# Many-Body Localization: Transitions in Spin Models

John Schliemann1, Jo$\tilde{\rm a}$o Vitor I. Costa2, Paul Wenk1, and J.
Carlos Egues2 1Institute for Theoretical Physics, University of Regensburg,
Regensburg, Germany
2Instituto de Fısica de Sao Carlos, Universidade de Sao Paulo, Sao Carlos,
Brazil

(August 27, 2024)

###### Abstract

We study the transitions between ergodic and many-body localized phases in
spin systems, subject to quenched disorder, including the Heisenberg chain and
the central spin model. In both cases systems with common spin lengths $1/2$
and $1$ are investigated via exact numerical diagonalization and random matrix
techniques. Particular attention is paid to the sample-to-sample variance
$(\Delta_{s}r)^{2}$ of the averaged consecutive-gap ratio $\langle r\rangle$
for different disorder realizations. For both types of systems and spin
lengths we find a maximum in $\Delta_{s}r$ as a function of disorder strength,
accompanied by an inflection point of $\langle r\rangle$, signaling the
transition from ergodicity to many-body localization. The critical disorder
strength is found to be somewhat smaller than the values reported in the
recent literature. Further information about the transitions can be gained
from the probability distribution of expectation values within a given
disorder realization.

##  I Introduction

Many-body localization has become in the last years one of the most
intensively growing areas of research in condensed matter physics and beyond
Nandkishore15 ; Altman15 ; Imbrie16a ; Agarwal17 ; Luitz17 ; Haldar17 ;
Abanin17 ; Abanin18 . It denotes the absence of thermalization in an isolated
interacting quantum system in the presence of typically strong disorder. In
the opposite ergodic phase the eigenstate thermalization hypothesis is
fulfilled stating that any appropriate subsystem of the isolated total system
(being in a pure state) is accurately described by equilibrium statistical
mechanics Deutsch91 ; Srednicki94 . On the other hand, the presence of
interactions distinguishes many-body localization from traditional Anderson
localization Anderson58 ; Evers08 .

In this work we revisit the transition from the ergodic to the many-body
localized phase in disordered Heisenberg spin chains and compare it with the
behavior of central spin models also subject to quenched disorder. For both
types of systems we consider the spin lengths $1/2$ and $1$. We introduce a
novel and very useful tool to quantify the transition, namely, the sample-to-
sample variance of the averaged consecutive-gap ratio and the underlying
probability distribution. As we will see in the following, a maximum of this
variance signals, for both of the above systems and both spin lengths, the
transition between ergodicity and many-body localization. This maximum is
accompanied by an inflection point of of the averaged consecutive gap ratio
$\langle r\rangle$, suggesting a close analogy to classic phase transitions
with $\langle r\rangle$ being an order parameter. These central observations
are summarized in Figs. 2, 7 for the Heisenberg chain and the central spin
model, respectively.

An additional tool in the analysis of the transition from the ergodic to the
many-body localized phase is the probability distribution of expectation
values within a given disorder realization. The corresponding data is
contained in Figs 5 and 9.

This paper is further organized as follows: In section II we introduce the
spin models to be studied and summarize the underlying theoretical techniques.
Our numerical results are presented in section III, and we close with a
summary and an outlook in section IV.

##  II Model and Approach

###  II.1 Spin Hamiltonian

We study a periodic Heisenberg spin chain interacting with an additional
central spin and being subject to an uniaxial quenched disorder field on each
site of the chain,

$H=J\sum_{i=1}^{K}\vec{I}_{i}\cdot\vec{I}_{i+1}+\frac{A}{K}\vec{S}\cdot\sum_{i=1}^{K}\vec{I}_{i}+2S\sum_{i=1}^{K}h_{i}I^{z}_{i}\,,$
(1)

where the parameter $J$ describes the coupling of the $K$ chain (or bath)
spins $\vec{I}_{i}=\vec{I}_{i+K}$ to their nearest neighbors. The coupling to
the central spin $\vec{S}$ is parametrized by $A$, and the random magnetic
field $h_{i}$ is chosen from a uniform distribution within the interval
$[-h,h]$. In what follows all spins will have length $S=I=1/2$ or $S=I=1$.

For $S=I=1/2$, the Hamiltonian (1) with $A=0$, i.e. the Heisenberg spin-$1/2$
chain with quenched on-site disorder, is a workhorse of numerical studies of
many-body localization Santos04 ; Pal10 ; Badarson12 ; Luitz15 ; Chandran15 ;
Agarwal15 ; Baygan15 ; Devakul15 ; Luitz16a ; Luitz16b ; Geraedts16 ;
Khemani16 ; Lim16 ; Serbyn16 ; Enss17 ; Acevedo17 ; Khemani17 ; Filho17 ;
Geraedts17 ; Xu18 ; Doggen18 ; Suntajs19 ; Panda20 ; Chanda20a ; Suntajs20 ;
Laflorencie20 ; Sierant20 ; Dhara20 ; Chanda20b ; Throckmorton20 . The factor
$2S$ in front of the disorder term makes contact to the usual parametrization
for $S=1/2$ and ensures an appropriate scaling behavior of the Hamiltonian for
larger spin lengths. Specifically, when considering all spins as classical
vectors of constant length (and not as operators), the transformation
$S\mapsto qS$, $I\mapsto qI$ leads to $H\mapsto q^{2}H$. Thus, disorder-
induced effects on the dynamics should occur at the same disorder strength
$h$. As we shall see in section III, this remains approximately true when
switching between $S=I=1/2$ and $S=I=1$.

Recent work by Hetterich et al. Hetterich18  studied the full model (1) for
$S=I=1/2$ concentrating on the case $J=1$. As these authors argue, dividing
the interaction parameter $A$ by the number of central spins $K$, as done in
Eq. (1), ensures that the spectral bandwidth of that coupling term is
approximately independent of the system size. We will see in the following
that this stipulation has indeed advantages when comparing data for different
numbers of spins.

###  II.2 Random Matrix Theory

An important method to distinguish a many-body localized phase from an ergodic
phase is random matrix theory, which is a theory for statistical fluctuations
of energy levels of a given quantum system Guhr98 . A modern tool to analyze
the energy level statistics is the consecutive gap ratio Oganesyan07  defined
as

$r_{n}=\frac{\min\\{s_{n},s_{n-1}\\}}{\max\\{s_{n},s_{n-1}\\}}=\min\left\\{\bar{r}_{n},\frac{1}{\bar{r}_{n}}\right\\}\quad,\quad\bar{r}_{n}=\frac{s_{n}}{s_{n-1}}\,,$
(2)

where $s_{n}=e_{n+1}-e_{n}$ is the difference between two neighboring energy
levels $e_{n+1}$, $e_{n}$. In the strictly many-body localized (or integrable)
phase, characterized by an extensive number of independent conserved
quantities, the differences $s_{n}$ obey Poisson statistics, and the
probability distribution for the random variable $r=r_{n}$ can easily be
determined to be Oganesyan07

$p(r)=\frac{2}{(1+r)^{2}}\,.$ (3)

On the other hand, in the fully ergodic phase, a system of the type (1) is
generally assumed to be described by the Gaussian orthogonal ensemble (GOE) of
random matrices Guhr98 . Here the analysis of small random matrices mimicking
the Hamiltonian predicts the pertaining probability distribution to be Atas13

$p(r)=\frac{27}{4}\frac{r+r^{2}}{(1+r+r^{2})^{5/2}}\,,$ (4)

which can be seen as an analog of the classic Wigner surmises for probability
distributions governing the traditional random variable $s=s_{n}$ Guhr98 .

As a consequence, the lowest moments of the probability distribution (3) in
the integrable case are given by

$\displaystyle\langle r\rangle_{p}$ $\displaystyle=$ $\displaystyle 2\ln
2-1\approx 0.3863\,,$ (5) $\displaystyle\langle r^{2}\rangle_{p}$
$\displaystyle=$ $\displaystyle 3-4\ln 2\approx 0.2274\,,$ (6)
$\displaystyle\Delta_{p}r$ $\displaystyle=$ $\displaystyle\sqrt{\langle
r^{2}\rangle_{p}-\langle r\rangle^{2}_{p}}\approx 0.2796\,,$ (7)

with

$\langle\cdot\rangle_{p}=\int_{0}^{1}dr(p(r)\cdot)\,,$ (8)

whereas in the ergodic situation (4) we have

$\displaystyle\langle r\rangle_{p}$ $\displaystyle=$ $\displaystyle
4-2\sqrt{3}\approx 0.5359\,,$ (9) $\displaystyle\langle r^{2}\rangle_{p}$
$\displaystyle=$
$\displaystyle\frac{27}{4}\ln\left(1+\frac{2}{\sqrt{3}}\right)-\frac{1}{2}-\frac{5}{2}\sqrt{3}\approx
0.3515\,,$ (10) $\displaystyle\Delta_{p}r$ $\displaystyle\approx$
$\displaystyle 0.2536\,,$ (11)

###  II.3 Statistical Data Analysis

####  II.3.1 Disorder ensemble

Consider an ensemble of $Q$ realizations of the local disorder field $h_{i}$,
$i\in\\{1,\dots,K\\}$. Each disorder realization, or sample, labeled by
$\alpha\in\\{1\dots,Q\\}$ leads to a probability distribution $p_{\alpha}(r)$
for the consecutive gap ratio $r\in[0,1]$. Given an arbitrary function $f(r)$,
these distributions determine the realization-dependent averages (or
expectation values)

$\langle f\rangle_{\alpha}=\int_{0}^{1}drp_{\alpha}(r)f(r)$ (12)

with variances

$(\Delta_{\alpha}f)^{2}=\langle f^{2}\rangle_{\alpha}-\langle
f\rangle_{\alpha}^{2}\,.$ (13)

The disorder-averaged probability distribution $p(r)$ at given disorder
strength (and other system parameters) is

$p(r)=\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}p_{\alpha}(r)\,,$ (14)

and a good numerical estimate for this quantity is

$p(r)\approx\frac{1}{Q}\sum_{\alpha=1}^{Q}p_{\alpha}(r)\quad,\quad Q\gg 1\,,$
(15)

for sufficiently large $Q$. The disorder-averaged expectation values of $f(r)$
reads

$\langle
f\rangle_{p}=\int_{0}^{1}drp(r)f(r)=\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}\langle
f\rangle_{\alpha}\,.$ (16)

####  II.3.2 Probability distribution for average within sample

On the other hand, we can view the numbers $x=\langle f\rangle_{\alpha}$ as
random variables according to the distribution

$\displaystyle s(x)$ $\displaystyle=$
$\displaystyle\frac{1}{(2h)^{K}}\int_{-h}^{h}dh_{1}\cdots\int_{-h}^{h}dh_{K}$
(17)
$\displaystyle\qquad\cdot\delta\left(x-\int_{0}^{1}drp(r;h_{1},\dots,h_{K})f(r)\right)$

where $p(r;h_{1},\dots,h_{K})=p_{\alpha}$ is the probability distribution
within a system with local disorder fields $h_{1},\dots,h_{K}$ forming the
disorder realization $\alpha$. Thus, the disorder-averaged expectation value
of $f(r)$ can be formulated as

$\langle f\rangle_{s}=\int
dxs(x)x=\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}\langle
f\rangle_{\alpha}=\langle f\rangle_{p}\,,$ (18)

where the integration goes over all values of $x=f(r)$ for $r\in[0,1]$. The
distribution $s(x)$ does in general not coincide with $p(r)$ even for
$x=f(r)=r$. Moreover $s(x)$ will of course have a dependence on the function
$f$, which we, however, shall suppress in the notation.

