# Boosting information transfer in a quantum correlated medium

Finn Schmolke  Etienne Springer  Eric Lutz  Institute for Theoretical Physics
I, University of Stuttgart, D-70550 Stuttgart, Germany

###### Abstract

Sharing and receiving information plays a pivotal role in science and
technology. Quantum communication relies on the principles of quantum
mechanics to transmit information in a nonclassical manner. Existing quantum
communication protocols are commonly based on shared entangled states between
sender and receiver, while the transmitting medium is classical. We here
demonstrate that information transfer may be enhanced in a quantum correlated
medium without entanglement distribution. We concretely show that nonclassical
correlations, with nonzero discord, between the first two spins of a spin
chain that acts as a quantum wire can increase the information flow and reduce
the propagation time. We relate this effect to the breaking of the spatial
symmetry of the out-of-time-order correlator that characterizes the spread of
information through the medium.

Communication is based on the transfer of information between one point and
another bri65 ; ger00 ; yu01 ; fra18 . A notable observation is that both
communication and information are inherently physical, since information is
transmitted by physical means, through a physical medium, for instance, in the
form of an electric current or a light wave bri65 ; ger00 ; yu01 ; fra18 .
They are therefore subjected to the laws of physics, in particular, to those
of quantum theory. Quantum mechanics has been shown to restrict the flow of
entropy/information, that is, the number of bits sent per unit time leb66 ;
bow67 ; pen83 ; bek88 ; bek90 ; yue92 ; cav94 ; ble00 . For a single channel,
the entropy flux $\dot{I}$ is upper bounded by the energy current $\dot{E}$
according to $\dot{I}^{2}\leq{\pi\dot{E}/3\hbar}$ leb66 ; bow67 ; pen83 ;
bek88 ; bek90 ; yue92 ; cav94 ; ble00 . There is hence a minimum energy cost
per bit associated with communication. This bound is believed to be
independent of the physical properties of the transmitting medium, and thus
universal ble00 . It is intimately related to the existence of quantum speed
limits that impose strong constraints on the evolution time of any quantum
system def17 .

We here show that nonclassical correlations may boost entropy currents in a
medium. Quantum physics, thus, not only fundamentally constrains information
flow, it can also enhance it. We focus in the following on information
transfer along a spin chain that acts as a quantum wire bos23 ; chr04 ; fit06
; kay07 ; cap07 ; fra08 ; ban11 ; yao11 ; god12 ; yao12 ; ajo13 ; sah15 ;
mar16 ; bos07 ; nik14 ; cha16 . Whereas photons are ideal carriers of
information over long distances, it is not straightforward to convert
information between two physically different photonic qubits. Spin chains are
a prominent, solid-state alternative for short- and mid-range communication,
for example, to connect quantum processors or quantum registers bos23 ; chr04
; fit06 ; kay07 ; cap07 ; fra08 ; ban11 ; yao11 ; god12 ; yao12 ; ajo13 ;
sah15 ; mar16 ; bos07 ; nik14 ; cha16 . In these systems, information is
transmitted dynamically from one end of a chain to the other, without the
requirement for any external control. We consider for concreteness the perfect
state transfer protocol introduced in Ref. chr04 , and implemented
experimentally in Ref. cha16 , where the magnetic interactions between
neighboring spins are engineered such that information is transferred with
unit fidelity. We demonstrate that the transmission of a qubit end-state along
the spin chain is significantly improved in the presence of nonclassical
correlations, with nonzero discord oll01 ; hen01 ; mod12 ; ber17 ; hu18 ,
between only the first two spins of the chain, that is, sender and receiver
are initially uncorrelated. This enhancement is associated with a reduced
arrival time and a larger value of the maximum entropy flux. We additionally
provide a sufficient condition for this increase of communication rate to
occur, and illustrate it with a general class of two-qubit $X$ states that can
be treated analytically yu07 ; ali10 ; che11 ; que12 . It is worth emphasizing
that this quantum improvement strongly differs from usual quantum
communication protocols, such as quantum key distribution, quantum
teleportation and super-dense coding, that usually rely on a shared entangled
state between sender and receiver gis07 .

(a)(c)(b)(d)$\times 10^{-2}$ Figure 1: Quantum enhanced information transfer
along an $XY$ spin chain. a) Perfect state transfer with unit fidelity
$F(\rho^{0}_{1},\rho^{t}_{N})$ between the edge spins, 1 and $N$, happens
faster with local quantum correlations between the first two spins (time
$\tau_{\mathrm{c}}$, red) than without (time $\tau_{\mathrm{u}}$, blue). b)
The information-energy flow inequality, $\dot{I}^{2}\leq{\pi\dot{E}/3}$, is
obeyed in both cases, and the maximum currents, ${\dot{I}}_{\text{max}}$ and
$\dot{E}_{\text{max}}$, are increased by the nonclassical correlations
(arrows). c) In the absence of correlations, information and energy of the
receiving qubit increase monotonically to the values of the sender qubit,
$I=\ln 2$ and $E=0$, at the same time $\tau_{\mathrm{u}}$. In the presence of
correlations, information and energy reach their maxima more rapidly. However,
energy is delayed and overshoots the sender value before attaining it at time
$\tau_{\mathrm{u}}$, after reflection at the end of the chain. d) Out-of-time-
order correlator $C(x,3,t)$ that characterizes the propagation of information
between spins 3 and $x$ at three different times, $\tau_{\mathrm{u}}/4$,
$\tau_{\mathrm{u}}/2$ and $3\tau_{u}/4$ (arrows). Nonclassical correlations
break its spatial symmetry, shifting it toward the wave front that moves with
the Lieb-Robinson velocity, therefore boosting the entropy flux. Parameters
are $\omega=1$, $\beta=0$, $J=1$ and $N=10$.

_Quantum correlated spin chain._ We consider a quantum wire consisting of a
linear $XY$ chain with $N$ qubits and inhomogeneous nearest-neighbor
interactions chr04

$\displaystyle H$
$\displaystyle=\omega\sum\limits_{j=1}^{N}\sigma^{z}_{j}+\sum\limits_{k=1}^{N-1}{J_{k}}(\sigma^{x}_{k}\sigma^{x}_{k+1}+\sigma^{y}_{k}\sigma^{y}_{k+1}),$
(1)

where $\omega$ is the level spacing and
$J_{k}=({\lambda}/{4})\sqrt{k\cdot(N-k)}$ are mirror-symmetric interaction
constants between adjacent qubits, with coupling strength $\lambda$ (we put
$\hbar=1$). We choose $\lambda=4J/N$, such that the maximum coupling constant
is $\mathrm{max}\\{J_{k}\\}=J$, independent of $N$, where $J$ sets the
characteristic energy scale chr05 ; com . The first spin $(j=1)$ acts as the
sender and the last one $(j=N)$ as the receiver. Information is transmitted
along the wire by dynamically transferring the state $\rho^{t=0}_{1}$ of the
first spin to the state $\rho^{t=\tau_{\mathrm{u}}}_{N}$ of the last spin in
time $\tau_{\mathrm{u}}$. The couplings $J_{k}$ are engineered such that after
time $\tau_{\mathrm{u}}=\pi/\lambda$, state transfer is perfect, with unit
fidelity,
$F(\rho^{0}_{1},\rho^{\tau_{\mathrm{u}}}_{N})=\text{Tr}\big{[}{\scriptstyle\sqrt{\sqrt{\rho^{0}_{1}}\rho^{\tau_{\mathrm{u}}}_{N}\sqrt{\rho^{0}_{1}}}}\big{]}^{2}=1$
chr04 . For instance, if the first spin is initially in the excited state
$|1\rangle\langle 1|$ and all the remaining spins are in their ground state
$|0\rangle\langle 0|$, then, after time $\tau_{\mathrm{u}}$, all the qubits
will be in their ground state, except the last one which will be in the
excited state. The time $\tau_{\mathrm{u}}$ has been shown to correspond to
the minimal possible evolution time, hence, to the quantum speed limit of
perfect state transfer in one dimension yun06 .

