# A scheme to create and verify scalable entanglement in optical lattice

You Zhou  These authors contributed equally to this work  Department of Modern
Physics, University of Science and Technology of China, Hefei, Anhui 230026,
China  Key Laboratory for Information Science of Electromagnetic Waves
(Ministry of Education), Fudan University, Shanghai 200433, China  Bo Xiao
These authors contributed equally to this work  Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China  CAS
Center for Excellence and Synergetic Innovation Center in Quantum Information
and Quantum Physics, University of Science and Technology of China, Hefei,
Anhui 230026, China  Meng-Da Li  Department of Modern Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China  CAS Center for
Excellence and Synergetic Innovation Center in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei, Anhui 230026,
China  Qi Zhao  Joint Center for Quantum Information and Computer Science,
University of Maryland, College Park, Maryland 20742, USA  Zhen-Sheng Yuan
<EMAIL_ADDRESS>Department of Modern Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China  CAS Center for Excellence and
Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Xiongfeng Ma<EMAIL_ADDRESS>Center for Quantum Information, Institute
for Interdisciplinary Information Sciences, Tsinghua University, Beijing
100084, China  Jian-Wei Pan  Department of Modern Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China  CAS Center for
Excellence and Synergetic Innovation Center in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei, Anhui 230026,
China

##  I Abstract

To achieve scalable quantum information processing, great efforts have been
devoted to the creation of large-scale entangled states in various physical
systems. Ultracold atom in optical lattice is considered as one of the
promising platforms due to its feasible initialization and parallel
manipulation. In this work, we propose an efficient scheme to generate and
characterize global entanglement in the optical lattice. With only two-layer
quantum circuits, the generation utilizes two-qubit entangling gates based on
the superexchange interaction in double wells. The parallelism of these
operations enables the generation to be fast and scalable. To verify the
entanglement of this non-stabilizer state, we mainly design three
complementary detection protocols which are less resource-consuming compared
to the full tomography. In particular, one just needs two homogenous local
measurement settings to identify the entanglement property. Our entanglement
generation and verification protocols provide the foundation for the further
quantum information processing in optical lattice.

##  II Introduction

Quantum information and quantum computation [1], which harvest the intrinsic
quantum features, like superposition and entanglement [2], can show advantages
against their classical counterparts. To build a practical quantum information
processor, the computation platform with high scalability is preferred and the
qubits should be coupled to each other to form a large-scale entanglement.
Hence, many researches have been focusing on the generation of scalable
entangled states in different physical systems, e.g, ion trap [3, 4, 5],
photons [6, 7], Rydberg atoms [8] and superconducting circuits [9, 10, 11].
Even though there are significant progresses of the qubit number in various
systems [12, 13], the generation of large-scale entanglement is still
challenging for the Noisy Intermediate Scale Quantum Devices [14].

Ultracold atoms in optical lattice [15] could be a practicable system to
overcome this challenge due to its feasible initialization and parallel
manipulation. By adiabatically increasing the lattice depth, the phase of
ultracold atom can be tuned from superfluid (SF) to Mott insulator (MI) [16,
17]. Under an unit filling rate, numerous atoms can be confined in the lattice
and serves as qubits. Based on this initialization, entangled states in
optical lattice have been demonstrated experimentally, for instance, the
generation of cluster state with controlled collision gate induced by the
spin-dependent lattice [18, 19], which acts as the resource state for the
measurement-based quantum computing [20, 21]. The development of superlattice
further improves the ability to control ultracold atoms, which is formed by
overlapping two different optical lattices to generate a series of double
wells. The structure of the double well can be modified to induce different
kinds of atomic dynamics, such as superexchange coupling [22] and controlled
exchange interaction [23], which can be both used to realize
$\sqrt{\mathrm{SWAP}}$ gate and entangle two atoms in a double well [24, 23].
Besides, the entangling operations in the superlattice can be performed in a
parallel way based on the periodicity of the lattice system, which is suitable
for the fast generation of large amount of entangled pairs and even large-
scale entangled states. Note that this kind of parallel operation can also be
implemented with multi-tweezer in Rydberg atom experiments [25, 26].

Every coin has two sides. The periodicity property also induces some
restrictions on the quantum operation and measurement. The small lattice
spacing, required by the large tunnelling between neighbour sites, creates a
challenge for the individual control of atoms, say, some local basis
rotations. The tight-focused optical tweezer [27] created with the recently
developed high-resolution imaging [28, 29, 30, 31, 32] could be a solution.
However, it is still challenging to perform a few of different single-qubit
operations on multiple qubits under a realistic time-scale to ensure the
system coherence. As a result, homogeneous operations and measurements on all
atoms are preferred in optical lattice experiments.

In this work, we propose a scheme to generate scalable entanglement of
ultracold atoms which is suitable for the implementation in the optical
superlattice system. It mainly contains two entangling steps: first, entangle
the atom-pair in each double well by the $\sqrt{\mathrm{SWAP}}$ gate; second,
shift the position of double wells to a single site by changing the phase of
superlattice, and then entangle the the new atom-pair with
$\sqrt{\mathrm{SWAP}}$ again. In this way, all atoms in the superlattice can
be connected with neighbours to form a global entangled state. Our theoretical
analysis shows that the final state possesses genuine multipartite
entanglement (GME). In addition, the final state is also less sensitive to
magnetic noises which can cause decoherence, since it only owns amplitude on
the computational basis whose total spin is zero.

In actual experiments, the inevitable noise may degrade the entanglement,
which should be verified further. Compared with quantum tomography [33, 34],
entanglement witness is a more efficient way [35, 36, 37] to realize this task
based on the pre-knowledge of the preparation. Current entanglement witnesses
are usually designed for some structured states, such as permutation-invariant
states [38, 39, 40] and stabilizer states [41, 42, 43]. However, in our
protocol the state generated by the non-Clifford gate $\sqrt{\mathrm{SWAP}}$
is not a standard stabilizer state, which makes the entanglement verification
challenging. To overcome this challenge, we first construct an entanglement
witness based on the preparation fidelity. By adopting the decomposition using
stabilizer-like method, we can lower bound the fidelity with a few spin-
correlation measurements. Some correlations among them are inhomogenous and
thus require the individual atom addressing whose implementation could be
challenging with current techniques. To further ease the experiment
realization, we show another complementary protocol which only requires
homogenous spin-correlations and only two measurement settings. At last, by
reversely evolving the state with the conjugate quantum gates, we provide an
intuitive method to indirectly bound the preparation fidelity and thus
qualitatively verify the entanglement.

##  III Results

###  III.1 Entangling Gates in Optical Superlattices

The behavior of ultracold atoms in optical lattices can be described by the
Hubbard model which is characterized by the tunnel coupling between
neighbouring sites $J$, the on-site interaction $V$, and the effective
chemical potential $\mu$. By increasing the depth of the lattice, a phase
transition from SF to MI can occur and the atoms start to be localized in each
site. To extend the ability to manipulate atoms, a more complicated periodic
potential, named superlattice, was proposed and applied in many experiments
[44, 45, 46, 47]. It is normally formed by overlapping two distinct optical
lattices. The period of the first lattice (denoted as the long lattice) is
twice as that of the second (denoted as the short lattice), which induces an
array of double wells steadily. The resulting potential shows
$V_{\mathrm{t}}(x,\phi)=V_{\mathrm{l}}\text{cos}^{2}(\uppi
x/a+\phi)+V_{\mathrm{s}}\text{cos}^{2}(2\uppi x/a)$ with $a$ being the lattice
spacing of the long lattice. The structure of such potential is dependent on
both the relative strength of two lattices and the relative phase $\phi$. When
the phase $\phi$ is not equal to $n\uppi/2$ ($n$ is an integer), all the
double wells are biased with a non-zero tilt $\Delta$ between the subsites. In
addition, the center of each double well can shift a period of short lattice
via changing n by an odd number, which is illustrated in Figure 1 (b).

