Critical Relaxed Stable Matchings with Two-Sided Ties

Critical Relaxed Stable Matchings with Two-Sided Ties

Meghana Nasre1 Prajakta Nimbhorkar2 Keshav Ranjan1

M. Nasre et al.
IIT Madras, Chennai, India Chennai Mathematical Institute and UMI ReLaX, India

We consider the stable marriage problem in the presence of ties in preferences and critical vertices. The input to our problem is a bipartite graph $G = (\AA \cup \BB, E)$ where $\AA$ and $\BB$ denote sets of vertices which need to be matched. Each vertex has a preference ordering over its neighbours  possibly containing ties. In addition, a subset of  vertices in $\AA \cup \BB$ are marked as critical and the goal is to output a matching that matches as many critical vertices as possible. Such matchings are called critical matchings in the literature and in our setting, we seek to compute a matching that is critical as well as optimal with respect to the preferences of the vertices.

Stability, which is a well-accepted notion of optimality in the presence of two-sided preferences, is generalized to weak-stability in the presence of ties. It is well known that in the presence of critical vertices, a matching that is critical as well as weakly stable may not exist.
Popularity is another well-investigated notion of optimality for the two-sided preference list setting, however,  in the presence of ties (even with no critical vertices), a popular matching need not exist.
We, therefore, consider the notion of relaxed stability which was introduced  and studied by  Krishnaa et. al. (SAGT 2020). We show that a critical matching which is relaxed stable always exists in our setting although computing a maximum-sized relaxed stable matching turns out to be NP-hard. Our main contribution is a 3/2 approximation to the maximum-sized critical relaxed stable matching for the stable marriage problem with two-sided ties and critical vertices.

[1]
Biró, P., Irving, R.W., Manlove, D.F.: Popular matchings in the marriage
and roommates problems. In: International Conference on Algorithms and
Complexity. pp. 97–108. Springer (2010)

[2]
Dean, B., Jalasutram, R.: Factor revealing lps and stable matching with ties
and incomplete lists. In: Proceedings of the 3rd International Workshop on
Matching Under Preferences. pp. 42–53 (2015)

[3]
Gale, D., Shapley, L.S.: College admissions and the stability of marriage. The
American Mathematical Monthly  69(1),  9–15 (1962),

[4]
Gärdenfors, P.: Match making: Assignments based on bilateral preferences.
Behavioral Science  20(3),  166–173 (1975)

[5]
Goko, H., Makino, K., Miyazaki, S., Yokoi, Y.: Maximally satisfying lower
quotas in the hospitals/residents problem with ties. In: 39th International
Symposium on Theoretical Aspects of Computer Science (2022)

[6]
Halldórsson, M.M., Iwama, K., Miyazaki, S., Yanagisawa, H.: Improved
approximation results for the stable marriage problem. ACM Transactions on
Algorithms (TALG)  3(3),  30–es (2007)

[7]
Hamada, K., Iwama, K., Miyazaki, S.: The hospitals/residents problem with lower
quotas. Algorithmica  74(1),  440–465 (2016),

[8]
Hamada, K., Miyazaki, S., Yanagisawa, H.: Strategy-proof approximation
algorithms for the stable marriage problem with ties and incomplete lists.
In: International Symposium on Algorithms and Computation. Schloss
Dagstuhl-Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing (2019)

[9]
Huang, C.C., Kavitha, T.: Improved approximation algorithms for two variants of
the stable marriage problem with ties. Mathematical Programming
154,  353–380 (2015)

[10]
Iwama, K., Miyazaki, S., Yanagisawa, H.: A 25/17-approximation algorithm for
the stable marriage problem with one-sided ties. Algorithmica
68(3),  758–775 (2014)

[11]
Kavitha, T.: Matchings, critical nodes, and popular solutions. In: 41st IARCS
Annual Conference on Foundations of Software Technology and Theoretical
Computer Science (FSTTCS 2021) (2021)

[12]
Király, Z.: Better and simpler approximation algorithms for the stable
marriage problem. Algorithmica  60(1),  3–20 (2011)

[13]
Király, Z.: Linear time local approximation algorithm for maximum stable
marriage. Algorithms  6(3),  471–484 (2013),

[14]
Krishnaa, P., Limaye, G., Nasre, M., Nimbhorkar, P.: Envy-freeness and relaxed
stability: Hardness and approximation algorithms. In: Algorithmic Game
Theory: 13th International Symposium, SAGT 2020, Augsburg, Germany, September
16–18, 2020, Proceedings 13. pp. 193–208. Springer (2020)

[15]
Krishnaa, P., Limaye, G., Nasre, M., Nimbhorkar, P., et al.: Envy-freeness and
relaxed stability: hardness and approximation algorithms. Journal of
Combinatorial Optimization  45(1),  1–30 (2023)

[16]
Lam, C.K., Plaxton, C.G.: A (1+ 1/e)-approximation algorithm for maximum stable
matching with one-sided ties and incomplete lists. In: Proceedings of the
Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 2823–2840.
SIAM (2019)

[17]
Manlove, D.F., Irving, R.W., Iwama, K., Miyazaki, S., Morita, Y.: Hard variants
of stable marriage. Theoretical Computer Science  276(1-2),
261–279 (2002)

[18]
McDermid, E.: A 3/2-approximation algorithm for general stable marriage. In:
International Colloquium on Automata, Languages, and Programming. pp.
689–700. Springer (2009)

[19]
Nasre, M., Nimbhorkar, P.: Popular matchings with lower quotas. In: 37th
IARCS Annual Conference on Foundations of Software Technology and
Theoretical Computer Science (FSTTCS 2017). pp. 44:1–44:15 (2017),

[20]
Nasre, M., Nimbhorkar, P., Ranjan, K., Sarkar, A.: Popular Matchings in the
Hospital-Residents Problem with Two-Sided Lower Quotas. In: 41st IARCS
Annual Conference on Foundations of Software Technology and Theoretical
Computer Science (FSTTCS 2021). Leibniz International Proceedings in
Informatics (LIPIcs), vol. 213, pp. 30:1–30:21 (2021)

[21]
Nasre, M., Nimbhorkar, P., Ranjan, K., Sarkar, A.: Popular critical matchings
in the many-to-many setting. arXiv  (2023)

[22]
Paluch, K.: Faster and simpler approximation of stable matchings. Algorithms
7(2),  189–202 (2014)

[23]
Robards, P.A.: Applying two-sided matching processes to the united states navy
enlisted assignment process. Tech. rep., NAVAL POSTGRADUATE SCHOOL MONTEREY
CA (2001)

[24]
Roth, A.E.: Stability and polarization of interests in job matching.
Econometrica: Journal of the Econometric Society pp. 47–57 (1984)

[25]
Roth, A.E.: On the allocation of residents to rural hospitals: a general
property of two-sided matching markets. Econometrica: Journal of the
Econometric Society pp. 425–427 (1986)

[26]
Tan, S.J., Yeong, C.M.: Designing economics experiments to demonstrate the
advantages of an electronic employment market in a large military
organization. Tech. rep., NAVAL POSTGRADUATE SCHOOL MONTEREY CA (2001)

[27]
Yang, W., Sycara, K.: Two-sided matching for the us navy detailing process with
market complication. Tech. rep., Technical Report CMU-RI-TR-03-49, Robotics
Institute, Carnegie-Mellon University (2003)

[28]
Yokoi, Y.: Envy-free matchings with lower quotas. Algorithmica  82(2),
188–211 (2020), <https://doi.org/10.1007/s00453-018-0493-7>

§ RELAXED STABILITY VERSUS POPULARITY

Relaxed stability and popularity do not define the same set of matchings even when preferences are strict and  critical vertices are restricted to only one side of the bipartition. That is, neither one implies the other. We give simple examples (from [15]).

Consider the example shown in Figure <ref>. Notice that the popular matching $M_1= \{(a_1,b_2),(a_2,b_1)\}$ is not relaxed stable because the blocking pair $(a_1,b_1)$ is not justified (there are no critical vertices). Observe that in the absence of critical vertices, relaxed stability is the same as stability.

Now, consider the example shown in Figure <ref>. Notice that the matching  $M_2=\{(a_1,b_2)\}$ is relaxed stable as the only blocking pair $(a_1,b_1)$ is justified by $a_1$. But $M_3=\{(a_1,b_1)\}$ is more popular than $M_2$. Thus a relaxed stable matching $M_2$ is not popular.

[thick,  ncnode/.style=draw=black,,circle,fill=black,scale=0.5,  cnode/.style=draw=red,circle,fill=red,scale=0.5,
every fit/.style=ellipse,draw,inner sep=-2pt,text width=0.5cm,  -,shorten >= 3pt,shorten <= 3pt, scale=0.7]

[ncnode] (a1) at (-1,3) ;
[ncnode] (a2) at (-1,1.5) ;
[ncnode] (b1) at (2.5,3) ;
[ncnode] (b2) at (2.5,1.5) ;
at (-1.8,3) $a_1$;
at (-1.8,1.5) $a_2$;
at (3.2,3) $b_1$;
at (3.2,1.5) $b_2$;

[fit=(a1) (a2),label=above:$\AA$] ;

[fit=(b1) (b2),label=above:$\BB$] ;

(a1) – (b1) node[pos=0.15,draw=white, fill=white,inner sep=0.1pt] 1 node[pos=0.8,draw=white, fill=white,inner sep=0.1pt] 1;
(a1) – (b2) node[pos=0.2,draw=white,fill=white,inner sep=0.1pt]2 node[pos=0.85,draw=white, fill=white,inner sep=0.1pt] 1;
(a2) – (b1) node[pos=0.2,draw=white,fill=white,inner sep=0.1pt]1 node[pos=0.85,draw=white, fill=white,inner sep=0.1pt] 2;

[thick,  ncnode/.style=draw=black,,circle,fill=black,scale=0.5,  cnode/.style=draw=red,circle,fill=red,scale=0.5,
every fit/.style=ellipse,draw,inner sep=-2pt,text width=0.5cm,  -,shorten >= 3pt,shorten <= 3pt, scale=0.7]

[ncnode] (a1) at (-1,2.25) ;
[cnode] (b1) at (2.5,3) ;
[cnode] (b2) at (2.5,1.5) ;
at (-1.8,2.25) $a_1$;
at (3.2,3) $b_1$;
at (3.2,1.5) $b_2$;

at (-1,3) $\AA$;

[fit=(b1) (b2),label=above:$\BB$] ;

(a1) – (b1) node[pos=0.15,draw=white, fill=white,inner sep=0.1pt] 1 node[pos=0.8,draw=white, fill=white,inner sep=0.1pt] 1;
(a1) – (b2) node[pos=0.2,draw=white,fill=white,inner sep=0.1pt]2 node[pos=0.85,draw=white, fill=white,inner sep=0.1pt] 1;

The red vertices are critical and black vertices are non-critical. The numbers on the edges denote the ranking of the other endpoint in the preference list of that vertex.


