site$-j$). The operator spreading picture is
unique to quantum systems, where in working with pure states, no information
is truly lost but transforms into many-body degrees of freedom that become
increasingly inaccessible provided control over an initial localized region.

Studying OTOCs and the timescales associated with scrambling dynamics provides
a conjugate perspective as compared to entanglement entropy measures and
transitions. Where OTOCs and specifically infinite temperature OTOCs examine
the light cone established by Heisenberg evolution and depend more strongly on
the commutivity graph, entanglement entropy examines how the wavefunction over
a bipartition of Hilbert space spreads throughout. Here we specifically focus
on the von Neumann bipartite entanglement entropy given as

$S_{vN}=-\sum_{k}\lambda_{k}\log(\lambda_{k}),$ (2)

where $\lambda_{k}$ are eigenvalues of the reduced density matrix $\rho_{A|B}$
(RDM) obtained by integrating out subsystem $\mathcal{A}\text{ or
}\mathcal{B}$ with corresponding Hilbert spaces
$\mathcal{H}_{A},\mathcal{H}_{B}$. OTOCs and entanglement identify similar
physics and previous colloquial conceptions of the two established quantum
scrambling as a unifying framework behind them; where, scrambling represents
the time for an OTOC between arbitrary sites to become $O(1)$ and entanglement
entropy to become $O(L)$. In the case of OTOCs, this limit is not rigorous
enough and only provides a best-case scenario for operators traversing the
system rather than providing a timescale for nontrivial operator strings to
span the system [47] (extensive operator entanglement). Rigorous relationships
between OTOCs and Renyi$-2$ entropy have been established [48, 49, 50] and
special cases have been studied in particular optical Hamiltonians [51, 52,
24].

Figure 1: (a) Illustration of the ring-star model achieved in (b) optically
dressed trapped-ion experiments and (c) through nonlocal unitary gates in a
circuit realization. (d) Central qubit mediated dynamics: local interaction
(black), c-qubit induced fast scrambling (green), and c-qubit inhibited
scrambling (red). Information scrambling approximated as wavefunction spread
in Hilbert space or $\sim e^{S_{vN}(t)}$ to show dramatic prethermal-like
approach to full thermalization. (e) Qualitative depiction of (d) or how a
well defined initial state grows to fill state/operator Hilbert space with a
varying rate depending on dynamical phase. (f) von Neumann entanglement
entropy $S_{vN}(t)$ following a quench from polarized state $|+y\rangle$ for
the ring-star Ising model under varying $\lambda$.

Unitary scrambling physics generally falls into two categories: systems that
thermalize rapidly and those that fail to do so. The former are known as fast-
scramblers; ergodic systems typically with variable all-to-all range
interactions that spread quantum information throughout the full Hilbert space
in $t_{sc}\sim\log(N)$. Models such as Sachdev-Ye-Kitaev (SYK) and non-
integrable infinite range Ising and XY models are known to exhibit fast-
scrambling physics [53, 54, 45, 46, 55]. Systems that fail to thermalize with
$t_{sc}\sim e^{L}$ are slow-scramblers, non-ETH obeying systems and candidates
for highly coherent quantum information storage. There are multiple vectors
through which non-ETH physics occurs: integrability, disorder-free
localization [56, 57], quantum scarring [58, 59, 60, 61], and/or higher order
exact or proximate conservation laws [62, 63, 64]. The origins of much of this
work stems from the dramatic quantum correlations observed in quantum
simulation experiments.

Typical models accessible to simulation and experiment are semiclassical in
nature with infinite or long-range interactions. These models exhibit the
characteristic unitary scrambling features we detailed previously, yet
continue to enrich the discussion with new puzzling results. In Lipkin-
Meshkov-Glick (LMG) model or the Dicke model exhibit strict conservation of
the total spin moment $\hat{S}^{2}$ such that the effective number of degrees
of freedom is $\mathcal{O}(L)$, compared to $\mathcal{O}(2^{L})$ [24, 65, 66].
These systems have been observed to spread information rapidly, while the
complexity saturation value remains low. This is in stark contrast to fast
scrambling models like SYK where infinite-range connectivity allows for rapid
and complex quantum information scrambling. One immediately puzzling question
is: how do long-range interactions, tending toward generating semiclassical
behavior, compete with local chaotic quantum dynamics to allow a fast-to-slow
scrambling transition? A complete understanding of quantum information physics
not only hinges on understanding the unitary dynamic contribution to
experimental results, but similarly understanding non-Hermitian processes.
These are inherent to quantum simulation platforms and represent the an exotic
next frontier for theory and experiment as we move toward fully expressive
quantum circuits and computation.

More generalized quantum dynamical behavior has been explored in recent
studies consisting of non-Hermitian operations: composite system-environment
undergoing quantum measurement [67, 68, 69, 70, 71, 72], light-matter
interactions [24], dissipative and driven systems [73, 74, 75, 76]. The most
extraordinary findings reveal that these non-unitary dynamics generate
effective inter-system interactions and impose effective static long-lived
symmetries: Floquet periodically driven systems are akin to various unitary
scrambling phases.

##  III Model

We consider the Hamiltonian for the c-qubit or ring-star Ising model:

$H=\sum_{i=0}^{L-1}\lambda\sigma^{z}_{i}\sigma^{z}_{c}-J\sigma^{z}_{i}\sigma^{z}_{i+1}+h\sigma^{x}_{i}+g\sigma^{z}_{i}+h_{c}\sigma^{x}_{c}+g_{c}\sigma^{z}_{c},$
(3)

where $\lambda$ represents the uniform spin-c-qubit interaction, and $J,h,g$
represent the much-studied nonintegrable mixed-field Ising model. Here we take
$J=1.0,h=h_{c}=1.05,g=g_{c}=0.45$ unless otherwise noted. This is a well
characterized nonintegrable point for polarized state evolution and operator
dynamics allowing us to benchmark the impact of the c-qubit [46, 77].

Previous works examined operator dynamics and entanglement growth in random
unitary circuits (RUCs) applied in a star-graph network. RUCs on the star-
graph generated OTOC dynamics that saturated at $t\propto\log(L)$ [78, 47]. In
contrast, an interesting section of [78], examines dynamics of time-
independent, star-Ising Hamiltonians which consist of Ising interactions and a
non-commuting field $h_{c}$ strictly on the central qubit. In contrast to fast
scrambling RUC networks, the time-independent c-qubit Hamiltonian dynamics
generate novel operator confinement on the c-qubit with a coherent lifetime
$\tau\propto\frac{h^{2}}{\lambda L}$. This defines the time for significant
operator weight to decohere into operator subspaces that no longer commute
with the Ising interaction, subsequently allowing slow scrambling to all other
sites. Provided this contrast between the fast-scrambling RUCs and the slow
Hamiltonian dynamics, we ask: can the c-qubit coupling mediate rapid
scrambling in a truly time-independent nonintegrable model and, if so,
when/how does this picture break down?

##  IV Analytic Insights

Before discussing the exact numerics on the full nonintegrable ring-star Ising
model, we first motivate the reasons underlying a dynamical transition from
fast-to-slow scrambling and the corresponding mechanism. We first summarize
typical operator growth in local spin models and then review the Ising star
graph dynamics.