Following Refs. leb66 ; bow67 ; pen83 ; bek88 ; bek90 , we begin by taking the
state of the first qubit to be thermal at inverse temperature $\beta$,
$\rho_{1}^{0}=\rho^{\mathrm{th}}_{1}=\exp(-\beta\omega\sigma^{z}_{1})/Z$, with
partition function $Z$ (we shall relax this assumption later on). In this
case, the information current is given by the von Neumann entropy rate (in
bits) leb66 ; bow67 ; pen83 ; bek88 ; bek90 . Instantaneous information and
energy fluxes at the receiver are accordingly

$\displaystyle\\!\dot{I}=-\partial_{t}\Tr[\rho^{t}_{N}\log(\rho^{t}_{N})]\,\,\,\text{
and }\,\,\,\dot{E}=\partial_{t}\Tr[\rho^{t}_{N}\omega\sigma^{z}_{N}].$ (2)

We further prepare the second spin $(j=2)$ in a thermal state at the same
inverse temperature,
$\rho_{2}^{0}=\rho^{\mathrm{th}}_{2}=\exp(-\beta\omega\sigma^{z}_{2})/Z$ (we
shall again relax this assumption in the following), and initialize the
remaining qubits in their respective ground states. The total initial state is
then of the form
$\rho(0)=\rho^{0}_{12}\otimes\outerproduct{0}{0}^{\otimes(N-2)}$.

We examine two different scenarios: (i) in the first case, the first two
qubits are uncorrelated,
$\rho^{0}_{12,\mathrm{u}}=\rho_{1}^{\mathrm{th}}\otimes\rho_{2}^{\mathrm{th}}$,
whereas (ii) in the second case, they are quantum correlated,
$\rho^{0}_{12,\mathrm{c}}=\rho^{\mathrm{th}}_{1}\otimes\rho^{\mathrm{th}}_{2}+\chi$,
with a correlation term
$\chi=-i\alpha(\sigma^{+}_{1}\sigma^{-}_{2}-\sigma^{-}_{1}\sigma^{+}_{2})$,
with amplitude $\alpha$ and spin ladder operators
$\sigma_{j}^{\pm}=\sigma_{x}\pm i\sigma_{y}$. This correlated state is such
that the respective reduced density operators coincide with the thermal
density matrices of the uncorrelated state mic19 ; this allows one to directly
compare the two situations. The parameter $\alpha$ controls the amount of
quantum correlations; it is upper bounded,
$\absolutevalue{\alpha}\leq{1}/{(4\cosh^{2}\beta)}\leq{1}/{4}$, to ensure
positivity of the density matrix. The state $\rho^{0}_{12,\mathrm{c}}$ has
nonzero geometric quantum discord defined as
$D_{\mathrm{g}}(\rho)=\min_{\sigma\in\mathcal{C}}2\|\rho-\sigma\|^{2}$, where
$\mathcal{C}$ is the set of all classically correlated states mod12 ; ber17 ;
hu18 . A nonzero value of the geometric discord indicates nonclassical
correlations dak10 ; gir12 . We find that the geometric quantum discord
increases quadratically with the amplitude of the correlation term,
$D_{\mathrm{g}}(\rho^{0}_{12,\mathrm{c}})=2\alpha^{2}$.

The presence of initial nonclassical correlations strongly affects the
propagation of information and energy in the medium. Figure 1 examines the
impact of local quantum correlations on information transmission along the
$XY$ spin chain (1). Figure 1a displays the fidelity
$F(\rho^{0}_{1},\rho^{t}_{N})$ between emitter and receiver as a function of
time for $N=10$ qubits and $\alpha=1/4$. While the first spin state is exactly
transmitted to the final qubit in time $\tau_{\mathrm{u}}$ for uncorrelated
initial states (blue), perfect state transfer is achieved much faster, after
time $\tau_{\mathrm{c}}$, for correlated initial conditions (red). The time
$\tau_{\mathrm{u}}$, therefore, no longer corresponds to the ultimate quantum
speed limit, as for the uncorrelated state yun06 . Interestingly, faithful
transfer occurs a second time at time $\tau_{\mathrm{u}}$. This enhanced
transmission is accompanied by an increase of the maximum information flux
${\dot{I}}_{\text{max}}$, as well as of the maximum energy current
$\dot{E}_{\text{max}}$ (arrows in Fig. 1b). We will see below that those
boosts are directly related to the amount of quantum correlations in the
initial state. We also observe that the information-energy flow inequality,
$\dot{I}^{2}\leq\pi\dot{E}/3$, is obeyed in both instances for
$t\leq\tau_{\mathrm{u}}$, respectively $t\leq\tau_{\mathrm{c}}$ (Fig. 1b).
After full information transfer, the roles of sender and receiver are
switched, and the currents are reversed. We mention that the fidelity, as well
as the information and energy fluxes can be evaluated analytically for the
spin chain (1) with the help of the Jordan-Wigner transformation (Supplemental
Material).

Figure 1 further shows the temporal evolution of the information $I$ and of
the energy $E$ of the receiving qubit $\rho_{N}$. In the uncorrelated case,
the two quantities reach the initial values of the sender, $I=\ln 2$ and
$E=0$, simultaneously at time $\tau_{\mathrm{u}}$. By contrast, information
and energy display a different time dependence in the presence of
correlations. The von Neumann entropy behaves similarly to the fidelity
$F(\rho^{0}_{1},\rho^{t}_{N})$, attaining its initial, and maximum, value at
time $\tau_{\mathrm{c}}$. However, the energy overshoots the initial energy of
the sender, reaching its maximum after $\tau_{\mathrm{c}}$, until it gets
reflected at the end of the chain and flows back such that it coincides with
the initial energy of the sender at time $\tau_{\mathrm{u}}$, like in the
uncorrelated case. This remarkable behavior is an immediate consequence of the
initial quantum correlations.

_Enhanced information transmission_. In order to quantitatively analyze the
boosted transfer of information along the quantum wire, we next use the out-
of-time-order correlator (OTOC) swi18 ; lew19 ; gar23 . We specifically
consider the expectation value of the squared commutator of two local,
spatially separated (Hermitian) operators $W(x,t)$ and $V(y,0)$,
$C(x,y,t)={\langle[W(x,t),V(y,0]^{2}\rangle}=2-2\mathrm{Re}[F(x,y,t)]$ for
unitary dynamics lar69 ; swi18 ; lew19 ; gar23 . The out-of-time-order
correlator, $F(x,y,t)=\langle W(x,t)V(y,0)W(x,t)V(y,0)\rangle$, characterizes
the spread of quantum information along the spin chain, commonly referred to
as information scrambling swi18 ; lew19 ; gar23 . For concreteness, we choose
the two single-site Pauli operators $W(x,t)=\sigma^{z}_{x}(t)$ and
$V(y=3,0)=\sigma^{z}_{3}(0)$; the time-evolved Heisenberg operator $W(x,t)$
can thus be regarded as a local probe (at site $x$ and time $t$) of the
information emitted by the (correlated) sender at $t=0$.

Figure 1d compares the stroboscopic, wave-like evolution of the correlation
function $C(x,3,t)$ at three different times, $\tau_{\mathrm{u}}/4$,
$\tau_{\mathrm{u}}/2$ and $3\tau_{\mathrm{u}}/4$, for uncorrelated (blue) and
correlated (red) initial states; we have evaluated $C(x,3,t)$ numerically for
a chain of $N=250$ spins lui17 ; lin18 ; bao20 . The correlator exhibits three
peaks that travel from left to right with, in both instances, the same
wavefront which moves close to the Lieb-Robinson velocity
$v_{\mathrm{LR}}=2J$. The latter velocity provides a fundamental upper bound
to the spread of information in locally interacting quantum systems lie72 .
Information can hence not propagate faster than $v_{\mathrm{LR}}$. However, we
note that the correlator is no longer spatially symmetric in the presence of
quantum correlations (red). The weight of the distribution $C(x,3,t)$ is
indeed shifted towards the wave front, with higher peaks near the wave front
compared to the case without initial correlations. For example, at
$t=\tau_{\mathrm{u}}/4$, quantum correlations increase the mean
$\overline{x}=\int dx\,x\,C(x,3,\tau_{\mathrm{u}}/4)/\int
dx\,C(x,3,\tau_{\mathrm{u}}/4)$ by $8.15\%$ from 37.41 to 40.46 for the
parameters of Fig. 1d. As a consequence, more information flows per unit time
towards the receiving end of the chain, giving rise to the faster state
transfer seen in Fig. 1abc. Symmetries of the out-of-time-order correlator are
further elaborated on in the Supplemental Material.