# Rényi entropy of the permutationally invariant part of the ground state
across a quantum phase transition

Yuki Miyazaki1 Giacomo Marmorini1,2 Nobuo Furukawa1 Daisuke Yamamoto2
1Department of Physical Science, Aoyama Gakuin University, Kanagawa 252-5258,
Japan  2Department of Physics, College of Humanities and Sciences, Nihon
University, Tokyo 156-8550, Japan

###### Abstract

We investigate the role of the permutationally invariant part of the density
matrix (PIDM) in capturing the properties of the ground state of the system
during a quantum phase transition. In the context of quantum state tomography,
PIDM is known to be obtainable with only a low number of measurement settings,
namely $\mathcal{O}(L^{2})$, where $L$ is the system size. Considering the
transverse-field Ising chain as an example, we compute the second-order Rényi
entropy of PIDM for the ground state by using the density matrix
renormalization group algorithm. In the ferromagnetic case, the ground state
is permutationally invariant both in the limits of zero and infinite field,
leading to vanishing Rényi entropy of PIDM. The latter exhibits a broad peak
as a function of the transverse field around the quantum critical point, which
gets more pronounced for larger system size. In the antiferromagnetic case,
the peak structure disappears and the Rényi entropy diverges like
$\mathcal{O}(L)$ in the whole field range of the ordered phase. We discuss the
cause of these behaviors of the Rényi entropy of PIDM, examining the possible
application of this experimentally tractable quantity to the analysis of phase
transition phenomena.

††preprint: APS/123-QED

##  I introduction

Historically, in the field of solid state physics, macroscopic observables
have been measured to detect a phase transition, e.g., magnetization for
magnetic phase transitions [1, 2] and electrical resistance for
superconductivity [3, 4]. Recently topological phases such as quantum Hall
effect [5, 6] and highly entangled phases including spin liquid phases in the
Kitaev model [7] have been attracting greater attention since they evade the
paradigm of symmetry breaking and conventional order parameters. Quantum
information theory offers an insightful perspective on this issue. For
example, the so-called topological entanglement entropy, which appears in the
von Neumann entanglement entropy as a universal constant, characterizes
topologically ordered phases [8, 9]; in other cases, the structure of the
entanglement spectrum helps us distinguish spin liquid phase from magnetically
ordered phases [10, 11]. In modern condensed matter physics, there is a
growing demand for both theoretical analysis and experimental measurement of
quantities related to quantum information.

In artificial quantum systems such as cold atomic and molecular gases in an
optical lattice [12, 13, 14, 15], photonic systems [16, 17], trapped-ion
systems [18, 19], Rydberg atom arrays in optical tweezers [20, 21, 22, 23,
24], and superconducting circuits [25], the simulation of quantum many-body
physics can be conducted in an ideal environment, taking advantage of its
cleanness and high controllability. Using these artificial quantum systems,
the reconstruction of a density matrix has been conducted by preparing the
identical state repeatedly and measuring the expectation value of various
observables [26, 27, 28]. The procedure to reconstruct a density matrix is
known as quantum state tomography (QST) [29, 30, 31, 32].

In the QST for $L$-qubit system, an arbitrary density matrix $\rho$, which is
a $2^{L}$ by $2^{L}$ matrix, can be expanded in the basis
$\\{\bigotimes_{i=1}^{L}\mathcal{\sigma}_{\kappa_{i}}\\}$ as

$\rho=\frac{1}{2^{L}}\sum_{\kappa_{1}=0}^{3}\cdots\sum_{\kappa_{L}=0}^{3}\left\langle\bigotimes_{i=1}^{L}\sigma_{\kappa_{i}}\right\rangle\left(\bigotimes_{i=1}^{L}\sigma_{\kappa_{i}}\right),$
(1)

where

$\displaystyle\sigma_{0}$ $\displaystyle=$
$\displaystyle\mathbbm{1}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\
\sigma_{1}=X=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},$
$\displaystyle\sigma_{2}$ $\displaystyle=$ $\displaystyle
Y=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix},\ \sigma_{3}=Z=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$ (2)

The basis elements are mutually orthogonal and span a $4^{L}$-dimensional
linear space $\mathbb{D}$ (see Fig. 1). The coefficient in Eq. (1) are
expectation values on the state $\rho$, which ideally can be the results of
experimental measurements. Eq. (1) tells us that QST is experimentally
challenging due to the the necessity (i) to measure different correlation
functions whose number increase exponentially with respect to the system size,
and (ii) to rotate the local spins (local quantization axes) site by site. QST
has been limited to small systems (at most 8 qubits so far), leaving room for
the development of scalable protocols for the analysis of phase transition
phenomena.

One considerable advancement against the above issues is QST via compressed
sensing [33, 34], in which $\rho$ is reconstructed with the condition that
$\rho$ is a low-rank state. This is a relatively feasible protocol requiring
$\mathcal{O}(r2^{L}(\log 2^{L})^{2})$ measurement settings, where $r={\rm
rank}\rho\ll 2^{L}$, but the exponential factor remains. On the other hand,
one may consider permutationally invariant (PI) QST [35, 36, 37], in which we
recover the PI part of a density matrix (PIDM) $\mathit{\Pi}(\rho)$. In PIQST,
we only need $\mathcal{O}(L^{2})$ measurement settings and only global
rotations of the quantization axes. Besides, it is equivalent to full QST in
case of PI states, defined by $\mathit{\Pi}(\rho)=\rho$, such as the GHZ
states [38, 39], the W states [40, 41, 42], and the Dicke states [43].

Figure 1: Schematic sketch of the linear spaces $\mathbb{D}$ and $\mathbb{P}$
spanned by the orthogonal operators
$\\{\bigotimes_{i=1}^{L}{\sigma}_{\kappa_{i}}\\}$ and by the orthogonal PI
operators $\\{\mathcal{B}_{d}^{(n)}\\}$ or the PI random operators
$\\{\mathcal{A}_{d}^{(n)}\\}$, respectively. Any density matrix $\rho$ is an
element of the space $\mathbb{D}$. The PI part of an arbitrary density matrix
of an $L$-qubit system, $\mathit{\Pi}(\rho)$, exist in
$\mathbb{P}\subset\mathbb{D}$.

To obtain a scalable solution in the quantum simulations, we focus on the
PIQST owing to the drastically reduced experimental cost. Although PIQST has
been demonstrated experimentally [35], it has not yet been performed for
large-size systems, especially for the application to the analysis of phase
transition phenomena. From the theoretical side, though we generally cannot
reconstruct $\rho$ via PIQST, it has been reported that $\mathit{\Pi}(\rho)$
can encode the nonseparability of $\rho$ [37]. Also, any entanglement measure
of $\mathit{\Pi}(\rho)$ is suggested to be the lower bound of that of $\rho$
[37]. However, none has been explored for the analysis of phase transition
phenomena yet. In this work, we study how the second-order Rényi entropy of
$\mathit{\Pi}(\rho)$ reflects the properties of the ground state of the system
when it undergoes a quantum phase transition. We establish a scalable
procedure, in which we directly compute the Rényi entropy of
$\mathit{\Pi}(\rho)$ from only $\mathcal{O}(L^{3})$ expectation values. We
calculate it for the ground state of the one-dimensional transverse-field
Ising model [44] using the density matrix renormalization group (DMRG)
algorithm [45, 46, 47].

For the case of ferromagnetic (FM) interaction, the Rényi entropy is zero both
in the limits of zero and infinite field because of the permutational
invariance of the ground states. It shows a broad peak around the quantum
critical point (QCP), getting more pronounced for larger-size systems. In the
antiferromagnetic (AFM) case, the peak structure disappears and the Rényi
entropy tends to diverge to infinity in the whole field range of the ordered
phase. By looking more closely at the scaling of the Rényi entropy versus
system size, we find that in AFM case the Rényi entropy grows almost linearly
and its slope sharply changes around the QCP, while for the FM case the slopes
around the QCP tend to take large values. We also analyze the behavior of the
Rényi entropy at low and high fields, according to the second-order
perturbation theory. We conjecture that the FM, AFM, and paramagnetic (PM)
phases, which are the ground states of the transverse-field Ising model, are
characterized by the different scaling law of the Rényi entropy of
$\mathit{\Pi}(\rho)$. We expect that origin of these behaviors will be
clarified by the experimental measurement and theoretical analysis of larger-
size systems, but we consider this study an important step in the analysis of
phase transition phenomena utilizing quantum information.

The remainder of this paper is organized as follows. In Sec. II, we review the
PIQST and its performance. In Sec. III, an analytical form of the second-order
Rényi entropy of $\mathit{\Pi}(\rho)$ is given. In Sec. IV, we show the
results of the Rényi entropy of $\mathit{\Pi}(\rho)$ of the ground state of
the transverse-field Ising chain across the quantum phase transition. We also
analyze its behavior in low- and high-field region according to second-order
perturbation theory. In Sec. V, we discuss the significant change of the Rényi
entropy of PIDM around the QCP, and conjecture that the FM, AFM, and PM phases
are characterized by its scaling law. We also present an open question about
the role of Rényi entropy of PIDM, and propose interesting future works to
obtain deeper insights. Section VI is devoted to the conclusion.