###  IV.1 Local Ising Model

In the completely local Ising chain limit $\lambda=0$, we can gain a heuristic
understanding of the characteristic light cone spreading by performing an
early-time expansion of the dynamical correlator
$\langle\sigma^{z}_{i}(t)\sigma^{z}_{j}\rangle$. Using the early-time
expansion of the Heisenberg evolution we can write
$e^{i\hat{H}t}\sigma^{z}_{i}e^{-i\hat{H}t}$ using the Baker–Campbell–Hausdorff
(BCH) expansion

$\\\
e^{-i\hat{H}t}\sigma^{z}_{i}e^{-i\hat{H}t}=\sum_{m}^{\infty}\frac{(it)^{m}}{m!}[H,S^{z}_{i}]_{m}\\\
\\\ [A,B]_{m}=[A,[A,B]_{m-1}];[A,B]_{0}=B\\\ \\\
[H,S^{z}_{i}]_{1}=it[hS^{x}_{i},S^{z}_{i}]=-ithS^{y}_{i}\\\ \\\
[H,S^{z}_{i}]_{2}=\frac{(it^{2})}{2}[H,S^{y}_{i}]\\\
=\frac{-t^{2}}{2}(-\lambda
S^{z}_{0}S^{x}_{i}+hS^{z}_{i}-gS^{x}_{i}-J(S^{z}_{i-1}S^{x}_{i}+S^{z}_{i+1}S^{x}_{i})).$
(4)

Before continuing this expansion to higher orders we observe a trend in
operator growth and can approximate the weight as

$[S^{z}_{i}(t),S^{z}_{j}]\propto\frac{t^{|i-j|}}{|i-j|!}\mathcal{O}(1)$ (5)

Naively, we then expect the OTOC, or squared commutator, to generically follow

$C_{zz}(i-j,t)\sim\frac{t^{2|i-j|}}{(|i-j|!)^{2}}\mathcal{O}(1)$ (6)

This approximation is valid for times $t\sim|i-j|/v_{B}$, where $v_{B}$ is the
characteristic butterfly velocity. Before this time, operator growth is
suppressed with an exponent that grows with distance, and allows for optimized
simulations using matrix product operator dynamics (MPO) by just keeping track
of the operator wavefront [79]. Though many extensive theoretical works
provide rigorous estimates on the form of operator growth for translationally
invariant models, here we simply examine the asymptotic form. This
exponentially growing operator weight with exponent like $r=|i-j|$ leads to
the development of a linear light cone with butterfly velocity exactly
calculated as $2eJ$ ($e$ being Euler’s number) for $h/J>1$ [80]. Following
this light cone in the nonintegrable Ising model, any localized operator spans
the full operator Hilbert space $\\{\hat{X},\hat{Y},\hat{Z},\hat{I}\\}$ within
region $r$.

An interesting extension of the OTOC is the integrated OTOC (iOTOC), which is
the integral over $r$ and the bipartite OTOC [81, 50]. Both provide a novel
characterization of the operator complexity within a region $r$. This also
provides an insightful way to then understand the corresponding entanglement
dynamics for pure state wavefunctions with energy density $k_{B}T$ as in
accordance with ETH the states and operators should exhibit ergodic
equilibration. In the nonintegrable Ising model, an operator saturates the
reduced Hilbert space $4^{|i-j|}$ after a time $t_{sat}=\frac{|i-j|}{v_{B}}$,
so simply integrating over Eq.6 provides an exponentially growing iOTOC for
all $C_{vw}$. This operator growth then guarantees ballistically growing
entanglement entropy $S_{vN}\sim\sum_{v,w}\log[\text{iOTOC}(v,w)]$.

###  IV.2 Star-Ising model

In the star limit, we set $J=0$ and tune external fields $h,g$ to allow more
expansive operator evolution. First working with only $\lambda\neq 0$, we then
add complexity to arrive at the full, nonintegrable ring-star Ising model. For
$|\lambda|>0$, operator growth between leaves of the graph is trivial. The
dynamics are exactly solvable as $[\sigma^{z}_{i},\hat{H}]=0$ for all sites
$i$. Similarly, the commutivity graph representing the Hamiltonian and how
local operators propagate is completely disconnected with vertices
$\sigma^{z}_{i},\sigma^{z}_{L}$ for all $i\in L$ and no bonds in between.
Operators evolve simply under the central Ising interaction and the two-time
correlator behaves as:

$\langle S^{x}_{i}(t)|S^{x}_{i}\rangle=\cos{2\lambda t}.$ (7)

And, again, exactly in this case, the OTOC goes as:

$C_{xx}(i,i,t)=2\sin^{2}(2\lambda t),$ (8)

Though this limit admits a trivial result, it allows us to understand the key
coherent property of the c-qubit qubit. We see that for all times, operators
orthogonal to the Ising interaction $\lambda$, initially prepared on the
leaves, propagate to the c-qubit after a time $t=\pi/\lambda$. Because all
terms in the Hamiltonian commute with this interaction
$[\sigma^{z}_{i}\sigma^{z}_{0},\sigma^{z}_{j}\sigma^{z}_{0}]=0$, the action of
operator development from leaves to c-qubit does not decohere into nontrivial,
orthogonal operators $\\{\hat{X},\hat{Y}\\}$. Due to this coherence, operators
oscillate between the initial node and the c-qubit. If the operator was
initially prepared on the c-qubit, it fluctuates onto all nodes, but instead
of becoming a many-body operator, it is a collective superposition of $L$
unique two-body operators
$\sum_{j}\sigma^{x,y}_{0}\sigma^{z}_{j}\in\mathcal{H}(4^{\otimes L})$. This
superposition, though nonlocal on a timescale $\tau=\mathcal{O}(1)$, has
minimal operator entanglement and the complexity of operators strings is fixed
to be a maximum of $2$. In this scenario, similarly, minimal entanglement
entropy develops as the number of unique operator states is of $\mathcal{L}$
in a Hilbert space of $4^{L}$. Quench experiments on such systems with nonzero
homogeneous magnetic fields will only permit half-system entanglement to grow
like $\log{L}$ since we have effectively a large semiclassical spin system
coupled to a single two-level qubit [82].

The simplest extension we can make is to now introduce a transverse field on
the c-qubit spin, $h_{c}\neq 0$. This was similarly analyzed in a disordered
star-Ising system for small values of $h/\lambda L$ [78]. For $|h_{c}|>0$, the
model retains the same integrability, as any $\\{z_{1},...z_{L}\\}$ is an
eigenstate of the system and will not evolve under unitary dynamics. We can
then reduce the problem to solving for the evolution of the central qubit in a
mixed $x-z$ field where the effective longitudinal strength given by
$\sum_{i}\sigma^{z}_{i}$. Though fully solvable, we find illuminating operator
dynamics in the infinite temperature limit. Solving exactly for
$\langle\sigma^{x}_{i}(t)\sigma^{x}_{i}\rangle$ as was first provided in [78],
we observe how the coherent oscillation
$\langle\sigma^{x}_{i}(t)\sigma^{x}_{i}\rangle=\cos(2Jt)$ for $h_{c}=0$,
evolves for increasing $h_{c}$ and leads to operator decoherence. Operator
decoherence, in this sense, is that as auxiliary operators such as
$\sigma^{z}_{c}\rightarrow\sigma^{x}_{c},\sigma^{y}_{c}$, they then no longer
commute with the c-qubit Ising interaction and operator weight then grows on
sites $j\neq i$. The superposition of these growing dynamical correlations on
sites $j\neq i$ similarly decreases the probability of finding
$\sigma^{x}_{i}$ on site$-i$ after time $t$, leading to a decay in the
autocorrelation function on site $i$.