Let us now investigate the role of initial nonclassical correlations in more
detail. Figure 2 clearly indicates that both the amplified maximum information
flow, $\dot{I}_{\text{c,max}}/\dot{I}_{\text{u,max}}$ (Fig. 2a) and the
enhanced transfer time, $\tau_{\mathrm{u}}/\tau_{\mathrm{c}}$ (Fig. 2b),
monotonically increase with the amount of quantum correlations, quantified by
the geometric discord of the initial state
$D_{\mathrm{g}}(\rho^{0}_{12,\mathrm{c}})$ (dark red, dashed). We may hence
conclude that nonclassical correlations are a local physical resource that can
boost communication rates in a wire. Interestingly, the state
$\rho^{0}_{12,\mathrm{c}}$ is separable with zero concurrence woo98  for all
values of $\alpha$. We observe, in fact, no information transmission boost
when a maximally entangled Bell state is used (Supplemental Material). This
quantum advantage is therefore closer to the deterministic quantum computation
with one qubit (DQC1) model kni98 , where the quantum resource for
computational speedup is associated with discord, and not with entanglement
dat08 ; lan08 ; wan19 .

(a)(b) Figure 2: Boosted information flow and transfer time. a) Maximum
information current ratio, $\dot{I}_{\text{c,max}}/\dot{I}_{\text{u,max}}$,
and b) transfer time ratio, $\tau_{\mathrm{u}}/\tau_{\mathrm{c}}$, as a
function of the geometric quantum discord $D_{\mathrm{g}}$ of the $X$ state,
Eq. (4), for various values of the chain length $N$ (solid lines). Both
quantities increase monotonically with the amount of quantum correlations,
which does not need to be entangled. The dashed line represents the correlated
thermal state of Fig. 1. Purple diamonds display the shift toward the wave
front, $\Delta\bar{x}=\bar{x}_{\mathrm{c}}-\bar{x}_{\mathrm{u}}$, of the
averaged out-of-time-order correlator $C(x,3,\tau_{\mathrm{u}}/4)$ induced by
the nonclassical correlations. Parameters for the $X$ state are $a=d=w=0$,
$b=c=1/2$ and $z=i\alpha$.

_Criterion for enhanced transmission_. The above effect is not restricted to
correlated thermal states or to $XY$ spin chains. We next consider a quantum
spin chain of the form $H=\omega\sum_{j}\sigma^{z}_{j}+H_{\text{int}}$ with
general coupling Hamiltonian $H_{\text{int}}$, and an initial correlated state
$\rho^{0}_{12,\mathrm{c}}={\rho_{\mathrm{u}}+\chi}$ with diagonal part
$\rho_{\mathrm{u}}$. We furthermore separate the unitary time evolution into
uncorrelated and correlated parts,
$\rho_{\mathrm{c}}(t)=\rho_{\mathrm{u}}(t)+U\big{(}\chi\otimes\outerproduct{0}{0}^{\otimes(N-2)}\big{)}U^{\dagger}$
with $U=\exp(-iHt)$. In order to boost state transfer, we demand that, at
short times, more probability is shifted towards the state
$\ket{\psi_{2}}=\ket{01}\otimes\ket{0}^{\otimes(N-2)}$ than in the
uncorrelated case, that is,
$\bra{\psi_{2}}\dot{\rho}_{\mathrm{c}}(\differential{t})\ket{\psi_{2}}>\bra{\psi_{2}}\dot{\rho}_{\mathrm{u}}(\differential{t})\ket{\psi_{2}}$.
Since the local initial states are equal in both cases, this condition implies
that $\bra{01}[H_{\mathrm{int}},\chi]\ket{01}>0$ (Supplemental Material). Note
that only the interaction Hamiltonian matters because the diagonal part does
not connect different sites of the chain. A necessary requirement for enhanced
transmission is therefore

$\displaystyle\left[H_{\mathrm{int}},\chi\right]\neq 0.$ (3)

Boosted entropy flux is thus generally possible whenever the correlation term
$\chi$ does not commute with the interaction Hamiltonian $H_{\text{int}}$. The
latter criterion underscores the quantum origin of the phenomenon.

As an illustration, we consider the quantum $XY$ chain (1) with a correlated
state given by an $X$ state yu07 ; ali10 ; che11 ; que12

$\displaystyle\rho^{0}_{12,\mathrm{c}}=\rho_{X}=\begin{pmatrix}a&0&0&w\\\
0&b&z&0\\\ 0&z^{\ast}&c&0\\\ w^{\ast}&0&0&d\end{pmatrix},$ (4)

with $|z|\leq\sqrt{bc}$ and $|w|\leq\sqrt{ad}$. Such X states define a general
class of quantum correlated bipartite qubit states that, for instance, include
Werner states and Bell states yu07 ; ali10 ; che11 ; que12 . The corresponding
reduced state of the first spin to be transferred is
$\rho^{0}_{1}=\Tr_{2}[\rho_{X}]=\text{diag}(a+b,c+d)$. Condition (3) requires
that $|z|\neq 0$, while the entries ($w,w^{\ast},a,d$), that belong to the
commuting part, may be chosen arbitrarily. The concurrence of the $X$ states
is given by $C(\rho_{X})=2\,\mathrm{max}\\{0,z-\sqrt{ad},w-\sqrt{bc}\\}$ yu07
; ali10 ; che11 ; que12 , whereas their geometric discord reads
$D_{\mathrm{g}}(\rho_{X})=\min\\{4(w^{2}+z^{2}),(a-c)^{2}+(b-d)^{2}+2(w+z)^{2}\\}/2$
yu07 ; ali10 ; che11 ; que12 . $X$ states are hence entangled if and only if
$bc<|w|^{2}$ or $ad<|z|^{2}$, but both conditions cannot be satisfied
simultaneously ali10 . States with $w=w^{\ast}=0$ and $ad=|z|^{2}=\alpha^{2}$,
with $\alpha<1/4$ (like the thermal correlated states considered before) are
thus not entangled. However, in general $D_{\mathrm{g}}(\rho_{X})=2\alpha^{2}$
for $z=i\alpha$, $\alpha\leq 1/2$, revealing the presence of nonclassical
correlations.

Figure 2 presents the amplified maximum information flow,
$\dot{I}_{\text{c,max}}/\dot{I}_{\text{u,max}}$ (Fig. 2a), and the enhanced
transfer time, $\tau_{\mathrm{u}}/\tau_{\mathrm{c}}$ (Fig. 2b), for the X
state (4) (solid lines) as a function of the geometric quantum discord
$D_{\mathrm{g}}(\rho_{X})$, for $N=5,20$ and $100$ spins, when $\alpha$ is
varied. As for the thermal correlated state,
$\rho^{0}_{12,\mathrm{c}}=\rho^{\mathrm{th}}_{1}\otimes\rho^{\mathrm{th}}_{2}+\chi$,
both quantities grow monotonically with the amount of nonclassical
correlations. However, since $D_{\mathrm{g}}(\rho_{X})$ can take larger values
when the $X$ state is entangled, the quantum enhancement is more pronounced.
For example, for the $N=11$ sites of the state transfer experiment reported in
Ref. cha16 , a maximal geometric discord of 1/2 would increase the maximal
information flux by $32.6\%$ and decrease the transfer time by $18.7\%$.
Figure 2b additionally shows that these boosts are accompanied by an increased
shift toward the wavefront,
$\Delta\bar{x}=\bar{x}_{\mathrm{c}}-\bar{x}_{\mathrm{u}}$, of the averaged
out-of-time-order correlator $C(x,3,\tau_{\mathrm{u}}/4)$ (purple diamonds),
confirming that this effect finds its origin in the breaking of the spatial
symmetry of the out-of-time-order correlator by the quantum correlations. The
increased maximum entropy current and transfer time both exhibit a square-root
dependence on the geometric quantum discord $D_{\mathrm{g}}$, that is, a
linear dependence of the correlation strength $\alpha$. We additionally note
that due to the singular derivative of the square root at the origin, only a
small amount of quantum discord is required to achieve a significant
transmission improvement.

