# Bounding Entanglement Entropy Using Zeros of Local Correlation Matrices

Zhiyuan Yao  Institute for Advanced Study, Tsinghua University, Beijing
100084, China  Lei Pan  Institute for Advanced Study, Tsinghua University,
Beijing 100084, China  Shang Liu<EMAIL_ADDRESS>Kavli Institute for
Theoretical Physics, University of California, Santa Barbara, California
93106, USA  Pengfei Zhang<EMAIL_ADDRESS>Institute for
Quantum Information and Matter and Walter Burke Institute for Theoretical
Physics, California Institute of Technology, Pasadena, California 91125, USA

###### Abstract

Correlation functions and entanglement are two different aspects to
characterize quantum many-body states. While many correlation functions are
experimentally accessible, entanglement entropy (EE), the simplest
characterization of quantum entanglement, is usually difficult to measure. In
this Letter, we propose a protocol to bound EE by local measurements. This
protocol utilizes local correlation matrices and focuses on their
(approximate) zero eigenvalues. Given a quantum state, each (approximate) zero
eigenvalue can be used to define a set of local projection operators. An
auxiliary Hamiltonian can then be constructed by summing these projectors.
When the construction only involves projectors of zero eigenvalues, we prove
the EE of a subsystem is bounded by the ground-state degeneracy of the
auxiliary Hamiltonian on this subsystem. When projectors from nonzero
eigenvalues are included, we show the EE can be bounded by a thermal entropy
of the subsystem. Our protocol can be applied experimentally to investigate
exotic quantum many-body states prepared in quantum simulators.

Introduction.– Entanglement not only plays an essential role in quantum
information science but also becomes more and more important in modern studies
of quantum matters [1, 2, 3]. Entanglement entropy (EE) can characterize
quantum thermalization [4], describe quantum critical [5, 6, 7, 8, 9] and
topologically ordered states [10, 11], and bridge quantum correlation and
geometric metric in holographic quantum matter [12, 13, 14]. On the other
hand, conventional characterizations of quantum matters are based on
correlation functions and physical observables [15]. It is, therefore, natural
to seek connections between EE and correlation functions or physical
observables.

There are several well-established connections. It is known that the mutual
information can be used to bound correlation functions [16, 17]. The
scrambling of entanglement is also directly related to the out-of-time-ordered
correlator [18, 19, 20, 21, 22, 23]. Experimentally, the Rényi entropy can be
determined by measuring the shift operator between multiple copies of the
target quantum state [24, 25, 26, 27, 28] or averaging over randomized
measurements on single copies of a given quantum state [29, 30, 31, 32, 33,
34]. Nevertheless, these protocols usually suffer from large statistical
fluctuations as the subsystem size increases. Hence, experimental measurements
of EEs so far are limited to small subsystem sizes.

In this Letter, we bring out a connection between the bound of the EE and the
(approximate) zeros in the correlation matrix. The correlation matrix is first
introduced [35] in the Hamiltonian reconstruction problem of “recovering” the
Hamiltonian from one of its excited states [36]. Here we introduce a slightly
generalized local version of the correlation matrix and use it to bound the EE
of a state. The local correlation matrix is Hermitian and positive
semidefinite with non-negative eigenvalues. We focus on (approximate) zeros
among all eigenvalues of the correlation matrix. Each (approximate) zero gives
rise to a set of projection operators $\hat{P}_{i}$ which (approximately)
satisfies $\hat{P}_{i}\ket{\psi}=0$ and constrains $\ket{\psi}$. To quantify
the extent of these constraints and give an upper bound of the EE, we
introduce a positive semidefinite auxiliary Hamiltonian as the sum of these
projectors multiplied by positive prefactors. When only projectors of zero
eigenvalues are included, we prove that the von Neumann EE is upper bounded by
the logarithm of the ground-state degeneracy of the auxiliary Hamiltonian
constructed on the subsystem. When projectors of approximate zeros are
included, the EE of a subsystem can be bounded by the thermal entropy at a
temperature determined by the subsystem energy of the state.

Figure 1: Correlation functions and entanglement entropy are two different
tools to characterize a quantum many-body state. This work establishes a
connection between correlation and entanglement, based on the three statements
discussed in the text.

Protocol.– We first present our protocol, schematically shown in Fig. 1, of
including only projectors from zero eigenvalues in the auxiliary Hamiltonian.
It follows step-wisely from the following three statements and connects
correlation functions and EE. It is schematically shown in Fig. 1 and is
discussed in detail below.

Statement 1: from Correlation Matrix to Eigenoperators. We consider a
continuous region of $k$ sites and a set of linearly-independent $k$-local
operators $\\{\hat{L}_{i}\\}$ acting only in this region. For a given quantum
many-body state $|\psi\rangle$, we introduce its hermitian correlation matrix
$\mathcal{M}$ with element $\mathcal{M}_{ij}$ defined as

$\mathcal{M}_{ij}=\langle\psi|\hat{L}^{\dagger}_{i}\hat{L}_{j}|\psi\rangle-\langle\psi|\hat{L}^{\dagger}_{i}|\psi\rangle\langle\psi|\hat{L}_{j}|\psi\rangle.$
(1)

Denote the eigenvalues of $\mathcal{M}$ as $\lambda_{\alpha}$ and the
corresponding eigenvectors as $\bf{e}_{\alpha}$. For each zero eigenvalue
$\lambda_{\alpha}=0$, if exists, one can use $\mathbf{e}_{\alpha}$ to
construct an eigenoperator $\hat{O}$ of $\ket{\psi}$ where
$\hat{O}\ket{\psi}=\xi\ket{\psi}$ for some constant $\xi$.

###### Proof.

Let us denote ${\bf e}_{\alpha}=(w_{1},w_{2},\dots,w_{n})$ and construct an
operator $\hat{O}=\sum\limits_{i}w_{i}\hat{L}_{i}$. Then, if
$\lambda_{\alpha}=0$ we have

$\langle\psi|\hat{O}^{\dagger}\hat{O}|\psi\rangle-\langle\psi|\hat{O}^{\dagger}|\psi\rangle\langle\psi|\hat{O}|\psi\rangle=\sum\limits_{ij}w_{i}^{*}\mathcal{M}_{ij}w_{j}=0.$
(2)

On the other hand, let
$\hat{O}|\psi\rangle=\xi|\psi\rangle+|\psi_{\perp}\rangle$ with
$\langle\psi_{\perp}|\psi\rangle=0$, the above relation translates to

$0=\langle\psi|\hat{O}^{\dagger}\hat{O}|\psi\rangle-\langle\psi|\hat{O}^{\dagger}|\psi\rangle\langle\psi|\hat{O}|\psi\rangle=\langle\psi_{\perp}|\psi_{\perp}\rangle\,,$
(3)

meaning $\hat{O}|\psi\rangle=\xi|\psi\rangle$. Clearly, the number of
independent eigenoperators $\hat{O}$ one can construct in this way equals the
geometric/algebraic multiplicity of eigenvalue zero. ∎

Statement 2: from Eigenoperators to Auxiliary Hamiltonian. From each
eigenoperator $\hat{O}$, one can construct a set of projection operators
$\\{\hat{P}_{i}\\}$ each of which annihilates the state,
$\hat{P}_{i}|\psi\rangle=0$.

###### Proof.

Let us denote $\hat{\Delta}$ the positive semidefinite operator defined as
($\hat{I}$ being the identity operator)

$\hat{\Delta}=(\hat{O}^{\dagger}-\xi^{*}\hat{I})(\hat{O}-\xi\hat{I}).$ (4)

and $\ket{i}$ the basis that diagonalizes $\hat{\Delta}$,

$\hat{\Delta}=\sum\limits_{i}\alpha_{i}|i\rangle\langle
i|\equiv\sum\limits_{i}\alpha_{i}\hat{P}_{i},$ (5)

where $\alpha_{i}>0$. Since $\hat{\Delta}$ annihilates the state and
$0=\bra{\psi}\hat{\Delta}\ket{\psi}=\sum_{i}\alpha_{i}\bra{\psi}\hat{P}_{i}\ket{\psi}=\sum_{i}\alpha_{i}\bra{\psi}\hat{P}_{i}^{2}\ket{\psi}=\sum_{i}\alpha_{i}||\hat{P}_{i}\ket{\psi}||^{2}$,
it follows that each $\hat{P}_{i}$ annihilates the state,
$\hat{P}_{i}|\psi\rangle=0$. Moreover, the projectors $\hat{P}_{i}$ are
$k$-local operators since $\hat{O}$ is $k$-local. ∎

Intuitively, the existence of a set of such projectors $\hat{P}_{i}$ means the
wave function $\ket{\psi}$ tends to have a simple form and be low entangled.
This is in contrast to a generic volume-law eigenstate where such local
projectors $\hat{P}_{i}$ do not exist as implied by the uniqueness of the
Hamiltonian reconstruction process [35]. This also means we should find as
many such projectors as we can if we want to give a good upper bound on the EE
of $\ket{\psi}$ of a subsystem $A$. To this end, we first exhaust all
continuous regions of $k$ sites in $A$ and collect all $k$-local projection
operators $\hat{P}_{i}$ constructed above 111Refer to Appendix A for a visual
demonstration.. We then construct a positive semidefinite auxiliary
Hamiltonian on $A$ by summing these projectors multiplied by positive numbers,

$\hat{H}^{A}_{\text{aux}}=\sum\limits_{\alpha}\tilde{c}_{\alpha}\hat{\Delta}_{\alpha}=\sum\limits_{i}c_{i}\hat{P}_{i},\quad\tilde{c}_{\alpha}>0,\,c_{i}>0\,,$
(6)

where $\alpha$ enumerates $\hat{\Delta}_{\alpha}$ operators of zero
eigenvalues of local correlation matrices on all $k$-site regions and
$c_{i}=\sum_{\alpha}\tilde{c}_{\alpha}\alpha_{i}$ according to Eq. (5). The
introduction of $\hat{H}^{A}_{\text{aux}}$ will prove to be convenient for us
to discuss the bound of EE.