The finite average

$\bar{f}=\frac{1}{Q}\sum_{\alpha=1}^{Q}\langle f\rangle_{\alpha}\approx\langle
f\rangle_{s}\,,$ (19)

is, again for appropriately large $Q$, an approximation to the expression
(18). On the other hand, it is a sum of stochastically independent (and
therefore uncorrelated) random variables with identical probability
distribution, so that the resulting joint distribution is

$\pi(x_{1},\dots,x_{Q})=\prod_{\alpha=1}^{Q}s(x_{\alpha})\,.$ (20)

Thus, the expectation value of $\bar{f}$ is

$\langle\bar{f}\rangle_{\pi}=\frac{1}{Q}\sum_{\alpha=1}^{Q}\langle
f\rangle_{s}=\langle f\rangle_{s}\,.$ (21)

####  II.3.3 Sample-to-sample variance

For the variance pertaining to the expectation value (21) one finds

$\displaystyle\left\langle\left(\bar{f}-\langle
f\rangle_{s}\right)^{2}\right\rangle_{\pi}=\sum_{\alpha,\beta=1}^{Q}\frac{\left\langle\left(\langle
f\rangle_{\alpha}-\langle f\rangle_{s}\right)\left(\langle
f\rangle_{\beta}-\langle f\rangle_{s}\right)\right\rangle_{\pi}}{Q^{2}}$
$\displaystyle\qquad=\frac{1}{Q^{2}}\sum_{\alpha=1}^{Q}\left\langle\left(\langle
f\rangle_{\alpha}-\langle
f\rangle_{s}\right)^{2}\right\rangle_{s}=\frac{(\Delta_{s}f)^{2}}{Q}$ (22)

with

$\displaystyle(\Delta_{s}f)^{2}$ $\displaystyle=$
$\displaystyle\langle(f-\langle f\rangle_{s})^{2}\rangle_{s}$ (23)
$\displaystyle=$
$\displaystyle\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}(\langle
f\rangle_{\alpha}-\langle f\rangle_{s})^{2}\,.$

Therefore, the fluctuations of the finite average (18) around its expectation
value (21) are characterized by the standard deviation

$\Delta_{\pi}\bar{f}=\sqrt{\left\langle\left(\bar{f}-\langle
f\rangle_{s}\right)^{2}\right\rangle_{\pi}}=\frac{\Delta_{s}f}{\sqrt{Q}}$ (24)

and shows the familiar decay $\propto 1/\sqrt{Q}$. This result follows of
course also from the Lindeberg-Levy central limit theorem applied to the sum
(19) of random variables. An approximate expression for the variance (23) is

$(\Delta_{s}f)^{2}\approx\frac{1}{Q}\sum_{\alpha=1}^{Q}(\langle
f\rangle_{\alpha}-\bar{f})^{2}\quad,\quad Q\gg 1\,.$ (25)

Note that the variance (23) also occurs as a contribution to the variance of
$f(r)$ calculated from the disorder-averaged probability distribution (14),

$\displaystyle(\Delta_{p}f)^{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}drp(r)(f(r)-\langle f\rangle_{p})^{2}$
$\displaystyle=$
$\displaystyle\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}\int_{0}^{1}drp_{\alpha}(r)(f(r)-\langle
f\rangle_{p})^{2}$ $\displaystyle=$
$\displaystyle\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}(\langle
f^{2}\rangle_{\alpha}-\langle f\rangle_{\alpha}^{2})$
$\displaystyle\qquad+\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}(\langle
f\rangle_{\alpha}^{2}-\langle f\rangle_{p}^{2})$ $\displaystyle=$
$\displaystyle\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}(\Delta_{\alpha}f)^{2}+(\Delta_{s}f)^{2}\,.$
(27)

Hence, the variance (II.3.3) with respect to the averaged probability
distribution (14) is the average of all variances within the disorder
realizations around their individual expectation value of $f$, plus the
variance (23) describing the fluctuations of these expectation values around
their mean. An approximate expression for the above result is

$\displaystyle(\Delta_{p}f)^{2}$ $\displaystyle\approx$
$\displaystyle\frac{1}{Q}\sum_{\alpha=1}^{Q}\int_{0}^{1}drp_{\alpha}(r)(f(r)-\bar{f})^{2}$
(28) $\displaystyle=$
$\displaystyle\frac{1}{Q}\sum_{\alpha=1}^{Q}(\Delta_{\alpha}f)^{2}+\frac{1}{Q}\sum_{\alpha=1}^{Q}(\langle
f\rangle_{\alpha}-\bar{f})^{2}$ (29)

where again $Q\gg 1$.

####  II.3.4 Variance of the variance

Let us now analyze the statistical fluctuations of the r.h.s. of the
approximate quantity (25) for finite $Q$ and define

$\bar{g}:=\frac{1}{Q}\sum_{\alpha=1}^{Q}(\langle
f\rangle_{\alpha}-\bar{f})^{2}=\frac{1}{Q}\sum_{\alpha=1}^{Q}\left(\langle
f\rangle^{2}_{\alpha}-\bar{f}^{2}\right)\,.$ (30)

An important difference between the above expression and the quantity (19) is
that above the summands depend, again via Eq. (19), on all expectation values
$\langle f\rangle_{\alpha}$, $\alpha\in\\{1,\dots,Q\\}$, and are therefore
distributed according to the joint probability distribution (20). The
expectation value of the random variable (30) with respect to the latter
distribution is

$\displaystyle\langle\bar{g}\rangle_{\pi}$ $\displaystyle=$
$\displaystyle\frac{1}{Q}\left(\sum_{\alpha=1}^{Q}\left\langle\langle
f\rangle^{2}_{\alpha}\right\rangle_{\pi}-\frac{1}{Q}\sum_{\alpha,\beta=1}^{Q}\left\langle\langle
f\rangle_{\alpha}\langle f\rangle_{\beta}\right\rangle_{\pi}\right)$ (31)
$\displaystyle=$ $\displaystyle\frac{Q-1}{Q}\left(\langle
f^{2}\rangle_{s}-\langle
f\rangle^{2}_{s}\right)=\frac{Q-1}{Q}(\Delta_{s}f)^{2}$

which approaches the variance (23) for large $Q$. Note also that, in contrast
to Eq. (18), it holds

$\langle f^{2}\rangle_{s}=\int
dxs(x)x^{2}=\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}\langle
f\rangle^{2}_{\alpha}\neq\langle f^{2}\rangle_{p}\,.$ (32)

It is now straightforward to establish that

$\displaystyle\left\langle(\langle
f\rangle^{2}_{\alpha}-\bar{f}^{2}-\langle\bar{g}\rangle_{\pi})(\langle
f\rangle^{2}_{\beta}-\bar{f}^{2}-\langle\bar{g}\rangle_{\pi})\right\rangle_{\pi}$
$\displaystyle\qquad\qquad=\delta_{\alpha\beta}\left(\langle
f^{4}\rangle_{s}-\langle f^{2}\rangle^{2}_{s}\right)+{\cal
O}\left(\frac{1}{Q}\right)\,,$ (33)

and therefore, analogously as in Eq. (22),

$\displaystyle\left(\Delta_{\pi}\bar{g}\right)^{2}$
$\displaystyle\quad=\sum_{\alpha,\beta=1}^{Q}\frac{\left\langle(\langle
f\rangle^{2}_{\alpha}-\bar{f}^{2}-\langle\bar{g}\rangle_{\pi})(\langle
f\rangle^{2}_{\beta}-\bar{f}^{2}-\langle\bar{g}\rangle_{\pi})\right\rangle_{\pi}}{Q^{2}}$
$\displaystyle\quad=\frac{(\Delta_{s}f^{2})^{2}}{Q}+{\cal
O}\left(\frac{1}{Q^{2}}\right)$ (34)

where

$\displaystyle(\Delta_{s}f^{2})^{2}$ $\displaystyle=$ $\displaystyle\langle
f^{4}\rangle_{s}-\langle f^{2}\rangle^{2}_{s}=\left\langle\left(f^{2}-\langle
f^{2}\rangle_{s}\right)^{2}\right\rangle_{s}$ (35) $\displaystyle=$
$\displaystyle\left(\Delta_{s}\left((\Delta_{s}f)^{2}\right)\right)^{2}\,.$

####  II.3.5 Consecutive gap ratio

In what follows we will be mainly concerned with the function $f(r)=r$ where
we have

$\displaystyle(\Delta_{s}r)^{2}$ $\displaystyle=$
$\displaystyle\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}(\langle
r\rangle_{\alpha}-\langle r\rangle_{s})^{2}$ (36) $\displaystyle=$
$\displaystyle\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}\left(\langle
r\rangle^{2}_{\alpha}-\langle r\rangle^{2}_{s}\right)$

with

$\langle
r\rangle_{s}=\int_{0}^{1}dxs(x)x=\lim_{Q\to\infty}\frac{1}{Q}\sum_{\alpha=1}^{Q}\langle
r\rangle_{\alpha}=\langle r\rangle_{p}\,,$ (37)

where $s(x)$ is, in accordance with Eq. (17), the probability distribution for
the random variable $x=\langle r\rangle_{\alpha}$. Finally, as seen in Eq.
(35), the variance of the variance (36) is determined by

$\displaystyle(\Delta_{s}r^{2})^{2}$ $\displaystyle=$ $\displaystyle\langle
r^{4}\rangle_{s}-\langle r^{2}\rangle^{2}_{s}=\left\langle\left(r^{2}-\langle
r^{2}\rangle_{s}\right)^{2}\right\rangle_{s}$ (38) $\displaystyle=$
$\displaystyle\left(\Delta_{s}\left((\Delta_{s}r)^{2}\right)\right)^{2}\,.$

##  III Numerical Results

The Hamiltonian (1) obviously conserves the $z$-component of the total spin,

$\vec{J}=\vec{S}+\sum_{i=1}^{K}\vec{I}\quad,\quad\left[H,J^{z}\right]=0\,.$
(39)

Thus, in order to apply random matrix theory, the spectra of each invariant
subspace of $J^{z}$ have to be analyzed separately Guhr98 . In this section we
present accumulated exact-diagonalization data from a separate evaluation of
all subspaces of $J^{z}$ except for the four subspaces of smallest dimension
where $|J^{z}|$ is maximal or differs from its maximal value by $1$. The
number of disorder realizations varies, depending on system size, between
several hundreds and $2\cdot 10^{5}$.