Figure 1: The features of superlattice. (a) Double-well system characterized
by the parameters $V$, $J_{\mathrm{inner}}$ and $\Delta$. (b) The shift of the
double-well center.

Subsites in each double well are separated by a low barrier, while neighbour
double wells are separated by a high barrier. We denote corresponding
tunneling strengths as $J_{\mathrm{inter}}$ and $J_{\mathrm{inner}}$
respectively. As $J_{\mathrm{inter}}$ is much smaller than
$J_{\mathrm{inner}}$, the hopping event between different double wells can be
ignored and the movement of atoms is restricted in each double well. In this
case, for two-level bosonic atoms, their dynamics in double wells can be
described well with a two-site mode Bose-Hubbard Hamiltonian characterized
with $V$, $J_{\mathrm{inner}}$ and $\Delta$ as follows.

$\displaystyle\hat{H}=$ $\displaystyle-
J_{\mathrm{inner}}\sum_{\sigma=\uparrow,\downarrow}(\hat{a}_{\mathrm{L},\sigma}^{{\dagger}}\hat{a}_{\mathrm{R},\sigma}+\hat{a}_{\mathrm{R},\sigma}^{{\dagger}}\hat{a}_{\mathrm{L},\sigma})$
(1)
$\displaystyle+\frac{V}{2}\sum_{i=\mathrm{R},\mathrm{L}}\hat{n}_{i,\uparrow}(\hat{n}_{i,\uparrow}-1)+\frac{V}{2}\sum_{i=\mathrm{R},\mathrm{L}}\hat{n}_{i,\downarrow}(\hat{n}_{i,\downarrow}-1)$
$\displaystyle+V\sum_{i=\mathrm{R},\mathrm{L}}(\hat{n}_{i,\uparrow}\hat{n}_{i,\downarrow}+\hat{n}_{i,\downarrow}\hat{n}_{i,\uparrow})+\frac{\Delta}{2}(\hat{n}_{\mathrm{L},\uparrow}-\hat{n}_{\mathrm{R},\uparrow}-\hat{n}_{\mathrm{L},\downarrow}+\hat{n}_{\mathrm{R},\downarrow}).$

Here, $\mathrm{R}(L)$ denotes right(left) subsite,
$\hat{a}_{i,\sigma}^{{\dagger}}$($\hat{a}_{i,\sigma}$) is the
creation(annihilation) operator for the boson on site i with the inner state
$\sigma$, and $\hat{n}_{i,\sigma}$ is the number operator. We consider a non-
biased double-well system starting with unit filling—one atom per site. Note
that the chemical potential $\mu$ is fixed here for unit filling and thus has
no effect on the dynamics, and it is ignored in Eq. (1). In the limit $V\gg
J_{\mathrm{inner}}$, the large $V$ prevents multiple occupation, so that the
system would evolve in a subspace consists of four basis states labeled by
inner states of the bosons $\ket{\uparrow,\uparrow}$,
$\ket{\downarrow,\uparrow}$, $\ket{\uparrow,\downarrow}$ and
$\ket{\downarrow,\downarrow}$. By using perturbation theory, the above model
in this subspace is equivalent to the isotropic Heisenberg Hamiltonian [48]:

$\hat{H}_{\mathrm{eff}}=-\frac{J_{\mathrm{ex}}}{2}(\hat{X}_{\mathrm{L}}\hat{X}_{\mathrm{R}}+\hat{Y}_{\mathrm{L}}\hat{Y}_{\mathrm{R}}+\hat{Z}_{\mathrm{L}}\hat{Z}_{\mathrm{R}}),$
(2)

where $\hat{X}_{i}$, $\hat{Y}_{i}$, $\hat{Z}_{i}$ are Pauli operators on
subsite $i=\mathrm{R},\mathrm{L}$, and $J_{\mathrm{ex}}\approx
2J_{\mathrm{inner}}^{2}/V$ is the superexchange coupling between subsites.

In the remaining of this work, we denote spin configuration $\ket{\downarrow}$
and $\ket{\uparrow}$ as $\ket{0}$ and $\ket{1}$. Initialized at $\ket{0,1}$
and driven by this effective Hamiltonian, the state of the system can
oscillate between $\ket{0,1}$ and $\ket{1,0}$ with a period of
$T=2\uppi\hbar/J_{\mathrm{ex}}$ while global phase is also recovered, named
the superexchange process. Taking an evolution time of $T/8$, the product
states $\ket{0,1}$ and $\ket{1,0}$ are prepared into two-qubit maximally
entangled states $1/\sqrt{2}(\ket{0,1}+i\ket{1,0})$ and
$1/\sqrt{2}(\ket{0,1}-i\ket{1,0})$ respectively while $\ket{0,0}$ and
$\ket{1,1}$ remain unchanged due to the high energy gap of $V$. The effective
unitary transformation corresponds to a $\sqrt{\mathrm{SWAP}}^{\dagger}$ gate
operation, i.e.,

$U_{\sqrt{\mathrm{SWAP}}^{\dagger}}=\begin{pmatrix}1&0&0&0\\\
0&(1-i)/2&(1+i)/2&0\\\ 0&(1+i)/2&(1-i)/2&0\\\ 0&0&0&1\end{pmatrix}.$ (3)

with respect to the basis $\ket{0,0}$, $\ket{1,0}$, $\ket{0,1}$, $\ket{1,1}$.
Moreover, with an evolution time of $3T/8$, one can realize the corresponding
$\sqrt{\mathrm{SWAP}}$ gate. In particular, when
$J_{\mathrm{inter}}/J_{\mathrm{ex}}\geq 25$, the infidelity of the
$\sqrt{\mathrm{SWAP}}^{\dagger}$ operation caused by the tunneling between
neighbour double wells would be smaller than $0.1\%$.

For the current gate generation scheme, due to the large ratio
$V/J_{\mathrm{inner}}$, the effective $J_{\mathrm{ex}}$ is far less than
$J_{\mathrm{inner}}$. In addition, such high $V/J_{\mathrm{inner}}$ requires a
deep potential in each site thus leads to a small $J_{\mathrm{inner}}$. As a
result, the period of the superexchange process would be long, e.g, tens of
milliseconds for 87Rb atoms in superlattice [22, 24], which would aggravate
the decoherence effect. To ease this problem, an alternative and faster gate
generation scheme can be adopted here [49], which is performed with a small
$V/J_{\mathrm{inner}}$. Instead of introducing the large energy ratio, this
scheme utilizes the coherent competition of superexchange and atom tunneling
to decrease the component with double occupation in state. As
$V/J_{\mathrm{inner}}$ is set to be a finite value $4/\sqrt{3}$, the undesired
component can be eliminated completely at specific time intervals. In
particular, with an evolution time of $\uppi\hbar/V$, such elimination leads
to a fast $\sqrt{\mathrm{SWAP}}^{\dagger}$ gate realization with high
fidelity.

Besides, the spin-dependent effect further improves the ability to manipulate
the atoms in optical lattices. By adding circular polarization components into
one lattice light field of superlattice [45, 50], the tilt $\Delta$ can be
different for different inner states. Therefore, the energy gap between inner
states on right subsite would be different from that on the left which induce
two applications here. One is to spin flip atoms in one subsite of every
double well without affecting atoms in the other subsite, enabling the state
transfer between the four basis states mentioned above [45]. The other is to
generate a relative phase between $\ket{0,1}$ and $\ket{1,0}$ and then
transfer the states $1/\sqrt{2}(\ket{0,1}-i\ket{1,0})$,
$1/\sqrt{2}(\ket{1,0}-i\ket{0,1})$ to the standard Bell states
$1/\sqrt{2}(\ket{1,0}+\ket{0,1})$ , $1/\sqrt{2}(\ket{1,0}-\ket{0,1})$ [24],
respectively.