Statement 3: from Auxiliary Hamiltonian to the EE bound—the Ground-State
Degeneracy and the Entanglement Entropy (GSD-EE) Theorem. If $D^{A}$ denotes
the ground-state degeneracy of $\hat{H}^{A}_{\text{aux}}$ under the open
boundary condition (OBC), the von Neumann EE of $\ket{\psi}$ between this
subsystem $A$ and the rest of the system $B$ is upper bounded by $\log D^{A}$.

###### Proof.

Because $\hat{P}_{i}|\psi\rangle=0$ and $\hat{P}_{i}$ only acts on the
subsystem $A$, the reduced density matrix
$\hat{\rho}_{A}=\text{Tr}_{\text{B}}|\psi\rangle\langle\psi|$ satisfies

$\text{Tr}_{\text{A}}(\hat{P}_{i}\hat{\rho}_{A})=\text{Tr}_{\text{A}}(\text{Tr}_{\text{B}}\hat{P}_{i}|\psi\rangle\langle\psi|)=0.$
(7)

Diagonalizing $\hat{\rho}_{A}$ with a set of orthonormal basis
$\\{\ket{\psi_{k}}\\}$,
$\hat{\rho}_{A}=\sum\limits_{k}p_{k}|\psi_{k}\rangle\langle\psi_{k}|$ where
$p_{k}>0$, we can cast the above relation into the following form

$\text{Tr}_{\text{A}}(\hat{P}_{i}\hat{\rho}_{A})=\sum\limits_{k}p_{k}||\hat{P}_{i}|\psi_{k}\rangle||^{2}\,.$
(8)

It becomes clear that $\hat{P}_{i}|\psi_{k}\rangle=0$ for every
$\ket{\psi_{k}}$ and $\hat{P}_{i}$. Therefore, $\ket{\psi_{k}}$ is the ground
state of $\hat{H}^{A}_{\text{aux}}$ with zero energy since
$\hat{H}^{A}_{\text{aux}}$ is positive semidefinite. Consequently, the number
of non-zero $p_{k}$ is no bigger than the ground-state degeneracy $D^{A}$ of
the $\hat{H}^{A}_{\text{aux}}$, and the von Neumann EE is upper bounded by
$\log D^{A}$. ∎

Bounding EE of a large subsystem $A$ requires the determination of $D^{A}$ on
the same large subsystem $A$, and to this end we have specially designed an
algorithm 222Refer to Appendix D for details.. For a one-dimensional (1D)
system, as a corollary of our GSD-EE theorem, if $D^{A}$ is upper bounded by a
power-law of the subsystem size $\sim L_{\text{A}}^{\alpha}$, the EE is then
upper bounded by $\sim\log L_{\text{A}}$, indicating the state is sub-volume
entangled. A few remarks are in order before we move on to specific examples.
Firstly, the operator space that the set of eigenoperators
$\\{\hat{O}_{\alpha}\\}$ spans does not depend on the form of $\hat{L}_{i}$ as
long as they span the same operator space $\mathcal{V}$. Secondly, the EE
upper bound does not depend on the form of $\hat{O}_{\alpha}$, i.e., the
choice of $\\{\mathbf{e}_{\alpha}\\}$, in the case of the zero eigenvalue of
$\mathcal{M}$ being degenerate 333see Appendix B for a proof. Finally, larger
$\mathcal{V}$ in general leads to a better upper bound with however, more
numerical/experimental effort to obtain $\mathcal{M}$. Consequently, we should
make a balance and choose a moderate number of operators. Nevertheless, using
1-local operator basis, our protocol already gives the perfect EE upper bound
of zero for product states 444This is proved in Appendix C..

Examples.– Below we shall apply the GSD-EE theorem to the scar states in the
generalized Affleck–Kennedy–Lieb–Tasaki (AKLT) model and the ground state of
the toric code model.

Example 1: Scar states in the AKLT model. The 1D AKLT model and its
generalizations are known to host a series of quantum many-body scar states
described the spectrum generating algebra [41, 42]. These scar states admit
the following analytical expression,

$\ket{\psi_{n}}=(\hat{Q}^{\dagger})^{n}\ket{G},$ (9)

where $\hat{Q}^{\dagger}=\sum_{i}(-1)^{i}(\hat{S}_{i}^{+})^{2}$ with $i$ being
the site index and $\hat{S}_{i}^{+}$ being the spin raising operator at site
$i$, and $\ket{G}$ is the ground state of the AKLT model under the periodic
boundary condition (PBC). Here the system size $L$ is assumed to be even,
$n=0,1,\cdots,L/2-1$ for odd $L/2$, and $n=0,1,\cdots,L/2$ for even $L/2$.

To construct the correlation matrices, we choose 3-local operators
$\\{\hat{L}_{i}\\}$ that are constructed as tensor products of the Gell-Mann
matrices,

$\hat{L}_{i}=\lambda_{a}^{(1)}\otimes\lambda_{b}^{(2)}\otimes\lambda_{c}^{(3)}\,,$
(10)

where $a,b,c=1,\cdots,8$ are the indices of the eight standard Gell-Mann
matrices and the superscripts $1,2,3$ denote three successive sites. As an
illustrative example, we choose the state
$\ket{\psi_{1}}=\hat{Q}^{\dagger}\ket{G}$ and construct the auxiliary
Hamiltonian, following the general protocol described above. Since our target
state is translational invariant up to a sign, we just need to construct the
local correlation matrix of arbitrary three neighboring sites: all the other
projectors entering the auxiliary Hamiltonian are obtained by simply
translating the local projectors of this correlation matrix. It turns out that
projectors constructed from $\ket{\psi_{1}}$ also annihilate all the scar
states $\ket{\psi_{n}}$ for all system sizes $L>3$ 555Refer to Appendix E for
details.. Thus the EE scaling behavior obtained for our target state
$\ket{\psi_{1}}$ will also bound all other scar states $\ket{\psi_{n}}$. Using
our specially designed algorithm, we are able to determine the ground-state
degeneracy $D^{A}$ of the auxiliary Hamiltonian up to system size
$L_{\text{A}}$ of a few hundred. We numerically find that
$D^{A}=2L_{\text{A}}+2$, which means the EE at most scales with the logarithm
of the subsystem size.

Figure 2: Comparison between the upper bounds of the bipartite EE obtained by
the ground-state degeneracy of the auxiliary Hamiltonian (green dots) and the
computed maximum bipartite EEs of all the scar states (blue dots), Eq. (9), in
the AKLT model.

To benchmark the effectiveness of our EE bounds, we numerically calculate the
exact values of the bipartite EEs of all the scar states. Since our result
bounds all the $\ket{\psi_{n}}$, we compare our bound with the maximum value
of the numerically calculated values. As shown in Fig. 2, the numerical EE
maximum is approaching $(\log L_{A})/2$, and our bound is asymptotically $\log
L_{A}$ 666Similar results are also obtained for scar states in the extended
Fermi-Hubbard model, see Appendix F that also includes Refs. [61, 75, 63,
64].. Further increasing the operator range to $k=4$ and $k=5$, we find that
$D^{A}$ remains the same as that of $k=3$ in both cases and that our upper
bound can not be improved anymore. A natural guess is that our EE bound using
$k=3$ is already tight. The reason behind our conjecture is that our bound
actually applies not only to pure states $(Q^{\dagger})^{n}\ket{G}$, but also
to linear superpositions of them and even to mixed states constructed from
them.