For $\lambda L>>h$, it was shown numerically that early time operator dynamics
go as

$\langle\sigma^{x}_{i}(t)\sigma^{x}_{i}\rangle\sim\cos(2\lambda
t)e^{-\sqrt{\frac{\pi}{2L}}\frac{h^{2}}{\lambda}t}$ (9)

using the memory matrix formalism [78]. Regardless of whether $\lambda$ is a
Gaussian random variable or homogeneous, we arrive at the same exponential
dependence on system size. In the case of Gaussian random variables, this
approximation was observed to be faster than the true decay rate. The key
physics being that the $L-$site interactions with the c-qubit lead to an
extensive coherence time, which we can evaluate explicitly by tracing over the
set of eigenstates $\mathbb{Z}$:

$\displaystyle\langle\sigma^{x}_{i}(t)\sigma^{x}_{i}\rangle=\frac{1}{2^{L}}\text{Tr}_{\mathbb{Z}}[\langle
z_{c},...z_{L}|e^{i\hat{H}t}\sigma^{x}_{i}e^{-i\hat{H}t}\sigma^{x}_{i}|z_{c},...z_{L}\rangle]$
(10) $\displaystyle=\langle
z_{c}|\otimes\langle\mathbb{Z}_{m}|e^{i\hat{H}t}\sigma^{x}_{i}e^{-i\hat{H}t}|\mathbb{Z}_{m}\rangle\otimes|z_{c}\rangle$
(11)
$\displaystyle\hat{H}|\mathbb{Z}_{m}\rangle\otimes|z_{c}\rangle=e^{i(h_{c}\sigma^{x}_{c}+\lambda\sum_{i>0}z_{i}\sigma^{z}_{c})t}|z_{c}\rangle$
(12)
$\displaystyle=\mathcal{I}\cos(2\omega_{Z_{m}}t)+\frac{(h_{c}\sigma^{x}_{c}+\lambda
Z_{m}\sigma^{z}_{c})}{\omega_{Z_{m}}}\sin(2\omega_{Z_{m}}t)|z_{0}\rangle.$
(13)

We use same simplified notation as [78], where
$\omega_{\mathbb{Z_{m}}}=\sqrt{h^{2}+\lambda^{2}Z_{m}^{2}}$, with
$Z_{m}=\sum_{i>0}z_{i}$ and $\mathbb{Z_{m}}=\mathbb{Z}[m]$ is just
$z-$eigenstate with magnetization $m\in[-L,L]$. As the set of states
$\\{z_{1}...z_{L}\\}$ commute with the Hamiltonian, we reduce the Heisenberg
evolution to that of a 2-level system and averaging over the respective
density of states, which is simply a binomial distribution in the infinite
temperature limit. Going back to the full evolution we then have

$\displaystyle\langle\sigma^{x}_{i}(t)\sigma^{x}_{i}\rangle=\frac{1}{2^{L}}\sum_{z_{0}}\sum_{m}\left(\begin{array}[]{c}L\\\
|m|\end{array}\right)\langle
z_{c}|[\mathcal{I}\cos(2\omega_{Z_{m}}t)+\frac{(h_{c}\sigma^{x}_{c}+\lambda
Z_{+}\sigma^{z}_{c})}{\omega_{Z_{m}}}\sin(2\omega_{Z_{m}}t)]\times$ (16)
$\displaystyle[\mathcal{I}\cos(2\omega_{Z^{-}_{m}}t)+\frac{(h_{c}\sigma^{x}_{c}+\lambda
Z_{-}\sigma^{z}_{c})}{\omega_{Z^{-}_{m}}}\sin(2\omega_{Z^{-}_{m}}]t)]|z_{c}\rangle$
(17) $\displaystyle=\frac{1}{2^{L-1}}\sum_{m}\left(\begin{array}[]{c}L\\\
|m|\end{array}\right)\cos(2\omega_{Z_{m}}t)\cos(2\omega_{Z^{-}_{m}}t)+\frac{(h_{c}^{2}+\lambda^{2}Z_{m}Z^{-}_{m})}{\omega_{Z_{m}}\omega_{Z^{-}_{m}}}\sin(2\omega_{Z_{m}}t)\sin(2\omega_{Z^{-}_{m}}t).$
(20)

$\sigma^{x}_{i}$ flips the single spin state $z_{i}$ and leads to two unique
frequencies $w_{Z_{m}}$ and $w_{Z^{-}_{m}}$, separated like $\lambda$ for
$h=0$, $Z^{-}_{m}=\sum_{j\neq i>0}z_{j}-z_{i}$. $h_{c}=0$ provides the exact,
non-scrambling result
$\langle\sigma^{x}_{i}(t)\sigma^{x}_{i}\rangle=\cos(2\lambda t)$, and for
$\lambda=0$ we have
$\langle\sigma^{x}_{i}(t)\sigma^{x}_{i}\rangle=\cos(2\lambda
t)^{2}+\sin(2\lambda t)^{2}=1$. Starting in the $h_{c}=0$ limit and moving
toward high transverse magnetic field $h_{0}>>\lambda L$, we numerically
integrate Eq.20 and study the autocorrelation function on site$-i$ (Fig.2). In
the low field limit, we expect exponentially small modifications to pure
cosine oscillations as we have recreated from [78], and in the high field
limit, $\sigma^{x}_{i}$ should completely break down as $\sigma^{z}_{c}$
decoheres and operator weight spreads to all sites rapidly. As
$\sigma^{z}_{c}\rightarrow[\sigma^{y}_{0},\sigma^{x}_{0}]$, operator weight
can be distributed to all sites $j\neq i$, and the rapid rotation of operators
on the c-qubit aliases away coherent growth on site-i.

In Fig.2(a) we see that coherent oscillations are apparent for $h/\lambda
L~{}\mathcal{O}(1)$ out to $t\lambda\sim 30$ and slow decay with increasing
$h$. Above $\log[h/\lambda L]>0$, oscillations are barely visible and dynamics
are dominated by exponential decay with no revival even out to time
$t\lambda\in[0,200]$. This is more clearly depicted in Fig.2(b), where we take
the long-time average of the autocorrelation function:

$A(i,t_{0})=\int_{t>0}^{t_{0}}|\langle\sigma^{x}_{i}(t)\sigma^{x}_{i}\rangle|dt.$
(21)

The autocorrelation function transitions sharply at
$h/L\sim\mathcal{O}(10^{-1})$, where $\sigma^{x}_{i}$ no longer has
significant weight on the initial site. For small $h$, the autocorrelation
function is a nearly pure $\cos{2\lambda t}$ and decay that grows with $h$.
Near the transition, $A(i,t_{0})$ sharply decays to $\sim 1/L$ for $h/\lambda
L>\mathcal{O}(10^{-1})$. We provide a Fourier analysis of Fig.2(a) in (c),
where the transition is more clearly resolved. The oscillatory part
($\epsilon_{0}$) decreases (elongates in time) above $h/\lambda L=1$ with an
exponential decay rate that peaks at the transition point. Above the
transition point, the dynamics are no longer captured by a decaying cosine
function, but crossover to exponentially damped operator dynamics. The large
magnetic field on the central site initially prevents operator weight from
leaving site$-i$ as shown by the elongating decay in (a). At this point the
magnetic field is dominating the operator dynamics and rapidly rotates
$\sigma^{z}_{c}$ into eventual two-body operators that live in a superposition
on all sites. In this regime, operator weight on the initial site saturates
with a scaling like $1/L$.

Adding a mix of on-site frustrating magnetic fields is not sufficient to fully
scramble information. Applying fields along the $x,z$-direction, local
operators do not spread throughout full operator Hilbert space as the system
can still be described as a collective $S=L/2$-spin interacting with a qubit.
Similarly the steady-state entanglement entropy remains independent of
$\lambda$ and system size, while the growth rate is determined by
$\text{min}[1/\lambda,1/h_{c}]$ and does not slow when when deep into the
coherent regime $\lambda L>h_{c}$ [see Supplemental for greater details].