Conclusions. Entanglement is a fundamental resource for quantum applications
that outperform their classical counterparts gis07 . However, entangled states
are a costly and fragile resource that is difficult to prepare and maintain
over long distances, owing to the detrimental effect of decoherence sch07 . We
have shown that information propagation can be enhanced in a quantum
correlated medium without requiring entanglement distribution between emitter
and receiver. Remarkably, local nonclassical correlations with nonzero discord
between only the first two qubits at the emitter suffice to boost the
information flow for spin chains of arbitrary length. While information cannot
propagate faster than the Lieb-Robinson velocity, the breaking of the spatial
symmetry of the out-of-time-order correlator induced by the quantum
correlations leads to an augmented entropy current. Our results indicate that
quantum correlated media are a useful resource for quantum enhanced
communication.

Acknowledgements. We acknowledge financial support from the Vector Foundation
and from the German Science Foundation (DFG) under project FOR 2724.

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Supplemental Material: Boosting information transfer in a quantum correlated
medium

In this Supplemental Material, we will (I) present the analytical evaluation
of the time evolution of the local populations using the Jordan-Wigner
transformation, (II) derive the condition for faster state transfer, (III)
analyze the out-of-time-order correlator of the perfect state transfer and the
uniformly coupled quantum spin chain and (IV) show that there is no
information boost for Bell states.

##  I Analytical results

###  I.1 Jordan-Wigner transformation

In this section we compute the local populations of the spin chain and from
them derive the expressions for the state transfer fidelity and the von
Neumann entropy. Our starting point is the perfect state transfer Hamiltonian
for a chain of $N$ qubits with nearest neighbor interaction, Eq. (2) of the
main text,

$\displaystyle
H=\omega\sum_{j=1}^{N}\sigma^{z}_{j}+\sum_{k=1}^{N-1}{J_{k}}(\sigma^{x}_{k}\sigma^{x}_{k+1}+\sigma^{y}_{k}\sigma^{y}_{k+1})\quad\mbox{\quad
with\quad}J_{k}$ $\displaystyle=\frac{\lambda}{4}\sqrt{k\cdot(N-k)},$ (S1)

where $\omega$ is the level spacing of the qubits and the $J_{k}$ are the
interaction constants between adjacent qubits. Let $\rho(0)$ be the composite
initial state of the full qubit chain. The interactions $J_{k}$ are engineered
in such a manner that after a constant time $\tau_{\mathrm{u}}=\pi/\lambda$
(which we consider the final time of the process), the state of the chain
corresponds to the mirror image of its initial configuration (the initial
local states of the individual qubits are swapped with their mirror image
states) Schr04 ; Schr05 . To avoid unphysical Hamiltonians and violation of
causality, we normalize the interaction constants such that their maximum
value is bounded. Specifically, we set $\lambda=4J/N$, so that
$\mathrm{max}\\{J_{k}\\}=J$, independent of $N$ Schr05 . The perfect transfer
time then scales linearly with the system size $N$, and we obtain
$\tau_{\mathrm{u}}=N\pi/(4J)$.

The initial states considered in the main text can be written in the form

$\displaystyle\rho(0)=\rho^{0}_{12}\otimes\outerproduct{0}{0}^{\otimes(N-2)},$
(S2)

where

$\displaystyle\rho^{0}_{12}=\rho_{X}=\begin{pmatrix}a&0&0&w\\\ 0&b&z&0\\\
0&z^{\ast}&c&0\\\ w^{\ast}&0&0&d\end{pmatrix},$ (S3)

is an $X$ state Syu07 ; Sali10 ; Sche11 ; Sque12 . Concretely, we consider
initial states that can be parametrized as

$\displaystyle\rho^{0}_{12}=\rho_{1}\otimes\rho_{2}+\chi,$ (S4)

with

$\displaystyle\rho_{j}=\begin{pmatrix}p_{j}&0\\\
0&1-p_{j}\end{pmatrix}\quad\text{and}\quad\chi=-i\alpha(\sigma^{+}_{1}\sigma^{-}_{2}-\sigma^{-}_{1}\sigma^{+}_{2}),$
(S5)

with real parameter $\alpha\in[0,1/4]$. For any $\alpha\neq 0$, these states
have non-zero geometric quantum discord Syu07 ; Sali10 ; Sche11 ; Sque12

$\displaystyle
D(\rho_{X})=\frac{1}{2}\min\\{4(w^{2}+z^{2}),(a-c)^{2}+(b-d)^{2}+2(w+z)^{2}\\},$
(S6)

but are not necessarily entangled, since their concurrence is given by Syu07 ;
Sali10 ; Sche11 ; Sque12

$\displaystyle C(\rho_{X})=2\mathrm{max}\\{0,z-\sqrt{ad},w-\sqrt{bc}\\}.$ (S7)

For initial states of the form Eq. S5, the evolution of the local populations
can be conveniently calculated by performing a Jordan-Wigner transformation
Stak99 . The transformed Hamiltonian reads

$\displaystyle
H_{\mathrm{JW}}=\omega\sum_{k=1}^{N}(2c^{\dagger}_{k}c_{k}-\mathds{1})+2\sum_{k=1}^{N-1}J_{k}(c^{\dagger}_{k}c_{k+1}+c^{\dagger}_{k+1}c_{k}),$
(S8)

where $c^{\dagger}_{k}$ and $c_{k}$ are the respective fermionic ladder
operators. They are related to the usual Pauli operators by

$\displaystyle
c_{k}=\exp(i\pi\sum^{k-1}_{n=1}c^{\dagger}_{n}c_{n})\sigma^{-}_{k},\qquad
c_{j}^{\dagger}=\exp(i\pi\sum^{k-1}_{n=1}c^{\dagger}_{n}c_{n})\sigma^{+}_{k}.$
(S9)

If we restrict ourselves only to the evolution of the local populations and
nearest-neighbor correlations, we obtain an effective von Neumann equation

$\displaystyle\dot{Z}(t)=i[\Omega,Z(t)],$ (S10)

for the correlations
$Z_{jk}(t)=\langle\sigma^{+}_{j}\sigma^{-}_{k}\rangle(t)$, representing the
populations on the diagonal $Z_{jj}(t)=p_{j}(t)$, and nearest-neighbor
correlations $Z_{j,j\pm 1}=\langle\sigma^{+}_{j}\sigma^{-}_{j\pm 1}\rangle(t)$
on the off-diagonals. The matrix $\Omega=\mathrm{diag}(2{\bf\it
J},\omega,2{\bf\it J})$, where ${\bf\it
J}=(J_{1},J_{2}\ldots,J_{N-1})^{\mathrm{T}}$, acts as an effective tridiagonal
Hamiltonian with the interaction strengths on the off-diagonals and the qubit
level spacing $\omega$ on the diagonal.

###  I.2 One-excitation subspace

The class of initial states can be further extended beyond Eq. S4 by
considering initial states that lie in the single excitation sector, i.e. in
the subspace that contains only one spin excitation. Since the total
magnetization in the $z$-direction is conserved,
$[H,\sum_{k}\sigma^{z}_{k}]=0$, the Hamiltonian can be decomposed into
mutually orthogonal subspaces corresponding to a fixed number of excitations
and the evolution inside each subspace is consequently independent from the
remaining Hilbert space. If we denote by

$\displaystyle P_{1}=\sum_{k}\outerproduct{1_{k}}{1_{k}}$ (S11)

the projector on the single excitation sector with

$\displaystyle\ket{1_{k}}\in\\{\ket{10\cdots 0},\ket{010\cdots
0},\ket{0010\cdots 0},\ldots,\ket{0\cdots 01}\\},$ (S12)

then the projection of the initial $X$ state Eq. S3 becomes

$\displaystyle P_{1}\rho_{X}P_{1}=\begin{pmatrix}b&z\\\
z^{\ast}&c.\end{pmatrix}$ (S13)

We choose $z=i\alpha$, where $\alpha\in[0,1/2]$ can now be twice as large as
before. The one excitation sector thus supports only $X$ states for which
$a=d=w=0$.