Example 2: Ground states of the toric code model. Our formalism can be easily
applied to stabilizer states [45, 46, 47]. Let us take the ground state of the
toric code model [48] as a concrete example. Consider a 2D square lattice with
qubits living on links and, say, with the PBC. The toric code model is defined
by the following Hamiltonian,

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Each term in the first (second) sum is a product of four Pauli-$X$ (Pauli-$Z$)
operators around a vertex (face), and is dubbed a star (plaquette) term. All
the terms in $H_{\rm TC}$ commute with each other, and the ground state of the
Hamiltonian is a simultaneous eigenstate of all the star and plaquette terms
with eigenvalue $+1$. Now consider a rectangular subsystem as shown in Fig.
3(a) or 3(b), and let us try to evaluate the EE for such a ground state. Since
each star or plaquette term is proportional to a projector up to a constant
shift, we may skip the initial steps of our formalism and directly construct
the auxiliary Hamiltonian $\hat{H}^{A}_{\rm aux}$ as the negative sum of all
the star and plaquette terms that are completely inside $A$, and the EE is
bounded from above by $\log D^{A}$. To compute $D^{A}$, notice that terms in
$\hat{H}^{A}_{\rm aux}$ form an independent set of stabilizers [45, 46, 47],
which means (i) they are proportional to tensor products of the identity
operator $\hat{I}$ and Pauli operators $\hat{X},\hat{Y},\hat{Z}$, (ii) they
square to the identity and mutually commute, and (iii) any product of a
nonempty subset of those operators is not proportional to the identity
(independence). These three properties imply that specifying the eigenvalue of
each stabilizer will reduce the Hilbert space dimension by a half. Therefore,
$D^{A}=2^{n_{A}-n_{s}}$ where $n_{A}$ is the number of qubits in the
subsystem, and $n_{s}$ is the number of stabilizers contained in
$\hat{H}^{A}_{\rm aux}$. A simple counting shows $S_{v}\leq\log
D^{A}=(|\partial A|-1)\log 2$ where $|\partial A|$ is the perimeter of the
rectangular. This upper bound turns out to be exact. The first term
proportional to $|\partial A|$ indicates the area law, and the subleading
constant term is known as the topological entanglement entropy [10, 11]
signifying topological order.

Figure 3: Two examples of a rectangular subsystem. The EE of the ground state
of the toric code model takes the form $S_{v}=(|\partial A|-1)\log 2$ in both
cases where $|\partial A|$ is the perimeter of the shaded area.

Approximate Zeros.– In previous discussions, we have demonstrated we can
obtain useful upper bound on EE when the local correlation matrix contains
exact zero eigenvalues. However, this protocol can easily fail. For one thing,
the local correlation matrices of low entangled states, such as perturbed
states of the ones considered in the previous section and many-body localized
states, may contain only approximate zero eigenvalues. For another, errors are
inevitable in experimentally measured correlation matrices. In this section,
we extend previous results by providing an upper bound of EE when the
correlation matrix contains approximate zero eigenvalues.

Figure 4: Comparision between our upper bounds (symbols shown in the legend)
and the exact bipartite EEs of the scar states (black crosses) in the PXP
model. The computation is perform for $L=20$ in the zero momentum and
inversion even sector. For completeness, the exact EEs of all the energy
eigenstates are shown.

Statements 1 and 2 require minor revision: instead of focusing only on zero
eigenvalues, we also include eigenvalues $\lambda_{\alpha}\leq c$ with a
finite cutoff $c$. Then following (4) and (5), we can define an additional set
of projection operators $\hat{P}_{i}$. Except for including these extra
projectors, the form of the auxiliary Hamiltonian (6) remains the same. Now,
Statement 3 needs to be modified.

Statement 3′: from Auxiliary Hamiltonian to the EE bound. Given the auxiliary
Hamiltonian $\hat{H}^{A}_{\text{aux}}$, we first compute its energy
$E^{A}_{\text{aux}}$ of the state $|\psi\rangle$ as
$E^{A}_{\text{aux}}=\text{Tr}_{A}(\hat{H}^{A}_{\text{aux}}\hat{\rho}_{A})=\langle\psi|\hat{H}^{A}_{\text{aux}}|\psi\rangle$.
We then consider the thermal reduced density matrix

$\hat{\rho}^{A}_{\text{aux}}(\beta)=\exp(-\beta\hat{H}^{A}_{\text{aux}})/\mathcal{Z}^{A}_{\text{aux}}(\beta).$
(12)

where $\mathcal{Z}^{A}_{\text{aux}}$ is the partition function and determine
the inverse temperature $\beta^{*}$ by matching the energy
$\text{Tr}_{\text{A}}(\hat{H}^{A}_{\text{aux}}\hat{\rho}^{A}_{\text{aux}}(\beta^{*}))=E^{A}_{\text{aux}}$.
Since for all density matrices with the same energy, the thermal density
matrix maximizes the entropy [49], the von Neumann EE of $\ket{\psi}$ is upper
bounded by the thermal entropy
$S_{\text{th}}(\beta^{*})=-\text{Tr}_{\text{A}}(\hat{\rho}^{A}_{\text{aux}}(\beta^{*})\log\hat{\rho}^{A}_{\text{aux}}(\beta^{*}))$.

When we only include projectors of exact zeros in the auxiliary Hamiltonian,
the statement 3’ can be reduced to the statement 3. In this case, we have
$E^{A}_{\text{aux}}=0$ and $\beta^{*}=\infty$, and consequently
$S_{\text{th}}(\infty)=\log D^{A}$.

Example 3: Scar states in the PXP model. We apply our extended protocol to the
scar states in the PXP model under the PBC [50, 51, 52]. We choose range-$k$
operator basis $\hat{L}_{i}$ as

$\hat{L}_{i}=\lambda_{a_{1}}^{(1)}\otimes\lambda_{a_{2}}^{(2)}\otimes...\otimes\lambda_{a_{k}}^{(k)}\,,$
(13)

with $\\{a_{k}\\}=\\{0,1,2,3\\}$ being the indices of the identity matrix and
Pauli matrices $\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}$ (the identity operator
where all $a_{k}$ are zero is excluded). In our numerical study, we choose the
system size $L=20$ and $k\in\\{3,4,5\\}$. Due to the Hilbert space
restriction, the correlation matrices for both scar states and thermal
eigenstates have the same number of trivial zero eigenvalues, but those for
scar states have more approximate zeros. We only include the
$\hat{\Delta}_{\alpha}$ operators with a certain cutoff $\lambda_{\alpha}<c$
in the auxiliary Hamiltonian, and at the same time enforce the Hilbert space
restriction, effectively setting $\tilde{c}_{\alpha}=\infty$ for the
$\hat{\Delta}_{\alpha}$ operators of trivial zeros. We then follow our
protocol to compute the upper bounds 777See Appendix G for details. As shown
in Fig. 4, for $k=4,5$, the upper bounds of EE from the thermal entropy
$S(\beta^{*})$ lead to a well separation between thermal states and scar
states.

Summary.– In summary, we have developed a protocol that connects correlation
and entanglement in a quantitative way, and tested it with exotic quantum
states. Our method can be readily applied to experiments as an economic way of
bounding the entanglement entropy of a quantum state. All one need to measure
are local physical observables that give rise to the matrix elements of the
local correlation matrices 888See Appendix H for an experimental protocol in
cold atom systems, which also includes Refs. [66, 67, 68, 69, 70, 71, 72, 73,
74]. Different from methods that focus directly on the reduced density matrix
[55, 56, 57, 58], our method is divide-and-conquer in nature. This means our
measurement cost to bound the EE of a subsystem $A$ only scales linearly with
the subsystem size $L_{\mathrm{A}}$. Moreover, for the exact zero case of
eigenvalues of local correlation matrices, the numerical effort can be
significantly reduced by our special algorithm. For the case of approximate
zeros, however, using thermal entropy to bound entanglement entropy is costly
and deserves further studies. Still, in this Noisy Intermediate-Scale Quantum
era [59], we envision our protocol to be particular useful.

Acknowledgements. We especially thank Hui Zhai for many invaluable
discussions. We are grateful to Xiao-Liang Qi for his interest and an
insightful discussion. We also thank the referees for their many comments and
suggestions that increase the clarity of this work. S. L. is supported by the
Gordon and Betty Moore Foundation under Grant No. GBMF8690 and the National
Science Foundation under Grant No. NSF PHY-1748958. P. Z. acknowledges support
from the Walter Burke Institute for Theoretical Physics at Caltech.

##  Appendix A Illustration of Our Protocol of Exact Zeros Using $2$-local
Operator Basis

We shall further elucidate our protocol of bounding the entanglement entropy
of a subsystem $A$ using $k=2$-local operator basis $\\{\hat{L}_{i}\\}$ here.
To facilitate understanding, we use a concrete example depicted in Fig. [5].
The subsystem $A$ is shaded red, say with sites $2,3,4,5$, and of size
$L_{A}=4$. Our protocol is first to measure the correlation matrix on sites
$2$ and $3$ (dotted blacks in the second row of Fig. [5), collect the
$\hat{\Delta}$ operators (equivalently projection operators) following
statements 1 and 2 in the main text. We then move to sites $3,4$ to repeat the
above procedure, and then to sites $4,5$ for the same task. After exhausting
all these continuous $k=2$ subsystems inside $A$, we collect all those
$\hat{\Delta}_{\alpha}$ to construct the auxiliary Hamiltonian
$\hat{H}_{\text{aux}}^{A}$ following equation (6) in the main text. The final
step is to utilize the GSD-EE theorem to calculate the ground-state degeneracy
$D^{A}$ that yields an upper bound of $\log D^{A}$ of the entanglement entropy
of $A$.