###  III.1 Heisenberg Chain

Figure 1: The probability distribution (15) for the consecutive gap ration $r$
of a Heisenberg chain ($A=0$) of $K=18$ spins of length $I=1/2$ (left) and of
$K=11$ spins of length $I=1$ (right) for different disorder strength $h$
obtained from exact-diagonalization data. At small disorder the system is
ergodic and well described by the distribution (4) (red), while with
increasing $h$ a transition to the Poisson-typed distribution (3) (green) sets
in.

For $A=0$ the central spin $\vec{S}$ becomes obsolete, and for $I=1/2$ and
vanishing disorder $h=0$ the resulting Heisenberg chain is integrable via the
Bethe ansatz Bethe31 . However, this is a rather isolated point in the phase
diagram as seen in Fig. 1 showing the disorder-averaged probability
distribution (15) obtained from exact-diagonalization data of Heisenberg chain
with spin lengths $I=1/2$ and $I=1$ at different disorder strengths. For even
small disorder such as $h=0.1J$ the system shows ergodic statistics (4) while
upon increasing $h$ it changes to the Poisson-type distribution (3). This
transition occurs for both spin lengths at about the same disorder strength,
which is a consequence of the scaling factor $2S=2I$ in the disorder term of
the Hamiltonian (1). The fact that both spin lengths show such a similar
behavior is good news for semiclassical approaches to many-body localization
in spin chains Acevedo17 ; Craps20 .

Figure 2: Top panels: The expectation value $\langle r\rangle_{p}$ as a
function of disorder strength in Heisenberg chains of spin length $I=1/2$
(left) and $I=1$ (right) for various system sizes and pertaining numbers of
disorder realizations. The error bars are determined by Eqs. (24), (25).
Inset: The standard deviation $\Delta_{p}r$ as a function of disorder
strength. The horizontal lines indicate the expected values for GOE and
Poissonian statistics as given in Eqs. (5)-(11).
Bottom panels: The standard deviation $\Delta_{s}r$ according to Eq. (25) for
the same parameters as in the top panels. For a more detailed view on the data
at large system sizes see also Fig. 4. The insets show the maximum of the
standard deviation as a function of system size where the error bars follow
Eqs. (35), (38).

In Fig. 2 we show the expectation value $\langle r\rangle_{p}=\langle
r\rangle_{s}$ as a function of disorder strength for Heisenberg chains of
different sizes along with the standard deviation $\Delta_{p}r$ (top panels).
The data shows a transition between the ergodic phase at small $h$
characterized by Eqs. (9), (11) to the values (5), (7) of the many-body
localized phase. The bottom panels display the sample-to-sample standard
deviation $\Delta_{s}r$ according to Eqs. (25), (36). As seen from the
figures, $\Delta_{s}r$ amounts only to about ten percent of $\Delta_{p}r$,
which demonstrates via Eq. (27) that $(\Delta_{s}r)^{2}$ is only a tiny
contribution to the variance $(\Delta_{p}r)^{2}$. On the other hand,
$\Delta_{s}r$ shows a pronounced maximum $(\Delta_{s}r)_{\rm max}$ which grows
rapidly with system size, as displayed in the insets of the lower panels.
Moreover, in close vicinity to the corresponding position $h=h_{\rm max}$,
$\langle r\rangle_{s}$. has an inflection point at $h=h_{\rm inf}$. In Fig. 3
we have plotted both disorder strengths for $I\in\\{1/2,1\\}$ as functions of
systems size $K$, which shows that both quantities seem to converge to a
common value for large $K$. Thus, the expectation value $\langle r\rangle_{s}$
and the standard deviation $\Delta_{s}r$ show as a function of disorder
strength typical features of a phase transition with the former quantity
playing the role of an order parameter.

Figure 3: Finite-size transition data for the Heisenberg chain of spin length
$I=1/2$ (left) and $I=1$ (right). The panels show as a function of system size
$K$ the position $h=h_{\rm max}$ of $(\Delta_{s}r)_{\rm max}$, the position
$h=h_{\rm inf}$ of the inflection point of $\langle r\rangle_{s}$, and the
crossing point $h=h_{\rm cr}(K)$ of two curves of the latter quantity with
consecutive systems sizes $K$ and $K+1$. The dashed lines are linear fits to
$h_{\rm max}$ and $h_{\rm inf}$, and the shaded regions estimate the
transition where $h_{\rm max}\approx h_{\rm inf}$

The crossing points of the data shown in the top panels of Fig. 2 are also
often considered as indications for a phase transition. Therefore, following
Refs. Pal10 ; Tikhonov16 , we also plot in Fig. 3 the positions $h=h_{\rm
cr}(K)$ where two curves of $\langle r\rangle_{s}$ with consecutive system
sizes $K$ and $K+1$ cross. This data set clearly deviates from $h_{\rm max}$,
$h_{\rm inf}$ and grows to larger disorder strengths, an observation known as
the “drifting of the critical disorder strength” with system site Pal10 ;
Enss17 . It is an interesting speculation whether $h_{\rm max}\approx h_{\rm
inf}$ and $h_{\rm cr}$ correspond, for large systems, to two distinct
transitions occurring in the same systems.

For the finite-size data depicted in Fig. 3 we estimate the transition point
to $h_{\rm max}\approx h_{\rm inf}\approx 2.6J\dots 3.0J$ for spin length
$I=1/2$, and $h_{\rm max}\approx h_{\rm inf}\approx 4.0J\dots 4.5J$ for $I=1$.
These values for the critical disorder strength for the transition from the
ergodic to the many-body lpcalized phase are somewhat smaller than the ones
reported in other works Pal10 ; Luitz15 ; Devakul15 ; Doggen18 ; Chanda20a ,
which favor, for $I=1/2$, values of $h/J\approx 4$ or larger. However, some of
these works Pal10 ; Luitz15 ; Doggen18  concentrate on chains with an even
number of spins and the subspace with total spin $J^{z}=0$, whereas here we
also take into account odd numbers of spins and all subspaces except for those
with $|J^{z}|\in\\{IK,IK-1\\}$. Moreover, we introduce a new and different
criterion to locate the transition given by the position of
$(\Delta_{s}r)_{\rm max}$.

Figure 4: The sample-to-sample standard deviation $\Delta_{s}r$ along with the
square root of “variance of the variance” (38) as a function of disorder
strength $h$ for Heisenberg chains of spin length $I=1/2$ (top panels) and
$I=1$ (bottom panels). The data sets are remarkably close to each other, in
particular at small disorder strength $h$

Fig. 4 shows the standard deviation $\Delta_{s}r$ along with the square root
of “variance of the variance” (38) as a function of disorder strength for both
spin lengths $I=1/2$ and $I=1$. Remarkably, both quantities are very close to
each other, especially at small disorder strength $h$. This observation should
be taken as an indication that the underlying probability distribution $s(x)$
is rather narrow since both quantities become strictly equal,
$\Delta_{s}r=\Delta_{s}r^{2}=0$, for a $\delta$-type distribution. This
conjecture is confirmed by the data of Fig. 5 which displays the probability
distribution (17) for the realization-specific average $\langle
r\rangle_{\alpha}$ with the disorder strengths being the same as in Fig. 1.
The probability distribution is much narrower than the distribution $p(r)$ and
broadens significantly in the transition region. The latter result is similar
to an obsevation by Pal and Huse Pal10  who found near the transition a
maximum in the width of the probability distribution of a long-ranged spin
correlator.

Figure 5: The probability distribution (17) for the realization-specific
average $\langle r\rangle_{\alpha}$ for disordered Heisenberg chains of spin
length $I=1/2$ (left) and $I=1$ (right). The disorder strengths $h$ in both
panels are the same as in Fig. 1.

We also note that a closer analysis of the data of the lower panels of Fig. 4
suggests that $\Delta_{s}r$ extrapolates to zero for $K\to\infty$ deep in the
ergodic phase ($h/J\approx 1$) as well as deep in the many-body localized
phase ($h/J\gtrsim 6$). This would mean that the probability distribution (17)
would develop into a $\delta$-function in this limit and the above range of
disorder strengths, which is consistent with the data of Fig. 5. Note that the
statement $\lim_{K\to\infty}\Delta_{s}r=0$ implies, according to Eq. (27),
that the variance $(\Delta_{p}r)^{2})$ is entirely given by the averaged
variances within the individual disorder realizations.

###  III.2 Central Spin Model

For $J=0$ the central spin model resulting from the Hamiltonian (1) is also
integrable via an appropriate Bethe ansatz and known as the Gaudin model
Gaudin76 ; Erbe10 . Compared to the Heisenberg chain, the coupling to the
central spin provides an alternative mechanism of introducing interaction
among the bath spins, which are subject to a random magnetic field.

Fig. 6 shows data analogous to Fig. 1 now for central spin models of spin
length $S=I=1/2$ and $S=I=1$. For small disorder, the system clearly deviates
from the Poisson-type distribution (3) and shows level repulsion. However,
differently from the case of the Heisenberg chain, the level statistics do not
fully reach the Gaussian orthogonal ensemble but change back before to an
integrable or many-body localized phase.