###  III.2 Entanglement generation protocol

Our protocol is expected to be performed on one-dimensional atomic chains
along X direction with a short lattice isolating the atoms. On each site, the
filling is initialized into unit through the SF-MI phase transition. Assuming
the inner state of atoms are prepared into $\ket{0}$ and taking 10-qubit
system as example, the protocol can be divided into the following steps which
are illustrated in Fig. 2.

  1. 1.

Turn on the spin-dependent superlattice [50] by ramping up a spin-dependent
long lattice along X. With selective spin-flip by coupled field, the Néel
state is prepared:

$\ket{\Phi_{1}}=\ket{1010101010}$ (4)

  2. 2.

Turn off the spin-dependent effect and perform
$\sqrt{\mathrm{SWAP}}^{\dagger}$ gate on the atom pairs
$(1,2),(3,4),\cdots,(9,10)$ in each double well,

$\displaystyle\ket{\Phi_{2}^{\prime}}=$
$\displaystyle\bigotimes_{k=1}^{5}\sqrt{\mathrm{SWAP}}^{\dagger}_{\\{2k-1,2k\\}}\ket{\Phi_{1}}$
(5) $\displaystyle=$
$\displaystyle\bigotimes_{k=1}^{5}\frac{1}{\sqrt{2}}(\ket{10}+i\ket{01})_{\\{2k-1,2k\\}}.$

  3. 3.

Turn on the spin-dependent effect again and introduce a relative phase
$\uppi/2$ on $\ket{10}$. As a result, the state of the system is the tensor
product of five Bell states,

$\displaystyle\ket{\Phi_{2}}=U_{\mathrm{phase}}\ket{\Phi_{2}^{\prime}}=\bigotimes_{k=1}^{5}\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})_{\\{2k-1,2k\\}}.$
(6)

  4. 4.

Change the relative phase $\phi$ by $\uppi/2$ to trap the atom pair
$(2,3),(4,5),\cdots,(8,9)$ in different double wells.

  5. 5.

Perform $\sqrt{\mathrm{SWAP}}^{\dagger}$ gates on the atom pairs
$(2,3),(4,5),\cdots,(8,9)$ in each double well, the final state is

$\displaystyle\ket{\Psi}=\bigotimes_{k=1}^{4}\sqrt{\mathrm{SWAP}}^{\dagger}_{\\{2k,2k+1\\}}\ket{\Phi_{2}}.$
(7)

Figure 2: Illustration of the preparation protocol. (a) Generate Néel state
with selective spin-flip. (b) Entangle and prepare the atom pairs into Bell
state with $\sqrt{\mathrm{SWAP}}^{\dagger}$ gate. (c) Shift the position of
double wells by changing the relative phase $\phi$. (d) Entangle the neighbour
atoms which are not entangled in (b).

During the entire process atomic motions along other directions are frozen by
deep lattice potentials. Finally, one-dimensional entangled systems are
generated. The gate operations entangle every pair of neighbour atoms so that
the target state is GME by itself. As the gate operation is not present, the
depth of each site should be deep enough to avoid any undesirable hopping
events.

The target state only owns non-zero projection on a few bases whose total spin
numbers are zero, and the magnetic field fluctuation can only introduce a
global phase. Consequently, our target can be robust to the magnetic field
fluctuation, and thus has longer coherence time, compared to for example the
GHZ state. In this basis collection, each basis can find another basis whose
spin on each qubit is anti-parallel to its own, e.g. $\ket{0,1,1,0,...}$ and
$\ket{1,0,0,1,...}$, and the projections on each pair of bases show same
modules but different arguments.

According to the definition of GME [2], it is inferred that a pure state whose
subsystems are all mixed states must be a GME state. Based on this, the purity
of all subsystems of the target state in the case of 10-qubit are calculated
and found to be less than 1, which verifies that this 10-qubit target state
possesses GME. This conclusion can be extended to the target state with a
different number of qubits.

We remark that Ref. [51] also studies the generation of entanglement by
$\sqrt{\mathrm{SWAP}}$ in the cold atom system. The state generated there is
like a GHZ state, which needs much more gate operations, compared to the
cluster-like state here. In addition, there is no valid method given there to
verify the entanglement.

In the following sections, we exhibit theoretical methods to detect the
multipartite entanglement in the cold atom system. The prepared state shown in
the Sec. III.2 is generated by parallel $\sqrt{\mathrm{SWAP}}^{\dagger}$ gates
and thus not a stabilizer state, such as the GHZ state and the cluster state.
This fact makes the entanglement detection task challenging and the methods
proposed in literature are not suitable in this scenario.

To overcome this challenge, we propose three complementary methods. The first
one is based on the fidelity estimation to the target state to detect the
strongest form of entanglement—genuine multipartite entanglement. By
evaluating the non-Pauli stabilizer after the entangling gate evolution, the
method only needs constant number of measurement setting with respect to the
system size to lower bound the fidelity. To further release the experiment
efforts, the second method adopts homogeneous measurements, with only two
measurement settings. Thus it is very efficient to realize and can tell
whether the prepared state is separable or not considering any bipartition of
the whole system. The third one is more intuitional but can reveal GME with
less experiment requirements. In particular, we indirectly estimate the
fidelity between the prepared state and the target state by evolving the
entangling process forward and backward.

All these methods need the spin measurement on single atom, however the
fluorescence imaging used in high-resolution experiments can not resolve
different spins. As the state we consider here only has one atom on each site,
it could be done by measuring the atom distribution after removing one spin
component in which the occupied site and unoccupied site represent different
spins [52] or splitting different spin components with gradient along
perpendicular direction [53].

Before showing these three methods, we give some related definitions about
multipartite entanglement.

A pure state is (bi-)separable if it is in a tensor product form
$\ket{\Psi_{b}}=\ket{\Phi_{A}}\otimes\ket{\Phi_{\bar{A}}}$, where
$\mathcal{P}_{2}=\\{A,\bar{A}\\}$ is a bipartition of the qubits in the
system. A mixed state is separable if it can be written as a mixing of pure
separable states,
$\rho_{\mathrm{b}}=\sum_{i}p_{i}\ket{\Psi_{\mathrm{b}}^{i}}\bra{\Psi_{\mathrm{b}}^{i}}$.
Note that each separable state $\ket{\Psi_{\mathrm{b}}^{i}}$ in the summation
can have different bipartitions. The separable state set is denoted as
$S_{\mathrm{b}}$. There is another restricted way for the extension to mixed
states. A state is $\mathcal{P}_{2}$-separable, if it is a mixing of pure
separable states with a same partition $\mathcal{P}_{2}$, and we denote the
state set as $S_{\mathrm{b}}^{\mathcal{P}_{2}}$. It is clear that
$S_{\mathrm{b}}^{\mathcal{P}_{2}}\subset S_{\mathrm{b}}$, and $S_{\mathrm{b}}$
can be generated by the convex mixture of all possible
$S_{\mathrm{b}}^{\mathcal{P}_{2}}$.