Figure 5: Illustration of steps of our protocol to bound entanglement entropy
(EE) of subsystem $A$ (shaded red) using 2-local operator basis
$\\{\hat{L}_{i}\\}$. For every consecutive two sites (dotted black) in $A$, we
use a set of 2-local operator basis $\\{\hat{L}_{i}\\}$ acting on these two
sites to construct the correlation matrix and determine the corresponding
positive semidefinite operators $\\{\hat{\Delta}_{\alpha}^{(n)}\\}$ where $n$
labels the choices of these two sites. The $\hat{\Delta}_{\alpha}$ operators
that enters the auxiliary Hamiltonian comes from the union of these
$\\{\hat{\Delta}_{\alpha}^{(n)}\\}$.

##  Appendix B Equivalence between Different Choices of Eigenvectors
$\\{\mathbf{e}_{\alpha}\\}$ with Degeneracy

We shall show in the degenerate case where
$\operatorname{dim}(\operatorname{Ker}(\mathcal{M}))=n>1$ the final result of
$D^{A}$ does not depend on the choice of eigenvectors of zero eigenvalues (our
Lemma 1 below). Let us denote the unitary matrix connecting two different sets
of zero-eigenvalue basis $\\{\mathbf{e}_{\alpha}\\}$ and
$\\{\mathbf{\tilde{e}}_{\alpha}\\}$ as $U$,

$\begin{pmatrix}\mathbf{\tilde{e}}_{1}\\\ \mathbf{\tilde{e}}_{2}\\\ \vdots\\\
\mathbf{\tilde{e}}_{n}\end{pmatrix}=\begin{pmatrix}U_{11}&U_{12}&\cdots&U_{1n}\\\
U_{21}&U_{22}&\cdots&U_{2n}\\\ \vdots\\\
U_{n1}&U_{n2}&\cdots&U_{nn}\end{pmatrix}\begin{pmatrix}\mathbf{e}_{1}\\\
\mathbf{e}_{2}\\\ \vdots\\\ \mathbf{e}_{n}\end{pmatrix}\,,$ (14)

and the state from which the correlation matrix is constructed as
$|\psi\rangle$. From

$\hat{O}_{\alpha}\ket{\psi}=\xi_{\alpha}\ket{\psi}\,,$ (15)

we can easily write down the eigenoperator equation for
$\hat{\widetilde{O}}_{\alpha}$,

$\hat{\widetilde{O}}_{\alpha}\ket{\psi}=\sum_{\beta}U_{\alpha\beta}\hat{O}_{\beta}\ket{\psi}=\sum_{\beta}U_{\alpha\beta}\xi_{\beta}\ket{\psi}\equiv\widetilde{\xi}_{\alpha}\ket{\psi}\,,$

where we have identified
$\widetilde{\xi}_{\alpha}=\sum_{\beta}U_{\alpha\beta}\xi_{\beta}$. The
corresponding positive semidefinite operator
$\hat{\widetilde{\Delta}}_{\alpha}$ for the tilde version then reads

$\displaystyle\hat{\widetilde{\Delta}}_{\alpha}$ $\displaystyle=$
$\displaystyle(\hat{\widetilde{O}}_{\alpha}^{\dagger}-\tilde{\xi}^{*}_{\alpha}\hat{I})(\hat{\widetilde{O}}_{\alpha}-\tilde{\xi}_{\alpha}\hat{I})$
(16) $\displaystyle=$
$\displaystyle\sum_{\beta,\gamma}U_{\alpha\beta}U_{\alpha\gamma}^{*}(\hat{O}_{\gamma}^{\dagger}-\xi^{*}_{\gamma}\hat{I})(\hat{O}_{\beta}-\xi_{\beta}\hat{I})\,.$

######  Theorem 1.

The value of ground-state degeneracy of the auxiliary Hamiltonian, $D^{A}$, is
the same for different basis choices $\\{\mathbf{e}_{\alpha}\\}$ and
$\\{\tilde{\mathbf{e}}_{\alpha}\\}$.

###### Proof.

It suffices to show if a ground state $\ket{G}$ of the auxiliary Hamiltonian
$\hat{H}^{A}_{\text{aux}}$ satisfy
$\hat{\widetilde{\Delta}}_{\alpha}\ket{G}=0,\;\forall\alpha$, then
$\hat{\Delta}_{\alpha}\ket{G}=0,\;\forall\alpha$ since we can reverse the role
$\\{\mathbf{e}_{\alpha}\\}$ and $\\{\tilde{\mathbf{e}}_{\alpha}\\}$ and repeat
the the whole process. From our condition
$\hat{\widetilde{\Delta}}_{\alpha}\ket{G}=0,\;\forall\alpha$ and Eq. (16), we
have

$\bra{G}\sum_{\beta,\gamma}U_{\alpha\beta}U_{\alpha\gamma}^{*}(\hat{O}_{\gamma}^{\dagger}-\xi^{*}_{\gamma}\hat{I})(\hat{O}_{\beta}-\xi_{\beta}\hat{I})\ket{G}=0\,.$
(17)

Summing the $\alpha$ index and noting
$\sum_{\alpha}U_{\alpha\beta}U_{\alpha\gamma}^{*}=\delta_{\beta\gamma}$ (delta
function), we have

$\bra{G}\sum_{\beta}(\hat{O}_{\gamma}^{\dagger}-\xi^{*}_{\beta}\hat{I})(\hat{O}_{\beta}-\xi_{\beta}\hat{I})\ket{G}=\bra{G}\sum_{\alpha}\hat{\Delta}_{\alpha}\ket{G}=0\,,$

where we have changed the dummy summation index from $\beta$ to $\alpha$.
Since $\hat{\Delta}_{\alpha}$ is positive semidefinite, the above equation
implies

$\hat{\Delta}_{\alpha}\ket{G}=0\,,\quad\forall\alpha\,.$ (18)

Therefore, $D^{A}$ does not depend on the choice of eigenvector basis in the
degenerate space of zero eigenvalues. ∎

##  Appendix C EE Bounds of Product States

In this section, we take product states as an example for demonstrating the
effectiveness of our GSD-EE theorem. We shall prove the local correlation
matrix of 1-local operator basis $\\{\hat{L}_{i}\\}$ is enough to give an EE
upper bound of zero for produce state.

For definiteness, consider a spin-1/2 system and denote the local bases on a
site as $\ket{0}$ and $\ket{1}$. Without loss of generality, we can write the
product state as

$\ket{\psi}=\prod_{i}\otimes\ket{0}_{i}=\ket{00\cdots 0}\,,$ (19)

since one can always rotate local basis on every site freely. Now, choose the
nontrivial 1-local operator basis as the standard Pauli operators
$\hat{X},\hat{Y}$, and $\hat{Z}$. The correlation matrix, defined in Eq. (1)
in our manuscript, is thus

$\mathcal{M}_{ij}=\bra{\psi}\hat{\sigma}_{i}\hat{\sigma}_{j}\ket{\psi}-\bra{\psi}\hat{\sigma}_{i}^{\dagger}\ket{\psi}\bra{\psi}\hat{\sigma}_{j}\ket{\psi}$
(20)

where we have used the shorthand notation
$\hat{\bm{\sigma}}\equiv(\hat{X},\hat{Y},\hat{Z})$. The explicit form of
$\mathcal{M}$ can be easily written down

$\mathcal{M}=\begin{pmatrix}1&-i&0\\\ i&1&0\\\ 0&0&0\end{pmatrix}\,.$ (21)

The eigenvalues of $\mathcal{M}$ are $0$, $0$, and $2$, and two linear
independent eigenvectors of $\mathcal{M}$ with zero eigenvalues can be taken
as $\mathbf{e}_{1}=(1/\sqrt{2},-i/\sqrt{2},0)^{T}$ and
$\mathbf{e}_{2}=(0,0,1)^{T}$. The corresponding two eigenoperators are
$\hat{O}_{1}=(\hat{X}-i\hat{Y})/\sqrt{2}$ with $\xi_{1}=0$ and
$\hat{O}_{2}=\hat{Z}$ with $\xi_{2}=-1$. The associated positive semidefinite
operators, defined in Eq. (4) in the main text, $\hat{\Delta}_{1}$ and
$\hat{\Delta}_{2}$ are