Figure 6: The probability distribution (15) for the consecutive gap ration $r$
of a central spin model ($J=0$) of $N=K+1=11$ spins of length $S=I=1/2$ (left)
and of $N=K+1=10$ spin of length $S=I=1$ (right) for different disorder
strength $h$. Similarly as for the Heisenberg chain, the data follows for
small but finite disorder approximately the GOE distribution (4) (red), while
with increasing $h$ a transition to the Poisson-typed distribution (3) (green)
sets in.

Fig. 7 displays data for the central spin model analogous to Fig. 2 for the
Heisenberg chain. Comparing the top panels of both figures suggests that a
transition from the (approximately) ergodic to the many-body localized phase
at the inflection point $h=h_{\rm inf}\lesssim 1$, which is consistent with
the findings of Ref. Hetterich18  for $A,h\gg J$. Here the fact that the
transition occurs at about the same disorder strength for different system
sizes depends on the scaling factor $1/K$ in front of the second term in the
Hamiltonian (1).

Figure 7: Top panels: The expectation value $\langle r\rangle_{p}$ as a
function of disorder strength in central spin modelss of spin length $S=I=1/2$
(left) and $S=I=1$ (right) for various system sizes and pertaining numbers of
disorder realizations. The error bars are determined by Eqs. (24), (25).
Inset: The standard deviation $\Delta_{p}r$ as a function of disorder
strength. The horizontal lines indicate the expected values for GOE and
Poissonian statistics as given in Eqs. (5)-(11).
Bottom panels: The standard deviation $\Delta_{s}r$ according to Eq. (25) for
the same parameters as in the top panels. For a more detailed view on the data
at large system sizes see also Fig. 8. The insets show the maximum of the
standard deviation as a function of system size.

Also the sample-to-sample standard deviation $\Delta_{s}r$ plotted in the
bottom panels of Fig. 7 behaves similarly as for the Heisenberg chain: For
large enough systems sizes this quantity developes a maximum
$(\Delta_{s}r)_{\rm max}$ near $h=h_{\rm inf}$ whose value increases
monotonously with system size, as shown in the insets. Thus, we have
qualitatively the same situation as for the Heisenberg chain.

Figure 8: The sample-to-sample standard deviation $\Delta_{s}r$ along with the
square root of “variance of the variance” (38) as a function of disorder
strength $h$ for central spin models of spin length $S=I=1/2$ (top panels) and
$S=I=1$ (bottom panels).

Moreover, as seen in Fig. 8, the square root of the “variance of the variance”
(38) follows, similarly as for the Heisenberg chain, closely the standard
deviation $\Delta_{s}r$ as a function of disorder strength for both spin
lengths $I=1/2$ and $I=1$.

Figure 9: The probability distribution (17) for the realization-specific
average $\langle r\rangle_{\alpha}$ for disordered cenral spin systems of spin
length $S=I=1/2$ (left) and $S=I=1$ (right). The disorder strengths $h$ in
both panels are the same as in Fig. 6.

This is consistent with the probability distribution (17) for the realization-
specific average $\langle r\rangle_{\alpha}$ shown in Fig. 9. As already seen
in the case of the Heisenberg chain, the probability distribution is much
narrower than the distribution $p(r)$ and becomes significantly broader in the
transition region.

##  IV Summary and Outlook

We have compared the transitions between ergodic and many-body localized
phases in disodered Heisenberg chains as well as central spin models composed
of spins of length $1/2$ and $1$. A useful new tool we introduce is the
sample-to-sample standard deviation $\Delta_{s}r$ of the expectation value
$\langle r\rangle_{\alpha}$ of the consecutive-gap ratio in an individual
disorder realization (sample) $\alpha$. This quantity assumes, for both types
of systems and spin lengths, a maximum as a function of disorder strength,
accompanied by an inflection point of $\langle r\rangle$. These are typical
features of a phase transition where the latter quantity play the role of an
order parameter. The critical disorder strength deduced from these
observations turn out to be smaller than those reported in the recent
literature.

Further information about the transitions is contained in the probability
distribution of the expectation values within a given disorder realization. We
expect the study of this probability distribution and its moments to be a
useful tool in the investgation of phenomena related to many-body localization
also in other systems.

###### Acknowledgements.

We thank F. Evers and F. Göhmann for useful discussions, and M. Trivelato for
collaboration on an earlier stage of this project. J.S. acknowledges support
by FAPESP and the hospitaliy of the University of Sao Paulo at Sao Carlos and
of IIP Natal. J.C.E. acknowledges support from the Sao Paulo Research
Foundation (FAPESP) Grants No. 2016/08468-0, No. 2018/19017-4, No.
2020/00841-9, and and from Conselho Nacional de Pesquisas (CNPq), Grant No.
306122/2018-9.

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# Genuinely Multipartite Entanglement vias Quantum Communication

Ming-Xing Luo<EMAIL_ADDRESS>School of Information Science and
Technology, Southwest Jiaotong University, Chengdu 610031, P.R. China  Shao-
Ming Fei  School of Mathematical Sciences, Capital Normal University, Beijing
100048, P.R. China  Max-Planck-Institute for Mathematics in the Sciences,
04103 Leipzig, Germany

###### Abstract

Multipartite entanglement is of important resources for quantum communication
and quantum computation. Our goal in this paper is to characterize general
multipartite entanglement according to quantum biseparable channels. We
firstly prove a no-go result for fully genuinely multipartite entanglement on
finite-dimensional spaces with the biseparable channel model. We further
propose a semi-device-independent entanglement model depending on the
connection ability in quantum circuit. This implies a complete hierarchy of
genuinely multipartite entanglements. It also provides a completely different
multipartite nonlocality from quantum network entanglements. These results
show new insights for multipartite entanglement, quantum network, and
measurement-based quantum computation.

_Introduction._ -Entanglement is an important quantum property of two or more
systems in quantum mechanics associated with Schrödinger evolution equations
Schr . There is only one bipartite entanglement model on finite-dimensional
spaces for characterizing nonlocal correlations, that is, it cannot be
decomposed into an ensemble of separable states. This allows to witness any
bipartite entanglement beyond all separable states using entanglement witness
HHH . Another device-independent method is inspired by Einstein-Podolsky-Rosen
(EPR) steering EPR  or Bell inequality Bell ; CHSH  which can witness stronger
quantum nonlocality of entanglement Wise .

Although the fully separable model is easily extended for multi-particle
systems, it cannot justify the genuinely multipartite nonlocality. Instead,
different models are constructed for special multipartite goals. One is the
so-called biseparable model Sv  which can verify the genuinely multipartite
entanglement beyond biseparable states. This allows to witness stronger
nonlocality than its from a fully separable model. If the local tensor
decomposition is considered, a high-dimensional model Kraft18 ; Des ; EKZ  or
quantum network entanglement rules out any network separable state consisting
of small entanglements which are shared by partial parties NWR ; Kraft ;
Luo2021 . Different from the biseparable model Sv  the new model provides a
device-independent verification of unknown entanglement devices. Another is
from the particle-losing model for featuring the entanglement robustness of
local systems QABZ ; BPB ; Luo2022 . It can follow a different hierarchy of
well-known entanglements including GHZ state and Dicke states beyond other
models Sv ; NWR ; Kraft ; Luo2021 . All of these entanglement models can only
justify special multipartite systems. A natural problem is to explore new
model for general many-body systems.

Figure 1: Schematic biseparable quantum channel model. (a) A bipartite
separable system. $\mathcal{E}(\cdot)$ is a bipartite separable channel
defined by $\mathcal{E}(\cdot)=\sum_{i}K_{i}\otimes
S_{i}\left(\cdot\right)K_{i}^{\dagger}\otimes S_{i}^{\dagger}$, where Kraus
operators $K_{i}$ and $S_{i}$ satisfying
$\sum_{i}K_{i}^{\dagger}{}K_{i}\otimes S_{i}^{\dagger}S_{i}=\openone$ with the
identity operator $\openone$. (b) A tripartite biseparable system. Each
$\mathcal{E}_{j}(\cdot)$ is a biseparable quantum channel over one bipartition
of three particles. (c) A tripartite entanglement in adversarial scenarios.
Here, two parties who share two particles $B$ and $C$ may perform local joint
operations defined by a biseparable quantum channel
$\mathcal{E}_{A|BC}(\cdot)$ over the bipartition $A$ and $\\{B,C\\}$.

In general, a bipartite entanglement can be generated by quantum communication
model, as shown in Fig.1(a). Each separable state can be obtained by passing a
product state through quantum separable channel Werner . This suggests a novel
model for multipartite entanglement by ruling out any outputs of fully
separable quantum channels. A natural extension is to explore biseparable
quantum channel Sv  in multipartite scenarios, as shown in Fig.1(b). It can be
further regarded as an adversarial model in cryptographic applications of
quantum secret sharing HB , as shown in Fig.1(c). This intrigues a new way for
exploring many-body systems.

Our goal in this work is to characterize multipartite entanglement in terms of
biseparable quantum channels. We first propose a bipartite-generation model to
characterize all the states generated by passing proper biseparable state
through biseparable channel. This implies the first no-go result for fully
genuinely multipartite entanglement. It further implies a complete hierarchy
for characterizing any multipartite entanglement according to the connection
ability in generation circuits. The present model provides a simple standard
to verify general multipartite entanglement using Schmidt numbers of reduced
density matrices. This shows a complete different multipartite nonlocality
from previous models Luo2022 ; NWR ; Kraft ; Luo2021 .

