††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Long-Range Bell States from Local Measurements and Many-Body Teleportation
without Time-Reversal

Lakshya Agarwal  Department of Physics & Astronomy, Texas A&M University,
College Station, Texas 77843, USA  Christopher M. Langlett  Department of
Physics & Astronomy, Texas A&M University, College Station, Texas 77843, USA
Shenglong Xu  Department of Physics & Astronomy, Texas A&M University, College
Station, Texas 77843, USA

###### Abstract

In this work, we study quantum many-body teleportation, where a single qubit
is teleported through a strongly-interacting quantum system, as a result of a
scrambling unitary and local measurements on a few qubits. While usual many-
body teleportation protocols require rewinding time and a double copy, we
demonstrate that teleportation is possible in more general systems, including
the 2D spin-$1/2$ XY model, without these constraints. Teleportation success
requires an entanglement resource, such as long-ranged entangled Bell states.
We outline a specific protocol for preparing this highly entangled many-body
state by evolving a product state and performing iterative measurements on
only two qubits with feedback control.

Rapid advancement in coherent control over quantum many-body systems provides
an exciting opportunity to study rich non-equilibrium phenomena [1, 2, 3].
Generic initial states thermalize under their own unitary dynamics [4, 5, 6].
Despite the global time-evolved state remaining pure, locally the state
becomes mixed and approaches the thermal ensemble from the rapid entanglement
growth. If one prepares a specific initial state on a qubit that is a part of
a larger system and time-evolves the entire state, then the information
contained in the qubit is not retrievable via measurements on the prepared
qubit, or any local subset of qubits, due to thermalization. While that bit of
quantum information is never lost – one can always trivially act $U^{\dagger}$
on the entire output state to get it back, the information spreads to non-
local degrees of freedom, making retrieval difficult, known as scrambling [7,
8, 9]. In other words, when Alice tosses a qubit into a black hole, the
information is not expected to be recovered by her or Bob, who is far away.

Remarkably, Hayden and Preskill [8] demonstrated that the black hole instead
acts as a mirror and Alice’s qubit reappears on Bob’s side under the right
conditions [10]. While the original motivation of the work is to study quantum
information dynamics inside a black hole, the Hayden-Preskill protocol and its
generalizations [11, 12, 13, 14, 15] extend to generic quantum many-body
systems [16, 17, 18, 19, 20], and have been realized experimentally [21, 22,
23]. Given Alice’s qubit is never lost due to unitary dynamics, these many-
body teleportation protocols aim at designing simple quantum operations,
including unitary evolution and measurements, to extract and teleport Alice’s
original qubit to Bob through a strongly interacting quantum many-body system.

Figure 1: Summary of the main results. (a) The teleportation protocol. The
initial quantum state on Alice’s qubit (red), scrambled by the unitary
dynamics, is teleported to the auxiliary qubit (white) on the other side of
the system by a few Bell measurements. (b) The state engineering protocol. The
initial state required in (a) can be engineered from any initial state by the
scrambling unitary dynamics and periodic measurements on two central qubits
with feedback control. In both protocols, only $\mathcal{O}(1)$ qubits are
measured.

Many-body teleportation is a generalization of conventional teleportation that
is based on the entangled pair of qubits shared by Alice and Bob (see the SM
for a brief description). Many-body teleportation involves two steps, encoding
and decoding the information. In the encoding step, a single bit of quantum
information spreads across the complex degrees of freedom of the many-body
Hilbert space through the application of the unitary $U$. On the other hand,
the decoding step concentrates the information at another qubit across the
system. As in the case of single-body teleportation, many-body teleportation
requires a long-range pair-wise entangled initial state between the system and
a double copy. In addition, the usual decoder [10] contains a unitary
operation $U^{*}$ which effectively rewinds time. Crucially, $U^{*}$ does not
trivially act on the entire output state but rather on the second copy. The
teleportation fidelity is closely related to the out-of-time ordered
correlator, which is measured by directly engineering $U^{*}$ [24, 25, 26, 27,
28, 29] or using randomized measurements without time reversal [30, 31, 32].
However, the current scheme of directly teleporting quantum information
through a many-body system requires the long-range entangled state, a double
copy of the system and time reversal, and is so far limited to small systems.