$\displaystyle\hat{\Delta}_{1}$
$\displaystyle=(\hat{X}+i\hat{Y})(\hat{X}-i\hat{Y})/2=\hat{I}+\hat{Z}=2\ket{1}\bra{1}\,,$
(22) $\displaystyle\hat{\Delta}_{2}$
$\displaystyle=(\hat{Z}+\hat{I})(\hat{Z}+\hat{I})=2(\hat{Z}+\hat{I})=4\mbox{$|1\rangle\langle
1|$}\,.$

This means the auxiliary Hamiltonian $\hat{H}^{A}_{\text{aux}}$, defined in
Eq. (6) in the main text, takes the form

$\hat{H}^{A}_{\text{aux}}=\sum_{i\in A}c_{i}\ket{1}_{i}\bra{1}_{i}\,,\quad
c_{i}>0\,,$ (23)

where $i$ labels the sites in subsystem A. Clearly, _regardless of the value
of $c_{i}$ as long as $c_{i}>0$_, the ground state of
$\hat{H}^{A}_{\text{aux}}$ under the open boundary condition is unique:
$\ket{G}=\prod_{i\in A}\otimes\ket{0}_{i}$. From our GSD-EE theorem, the von
Neumann entropy $S_{v}$ is upper bounded by $\log 1=0$. On the other hand,
$S_{v}\geq 0$, and, therefore, our GSD-EE theorem yields the exact result
$S_{v}=0$ for product states.

A remark is in order here. In practice we do not know the rotated basis,
$\ket{0}$ and $\ket{1}$, and hence do not know $\hat{X},\hat{Y}$, and
$\hat{Z}$ defined above. However, the choice of bases $\\{\hat{L}_{i}\\}$ is
immaterial. Indeed, if one repeats the above process for a general product
state,

$\ket{\psi}=\prod_{i}\otimes\ket{\phi_{i}}=\prod_{i}\otimes(\alpha_{i}\ket{0}+\beta_{i}\ket{1})\,,$
(24)

one will again find two zero eigenvalues of the correlation matrix. The two
eigenvectors, up to some normalization constants, can be taken as

$\displaystyle\mathbf{e}_{1}$
$\displaystyle=\big{(}\dfrac{\alpha\beta^{*}+\alpha^{*}\beta}{|\beta|^{2}-|\alpha|^{2}},\dfrac{i\alpha^{*}\beta-i\alpha\beta^{*}}{|\beta|^{2}-|\alpha|^{2}},1\big{)}^{T}\,,$
(25) $\displaystyle\mathbf{e}_{2}$
$\displaystyle=\big{(}\dfrac{\alpha^{2}-\beta^{2}}{2\alpha\beta},-i\dfrac{\alpha^{2}-\beta^{2}}{2\alpha\beta},1\big{)}^{T}\,.$

The eigenvalues $\xi_{1}$ and $\xi_{2}$ for $\hat{O}_{1}$ and $\hat{O}_{2}$
are $\xi_{1}=1/(|\beta|^{2}-|\alpha|^{2})$ and $\xi_{2}=0$, respectively. The
two $\hat{\Delta}$ operators, up to some constant normalization factors,
constructed for site $i$ are

$\hat{\Delta}_{1}=\hat{\Delta}_{2}=\begin{pmatrix}|\alpha_{i}|^{2}&-\alpha_{i}^{*}\beta_{i}\\\
-\alpha_{i}\beta_{i}^{*}&|\beta_{i}|^{2}\end{pmatrix}=\mbox{$|\phi_{i}^{\perp}\rangle\langle\phi_{i}^{\perp}|$}\,,$
(26)

where $\ket{\phi_{i}^{\perp}}=\beta_{i}^{*}\ket{0}-\alpha_{i}^{*}\ket{1}$ is
orthogonal (zero inner product) to $\ket{\phi_{i}}$. The resulting auxiliary
Hamiltonian then reads (up to some normalization constants)

$\hat{H}^{A}_{\text{aux}}=\sum_{i}c_{i}\mbox{$|\phi_{i}^{\perp}\rangle\langle\phi_{i}^{\perp}|$}\,,\quad
c_{i}>0\,,$ (27)

and its ground state is unique with $D^{A}=1$.

Having showed the special case of spin-1/2 systems where the local Hilbert
space dimension $q$ equals two, we shall continue to give a formal
constructive proof for the case of general $q$. Denote the local Hilbert basis
as $\ket{i},i=0,1,\cdots,q-1$ and still, without loss of generality, the
general form a product state $\ket{\psi}$ is taken to be

$\ket{\psi}=\prod_{i}\otimes\ket{0}_{i}=\ket{00\cdots 0}\,,$ (28)

as before. And we shall take the operator basis $\\{\hat{L}_{i}\\}$ to be the
set of complete 1-local operators (still the trivial identity operator can be
neglected as before). Then, a general 1-body eigenoperator $\hat{O}$ where
$\hat{O}\ket{\psi}=\xi\ket{\psi}$ (they exist, for example we can construct
$\ket{0}\bra{1}$ as one such operator) can be formally written as

$\hat{O}=\xi\mbox{$|0\rangle\langle
0|$}+\sum_{i=1}^{q-1}\mbox{$|\phi_{i}\rangle\langle i|$}\,,$ (29)

where $\ket{\phi_{i}}=\hat{O}\ket{i}$. The positive semidefinite operator
$\hat{\Delta}$ then reads

$\hat{\Delta}=(\hat{O}^{\dagger}-\xi^{*}\hat{I})(\hat{O}-\xi\hat{I})=\sum_{i,j=1}^{q-1}C_{ji}\mbox{$|j\rangle\langle
i|$}$ (30)

where

$C_{ji}=\left(\bra{\phi_{j}}-\xi^{*}\bra{j}\right)\left(\ket{\phi_{i}}-\xi\ket{i}\right)$
(31)

is positive semidefinite (can be proven by showing
$\sum_{i,j}x_{j}^{*}C_{ji}x_{i}\geq 0$ since it is the squared norm of
$x_{i}(\ket{\phi_{i}}-\xi\ket{i})$) as expected. Therefore, we can diagonalize
$C_{ij}$ and bring $\hat{\Delta}$ into diagonal form

$\hat{\Delta}=\sum_{i,j=1}^{q-1}C_{ji}\mbox{$|j\rangle\langle
i|$}=\sum_{i}a_{i}\mbox{$|i\rangle\langle
i|$}\equiv\sum_{i}a_{i}\hat{P}_{i}\,,$ (32)

where $a_{i}\geq 0$ are the eigenvalues of matrix $C_{ij}$. The important
thing to note is that
$\hat{P}_{i}\ket{0}=\ket{\tilde{i}}\braket{\tilde{i}}{0}=0$ since
$\ket{\tilde{i}}$ is a linear combination of $\ket{i}$ with
$i=1,2,\cdots,q-1$. Moreover, by taking a set of states $|\phi_{j}\rangle$
where the matrix $C_{ij}$ has rank $q-1$ (for example, setting
$\ket{\phi_{i}}=(\xi+1)\ket{i}$ makes $C$ an identity matrix), $|0\rangle$
becomes the only state in the kernel of $\hat{\Delta}$. As a result, the
ground state of $\hat{H}^{A}_{\text{aux}}$ under the open boundary condition
is unique as in the spin-1/2 case: $\ket{G}=\prod_{i\in A}\otimes\ket{0}_{i}$,
and our upper bound of the von Neumann entropy is exact, $S_{v}=\log 1=0$.

##  Appendix D Special Algorithm for the Ground-State Degeneracy of the
Auxiliary Hamiltonian

Our GSD-EE theorem states that the EE of a state $\ket{\psi}$ on a subsystem
of size $L_{A}$ is upper bounded by the ground-state degeneracy $D^{A}$ of the
auxiliary Hamiltonian $\hat{H}^{A}_{\text{aux}}$ under the open boundary
condition. Because of the exponential increase of the Hilbert space dimension,
the maximum size $L_{A}$ can be reached by directly attacking this problem
using the exact diagonalization method is quite limited. We now explain a
special algorithm that overcomes the difficulty. For the simplicity of
notations, consider the concrete example of AKLT scar states mentioned in the
main text. Here, due to the translation symmetry, we obtain the same set of
projectors from the local correlation matrix of any three consecutive sites.
Let $\ket{G_{m}^{(L_{A})}}$ denote the orthonormal ground states of the
auxiliary Hamiltonian of system size $L_{A}$ with $m=1,2,\cdots,D^{A}$ as the
index, and let $\hat{P}^{(L)}_{i}$ denote the $i$-th projector acting on the
sites $L-2,L-1$ and $L$, so that