######  Definition 1.

An $N$-qubit quantum state $\rho$ is fully entangled, if it is outside of the
separable state set $S_{\mathrm{b}}^{\mathcal{P}_{2}}$ for any bipartition,

$\rho\notin
S_{\mathrm{b}}^{\mathcal{P}_{2}},\forall\mathcal{P}_{2}=\\{A,\bar{A}\\}.$ (8)

######  Definition 2.

An $N$-qubit quantum state $\rho$ possesses genuine multipartite entanglement,
if it is outside of the separable state set $S_{\mathrm{b}}$,

$\rho\notin S_{\mathrm{b}}.$ (9)

Since $S_{\mathrm{b}}^{\mathcal{P}_{2}}\subset S_{\mathrm{b}}$, GME is a
stronger claim than full entanglement. We also remark that the recently
demonstrated entanglement in the IBM cloud quantum computing [54] is actually
the full entanglement defined here. By Def. 1, for a state with full
entanglement, it is possible to prepare it by mixing bi-separable states with
different bipartitions [55]. On the other hand, GME describes the strongest
form of quantum entanglement, that is, all the qubits in the system are indeed
entangled with each other. GME is essential in various multipartite quantum
information tasks, such as quantum cryptography [56], quantum nonlocality
[57], quantum networks [58, 59], quantum metrology [60] and measurement-based
quantum computing [61].

###  III.3 Entanglement detection based on fidelity estimation

In this section, we show an entanglement detection protocol based on the
fidelity value between the prepared state and the target state.

######  Proposition 1.

The operator $\mathcal{W}_{\Psi}$ can witness genuine multipartite
entanglement near $\ket{\Psi}$,

$\mathcal{W}_{\Psi}=\frac{5}{8}\mathbb{I}-\ket{\Psi}\bra{\Psi},$ (10)

with $\langle\mathcal{W}_{\Psi}\rangle\geq 0$ for any separable state in
$S_{\mathrm{b}}$.

According to Proposition 1, if the fidelity of the prepared state
$\rho_{\mathrm{pre}}$ with the target state $\ket{\Psi}$, i.e.,
$\mathrm{Tr}(\rho_{\mathrm{pre}}\ket{\Psi}\bra{\Psi})$, exceeds $\frac{5}{8}$,
$\rho_{\mathrm{pre}}$ possesses GME. However, it is generally difficult to
evaluate the quantity $\mathrm{Tr}(\rho_{\mathrm{pre}}\ket{\Psi}\bra{\Psi})$
by the direct projection on $\ket{\Psi}$, as it is an entangled state.

Alternatively, one needs to decompose the density operator $\Psi$ into the
summation of many local measurements in the form $\otimes_{i=1}^{N}O_{i}$,
which is easier to implement in experiments. Here $O_{i}$ is Hermitian
operator of the i-th qubit. The number of local measurements characterizes the
experiment effort to estimate the fidelity.

Here, in order to reduce the measurement effort, instead of direct decomposing
$\Psi$, we give a lower bound of the fidelity by using the stabilizer-like
operator for the non-stabilizer state $\Psi$.

######  Proposition 2.

For the target state $\ket{\Psi}\bra{\Psi}$ and its $N$ independent
stabilizers $S_{i}^{\prime}$, the following inequality holds,

$\ket{\Psi}\bra{\Psi}\geq\frac{1}{2}\sum_{i=1}^{N}S_{i}^{\prime}-\left(\frac{N}{2}-1\right)\mathbb{I},$
(11)

where $A\geq B$ indicates that $(A-B)$ is positive semidefinite. The
stabilizer is determined by the evolution of $\sqrt{\mathrm{SWAP}}$ gates
$S_{i}^{\prime}=US_{i}U^{{\dagger}}$ with
$U=\bigotimes_{k=1}^{\frac{N}{2}-1}\sqrt{\mathrm{SWAP}}^{\dagger}_{\\{2k,2k+1\\}}$,
where

$\displaystyle S_{2k-1}=X_{k}X_{k+1},S_{2k}=-Z_{k}Z_{k+1},$ (12)

for $k=1,\cdots N/2$, are the stabilizers for the Bell pairs.

Due to $\sqrt{\mathrm{SWAP}}^{{\dagger}}$ is not a Clifford gate, the
corresponding stabilizer $S_{i}^{\prime}$ is not in a tensor product of Pauli
operators, but the summation of them. One can directly get $S_{i}^{\prime}$ by
the evolution of the parallel $\sqrt{\mathrm{SWAP}}^{\dagger}$ on $S_{i}$. Due
to the locality of the evolution, $S_{i}$ can only be transformed by the gate
with overlapping support. For example,

$\begin{split}S_{1}^{\prime}=U_{2,3}X_{1}X_{2}U^{\dagger}_{2,3}&=\frac{1}{2}X_{1}(X_{2}I_{3}+I_{2}X_{3}-Y_{2}Z_{3}+Z_{2}Y_{3}),\\\
S_{2}^{\prime}=-U_{2,3}Z_{1}Z_{2}U^{\dagger}_{2,3}&=-\frac{1}{2}Z_{1}(Z_{2}I_{3}+I_{2}Z_{3}+Y_{2}X_{3}-X_{2}Y_{3})\end{split}$
(13)

where each new stabilizer is the summation of four Pauli operator. For the
stabilizers in the bulk of the 1-D chain, there are two
$\sqrt{\mathrm{SWAP}}^{\dagger}$ on them. Take $X_{3}X_{4}$ and $-Z_{3}Z_{4}$
as example, $U_{2,3}$ and $U_{4,5}$ are performed on them. As a result, the
corresponding stabilizers are:

$\begin{split}&(U_{2,3}\otimes U_{4,5})X_{3}X_{4}(U_{2,3}\otimes
U_{4,5})^{\dagger}=\\\
&(X_{2}I_{3}+I_{2}X_{3}+Y_{2}Z_{3}-Z_{2}Y_{3})\otimes\\\
&(X_{4}I_{5}+I_{4}X_{5}-Y_{4}Z_{5}+Z_{4}Y_{5}),\\\ -&(U_{2,3}\otimes
U_{4,5})Z_{3}Z_{4}(U_{2,3}\otimes U_{4,5})^{\dagger}=\\\
-&(Z_{2}I_{3}+I_{2}Z_{3}+Y_{2}X_{3}-X_{2}Y_{3})\otimes\\\
&(Z_{4}I_{5}+I_{4}Z_{5}-Y_{4}X_{5}+X_{4}Y_{5}).\end{split}$ (14)

Both are the linear combinations of 16 Pauli tensors. For other stabilizers,
similar forms could be obtained in this way.

In total, the number of these Pauli tensors is almost $32N$ which scales
linearly with the qubit number. In fact, the measurement effort can be further
reduced, as several Pauli tensors can be grouped and measured in one local
measurement setting (LMS) simultaneously. For the Pauli tensors from
$S_{2k-1}^{\prime}=UX_{2k-1}X_{2k}U^{\dagger}$, we introduce the following
expression for the example case $N=10$,

$\displaystyle
X_{1}\otimes(X_{2}X_{3},Y_{2}Z_{3},Z_{2}Y_{3})\otimes(X_{4}X_{5},Y_{4}Z_{5},Z_{4}Y_{5})$
(15)
$\displaystyle\otimes(X_{6}X_{7},Y_{6}Z_{7},Z_{5}6_{7})\otimes(X_{8}X_{9},Y_{8}Z_{9},Z_{8}Y_{9})\otimes
X_{10},$

where one can periodically select one of the three Pauli tensors in every
brace to construct one LMS, such that these LMSs cover all the possible Pauli
operators from $S_{2k-1}^{\prime}$. That is, select same tensor in (2,3) and
(6,7) and the same Pauli operator in (4,5) and (8,9). Thus, only 9 LMSs are
needed here. Following the same way, we can also find another 9 LMSs for the
Pauli operators from $S_{2k}^{\prime}=UZ_{2k-1}Z_{2k}U^{\dagger}$. As a
result, totally only a constant number of $18$ LMSs is needed to obtain all
the expectation values of the stabilizers.