In this work, we circumvent the aforementioned obstacles. First, we show that
many-body teleportation can occur in much more general quantum many-body
systems without a double copy or time-reversal. The simplest example is the
homogeneous spin-$1/2$ XY model on a 2D lattice, which has been realized in
various quantum simulators [33, 34, 35, 36]. We show that the scrambling
unitary dynamics and $\mathcal{O}(1)$ Bell measurements lead to successful
teleportation. Second, we demonstrate that the required long-ranged entangled
initial state, sometimes referred to as the “rainbow” state, can be engineered
in the same system by repetitively measuring only two qubits along with the
unitary dynamics. In contrast to other state engineering protocols [37, 38,
39, 40, 41], ours does not require control of the coupling parameters,
additional dissipators, or the non-interacting properties of the 1D XY chain,
and is even applicable to 2D lattices. These results are based on the
observation that the 2D XY model, although strongly-interacting, hosts special
non-thermal eigenstates that are products of long-range Bell states. These
special eigenstates belong to a special class of quantum many-body scars
(QMBS) [42, 43, 44] referred to as rainbow scars [45]. This work, summarized
in Fig. 1, provides a simple protocol to realize quantum many-body
teleportation by taking advantage of unitary dynamics and local measurement.
Recently, there has been an increasing interest in the complex role of
measurements in quantum many-body systems [46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61], most work considering local measurements on an
extensive portion of the system. Our work demonstrates that measuring
$\mathcal{O}(1)$ qubits, combined with scrambling unitary dynamics and scars,
can also lead to rich non-equilibrium phenomena.

We first briefly describe the Hayden-Preskill protocol and the decoder [10].
Consider a system of $N$ qubits $q_{1},\ldots,q_{N}$ maximally entangled with
another $N$ qubits $q_{\tilde{1}},\ldots,q_{\tilde{N}}$. Bob has full control
of $q_{\tilde{1}},\ldots,q_{\tilde{N}}$ plus an $\mathcal{O}(1)$ number of
qubits in the system, denoted by $q_{E}$. On the other hand, Alice strictly
has one qubit, say $q_{1}$, in the system that does not belong to Bob, where
Alice prepares an initial state $\ket{A}$ by projective measurement and
single-qubit rotation and breaks the entanglement between $q_{1}$ and
$q_{\overline{1}}$. From Alice’s point of view, the entire system is initially
in a fully mixed state and her prepared state on $q_{1}$ will be lost due to
thermalization and become unrecoverable. However, it turns out that Bob can
recover Alice’s state through operations only on his qubits. In Ref. [10], two
decoding schemes are presented, a stochastic one based on measurement and
post-selection, and a deterministic one based on Grover’s search. In this
work, we focus on the stochastic scheme which we now outline. Bob takes a new
qubit $q_{b}$ and entangles it with $q_{\tilde{1}}$ as an EPR pair, he then
engineers a $u^{*}$ and acts it on $q_{\overline{1}},\ldots,q_{\overline{N}}$.
Finally he performs Bell measurements on $q_{E}$ and the partners
$q_{\overline{E}}$ and then post-selects the results to be EPR states.
Graphically, the protocol is depicted by

$\cdots$$\cdots$$\cdots$$\cdots$$u$$u^{*}$ (1)

The initial state of $q_{1}$, its mirror partner $q_{\overline{1}}$ and the
auxiliary qubit $q_{b}$ is
$\ket{A}_{1}(\ket{00}+\ket{11})_{\overline{1}b}/\sqrt{2}$. The other qubits
form EPR pairs with their mirror partners. The initial state in the Bell basis
of $q_{1}$ and $q_{\overline{1}}$ is