$\hat{H}^{A}_{\text{aux}}=\hat{H}^{A^{\prime}}_{\text{aux}}+\sum_{i}c_{i}\hat{P}^{(L_{A})}_{i}\equiv\hat{H}^{A^{\prime}}_{\text{aux}}+\hat{H}^{(L_{A})}$
(33)

where $A^{\prime}$ is the subsystem of sites the $1,2,\cdots,L_{A}-1$, and
$\hat{H}^{(L_{A})}$ is the part that acts only on the three sites
$L_{A}-2,L_{A}-1$ and $L_{A}$. Since $\ket{G^{(L_{A})}_{m}}$ is annihilated by
every projector acting on the sites $1,2,\cdots,L_{A}-1$, we can expand it
using the ground states $\ket{G^{(L_{A}-1)}_{k}}$ of the auxiliary Hamiltonian
with system size $L_{A}-1$ and the basis $\ket{e_{n}}$ on the $L_{A}$-th site,

$\ket{G_{m}^{(L_{A})}}=\sum_{n,k}C_{kn,m}^{(L_{A})}\ket{G^{(L_{A}-1)}_{k}}\otimes\ket{e_{n}}$
(34)

where $n$ and $k$ are the indices of the corresponding orthonormal states and
$C_{kn,m}^{(L_{A})}$ is the unknown coefficient. These coefficients and the
degeneracy $D^{A}$ can be obtained by diagonalizing $\hat{H}^{A}_{\text{aux}}$
or equivalently $\hat{H}^{(L_{A})}$ under the basis
$\ket{G^{(L_{A}-1)}_{k}}\otimes\ket{e_{n}}$. Moreover, we do not need to store
the memory-consuming part $\ket{G^{(L_{A}-1)}_{k}}$ in this process.
Recursively using (34), we can expand $\ket{G^{(L_{A}-1)}_{k}}$ as

$\sum_{l,n}C_{ln,k}^{(L_{A}-1)}\left(\sum_{i,j}C_{ij,l}^{(L_{A}-2)}\ket{G_{i}^{(L_{A}-3)}}\otimes\ket{e_{j}}\right)\otimes\ket{e_{n}}\,.$

Since $\hat{H}^{(L_{A})}$ acts trivially on $\ket{G_{i}^{(L_{A}-3)}}$, we just
need to store the coefficients $C_{ij,l}^{(L_{A}-2)}$ and
$C_{ln,m}^{(L_{A}-1)}$ to calculate the matrix elements of $\hat{H}^{(L_{A})}$
in the basis $\ket{G^{(L_{A}-1)}_{k}}\otimes\ket{e_{n}}$, diagonalize the
resulting matrix, and obtain the ground state degeneracy $D^{A}$ as well as
$C_{kn,m}^{(L_{A})}$. Since this recursive method is memory-cheap and we just
need to diagonalize a matrix of dimension $D^{A^{\prime}}\times q$ (where $q$
is the Hilbert space dimension of a single site) to get $D^{A}$, calculating
the ground state degeneracy of system size $L_{A}$ of a few hundreds is very
easy.

##  Appendix E Projector Space of Scar States in the AKLT Model

To employ our GSD-EE theorem to upper bound the bipartite entanglement
entropies of all the scar states,

$\ket{\psi_{n}}=(\hat{Q}^{\dagger})^{n}\ket{G},$ (35)

we need to find a set of projection operators $\\{\hat{P}_{i}\\}$ such that
each of them annihilates every $\ket{\psi_{n}}$,

$\hat{P}_{i}\ket{\psi_{n}}=0\,,\quad\forall\,i,n\,.$ (36)

Let $\hat{H}_{\text{aux},n}^{A}$ denotes an auxiliary Hamiltonian of
$\ket{\psi_{n}}$, let $\hat{P}_{i}^{(n)}=\ket{i^{(n)}}\bra{i^{(n)}}$ denotes
the projectors entering $\hat{H}_{\text{aux},n}^{A}$. Since, as implied by
Lemma 1, the kernel (ground-state manifold) of the auxiliary Hamiltonian is
invariant for fixed operator space
$\mathcal{V}=\operatorname{Span}\\{\hat{L}_{i}\\}$, the complement of the
kernel, the state space $V^{(n)}\equiv\operatorname{Span}\\{\ket{i^{(n)}}\\}$,
is also invariant. Therefore, the direct way to identify $\\{\hat{P}_{i}\\}$
annihilating all the scar states is to first calculate the common intersection
$V_{\cap}\equiv\bigcap\limits_{n}\,V^{(n)}$ and then form
$\\{\mbox{$|i\rangle\langle i|$}\\}$ using a set of bases $\\{\ket{i}\\}$ in
$V_{\cap}$. This is, however, computationally too expensive, for it requires
calculating $V^{(n)}$ for every $\ket{\psi_{n}}$. Fortunately, making use of
the special structure of the AKLT Hamiltonian and the ground state $\ket{G}$,
we typically only need to deal with one $\ket{\psi_{n}}$, thus significantly
reducing the computational cost. First note that the AKLT Hamiltonian under
the periodic boundary condition (PBC) is a sum of projectors,

$\hat{H}_{\text{AKLT}}=\sum_{i=1}^{L}\hat{P}_{i,i+1}^{(2)}\,,$ (37)

where $\hat{P}_{i,i+1}^{(2)}$ is the projection operator of two spin-1’s at
site $i$ and $i+1$ onto total spin-2, and that the ground state $\ket{G}$ is
annihilated by every $\hat{P}_{i,i+1}^{(2)}$,

$\hat{P}^{(2)}_{i,i+1}\ket{G}=0\,,\quad\forall\;i\,.$ (38)

If we partite the system into two parts, one consisting of $k$ consectutive
sites and the other part containing the rest $L-k$ sites as shown in Fig. [6],
and denote the part of the Hamiltonian acting only in the $k$-site region as
$\hat{H}_{k}$, i.e.

$\hat{H}_{k}=\sum_{i=1}^{k-1}\hat{P}_{i,i+1}^{(2)}\,,$ (39)

it follows that the ground state is annihilated by $\hat{H}_{k}$,

$\hat{H}_{k}\ket{G}=0\,.$ (40)

Figure 6: Partition of the system into $k$-site region and $(L-k)$-site
region.

Since $\hat{H}_{k}$ is nothing but the AKLT Hamiltonian of the $k$-site region
under the open boundary condition (OBC), Eq. (40) implies $\ket{G}$ can be
decomposed in the following way,

$\ket{G}=\sum_{\eta=1}^{4}\ket{G_{k,\eta}}\otimes\ket{\Psi_{\eta}}$ (41)

where $\ket{{G}_{k,\eta}}$ is the ground state of the $k$-site AKLT
Hamiltonian under the OBC with $\eta=1,2,3,4$ as labels of the four-fold
degeneracy, and $\ket{\Psi_{\eta}}$ is a state vector in the Hilbert space of
the rest $L-k$ sites. Similarly, we can write $\hat{Q}^{\dagger}$ as a sum of
two terms, denoted by $\hat{Q}_{k}$ and $\hat{Q}_{L-k}^{\dagger}$, that act
only in the $k$\- and $(L-k)$-site regions respectively,

$\hat{Q}^{\dagger}=\sum_{i=1}^{k}(-1)^{i}\left(\hat{S}_{i}^{+}\right)^{2}+\sum_{i=k+1}^{L}(-1)^{i}\left(\hat{S}_{i}^{+}\right)^{2}\equiv\hat{Q}_{k}^{\dagger}+\hat{Q}_{L-k}^{\dagger}\,.$

The key observation is that, using the decomposition of $\ket{G}$ and
$\hat{Q}^{\dagger}$, we can decompose a general scar state, Eq. (35), as

$\displaystyle\ket{\psi_{n}}$ $\displaystyle=$
$\displaystyle\left(\hat{Q}_{k}^{\dagger}+\hat{Q}_{L-k}^{\dagger}\right)^{n}\ket{G}$
(42) $\displaystyle=$
$\displaystyle\sum_{m=0}^{n}\sum_{\eta=1}^{4}(\hat{Q}_{k}^{\dagger})^{m}\ket{G_{k,\eta}}\otimes\ket{\tilde{\Psi}_{m,\eta}}\,,$

where $\ket{\tilde{\Psi}_{m,\eta}}$ is a state vector in the Hilbert space the
$L-k$ sites. Since $\ket{G_{k,\eta}}$ only contain $z$-component of the total
spin, $S_{z}$, greater or equal than $-1$ and each $\hat{Q}_{k}^{\dagger}$
raises $S_{z}$ by two [60], the maximum value of $m$ we need to address is
$\left\lfloor\frac{k+1}{2}\right\rfloor$ where $\left\lfloor x\right\rfloor$
denotes the maximum integer no bigger than $x$. Let $\\{\ket{\Phi_{i}}\\}$
denotes a set of orthonormal bases where each basis has zero inner product
with $(\hat{Q}_{k}^{\dagger})^{m}\ket{G_{k,\eta}}$ for every possible values
of $m$ and $\eta$,

$\langle\Phi_{i}|(\hat{Q}_{k}^{\dagger})^{m}\ket{G_{k,\eta}}=0\,,\quad\forall\;i,m,\eta\,.$
(43)