##  II permutationally invariant quantum state tomography

PIQST was introduced as a scalable protocol to reconstruct the density matrix
of a PI state, which has been mainly realized in photonic systems [35, 36]. In
PIQST, we construct PIDM as

$\mathit{\Pi}(\rho)=\frac{1}{L!}\sum_{s\in\mathfrak{S}_{L}}\hat{\mathcal{P}}_{s}\rho\hat{\mathcal{P}}_{s}^{\dagger},$
(3)

where $\mathfrak{S}_{L}$ denotes the symmetric group of degree $L$ and
$\hat{\mathcal{P}}_{s}$ is the unitary operator representing the permutation
$s$ which rearranges $L$ qubits in a certain order:
$\hat{\mathcal{P}}_{s}\ket{q_{1}q_{2}\cdots q_{L}}=\ket{q_{s(1)}q_{s(2)}\cdots
q_{s(L)}}$. We can easily confirm that $\mathit{\Pi}(\rho)$ has the
fundamental properties of a density matrix: Hermicity, unit trace, and
positive semi-definiteness. An arbitrary PIDM exists in a linear space
$\mathbb{P}$ (see Fig. 1), and it can be expanded in the orthogonal PI basis
$\\{\mathcal{B}^{(k,l,m,n)}\\}$ as

$\mathit{\Pi}(\rho)=\frac{L!}{2^{L}}\sum_{k+l+m+n=L}\frac{\braket{\mathcal{B}^{(k,l,m,n)}}}{k!l!m!n!}\mathcal{B}^{(k,l,m,n)},$
(4)

where

$\mathcal{B}^{(k,l,m,n)}=\mathit{\Pi}\left(X^{\otimes k}\otimes Y^{\otimes
l}\otimes Z^{\otimes m}\otimes\mathbbm{1}^{\otimes n}\right)$ (5)

and the sum $\sum_{k+l+m+n=L}$ runs over all possible combinations of non-
negative integers, $k$, $l$, $m$, and $n$, which are the number of $X$, $Y$,
$Z$, and $\mathbbm{1}$ in $\mathcal{B}^{(k,l,m,n)}$, respectively, subject to
$k+l+m+n=L$. By definition of the function $\mathit{\Pi}$, the PI basis
elements which contain exactly same number of $X$, $Y$, $Z$, and $\mathbbm{1}$
are regarded as the same; e.g., $\mathit{\Pi}(X\otimes
Y)=\mathit{\Pi}(Y\otimes X)$ for $L=2$, $\mathit{\Pi}(X\otimes X\otimes
Y)=\mathit{\Pi}(X\otimes Y\otimes X)=\mathit{\Pi}(Y\otimes X\otimes X)$ for
$L=3$, etc. Naively this equality of correlation functions is the reason of
the cost reduction in PIQST, which can be performed with merely $D_{L}$
measurement settings with

$D_{L}\equiv\binom{L+2}{2}=\frac{L^{2}+3L+2}{2}.$ (6)

Thus the number of different expectation values to construct
$\mathit{\Pi}(\rho)$ is

$\sum_{n=0}^{L}D_{L-n}=\frac{L^{3}+6L^{2}+11L+6}{6}$ (7)

in total. This is a powerful fact, since the exponential factor, which remains
even in QST via compressed sensing [33, 34], is absent here. Moreover, there
exists a linear transformation between the PI basis
$\\{\mathcal{B}^{(n)}_{\mu}\\}_{\mu=1}^{D_{L-n}}$ and another basis
$\\{\mathcal{A}_{\nu}^{(n)}\\}_{\nu=1}^{D_{L-n}}$, namely

$\mathcal{B}^{(n)}_{\mu}=\sum_{{\nu}=1}^{D_{L-n}}C_{\mu\nu}^{(n)}\mathcal{A}_{\nu}^{(n)},$
(8)

where

$\mathcal{A}_{\nu}^{(n)}=\mathit{\Pi}\left(A_{\nu}^{\otimes(L-n)}\otimes\mathbbm{1}^{\otimes
n}\right)$ (9)

and $A_{\nu}=\alpha_{\nu}X+\beta_{\nu}Y+\gamma_{\nu}Z$, subject to
$(\alpha_{\nu})^{2}+(\beta_{\nu})^{2}+(\gamma_{\nu})^{2}=1\
(\alpha_{\nu},\beta_{\nu},\gamma_{\nu}\in\mathbb{R})$, is a random operator
acting on a local spin. Here
$\mathcal{B}_{\mu}^{(n)}=\mathcal{B}^{(k_{\mu},l_{\mu},m_{\mu},n)}$ and $k$,
$l$, and $m$ are specified by the index $\mu$ (hence $k_{\mu}$, $l_{\mu}$,
$m_{\mu}$). The nontrivial single-site operators $A_{\nu}$ that compose
$\mathcal{A}_{\nu}^{(n)}$ are identical, whereas those of
$\mathcal{B}_{\mu}^{(n)}$ are generally different. Through Eq. (8), we only
need to measure the spins at all sites in the same direction to reconstruct
$\mathit{\Pi}(\rho)$ in Eq. (4). Note that $\\{\mathcal{A}_{\nu}^{(n)}\\}$ are
not required to be orthogonal as long as they are linearly independent,
otherwise the linear space $\mathbb{P}$ spanned by
$\\{\mathcal{A}_{\nu}^{(n)}\\}$ is not sufficiently large to express
$\mathit{\Pi}(\rho)$. In the experiments, for a four-photon Dicke state (PI
state), it has been reported that $\mathit{\Pi}(\rho)$ gives high fidelity
when taking an inhomogeneous distribution of the vectors
$\mbox{\boldmath$a$}_{\nu}=(\alpha_{\nu},\beta_{\nu},\gamma_{\nu})^{\top}$
[35]; This can be understood by taking the effect of noise into account. The
coefficients $C_{\mu\nu}^{(n)}$ are analytically determined by the sole
information of the measurement directions $\\{\mbox{\boldmath$a$}_{\nu}\\}$.
Let us define the vectors $\mbox{\boldmath$\mathcal{A}$}^{(n)}$ and
$\mbox{\boldmath$\mathcal{B}$}^{(n)}$ as

$\displaystyle\mbox{\boldmath$\mathcal{A}$}^{(n)}$ $\displaystyle=$
$\displaystyle\left(\mathcal{A}_{1}^{(n)},\mathcal{A}_{2}^{(n)},\cdots,\mathcal{A}_{D_{L-n}}^{(n)}\right)^{\top},$
(10) $\displaystyle\mbox{\boldmath$\mathcal{B}$}^{(n)}$ $\displaystyle=$
$\displaystyle\left(\mathcal{B}_{1}^{(n)},\mathcal{B}_{2}^{(n)},\cdots,\mathcal{B}_{D_{L-n}}^{(n)}\right)^{\top}.$
(11)

Having fixed the number of identity matrices $n$, we decompose the matrix
$\mathcal{A}_{\nu}^{(n)}$ into orthogonal basis elements
$\\{\mathcal{B}^{(n)}_{\mu}\\}_{{\mu}=1}^{D_{L-n}}$, namely
$\mbox{\boldmath$\mathcal{A}$}^{(n)}=(C^{(n)})^{-1}\mbox{\boldmath$\mathcal{B}$}^{(n)}$
or
$\mathcal{A}^{(n)}_{\nu}=\sum_{{\mu}=1}^{D_{L-n}}((C^{(n)})^{-1})_{\nu\mu}\mathcal{B}_{\mu}^{(n)}$,
where $(C^{(n)})^{-1}$ is a $D_{L-n}$ by $D_{L-n}$ matrix in which the entries
are given by

$\displaystyle((C^{(n)})^{-1})_{\nu\mu}$ $\displaystyle=$
$\displaystyle\frac{{\rm
Tr}\left[\mathcal{A}^{(n)}_{\nu}\mathcal{B}^{(n)}_{\mu}\right]}{{\rm
Tr}\left[\left(\mathcal{B}^{(n)}_{\mu}\right)^{2}\right]}$ (12)
$\displaystyle=$
$\displaystyle\frac{\left(2\alpha_{\nu}\right)^{k_{\mu}}\left(2\beta_{\nu}\right)^{l_{\mu}}\left(2\gamma_{\nu}\right)^{m_{\mu}}(L-n)!}{k_{\mu}!l_{\mu}!m_{\mu}!2^{L-n}}.$

Thus we can calculate
$\\{\braket{\mathcal{B}^{(n)}_{\mu}}\\}_{\mu=1}^{D_{L-n}}$ from
$\\{\braket{\mathcal{A}^{(n)}_{\nu}}\\}_{\nu=1}^{D_{L-n}}$ using Eqs. (8) and
(12) for each $n$.

In this work, we focus on the second-order Rényi entropy [48] of PIDM, which
is directly computed from a set of expectation values as shown in the next
section.

##  III Second-order Rényi entropy of the permutationally invariant part of a
density matrix

Before studying the second-order Rényi entropy, let us review the purity,
$P(\rho)={\rm Tr}(\rho^{2})$. Let us recall that it takes values between
$1/2^{L}$ and $1$, in particular $P(\rho)=1$ for a pure states and
$P(\rho)=1/2^{L}$ for a maximally mixed $L$-qubit state. In this work, we
consider the purity of PIDM $P(\mathit{\Pi}(\rho))={\rm
Tr}\left[\mathit{\Pi}(\rho)^{2}\right]$. Since $\mathit{\Pi}(\rho)$ has the
properties of a density matrix, $P(\mathit{\Pi}(\rho))$ also takes values
between $1/2^{L}$ and $1$. $P(\mathit{\Pi}(\rho))$ can be expressed as follows
(see also Appendix A for the derivation):

$P(\mathit{\Pi}(\rho))=\frac{L!}{2^{L}}\sum_{k+l+m+n=L}\frac{\braket{\mathcal{B}^{(k,l,m,n)}}^{2}}{k!l!m!n!}.$
(13)

The second-order Rényi entropy is easy to calculate from
$P(\mathit{\Pi}(\rho))$ as $S_{2}(\mathit{\Pi}(\rho))=-\log
P(\mathit{\Pi}(\rho))$ [48], satisfying $0\leq S_{2}(\mathit{\Pi}(\rho))\leq
L\log 2$.

We indicate a procedure for evaluating $S_{2}(\mathit{\Pi}(\rho))$ in
experiments and numerical simulations (this work) in Fig. 2. The numerical
calculation steps to obtain $P(\mathit{\Pi}(\rho))$ are the following: (i) we
obtain the ground state of a physical Hamiltonian as a MPS [49, 50] using the
DMRG algorithm [45, 46, 47], (ii) we calculate the expectation values
$\\{\braket{\mathcal{A}_{\nu}^{(n)}}\\}$, (iii) we compute the purity of PIDM
by Eq. (13), converting $\\{\braket{\mathcal{A}_{\nu}^{(n)}}\\}$ into
$\\{\braket{\mathcal{B}^{(n)}_{\mu}}\\}$ via Eq. (8).