$\displaystyle\ket{\psi}=\frac{1}{2}\ket{\text{EPR}}\ket{A}_{b}+\sum\limits_{s}\frac{1}{2}\sigma_{1}^{s}\ket{\text{EPR}}\sigma_{b}^{s}\ket{A}_{b},$
(2)

where $\ket{\text{EPR}}$ is a product of $N$ EPR states between $q_{i}$ and
its mirror partner $q_{\overline{i}}$, and $\sigma^{s}$ are the Pauli
operators. The first term, where the state $\ket{A}$ appears over $q_{b}$, is
an eigenstate of the composite unitary $u\otimes u^{*}$ [62]. However, the
second term is not, and the perturbation caused by the Pauli operator
$\sigma_{1}^{s}$ on the first qubit spreads across the system disrupting the
pairwise entanglement structure. Therefore in the late-time regime, Bob can
separate the first term from the others and acquire the state $\ket{A}$ by
performing Bell measurements on only a few pairs of qubits and then post-
selecting the outcome to be EPR states. Given that the second term thermalizes
to the infinite-temperature ensemble at late times, the probability of
successful post-selection on $E$-pair measurements is $4^{-1}+3\times
4^{-E-1}$. Only a small number of post-selections is required to select the
first term of Eq. (2) where the state $\ket{A}$ is teleported to $q_{b}$. The
local measurement with post-selection acts effectively as a long-range
projection on the time evolved state to force $q_{1}$ and $q_{\overline{1}}$
into an EPR state, leading to teleportation just like the usual single-body
teleportation.

From the discussion above, it is clear that the choice of a factorizable
unitary $U=u\otimes u^{*}$ is to ensure that it thermalizes a typical state
but not the global EPR state, which is later stabilized by the Bell
measurement with post-selection. However, this is not the only choice. We now
arrive at the general condition for the unitary dynamics $U$ to achieve many-
body teleportation:

  1. 1.

$U$ thermalizes a typical initial state other than the special eigenstate,
i.e., $U$ is neither free nor integrable.

  2. 2.

$U$ hosts a special eigenstate exhibiting long-range and pair-wise
entanglement structure, such as the EPR state.

Obviously, $U=u\otimes u^{*}$ meets both conditions above, however such a
unitary is challenging to engineer experimentally, especially in analog
quantum simulators, because it requires a duplicate copy and time rewinding
the second copy.

Teleporting without time reversal.— Now we demonstrate that there exist simple
systems whose dynamics satisfy both the conditions without requiring a
duplicate copy or rewinding time. In particular, examples include the XY
model, one of the prototypical quantum many-body models realized in various
quantum simulators [33, 34, 35, 36]. Let us consider the XY model on a square
lattice of size $L_{x}\times L_{y}$,

$\displaystyle
H_{xy}=\sum_{\braket{ij}}\left(J_{x}\sigma^{x}_{i}\sigma^{x}_{j}+J_{y}\sigma^{y}_{i}\sigma^{y}_{j}\right),$
(3)

where $\braket{ij}$ stands for nearest neighbor. We choose $J_{x}\neq J_{y}$
to avoid complications due to charge conservation, and set $J_{y}=1.2J_{x}$.

Now we demonstrate that this model satisfies both conditions for
teleportation. First, unlike the 1D chain, the 2D XY model does not become
non-interacting under the Jordan-Wigner transformation, and therefore
thermalizes a typical state, satisfying the first condition. Second, we show
that the model hosts a special long-range pair-wise entangled eigenstate in
which a qubit $q_{i,j}$ forms a Bell state with its reflection partner
$q_{\overline{i},j}$, where $\overline{i}=L_{x}-i$. Denote two of the Bell
states $\frac{1}{\sqrt{2}}(\ket{00}\pm\ket{11})$ as $\ket{I}$ and $\ket{Z}$
respectively. The eigenstate is written as,

$\displaystyle\ket{\text{EIG}}=\prod_{i+j\in\text{even}}\ket{I_{(i,j),(\overline{i},j)}}\prod_{i+j\in\text{odd}}\ket{Z_{(i,j),(\overline{i},j)}}.$
(4)

Notice the checkerboard pattern between $\ket{I}$ and $\ket{Z}$, depicted in
the left panel of Fig. 1(b).

One can straightforwardly verify the state is an eigenstate of the XY model
using the property of the EPR state and Pauli commutation relations (See
details in SM). We note that it is known that the XY ladder ($L_{y}=2$) hosts
special eigenstates built from short-range Bell pairs on the rungs [63, 64],
which is different from Eq. (4). The special eigenstate identified here exists
for arbitrary $L_{y}$ and even $L_{x}$ beyond the ladder geometry, built from
long-range Bell states within each row. There are also other similar special
eigenstates in the XY model which we discuss in detail in SM. The special
eigenstate $\ket{\text{EIG}}$ is related to $\ket{\text{EPR}}$ by single qubit
rotations, therefore the XY model meets both conditions for many-body
teleportation.