Figure 2: Autocorrelation $\langle\sigma^{x}_{i}(t)\sigma^{x}_{i}\rangle$:
(a) Eq.20 vs. time ($t\lambda)$ as a function of $h,L=40,\lambda=1.0$. (b)
Long-time average of (a) $A(i,t_{0}=200)$ as a function of $h$ and system size
$L$. (c) Curve fit of smoothed results in (a) consisting of coherent
oscillating component with frequency $\epsilon_{0}$ and exponential decay
parameter $\epsilon_{1}$. $\epsilon_{0}$ shown in the top, solid color palette
remains equal to $2\lambda$ roughly until reaching the critical point
$\lambda=h/L$. Vertical lines in (c) correspond to critical point
$h_{c}=L\lambda=[0.5,1.0,1.5,2.0]$.

##  V Numerical Results

After discussing the nearest neighbor mixed-Ising chain and the star Ising
model, we now seek to understand the dynamical behavior in the full, ring-star
system. Here we study the dynamics of the star-local model using exact
diagonalization for systems up to $L+1=13$ and Krylov subspace expansion
techniques for evaluating the Schrodinger ODE for sizes up to $L+1=22$. We
employ periodic boundary conditions $i=L=0$. In the previous section we
revealed that a dynamical operator growth transition occurs as a function of
$h_{c}/\lambda$ on the Ising star graph that allows for rapid operator growth
from the central qubit or coherently protected operator dynamics on the
central qubit. When the operator dynamics are coherently oscillating on the
central site, this leads to slow two-time correlator growth. Here we
investigate the fate of this transition, its effect on the secondary, local
Ising channel for quantum scrambling, and whether fast-scrambling is
achievable in simple local-nonlocal construction. We know that the mixed-field
Ising model is nonintegrable and capable of scrambling information throughout
the full operator Hilbert space, but does a simple nonlocal qubit rapidly
enhance this process?

Firstly, we calculate the average adjacency level ratio
$\langle\bar{r}\rangle$ of the full Hamiltonian Eq. 3 [see Appendix Fig.S4].
We solve the full spectrum exactly considering the parity conserving and $k=0$
sector of the Ising chain $(L=15)$ and confirm that for the ring-star model
and nonintegrable Ising model that all points in phase space we consider
indeed follow GOE random matrix statistics with $\langle\bar{r}\rangle\approx
0.53$. Though the level spacing provides a first check of nonintegrable
dynamics, it is not sufficient in capturing strongly coherent effects inherent
to fractionalized regions of the Hilbert space or rare states that exhibit
confinement or slow growth [83].

Figure 3: Entanglement spreading in the ring-star model:  (a) $S_{vN}(t)$ as a
function of $\lambda$, (b) plotted on a semilog axis in time ($tJ$), and (c)
$\lambda$ reparameterized to $\lambda/\sqrt{L}$, system size $L$. (d)
Polynomial fit of the intermediate time entanglement growth behavior
$S_{vN}\propto t^{\alpha}$. System size $L+1=22$ and $J,h,g=[1.0,1.05,0.45]$.

###  V.1 Entanglement Growth

The first test as to the general information scrambling capacity of this model
is to understand the entanglement entropy dynamics. We numerically investigate
entanglement growth $S_{vN}(t)$ as a function of c-qubit coupling. To
understand the infinite temperature information dynamics of the system, we
work with a product state with energy density equivalent to that of an
infinite temperature state $\langle H_{i}\rangle=0$. As the system exhibits
GOE random matrix statistics for all parameters of the Hamiltonian tuned here,
we expect that any such effective infinite temperature pure state will indeed
obey ETH. The state we work with here is the polarized $|+y\rangle$ spin-
state, which has been used to exemplify high-energy quench dynamics of the
mixed-field Ising chain previously [46].

In the limit $\lambda=0$, we have the local, nonintegrable Ising model which
exhibits ballistic entanglement growth for times up to $L/v_{k}$, where
$v_{k}$ is the maximal dispersion associated with quasiparticle momentum $k$.
This timescale is upper-bounded by $L/(2J)$ which provides the characteristic
timescale of nearest-neighbor interactions multiplied by the length of the
system. The factor of two comes from periodic boundary conditions and simply
captures the longest path between spins. In the $\lambda=0$ curve in Fig.3(c)
we see that for increasing system size, entanglement saturation occurs on a
timescale that scales linearly with $L$ and the maximal quasiparticle velocity
remains independent of system size. Entanglement saturates like $L\log(2)$ as
the effective infinite temperature state explores the full Hilbert space of
the system. With the structure of correlations reaching the scale of the
Hilbert space $O(2^{L})$ we then expect entanglement to be similar to the log
of full operator space complexity.

As we slowly increase the nonlocal coupling to the central c-qubit, we expect
that nonintegrability is maintained and now the shortest path between sites is
next-nearest neighbor, mediated by the central qubit. The timescale for the
c-qubit mediated entanglement spread is $2/\lambda$, where the factor of $2$
comes from the second order interaction with the central c-qubit that allows
correlations to develop between sites $i,j\neq 0$. In the extremely weak
regime $\lambda<<J,h,g$, the c-qubit qubit can similarly be thought of as a
cavity, where the finite size Hilbert space can be disregarded.

Including the c-qubit modifies the early-time entanglement growth, as
operators grow ballistically due to local transport and super-ballistically
with an additional nearest-neighbor c-qubit mediated contribution. We see in
Fig.3(a) that for $L+1=20$ as $\lambda\rightarrow h$ the linear coefficient of
entanglement growth grows continuously and the saturation value is modified by
the addition of a single qubit. In this enhanced rapid scrambling regime, if
we reparameterize the central qubit coupling
$\lambda\rightarrow\lambda/\sqrt{L}$ such that the effective spin-spin
interaction is not extensive $J_{\text{eff}}\sim\lambda^{2}.0/L$, we find that
the coefficient for linear growth is roughly independent of system size
(Fig.3(b) $\lambda=\frac{2.5}{\sqrt{L}}$). Entanglement entropy growth rate
that increases with system size is indicative of fast-scrambling behavior
[46].

For $\lambda>h$, we see a surprising entanglement transition. The early time
growth defined as $t<2\frac{1}{\lambda}$ exhibits rapid entanglement growth
but becomes sub-ballistic at intermediate times
$2\frac{1}{\lambda}<t<t_{sat}$. In Fig.3(b) the width of this intermediate
timescale grows exponentially with increasing $\lambda$ as exhibited by the
logarithmic scale in time, while $S_{vN}(t\rightarrow\infty)$ remains largely
unchanged. We perform a polynomial fit of the entanglement growth in Fig.3(d)
over the region $\log[2]<S_{vN}(t)<S_{vN;sat}$ and find that in the local and
fast-scrambling regime, entanglement continues to grow ballistically with
exponent $\alpha\sim 1.0-1.1$. For $\lambda>\lambda_{c}$, $\alpha$
monotonically decreases with $\lambda$ as expected if
$S_{vN}\propto\frac{1}{\lambda}\log[t]$.

In order to determine the location of the phase transition and relevant
scalings, we analyze the entanglement entropy at a fixed time $t^{*}$. We
choose a time such that the $S_{vN}(t^{*},\lambda=0)$ is roughly
$\frac{1}{2}S_{vN}(t\rightarrow\infty)$. In Fig.4(a) we plot $S_{vN}(t^{*})$
as a function of two magnetic fields $h$ depicted by solid/dashed curves and
as a function of system size $L$. We extract the maxima of Fig.4(a) and
perform a size and field scaling and find that the fast-to-slow scrambling
transition value occurs like $\lambda_{c}\propto L^{(-0.52\pm
0.05)}h_{c}^{(0.52\pm 0.02)}$, (b,c) respectively. In the high $h_{c}$ limit,
the transition becomes nearly independent of system size ($h_{c}=2.5$ curve in
Fig.4(c)) and in the low field limit, is dominated by system size:
$h_{c}=0.05-0.75$ curves in Fig.4(c). Greater details on identifying the
entanglement transition are included in the Supplemental Information. This
contrasts the dynamical transition observed in the star Ising model with a
transition that occurs at $\lambda_{c}=h_{c}/L$.