The Hamiltonian inside this subspace becomes

$\displaystyle P_{1}HP_{1}=\Omega,$ (S14)

which is equal to the adjacency matrix of the physical graph corresponding to
the spin chain and it coincides with the effective Hamiltonian of the Jordan-
Wigner transform, Eq. S10. The local populations on the full Hilbert space can
be written as

$\displaystyle p_{j}(t)=\Tr[\rho(t)\Pi_{j}],\quad\text{with
projectors}\quad\Pi_{j}=\mathds{1}^{\otimes j-1}\otimes\begin{pmatrix}1&0\\\
0&0\end{pmatrix}\otimes\mathds{1}^{\otimes N-j}.$ (S15)

In the single excitation sector, the projector $\Pi_{j}$ reduces to

$\displaystyle P_{1}\Pi_{j}P_{1}=\outerproduct{j}{j},$ (S16)

such that the populations coincide with the ones obtained from the Jordan-
Wigner transform

$\displaystyle p_{j}(t)=Z_{jj}(t),$ (S17)

following from the same effective von Neumann equation (S10). Both approaches,
the Jordan-Wigner transformation and projecting onto the single excitation
sector, thus lead to the same evolution equation for the local populations
differing only in the permitted values for $\alpha$.

###  I.3 Local populations

For initial states that either lie in the single excitation subspace, Eq. S13,
or can be parametrized as Eq. S4, we accordingly only need to consider the
effective von Neumann equation Eq. S10. The solution reads

$\displaystyle Z(t)={e^{i\Omega t}}Z(0)e^{-i\Omega t},$ (S18)

where the time evolution operator Schr04

$\displaystyle U(t)={e^{i\Omega t}}=\exp(i\lambda S_{x}t)$ (S19)

represents the rotation of a spin $(N-1)/2$ particle around the $x$-axis.

In this description, the initial states take the form

$\displaystyle Z(0)=\begin{bmatrix}p_{1}(0)&i\alpha&\cdots&0\\\
-i\alpha&p_{2}(0)&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&0\end{bmatrix}$ (S20)

corresponding to the initial $X$ state in Eq. (14) of the main text. The
populations then follow from the diagonal elements of $Z(t)$, which can be
straightforwardly computed in terms of the matrix elements of the unitary
(S19)

$\displaystyle
p_{k}(t)=Z_{kk}(t)=p_{1}|U_{k1}|^{2}+p_{2}|U_{k2}|^{2}-2\alpha\mathrm{Im}[U_{k1}^{\ast}U_{k2}].$
(S21)

We are in particular interested in the population of the last spin

$\displaystyle
p_{N}(t)=Z_{NN}(t)=p_{1}|U_{N1}|^{2}+p_{2}|U_{N2}|^{2}-2\alpha\mathrm{Im}[U_{N1}^{\ast}U_{N2}].$
(S22)

The matrix elements $U_{kl}$ follow from the Wigner $d$-matrix Swig12

$\displaystyle
d^{j}_{m^{\prime}m}(\theta)=[(j+m^{\prime})!(j-m^{\prime})!(j+m)!(j-m)!]^{1/2}\sum_{s=s_{\mathrm{min}}}^{s_{\mathrm{max}}}\left[\frac{(-1)^{s}i^{m-m^{\prime}}\cos(\frac{\theta}{2})^{2j+m-m^{\prime}-2s}\sin(\frac{\theta}{2})^{m^{\prime}-m+2s}}{(j+m-s)!s!(m^{\prime}-m+s)!(j-m^{\prime}-s)!}\right],$
(S23)

where $s_{\mathrm{min}}=\mathrm{max}\\{0,m-m^{\prime}\\}$ and
$s_{\mathrm{max}}=\mathrm{min}\\{j+m,j-m^{\prime}\\}$ and $\theta=\lambda t$.
Equation S19 represents the elements of the rotation matrix of a spin
$j=(N-1)/2$ particle with magnetizations running over
$m=-j,-j+1,\ldots,j-1,j$. We therefore have

$\displaystyle U_{kl}$
$\displaystyle=d^{(N-1)/2}_{\frac{-(N-1)}{2}+k-1,\frac{-(N-1)}{2}+l-1}(\theta),$
(S24)
$\displaystyle=[(k-1)!(N-k)!(l-1)!(N-l)!]^{1/2}\sum_{s=s_{\mathrm{min}}}^{s_{\mathrm{max}}}\left[\frac{(-1)^{s}i^{l-k}\cos(\frac{\theta}{2})^{(N-1)+l-k-2s}\sin(\frac{\theta}{2})^{k-l+2s}}{(l-1-s)!s!(k-l+s)!(N-k-s)!}\right],$
(S25)

with $s_{\mathrm{min}}=\mathrm{max}\\{0,l-k\\}$ and
$s_{\mathrm{max}}=\mathrm{min}\\{l-1,N-k\\}$. Plugging in $(k,l)=(1,N)$ and
$(k,l)=(2,N)$, we get for the individual contributions

$\displaystyle[(k-1)!(N-k)!(l-1)!(N-l)!]^{1/2}=\begin{dcases}(N-1)!,&(k,l)=(1,N)\\\
\sqrt{(N-1)!(N-2)!},&(k,l)=(2,N)\end{dcases}$ (S26) $\displaystyle
s_{\mathrm{min}}$
$\displaystyle=\mathrm{max}\\{0,l-k\\}=\begin{dcases}N-1,&(k,l)=(1,N)\\\
N-2,&(k,l)=(2,N)\end{dcases}$ (S27) $\displaystyle s_{\mathrm{max}}$
$\displaystyle=\mathrm{min}\\{l-1,N-k\\}=\begin{dcases}N-1,&(k,l)=(1,N)\\\
N-2,&(k,l)=(2,N)\end{dcases}$ (S28)
$\displaystyle(-1)^{s}i^{l-k}=\begin{dcases}(-1)^{N-1}i^{N-1},&(k,l)=(1,N)\\\
(-1)^{N-2}i^{N-2},&(k,l)=(2,N)\end{dcases}$ (S29)
$\displaystyle(l-1-s)!s!(k-l+s)!(N-k-s)!=\begin{dcases}(N-1)!,&(k,l)=(1,N)\\\
(N-2)!,&(k,l)=(2,N)\end{dcases}$ (S30) $\displaystyle(N-1)+l-k-2s$
$\displaystyle=\begin{dcases}0,&(k,l)=(1,N)\\\ 1,&(k,l)=(2,N)\end{dcases},$
(S31) $\displaystyle k-l+2s$ $\displaystyle=\begin{dcases}N-1,&(k,l)=(1,N)\\\
N-2,&(k,l)=(2,N)\end{dcases}.$ (S32)

Finally, using Eqs. S26, S27, S28, S29, S30, S31 and S32 together with Eq.
S33, we obtain the time evolution of the local population of the $N$th qubit
of the chain as

$\displaystyle
p_{N}(t)=p_{1}(0)\left[\sin(\frac{\theta}{2})\right]^{2(N-1)}+p_{2}(0)(N-1)\left[\sin(\frac{\theta}{2})\right]^{2(N-2)}\cos(\frac{\theta}{2})^{2}+2\sqrt{N-1}\alpha\left[\sin(\frac{\theta}{2})\right]^{2N-3}\cos(\frac{\theta}{2}).$
(S33)

The last term in the above equation is always greater or equal to zero for
$\theta\in[0,\pi]$ which is equivalent to $t\in[0,\tau_{\mathrm{u}}]$, where
$\tau_{\mathrm{u}}=\pi/\lambda=N\pi/(4J)$ is the transfer time of the
uncorrelated chain.