Now we are ready to discuss the specific protocol. In the teleportation
protocol, we begin with the eigenstate in Eq. (4). We will discuss how to
prepare the initial state using the same model later. First, we choose the
first row, break the long range entanglement between $q_{1,1}$ and
$q_{\overline{1},1}$ and inject the state $\ket{A}$ to be teleported into
$q_{1,1}$. Second, we take a new auxiliary qubit $q_{b}$ and entangle it with
$q_{\overline{1},1}$. After this setup, the state, similar to Eq. (2),
becomes,

$\displaystyle\ket{\psi}=\frac{1}{2}\ket{\text{EIG}}\ket{A}_{b}+\sum\limits_{s}\frac{1}{2}\sigma_{(1,1)}^{s}\ket{\text{EIG}}\sigma_{b}^{s}\ket{A}_{b}.$
(5)

Then the state is evolved for a time $t$ under the Hamiltonian which does not
affect the auxiliary qubit $q_{b}$. Afterwards, local bell measurements are
performed on a few qubits and their mirror partners, and results are post-
selected to match the pattern in $\ket{\text{EIG}}$ ($\ket{I}$ or $\ket{Z}$)
to stabilize the eigenstate. Because of thermalizing dynamics, the pairwise
entanglement structure in the second term is disrupted by the perturbation
$\sigma^{s}_{(1,1)}$ over time. Therefore, only the first term can survive the
post-selection given a few number of measurements. Consequently, the initial
state on $q_{1}$ is teleported to the auxiliary qubit $q_{b}$, due to the
combined effect of the thermalization dynamics, the special eigenstate, and
the Bell measurements.

Figure 2: (a) The teleportation protocol of XY model in a geometry of
$10\times 2$ qubits. The solid line and dashed line represent the fidelity and
post-selection probability, respectively. In the first case, two pair center
qubits, shown in the inset, are measured and post-selected. In the second
case, two additional pairs are measured. As more pairs of qubit are measured,
the fidelity approaches to 1 and the post-selection probability approaches to
$1/4$. (b) We repeat the simulation for geometry of $20\times 1$ and $6\times
3$. The teleportation fails in the 1D case because it is non-interacting.

For simplicity, we numerically simulate the teleportation on a ladder
($L_{y}=2$). We emphasize that the teleportation occurs from one side to the
other side within a leg, and the ladder setup is simply to make the model
interacting. As shown in Fig. 2(a), the probability $P$ of successful post-
selection decreases as time proceeds but saturates to a finite value. In the
late-time regime, as the number of measured pair increases, $P$ exponentially
approaches to $1/4$, indicating that only the first term is selected. The
quality of the teleportation is characterized by the fidelity
$F=\bra{A}\rho_{b}\ket{A}$ where $\rho_{b}$ is the density matrix of $q_{b}$.
Fig. 2(b) plots $F$ as a function of time of the unitary evolution before the
measurement (inset: the number of measured pairs). The fidelity stays at the
initial value $1/2$ until the perturbation of $\sigma^{s}_{(1,1)}$ reaches the
measured qubits, and quickly saturates to the maximal value $F\sim 1$ for a
random initial state $\ket{A}$, confirming the Hayden-Preskill like
teleportation in the XY model without explicit time reversal. The Bell
measurements are essential for the protocol, without which $\rho_{b}$ would
stay fully mixed and the fidelity would stay $1/2$.

To emphasize the necessary condition of thermalization dynamics for the
teleportation, we repeat the protocol for the free 1D XY chain in Fig 2(b),
showing that the fidelity is significantly below 1, in contrast to the
interacting case. This is because the post-selection of the Bell measurement
cannot completely rule out the second term which does not thermalize in the
free case. We also repeat the protocol for $6\times 3$ qubits, showing that it
does not rely on the ladder geometry.