## Main Results

_Fully genuinely multipartite entanglement._ A general isolated
$d$-dimensional quantum system is represented by a normalized vector
$|\phi\rangle$ in Hilbert space $\mathcal{H}_{d}$. Instead, an open system is
described by probabilistically mixing an ensemble of pure states
$\\{|\phi_{i}\rangle\\}$, that is,
$\rho=\sum_{i}p_{i}|\phi_{i}\rangle\langle\phi_{i}|$, where $\\{p_{i}\\}$ is a
probability distribution. An $n$-particle pure state $|\Phi\rangle$ on Hilbert
space $\otimes_{i=1}^{n}\mathcal{H}_{A_{i}}$ is biseparable Sv  if
$|\Phi\rangle=|\phi\rangle_{I}|\psi\rangle_{\overline{I}}$ with pure states
$|\phi\rangle$ and $|\psi\rangle$, where $I$ and $\overline{I}$ are
bipartition of $\\{A_{1},\cdots,A_{n}\\}$. Here, $|\Phi\rangle$ can be
generated from a fully separable state $|0\rangle^{\otimes n}$ passing through
a biseparable quantum channel $\mathcal{E}(\cdot):=U_{I}\otimes
V_{\overline{I}}(\cdot)U_{I}^{\dagger}\otimes V_{\overline{I}}^{\dagger}$,
that is, $|\Phi\rangle=(U_{I}\otimes V_{\overline{I}})|0\rangle^{\otimes n}$,
where $U_{I}(\otimes_{A_{i}\in I}|0\rangle_{A_{i}})=|\phi\rangle_{I}$ and
$V_{\overline{I}}(\otimes_{A_{j}\in\overline{I}}|0\rangle_{A_{j}})=|\psi\rangle_{\overline{I}}$.
This intrigues a multipartite entanglement model. Define a biseparable
complete positive trace-preserving (BCPTP) channel Werner ; Sv  on Hilbert
space $\mathcal{H}_{I}\otimes\mathcal{H}_{\overline{I}}$ as

$\displaystyle\mathcal{E}_{I}(\rho)=\sum_{i}(K_{i}\otimes
S_{i})\rho(K_{i}^{\dagger}\otimes S_{i}^{\dagger})$ (1)

where $K_{i},S_{i}$ are respective Kraus operators on Hilbert space
$\mathcal{H}_{I}$ and $\mathcal{H}_{\overline{I}}$ and satisfy
$\sum_{i}K_{i}^{\dagger}K_{i}\otimes S_{i}^{\dagger}S_{i}=\openone$ with the
identity operator $\openone$, $\mathcal{H}_{I}=\otimes_{A_{i}\in
I}\mathcal{H}_{A_{i}}$, and
$\mathcal{H}_{\overline{I}}=\otimes_{A_{j}\in\overline{I}}\mathcal{H}_{A_{j}}$.
It is forward to show any biseparable state Sv
$\rho_{bs}=\sum_{i}p_{i}\rho_{i|I}\otimes\varrho_{i|\overline{I}}$ over a
given bipartition $I$ and $\overline{I}$ can be represented by using BCPTP
channel as $\rho_{bs}=\mathcal{E}_{I}(|0\rangle\langle 0|^{\otimes n})$, where
$K_{i}$ is defined by $|0\rangle\mapsto\rho_{i|I}$, and $S_{i}$ is defined by
$|0\rangle\mapsto\varrho_{i|\overline{I}}$. This means that any biseparable
state Sv  can be generated by a mixture of BCPTP channels, that is,

$\displaystyle\rho_{bs}=\sum_{I}q_{I}\mathcal{E}_{I}(|0\rangle\langle
0|^{\otimes n})$ (2)

where the summation is over any proper subset of $\\{A_{1},\cdots,A_{n}\\}$.
Thus BCPTP channel provides an equivalent representation of the biseparable
model Sv . Instead of fully separable states, in what follows we consider any
biseparable state as the input of BCPTP channel.

Definition 1. An $n$-partite state is fully genuinely-generation entanglement
if it is not bipartite-generation state given by

$\displaystyle\rho_{bg}=\sum_{I}p_{I}\mathcal{E}_{I}(\rho_{bs})$ (3)

where $I\subset\\{A_{1},\cdots,A_{n}\\}$, $\mathcal{E}_{I}$ are BCPTP channels
in terms of the bipartition $I$ and $\overline{I}$, $\\{p_{I}\\}$ is a
probability distribution, and $\rho_{bs}$ is any biseparable state.

In Definition 1, $\rho_{bs}$ may be entanglement for different bipartitions
$I$ and $\overline{I}$. This is different from the standard biseparable model
input a fully separable state in Eq.(2). For any genuinely $n$-partite
entanglement Sv , assume that the Schmidt decomposition respect to the
bipartition $I=\\{A_{1}\\}$ and $\overline{I}=\\{A_{2},\cdots,A_{n}\\}$ is
given by

$\displaystyle|\Phi\rangle=\sum_{i=1}^{d}\sqrt{\lambda_{i}}|\phi_{i}\rangle_{I}|\psi_{i}\rangle_{\overline{I}},$
(4)

where $\lambda_{i}$ are Schmidt coefficients satisfying
$\sum_{i=1}^{d}\lambda_{i}=1$, $\\{|\phi_{i}\rangle\\}$ are orthogonal states
of the particle $A_{1}$, and $\\{|\psi_{i}\rangle\\}$ are orthogonal states of
the particles in $\overline{I}$. There is a unitary transformation $U$ on
Hilbert space $\mathcal{H}_{\overline{I}}$ satisfying (Appendix A):

$\displaystyle
U:|\psi_{i}\rangle_{\overline{I}}\mapsto|0\rangle_{A_{2}}|\hat{\psi}_{i}\rangle_{J},i=1,\cdots,d$
(5)

where $\\{|\hat{\psi}_{i}\rangle\\}$ are orthogonal states of the particles in
$J:=\\{A_{3},\cdots,A_{n}\\}$. Thus $|\Phi\rangle$ is a bipartite-generation
state from Definition 1. This implies a general no-go result for any finite-
dimensional state.

Result 1. There is no fully genuinely-generation multipartite entanglement on
any finite-dimensional Hilbert space.

Figure 2: Schematic quantum circuit for generating a general state. (a)
Standard quantum circuit model with different depths. Here, $U_{i}$ are two-
particle universal logical gates Luo2014a ; Luo2014 . (b) The present circuit
model with two-depth. There are two BCPTP channels
$\sum_{I}p_{I}\mathcal{E}_{I}(\cdot)$ and
$\sum_{J}q_{J}\mathcal{E}_{J}(\cdot)$ over different bipartitions
$I,\overline{I}$ and $J,\overline{J}$.

Result 1 holds for any pure or mixed state on finite dimensional Hilbert
space. It means any state can be generated by passing a biseparable state
through biseparable channels. Combing with Eqs.(1) and (2), Result 1 implies
any state $\rho$ on Hilbert space $\otimes_{i=1}^{n}\mathcal{H}_{A_{i}}$ has
the following decomposition

$\displaystyle\rho=\sum_{I,J}p_{I}q_{J}\mathcal{E}_{J}\circ\mathcal{E}_{I}(|0\rangle\langle
0|^{\otimes n})$ (6)

where $\\{\mathcal{E}_{I},\forall I\\}$ and $\\{\mathcal{E}_{J},\forall J\\}$
are BCPTP channels, and $\\{p_{I}\\}$ and $\\{q_{J}\\}$ are probability
distributions. Eq.(6) implies a universal circuit with two-depth for building
any state from a fully product state, as shown in Fig.2. The present model is
stronger than the standard quantum circuit model with unfixed depth Luo2014a ;
Luo2014 . The present biseparable quantum channels may activate entanglement
from biseparable states. This kind of entanglement swapping is the core of
quantum networks BBC ; Kimble .

_Partial-connection multipartite entanglement._ The fully genuinely-generation
multipartite entanglement requires ruling out all biseparable states and
biseparable quantum channels. This may be too strong both in theory and
applications. Instead of partial joint operations in Definition 1, we consider
a one-side biseparable channel $\mathcal{E}_{I}$, where joint operations may
be performed on $I$ while fully local operations are on $\overline{I}$. This
can be regarded as a semi-device-independent scenario as shown in Fig.1(c),
where partial adversaries in $I$ may perform joint operations while others are
not allowed. Denote $\ell_{I}$ as the number of particles in $I$. Define a
$k$-connection BCPTP channel $\mathcal{E}_{I}^{(k)}$ on Hilbert space
$\mathcal{H}_{I}\otimes\mathcal{H}_{\overline{I}}$ as

$\displaystyle\mathcal{E}_{I}^{(k)}(\rho)=\sum_{i}(K_{i}\otimes(\otimes_{j\in\overline{I}}S_{i;j}))\rho(K_{i}^{\dagger}\otimes(\otimes_{j\in\overline{I}}S_{i;j}^{\dagger}))$
(7)

where $K_{i}$ and $S_{i;j}$ are respective Kraus operators on Hilbert space
$\mathcal{H}_{I}$ and $\mathcal{H}_{A_{j}}$, and satisfy
$\sum_{i}(K_{i}^{\dagger}K_{i}\otimes(\otimes_{j\in\overline{I}}S_{i;j}^{\dagger}S_{i;j}))=\openone$.
The present $k$-connection BCPTP channel is of state-dependent. Our goal here
is for featuring genuinely multipartite entanglement with the present quantum
channel (7).

Definition 2. An $n$-partite state is $k$-connection genuinely entanglement
($k$-CGE) if it is not a $k$-connection biseparable state given by

$\displaystyle\rho_{bg}^{(k)}=\sum_{\ell_{I}\leq
k}p_{I}\mathcal{E}_{I}^{(\ell_{I})}(\rho_{bs})$ (8)

where $\mathcal{E}_{I}^{(\ell_{I})}$ are $\ell_{I}$-connection BCPTP channels
in terms of bipartition $I$ and $\overline{I}$, $\\{p_{I}\\}$ is a probability
distribution, and $\rho_{bs}$ is any biseparable state.

Figure 3: Schematic hierarchy of genuinely $n$-partite entanglement. The set
consisting of $k$-CGE is included in the set consisting of $k-1$-CGE. The
largest set is consisting of 1-CGE while the smallest set contains
$k_{\max}$-CGE with $k_{\max}=\lfloor\frac{n}{2}\rfloor$, where $\lfloor
x\rfloor$ denotes the maximal integer no more than $x$.

Similar to the proof of Theorem 1, the Schmidt decomposition of a given
$n$-particle pure state $|\Phi\rangle$ on $d^{n}$-dimensional Hilbert space
$\otimes_{i=1}^{n}\mathcal{H}_{A_{i}}$ is given by

$\displaystyle|\Phi\rangle=\sum_{i=1}^{N}\sqrt{\lambda_{i}}|\phi_{i}\rangle_{I}|\psi_{i}\rangle_{\overline{I}}$
(9)

where $\lambda_{i}$ are Schmidt coefficients satisfying
$\sum_{i=1}^{N}\lambda_{i}=1$, $\\{|\phi_{i}\rangle\\}$ are orthogonal states
of the particles in $I$, and $\\{|\psi_{i}\rangle\\}$ are orthogonal states of
the particles in $\overline{I}$. For any integer $\ell_{I}$ with $\ell_{I}\geq
n/2+1$, it follows $N\leq d^{n-\ell_{I}}<d^{n/2-1}$. There is a unitary
transformation $U$ on Hilbert space $\mathcal{H}_{I}$ satisfying

$\displaystyle
U:|\psi_{i}\rangle_{I}\mapsto|0\rangle_{A_{j}}|\hat{\psi}_{i}\rangle_{J},i=1,\cdots,N$
(10)

where $A_{j}\in\overline{I}$ and $\\{|\hat{\psi}_{i}\rangle\\}$ are orthogonal
states of the particles in $J:=\overline{I}-\\{A_{j}\\}$. So, the state
$|\Phi\rangle$ is an $\ell_{I}$-connection biseparable. This implies another
no-go result as follows.