###  I.4 Fidelity and von Neumann entropy

The Hamiltonian (1) does not create coherences. Therefore, the reduced density
matrices of the local qubits are always diagonal. Knowledge of the local
populations is hence sufficient to compute the fidelity and the von Neumann
entropy as will be illustrated below. For two single-qubit states $\rho$ and
$\sigma$, the fidelity can be computed according to Sbar13

$\displaystyle
F(\rho,\sigma)=O(\rho,\sigma)+\sqrt{[1-O(\rho,\rho)][1-O(\sigma,\sigma)]},$
(S34)

where $O(\rho,\sigma)=\Tr[\rho\sigma]$. To detect the arrival of a state at
the end of the chain, we compute the fidelity of the initial state at the
first site with the local state at the last site. In the absence of
coherences, the fidelity reduces to

$\displaystyle
F(\rho_{1}(0),\rho_{N}(t))=\left[\sqrt{(1-p_{1}(0))(1-p_{N}(t))}+\sqrt{p_{1}(0)p_{N}(t)}\right]^{2}.$
(S35)

The initial local state is hence transferred with unit fidelity when
$p_{N}(t)=p_{1}(0)$. Initially, we have $p_{N}(0)=0$ (the last spin starts in
the ground state). In the absence of quantum correlations, for $t>0$, there
will then be a monotonic increase in the population until the state is
transferred (cf. Eq. S33). For every positive value of $\alpha$ the last term
in Eq. S33 is always positive during the transfer time, any amount of nonzero
quantum discord can lead to faster state transfer. Note that even for
$\alpha<0$, the Hamiltonian Eq. S1 still guarantees a transfer time of at most
$\tau_{\mathrm{u}}=\pi/\lambda$ where the correlation term in Eq. S33
vanishes.

In order to evaluate the information received at the end of the chain, we use
the von Neumann entropy. Because local states always remain diagonal, the von
Neumann entropy is simply given by

$\displaystyle
I=-\Tr[\rho_{N}(t)\log(\rho_{N}(t))]=-p_{N}(t)\log(p_{N}(t))-(1-p_{N}(t))\log(1-p_{N}(t)).$
(S36)

The information flow can accordingly be conveniently expressed as

$\displaystyle\dot{I}=-\dot{p}_{N}(t)\log(\frac{p_{N}(t)}{1-p_{N}(t)}).$ (S37)

where $p_{N}(t)$ is given in Eq. S33.

##  II Condition for faster state transfer

In quantum systems with local interactions like the perfect state transfer
model Eq. S1, every order of the power series of the time evolution operator
$U=\exp(-iHt)$ can only connect adjacent sites. For instance, the evolution of
a state $\ket{\psi_{1}}=\ket{1}\otimes\ket{0}^{\otimes(N-1)}$ is given by

$\displaystyle
U\ket{\psi_{1}}=\exp(-iHt)\ket{\psi_{1}}=\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}t^{n}H^{n}\ket{1}\otimes\ket{0}^{\otimes(N-1)}.$
(S38)

Every order of the Hamiltonian can shift the excitation through the chain by
at most one site such that after applying the Hamiltonian $n$ times, the state
is generally in a superposition of the form

$\displaystyle
H^{n}\ket{1}\otimes\ket{0}^{N-1}=\sum_{s=1}^{n}\sum_{j=1}^{M(n)}c_{j}(n,s)\ket{s_{\sigma(j)}}\otimes\ket{0}^{N-1-n}$
(S39)

where $s\leq n$ corresponds to a fixed number of excitations and
$\ket{s_{\sigma(j)}}$ denotes all possible permutations of ones and zeros for
a given $s$ (all the possibilities to distribute $s$ excitations over $n$
sites) with

$\displaystyle M(n)=\begin{pmatrix}n\\\ s\end{pmatrix}=\frac{n!}{s!(n-s)!}$
(S40)

elements. The $c_{j}(n,s)$ are arbitrary coefficients that depend on the
specifics of the Hamiltonian.

Every order of $H$ is accompanied by a corresponding order of $t$ such that,
at each step in time, any excitation can at most reach its immediate neighbor.
Information propagation is moreover upper bounded by the Lieb-Robinson
velocity $v_{\mathrm{LR}}=2J$ and can thus spread at most linearly in time.
This limitation cannot be overcome even in the presence of correlations.
However, transmission of a qubit end-state along the spin chain is still
significantly improved in the presence of nonclassical correlations. To show
this, we consider a quantum spin chain of the general form

$\displaystyle H=\sum_{j}\omega_{j}\sigma^{z}_{j}+H_{\text{int}}$ (S41)

with possibly different level spacings $\omega_{j}$ and general nearest-
neighbor coupling Hamiltonian $H_{\text{int}}$. We assume that the initial
correlated state $\rho^{0}_{12,\mathrm{c}}=\rho_{\mathrm{u}}+\chi$ can be
decomposed into a diagonal part $\rho_{\mathrm{u}}$ with
$\Tr[\rho_{\mathrm{u}}]=1$ and correlations $\chi$ with $\Tr[\chi]=0$. We
further assume that the local initial states are diagonal and that the
Hamiltonian does not generate coherences during the evolution, at least not in
the subspace the evolution is restricted to. We separate the unitary time
evolution into uncorrelated and correlated parts

$\displaystyle\rho_{\mathrm{c}}(t)=\rho_{\mathrm{u}}(t)+U\chi U^{\dagger}.$
(S42)

In order to boost state transfer, only the local populations matter and we
demand that, at short times, more probability is shifted towards the state
$\ket{\psi_{2}}=\ket{01}\otimes\ket{0}^{\otimes(N-2)}$ than in the
uncorrelated case, that is,

$\displaystyle\bra{\psi_{2}}\dot{\rho}_{\mathrm{c}}(\differential{t})\ket{\psi_{2}}>\bra{\psi_{2}}\dot{\rho}_{\mathrm{u}}(\differential{t})\ket{\psi_{2}},$
(S43)

where

$\displaystyle\bra{\psi_{2}}\dot{\rho}_{\mathrm{c}}(\differential{t})\ket{\psi_{2}}=\bra{\psi_{2}}\rho_{c}(0)\ket{\psi_{2}}-i\bra{\psi_{2}}[H,\rho_{\mathrm{u}}(0)]\ket{\psi_{2}}-i\bra{\psi_{2}}[H,\chi]\ket{\psi_{2}}.$
(S44)

Since the local initial states are equal and the Hamiltonian appears only in
first order (thus connecting only the first and second site), condition (S43)
implies that $-i\bra{01}[H_{\mathrm{int}},\chi]\ket{01}>0$. Note that only the
interaction Hamiltonian matters because the diagonal part does not connect
different sites of the chain. A minimal requirement for enhanced transmission
is therefore that the commutator

$\displaystyle\left[H_{\mathrm{int}},\chi\right]\neq 0$ (S45)

does not vanish.

In the perfect state transfer model (1) condition Eq. S45 ensures that local
initial correlations on the sender’s side, simultaneously reduce the arrival
time and increase the maximum information flow $\dot{I}_{\mathrm{max}}$ (see
Figs. 1 and 2 of the main text) for, in principle, arbitrary system size $N$
(see Eq. S33).

##  III Out-of-time-order correlator

In this section, we provide additional information on the out-of-time-order
correlator in the spin chain (1). We numerically compute the squared
commutator

$\displaystyle
C(x,y,t)=\langle[W(x,t),V(y,0)]^{\dagger}[W(x,t),V(y,0)]\rangle,$ (S46)

of the time evolved Heisenberg operator $W(x,t)$ localized at $x$ and the
operator $V(y,0)$ localized at $y$ at the initial time. For operators that are
both unitary and Hermitian, the correlator conveniently reduces to

$\displaystyle C(x,y,t)=2-2\mathrm{Re}[F(x,y,t)],$ (S47)

where

$\displaystyle
F(x,y,t)=\langle\sigma^{z}(x,t)\sigma^{z}(y,0)\sigma^{z}(x,t)\sigma^{z}(y,0)\rangle$
(S48)

is the out-of-time-order correlator. The action of $F(x,y,t)$ can be
decomposed into consecutive steps: the local Pauli operator at site $y$ acts
on the initial state $\ket{\psi(y,0)}=\sigma^{z}(y,0)\ket{\psi}$. Then, the
time evolved operator located at another site $x$ acts on the resulting state
yielding $\ket{\psi(y,0;x,t)}=\sigma^{z}(x,t)\ket{\psi(y,0)}$. The out-of-
time-order correlator evaluates the overlap,
$F(x,y,t)=\innerproduct{\psi(x,t;y,0)}{\psi(y,0;x,t)}$, between the perturbed
state $\ket{\psi(y,0;x,t)}$ and a state $\ket{\psi(x,t;y,0)}$, where both
actions are performed in reverse order. Initially, all Heisenberg operators
commute and the overlap is one, yielding a zero correlator $C(x,y,t)=0$. As
time increases, the time evolution will gradually connect Heisenberg operators
at increasingly distant sites. As soon as two sites $j$ and $k$ have
overlapping support, the out-of-time-order correlator will no longer be one,
and the correlator $C(x,y,t)\neq 0$ will be different from zero. The latter
therefore provides a measure of operator spreading, and is related to
information propagation Sxu24 .