We can then easily construct a set of $k$-local projection operators
$\\{\mbox{$|\Phi_{i}\rangle\langle\Phi_{i}|$}\\}$ and each
$|\Phi_{i}\rangle\langle\Phi_{i}|$ annihilates all the scar states, Eq. (35),
for system size $L>k$. From construction, it is clear that the state space
spanned by $\\{\ket{\Phi_{i}}\\}$, denoted as $V_{\Phi}$, is a subset of the
common intersection $V_{\cap}$ we are looking for,

$V_{\Phi}\equiv\operatorname{Span}\\{\,\ket{\Phi_{1}},\ket{\Phi_{2}},\cdots\,\\}\subseteq
V_{\cap}\,.$ (44)

On the other hand, for a scar state $\ket{\psi_{n}}$ with fixed
$n\geq\lfloor\frac{k+1}{2}\rfloor$ of a large enough system size $L$, the
states $\ket{\tilde{\Psi}_{m,\eta}}$ in the decomposition of $\ket{\psi_{n}}$,
Eq. (42), are in general linearly independent. This implies that for such a
scar state $\ket{\psi_{n}}$,

$\hat{P}_{i}^{(n)}\ket{\psi_{n}}=0\,,\;\Leftrightarrow\;\hat{P}_{i}^{(n)}(\hat{Q}_{k}^{\dagger})^{m}\ket{G_{k,\eta}}=0\,,\;\forall\;m,\eta\,.$

In other words, $V^{(n)}=V_{\Phi}$ for such $\ket{\psi_{n}}$, which further
implies $V_{\cap}\subseteq V_{\Phi}$. Together with Eq. (44), we conclude

$V_{\Phi}=V_{\cap}\,,$

and $V_{\cap}$ can be determined by such a scar state $\ket{\psi_{n}}$.

The above consideration holds generally for $k\geq 2$, and further
simplification can happen for a given $k$. For example, for $k=3$, a simple
calculation shows that

$\operatorname{Span}\\{\,(\hat{Q}_{3}^{\dagger})^{2}\ket{G_{3,\eta}}\,\\}\subset\operatorname{Span}\\{\,\ket{G_{3,\eta}},\hat{Q}_{3}^{\dagger}\ket{G_{3,\eta}}\,\\}\,.$

This means that a $3$-local projector that annihilates
$\ket{\psi_{1}}=\hat{Q}^{\dagger}\ket{G}$ will also annihilates all the other
scar states $\ket{\psi_{n}}$ for all system sizes $L>3$. Moreover, our
$3$-local operator basis

$\hat{L}_{i}=\lambda_{a}^{(1)}\otimes\lambda_{b}^{(2)}\otimes\lambda_{c}^{(3)}\,,\quad
a,b,c=1,2,\cdots,8$ (45)

is general enough for the constructed state space
$V=\operatorname{Span}\\{\ket{i}\\}$ from $\\{\hat{P}_{i}\\}$ to contain
$V_{\Phi}$. Similar result holds both for 4-local operator basis

$\hat{L}_{i}=\lambda_{a}^{(1)}\otimes\lambda_{b}^{(2)}\otimes\lambda_{c}^{(3)}\otimes\lambda_{d}^{(4)}\,,$
(46)

and 5-local operator basis

$\hat{L}_{i}=\lambda_{a}^{(1)}\otimes\lambda_{b}^{(2)}\otimes\lambda_{c}^{(3)}\otimes\lambda_{d}^{(4)}\otimes\lambda_{e}^{(5)}\,,$
(47)

where $a,b,c,d,e=1,2,\cdots,8$.

##  Appendix F EE Bounds of Scar States in the Fermi–Hubbard Model

The scar states in the 1D extended Fermi–Hubbard model for spin-$1/2$ fermions
take a similar form to those of the AKLT model [61, 62]. They are known as the
$\eta$-pairing states [63] and take the form of

$\ket{\psi_{n}}=(\hat{\eta}^{\dagger})^{n}\ket{0}\,,$ (48)

where $\ket{0}$ is the vacuum state with no fermions and
$\hat{\eta}^{\dagger}=\sum_{i}(-1)^{i}\hat{c}_{i\uparrow}^{\dagger}\hat{c}_{i\downarrow}^{\dagger}$
is the so-called $\eta$-pairing operator with $\hat{c}_{i\uparrow}^{\dagger}$
($\hat{c}_{i\downarrow}^{\dagger}$) being the fermionic creation operator for
spin up (down) at site $i$. The procedure to construct the local correlation
function parallels that of the AKLT model. We choose our target scar state as
the half-filled state $\ket{\psi}_{L/2}$ ($L$ is assumed to be even), and
operators $\hat{L}_{i}$ as range-$3$ operators with a similar tensor product
form,
$\hat{L}_{i}=\lambda_{a}^{(1)}\otimes\lambda_{b}^{(2)}\otimes\lambda_{c}^{(3)}$
on three neighboring sites $1$, $2$ and $3$. The Hilbert space is four-
dimensional for each site, spanned by an unoccupied state, two singly occupied
states with different spins, and one doubly occupied state. Thus, here we take
$\lambda$ as the fifteen generators of SU(4) group and the identity
($a,b,c=1,\cdots,16$). The local correlation matrix is then calculated and the
auxiliary Hamiltonian is constructed in a similar fashion. We find ground
state degeneracy $D^{A}$ scales with the subsystem size $L_{\text{A}}$ as
$D^{A}=L_{A}+1$, meaning the EE of this state is bounded by the logarithm of
the subsystem size. For reasons identical to the AKLT case, this bound is also
valid for all other scar states of $n\neq L/2$. Finally, we compare the exact
value of EE with our bound of the scar state $\ket{\psi_{L/2}}$. The exact
value of EE is analytically known [64] to be asymptotically $S_{v}\propto(\log
L_{A})/2$. Therefore, apart from a ratio of two, our bound gives the correct
log-volume scaling of the EE.

Figure 7: Comparison between the EE bounds (green dots) from the ground state
degeneracies of the auxiliary Hamiltonian $\hat{H}^{A}_{\text{aux}}$ and the
exact values (blue dots) for $|\psi_{L/2}\rangle$ for various subsystem size
$L_{\text{A}}$.

##  Appendix G Numerical Details for Bounding Entanglement Entropies in PXP
Model

In this section, we shall elaborate on the numerical procedure we use to bound
the EEs of the scar states in the PXP model. We start to construct the
correlation matrix for given range-$k$ operator basis of the form (still the
identity operator is excluded)

$\hat{L}_{i}=\lambda_{a_{1}}^{(1)}\otimes\lambda_{a_{2}}^{(2)}\otimes...\otimes\lambda_{a_{k}}^{(k)}\,,$
(49)

where $\\{a_{j}\\}=\\{0,1,2,3\\}$ for $j=1,2,\cdots,k$ are the indices of the
$2\times 2$ identity matrix $\mathbb{I}$ (where $a_{j}=0$) and the three Pauli
matrices $\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}$. We calculate the
$(4^{k}-1)$-dimensional local correlation matrix for every eigenstate and
compare the correlation spectrum of a scar state with surrounding thermal
states. As shown in Fig. 8, the correlation spectra of scar states are
markedly different with more smaller eigenvalues. From such plot, we can
determine a position $\alpha=N_{1}$ (labeled by pink arrow in Fig. 8) where
the eigenvalue plot of the scar state is about to cross those of the
surrounding thermal states.

Figure 8: Plots of correlation spectra for a group of eigenstates (eigenstate
indices $n$ shown in the legends) around two quantum many-body scar states
(marked as black) for the PXP model in the zero momentum and inversion even
sector. The system size is $L=20$ and the range of the operator basis
$\\{\hat{L}_{i}\\}$ is $k=5$. In both groups, the eigenvalues of the
correlation matrix of the scar state noticeably fall below (excluding the
first $N_{0}=608$ trivial zeros) those of the surrounding thermal eigenstates
when $\alpha\leq N_{1}$ (corresponding points marked by pink arrows). The
$N_{1}-N_{0}$ positive semidefinite $\hat{\Delta}_{\alpha}$ operators
associated with these eigenvalues are then taken as the starting point to
construct $\hat{H}^{A}_{\text{aux}}$ to bound EE.