To evaluate the robustness of our witness, we apply the white noise model
$\rho_{\mathrm{pre}}=p\mathbb{I}/2^{N}+(1-p)\ket{\Psi}\bra{\Psi}$, and it
shows that the noise tolerance is $p=\frac{3}{4N}$. As $N=10$, it equals
$7.5\%$. In fact, by utilizing the trade-off between the robustness and the
measure budget [62, 63], one can enhance the noise tolerance of the witness
further. In Ref. [64], some of us further generalize the witness here by
utilizing more local measurement settings, and the witnesses constructed there
would be more suitable to realize in other quantum systems, such as
superconducting-qubit. In particular, a numerical algorithm is developed in
Ref. [64] to search for the witness with the optimal noise tolerance under
given measurement settings. For instance, one can reach the noise tolerance
$p=\frac{3}{16(1-2^{-N/2})}$ with about $2\cdot 3^{(N/2-1)}$ settings.

###  III.4 Entanglement detection with homogeneous measurements

The entanglement witness shown in Sec. III.3 based on the fidelity estimation
can detect GME, with a few number of LMSs. However, in each of the LMS, the
Pauli operators may be not the same, such as $X_{1}Y_{2}Z_{3}\cdots$. This
kind of inhomogeoneus measurement needs additional local basis rotation, which
is challenging for the current cold atom system.

In this section, we reduce the measurement efforts further by only considering
the homogenous measurement, say $O^{\otimes N}$. In particular, we only need
two LMSs, $X^{\otimes N}$ and $Z^{\otimes N}$. Note that $X^{\otimes N}$ and
$Y^{\otimes N}$ are symmetric for our generation, thus we only need to measure
one of them. Detailed illustration on this point is given in Method.

Instead of detecting GME, the protocol here aims to detect the full
entanglement property defined in Eq. (8), i.e., not separable with respective
to any bipartition. First, let us show the following Lemma which plays an
important role of the detection.

######  Lemma 1.

For a $k$-qubit quantum state $\rho$ with $k$ being even, if it is
$\mathcal{P}_{2}$-separable, with $\mathcal{P}_{2}=\\{A,\bar{A}\\}$ and the
number of qubits contained in $A$ and $\bar{A}$ being odd, the following
inequality holds,

$\displaystyle|\langle X^{\otimes k}\rangle|+|\langle Y^{\otimes
k}\rangle|+|\langle Z^{\otimes k}\rangle|\leq 1,$ (16)

where $\langle O\rangle=\mathrm{Tr}(\rho O)$ is the expectation value.

The proof of Lemma is based on the anticommutative relation [65, 66] on
subsystems $A$ and $\bar{A}$ respectively. As $k=2$, it becomes to the common
criterion for the Bell state. As a result, the violation of the bound in Eq.
(16) indicates that the underlying even-qubit state is non-separable regarding
any odd-odd bipartitions.

Second, we give the following observation which shows the relation of the
entanglement property of the state and its reduced density matrix (RDM).

######  Observation 1.

Suppose a quantum state $\rho^{\prime}_{B,\bar{B}}$ is created from
$\rho_{A,\bar{A}}$ by local operation and classical communications(LOCC)
$\Lambda$ with respective to the bipartiton $\\{A,\bar{A}\\}$, i.e.,

$\displaystyle\rho^{\prime}_{B,\bar{B}}=\Lambda_{A\rightarrow
B,\bar{A}\rightarrow\bar{B}}(\rho_{A,\bar{A}}),$ (17)

If $\rho_{A,\bar{A}}$ is separable for the bipartition $\\{A,\bar{A}\\}$,
$\rho^{\prime}_{B,\bar{B}}$ is also separable for the bipartition
$\\{B,\bar{B}\\}$; in other words, if $\rho^{\prime}_{B,\bar{B}}$ is entangled
for $\\{B,\bar{B}\\}$, $\rho_{A,\bar{A}}$ is also entangled for
$\\{A,\bar{A}\\}$.

The observation holds as LOCC can not create entanglement. Specifically, the
partial trace operation is an example of LOCC, where $B,\bar{B}$ are
subsystems of $A,\bar{A}$ respectively.

By Definition 1, an $N$-qubit quantum state is fully entangled, if it cannot
been written in the following form

$\displaystyle\rho\neq\sum_{i}p_{i}\rho_{A}^{i}\otimes\rho_{\bar{A}}^{i},$
(18)

for all possible bipartitions $\\{A,\bar{A}\\}$ of the whole system. In the
following, we employ Lemma 1 and Observation 1 time and time again on RDMs of
the 1-D prepared state, and finally certify its full entanglement. The RDM of
the prepared state $\rho_{\mathrm{pre}}$, for example, on the first two
qubits, is denoted by $\rho_{12}$. Note that the given expectation values are
from the perfect target state. The practical value in the experiment may
deviate from it, but can still reveal the entanglement property if the
deviation is not too large. Detailed noise analysis are shown in Method.

To certify the full entanglement, we prove it by contradiction. Specifically,
we first assume that the state can be written as right hand side of Eq. (18)
for some $\\{A,\bar{A}\\}$, and then show that all the qubit are either in $A$
or in $\bar{A}$, which is actually not separable and leads to the
contradiction.

Here we take $N=6$ for simplicity, and it can be directly generalized to any
even qubit number $N$. The procedure is listed as follows.

  1. 1.

For the RDM $\rho_{12}$, one has

$\displaystyle|\langle X_{1}X_{2}\rangle|+|\langle Y_{1}Y_{2}\rangle|+|\langle
Z_{1}Z_{2}\rangle|=\gamma_{2}>1,$ (19)

with $\gamma_{2}=1.5$ for the target state. It indicates that $\rho_{12}$ is
non-separable on account of Lemma 1. Thus qubits $1,2$ should both be
contained in $A$ or $\bar{A}$, otherwise it violates Observation 1. Without
loss of generality, we assume $1,2$ are in $A$. Similarly, $\rho_{13}$ also
satisfies the inequality in Eq. (19), and thus $1,2,3$ are all in $A$.

  2. 2.

In fact, one can not proceed along the chain, since the RDMs
$\rho_{23},\rho_{34}\cdots$ are not entangled. Thus we consider correlations
involving more qubits. For the RDM $\rho_{1234}$, one has

$\displaystyle\left|\left\langle\bigotimes_{i=1}^{4}X_{i}\right\rangle\right|+\left|\left\langle\bigotimes_{i=1}^{4}Y_{i}\right\rangle\right|+\left|\left\langle\bigotimes_{i=1}^{4}Z_{i}\right\rangle\right|=\gamma_{4}>1,$
(20)

with $\gamma_{4}=1.5$ for the target state. It indicates that $\rho_{1234}$ is
non-separable for the bipartition $\\{123,4\\}$ due to Lemma 1. Thus qubits
$1,2,3,4$ should all be contained in $A$, due to Observation 1 and Step 1.

  3. 3.

Same as Step 1, by applying Eq. (19) for $\rho_{46}$ and $\rho_{56}$, one can
proceed to conclude that qubits $4,5,6$ should all be contained in $A$ or
$\bar{A}$. Due to the result $\\{1,2,3,4\\}\subset A$ got in Step 2, we can
arrive at the result that all the qubits are contained in $A$. This is
contradict to the separable state on the right hand side of Eq. (18). Thus the
prepared state owns full entanglement.

###  III.5 Entanglement detection with reverse evolution

Compared with the previous two entanglement detection protocols, the third one
is more intuitional and aims to reveal GME. As shown in Sec. III.3, it is not
easy to obtain the fidelity to the target non-stabilizer state. The first
method should use several non-homogenous LMSs to lower bound the fidelity.

In this section, we indirectly estimate the fidelity between the prepared
state and the target state by evolving the entangling process backwards. Since
the fidelity to both the Bell state and product state can be measured in the
experiment, we can indirectly estimate the fidelity to $\ket{\Psi}$ and
certify GME according to Lemma 1. The reverse evolution follows the state
preparation process shown in Sec. III.2.