Figure 4: Identifying entanglement dynamics crossover: (a) Entanglement
entropy calculated at time $t^{\star}\sim 3.5tJ$ plotted as a function of
c-qubit coupling $\lambda$ ($S_{vN}(t^{\star})/t^{\star}\sim v_{b}$) for three
transverse magnetic fields and various system sizes
($h_{c}\in[1.05,2.5];L\in[11-21,2]$. For nonzero $\lambda$, entanglement
growth rate varies as a function of $h_{c},L$. Crossover point $\lambda_{c}$
determined as maxima of the curves in (a) with $\lambda_{c}(L)$ and
$\lambda_{c}(h_{c})$ plotted in (b,c), respectively. In the respective $h_{c}$
dominant and $L$ dominant regimes, the critical point find $\lambda_{c}\propto
L^{\gamma}h_{c}^{\kappa}$ with $\gamma=-0.52\pm 0.02(1)$ and $\kappa=0.52\pm
0.05(1)$.

The novel entanglement transition mediated by the c-qubit dynamics is
extremely surprising in that not only does the nonlocal coupling provide a
secondary channel for distributing entanglement, but in the strong coupling
regime it inhibits growth of even the local Ising interactions with which it
commutes. The problem is similarly interesting in that it is completely
disorder-free, so the slow information growth in the system can be attributed
to purely coherent effects. In order to shed more light on how this central
coupling serves to rapidly/slowly scramble quantum information we examine the
transverse and longitudinal OTOCs ($C_{xx},C_{zz}$).

Figure 5: Space-time OTOC spreading:  (a-c) OTOC for longitudinal spin
component $C_{zz}$ for $\lambda={0,0.6,2.6}$, respectively. Initial linear
light cone spreads on top of c-qubit mediated operator weight that spreads
super-ballistically on all sites after an initial wait time that scales with
$min[1/h_{x},1/\lambda]$. For $(\lambda/\sqrt{L})/h\geq 1$, $C_{zz}$ is
suppressed on all sites beyond a weak initial light cone. (d) In contrast, for
$\lambda=2.6$, transverse OTOC $C_{xx}$ rapidly fluctuates on the initial
site-$i$ and spreads to nearest neighbors $i+1$ and the central site $L$ on
the order of $t=1/\lambda$. The light cone profile is no longer visible and
$C_{xx}$ rapidly oscillates with frequency $2\lambda$ for sites $|i-j|>1$.
System size $L+1=14$ and $J=1.0,h=1.05,g=0.45$.

###  V.2 Operator Spreading

Starting from an initial state selected from the Haar measure, as to
approximate infinite temperature state, we examine how operators initially
prepared on site $i=0$ spread under the influence of Ising interactions, where
site $c$ represents the central qubit. When calculating the OTOC we simplify
Eq.1, as the local $\hat{X},\hat{Z}$ operators are Hermitian and initially
commute on all sites at $t=0$. We then explicitly evaluate

$C_{VW}=1-\langle\hat{W}(j,t)\hat{V}(i,0)\hat{W}(j,t)\hat{V}(i,0)\rangle$ (22)

In Fig.5(a), when considering only local interactions we see that operator
weight develops according to a ballistic light cone with a bubble-like profile
that saturates on all sites following the wavefront:
$C_{zz}(r,t)\propto\frac{t^{2r}}{r!^{2}}$. With small, nonzero $\lambda$ (b)
the light cone remains apparent but now spreads on top of growing operator
weight distributed by the central qubit. The central qubit super-ballistically
distributes operator weight to all sites on a timescale like
$\frac{2}{\lambda}$. The growth timescale on the c-qubit is half that of the
bulk spins, as the $\sigma_{z}$ operator must decohere on the initial site and
again on the central site and hence no longer commutes with the Ising
interaction. For $\lambda<<h_{c}$, $1/\lambda$ sets the timescale for this
process. In the strong coupling regime (c), a weak light cone remains but
transfers a small fraction of the original operator weight. Operator spreading
becomes increasingly restricted where the lightcone profile is only visible
for few $tJ$ and $C_{z}z(j,t)$ grows increasingly slowly on sites far from the
initialized operator. This behavior is puzzling; the highly nonlocal coupling
c-qubit with increasing interaction leads to unintuitively slow local operator
spreading.

Here we can draw insight from the analytic results on the star-Ising model
(Fig.2). The $L-$body interaction on the central qubit acts to coherently
project operators into $\sigma^{z}$, while rapidly aliasing orthogonal
operators. Once $\sigma^{z}$ grows on the central qubit, it develops a
coherent lifetime with operator weight decaying like $\sim e^{-1/\lambda}$.
$C_{zz}$ characterizes how operators no longer commute with $\sigma^{z}$, and
in the same vain as the star-Ising model, we expect operators on the central
site to be strongly projected into the $z-$subspace and coherently oscillate
like $2\lambda$. As the central c-qubit is highly/fully entangled with the
remaining chain, the coherent projection on the central site then similarly
restricts how rapidly the many-body state of the $L-$spins decoheres from the
$z-$subspace. We expect $C_{zz}(t)$ to exhibit slow behavior as $z-$operators
are strongly driven on all sites. We then examine $C_{xx}(t)$, which captures
the rapid growth of $\hat{Z}$. In Fig.5(d) we see that $C_{xx}(t)$ continues
to grow like $\mathcal{O}(1)$ across all sites, with strong, oscillatory
behavior on the central qubit ($i-j=c$). $C_{xx}(j,t)$ on all sites similarly
oscillates at frequency $2\lambda$ but shifted by half a period compared to
$C_{xx}(c,t)$.

Figure 6: $C_{xx}(|i-j|,t)$ spreading in the ring-star Ising model: Real-time
evolution of $C_{xx}(i-j,t)$ for varying c-qubit coupling $\lambda$, plotted
on a log-log scale. $\lambda=0$ exhibits lightcone spreading
$C_{xx}\propto\frac{t^{2(i-j)}}{(i-j)!^{2}}$. In the fast-scrambling regime
($\lambda<\lambda_{c}$) (b,c), operator weight grows homogeneously across all
sites at half the rate as the on the c-qubit site ($C_{xx}(c,t)$, blue).
(b,c). (d) As $\lambda\sim\mathcal{O}(1)$, $C_{xx}(c,t)$ becomes
$\sim\mathcal{O}(1)$ at $t\propto\lambda^{-1}$. Once $C_{xx}(c,t)$ becomes
$\mathcal{O}(1)$, the rapid growth on spin sites $|i-j|>1$ slows and coherent
oscillations develop. For $\lambda=3.0$ and focusing on distant sites (purple)
and $tJ\in[1,5]$, the operator growth after c-qubit saturation becomes slower
than that observed for $\lambda=1.0$. System size $L+1=15$ and
$J=1.0,h=1.05,g=0.45$.

On the initial site $i$, as $\sigma^{z}_{i}$ decoheres under the transverse
magnetic field it no longer commutes with itself nor the Ising interaction.
This leads to a nonzero $C_{zz}(t)$ on site$-i$ and subsequent operator weight
growing on nearest neighbors and the central qubit. Once operator weight grows
on the c-qubit, the associated decoherence time of $\sigma^{z}_{c}$ is
exponential in $\lambda$ and as weak operator weight leaks onto bulk spins,
the same coherent oscillations and slow decay is imparted on the local spin-
chain. We gain further insight by examining the individual curves that make up
the density plot in Fig.5 as Fig.6. We see the polynomial growth associated
with the light-cone spreading on top of the exponential c-qubit mediated
growth (a-c).In the slow scrambling regime (Fig.6(d)), we find that for
$\lambda=3.0$ that $C_{xx}(c,t)$ becomes $\mathcal{O}(1)$ at
$t_{c}=\pi/2\lambda$ (dashed, red) and after time $t_{\lambda}=\pi/\lambda$
the rate of change of $C_{xx}(i-j,t)$ decreases dramatically (solid). Once
$\sigma^{z}$ exists on the central qubit, orthogonal operators on the c-qubit
and bulk sites rapidly fluctuate and lead to an effective wait time, slowing
the amount of operator growth under the action of the Ising$-zz$ interaction.