For locally interacting quantum systems (where interactions decay at least
exponentially as a function of the distance between sites), the above
statement is true in general. The operator growth rate is bounded from above
by the Lieb-Robinson velocity $v_{\mathrm{LR}}$ Slie72 . Generically, the
correlator $C(x,y,t)$ spreads through space with a wave front, whose tail has
a universal scaling behavior Skhe18 ; Sxu20 ; Sbao20

$\displaystyle C(r,t)\sim\exp(\frac{\Lambda(t-r(t)/v)^{1+p}}{t^{p}}),$ (S49)

where $r=|x-y|$ is the distance between sites and $\Lambda$ may in some cases
be seen as a quantum extension of the classical Lyapunov exponent Sbao20 ;
Smal16 ; Shas17 ; Sroz17 ; Sgar18 ; Sxu20 . In this regard, the speed of the
wave front $v$ effectively defines a causal light cone, and confines the
“connected” region. In chaotic systems $v$ is usually referred to as the
butterfly velocity. The parameter $p$ is model-dependent and controls the
broadening of the wave front. For a given time $t$, the correlator thus decays
exponentially in space. For sites located beyond the light cone $r>vt$, there
is also an exponential decay of $C(r,t)$. Note that the reverse statement is
also true: Lieb-Robinson type bounds imply locality of interactions Swil22 .

(a)(b)$C(x,N/2,t)$Perfect state transfer couplings
$J_{k}=\frac{\lambda}{4}\sqrt{k\cdot(N-k)}$ Supplementary Figure S1:
Correlator $C(x,N/2,t)$ (in arbitrary units) for a chain of $N=250$ qubits.
The average is taken over the infinite temperature maximally mixed state in
Eq. S47. The function $C(x,N/2,t)$ is shown for discrete times and the lines
are offset in $y$-direction Slin18 . Every fourth line is highlighted in blue
for better visibility. (a) At early times, the out-of-time-order correlator
decays to zero beyond a causally connected region restricted by the finite
speed of information propagation, the Lieb-Robinson velocity,
$v_{\mathrm{LR}}=2J$, which effectively defines a light-cone like structure.
The pink line indicates the first occurrence of nonzero correlator $C(x,y,t)$.
(b) The boundary deviates from the light cone at later times and operator
spreading slows down. Eventually the boundaries are reached at
$t=\tau_{\mathrm{u}}/2\approx 98$ where the causal region shrinks again to a
point at $t=\tau_{\mathrm{u}}$. Sites once causally connected become
disconnected again. This due to high degree of symmetry of the Hamiltonian
which imposes symmetries on the correlator, cf. Eqs. S58 and S63.
(a)(b)$C(x,N/2,t)$Uniform couplings $J_{k}=J$ Supplementary Figure S2:
Correlator $C(x,N/2,t)$ (in arbitrary units) for a uniformly coupled chain of
$N=250$ qubits with $J_{k}=J$ $\forall k$ in Eq. S1. The average is taken over
the infinite temperature maximally mixed state in Eq. S47. The function
$C(x,N/2,t)$ is shown for discrete times and the lines are offset in
$y$-direction Slin18 . Every fourth line is highlighted for better visibility.
(a) Information propagates at the Lieb-Robinson velocity, $v_{\mathrm{LR}}=2J$
(dashed line) until the boundaries of the chain are reached. The out-of-time-
order correlator is not periodic. Sites that are causally connected remain
connected. The pink line indicates the first occurrence of nonzero correlator
$C(x,y,t)$, cf. Fig. S1a. (b) Short time behavior of (a), the light cone is
clearly visible, cf. Fig. S1b.
(a)(b)(c)(d)(e)(f)$C(x,1,t)$$C(x,2,t)$$C(x,3,t)$uncorrelatedcorrelatedPerfect
state transfer couplings $J_{k}=\frac{\lambda}{4}\sqrt{k\cdot(N-k)}$
Supplementary Figure S3: Correlator $C(x,y,t)$ (in arbitrary units) for the
correlated (left, blue) and uncorrelated (right, red) initial $X$ state for
reference spins at different sites $y$. $C(x,y,t)$ is shown for discrete times
and the lines are offset in $y$-direction Slin18 . Every fourth line is
highlighted for better visibility. (a),(b) The reference spin is at the first
site $y=1$. On each site $x$, the correlator $C(x,y,t)$ is nonzero only for a
short time and has one peak. The uncorrelated and correlated correlators
coincide. (c),(d) The reference spin is at the second site $y=2$. There are
two peaks in $C(x,y,t)$ at each site. The uncorrelated and correlated
correlators coincide. (e),(f) The reference spin is at the third site $j=3$.
There are three peaks in $C(x,y,t)$ at each site. This situation corresponds
to the out-of-time-order correlator depicted in the main text in Fig. 1d. The
straight dashed line indicates the Lieb-Robinson velocity for this model given
by $v_{\mathrm{LR}}=2J$. The upper bound is attained near the center of the
chain where the out-of-time-order correlator runs parallel to
$v_{\mathrm{LR}}t$, cf. Section III.2.
(a)(b)(c)(d)(e)(f)$C(x,1,t)$$C(x,2,t)$$C(x,3,t)$uncorrelatedcorrelatedUniform
couplings $J_{k}=J$ Supplementary Figure S4: Correlator $C(x,y,t)$ (in
arbitrary units) for the correlated (left, blue) and uncorrelated (right, red)
initial $X$ state for reference spins at different sites $y$ in the uniformly
coupled spin chain with $(\omega_{k}=1,J_{k}=J)$. (a),(b) The reference spin
is at site $y=1$; the out-of-time-order correlators coincide. (c),(d) The
reference spin is at site $y=2$; the out-of-time-order correlator is visibly
enhanced near the left end of the chain for the correlated initial state. The
magnitude of the wavefront decays quickly and the effect of the correlations
gets less pronounced with the distance $x$. (e),(f) The reference spin is at
site $y=3$. Correlations have a notable impact on the out-of-time-order
correlator only up to a moderate distance in $x$. The straight dashed line
indicates the Lieb-Robinson velocity for this model given by
$v_{\mathrm{LR}}=2J$. Contrary to the perfect state transfer model, the bound
is attained at all times and information spreads linearly with
$v_{\mathrm{LR}}$. The information transfer is however incomplete across the
$N=250$ spins because information gets scrambled, i.e. the function $C(x,y,t)$
is different from zero at least in parts of the causal region.

###  III.1 Symmetries

We now proceed by investigating the symmetry properties of the out-of-time-
order correlator in time and space. For an arbitrary uncorrelated initial
condition where

$\displaystyle Z(0)=\mathrm{diag}(p_{1}(0),p_{2}(0),\ldots,p_{N}(0)),$ (S50)

(cf. Eq. S20), the evolution of the populations will be of the form

$\displaystyle p_{k}(t)=\sum_{m=1}^{N}|U_{km}|^{2}.$ (S51)

From the symmetry relations of the Wigner $d$-matrix elements (S23), it
follows that

$\displaystyle(-1)^{N-k}U_{km}(t)=U_{N-k+1,l}(T-t)$ (S52)

and thus $|U_{km}|^{2}(t)=|U_{N-k+1,m}|^{2}(T-t)$. This implies that spins
opposite to each other undergo time reversed dynamics, i.e.