Initial state engineering.— The teleportation protocol previously discussed
requires a special initial state $\ket{\text{EIG}}$ with long-range
entanglement, as in other teleportation protocols, which is often challenging
to engineer experimentally. The next main result of this work is preparing the
special initial state by utilizing unitary dynamics and periodic measurement
on only two neighboring qubits, $q_{c}$ and $q_{\overline{c}}$ that form an
EPR pair $\ket{I}$ in $\ket{\text{EIG}}$ (Fig. 1(a)). The protocol is as
follows:

  1. Step 1:

Initialize the system in a simple initial state such as, $\ket{0\cdots 0}$.

  2. Step 2:

From the last step, $q_{c}$ and $q_{\overline{c}}$ are in a product state.
Unitarily rotate the two-qubit state into an EPR pair.

  3. Step 3:

Evolve the state for a time $T$ under the Hamiltonian $H_{xy}$.

  4. Step 4:

Measure $q_{c}$ and $q_{\overline{c}}$ in the computational basis. If the
resulting state is $\ket{01}$ or $\ket{10}$, discard the entire state and
reset it with the initial state $\ket{0\cdots 0}$. Go to Step 2 and repeat.

The time-evolved state depends on the measurement outcomes in Step 4.
Averaging over all measurement outcomes, the system is described by a density
matrix. Since after Step 2, $q_{c}$ and $q_{\overline{c}}$ are always in the
EPR state and decouple from the system, we only need to consider the density
matrix $\rho$ of the rest of the system, which is updated by a quantum channel
of the form,

$\displaystyle\rho^{(n+1)}=$
$\displaystyle\mathcal{M}(T)\rho^{(n)}=K_{00}\rho^{(n)}K^{\dagger}_{00}+K_{11}\rho^{(n)}K^{\dagger}_{11}$
(6)
$\displaystyle+\text{tr}\left(K_{01}\rho^{(n)}K_{01}^{\dagger}\right)\rho_{0}+\text{tr}\left(K_{10}\rho^{(n)}K_{10}^{\dagger}\right)\rho_{0}.$

Here $K_{s}(T)$ is an operator $\bra{s}e^{-iH_{xy}T}\ket{I_{c\overline{c}}}$
with dimensions $2^{N-2}$. The last two terms account for resetting the state.
By construction, $\ket{\text{EIG}}$ in Eq. (4) is a steady state of the
quantum channel with unit eigenvalue. The efficiency of the protocol to reach
the steady state depends on the second largest eigenvalue $\lambda$ of the
quantum channel. Given a finite gap $\Delta=1-|\lambda|$, any initial state is
guaranteed to approach the target state. The fidelity between the density
matrix and the pure target state is determined by
$F=\bra{\text{EIG}}\rho\ket{\text{EIG}}$. To reach the final state with
fidelity $F=1-\epsilon$, the number of required measurements scales as
$-\log(\epsilon/\Delta)$ for small $\Delta$.

The above quantum channel only depends on a single parameter $T$, the time
interval between successive measurements. In Fig. 3(a), we plot the gap of
$\mathcal{M}$ as a function of $T$ for systems of $12$ qubits in two different
geometries. The gap is always finite, indicating a unique steady state, and
the maximal value for both geometries occurs at $T\sim 0.4$, which is chosen
for the following simulations. Fig. 3(b) shows the fidelity $F$ between $\rho$
and the target state as a function of the number of measurements by directly
simulating Eq. (6). The fidelity exceeds 0.99 at $N_{tot}=150$, confirming
that $\rho$ reaches the targeted pure state. We remark $N_{tot}$ includes
measurements discarded due to resetting, since the quantum channel includes
the resetting procedure. On the other hand, the coherence time required is
$N_{c}T$, where $N_{c}$ is the number of measurements required to reach the
steady state after the final resetting. One then obtains $N_{c}$ by post-
selecting the measurement result to be $\ket{00}$ and $\ket{11}$,
corresponding to the first two terms in Eq. (6) with proper normalization.
With post-selecting to not resetting, the fidelity reaches 0.99 in 20 steps,
shown in the inset of Fig. 3. This indicates that the required coherence time
is as short as $8J^{-1}$. There is a clear experimental signature when the
steady-state is reached with the outlined protocol because the measurements
will always result in $\ket{00}$ or $\ket{11}$ and no $\ket{01}$ and
$\ket{10}$ once the steady state is reached.