Result 2. There is no $k$-CGE on finite-dimensional space for any integer
$k\geq n/2+1$.

Result 2 rules out the possibility of $k$-CGE for large $k$. The situation is
different for small $k$. For special case of $n=2$, Definition 2 is reduced to
the standard separable model of bipartite systems Schr ; HHH . For each $n\geq
3$, from Definition 2 any biseparable state Sv  is $1$-connection biseparable
state. A further fact is that any $k$-connection biseparable state is
$s$-connection biseparable state for any $s\geq k$. This implies a complete
hierarchy for all the multipartite entanglements, as shown in Fig.3. The
largest set contains $1$-CGEs, that is, the genuinely multipartite
entanglement in the biseparable model Sv . Instead, the smallest set consists
of the strongest multipartite entanglement, that is, $k_{\max}$-CGE with
$k_{\max}=\lfloor\frac{n}{2}\rfloor$.

For any $n$-particle state $|\Phi\rangle$ on Hilbert space
$\otimes_{i=1}^{n}\mathcal{H}_{A_{i}}$, from Eq.(9) the orthogonality of
$\\{|\phi_{i}\rangle_{I}\\}$ allows for constructing a unitary transformation
(10) if and only if the Schmidt number satisfies $N\leq d^{k-1}$ with
$\ell_{I}=k$. This implies a directive way to verify a given pure $k$-CGE by
using the Schmidt number of reduced density matrices (Appendix B).

Result 3. An $n$-partite pure state on $d^{n}$ dimensional Hilbert space
$\otimes_{i=1}^{n}\mathcal{H}_{A_{i}}$ is $k$-CGE if and only if the Schmidt
number of the reduced density matrix of any $k$ particles is larger than
$d^{k-1}$.

From Definition 2 any genuinely multipartite entanglement in the biseparable
model Sv  is $1$-CGE. Moreover, the present $k$-CGE is stronger than robust
entanglement with the robustness-depth $k$ since the particle-losing channel
Luo2022  is local CPTP channel. From Result 3 it generally requires to
evaluate Schmidt numbers of almost all the reduced density matrices. This
yields to a NP hard problem for general entanglement because of exponential
number of reduced states. Instead, it is easy for some special states.

One example is $n$-partite Greenberger-Horne-Zeilinger (GHZ) states GHZ  given
by

$\displaystyle|GHZ\rangle=\sum_{i=1}^{d}a_{i}|i\cdots{}i\rangle_{A_{1}\cdots{}A_{n}}$
(11)

where $a_{i}$ satisfy $\sum_{i=1}^{d}a_{i}^{2}=1$. It easy to verify the state
(11) is $1$-CGE from its symmetry. Moreover, for $n$-partite W-type states Dur
:

$\displaystyle|W\rangle_{A_{1}\cdots{}A_{n}}=\sum_{i=1}^{d}a_{i}|1_{i}\rangle+a_{n+1}|1\cdots
1\rangle$ (12)

from Result 3 it is $2$-CGE for $a_{i}\not=0$, $i=1,\cdots,n+1$, where
$|1_{i}\rangle$ denotes the $i$-excitation defined by
$|1_{i}\rangle=|0\rangle^{\otimes i-1}|1\rangle|0\rangle^{\otimes n-i}$. This
can be extended for $n$-qudit Dicke state with $s$ excitations Dicke  given by

$\displaystyle|D_{s,n}\rangle=\frac{1}{\sqrt{L_{s}}}\sum_{i_{1}+\cdots{}+i_{n}=s}|i_{1}\cdots{}i_{n}\rangle_{A_{1}\cdots{}A_{n}}$
(13)

where $L_{s}$ is the normalization constant. It is $k$-CGE with
$k=\lfloor\log_{d}(s+1)\rfloor+1$ (Appendix C). Moreover, $|D_{s,n}\rangle$ is
equivalent to $|D_{(d-1)^{n}-s,n}\rangle$ under local unitary operations. This
implies the strongest nonlocality of Dicke state with
$s=\lfloor(d-1)^{n}/2\rfloor$.

Another are entangled quantum networks which show different nonlocalities
beyond single entanglement RBBB ; RPM . One is resource states for quantum
computation cluster . The so-called cluster states cluster  consist of any
generalized Einstein-Podolsky-Rosen (EPR) states EPR  is 1-CGE (Appendix D).
Similar result holds for graph states graph  consisting of generalized EPR EPR
and GHZ states GHZ . While some quantum networks show different generation
abilities. One example is $n$-partite complete-connected network
$\mathcal{N}_{c}$ where each pair shares one bipartite entanglement
$|\phi_{ij}\rangle$. Recent result shows the joint state of any $k$-partite
subnetwork is entangled for $k\geq 2$ Luo2022 ; Sv . Hence, from Results 2 and
3 the joint state of $\mathcal{N}_{c}$ is $k_{\max}$-CGE with
$k_{\max}=\lfloor\frac{n}{2}\rfloor$. This means $\mathcal{N}_{c}$ with any
bipartite entanglement shows stronger nonlocality than GHZ state (11) and W
state (12) for any $n>4$ in the present model. This is different from the
robust-entanglement model Luo2022  where both the W state and
$\mathcal{N}_{c}$ has the same robust-depth. Remarkably, it is converse to the
recent result NWR ; Kraft ; Luo2021  which shows both GHZ and W states are
stronger. This shows a surprising feature of the genuinely multipartite
nonlocality beyond bipartite scenarios, that is, it is model-dependent. For
general quantum network we provide a polynomial-time algorithm for estimating
the upper bound of $k$ (Appendix E).

Robustness of $k$-CGE. From Eq.(8) all the $k$-connection biseparable states
consist of a convex set $\mathcal{S}_{k}$. This allows for verifying a general
entanglement near to $k$-CGE $|\Phi\rangle$ by using linear entanglement
witness HHH96 ; HHH  defined as

$\displaystyle\mathtt{W}_{|\Phi\rangle}=r\openone-|\Phi\rangle\langle\Phi|$
(14)

where
$r=\max_{\rho_{cb}\in\mathcal{S}_{k}}D(\rho_{cb},|\Phi\rangle\langle\Phi|)$
for any distance function $D(\cdot,\cdot)$ of states HHH , and $\openone$ is
the identity operator. From Result 3 the entanglement witness for genuinely
multipartite entanglement HHH ; Sv  is useful for verifying any $1$-CGE. One
example is the GHZ state (11) with $r=\max\\{a_{1}^{2},\cdots,a_{d}^{2}\\}$.

Figure 4: Visibilities of noisy 4-particle W state. Here we consider a special
case of $a_{1}=a_{2}=a_{3}=a_{4}=\frac{\cos\theta}{2}$ with
$\theta\in(0,\frac{\pi}{2})$. In this case, the visibility for $2$-CGE is
given by $r=\max\\{\cos^{2}\theta,1-\frac{\cos^{2}\theta}{2}\\}$. Instead, the
visibility for $1$-CGE HHH ; Sv  is given by
$r_{0}=\max\\{\sin^{2}\theta,\frac{\cos^{2}\theta}{2}\\}$.

For the W-type state (12) with $n=4$, it is easy to get
$r=\max\\{1-a_{5}^{2},1-a_{i}^{2}-a_{j}^{2},1\leq i<j\leq 4\\}$. This implies
a different entanglement witness for verifying 2-CGE beyond the genuinely
multipartite entanglement (or 1-CGE) HHH ; Sv  with
$r=\max\\{a_{5}^{2},a_{i}^{2}+a_{j}^{2},1\leq i<j\leq 4\\}$. Both visibilities
are $v>\frac{16r+1}{17}$ for Werner state Werner  $\rho_{v}=v|W\rangle\langle
W|+\frac{1-v}{16}\openone$, as shown in Fig.4.

_Conclusions._ –Result 3 provides an efficient way for verifying special
multipartite entanglements. This intrigues a natural problem to explore new
way for general multipartite entanglements. While the entanglement is the
weakest nonlocality of quantum states, one may explore new hierarchies in
terms of the so-called multipartite steering HR ; JSU  or Bell nonlocality
Bell ; GHZ . This is of special importance for recovering novel multipartite
nonlocality beyond bipartite scenarios. Additionally, the present model shows
the first example which shows converse multipartite nonlocality to recent
quantum network model NWR ; Kraft ; Luo2021 . This intrigues a basic problem
to explore the intrinsic nonlocality or the most reasonable model for
multipartite systems.

In conclusion, we have investigated genuinely-generation multipartite
entanglement with the help of biseparable quantum channel. We proposed a
bipartite-generation model to characterize all the multipartite state
generated by using proper biseparable state. This has implied a no-go result
for the strongest multipartite entanglement in the present model. We further
defined a biseparable model with local-connection. The one-side local joint
operation has provided a general standard for characterizing the connecting
ability of multipartite entanglement. We obtained a simple hierarchy of
multipartite entanglement in terms of the connection ability. The new
entanglement witness is used to verify the entanglement robustness. These
results should be interesting in multipartite entanglement theory, quantum
communication, and quantum computation.

Acknowledgements

We thanks for Chen Jing-Ling and Bu Kaifeng. This work was supported by the
National Natural Science Foundation of China (Grants Nos. 62172341, 61772437,
and 12075159), Beijing Natural Science Foundation (Grant No.Z190005), Academy
for Multidisciplinary Studies, Capital Normal University, Shenzhen Institute
for Quantum Science and Engineering, Southern University of Science and
Technology (Grant Nos. SIQSE202001 and SIQSE202105), and the Academician
Innovation Platform of Hainan Province.