In optical-lattice experiments, a set of correlation functions
$\\{\braket{\mathcal{A}_{\nu}^{(n)}}\\}$ can possibly be measured using
quantum-gas microscope (QGM) [51, 52, 53, 54, 55, 56], after a series of
magnetic and optical operations that rotate the quantization axes to
$\\{\mbox{\boldmath$a$}_{\nu}\\}$. This experimental protocol is scalable
because we can directly calculate the purity of PIDM from only $D_{L}$
measurement settings without constructing PIDM.

In the context of the analysis of quantum entanglement, the Rényi entanglement
entropy $S_{2}(\rho_{\rm A})$, where $\rho_{\rm A}={\rm Tr_{\bar{A}}}\rho$ is
a reduced density matrix, is often used to evaluate the amount of quantum
entanglement between two complementary subsystems A and $\bar{\rm A}$ [57, 58,
59]. In this work, we treat the Rényi entropy of the whole system,
specifically regarding PIDM. Naively, the latter acts as a measure of how a
state $\rho$ is close to a PI state.

Figure 2: The protocols for evaluating the second-order Rényi entropy of PIDM
in the experiments and in the numerical simulation (this work). In this work,
we apply DMRG algorithm to obtain the ground state of a physical Hamiltonian
and simulate the observations of $\mathcal{A}_{\nu}^{(n)}$ by calculating the
expectation value of the matrix product operator $A_{\nu}$. The numerical
calculations after the measurements are the same in the experimental protocol
and on the simulations.

##  IV Model and Results

###  IV.1 The transverse-field Ising chain

As a case study, we compute the second-order Rényi entropy of the PIDM
$S_{2}(\mathit{\Pi}(\rho))$ for the ground state of the transverse-field Ising
chain [44] with open boundary conditions, whose Hamiltonian is given by

$\displaystyle\hat{\mathcal{H}}=J\sum_{i=1}^{L-1}\hat{Z}_{i}\hat{Z}_{i+1}-\mathit{\Gamma}\sum_{i=1}^{L}\hat{X}_{i},$
(14)

where $\hat{Z}_{i}$ and $\hat{X}_{i}$ are Pauli matrices acting on site $i$.
Here, we only deal with the case of even $L$. In absence of $\mathit{\Gamma}$,
the Hamiltonian is classical and features a trivial ground state. For the case
of FM interaction ($J<0$), the ground state is $\ket{\rm
FM}=c_{1}\ket{\uparrow}^{\otimes L}+c_{2}\ket{\downarrow}^{\otimes L}$, and
for the case of AFM interaction ($J>0$), the ground state is $\ket{{\rm
AFM}}=c_{1}\ket{\uparrow\downarrow}^{\otimes\frac{L}{2}}+c_{2}\ket{\downarrow\uparrow}^{\otimes\frac{L}{2}}$,
subject to $|c_{1}|^{2}+|c_{2}|^{2}=1$. When an infinitesimal field is added,
the ground state is symmetric, and we get $\ket{\rm FM}$ and $\ket{\rm AFM}$
with $c_{1}=c_{2}=1/\sqrt{2}$ [60]. In the limit of infinite field
(${\it\Gamma}\rightarrow\infty$), we can neglect the first term of Eq. (14),
yielding the PM phase which is given by the direct product of the eigenstates
of $X$ as

$\ket{\rightarrow}^{\otimes
L}=\frac{1}{2^{L/2}}\sum_{n=0}^{L}\sum_{s\in\mathfrak{S}_{L}}\frac{\hat{\mathcal{P}}_{s}\left(\ket{\uparrow}^{\otimes(L-n)}\otimes\ket{\downarrow}^{\otimes
n}\right)}{(L-n)!n!},$ (15)

where $\ket{\rightarrow}=(\ket{\uparrow}+\ket{\downarrow})/\sqrt{2}$. In the
intermediate field region, both terms contribute and a phase transition occurs
at the QCP (${\it\Gamma}/J=1$).

###  IV.2 Behaviors of the second-order Rényi entropy of the permutationally
invariant part for the ground state

We show the plots of $P(\mathit{\Pi}(\rho))$ and $S_{2}(\mathit{\Pi}(\rho))$
as a function of $\mathit{\Gamma}/|J|$ in Figs. 3 and 4 for the FM and AFM
case, respectively. In the DMRG calculation, we take the full bond dimension
(the maximum bond dimension of MPS $\chi_{\rm max}=2^{L/2}$) up to $L=10$; we
then set the cutoff of the maximum bond dimension $\chi_{\rm max}=16,10,4,4$
for $L=12,14,16,18$, respectively, due to the computational cost. This
truncation does not significantly affect the results because the purity of
PIDM has essentially converged at $\chi_{\rm max}=4$ (see also Appendix C for
the $\chi_{\rm max}$ dependence of $P(\mathit{\Pi}(\rho))$).

In the case of FM interaction, we obtain $S_{2}(\mathit{\Pi}(\rho))=0$
($P(\mathit{\Pi}(\rho))=1$) at zero field. For sufficiently large field, the
values of $S_{2}(\mathit{\Pi}(\rho))$ approach zero
($P(\mathit{\Pi}(\rho))\rightarrow 1$). These results are trivial because the
ground states are PI states in these two limits. For intermediate fields,
there appears a broad peak around the QCP (black dashed line), which gets more
pronounced for increasing system size. In the case of AFM interaction,
$S_{2}(\mathit{\Pi}(\rho))$ takes a finite value in the limit of zero field,
caused by the non-PI ground states $\ket{{\rm AFM}}$. In the limit of large
field, $S_{2}(\mathit{\Pi}(\rho))$ gets closer to zero, similar to the FM
case.

It is well known that the QCP in the transverse-field Ising model can be
characterized by various properties, such as the divergence of the
entanglement entropy [61] and the maximum of next-nearest-neighbor
entanglement [62]. Here a question arises: can $P(\mathit{\Pi}(\rho))$ or
$S_{2}(\mathit{\Pi}(\rho))$ characterize the QCP in the thermodynamic limit?
We show the plots of $S_{2}(\mathit{\Pi}(\rho))$ as a function of $L$ for FM
and AFM cases in Figs. 5(a) and 5(b), respectively. (see Appendix C for the
convergence of the numerical data as a function of $\chi_{\rm max}$). We find
that the data are well represented by fitting functions of the form

$f_{S_{2}}(L)=p_{1}L+p_{2}+\frac{p_{3}}{L}+p_{4}\log L,$ (16)

where $p_{1}$, $p_{2}$, $p_{3}$ and $p_{4}$ are fitting parameters (dashed
lines in Fig. 5). We show the values of the fitting parameters in Figs. 6 and
7, determined by using the data between $L=2$ and $22$. The error bars in
Figs. 6 and 7 represent plus/minus one standard deviation. For the FM case,
the behavior of the fitting functions are fundmentally different between FM
and PM regions. The largest slope of $S_{2}(\mathit{\Pi}(\rho))$ appears
around the QCP, leading to a peak structure of $p_{1}$. The peak is slightly
shifted from the QCP, but we expect that it will eventually coincide with the
QCP for larger-size systems. For the AFM case, the slope of
$S_{2}(\mathit{\Pi}(\rho))$ sharply decreases around the QCP for increasing
${\it\Gamma}$.

Figure 3: (a) The purity and (b) the second-order Rényi entropy of PIDM of the
ground state of the FM transverse field Ising model as a function of the
transverse-field for various system sizes. The black dashed line represents
the QCP ($\mathit{\Gamma}/|J|=1$).

Figure 4: (a) The purity and (b) the second-order Rényi entropy of PIDM of the
ground state of the AFM transverse-field Ising model as a function of the
transverse field for various system sizes. The black dashed line represents
the QCP ($\mathit{\Gamma}/J=1$).

.

Figure 5: The second-order Rényi entropy of PIDM for the ground state of the
transverse-field Ising chain as a function of $L$ in (a) FM case and (b) AFM
cases. The dashed lines represent the fitting functions $f_{S_{2}}(L)$ defined
by Eq. (16) with fitting parameters as shown in Figs. 6 and 7.

###  IV.3 Perturbative approach to low- and high-field regimes

Let us consider the behavior of the second-order Rényi entropy under low
(${\it\Gamma}/|J|\ll 1$) and high field ($|J|/{\it\Gamma}\ll 1$) according to
the second-order perturbation theory (see also Appendix B for detailed
calculations). In the low-field region, the second-order Rényi entropy of the
ground state is given by

$S_{2}(\mathit{\Pi}(\rho))\approx\left(\frac{\it\Gamma}{J}\right)^{2}\left(\frac{1}{4}-\frac{1}{2L}\right)$
(17)

for the FM case, and

$\displaystyle S_{2}(\mathit{\Pi}(\rho))$ $\displaystyle\approx$
$\displaystyle L\log 2-\frac{1}{2}\log L-\log\sqrt{2\pi}$ (18) $\displaystyle\
\ \ \ \ \ \ \ \
-\left(\frac{\it\Gamma}{J}\right)^{2}\left(\frac{1}{4}+\frac{1}{2L}\right)$

for the AFM case. Thus, the fitting parameters in Eq. (16) are determined as
$(p_{1},p_{2},p_{3},p_{4})=(0,\frac{1}{4}(\frac{\it\Gamma}{J})^{2},-\frac{1}{2}(\frac{\it\Gamma}{J})^{2},0)$
and $(p_{1},p_{2},p_{3},p_{4})=(\log
2,-\log\sqrt{2\pi}-\frac{1}{4}(\frac{\it\Gamma}{J})^{2},-\frac{1}{2}(\frac{\it\Gamma}{J})^{2},-\frac{1}{2})$
for FM and AFM cases, respectively (see also blue dashed lines in Figs. 6 and
7). In the high-field region, the second-order Rényi entropy of the ground
state is given by

$S_{2}(\mathit{\Pi}(\rho))\approx\left(\frac{J}{\it\Gamma}\right)^{2}\left(\frac{L}{8}-\frac{3}{8}\right)$
(19)

for the both FM and AFM cases, leading to fitting parameters
$(p_{1},p_{2},p_{3},p_{4})=(\frac{1}{8}(\frac{J}{\it\Gamma})^{2},-\frac{3}{8}(\frac{J}{\it\Gamma})^{2},0,0)$
(see also red dashed lines in Figs. 6 and 7).