We shall use the $N_{1}-N_{0}$ (we subtract $N_{0}$ to exclude the trivial
zeros) eigenvalues to construct an auxiliary Hamiltonian
$\hat{H}_{\text{aux}}^{A}$ to bound the EE. For such nonzero eigenvalue
$\lambda_{\alpha}\neq 0$, we still proceed to define an operator $\hat{O}$
using the corresponding eigenvector
$\mathbf{e}_{\alpha}=(w_{1},w_{2},\cdots,w_{n})$ in a similar fashion,
$\hat{O}=\sum_{i}w_{i}\hat{L}_{i}$. In this case, $\hat{O}$ is not an
eigenoperator of $\ket{\psi}$ anymore,

$\hat{O}\ket{\psi}=\xi\ket{\psi}+\ket{\psi_{\perp}}\,,\quad\text{where
}\langle\psi_{\perp}|\psi_{\perp}\rangle>0\,.$ (50)

We can, however, still define a positive semidefinite operator
$\hat{\Delta}_{\alpha}$ as before,

$\hat{\Delta}_{\alpha}=\left(\hat{O}^{\dagger}-\xi^{*}\hat{I}\right)\left(\hat{O}-\xi\hat{I}\right)\,.$
(51)

The constructed auxiliary Hamiltonian still takes the same form

$\hat{H}_{\text{aux}}^{A}=\sum_{\alpha}\tilde{c}_{\alpha}\hat{\Delta}_{\alpha}\,,\quad\tilde{c}_{\alpha}>0\,.$
(52)

For given $N_{1}$, optimizing the value of $\tilde{c}_{\alpha}$ to get the
best upper bound is beyond the scope of our paper. Therefore, for
demonstration purposes, we shall take the simplest case where
$\tilde{c}_{a}=1,\forall\alpha$. Since $N_{1}$ is handpicked, we choose to
vary the value of $N_{1}$ to get a comparatively good upper bound of EE. We
summarize the optimized result of the EE upper bounds of the scar states of
the PXP model in Table 1 (see also [65]).

| $n=1$ | $n=2$ | $n=7$ | $n=30$ | $n=102$
---|---|---|---|---|---
$k=3$ |  0.4038 |  10 |  1.6391 |  24 |  2.1314 |  12 |  2.6668 |  16 |  2.2858 |  19
$k=4$ |  0.4031 |  81 |  1.5718 |  93 |  2.0201 |  76 |  2.0227 |  86 |  1.9797 |  89
$k=5$ |  0.3868 |  302 |  1.4835 |  330 |  1.9588 |  203 |  2.0090 |  336 |  1.7840 |  315
Table 1: Optimization results of entanglement entropy upper bounds of scar
states (with eigenstate index $n$) of the PXP model in the zero momentum and
inversion even sector. Results of the range $k=3,4,5$ of operator basis
$\\{\hat{L}_{i}\\}$ are shown. In each cell, the left value is the optimized
EE upper bound (rounded with 4 decimal points) and the right number is the
corresponding optimal value of $N_{1}-N_{0}$. Due to the “particle-hole”
symmetry of the PXP model [52], results from the only the first half, with
eigenstate indices $n=1,2,7,30,102$, of all the scar states are shown.

As can be seen from either Fig. 4 in the main text or Table. I here, the
obtained optimal upper bound decrease with the increase of $k$. For $k=4,5$,
our bounds of the scar states are well separated from the thermal continuum,
signifying weak ergodicity breaking [51].

##  Appendix H Experimental Scheme to Measure the Correlation Matrix

Here we will give a detailed description of the experimental measurement of
correlation matrix $\mathcal{M}$ in the context of cold atom systems. We also
give an explicit example of expanding the product of two operators.

To be concrete, we consider a spin-1/2 chain in optical lattices, which can be
realized by ultracold atoms with two hyperfine states and large on-site
repulsion. The system is prepared in the state $|\psi\rangle$, of which the
entanglement property is to be bounded. As an example, we consider the
measurement of the correlation matrix $\mathcal{M}$ of the subsystem $A$ that
contains first two sites $n=1,2$. Let us choose our operator basis
$\\{\hat{L}_{i}\\}$ as
$\\{\hat{\sigma}_{a}^{1},\hat{\sigma}_{a}^{2},\hat{\sigma}_{a}^{1}\hat{\sigma}_{b}^{2}\\}$
where $a,b=x,y,z$ labels the three Pauli operators, and add identity operator
$\hat{I}$ to form the complete set of hermitian operator basis
$\\{\hat{\tilde{L}}_{i}\\}=\\{\hat{I},\hat{\sigma}_{a}^{1},\hat{\sigma}_{a}^{2},\hat{\sigma}_{a}^{1}\hat{\sigma}_{b}^{2}\\}$.
To obtain the matrix element $\mathcal{M}_{ij}$, we need to measure
$\langle\psi|\hat{L}_{i}^{\dagger}\hat{L}_{j}|\psi\rangle$ and
$\langle\psi|\hat{L}_{i}|\psi\rangle$. However, it is straightforward to see
that $\hat{L}_{i}^{\dagger}\hat{L}_{j}=\hat{L}_{i}\hat{L}_{j}$ is an element
in $\\{\hat{\tilde{L}}_{i}\\}$ using the fact that

$\hat{\sigma}^{n}_{a}\hat{\sigma}^{n}_{b}=\delta^{ab}\hat{I}+\sum_{c}i\epsilon_{abc}\hat{\sigma}_{c}^{n},$
(53)

where $\epsilon_{abc}$ is the Levi-Civita symbol. In more general cases, when
the set of operators $\\{\hat{L}_{i}\\}$ is orthogonal under the matrix trace
$\mathrm{Tr}(\hat{L}_{i}\hat{L}_{j})\propto\delta_{ij}$, we can write

$\hat{L}_{i}\hat{L}_{j}=\sum_{k}C_{ij}^{k}\hat{\tilde{L}}_{k},\qquad
C_{ij}^{k}=\frac{\mathrm{Tr}(\hat{L}_{i}\hat{L}_{j}\hat{\tilde{L}}_{k})}{\mathrm{Tr}(\hat{\tilde{L}}_{k}^{2})}.$
(54)

As a result, knowing $\\{\langle\psi|\hat{\tilde{L}}_{i}|\psi\rangle\\}$, or
$\\{\langle\psi|\hat{L}_{i}|\psi\rangle\\}$, is enough to determine the
correlation matrix $\mathcal{M}$. We first consider the measurement of
$\hat{\sigma}^{1}_{z}$, $\hat{\sigma}^{2}_{z}$, and
$\hat{\sigma}^{1}_{z}\hat{\sigma}^{2}_{z}$, which directly corresponds to the
(correlation of) atom occupation in different hyperfine states. The technique
of quantum gas microscopy can be directly applied to measure these quantities
by fluorescence imaging [66, 67, 68, 69, 70, 71]. Repeated measurements
produce the probability distribution of spin states in the $z$ direction
$p_{s_{1}s_{2}}$ where the subscript $s_{n}=\uparrow,\downarrow$ labels the
state of the $n$-th site. We have

$\displaystyle\langle\psi|\hat{\sigma}^{1}_{z}|\psi\rangle$
$\displaystyle=\sum_{s_{2}}(p_{\uparrow s_{2}}-p_{\downarrow s_{2}}),$ (55)
$\displaystyle\langle\psi|\hat{\sigma}^{2}_{z}|\psi\rangle$
$\displaystyle=\sum_{s_{1}}(p_{s_{1}\uparrow}-p_{s_{1}\downarrow}),$
$\displaystyle\langle\psi|\hat{\sigma}^{1}_{z}\hat{\sigma}^{2}_{z}|\psi\rangle$
$\displaystyle=p_{\uparrow\uparrow}+p_{\downarrow\downarrow}-p_{\downarrow\uparrow}-p_{\uparrow\downarrow},$

which can be measured at the same time. Moreover, for systems with translation
symmetry, one can use data on different sites to compute $p_{s_{1}s_{2}}$
efficiently in a single-shot measurement.

For other operators, the measurement can be done with an additional Raman
pulse that couples two hyperfine states. Such optical manipulation of a single
site in optical lattices has been realized in experiments (for example, in
[72, 73, 74]). We give a explicit example for the operator
$\hat{\sigma}_{x}^{1}\hat{\sigma}_{z}^{2}$. By tuning the relative phase
between Raman laser, we choose the coupling between two hyperfine states on
the first site as $\delta\hat{H}=\Omega\hat{\sigma}_{y}^{1}$. We consider
applying a $\pi/2$ pulse, which leads to an unitary evolution denoted by
$\hat{U}$. Using the fact that
$\hat{\sigma}_{x}^{1}=-\hat{U}^{\dagger}\hat{\sigma}_{z}^{1}\hat{U}$, we have

$\langle\psi|\hat{\sigma}_{x}^{1}\hat{\sigma}_{z}^{2}|\psi\rangle=-\langle\psi|\hat{U}^{\dagger}\hat{\sigma}_{z}^{1}\hat{\sigma}_{z}^{2}\hat{U}|\psi\rangle.$
(56)

As a result, after applying the $\pi/2$ pulse, the experimental protocol to
measure $\hat{\sigma}_{x}^{1}\hat{\sigma}_{z}^{2}$ directly follows the
protocol of measuring $\hat{\sigma}_{z}^{1}\hat{\sigma}_{z}^{2}$. Other
operators can be measured in the same spirit, and the generalization to
$k$-local operators is straightforward.

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