Figure 3: Illustration of reverse revolution. At point 3, the state
preparation is accomplished; At point 4 and point 5, the system is under
reverse evolution. In the perfect case, the state can return back to Bell
pairs and Néel state. Here, due to the noise and decoherence, we denote the
corresponding mixed state as $\rho_{\phi_{i}}$, and the fidelity is
$\mathrm{Tr}(\rho_{\phi_{i}}\Phi_{i})$ to the perfect case. Note that one can
estimate the fidelity to $\Phi_{1},\Phi_{5}$ with one measurement settings
$Z^{\otimes 10}$, and the one to $\Phi_{2},\Phi_{4}$ with two settings
$X^{\otimes 10},Z^{\otimes 10}$.

  * •

Operate $\sqrt{\mathrm{SWAP}}$ gates parallelly on the qubit pairs
$(2,3),(4,5),\cdots,(8,9)$.

$\displaystyle\ket{\Phi_{4}}=$
$\displaystyle\bigotimes_{k=1}^{4}\sqrt{\mathrm{SWAP}}_{\\{2k,2k+1\\}}\ket{\Psi}$
(21) $\displaystyle=$
$\displaystyle\bigotimes_{k=1}^{5}\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})_{\\{2k-1,2k\\}}.$

  * •

Adjust the relative phase between $\ket{01}$ and $\ket{10}$ in each of the
Bell pairs in $\ket{\Phi_{4}}$.

$\displaystyle\ket{\Phi_{4}^{\prime}}=U_{\mathrm{phase}}^{\dagger}\ket{\Phi_{4}}=\bigotimes_{k=1}^{5}\frac{1}{\sqrt{2}}(\ket{10}+i\ket{01})_{\\{2k-1,2k\\}}.$
(22)

  * •

Operate $\sqrt{\mathrm{SWAP}}$ gates parallelly on the qubit pairs
$(1,2),(3,4),\cdots,(9,10)$.

$\displaystyle\ket{\Phi_{5}}$
$\displaystyle=\bigotimes_{k=1}^{5}\sqrt{\mathrm{SWAP}}_{\\{2k-1,2k\\}}\ket{\Phi_{4}^{\prime}}$
(23) $\displaystyle=\ket{1010101010}.$

In the above reverse protocol, the state is evolved backwards finally to the
initial product state. Note that here we list the perfect reverse evolution,
that is, $\ket{\Phi_{4}}=\ket{\Phi_{2}}$ and $\ket{\Phi_{5}}=\ket{\Phi_{1}}$
shown in the generation protocol in Sec. III.2. The practical states actually
deviate from them due to noises, denoted by $\rho_{\phi_{i}}$ for
$i=1,2\cdots,5$, and the fidelity decreases due to the imperfection of the
underlying gates.

One can estimate the fidelity to product states
$\ket{\Phi_{1}},\ket{\Phi_{5}}$ with the $Z^{\otimes 10}$ measurement setting,
entangled-pair states $\ket{\Phi_{2}^{\prime}},\ket{\Phi_{4}^{\prime}}$ with
$XYXY\cdots XY$ and $Z^{\otimes 10}$, and Bell-pair states
$\ket{\Phi_{2}},\ket{\Phi_{4}}$ with $X^{\otimes 10}$ and $Z^{\otimes 10}$. In
Fig. 3, we show an illustration. By fitting the fidelity values from the
experiment, one can indirectly estimate the fidelity of the prepared state
$\rho_{\mathrm{pre}}$ to the target state $\ket{\Psi}$, which can be compared
with the theoretical bound for GME, say $\frac{5}{8}$, denoted by the dotted
horizon line in the figure. Note that
$\sqrt{\mathrm{SWAP}}=\sqrt{\mathrm{SWAP}}^{{\dagger}3}$, i.e., it takes three
times of evolution time, and one may needs to consider this point in the
fitting. We remark that this entanglement detection method is intuitive, and
one could make it more rigorous by assuming more specific noise models.

##  IV Discussion

In this work, we propose an experimental scheme to generate and characterize
large-scale entanglement of cold atoms confined in the optical superlattice.
The generation scheme utilizes the entangling gate induced by the
superexchange interaction, and is robust to decoherence. To characterize the
entanglement, we propose several complementary methods considering the
experimental implementation feasibility. Moreover, it is straightforward to
generate our scheme to the high dimension scenarios, and it is also
interesting to construct some efficient entanglement verification tools for
them. In summary, our entanglement generation and verification protocols are
well tailored for the cold atom system, and lay the foundation for the further
applications, such as measurement-based quantum computing.

##  V Methods

###  V.1 Proof of Proposition 1

###### Proof.

To prove that Eq. (10) is a legal GME witness, we need to show that the
maximal fidelity between the separable state and our target state is upper
bounded by $5/8$, that is

$F_{\mathrm{max}}:=\max_{\rho_{\mathrm{s}}\in
S_{\mathrm{b}}}\mathrm{Tr}(\rho_{\mathrm{s}}\Psi)\leq 5/8.$ (24)

By the convexity of $S_{\mathrm{b}}$, we can reduce the maximization to the
pure separable state

$\displaystyle F_{\mathrm{max}}=$
$\displaystyle\max_{\mathcal{P}_{2}}\max_{\Phi_{\mathrm{s}}\in
S_{\mathrm{b}}^{\mathcal{P}_{2}}}\mathrm{Tr}(\Phi_{\mathrm{s}}\Psi)$ (25)
$\displaystyle=$
$\displaystyle\max_{\mathcal{P}_{2}}\lambda_{1}(\Psi,\mathcal{P}_{2}),$

where we maximize over $\Phi_{\mathrm{s}}$ from a given bipartition
$\mathcal{P}_{2}$ and also all possible bipartitions. The second line is due
to the fact that the maximization result over $\Phi_{\mathrm{s}}$ is just the
square of the largest Schmidt coefficient of $\Psi$ with respective to
$\mathcal{P}_{2}$ [67], where the Schmidt decomposition shows
$\ket{\Psi}=\sum_{i=1}^{d=\min{\\{d_{A},d_{\bar{A}}\\}}}\sqrt{\lambda_{i}}\ket{\phi_{i}}_{A}\ket{\phi^{\prime}_{i}}_{\bar{A}}$
and $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{d}$.

In fact, we only need to consider the bipartitions where $A$ contains the
first $n_{A}$ and $\bar{A}$ contains the last $n_{\bar{A}}=N-n_{A}$ qubits on
the 1-D chain. That is, there is only one boundary between $A$ and $\bar{A}$.
Other choices with more boundaries can lead to a smaller $\lambda_{1}$.
Consider the one boundary scenario, there are two types of bipartations
depending on that the boundary is between $\\{2k-1,2k\\}$ or $\\{2k,2k+1\\}$,
as shown in Fig. 4.

Figure 4: Bipartitions with different choices of boundary. Atom pairs
connected with black line are Bell pairs. The dotted lines denote the boundary
between $A$ and $\bar{A}$.

Based on the causality of the quantum gate and the fact that quantum gates
inside the subsystem $A$ and $\bar{A}$ do not change the Schmidt number, we
can determine $\lambda_{1}$ for both of cases. For the $\\{2k-1,2k\\}$ case,
$\lambda_{1}=\frac{1}{2}$, since we can ignore the latter
$\sqrt{\mathrm{SWAP}}^{{\dagger}}$ and the entanglement spectrum is just the
same with that of the Bell state.