We have considered open boundary conditions so far. With periodic boundary
conditions, in the low-field region the Rényi entropy of PIDM is given by

$S_{2}(\mathit{\Pi}(\rho))\approx 0$ (20)

for the FM case and,

$S_{2}(\mathit{\Pi}(\rho))\approx L\log 2-\frac{1}{2}\log L-\log\sqrt{2\pi}$
(21)

for the AFM case, leading to fitting parameters
$(p_{1},p_{2},p_{3},p_{4})=(0,0,0,0)$ and $(p_{1},p_{2},p_{3},p_{4})=(\log
2,-\log\sqrt{2\pi},0,-\frac{1}{2})$, respectively. Unlike open boundary
conditions, the second-order term with respect to ${\it\Gamma}/J$ does not
contribute to $S_{2}(\mathit{\Pi}(\rho))$, and the $L^{-1}$ term vanishes. In
the high-field region, the Rényi entropy of PIDM is given by

$S_{2}(\mathit{\Pi}(\rho))\approx\left(\frac{J}{\it\Gamma}\right)^{2}\left(\frac{L}{8}-\frac{1}{4}\right)$
(22)

for the both FM and AFM cases, leading to
$(p_{1},p_{2},p_{3},p_{4})=(\frac{1}{8}(\frac{J}{\it\Gamma})^{2},-\frac{1}{4}(\frac{J}{\it\Gamma})^{2},0,0)$.
The coefficient of $(J/{\it\Gamma})^{2}$ in $p_{2}$ is slightly different from
the case with open boundary conditions, but the behavior is almost the same.

##  V discussion

In this section, we summarize the behavior of $S_{2}(\mathit{\Pi}(\rho))$ and
discuss the possible application of $S_{2}(\mathit{\Pi}(\rho))$ for
characterizing the phases of the system. First, we can see that some
properties of $S_{2}(\mathit{\Pi}(\rho))$ reflects the QCP. For example, in
Fig. 5 (a) the largest slope of $S_{2}(\mathit{\Pi}(\rho))$ appears around the
QCP, leading to a peak structure of $p_{1}$. In Fig. 5 (b), we can see that
the slope of $S_{2}(\mathit{\Pi}(\rho))$ sharply decreases around the QCP for
increasing ${\it\Gamma}$, bringing a sharp change of $p_{1}$. Not only
$p_{1}$, but also the other fitting parameters show significant changes, which
can be associated with the criticality. The critical behavior of the fitting
parameters in Figs. 6 and 7 is slightly shifted from the QCP, but we ascribe
this to the limitation of system size.

Secondly, we conjecture that the finite-size scaling law of
$S_{2}(\mathit{\Pi}(\rho))$ characterizes the FM, AFM, and PM phase. For
instance, in Fig. 6, $p_{1}$ is almost zero in the FM phase while it takes
finite values in the PM phase. In Fig. 7, $p_{4}$ takes non-zero values in the
AFM phase, but it almost vanishes in the PM phases. From the results of the
second-order perturbation theory, we expect that in the thermodynamic limit
$S_{2}(\mathit{\Pi}(\rho))$ in the FM, AFM, and PM phases scales as

$S_{2}(\mathit{\Pi}(\rho))=\begin{cases}\mathcal{O}(1)&\mbox{(FM)},\\\
p_{1}L+p_{4}\log L+\mathcal{O}(1)&\mbox{(AFM)},\\\
p_{1}L+\mathcal{O}(1)&\mbox{(PM)},\end{cases}$ (23)

respectively.

While in this work we investigate the behavior of $S_{2}(\mathit{\Pi}(\rho))$
across the quantum phase transition of the transverse-field Ising model, the
question arises whether this behavior appears for a generic quantum phase
transition. In this respect, we observe that $S_{2}(\mathit{\Pi}(\rho))$ can
be interpreted as a measure of how a state $\rho$ is close to a PI state,
which in fact vanishes for an exactly PI state. It is then reasonable to
expect that a quantum phase transition between two PI states will produce a
peak structure in $S_{2}(\mathit{\Pi}(\rho))$ similar to Fig. 3 (b), whereas
one between a PI and a non-PI state will bring about a sharp, step-like,
change like in Fig. 4 (b). In order to get insights into the case of a
transition between two non-PI state, the analysis of different models is
required; however, we may speculate that another kind of step-like behavior
occurs. We consider it particularly interesting to study whether
$S_{2}(\mathit{\Pi}(\rho))$ can distinguish between two states in which no
standard order parameter can be defined, such as quantum spin liquids and
topological phases.

By the definition Eq. (3), $\mathit{\Pi}(\rho)$ is the average over all
possible site permutations of a state $\rho$. We thus note that recovering
$\mathit{\Pi}(\rho)$ instead of $\rho$ leads to a loss of information about
the local properties of $\rho$. $S_{2}(\mathit{\Pi}(\rho))$ in some way
quantifies this loss.

Figure 6: The fitting parameters as a function of ${\it\Gamma}/|J|$ for FM
case. The blue and red dashed lines are analytical results by the second-order
perturbation theory at low-field and high-field regions, respectively. Black
dashed line represents the QCP. Figure 7: The fitting parameters as a function
of ${\it\Gamma}/J$ for AFM case. The blue and red dashed lines are analytical
results by the second-order perturbation theory at low-field and high-field
regions, respectively. Black dashed line represents the QCP.

##  VI conclusion

We have studied the second-order Rényi entropy of PIDM
$S_{2}(\mathit{\Pi}(\rho))$ for the ground state of the transverse-field Ising
chain. In the case of FM interaction, $S_{2}(\mathit{\Pi}(\rho))$ exhibits a
broad peak structure around the QCP, which becomes more pronounced for
increasing system size as shown in Fig. 3. In the AFM case, the peak structure
disappears and $S_{2}(\mathit{\Pi}(\rho))$ tends to diverge to infinity in the
whole range of the AFM phase as shown in Fig. 4. $S_{2}(\mathit{\Pi}(\rho))$
can be seen as a measure of permutational invariance of the (ground) state,
which takes zero value for a PI state. We have especially studied its
behaviors as a function of system size, varying the intensity of the
transverse field. We have fitted the data of $S_{2}(\mathit{\Pi}(\rho))$
versus $L$ using a function containing linear, constant, $L^{-1}$ and
logarithmic terms. For the FM case, the fitting function approximately at the
QCP has the largest slope. We ascribe this slight shift to the limitation of
system size and expect the fitting function at the QCP to be the upper bound
of $S_{2}(\mathit{\Pi}(\rho))$ for sufficiently large $L$. For the AFM case,
we have found that the fitting parameters sharply change around the QCP. We
expect that the the fitting function at the QCP will be a boundary between AFM
and PM phases. We also analyze the behavior of $S_{2}(\mathit{\Pi}(\rho))$
under low and high fields according to the second-order perturbation theory.
We conjecture that the FM, AFM, and PM phases are distinguished by the scaling
law of $S_{2}(\mathit{\Pi}(\rho))$ as shown in Eq. (23).

Lastly, let us emphasize that, while our simulations are practically limited
to $L=22$ by the explicit calculation of ${\it\Pi}(\rho)$ from $\rho$, the
prospective experimental protocols and subsequent calculation of purity via
Eqs. 8, 12, and 13 would be in principle scalable to a much larger system
size, being dependent upon the measurement of only $\mathcal{O}(L^{3})$
correlations (corresponding to only $\mathcal{O}(L^{2})$ experimental
settings). We hope that future efforts will clarify the properties of
$S_{2}(\mathit{\Pi}(\rho))$ through both experimental measurements and
theoretical analysis of other physical models, possibly acquiring some
insights into the connection with symmetry breaking, conformal field theory,
topological orders, etc.

###### Acknowledgements.

We thank T. Shimokawa, D. Kagamihara, S. Goto, R. Kaneko, and I. Danshita for
useful discussions. This work was supported by JSPS KAKENHI Grant Nos.
23KJ1867 (Y.M.), 18K03525, 21H05185, 23K25830 and 24K06890 (D.Y.), 22H01171
(N.F., D.Y.), JST PRESTO Grant No. JPMJPR2118, Japan (D.Y.), and Aoyama Gakuin
University Research Institute “Early Eagle” grant program for promotion of
research by early career researchers.

##  Appendix A ANALYTICAL CALCULATION OF THE PURITY OF THE PIDM EXPRESSED IN
THE $\mathcal{B}^{(k,l,m,n)}$ BASIS

In this appendix, we briefly give the derivation of Eq. (13). First, let us
consider the trace of $(\mathcal{B}^{(k,l,m,n)})^{2}=\mathit{\Pi}(X^{\otimes
k}\otimes Y^{\otimes l}\otimes Z^{\otimes m}\otimes\mathbbm{1}^{\otimes
n})^{2}$.