$\displaystyle\langle\sigma^{z}_{k}\rangle(t)=\langle\sigma^{z}_{N+1-k}\rangle(T-t).$
(S53)

In order to analyze the properties of the out-of-time-order correlator, we
need to investigate the Heisenberg operators $\sigma^{z}_{k}(t)$. The Pauli
operators in the single excitation sector, or the Jordan-Wigner transformation
respectively, are given by

$\displaystyle\sigma^{z}_{k}(0)\sim 2\outerproduct{k}{k}-\mathds{1},$ (S54)

where $\outerproduct{k}{k}$ projects onto the $k$th element and $\sim$
indicates the representation in single excitation sector or the Jordan-Wigner
transformation respectively. The corresponding Heisenberg operators evolve
according to

$\displaystyle\sigma^{z}_{k}(t)\sim
2U^{\dagger}(t)\outerproduct{k}{k}U(t)-\mathds{1}$
$\displaystyle=2\sum_{m,n=1}^{N}U^{\ast}_{km}(t)U_{kn}(t)\outerproduct{m}{n}-\mathds{1},$
(S55)
$\displaystyle\overset{\eqref{eq:uni}}{=}2\sum_{m,n=1}^{N}U^{\ast}_{N-k+1,m}(t-T)U_{N-k+1,n}(t-T)\outerproduct{m}{n}-\mathds{1},$
(S56) $\displaystyle\sim\sigma^{z}_{N+1-k}(T-t).$ (S57)

This immediately implies that the out-of-time-order correlator has the same
time-reversal mirror symmetry

$\displaystyle F(x,y,t)=F(N-x+1,N-y+1,t-T).$ (S58)

This is true irrespective of the initial condition. Again invoking symmetry
relations of the Wigner $d$-matrix, we further obtain

$\displaystyle U_{kl}(T-t)=U_{k,N-l+1}(t)(-1)^{l-1}$ (S59)

and therefore, proceeding as before, we obtain

$\displaystyle\sigma^{z}_{k}(T-t)$ $\displaystyle\sim
2\sum_{m,n=1}^{N}U^{\ast}_{km}(t)U_{kn}(t)\outerproduct{m}{n}-\mathds{1},$
(S60)
$\displaystyle=2\sum_{m,n=1}^{N}U^{\ast}_{k,N-m+1}(t-T)U_{k,N-n+1}(t-T)\outerproduct{m}{n}-\mathds{1},$
(S61) $\displaystyle\sim\sigma^{z}_{k}(t).$ (S62)

For symmetric initial states this implies that the out-of-time-order
correlator has the additional symmetry

$\displaystyle F(x,y,t)=F(x,y,T-t).$ (S63)

Depending on the (initial) state $\rho$, over which the quantum mechanical
average is performed, some of those symmetries might be absent. Figure S1
shows the out-of-time-order correlator for the perfect state transfer
Hamiltonian (1) for a chain of length $N=250$ with the reference site in the
middle of the chain, $y=N/2=125$. Here, the quantum mechanical average is
taken over the completely mixed state $\rho=\mathds{1}/d$, so both Eqs. S58
and S63 are valid, causing the high symmetry of the out-of-time-order
correlator. For small times (compared to the length of the chain), the out-of-
time-order correlator spreads linearly (cf. Section III.2) with the butterfly
velocity of a uniformly coupled chain, $v_{\mathrm{LR}}=2J$, Fig. S1b.
Considering the full perfect state transfer time up to
$\tau_{\mathrm{u}}\approx 200$ reveals how the engineered couplings conspire
to perfectly invert the dynamics. After the initial linear spread, the out-of-
time-order correlator bends and reaches the boundaries at
$\tau_{\mathrm{u}}/2\approx 98$ where the causal region starts to shrink to a
point again at the final time $\tau_{\mathrm{u}}$. The system size is
fundamentally encoded into the Hamiltonian and therefore the system knows its
size beforehand, explaining the curving of the out-of-time-order correlator
near the boundaries.

These features are specific to the particular form of the interactions and the
resulting symmetries and are thus not expected generically. To highlight these
special properties, we directly compare the behavior to a system with uniform
interactions while otherwise keeping the same parameters. Figure S2 displays
snapshots of the out-of-time-order correlator at the same times as in Fig. S1.
Since now the system is agnostic to its size, the out-of-time-order correlator
exhibits a linear growth at the butterfly velocity throughout until hitting
the boundaries. The operators hence spread to the boundaries faster than with
engineered couplings but information is transmitted only incompletely. Once a
region is causally connected it remains connected and after reflecting at the
boundaries information gets scrambled all across the chain and becomes
practically inaccessible at the receiving end.

In Fig. S3 the quantum mechanical average is taken over the local initial $X$
state which is not symmetric. Therefore only Eq. S58 applies and the out-of-
time-order correlator is invariant only upon inverting space and reversing
time. We compare the uncorrelated (blue) and correlated (red) states and find
that the out-of-time-order correlators coincide for $y=1,2$. The effect of the
correlations first becomes apparent for reference spin $y=3$, displayed in
Fig. S3e,f (corresponding to to Fig. 1d of the main text). Quantum
correlations break the time-reversal symmetry, Eq. S63, and shift the weight
towards the wavefront such that more information is transmitted at the same
time thereby increasing the energy flow and boosting state transfer. Because
of the small amplitude of the out-of-time-correlator and the large system
size, the effect is less pronounced in the illustration in Fig. S3f. As
before, we compare the results to the uniformly coupled chain in Fig. S4 which
shows linear growth and scrambling in accordance with Fig. S2.

###  III.2 Lieb-Robinson velocity

In this section, we show why the perfect state transfer Hamiltonian Eq. S1
leads to information propagation in a local region at maximum speed given by
the Lieb-Robinson velocity, as observed in Figs. S1 and S3. The off-diagonal
elements of the Hamiltonian in the single excitation sector are given by

$\displaystyle J_{k}=\frac{\lambda}{4}\sqrt{k\cdot(N-k)}.$ (S64)

For elements of $J_{k}$ close to the middle of the chain with $k=N/2+m$ where
$|m|\ll N$, we obtain (with $\lambda=4J/N$)

$\displaystyle
J_{k}=2J\left[1-\frac{2m^{2}}{N^{2}}+\mathcal{O}(m^{4})\right].$ (S65)

Thus, in the vicinity of the chain’s center, the interaction strengths are
constant up to second order in the distance $m$. Therefore, in a region where
the correction is still small, i.e. $2m^{2}/N^{2}\ll 1$, the out-of-time-order
correlator exhibits the same linear growth as the uniformly coupled chain. In
Fig. S1, the system size is $N=250$ and hence, to a good approximation, the
operators spread linearly over the first $m\approx 25$ sites (where
$2m^{2}/N^{2}=0.02$).

##  IV No boost for Bell states

In this section we show that using the canonical Bell states as initial states
for the perfect state transfer protocol does not lead to a boost. The initial
two-qubit state $\rho^{0}_{12}=\rho_{X}$ (4), is separable with zero
concurrence for all $\alpha\in[0,1/4]$. In particular, entanglement is thus
not necessary for faster state transfer. We observe, in fact, neither a boost
in the state transfer nor in the information flow when a maximally entangled
Bell state is used. The canonical Bell states belong to the class of $X$
states

$\displaystyle\rho^{0}_{12}=\rho_{X}=\begin{pmatrix}a&0&0&w\\\ 0&b&z&0\\\
0&z^{\ast}&c&0\\\ w^{\ast}&0&0&d\end{pmatrix},$ (S66)

where every parameter is a real number equal to either $0$ or $\pm 1/2$. In
this case the correlations take form

$\displaystyle\chi^{\mathrm{B}}=\begin{pmatrix}0&0&0&w\\\ 0&0&z&0\\\
0&z&0&0\\\ w&0&0&0\end{pmatrix},$ (S67)

where either $w=\pm 1/2$ and $z=0$ or vice versa. Most importantly, Bell
states do not satisfy the necessary condition (3), i.e.
$[H_{\mathrm{int}},\chi^{\mathrm{B}}]=0$ but leave the out-of-time-order
correlator invariant under time-reversal Eq. S63. Therefore Bell states do not
affect the dynamics of the system and entanglement is thus neither necessary
nor sufficient to boost information transfer. As demonstrated in Fig. 2 of the
main text, maximally entangled states indeed provide the highest enhancement
possible but only if the basis of the quantum correlations matches the
interaction Hamiltonian in the sense of Eq. S45.

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