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## Appendix A. Proof of Eq.(5)

The Schmidt decomposition of any $n$-partite pure state $|\Phi\rangle$ on
Hilbert space $\otimes_{i=1}^{n}\mathcal{H}_{A_{i}}$ is given by

$\displaystyle|\Phi\rangle=\sum_{i=1}^{d}\sqrt{\lambda_{i}}|\phi_{i}\rangle_{A_{1}}|\psi_{i}\rangle_{A_{2}\cdots{}A_{n}}$
(15)

where $\lambda_{i}$ are Schmidt coefficients satisfying
$\sum_{i=1}^{d}\lambda_{i}=1$, $\\{|\phi_{i}\rangle\\}$ are orthogonal states
of the particle $A_{1}$, and $\\{|\psi_{i}\rangle\\}$ are orthogonal states of
the particles $A_{2},\cdots,A_{n}$. For a given set of orthogonal states
$\\{|\psi_{i}\rangle,i=1,\cdots,d\\}$, there exists an orthogonal basis
$\\{|\psi_{j}\rangle,j=1,\cdots,d\\}$ of Hilbert space
$\otimes_{i=2}^{n}\mathcal{H}_{A_{i}}$. Similarly, for a given set of
orthogonal states $\\{|\hat{\psi}_{i}\rangle\\}$ of the particles
$A_{3},\cdots,A_{n}$, there exists an orthogonal basis
$\\{|\hat{\phi}_{j}\rangle\\}$ of Hilbert space
$\otimes_{i=2}^{n}\mathcal{H}_{A_{i}}$ such that
$|\hat{\phi}_{i}=|0\rangle_{A_{2}}|\hat{\psi}_{i}\rangle_{A_{3}\cdots{}A_{n}}$
for $i=1,\cdots,d$. This can be obtained by extending the set
$\\{|0\rangle_{A_{2}}|\hat{\psi}_{i}\rangle_{A_{3}\cdots{}A_{n}}\\}$ into an
orthogonal basis on Hilbert space $\otimes_{i=2}^{n}\mathcal{H}_{A_{i}}$. With
this basis, define the following mapping on Hilbert space
$\otimes_{i=2}^{n}\mathcal{H}_{A_{i}}$ as

$\displaystyle U:$
$\displaystyle|\psi_{i}\rangle_{A_{2}\cdots{}A_{n}}\mapsto|0\rangle_{A_{2}}|\hat{\psi}_{i}\rangle_{A_{3}\cdots{}A_{n}},i=1,\cdots,d$
(16)
$\displaystyle|\psi_{j}\rangle_{A_{2}\cdots{}A_{n}}\mapsto|\hat{\phi}_{j}\rangle_{A_{2}\cdots{}A_{n}},i=d+1,\cdots,d^{n-1}$

Note that $\\{|\psi_{i}\rangle_{A_{2}\cdots{}A_{n}}\\}$ and
$\\{|\hat{\phi}_{j}\rangle_{A_{2}\cdots{}A_{n}}\\}$ are orthogonal bases of
Hilbert space $\otimes_{i=2}^{n}\mathcal{H}_{A_{i}}$. Hence, $U$ is a unitary
transformation on Hilbert space $\otimes_{i=2}^{n}\mathcal{H}_{A_{i}}$. This
has completed the proof of Eq.(5).

## Appendix B. Proof of Result 3

For a given $n$-partite pure state $|\Phi\rangle$ on Hilbert space
$\otimes_{i=1}^{n}\mathcal{H}_{A_{i}}$, assume that the Schmidt decomposition
is given by

$\displaystyle|\Phi\rangle=\sum_{i=1}^{N}\sqrt{\lambda_{i}}|\phi_{i}\rangle_{I}|\psi_{i}\rangle_{\overline{I}}$
(17)

for each bipartition $I$ and $\overline{I}$ of $\\{A_{1},\cdots,A_{n}\\}$,
where $\lambda_{i}$ are Schmidt coefficients satisfying
$\sum_{i}\lambda_{i}=1$, $\\{|\phi_{i}\rangle_{I}\\}$ are orthogonal states of
particles in $I$, and $\\{|\psi_{i}\rangle_{\overline{I}}\\}$ are orthogonal
states of particles in $\overline{I}$. Here, $N$ is the Schmidt number of the
reduced density matrix
$\rho_{I}=\sum_{i}\lambda_{i}^{2}|\phi_{i}\rangle_{I}\langle\phi_{i}|$. If
$N\leq d^{k-1}$ with $\ell_{I}=k$, from Result 2, there exists a unitary
transformation $U$ on $k$ particles $A_{i_{1}},\cdots,A_{i_{k}}$ such that
$U|\Phi\rangle=|\hat{\Phi}\rangle|0\rangle_{A_{i_{1}}}$ for some
$n-1$-particle state $|\hat{\Phi}\rangle$. This means that $|\Phi\rangle$ is
$k$-connection biseparable state, that is, it is not $k$-CGE.

Moreover, $N>d^{k-1}$ with $\ell_{I}=k$, we show that $|\Phi\rangle$ is
$k$-CGE. The proof is completed by contradiction. Assume that $|\Phi\rangle$
is not $k$-CGE, that is, $|\Phi\rangle$ is $k$-connection biseparable state.
Hence, there exists a unitary operation $U$ on $k$ particles
$A_{i_{1}},\cdots,A_{i_{k}}$ such that
$U|\Phi\rangle=|\hat{\Phi}\rangle|0\rangle_{A_{i_{j}}}$. Let
$I=\\{A_{i_{1}},\cdots,A_{i_{k}}\\}$. From Eq.(17), we have

$\displaystyle|\hat{\Phi}\rangle|0\rangle_{A_{i_{j}}}$ $\displaystyle=$
$\displaystyle\sum_{t=1}^{N}\sqrt{\lambda_{t}}U|\phi_{t}\rangle_{I}|\psi_{t}\rangle_{\overline{I}}$
(18) $\displaystyle=$
$\displaystyle\sum_{t=1}^{N}\sqrt{\lambda_{t}}|\hat{\phi}_{t}\rangle_{I-\\{A_{i_{j}}\\}}|0\rangle_{A_{i_{j}}}|\psi_{t}\rangle_{\overline{I}}$

for some orthogonal states $\\{|\hat{\phi}_{t}\rangle_{I-\\{A_{i_{j}}\\}}\\}$
on the particles in $I-\\{A_{i_{j}}\\}$. However, the dimension of Hilbert
space $\otimes_{s\in I-\\{A_{i_{j}}\\}}\mathcal{H}_{A_{s}}$ is $d^{k-1}$. So,
there are at most $d^{k-1}$ orthogonal states. This is contradicted to the
orthogonal state set
$\\{|\hat{\phi}_{t}\rangle_{I-\\{A_{i_{j}}\\}},j=1,\cdots,N\\}$ with
$N>d^{k-1}$. It means that $|\Phi\rangle$ is $k$-connection biseparable state.
This has completed the proof.

## Appendix C. Cluster states

In this section, we firstly prove any cluster state generated by any
generalized EPR states EPR  is 1-CGE. And then, it will be extended for any
graph state generated by any generalized EPR states EPR  and generalized GHZ
states GHZ .

Consider an $n$-partite cluster state $|C\rangle$ generated by generalized EPR
states EPR  $|\phi_{1}\rangle,\cdots,|\phi_{m}\rangle$, where
$|\phi_{i}\rangle=\cos\theta_{i}|00\rangle+\sin\theta_{i}|11\rangle$ with
$\theta_{i}\in(0,\frac{\pi}{2})$, $i=1,\cdots,m$. Assume $|C\rangle$ is shared
by $n$ parties $\textsf{A}_{1},\cdots,\textsf{A}_{n}$. Each party
$\textsf{A}_{i}$ can perform local controlled-phase operation
$C_{j}(\theta_{i_{j}})$ on the shared two qubits, where
$C_{j}(\theta_{i_{j}})$ is defined by

$\displaystyle C_{j}(\theta_{i_{j}})$ $\displaystyle=$
$\displaystyle|00\rangle\langle 00|+|01\rangle\langle 01|+|10\rangle\langle
10|$ (19) $\displaystyle+e^{i\theta_{i_{j}}}|11\rangle\langle 11|$

One party may perform different $C_{j}(\theta_{i_{j}})$’s on each pair of
shared qubits. Assume that each party $\textsf{A}_{i}$ shares $k_{i}$ EPR
states with others. Here, all the $k_{i}$ qubits shared by the party
$\textsf{A}_{i}$ is jointly combined into a particle on
$2^{k_{i}}$-dimensional Hilbert space $\mathcal{H}_{A_{i}}$. Denote $\rho_{i}$
as the reduced density matrix of the party $\textsf{A}_{i}$. It is easy to
obtain the Schmidt number of $\rho_{i}$ is $2^{k_{i}}$. Moreover, any local
controlled-phase operation (19) does not change its Schmidt number. This has
proved that $|C\rangle$ is $1$-connection state. Moreover, it is genuinely
multipartite entanglement Luo2021 ; Sv . So, $|C\rangle$ is $1$-CGE.

Now, consider an $n$-partite graph state $|G\rangle$ generated by generalized
EPR states EPR  $|\phi_{1}\rangle,\cdots,|\phi_{m}\rangle$ and generalized GHZ
states GHZ  $|\psi_{1}\rangle,\cdots,|\psi_{\ell}\rangle$, where
$|\psi_{i}\rangle$ are $k_{i}$-qubit GHZ states defined by
$|\psi_{i}\rangle=\cos\vartheta_{i}|0\rangle^{\otimes
k_{i}}+\sin\vartheta_{i}|1\rangle^{\otimes k_{i}}$ with
$\vartheta_{i}\in(0,\frac{\pi}{2})$. Assume each party $\textsf{A}_{i}$ shares
$s_{i}$ EPR states and $t_{i}$ GHZ states with others. Each party
$\textsf{A}_{i}$ can perform local joint controlled-phase operation
$CC_{j}(\theta_{i_{j}})$ on the shared $u$ qubits, where
$CC_{j}(\theta_{i_{j}})$ is defined by

$\displaystyle CC_{j}(\theta_{i_{j}})$ $\displaystyle=$
$\displaystyle\sum_{i_{1}\cdots i_{u}\not=1\cdots 1}|i_{1}\cdots
i_{u}\rangle\langle i_{1}\cdots i_{u}|$ (20)
$\displaystyle+e^{i\theta_{i_{j}}}(|1\rangle\langle 1|)^{\otimes u}$

All the $s_{i}+t_{i}$ qubits shared by the party $\textsf{A}_{i}$ is jointly
combined into a particle in $2^{s_{i}+t_{i}}$-dimensional Hilbert space
$\mathcal{H}_{A_{i}}$. Denote $\varrho_{i}$ as the reduced density matrix of
the party $\textsf{A}_{i}$. It is easy to obtain the Schmidt number of
$\varrho_{i}$ is $2^{s_{i}+t_{i}}$. Moreover, any local operation (20) does
not change the Schmidt number. This has proved that $|G\rangle$ is
$1$-connection state. Moreover, it is genuinely multipartite entanglement
Luo2021 ; Sv . Hence, $|G\rangle$ is $1$-CGE.