$\displaystyle{\rm Tr}\left[(\mathcal{B}^{(k,l,m,n)})^{2}\right]$ (24)
$\displaystyle=$
$\displaystyle\frac{1}{\left(\frac{L!}{k!l!m!n!}\right)^{2}}{\rm
Tr}\left[\left(\sum_{s\in\tilde{\mathfrak{S}}_{L}}\hat{\mathcal{P}}_{s}(X^{\otimes
k}\otimes Y^{\otimes l}\otimes Z^{\otimes m}\otimes\mathbbm{1}^{\otimes
n})\hat{\mathcal{P}}_{s}^{\dagger}\right)\left(\sum_{s^{\prime}\in\tilde{\mathfrak{S}}_{L}}\hat{\mathcal{P}}_{s^{\prime}}(X^{\otimes
k}\otimes Y^{\otimes l}\otimes Z^{\otimes m}\otimes\mathbbm{1}^{\otimes
n})\hat{\mathcal{P}}_{s^{\prime}}^{\dagger}\right)\right]$ $\displaystyle=$
$\displaystyle\frac{1}{\left(\frac{L!}{k!l!m!n!}\right)^{2}}{\rm
Tr}\left[\sum_{s\in\tilde{\mathfrak{S}}_{L}}\hat{\mathcal{P}}_{s}\left((X^{2})^{\otimes
k}\otimes(Y^{2})^{\otimes l}\otimes(Z^{2})^{\otimes
m}\otimes(\mathbbm{1}^{2})^{\otimes
n}\right)\hat{\mathcal{P}}_{s}^{\dagger}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{\left(\frac{L!}{k!l!m!n!}\right)^{2}}\sum_{s\in\tilde{\mathfrak{S}}_{L}}{\rm
Tr}\left[\hat{\mathcal{P}}_{s}\left(\mathbbm{1}^{\otimes
L}\right)\hat{\mathcal{P}}_{s}^{\dagger}\right]\ =\
\frac{1}{\left(\frac{L!}{k!l!m!n!}\right)^{2}}\times\frac{L!}{k!l!m!n!}\times({\rm
Tr}\mathbbm{1})^{L}\ =\ \frac{k!l!m!n!}{L!}2^{L},$

where $\sum_{\sigma\in\tilde{\mathfrak{S}}_{L}}$ denotes the summation over
all the non-trivial permutations. Subsequently, we calculate
$\mathit{\Pi}(\rho)^{2}$ as follows:

$\displaystyle\mathit{\Pi}(\rho)^{2}$ $\displaystyle=$
$\displaystyle\left\\{\sum_{k+l+m+n=L}\frac{\braket{\mathit{\Pi}\left(X^{\otimes
k}\otimes Y^{\otimes l}\otimes Z^{\otimes m}\otimes\mathbbm{1}^{\otimes
n}\right)}}{{\rm Tr}\left[(\mathit{\Pi}\left(X^{\otimes k}\otimes Y^{\otimes
l}\otimes Z^{\otimes m}\otimes\mathbbm{1}^{\otimes
n}\right))^{2}\right]}\mathit{\Pi}\left(X^{\otimes k}\otimes Y^{\otimes
l}\otimes Z^{\otimes m}\otimes\mathbbm{1}^{\otimes n}\right)\right\\}$ (25)
$\displaystyle\times\left\\{\sum_{k^{\prime}+l^{\prime}+m^{\prime}+n^{\prime}=L}\frac{\braket{\mathit{\Pi}(X^{\otimes
k^{\prime}}\otimes Y^{\otimes l^{\prime}}\otimes Z^{\otimes
m^{\prime}}\otimes\mathbbm{1}^{\otimes n^{\prime}})}}{{\rm
Tr}\left[(\mathit{\Pi}\left(X^{\otimes k^{\prime}}\otimes Y^{\otimes
l^{\prime}}\otimes Z^{\otimes m^{\prime}}\otimes\mathbbm{1}^{\otimes
n^{\prime}}\right))^{2}\right]}\mathit{\Pi}(X^{\otimes k^{\prime}}\otimes
Y^{\otimes l^{\prime}}\otimes Z^{\otimes
m^{\prime}}\otimes\mathbbm{1}^{\otimes n^{\prime}})\right\\}.$

There are
$\left(\sum_{n=0}^{L}D_{L-n}\right)\times\left(\sum_{n^{\prime}=0}^{L}D_{L-n^{\prime}}\right)$
terms in total, but after taking the trace most of them do not contribute and
only $\sum_{n=0}^{L}D_{L-n}$ terms are left due to the orthogonality

$\displaystyle{\rm
Tr}\left[\mathcal{B}^{(k,l,m,n)}\mathcal{B}^{(k^{\prime},l^{\prime},m^{\prime},n^{\prime})}\right]$
(26) $\displaystyle=$ $\displaystyle{\rm
Tr}\left[(\mathcal{B}^{(k,l,m,n)})^{2}\right]\delta_{k,k^{\prime}}\delta_{l,l^{\prime}}\delta_{m,m^{\prime}}\delta_{n,n^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{k!l!m!n!}{L!}2^{L}\delta_{k,k^{\prime}}\delta_{l,l^{\prime}}\delta_{m,m^{\prime}}\delta_{n,n^{\prime}}.$

Eventually we obtain the following formula:

$\displaystyle{\rm Tr}\left[\mathit{\Pi}(\rho)^{2}\right]$ $\displaystyle=$
$\displaystyle\sum_{k+l+m+n=L}\frac{\braket{\mathcal{B}^{(k,l,m,n)}}^{2}}{{\rm
Tr}\left[(\mathcal{B}^{(k,l,m,n)})^{2}\right]}$ (27) $\displaystyle=$
$\displaystyle\frac{L!}{2^{L}}\sum_{k+l+m+n=L}\frac{\braket{\mathcal{B}^{(k,l,m,n)}}^{2}}{k!l!m!n!}.$

##  Appendix B SECOND-ORDER PERTURBATION THEORY

In this appendix, we show the analytical calculation of
$S_{2}(\mathit{\Pi}(\rho))$ at low and high field, according to the second-
order perturbation theory. We focus on the transverse-field Ising chain with
open boundary conditions for even $L$.

###  B.1 High-field regime

Let us consider the Hamiltonian of the transverse-field Ising chain with a
perturbation parameter $\lambda=J/{\it\Gamma}$ as

$\hat{\mathcal{H}}/{\it\Gamma}=\hat{\mathcal{H}}^{(0)}+\lambda\hat{\mathcal{H}}^{\prime},$
(28)

where

$\displaystyle\hat{\mathcal{H}}^{(0)}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{L}\hat{X}_{i},$ (29)
$\displaystyle\hat{\mathcal{H}}^{\prime}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{L-1}\hat{Z}_{i}\hat{Z}_{i+1}$ (30)

are the unperturbed Hamiltonian and the perturbation, respectively.
$\hat{\mathcal{H}}^{(0)}$ provides the nondegenerate ground state
$\ket{\rightarrow}^{\otimes L}$ , where
$\ket{\rightarrow}=(\ket{\uparrow}+\ket{\downarrow})/\sqrt{2}$ , and the
ground state of $\hat{\mathcal{H}}/{\it\Gamma}$, denoted $\ket{\rm HF}$, is
given by

$\ket{\rm HF}\simeq\frac{1}{\mathcal{N}}\left(\ket{\rightarrow}^{\otimes
L}-\frac{\lambda}{4}\sum_{m=0}^{L-2}\ket{\psi_{m}}\right),$ (31)

where $\ket{\psi_{m}}=\ket{\rightarrow}^{\otimes
m}\otimes\ket{\leftarrow}^{\otimes
2}\otimes\ket{\rightarrow}^{\otimes(L-m-2)}$ with
$\ket{\leftarrow}=\hat{Z}\ket{\rightarrow}=(\ket{\uparrow}-\ket{\downarrow})/\sqrt{2}$,
and $\mathcal{N}=\left(1+\frac{1}{16}\lambda^{2}(L-1)\right)^{1/2}$ is a
normalization factor. Note that the $\mathcal{O}(\lambda^{2})$ term of
$\ket{\rm HF}$ is zero. PIDM of $\ket{\rm HF}\bra{\rm HF}$ is given by

${\it\Pi}(\ket{\rm HF}\bra{\rm
HF})=\frac{1}{\mathcal{N}^{2}}\frac{1}{L!}\left[L!\ket{\rightarrow}^{\otimes
L}\bra{\rightarrow}^{\otimes
L}-\frac{\lambda}{4}\sum_{s\in\mathfrak{S}_{L}}\sum_{m=0}^{L-2}\left(\ket{\rightarrow}^{\otimes
L}\bra{\psi_{m}}\hat{\mathcal{P}}_{s}^{\dagger}+{\rm H.c.}\right)\right],$
(32)

which is sufficient to include up to $\mathcal{O}(\lambda^{2})$ terms in the
second-order Rényi entropy of PIDM. The purity and second-order Rényi entropy
of PIDM up to second order in the perturbation parameter are calculated as
follows:

$\displaystyle P({\it\Pi}(\ket{\rm HF}\bra{\rm HF}))$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{N}^{4}}\frac{1}{(L!)^{2}}\left[(L!)^{2}+2\times\frac{\lambda^{2}}{16}\sum_{s,s^{\prime}\in\mathfrak{S}_{L}}\sum_{m,m^{\prime}=0}^{L-2}{\rm
Tr}\left(\ket{\rightarrow}^{\otimes
L}\bra{\psi_{m}}\hat{\mathcal{P}}_{s}^{\dagger}\hat{\mathcal{P}}_{s^{\prime}}\ket{\psi_{m^{\prime}}}\bra{\rightarrow}^{\otimes
L}\right)\right]$ (33) $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{N}^{4}}\frac{1}{(L!)^{2}}\left[(L!)^{2}+2\times\frac{\lambda^{2}}{16}(L-2)!2!L!(L-1)^{2}\right]$
$\displaystyle\simeq$ $\displaystyle
1-\lambda^{2}\left(\frac{L}{8}-\frac{3}{8}\right),$ $\displaystyle
S_{2}({\it\Pi}(\ket{\rm HF}\bra{\rm HF}))$ $\displaystyle=$
$\displaystyle-\log\left[1-\lambda^{2}\left(\frac{L}{8}-\frac{3}{8}\right)\right]$
(34) $\displaystyle\simeq$
$\displaystyle\lambda^{2}\left(\frac{L}{8}-\frac{3}{8}\right).$

###  B.2 Low-field regime

In order to analyze the behavior of the Rényi entropy of PIDM in the low-field
region, we consider the following Hamiltonian

$\hat{\mathcal{H}}/|J|=\hat{\mathcal{H}}^{(0)}+\lambda\hat{\mathcal{H}}^{\prime},$
(35)

where

$\displaystyle\hat{\mathcal{H}}^{(0)}$ $\displaystyle=$ $\displaystyle{\rm
sgn}(J)\sum_{i=1}^{L-1}\hat{Z}_{i}\hat{Z}_{i+1},$ (36)
$\displaystyle\hat{\mathcal{H}}^{\prime}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{L}\hat{X}_{i}.$ (37)

Here we redefine the perturbation parameter as $\lambda={\it\Gamma}/|J|$. Let
us begin with the FM case ($J<0$). The ground state of
$\hat{\mathcal{H}}^{(0)}$ is two-fold degenerate, namely
$\ket{\uparrow}^{\otimes L}$ and $\ket{\downarrow}^{\otimes L}$, and the
$L$th-order perturbation gives a small energy split between symmetrized and
antisymmetrized combinations, which favors $(\ket{\uparrow}^{\otimes
L}+\ket{\downarrow}^{\otimes L})/\sqrt{2}$. Here we define