Figure 7: $C_{zz}(t)$ spreading in the ring-star Ising model: Longitudinal
OTOC $C_{zz}(t)$ on (a) site $j=10$ and (c) c-qubit $j=c$ with real-time
evolution plotted on a log-log scale. (b, d) Polynomial fit at intermediate
times $C_{zz}(t)\propto\alpha t^{\beta}$. As $\lambda$ increases, coefficient
of longitudinal operator growth becomes exponentially suppressed (top to
bottom curve on log scale) $\log{\alpha}\propto\lambda/\sqrt{L}h^{2}$.
Operator growth on distant spin sites is approximately linear $\beta\sim 1$ in
the slow scrambling regime and coefficient
$\alpha=e^{-\frac{3}{2}\frac{\lambda}{\sqrt{L}h^{2}}}$. (d) $C_{zz}$ the
c-qubit site grows with exponent $\beta_{c}\sim 0.5$ and coefficient
$\alpha_{c}=e^{-\frac{\lambda}{h^{2}}}$. Operator growth on bulk spin sites is
expected to behave like $C_{zz}(j,t)=C_{zz}(L,t)^{2}$, as the effective
c-qubit mediated spin-spin interaction is second order process in
$H_{\lambda}$.

A complementary perspective for operators orthogonal to $\hat{Z}$ is observed
by $C_{zz}$ (Fig.7. Looking at the OTOC in the slow scrambling regime,
operator weight on the c-qubit and bulk sites grows like $h^{2}t^{6}$, as
captured by early-time expansion, up to time $t_{\lambda}=\pi/2\lambda$ [see
Supplemental for greater details]. For $t>t_{\lambda}$, operators become
essentially projected onto $z-$operator subspace due to the coherent lifetime
of operators on the central qubit. Fitting the OTOC to $\alpha t^{\beta}$
shows that operator weight decoheres on the c-qubit with exponent
$\beta_{c}\sim 0.45$, while on bulk sites it is nearly linear with
$\beta_{j}=\sim 0.85$. On all sites, the coefficient of growth becomes
exponentially suppressed in $\lambda$ with $\log[\alpha]\propto-\lambda$. This
can be understood heuristically: once operator weight lies on the central
site, it coherently oscillates between sites like $2\lambda$ and leads to a
waiting time during which little to no operator weight has decohered from the
$z-$subspace as it oscillates between $z-I$. The amount of operator weight
that decoheres in-between these waiting periods is
$\int_{t_{c1}}^{t_{c2}}e^{-\lambda}$ so the rate of operator growth into this
orthogonal subspace is independent of time and leads generically to
$C_{xx}(j!=c,t)\propto e^{-\lambda}t$. This slow growth for simple two-body
body operators outside of the $z-$subspace continues to be exponentially
suppressed for greater complexity many-body operators. The sublinear growth of
OTOCs into the bulk of operator Hilbert space then generically provides
$\log[t]$ entanglement entropy growth, as we have observed for quantum
quenches from the $|+y\rangle$ state.

##  VI Discussion

Here we have shown how the collective interactions between many constituent
spins and a central auxiliary qubit is both able to rapidly scramble
information across all degrees of freedom and restrict state/operator growth
from exploring the full Hilbert space. From an entanglement entropy picture,
the central bit entangles rapidly with its surrounding environment, the state
space of the qubit simply being $q=2$. Treating it as a 2-level system in an
effective bath, the extensive bath interactions and on-site magnetic field
frustrate the central qubit and lead to a rapidly fluctuating spin-moment when
$N\lambda\sim h_{c}$. In the operator picture this leads to rapidly decohering
operators that quickly lie in a superposition of states
$\\{\hat{X},\hat{Y},\hat{I},\hat{Z}\\}$. For $\lambda>>h_{c}$, $\\{\hat{Z}\\}$
operators are strongly driven to the qubit and therefore do not fully decohere
while $\\{\hat{X},\hat{Y}\\}$ become essentially echoed-out.

This picture is more clearly developed when we again turn-off local
interactions, $J=0$. In this case only the central bit is able to mediate
entanglement throughout the system. As a function of $\lambda/h_{c}$ the half-
chain entanglement entropy grows with a velocity like $1/\lambda$ until
$\lambda~{}h_{c}$ and surprisingly saturates with an entropy that is
independent of system size. The entropy growth rate and saturation value do
not change when crossing over into the strong-coupling regime. Though the
half-system entropy value is identical, we know from an operator picture that
scrambling occurs slowly on the leaves of the star graph, leaving the central
qubit to scramble rapidly while other sites scramble slowly. From both the
operator and entropy picture, we know that the central bit is highly entangled
with the system and the saturation entropy remains fixed regardless of
coupling; therefore once the central bit becomes highly entangled, it
similarly cannot scramble quantum information throughout. The saturation in
information capacity of the star-graph remains fixed as a function of
$\lambda$, so by tuning $\lambda$ we tune whether entanglement is frozen in a
highly entangled auxiliary bit or distributed globally. Understanding the
information dynamics of the star-graph then informs the surprising behavior
observed in the fully interacting star-local Ising system. Once the central
qubit becomes highly entangled with it’s environment, it similarly acts to
project operators into the $z-$subspace, inhibiting the local, nonintegrable
Ising interactions from fully scrambling the system and reaching full
thermalization quickly.

As the commutivity graph that translates operators from site$-i$ to site$-j$
requires operation of the Ising$-zz$ interactions, which similarly commutes
with the central spin coupling, local operator growth spreading is inhibited
on the same decoherence timescale. The local channel is effectively turned off
by the slow operator decoherence. If the local interactions were orthogonal to
the central coupling, we expect different operator spreading results. Here we
take a simple extension of our model and change the form of the auxiliary
coupling to be Ising$-xx$ which we term a compass, ring-star Ising model. Now
the the auxiliary coupling projects the system into $\hat{X}$ subspace, which
no longer commutes with the local $-zz$ interactions. This should allow many-
body operators to propagate throughout but inhibit the growth of operator
strings containing $\hat{Y},\hat{Z}$. Translating the operator picture to the
entropy picture, we are allowing higher order many-body operators to develop
through local channels, so the intermediate time saturation value should be
significantly larger and scale extensively with system size, the
logarithmically slow growth of orthogonal operator strings should then extend
to long times as operators can decohere to larger regions of the Hilbert
space.

Figure 8: Information spreading in the transverse ring-star model
($\lambda\sum_{i}^{L-1}\sigma^{x}_{i}\sigma^{x}_{c},J=1.0$): (a) Early-time
entanglement entropy growth crosses over from ballistic to sub-ballistic with
increasing $\lambda$, plotted on a semi-log scale. Entropy depicts a
prethermal phase with entanglement that saturates around $t$, then exhibits
characteristic MBL, logarithmic growth $\propto\log(t)$ out to late times
$tJ=200$. Early-time entropy growth increases smoothly with $\lambda$ until
reaching a discontinuity at $\lambda\sim 1$. (c, d, e, f) OTOC for transverse
spin component for $\lambda={0,0.84,2.0,3.0}$, respectively. In contrast to
the parallel ring-star Ising model, the light cone becomes increasingly
visible as $\lambda$ increases. (c) In the nonintegrable, purely local case,
operator scrambling saturates following the light cone. As c-qubit coupling
grows, operators orthogonal to $\sigma^{x}$ rapidly fluctuate, so that
following the light cone, $\sigma^{x}$ operator weight dominates each site.
(e) The light cone boundary eventually becomes the only region where operators
are orthogonal to the c-qubit coupling and propagate under local Ising
interactions. System size $L=13$ and $J=1.0,h=1.05,g=0.45$.