During this hybrid evolution consisting of unitaries and measurements, the
density matrix, which is initially pure, becomes mixed by the measurements at
early times and is purified by the measurements at late times, which is
illustrated in the non-monotonic behavior of the second Rényi entropy
$S^{(2)}(\rho)=-\log\text{tr}(\rho^{2})$ shown in Fig. 3(c). On the other
hand, the mutual information between a qubit $q_{i}$ and its mirror partner
$q_{\overline{i}}$ increases to the maximal value 2, confirming the formation
of the Bell states. The farthest pair and nearest pair form the Bell states
simultaneously, indicating the non-local entangling power of the measurements,
which is the key mechanism of the entanglement SWAP protocol [65] and has been
discussed recently in the context of quantum circuits [54, 58].

We emphasize that $f(\rho)$ reaching the maximal value at $N_{tot}$ indicates
that most quantum trajectories converge to the steady state given these
measurements. What is more relevant in practice is the typical number of total
measurements required for a trajectory, whereby $N_{tot}$ is a loose bound. To
study the distribution of the required number of measurements, we simulate
quantum trajectories by sampling over all measurement outcomes, including
resetting, using standard Monte Carlo. Fig. 3(c) plots the histogram of
$N_{tot}$ for 200 simulations, which peaks around 50. Simulating quantum
trajectories allows us to probe larger system sizes because only a pure state
instead of the density matrix is evolved. Remarkably, for the system of 16
qubits, typically only 80 measurements are required to stabilize 8 pairs of
long-range Bell states (Fig. 3(d)). Furthermore, the number of measurements
required after previous resetting is only 25. Once the long-range entanglement
is established, the state may be utilized in the aforementioned many-body
teleportation protocol.

Figure 3: (a) The gap of the quantum channel in Eq. (6), as a function of
measurement period, $T$. In all cases, the target state is the unique steady
state of the quantum channel. (b) The fidelity between the density matrix
$\rho$ and the target state as a function of the number of measurements. The
density matrix approaches to the steady state after $\sim 150$ measurements.
Only considering the dynamics after the last reset, $\rho$ reaches the unique
steady state after $\sim 20$ measurements, shown in the inset. (c) As the
number of measurements increase, the density matrix becomes mixed first and is
then purified by late-time measurement, characterized by its Rényi entropy.
The mutual information between the two qubits in the farthest pair (blue) and
nearest pair (orange) have the same qualitative behavior and reach the maximal
value 2. (d) Histogram of the total number of measurements to reach fidelity
0.99, from simulating 200 quantum trajectories.

Discussion and outlook.— We demonstrated that many-body teleportation is
achievable in the XY model by utilizing a long-range entangled eigenstate and
local measurement, whereby the eigenstate is engineered through the interplay
between unitary dynamics and iterative measurement on a single pair of qubits
with feedback control. Modifying the orientation of the measured qubit pair
stabilizes various other eigenstates of the XY model which have different
global entanglement structures, suggesting enhanced control over the global
entanglement with local operations. Interestingly, when the Hamiltonian
possesses further symmetries such as, $U(1)$ for $J_{x}=J_{y}$ or translation
invariance, the special eigenstate is projected into each symmetry sector,
leading to even richer non-equilibrium phenomena, which is left for future
work. Another future direction will be teleportation using the finite-
temperature thermofield double state [66, 67, 68, 13]. Furthermore, QMBS with
interesting entanglement structure, e.g.,[69, 70, 71, 72, 73] have since been
studied in a host of models both theoretically and experimentally.
Generalizing the protocol of this work gives a roadmap to engineer these
states.

Acknowledgement.— The numerical simulations in this work were conducted with
the advanced computing resources provided by Texas A&M High Performance
Research Computing.

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##  Appendix A Conventional Quantum Teleportation

Conventional single-body teleportation begins with a qubit that encodes the
information, along with an EPR pair,