$\displaystyle\ket{\psi_{m}}$ $\displaystyle=$
$\displaystyle\ket{\uparrow}^{\otimes
m}\otimes\ket{\downarrow}\otimes\ket{\uparrow}^{\otimes(L-m-1)},$ (38)
$\displaystyle\ket{\phi_{m}}$ $\displaystyle=$
$\displaystyle\ket{\downarrow}^{\otimes
m}\otimes\ket{\uparrow}\otimes\ket{\downarrow}^{\otimes(L-m-1)}.$ (39)

The ground state of $\hat{\mathcal{H}}/|J|$, denoted $\ket{\rm LF_{FM}}$, is
given by

$\ket{\rm
LF_{FM}}\simeq\frac{1}{\mathcal{N}}\left(\frac{\ket{\uparrow}^{\otimes
L}+\ket{\downarrow}^{\otimes
L}}{\sqrt{2}}-\frac{\lambda}{2}\left(\frac{\ket{\psi_{0}}+\ket{\phi_{0}}}{\sqrt{2}}+\frac{\ket{\psi_{L-1}}+\ket{\phi_{L-1}}}{\sqrt{2}}\right)-\frac{\lambda}{4}\sum_{m=1}^{L-2}\frac{\ket{\psi_{m}}+\ket{\phi_{m}}}{\sqrt{2}}\right),$
(40)

where
$\mathcal{N}=\left[1+\lambda^{2}\left(\frac{1}{2}-\frac{1}{16}(L-2)\right)\right]^{1/2}$
is a normalization factor. Note that the $\mathcal{O}(\lambda^{2})$ term of
$\ket{\rm LF_{FM}}$ is zero. PIDM of $\ket{\rm LF_{FM}}\bra{\rm LF_{FM}}$ is
given by

${\it\Pi}(\ket{\rm LF_{FM}}\bra{\rm
LF_{FM}})=\frac{1}{\mathcal{N}^{2}}\frac{1}{L!}(A+(B+C+{\rm H.c.})),$ (41)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{L!}{2}\left(\ket{\uparrow}^{\otimes
L}+\ket{\downarrow}^{\otimes L}\right)\left(\bra{\uparrow}^{\otimes
L}+\bra{\downarrow}^{\otimes L}\right),$ (42) $\displaystyle B$
$\displaystyle=$
$\displaystyle-\frac{\lambda}{4}\sum_{s\in\mathfrak{S}_{L}}\left(\ket{\uparrow}^{\otimes
L}+\ket{\downarrow}^{\otimes
L}\right)\left(\bra{\psi_{0}}+\bra{\phi_{0}}+\bra{\psi_{L-1}}+\bra{\phi_{L-1}}\right)\hat{\mathcal{P}}^{\dagger}_{s},$
(43) $\displaystyle C$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{8}\sum_{s\in\mathfrak{S}_{L}}\sum_{m=1}^{L-2}\left(\ket{\uparrow}^{\otimes
L}+\ket{\downarrow}^{\otimes
L}\right)\left(\bra{\psi_{m}}+\bra{\phi_{m}}\right)\hat{\mathcal{P}}^{\dagger}_{s}.$
(44)

The purity and second-order Rényi entropy of PIDM within second-order
perturbation theory are calculated as follows:

$\displaystyle P({\it\Pi}(\ket{\rm LF_{FM}}\bra{\rm LF_{FM}}))$ (46)
$\displaystyle=$ $\displaystyle\frac{1}{\mathcal{N}^{4}}\frac{1}{(L!)^{2}}{\rm
Tr}(A^{2}+2BB^{\dagger}+2BC^{\dagger}+2CB^{\dagger}+2CC^{\dagger})$
$\displaystyle=$
$\displaystyle\frac{1}{\mathcal{N}^{4}}\frac{1}{(L!)^{2}}\left\\{(L!)^{2}+2\times\frac{\lambda^{2}}{4}4(L-1)!L!+4\times\frac{\lambda^{2}}{8}2(L-2)(L-1)!L!+2\times\frac{\lambda^{2}}{16}(L-2)^{2}(L-1)!L!\right\\}$
$\displaystyle\simeq$ $\displaystyle
1-\lambda^{2}\left(\frac{1}{4}-\frac{1}{2L}\right),$ $\displaystyle
S_{2}({\it\Pi}(\ket{\rm LF_{FM}}\bra{\rm
LF_{FM}}))=-\log\left[1-\lambda^{2}\left(\frac{1}{4}-\frac{1}{2L}\right)\right]\simeq\lambda^{2}\left(\frac{1}{4}-\frac{1}{2L}\right).$

For the AFM case ($J>0$), the ground state of the unperturbed Hamiltonian Eq.
(36) is two-fold degenerate, namely
$\ket{\uparrow\downarrow}^{\otimes\frac{L}{2}}$ and
$\ket{\downarrow\uparrow}^{\otimes\frac{L}{2}}$, and the $L$th-order
perturbation gives a small energy split between symmetrized and
antisymmetrized states of them, selecting
$(\ket{\uparrow\downarrow}^{\otimes\frac{L}{2}}+\ket{\downarrow\uparrow}^{\otimes\frac{L}{2}})/\sqrt{2}$.
Here let us define
$\ket{\psi_{m}^{(\uparrow)}},\ket{\psi_{m}^{(\downarrow)}},\ket{\phi_{m}^{(\uparrow)}}$,
and $\ket{\phi_{m}^{(\downarrow)}}$ as

$\displaystyle\ket{\psi_{m}^{(\uparrow)}}$ $\displaystyle=$
$\displaystyle\ket{\uparrow\downarrow}^{\otimes
m}\otimes\ket{\uparrow\uparrow}\otimes\ket{\uparrow\downarrow}^{\otimes(\frac{L}{2}-m-1)},$
(47) $\displaystyle\ket{\psi_{m}^{(\downarrow)}}$ $\displaystyle=$
$\displaystyle\ket{\uparrow\downarrow}^{\otimes
m}\otimes\ket{\downarrow\downarrow}\otimes\ket{\uparrow\downarrow}^{\otimes(\frac{L}{2}-m-1)},$
(48) $\displaystyle\ket{\phi_{m}^{(\uparrow)}}$ $\displaystyle=$
$\displaystyle\ket{\downarrow\uparrow}^{\otimes
m}\otimes\ket{\uparrow\uparrow}\otimes\ket{\downarrow\uparrow}^{\otimes(\frac{L}{2}-m-1)},$
(49) $\displaystyle\ket{\phi_{m}^{(\downarrow)}}$ $\displaystyle=$
$\displaystyle\ket{\downarrow\uparrow}^{\otimes
m}\otimes\ket{\downarrow\downarrow}\otimes\ket{\downarrow\uparrow}^{\otimes(\frac{L}{2}-m-1)}.$
(50)

The ground state of Eq. (35), denoted $\ket{\rm LF_{AFM}}$, is given by

$\displaystyle\ket{\rm LF_{AFM}}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\mathcal{N}}\left[\frac{\ket{\uparrow\downarrow}^{\otimes\frac{L}{2}}+\ket{\downarrow\uparrow}^{\otimes\frac{L}{2}}}{\sqrt{2}}+\frac{\lambda}{4\sqrt{2}}\left(\sum_{m=0}^{\frac{L}{2}-2}\left(\ket{\psi^{(\uparrow)}_{m}}+\ket{\phi^{(\downarrow)}_{m}}\right)+\sum_{m=1}^{\frac{L}{2}-1}\left(\ket{\psi^{(\downarrow)}_{m}}+\ket{\phi^{(\uparrow)}_{m}}\right)\right)\right.$
(51) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \
\left.{\color[rgb]{1,1,1}\frac{1}{1}}+\frac{\lambda}{2\sqrt{2}}\left(\ket{\psi^{(\uparrow)}_{\frac{L}{2}-1}}+\ket{\phi^{(\downarrow)}_{\frac{L}{2}-1}}+\ket{\psi^{(\downarrow)}_{0}}+\ket{\phi^{(\uparrow)}_{0}}\right)\right],$

where
$\mathcal{N}=\left[1+\lambda^{2}\left(\frac{L}{16}+\frac{3}{8}\right)\right]^{1/2}$
is a normalization factor. Note that $\mathcal{O}(\lambda^{2})$ term of
$\ket{\rm LF_{AFM}}$ is zero. PIDM of $\ket{\rm LF_{AFM}}\bra{\rm LF_{AFM}}$
is given by

${\it\Pi}(\ket{\rm LF_{AFM}}\bra{\rm
LF_{AFM}})=\frac{1}{\mathcal{N}^{2}}\frac{1}{L!}(A+(B+C+{\rm H.c.})),$ (52)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s\in\mathfrak{S}_{L}}\hat{\mathcal{P}}_{s}\left(\ket{\uparrow\downarrow}^{\otimes\frac{L}{2}}+\ket{\downarrow\uparrow}^{\otimes\frac{L}{2}}\right)\left(\bra{\uparrow\downarrow}^{\otimes\frac{L}{2}}+\bra{\downarrow\uparrow}^{\otimes\frac{L}{2}}\right)\hat{\mathcal{P}}_{s}^{\dagger},$
(53) $\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{\lambda}{8}\sum_{s\in\mathfrak{S}_{L}}\left[\hat{\mathcal{P}}_{s}\left(\ket{\uparrow\downarrow}^{\otimes\frac{L}{2}}+\ket{\downarrow\uparrow}^{\otimes\frac{L}{2}}\right)\left(\sum_{m=0}^{\frac{L}{2}-2}\left(\bra{\psi^{(\uparrow)}_{m}}+\bra{\phi^{(\downarrow)}_{m}}\right)+\sum_{m=1}^{\frac{L}{2}-1}\left(\bra{\psi^{(\downarrow)}_{m}}+\bra{\phi^{(\uparrow)}_{m}}\right)\right)\hat{\mathcal{P}}_{s}^{\dagger}\right],$
(54) $\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{\lambda}{4}\sum_{s\in\mathfrak{S}_{L}}\left[\hat{\mathcal{P}}_{s}\left(\ket{\uparrow\downarrow}^{\otimes\frac{L}{2}}+\ket{\downarrow\uparrow}^{\otimes\frac{L}{2}}\right)\left(\bra{\psi^{(\uparrow)}_{\frac{L}{2}-1}}+\bra{\phi^{(\downarrow)}_{\frac{L}{2}-1}}+\bra{\psi^{(\downarrow)}_{0}}+\bra{\phi^{(\uparrow)}_{0}}\right)\hat{\mathcal{P}}_{s}^{\dagger}\right].$
(55)