In Fig. 8 we observe exactly this behavior. Noting that the magnitude of the
orthogonal field on the c-qubit changes from $h\rightarrow g=0.45$, the
inhibited operator growth goes as $\lambda_{x}/g^{2}$. The entanglement
entropy (a) following a quench from $|+y\rangle$ transitions from local
spreading, fast-spreading, to confined for $\lambda_{x}>0.5$. For times
$tJ<10$, the system exhibits ballistic operator growth regardless of coupling
but reaches an intermediate saturation value $S_{vN}\approx 3$ with slow
growth extending out to late times. The OTOC agrees identically, where we have
the same ballistic spreading lightcone with fully scrambled operators
following the wavefront (b). With increasing coupling (c) we observe a small
amount of operator weight that spreads super-ballistically across all sites, a
well defined wave-front, and a suppression in the OTOC behind the wavefront.
This becomes more dramatic deeper into the confined regime (d,e), where the
wavefront becomes the only region where operators orthogonal to $\hat{X}$ may
be found. As the local Ising interaction is orthogonal to the operator
subspace in which the c-qubit protects, the wavefront may freely propagate but
operators behind the wavefront are aliased to predominantly span strings of
$\hat{X}$ and $\hat{I}$ operators.

##  VII Conclusion

We have interrogated the quantum entanglement and operator dynamics in a
central spin-bath design. The local Ising chain acts as a structured,
thermalizing bath that when coupled to a nonlocal central qubit, is able to
rapidly scramble quantum information across the system up until the
information capacity of the c-qubit is quenched. When the central bit is
rapidly quenched in the strong coupling regime, it leads to operator
confinement in sectors of the Hilbert space spanned by $\hat{I}$ and operators
parallel to the central qubit coupling. Entanglement entropy grows slowly out
to late times as observed in effective infinite temperature states.

This model admits unique limits; where, for weak coupling it exhibits super-
ballistic OTOC spreading and fast-scrambling as supported by previous works
examining a similar star-graph structure, and a confined quantum Zeno-like
limit in which the complexity of operator growth is inhibited by extensive,
coherent interaction with the two level central mode. This confined regime
mirrors a larger class of models which admit prethermal or localization
physics: disorder driven many-body localization, projective or weak quantum
measurement, floquet-driven prethermalization. Our star-local work is similar
in that the c-qubit drives operator fluctuations on the leaves on a timescale
like $1/\lambda$ once $\lambda$ is greater than noncommutting fields on the
central site. In contrast, the qubit retains its infinite-range information
sharing capacity compared to the external drive scenario in floquet physics.
In the measurement theory sense, once the central qubit becomes nearly
maximally entangled with the Ising spin chain bath, its as if the bath
projects the central spin which in turn freezes the information sharing
dynamics with which the bath is entangled. And finally in the MBL case,
instead of local integrals of motion that are nearly integrable with
exponentially small overlap, here we have global operators orthogonal to the
auxiliary coupling that have a logarthimically slow thermalization timescale.

This work provides a novel mechanism for exploring a host of quantum
information dynamics. Here we outline the existence of a dynamical transition
that occurs under static Hamiltonian dynamics, and in future work it would be
fruitful to investigate the nature of the transition and how similar physics
can be observed in RUCs. Tracing out the central qubit to produce non-
Hermitian physics may similarly illuminate how this model relates to strong
periodic driving prethermalization. As the inverse problem, it would be
interesting if exotic non-Hermitian physics has a hidden unitary quantum
system analog, where multiple ancillary qubit degrees of freedom produce the
observed non-Hermitian dynamics. Future work should also seek to understand
how this transition persists with larger c-qudit towards an infinite bosonic
fields or multiple dissipative modes. It would be an interesting engineering
application if uniform coupling to a bosonic mode or central-qubit can protect
against dephasing errors in the environmental qubit platform.

##  VIII Acknowledgements

J.C.S and N.T. would like to thank Sumilan Banerjee, Chandrasekhar Ramanathan,
Brian Skinner, Xiaozhou Feng, Shi Feng, and Sayantan Roy for useful
discussions. This material is based upon work supported by the U.S. Department
of Energy, Office of Science, Office of Basic Energy Sciences under Award
Number DE-FG02-07ER46423. Computations were done using the QuSpin python
package [84] on the Unity cluster at the Ohio State University.

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Supplemental Material for "Entanglement Dynamics between Ising Spins and a
Central Ancilla"

##  I Operator Spreading on the Star Graph

In addition to the exactly solvable star-Ising model presented in the main
text, we present how the dynamics extend with a magnetic field applied to the
c-qubit and all other sites.

$H=\lambda\sum_{i}\sigma^{z}_{c}\sigma^{z}_{i}+h\sigma^{x}_{i}+h\sigma^{x}_{c}.$
(23)

The early-time behavior using the BCH expansion of the commutator and for
general ($h/\lambda$).

$\displaystyle[\sigma^{x}_{j}(t),\sigma^{x}_{i}]$
$\displaystyle=\sum_{m}^{\infty}\frac{(it)^{m}}{m!}[[H,\sigma^{x}_{i}]_{m},\sigma^{x}_{j}]$
$\displaystyle[A,B]_{m}$ $\displaystyle=[A,[A,B]_{m-1}];[A,B]_{0}=B$
$\displaystyle[H,\sigma^{x}_{i}]_{1}$
$\displaystyle=it[\sigma^{z}_{i},\sigma^{x}_{i}]=it\lambda\sigma^{z}_{0}\sigma^{y}_{i}$
$\displaystyle[H,\sigma^{x}_{i}]_{2}=\frac{(it^{2})}{2}[H,\lambda\sigma^{z}_{0}\sigma^{y}_{i}]$
$\displaystyle=\frac{-t^{2}}{2}(-\lambda^{2}\sigma^{x}_{i}+h\lambda\sigma^{y}_{0}\sigma^{y}_{i}).$
$\displaystyle[H,\sigma^{x}_{i}]_{3}=\frac{(it^{3})}{6}[H,...]$
$\displaystyle=\frac{-it^{3}}{6}(-\lambda^{3}\sigma^{z}_{0}\sigma^{y}_{i}+h\lambda^{2}(\sum_{j\neq
i}\sigma^{x}_{0}\sigma^{z}_{j}\sigma^{y}_{i}+\sigma^{x}_{0}\sigma^{x}_{i})-h^{2}\lambda\sigma^{z}_{0}\sigma^{y}_{i})$

In the limit $h/\lambda=0,\lambda=1$ we have:

$\displaystyle[\sigma^{x}_{j}(t),\sigma^{x}_{i}]$
$\displaystyle=[\sum_{m}\frac{(it)^{2m}}{2m!}\sigma^{x}_{i}+\frac{(it)^{2m+1}}{(2m+1)!}\sigma^{z}_{0}\sigma^{y}_{i},\sigma^{x}_{j}]$
$\displaystyle=[\cos(2t)\sigma^{x}_{i}-i\sin(2t)\sigma^{z}_{0}\sigma^{y}_{i},\sigma^{x}_{j}].$

Then the two-time correlator behaves as:

$\langle\sigma^{x}_{i}(t)|\sigma^{x}_{i}\rangle=\cos{2t}.$ (24)

And the OTOC goes as:

$C_{xx}(i,1,t)=2\sin^{2}(2t),$ (25)

strictly for $i\in{c,1}$. Under the operation of the central Ising
interaction, the operator weight is completely preserved to live in the
projective space as we have seen before. When adding a non-commuting term that
rotates states away from this space, how does this then modify growth on the
remaining sites? Looking back, we see that

$\displaystyle\langle\sigma^{x}_{j}(t)|\sigma^{x}_{i}\rangle\sim\mathcal{O}(1)\propto\frac{t^{3}}{6}h^{2}$
(26) $\displaystyle C_{xx}(j,i,t)\propto h^{2}t^{6}.$ (27)

As exhibited in Fig.S1, we see that for fixed $\lambda=1$, that $h^{2}t^{6}$
provides an excellent approximation to the early-time growth behavior of OTOC
on next nearest neighbors to the initial site. This rapid growth on the
central site and corresponding neighbors at early-times in essence captures
the extreme early-time entanglement growth that is observed to grow steadily
with growing $h/\lambda$. The interesting phenomena occurs when
$t\sim\frac{\pi}{2\lambda}$, where in Fig.S1(a) the operator space $q$ is
completely preserved, allowing $C(0,i,t)$ to decay to zero. On this same
timescale we see in (b, c) and most clearly in (d) that this leads to an
effective wait time that is proportional $\lambda$ and similarly corresponds
to the amount of operator weight outside of the space $q$ on the c-qubit.
Also, any weight operator weight transferred to edge spins then undergoes
similar oscillations seen in (a) during this wait time. In the weakly
decoherent regime $h<<\lambda$, the wait time leads to essentially linear OTOC
growth with oscillations that similarly grow in time.

Figure S1: OTOC $C_{xx}(i,j,t)$ for (a) $h=0$, (b) $h_{c}>0$, (c, d) $h>0$
with $L+1=13$, $\lambda=1$. (a) OTOC is only nonzero on the edge and central
c-qubit as coherent evolution under the star-coupling does not allow operators
to propagate beyond the c-qubit. (b,c) early-time growth behavior is
accurately captured by BCH expansion of the two-time correlator. (d) At
intermediate times, $t$, as operator weight transitions back to the initial
site, there is then essentially a waiting time that occurs before more
operator weight may grow on sites $j\neq[i,0]$. This leads to a polynomial
growth in time of the two-time correlators.

##  II Entanglement Spreading on the Star Graph

Figure S2: Information spreading star model $(J=0)$: Real-time evolution of
the $1/2-$chain entanglement entropy initialized in $|+y\rangle$ (a, b), and
OTOC following quench from a Haar random initial state. Information spreading
in the star-Ising model $(J=0)$: Real-time evolution of the the $1/2-$chain
entanglement entropy initialized in $|+y\rangle$ (a,b) and $C_{zz}(t,i-j)$
with initial state chosen randomly from the Haar distribution (c,d). (a)
Entanglement entropy plotted as a function of length for $\lambda=1.0$. Growth
is independent of system size for fixed $\lambda$ and saturates to a value
independent of system size. (b) Entanglement growth rate and steady state
value for $\lambda>h$ saturate and become independent of $\lambda$. (c, d)
OTOC $C_{zz}$ for $\lambda/\sqrt{L}=\\{0,0.78\\}$, respectively. (c) Initial
operator simply fluctuates under mixed-fields and does not propagate. (d) OTOC
grows on the c-qubit at a rate that grows with $\lambda$ after an initial wait
time that scales with $\text{min}[1/h,1/\lambda]$. Operator weight then
spreads to the remaining leaves after a time $2\ \text{min}[1/h,1/\lambda]$
corresponding to two processes of operator transfer that occur on the c-qubit.

Here we examine the corresponding entanglement entropy growth on the Ising
star graph. Starting from the same polarized product state used in the main
results ($|+y\rangle$) we plot the half-system entanglement entropy following
a quantum quench. We treat $\lambda$ as the variable and set $h,h_{c}=1.05$
and $g,g_{c}=0.45$. The entanglement entropy saturates to a length independent
value Fig.S2(a), constant for system sizes $L=11-19$. Though the mixture of
onsite fields allows local operators to explore the operator space of single
sites, the collective magnetic field and collective c-qubit interaction still
limit the system to behave semiclassically. In Fig.S2(b) we note that the
entanglement entropy growth rate and saturation value become independent for
$\lambda>1$. This is an interesting finding as we know in the large
$\lambda>>h$ limit that operators simply scramble between the initial site and
the cqubit. The corresponding entanglement entropy between halves is dominated
by the system-qubit entanglement as entanglement generated between spins grows
slowly. Once the system rapidly entangles with the cqubit, the entanglement
entropy of the system remains fixed but becomes uniformly distributed between
all sites. In the OTOC picture operators are scrambled with the cqubit and
then leak into the remaining sites.

##  III OTOC Growth vs. System Size Scaling

Figure S3: Finite size scaling of the coefficient of linear growth $\alpha
t^{\beta}$ (see Main text Fig.7) in $C{zz}(|i-j|=6,t)$ for system sizes
$L=[8,10,12,14]$. (Top) $\log[\alpha]$ decreases linearly with $\lambda$ as
exponentially less operator weight decoheres on all sites and spreads
throughout the system. (Bottom) Plot of $1/\alpha$ for better resolution of
the system size dependence. We observe a small decrease in $\alpha$ with
increasing system size, corresponding to the weak coefficient in
$\lambda_{c}\sim\sqrt{L}$.

In the main text we examined the longitudinal OTOC and found that it goes like
$C_{zz}(t)\approx e^{-\lambda}t$. In Fig.S3 we perform the same analysis and
vary the system size. We see that close to the transition $\lambda\leq 1$ that
the coefficient on OTOC growth is roughly independent of system size. For
increasing $\lambda$ we see in Fig.S3(top) that there is weak decrease in
$\log(\alpha)$ with system size. Looking at the $1/\alpha$ to highlight the
slight difference in system size, we see that into the localized operator
phase $\lambda>3-4$ that larger system sizes exhibit increasingly slow
operator growth.

##  IV Spectral Statistics of the Star-Ising Model

We calculate the spectral statistics for the central qubit Ising model studied
in the main text. These statistics provide insight into the random nature like
behavior of the density matrix or whether there are conservation laws present.
Anything beyond random statistics manifests as prethermalization, MBL, or
integrable behavior when a system is quench from a highly energetic state. We
calculate the spectral statistics using 100 states surrounding the middle of
the energy spectrum. The ratio is between two levels $r_{i}$ is calculated as

$r_{i}=\text{min}[\frac{E_{i}-E_{i-1}}{E_{i}+E_{i-1}},\frac{E_{i+1}-E_{i}}{E_{i}+E_{i+1}}]$
(28)

We then take the average spacing between all levels. We work in the $k=0$
sector as well as the paritiy conserving sector, as the translational
invariance leads to degeneracies between $k-$sectors when only one particular
sector is meaningful under dynamics. We take the tensor product of this Ising
spin-chain Hilbert space with the central qubit. In Fig.S4 we observe that the
only point in phase space that does not exhibit GOE random matrix statistics
$\langle r\rangle\approx 0.53$ as for $\lambda=0,g=0$, where we have the
integrable TFIM. Here the system exhibits sub-Poissonian statistics.

Figure S4:  Average adjacency level spacing of energy eigenvalues
$r(E_{i},E_{i+1})$ for the ring-star Ising Hamiltonian for $L+1=16$ with
periodic boundary conditions for fixed momentum and Z-reflection symmetry
blocks of the Hamiltonian. (a) Spacing as a function of $\lambda,g$ for fixed
$h$, (b) h, lambda, g = 0, and (c) $h,g,\lambda=0$. We find strong evidence of
the integrable case of $h\neq 0,g=0,\lambda=0$ in (b,c).