$\displaystyle\ket{\psi_{i}}=\leavevmode\hbox to31.65pt{\vbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{} {}{}
{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{14.22638pt}{14.22638pt}\pgfsys@lineto{14.22638pt}{4.0pt}\pgfsys@curveto{14.22638pt}{1.79083pt}{16.01721pt}{0.0pt}{18.22638pt}{0.0pt}\pgfsys@lineto{24.45276pt}{0.0pt}\pgfsys@curveto{26.66193pt}{0.0pt}{28.45276pt}{1.79083pt}{28.45276pt}{4.0pt}\pgfsys@lineto{28.45276pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@moveto{23.33957pt}{0.0pt}\pgfsys@curveto{23.33957pt}{1.10458pt}{22.44415pt}{2.0pt}{21.33957pt}{2.0pt}\pgfsys@curveto{20.23499pt}{2.0pt}{19.33957pt}{1.10458pt}{19.33957pt}{0.0pt}\pgfsys@curveto{19.33957pt}{-1.10458pt}{20.23499pt}{-2.0pt}{21.33957pt}{-2.0pt}\pgfsys@curveto{22.44415pt}{-2.0pt}{23.33957pt}{-1.10458pt}{23.33957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.0pt}{0.0pt}\pgfsys@curveto{3.0pt}{1.65688pt}{1.65688pt}{3.0pt}{0.0pt}{3.0pt}\pgfsys@curveto{-1.65688pt}{3.0pt}{-3.0pt}{1.65688pt}{-3.0pt}{0.0pt}\pgfsys@curveto{-3.0pt}{-1.65688pt}{-1.65688pt}{-3.0pt}{0.0pt}{-3.0pt}\pgfsys@curveto{1.65688pt}{-3.0pt}{3.0pt}{-1.65688pt}{3.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \par
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (7)

A Bell measurement is performed on the first two qubits, where the outcome is
post-selected to be an EPR state. Graphically, the un-normalized state is,

$\displaystyle\ket{\psi_{f}}=\leavevmode\hbox to31.65pt{\vbox
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{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{10.22638pt}\pgfsys@curveto{0.0pt}{12.43555pt}{1.79083pt}{14.22638pt}{4.0pt}{14.22638pt}\pgfsys@lineto{10.22638pt}{14.22638pt}\pgfsys@curveto{12.43555pt}{14.22638pt}{14.22638pt}{12.43555pt}{14.22638pt}{10.22638pt}\pgfsys@lineto{14.22638pt}{4.0pt}\pgfsys@curveto{14.22638pt}{1.79083pt}{16.01721pt}{0.0pt}{18.22638pt}{0.0pt}\pgfsys@lineto{24.45276pt}{0.0pt}\pgfsys@curveto{26.66193pt}{0.0pt}{28.45276pt}{1.79083pt}{28.45276pt}{4.0pt}\pgfsys@lineto{28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{} {}{}
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It is clear that after the protocol, the initial state of the first qubit
appears on the last qubit. The factor of $1/2$ above is the normalization,
indicating that the probability of the successful post-selection is $1/4$. The
above simple protocol is the stochastic version of single-body teleportation,
which can be generalized to the many-body case. Of course, in the conventional
protocol, the post-selection can be avoided. There are 4 possible outcomes of
the Bell measurements, which tells how the state of the third qubit is related
to the initial state of the first qubit. Therefore, based on the measurement
outcome, one can apply the corresponding Pauli operator to the third qubit to
retrieve the state. In the original story, Alice owns the first two qubits and
Bob owns the third qubit which can be far way. Alice then performs the Bell
measurement on the first two qubits and tells Bob the result over telephone so
that Bob can get Alice’s state by applying the Pauli operator to his qubit.

There also exist many-body teleportation protocols that only include unitary
operations and thus is free of post-selection, which is based on the Grover’s
search and in the simplest case, can be understood as the SWAP gate.

##  Appendix B Special eigenstates of the XY model

In the main text, we focused on the non-equilibrium dynamics resulting from
one special eigenstate of the XY model, given by

$\displaystyle\ket{\text{EIG}}=\prod_{i+j\in\text{even}}\ket{I_{(i,j),(\overline{i},j)}}\prod_{i+j\in\text{odd}}\ket{Z_{(i,j),(\overline{i},j)}}.$
(9)

In this section, we explicitly show that this state is an eigenstate of the XY
model and also provide other closely related special eigenstates. To simplify
the notation, in the following, we will use $X$, $Y$ and $Z$ to represent the
three Pauli operators. The XY Hamiltonian on a 2D square lattice of size
$L_{x}\times L_{y}$ is written as