The purity and second-order Rényi entropy of PIDM within second-order
perturbation theory are calculated as follows:

$\displaystyle P({\it\Pi}(\ket{\rm LF_{AFM}}\bra{\rm LF_{AFM}}))$
$\displaystyle=$ $\displaystyle\frac{1}{\mathcal{N}^{4}}\frac{1}{(L!)^{2}}{\rm
Tr}(A^{2}+2BB^{\dagger}+2BC^{\dagger}+2CB^{\dagger}+2CC^{\dagger})$
$\displaystyle=$
$\displaystyle\frac{1}{\mathcal{N}^{4}}\frac{1}{(L!)^{2}}\left\\{2\left(\frac{L}{2}\right)!\left(\frac{L}{2}\right)!L!+2\times\frac{\lambda^{2}}{4}\left(\frac{L}{2}\right)!\left(\frac{L}{2}-1\right)!L!\left(\frac{L}{2}-1\right)^{2}\right.$
$\displaystyle\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \
+4\times\frac{\lambda^{2}}{2}\left(\frac{L}{2}\right)!\left(\frac{L}{2}-1\right)!L!\left(\frac{L}{2}-1\right)+2\times\lambda^{2}\left(\frac{L}{2}\right)!\left(\frac{L}{2}-1\right)!L!\right\\}$
$\displaystyle\simeq$
$\displaystyle\frac{2\left(\frac{L}{2}\right)!\left(\frac{L}{2}\right)!}{L!}+\lambda^{2}\left(-\frac{2\left(\frac{L}{2}\right)!\left(\frac{L}{2}\right)!}{L!}\left(\frac{L}{8}+\frac{3}{4}\right)+\frac{\left(\frac{L}{2}\right)!\left(\frac{L}{2}-1\right)!}{L!}\left(\frac{L^{2}}{8}+\frac{L}{2}+\frac{1}{2}\right)\right),$
$\displaystyle S_{2}({\it\Pi}(\ket{\rm LF_{AFM}}\bra{\rm LF_{AFM}}))$
$\displaystyle=$
$\displaystyle-\log\left[\frac{2\left(\frac{L}{2}\right)!\left(\frac{L}{2}\right)!}{L!}+\lambda^{2}\left(-\frac{2\left(\frac{L}{2}\right)!\left(\frac{L}{2}\right)!}{L!}\left(\frac{L}{8}+\frac{3}{4}\right)+\frac{\left(\frac{L}{2}\right)!\left(\frac{L}{2}-1\right)!}{L!}\left(\frac{L^{2}}{8}+\frac{L}{2}+\frac{1}{2}\right)\right)\right]$
$\displaystyle\simeq$ $\displaystyle L\log 2-\frac{1}{2}\log
L-\log\sqrt{2\pi}-\lambda^{2}\left(\frac{1}{4}+\frac{1}{2L}\right)$ (57)

Here we have used the Stirling’s formula to obtain Eqs. 57.

##  Appendix C MAXIMUM BOND DIMENSION DEPENDENCE OF THE PURITY OF PIDM

In this appendix, we show the plots of the purity of PIDM for the ground state
of the transverse-field Ising chain as a function of the maximum bond
dimension of MPS $\chi_{\rm max}$ in Fig. 8. We can see that the purity
sufficiently converges at $\chi_{\rm max}\simeq 4$.

Figure 8: The maximum bond dimension $\chi_{\rm max}$ dependence of the purity
of PIDM for the ground state of (a) the FM and (b) the AFM transverse-field
Ising models at the QCP.

## References

  * [1] M. Matsumoto, B. Normand, T. M. Rice, and Manfred Sigrist, Physical Review B 69, 054423 (2004).
  * [2] Z. Wang, D. L. Quintero-Castro, S. Zherlitsyn, S. Yasin, Y. Skourski, A. T. M. N. Islam, B. Lake, J. Deisenhofer, and A. Loidl, Physical Review Letters 116, 147201 (2016).
  * [3] H. K. Onnes, Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences 14, 113 (1911).
  * [4] D. A. Cardwell High-temperature superconducting materials (Springer, Boston, 1991).
  * [5] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Physical Review Letters 48, 1559 (1982).
  * [6] L. Fidkowski and A. Kitaev, Physical Review B 83, 075103 (2011).
  * [7] A. Kitaev, Annals of Physics 321, 2 (2006).
  * [8] A. Kitaev and J. Preskill, Physical Review Letters 96, 110404 (2006).
  * [9] M. Levin and X.-G. Wen, Physical Review Letters 96, 110405 (2006).
  * [10] F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Physical Review B 81, 064439 (2010).
  * [11] A. M. Turner, F. Pollmann, and E. Berg, Physical Review B 83, 075102 (2011).
  * [12] I. Bloch, Nature Physics 1, 23 (2005).
  * [13] I. Bloch, J. Dalibard, and S. Nascimbène, Nature Physics 8, 267 (2012).
  * [14] C. Gross and I. Bloch, Science 357, 995 (2017).
  * [15] Y. Takahashi, Proceedings of the Japan Academy, Series B 98, 141 (2022).
  * [16] A. Aspuru-Guzik and P. Walther, Nature Physics 8, 285 (2012).
  * [17] C. Noh and D. G. Angelakis, Reports on Progress in Physics 80, 016401 (2016).
  * [18] R. Blatt and C. F. Roos, Nature Physics 8, 277 (2012).
  * [19] C. Monroe, W. C. Campbell, L. M. Duan, Z. X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Reviews of Modern Physics 93, 025001 (2021).
  * [20] F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L.Béguin, A.Vernier, T. Lahaye, and A. Browaeys, Physical Review X 4, 021034 (2014).
  * [21] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, andM. D. Lukin, Nature (London) 551, 579 (2017).
  * [22] A. Browaeys and T. Lahaye, Nature Physics 16, 132 (2020).
  * [23] P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P.Henry, T. C. Lang, T. Lahaye, A. M. Läuchli, and A. Browaeys, Nature (London) 595, 233 (2021).
  * [24] S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuletić, and M. D. Lukin, Nature (London) 595, 227 (2021).
  * [25] A. A. Houck, H. E. Türeci, and J. Koch, Nature Physics 8, 292 (2012).
  * [26] N. Kiesel, C. Schmid, G. Tóth, E. Solano, and H. Weinfurter, Physical Review Letters 98, 063604 (2007).
  * [27] H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, Nature 438, 643 (2005).
  * [28] R. Barends, J. Kelly, A. Megrant, A. Veitia, D.Sank, E. Jeffrey, T. C.White, J. Mutus, A. G. Fowler, B.Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, Nature 508, 500 (2014).
  * [29] G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, Advances in Imaging and Electron Physics 128, 205 (2003).
  * [30] U. Fano, Reviews of Modern Physics 29, 74 (1957).
  * [31] M. Paris and J. Řeháček, Lecture Notes in Physics 649 Quantum State Estimation (Springer, Heidelberg, 2004).
  * [32] M. Hayashi, Asymptotic Theory in Quantum Statistical Inference: Selected Papers, World Scientific (2005).
  * [33] D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Physical Review Letters 105, 150401 (2010).
  * [34] A. Steffens, C. A. Riofrío,W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, Quantum Science and Technology 2, 025005 (2017).
  * [35] G. Tóth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, and H. Weinfurter, Physical Review Letters 105, 250403 (2010).
  * [36] T. Moroder, P. Hyllus, G. Tóth, C. Schwemmer, A. Niggebaum, S. Gaile, O. Gühne, and H. Weinfurter, New Journal of Physics 14, 105001 (2012).
  * [37] T. Gao, F. Yan, and S. J. van Enk, Physical Review Letters 112, 180501 (2014).
  * [38] D. M. Greenberger, M. A. Horne, A. Shimony, A. Zeilinger, American Journal of Physics 58, 1131 (1990).
  * [39] M. S. Kaznady and D. F. V. James, Physical Review A 79, 022109 (2009).
  * [40] W. Dür, G. Vidal, and J.I. Cirac, Physical Review A 62, 062314 (2000).
  * [41] C. F. Roos, M. Riebe, H. Häffner, W. Hänsel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, R. Blatt, Science 304 1478 (2004).
  * [42] H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, Nature (London) 438, 643 (2005).
  * [43] R. H. Dicke, Physical Review 93, 99 (1954).
  * [44] P.G. de Gennes, Solid State Communications 1, 132 (1963).
  * [45] S. R. White, Physical Review Letters 69, 2863 (1992).
  * [46] S. R. White, Physical Review B 48, 10345 (1993).
  * [47] U. Schollwöck, Annals of physics 326 96 (2011).
  * [48] A. Rényi, On Measures of Entropy and Information. (University of California Press, Berkeley, California) 1, 547 (1961).
  * [49] G. Vidal, Physical Review Letters 91, 147902 (2003).
  * [50] S. Östlund and S. Rommer, Physical Review Letters 75, 3537 (1995).
  * [51] W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner, Nature 462, 74 (2009).
  * [52] L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W. S. Bakr, T. Lompe, and M. W. Zwierlein, Physical Review Letters 114, 193001 (2015).
  * [53] E. Haller, J. Hudson, A. Kelly, D. A. Cotta, B. Peaudecerf, G. D. Bruce, and S. Kuhr, Nature Physics 11, 738 (2015).
  * [54] M. Miranda, R. Inoue, Y. Okuyama, A. Nakamoto, and M. Kozuma, Physical Review A 91, 063414 (2015).
  * [55] R. Yamamoto, J. Kobayashi, T. Kuno, K. Kato, and Y. Takahashi, New Journal of Physics 18 023016 (2016).
  * [56] M. Miranda, R. Inoue, N. Tambo, and M. Kozuma, Physical Review A 96, 043626 (2017).
  * [57] R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, Nature 528, 77 (2015).
  * [58] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science 353, 794 (2016).
  * [59] T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Science 364, 260 (2019).
  * [60] P. Pfeuty, Annals of Physics 57, 79 (1970).
  * [61] B. Schumacher, Physical Review A 54, 2614 (1996).
  * [62] T. J. Osborne and M. A. Nielsen, Physical Review A 66, 032110 (2002).