For the $\\{2k,2k+1\\}$ case, we just need to focus on the four-qubit state
after the ignorance of the other distant $\sqrt{\mathrm{SWAP}}^{{\dagger}}$
gates, that is,

$\displaystyle\sqrt{\mathrm{SWAP}}^{\dagger}_{\\{2,3\\}}\ket{\phi_{\mathrm{Bell}}}_{\\{1,2\\}}\otimes\ket{\phi_{\mathrm{Bell}}}_{\\{3,4\\}}.$
(26)

Here for denotation simplicity we take $k=1$ without loss of generality, and
$\ket{\phi_{\mathrm{Bell}}}=\frac{1}{\sqrt{2}}\ket{01}+\ket{10}$. By
calculation one finds that the reduced density matrix of the first two-qubit
is

$\displaystyle\rho_{12}=\frac{1}{2}\phi_{\mathrm{Bell}}+\frac{1}{2}\mathbb{I}^{\otimes
2}/4,$ (27)

i.e., the mixture of the Bell state and the maximal mixed state. As a result,
$\rho_{12}$ is a Bell-diagonal state and thus its four eigenvalues on the Bell
basis is $\\{\frac{5}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8}\\}$ with
$\lambda_{1}=\frac{5}{8}$. In summary, we prove that
$F_{\mathrm{max}}=\frac{5}{8}$. ∎

###  V.2 Proof of Proposition 2

###### Proof.

We first show that $S_{i}^{\prime}$ stabilizes the target $\ket{\Psi}$ by
definition. Before the final layer of $\sqrt{\mathrm{SWAP}}^{{\dagger}}$
gates, the state is the product of a few of Bell pairs, denoted by
$\ket{\Psi_{\mathrm{Bell}}}$, which is stabilized by $S_{i}$ defined in Eq.
(12). That is, $S_{i}\ket{\Psi_{\mathrm{Bell}}}=\ket{\Psi_{\mathrm{Bell}}}$.
Thus, one has
$S_{i}^{\prime}\ket{\Psi}=US_{i}U^{{\dagger}}U\ket{\Psi_{\mathrm{Bell}}}=U\ket{\Psi_{\mathrm{Bell}}}=\ket{\Psi}$.

Then we prove the inequality in Eq. (11). We remark that Eq. (11) holds for
any generalized stabilizer state. The $N$ independent stabilizers $S_{i}$
commute with each other and their common eigenstates together determine an
orthogonal basis denoted by $\ket{\Psi_{\vec{\mathbf{x}}}}$. Here
$\vec{\mathbf{x}}=x_{1}x_{2}\cdots x_{N}$ with each $x_{i}$ taking $1$ or $-1$
and
$S_{i}^{\prime}\ket{\Psi_{\vec{\mathbf{x}}}}=x_{i}\ket{\Psi_{\vec{\mathbf{x}}}}$.
It is clear our target state $\ket{\Psi}=\ket{\Psi_{11\cdots 1}}$. It is clear
that both operators on the left and right of Eq. (11) are diagonal in this
basis, thus we can prove it by checking for the matrix elements for every
$\Psi_{\vec{\mathbf{x}}}$. For $\vec{\mathbf{x}}=11\cdots 1$, one has
$1\geq\frac{1}{2}N-(N/2-1)=1$. For $\vec{x}$ which contains only one zero,
such as $\vec{\mathbf{x}}=01\cdots 1$, one has
$0\geq\frac{1}{2}(N-1-1)-(N/2-1)=0$. For the $\vec{\mathbf{x}}$ containing
more zeros, one can prove the inequality similarly. ∎

###  V.3 Symmetry of the target state

Remember that in principle we should apply three measurement settings
$\\{X^{\otimes N},Y^{\otimes N},Z^{\otimes N}\\}$ to detect the full
entanglement in Sec. III.4. Here we show rigorously as follows that one does
not need to worry about the direction on the $X-Y$ plane in every measurement.
In fact, one only needs to measure $\langle X^{\otimes N}\rangle$ by the
symmetry of the target state.

In the experiment, actually there is no reference pulse to decide the
direction of the measurement, that is, one chooses a random angle
$X_{\theta}=\cos(\theta)X+\sin(\theta)Y$ for the measurement. Note that there
is a corresponding unitary (rotation around $Z$),

$\displaystyle U_{\theta}=\mathrm{e}^{-\mathrm{i}\frac{Z}{2}\theta}$ (28)

such that $X_{\theta}=U_{\theta}XU_{\theta}^{{\dagger}}$.

As a result, suppose there is a standard $X$ direction, the actual measurement
shows,

$\displaystyle\int_{\theta=0}^{2\uppi}\mathrm{Tr}(X_{\theta}^{\otimes
N}\rho)d\theta$
$\displaystyle=\int_{\theta=0}^{2\uppi}\mathrm{Tr}(U_{\theta}^{\otimes
N}X^{\otimes N}U_{\theta}^{{\dagger}\otimes N}\rho)d\theta$ (29)
$\displaystyle=\int_{\theta=0}^{2\uppi}\mathrm{Tr}(X^{\otimes
N}U_{\theta}^{{\dagger}\otimes N}\rho U_{\theta}^{\otimes N})d\theta$
$\displaystyle=\mathrm{Tr}\left(X^{\otimes N}\rho_{\mathrm{sym}}\right).$

where $\rho$ is the prepared state in the experiment, and
$\rho_{\mathrm{sym}}\equiv\int_{\theta=0}^{2\uppi}U_{\theta}^{{\dagger}\otimes
N}\rho U_{\theta}^{\otimes N}d\theta$. In other words, measuring the state
$\rho$ in random direction is equivalent to “twirling” the state, and the
resulting state is symmetric with respective to the measurement direction.

$\displaystyle\mathrm{Tr}(X_{\theta}^{\otimes
N}\rho_{\mathrm{sym}})=\mathrm{Tr}(X_{\theta^{{}^{\prime}}}^{\otimes
N}\rho_{\mathrm{sym}}).$ (30)

Specifically, $X=X_{\theta=0}$ and $Y=X_{\theta=\frac{\uppi}{2}}$, and one
also has

$\displaystyle\mathrm{Tr}(Z^{\otimes
N}\rho_{\mathrm{sym}})=\mathrm{Tr}(Z^{\otimes N}\rho),$ (31)

since $Z$ commutes with $U_{\theta}$ defined in Eq. (28).

It is not hard to see that the entanglement of $\rho_{\mathrm{sym}}$ is not
stronger than $\rho$, since $\rho_{\mathrm{sym}}$ is obtained from $\rho$
using LOCC, that is, the twirling can not increase the entanglement. Thus if
one can detect the entanglement property of $\rho_{\mathrm{sym}}$, this
property should also holds for $\rho$.

On the other hand, the target state is a symmetric state on $X-Y$ plane, i.e.,
$U_{\theta}^{\otimes N}\Psi U_{\theta}^{{\dagger}\otimes
N}=\Psi,\forall\theta$. In fact, one can write the rotation unitary as

$\displaystyle U_{\theta}^{\otimes
N}=\mathrm{e}^{-\mathrm{i}\frac{\theta}{2}\left[\sum_{i=1}^{N}Z_{i}\right]}.$
(32)

Since our target state $\ket{\Psi}$ only has non-zero projection on the
computational bases whose total spin is zero, i.e., half number of 0 and 1,
the rotation unitary introduce the same phase for these bases, and the state
is unchanged.