$\displaystyle
H_{xy}=\sum_{\braket{rr^{\prime}}}J_{x}X_{r}X_{r^{\prime}}+J_{y}Y_{r}Y_{r^{\prime}}$
(10)

where $r$ and $r^{\prime}$ are coordinates of nearest neighbor sites. We focus
on even $L_{x}$ and denote the mirror reflection of coordinate $(i,j)$ as
$(\overline{i},j)\equiv(L_{x}+1-i,j)$. Let us first consider the term
$X_{(i,j)}X_{(i+1,j)}$ on the vertical bonds. This term, acting on
$\ket{\text{EIG}}$, affects two Bell pairs, one in $\ket{I}$ and the other one
in $\ket{Z}$. We get

$\displaystyle
X_{(i,j)}X_{(i,j+1)}\ket{\text{EIG}}=-X_{(\overline{i},j)}X_{(\overline{i},j+1)}\ket{\text{EIG}}.$
(11)

Therefore, the $XX$ terms on a vertical bond and its mirror partner cancel
each other when acting on $\ket{\text{EIG}}$, same for the $YY$ terms.
Demonstrating that $\ket{\text{EIG}}$ is an eigenstate of all the terms on the
vertical bonds with zero eigenvalue.

Now let us consider the $XX$ terms on the horizontal bond. We have

$\displaystyle
X_{(i,j)}X_{(i+1,j)}\ket{\text{EIG}}=-X_{(\overline{i},j)}X_{(\overline{i+1},j)}\ket{\text{EIG}}.$
(12)

except $i=L_{x}/2$, i.e., the central bonds on each row. Therefore, all terms
except those on the central bonds cancel out when acting on
$\ket{\text{EIG}}$. The terms on the central horizontal bonds obey the
relations,

$\displaystyle
X_{(L_{x}/2,j)}X_{(L_{x}/2+1,j)}\ket{\text{EIG}}=(-1)^{L_{x}/2+j}\ket{\text{EIG}},$
(13) $\displaystyle
Y_{(L_{x}/2,j)}Y_{(L_{x}/2+1,j)}\ket{\text{EIG}}=-(-1)^{L_{x}/2+j}\ket{\text{EIG}}.$

Upon which summing over $j$, we see that $\ket{\text{EIG}}$ is an eigenstate
of the XY Hamiltonian with eigenvalue
$-(J_{x}-J_{y})(-1)^{L_{x}/2}\text{mod}(L_{y},2)$.

There also exist other closely related eigenstates in the spectrum of the XY
model. They are

$\displaystyle\ket{\text{EIG}}_{IZ}=\prod_{i+j\in\text{even}}\ket{I_{(i,j),(\overline{i},j)}}\prod_{i+j\in\text{odd}}\ket{Z_{(i,j),(\overline{i},j)}},$
(14)
$\displaystyle\ket{\text{EIG}}_{ZI}=\prod_{i+j\in\text{even}}\ket{Z_{(i,j),(\overline{i},j)}}\prod_{i+j\in\text{odd}}\ket{I_{(i,j),(\overline{i},j)}},$
$\displaystyle\ket{\text{EIG}}_{XY}=\prod_{i+j\in\text{even}}\ket{X_{(i,j),(\overline{i},j)}}\prod_{i+j\in\text{odd}}\ket{Y_{(i,j),(\overline{i},j)}},$
$\displaystyle\ket{\text{EIG}}_{YX}=\prod_{i+j\in\text{even}}\ket{Y_{(i,j),(\overline{i},j)}}\prod_{i+j\in\text{odd}}\ket{X_{(i,j),(\overline{i},j)}}.$

The first is the one discussed above and the other three are related to first
one by single qubit rotations. Repeating the above analysis for other three
shows that they are indeed eigenstates with the corresponding eigenvalues;
$(J_{x}-J_{y})(-1)^{L_{x}/2}\text{mod}(L_{y},2)$,
$-(J_{x}+J_{y})(-1)^{L_{x}/2}\text{mod}(L_{y},2)$ and
$(J_{x}+J_{y})(-1)^{L_{x}/2}\text{mod}(L_{y},2)$. They are degenerate for even
$L_{y}$ but are distinct for odd $L_{y}$.

These eigenstates have maximal entanglement entropy within each row and zero
entanglement entropy between rows. For even $L_{y}$, one can also write down
four additional eigenstates containing the Bell states along the columns.