###  V.4 Proof of Lemma 1

###### Proof.

Without loss of generality, we assume that the subsystem $A$ contains the
first $k_{1}$ qubits, and $|A|=k_{1}$, $|\bar{A}|=k-k_{1}=k_{2}$ are both odd
numbers. Since the left hand of Eq. (16) is a convex function of the state,
thus we only need to consider the pure separable state in the form
$\ket{\Psi_{\mathrm{s}}}=\ket{\Psi_{k_{1}}}\otimes\ket{\Psi_{k_{2}}}$, and the
expectation value can be written apart as,

$\displaystyle|\langle X^{\otimes k_{1}}\rangle||\langle X^{\otimes
k_{2}}\rangle|+|\langle Y^{\otimes k_{1}}\rangle||\langle Y^{\otimes
k_{2}}\rangle|+|\langle Z^{\otimes k_{1}}\rangle||\langle Z^{\otimes
k_{2}}\rangle|$ (33) $\displaystyle\leq$ $\displaystyle\sqrt{|\langle
X^{\otimes k_{1}}\rangle|^{2}+|\langle Y^{\otimes k_{1}}\rangle|^{2}+|\langle
Z^{\otimes k_{1}}\rangle|^{2}}\times$ $\displaystyle\sqrt{|\langle X^{\otimes
k_{2}}\rangle|^{2}+|\langle Y^{\otimes k_{2}}\rangle|^{2}+|\langle Z^{\otimes
k_{2}}\rangle|^{2}},$

where the second line is due to Cauchy-Schwarz inequality. Since $k_{1}$ is an
odd number, one can check that $X^{\otimes k_{1}},Y^{\otimes k_{1}}$ and
$Z^{\otimes k_{1}}$ anticommute with each other, and thus $|\langle X^{\otimes
k_{1}}\rangle|^{2}+|\langle Y^{\otimes k_{1}}\rangle|^{2}+|\langle Z^{\otimes
k_{1}}\rangle|^{2}\leq 1$ [65, 66]. Similarly one has $|\langle X^{\otimes
k_{2}}\rangle|^{2}+|\langle Y^{\otimes k_{2}}\rangle|^{2}+|\langle Z^{\otimes
k_{2}}\rangle|^{2}\leq 1$ . As a result, we finish the proof. ∎

###  V.5 Proof of Observation 1

###### Proof.

The Observation is right by the definition of LOCC. Here we show the case for
the partial trace operation by contradiction. Suppose
$\rho_{A,\bar{A}}=\sum_{i}p_{i}\rho_{A}^{i}\otimes\rho_{\bar{A}}^{i}$ is
separable. The partial trace $\mathrm{Tr}_{A\rightarrow
B}(\rho_{A}^{i})=\rho_{B}^{i}$ and
$\mathrm{Tr}_{\bar{A}\rightarrow\bar{B}}(\rho_{\bar{A}}^{i})=\rho_{\bar{B}}^{i}$,
such that
$\rho_{B,\bar{B}}=\sum_{i}p_{i}\rho_{B}^{i}\otimes\rho_{\bar{B}}^{i}$ is still
separable, which contradicts with the assumption that it is entangled. ∎

###  V.6 Influence from Operation Errors

The fidelity of the state creation is affected by the error of each operation,
caused by the decoherence effect and noises. For the large-scale entangled
state, the degradation of fidelity and also the entanglement detection is more
apparent compared to few-body states. In the meantime, the read-out error also
affects the entanglement detection. As a result, it is meaningful to analyze
the influence of various errors in the experimental implementation.

Here we consider the errors in the following operations: the initial
preparation of Néel state; the final projective measurement of spins; and the
fidelity of the intermediate quantum gates. We show their influences on the
entanglement detection of a ten-qubit system, i.e., $N=10$, by using the
witness method in Sec. III.4, which is more practical for experiments. As
shown in the procedure of entanglement detection in Sec. III.4, we should
measure the witness value in Eq. (16) for a few of reduced density matrices of
subsystems and also the whole system. Here we take the subsystem as

$\displaystyle\\{1,2\\},\\{1,3\\};\\{1,2,3,4\\},\\{1,2,3,5\\};$ (34)
$\displaystyle\\{1-6\\},\\{1-5,7\\};\\{1-8\\},\\{1-7,9\\}$

which contain $n=2,4,6,8$ qubits.

Figure 5: The value of witness of Eq. (16) in Sec. III.4 by taking account of
errors in (a) spin-flip, (b) spin-measurement, and (c) entangling step. In (c)
$P_{\mathrm{SF}}$ and $P_{\mathrm{MS}}$ are taken to be 0.98 and 0.985,
respectively. Here the system contains $N=10$ qubits and the subsystems are
chosen according Eq. (34) with qubit number $n=2,4,6,8,10$. The blue dashed
line shows the lower bound of the witness that can certify the entanglement.

In the first step of Sec. III.2, starting from the all-zero state
$\ket{00\cdots 0}$, one should flip the spins on the odd sites to prepare the
Néel state in Eq. (4). There is the probability of a non-flip for the odd
site, and also the probability of a wrong flip for the even site. For
simplicity, we assume both error probabilities are equal, and denote the
probability of the correct flip as $P_{\mathrm{SF}}$ for each site. In this
case, the system is initially prepared into a mixture of different product
states weighted with corresponding probabilities. In Fig. 5 (a), we plot the
witness values of Eq. (16) for different subsystem size $n$ with respective to
$P_{\mathrm{SF}}$. Similarly, in the final measurement, the probability to
measure the spin of single atom correctly is denoted as $P_{\mathrm{MS}}$.
That is, there is probability $1-P_{\mathrm{MS}}$ to recognize $\ket{0}$ as
$\ket{1}$ or vice versa. The witness value with respective to
$P_{\mathrm{MS}}$ is shown in Fig. 5 (b). From these two figures, we found
that the witness values are the same for the subsystem with the same qubit
number, for example, $n=2$ case with $\rho_{1,2}$ and $\rho_{1,3}$. When there
is no error, say $P_{\mathrm{SF}}=1$ and $P_{\mathrm{MS}}=1$, all the witness
values return $1.5$ for all the subsystems, except the whole system with the
value $3$. One can see that even though there is some error, the witness can
detect the entanglement as the value is larger than $1$. For larger
subsystems, the values decay faster with $P_{\mathrm{SF}}$ and
$P_{\mathrm{MS}}$.

Similar as the six-qubit case discussed in Sec. III.4, here for total system
size $N=10$, one only needs to measure the witness for
$\\{\rho_{1,2},\rho_{1,3},\rho_{1,2,3,4},\rho_{1,2,3,5},\rho_{1,2,3,4,5,6},\\\
\rho_{1,2,3,4,5,7},\rho_{7,8,9,10},\rho_{8,10},\rho_{9,10}\\}$ to verify the
full entanglement. As a result, the result of the subsystems with at most
$n=6$ qubits decides the lower bound of the operation fidelity. According to
Fig. 5 (a) and (b), the bounds are 0.95 and 0.97 for $P_{\mathrm{SF}}$ and
$P_{\mathrm{MS}}$ respectively.

At last, in Fig. 5 (c), we study the influence of the fidelity of the
entangling step between the above two steps by taking $P_{\mathrm{SF}}=0.98$
and $P_{\mathrm{MS}}=0.985$. Here we assume that this operation has a
probability $P_{\mathrm{ES}}$ to be performed perfectly while it contributes
zero to the witness with probability $1-P_{\mathrm{ES}}$, that is, essentially
outputs a maximal mixed state. As shown in Fig. 5 (c), the full entanglement
could be verified when the fidelity of entangling step exceeds $0.85$.

##  VI CODE AVAILABILITY

The code that supports the findings of this study are available from the
corresponding author upon reasonable request.

##  VII DATA AVAILABILITY

Data sharing is not applicable to this article as no data sets were generated
or analyzed during the current study.

##  VIII Acknowledgments

This work was supported by the National Natural Science Foundation of China
(NNSFC) No. 12125409, the Anhui Initiative in Quantum Information
Technologies, and the Chinese Academy of Sciences. X. Ma acknowledges the
support by the NNSFC No. 11875173 and No. 12174216 and the National Key
Research and Development Program of China No. 2019QY0702 and No.
2017YFA0303903.

##  IX COMPETING INTERESTS

The authors declare that there are no competing interests.

##  X AUTHOR CONTRIBUTIONS

All authors contributed to the theory, numerical analysis, and writing up the
manuscript. Y. Z. and B. X. contributed equally to this work.

##  XI References

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