A further result for connection ability of general networks will be proved in
Appendix E.

## Appendix D. Connection ability of Dicke state

In this section, we prove any $n$-qudit Dicke state (14) is $k$-CGE with
$k=\lfloor\log_{d}(s+1)\rfloor+1$. Specially, for any given bipartition
$I=\\{A_{j_{1}},\cdots,A_{\ell_{I}}\\}$ and $\overline{I}$ with
$\ell_{I}=k+1$, the Schmidt decomposition of $|D_{s,n}\rangle$ is given by

$\displaystyle|D_{s,n}\rangle=\sum_{i=0}^{L_{I}}\sqrt{\gamma_{i}}|\phi_{i}\rangle_{I}|\psi_{i}\rangle_{\overline{I}}$
(21)

where $\\{|\phi_{i}\rangle_{I}\\}$ are orthogonal states of particles in $I$,
$\\{|\psi_{i}\rangle_{\overline{I}}\\}$ are orthogonal states of particles in
$\overline{I}$, $\gamma_{i}$ are Schmidt coefficients satisfying
$\sum_{i}\gamma_{i}=1$, and $L_{I}$ denotes the Schmidt number. In fact,
$|\phi_{i}\rangle_{I}$ are generalized $\ell_{I}$-qudit Dicke states defined
by

$\displaystyle|\phi_{i}\rangle_{I}=\sum_{j_{1}+\cdots{}+j_{\ell_{I}}=i}\alpha^{(i)}_{j_{1}\cdots{}j_{\ell_{I}}}|j_{1}\cdots{}j_{\ell_{I}}\rangle_{A_{j_{1}}\cdots{}A_{\ell_{I}}}$
(22)

where $\alpha^{(i)}_{j_{1}\cdots{}j_{\ell_{I}}}$ depend on proper coefficients
of $\gamma_{i}$’s and satisfy
$\sum_{j_{1}+\cdots{}+j_{\ell_{I}}=i}(\alpha^{(i)}_{j_{1}\cdots{}j_{\ell_{I}}})^{2}=1$.
Similarly, $|\psi_{i}\rangle_{\overline{I}}$ are generalized
$n-\ell_{I}$-qudit Dicke states defined by

$\displaystyle|\psi_{i}\rangle_{\overline{I}}=\sum_{\sum_{A_{r_{t}}\in\overline{I}}r_{t}=s-i}\beta^{(i)}_{r_{1}\cdots{}r_{n-\ell_{I}}}|r_{1}\cdots{}r_{n-\ell_{I}}\rangle_{A_{r_{1}}\cdots{}A_{r_{n-\ell_{I}}}}$
(23)

where $\beta^{(i)}_{r_{1}\cdots{}r_{n-\ell_{I}}}$ depends on some coefficients
of $\gamma_{i}$’s and satisfy
$\sum_{\sum_{A_{r_{t}}\in\overline{I}}r_{t}=s-i}(\beta^{(i)}_{r_{1}\cdots{}r_{n-\ell_{I}}})^{2}=1$.

Note that for each pair of $i,j$ with $i\not=j$, the states
$|\phi_{i_{1}}\rangle$ and $|\phi_{i_{2}}\rangle$ are orthogonal because
$\\{|j_{1}\cdots{}j_{\ell_{I}}\rangle\\}$ (with
$j_{1}+\cdots{}+j_{\ell_{I}}=i_{1}$) and
$\\{|j^{\prime}_{1}\cdots{}j^{\prime}_{\ell_{I}}\rangle\\}$ with
$j^{\prime}_{1}+\cdots{}+j^{\prime}_{\ell_{I}}=i_{2}$) are orthogonal. This
implies $\\{|\phi_{i}\rangle_{I}\\}$ defined in Eq.(22) are orthogonal.
Similarly, we can prove $\\{|\psi_{i}\rangle_{\overline{I}}\\}$ defined in
Eq.(23) are orthogonal. This has proved the Schmidt decomposition (21) with
the orthogonal states in Eqs.(22) and (23). Moreover, it implies the Schmidt
number $L_{I}=s+1$. From Result 3, it has proved the result.

## Appendix E. Estimating upper bound of $k$ for entangled quantum networks

In this section, we estimate the upper bound of $k$ for an $n$-partite
entangled quantum network using a polynomial-time algorithm. Here, for
simplicity we assume each pair shares at least one bipartite entanglement on
$d$-dimensional Hilbert spaces $\mathcal{H}_{d}\otimes\mathcal{H}_{d}$.

We firstly present a sufficient and necessary condition for verifying an
entangled quantum network. Consider an $n$-partite quantum network
$\mathcal{N}_{q}$ consisting of any bipartite entanglement on Hilbert space
$\mathcal{H}_{d}\otimes\mathcal{H}_{d}$. Denote
$\mathcal{N}_{q}^{(\mathcal{A})}$ as the subnetwork consisting of all parties
in $\mathcal{A}$. For a given subnetwork $\mathcal{N}_{q}^{(\mathcal{A})}$,
denote $s_{i;in}$ denotes the inner connectedness degree of the party
$\textsf{A}_{i}$, that is, the number of bipartite entangled states shared
with other parties in $\mathcal{A}$. While $s_{i;out}$ denotes the outer
connectedness degree of the party $\textsf{A}_{i}$, that is, the number of
bipartite entangled states shared with other parties out of $\mathcal{A}$.
Denote $t_{i}$ as all the other inner connectedness degrees in
$\mathcal{N}_{q}^{(\mathcal{A})}$, that is, the number of bipartite entangled
states shared by two parties in $\mathcal{A}-\\{\mathcal{A}_{i}\\}$. We
firstly prove the following lemma.

Lemma 1. An $n$-partite quantum network $\mathcal{N}_{q}$ is $k$-connection
biseparable if $s_{i;in}+2t_{i}\geq s_{i;out}$ for some $i$ and $k$-partite
subnetwork.

Proof of Lemma 1. The proof is completed by two steps. One step is to
transform all the entangled states shared by inner parties in a given
subnetwork $\mathcal{N}_{q}^{(\mathcal{A})}$. The other is to change all the
entangled states shared by one party into others in $\mathcal{A}$. For
simplicity, consider a $k$-partite subnetwork
$\mathcal{N}_{q}^{(\mathcal{A})}$ with
$\mathcal{A}=\\{\textsf{A}_{1},\cdots,\textsf{A}_{k}\\}$. In what follow, we
only prove the result for the party $\textsf{A}_{1}$. Similar proof holds for
other subnetworks and parties.

Denote all the entangled states shared by the party $\textsf{A}_{1}$ and
parties in $\mathcal{A}$ as (see red lines in Fig.5(a))

$\displaystyle\varrho_{1;in}=\otimes_{i\in I}\rho_{A_{1\to i}A_{i\to 1}}$ (24)

where $\rho_{A_{1\to i}A_{i\to 1}}$ are bipartite entangled states on Hilbert
space $\mathcal{H}_{d}\otimes\mathcal{H}_{d}$. Denote all the entangled states
shared by the party $\textsf{A}_{1}$ and parties out of $\mathcal{A}$ as (see
orange lines in Fig.5(a))

$\displaystyle\varrho_{1;out}=\otimes_{j\in J}\rho_{A_{1\to j}A_{j\to 1}}$
(25)

where $\rho_{A_{1\to j}A_{j\to 1}}$ are bipartite entangled states on Hilbert
space $\mathcal{H}_{d}\otimes\mathcal{H}_{d}$. Denote all the entangled states
shared by any two parties in
$\mathcal{A}_{i}:=\mathcal{A}-\\{\textsf{A}_{1}\\}$ as (see green lines in
Fig.5(a))

$\displaystyle\varrho_{\mathcal{A}_{i}}=\otimes_{A_{i}A_{j}\in\mathcal{A}_{i}}\rho_{A_{i\to
j}A_{j\to i}}$ (26)

where $\rho_{A_{i\to j}A_{j\to i}}$ are bipartite entangled states on Hilbert
space $\mathcal{H}_{d}\otimes\mathcal{H}_{d}$. There exists a local CPTP
mapping $\mathcal{E}_{\mathcal{A}_{i}}$ such that

$\displaystyle\mathcal{E}_{\mathcal{A}_{i}}:\varrho_{\mathcal{A}_{i}}\mapsto\otimes_{A_{i}A_{j}\in\mathcal{A}_{i}}|00\rangle_{A_{i}A_{j}}\langle
00|$ (27)

Figure 5: Schematic quantum network. (a) A general quantum network with
subnetwork consisting of all parties in $\mathcal{A}$. Here, the orange lines
denote entanglements shared by $\textsf{A}_{1}$ and others out of
$\mathcal{A}$. The red lines denote entanglements shared by $\textsf{A}_{1}$
and others in $\mathcal{A}$. The green lines denote entanglements shared by
parties (except for $\textsf{A}_{1}$) in $\mathcal{A}$. (b) Swapping two
particles (red dotted lines) in $\mathcal{A}$.

Denote $\mathcal{A}_{1}$ consists of all particles $A_{1\to j}$ in Eq.(25).
Denote $\mathcal{B}$ consists of all particles $A_{1\to j}$ in Eq.(24) and all
particles $A_{i\to j}$ and $A_{j\to i}$ in Eq.(25). Let $SWAP(A_{i}B_{i})$ be
swapping operation of two particles $A_{i}$ and $B_{i}$ defined by

$\displaystyle SWAP(A_{i}B_{i})=\sum_{i}|ii\rangle\langle
ii|+\sum_{i\not=j}|ij\rangle\langle ji|$ (28)

where $A_{i}\in\mathcal{A}_{1}$ and $B_{i}\in\mathcal{B}$, as shown in
Fig.5(b). After these swapping operations, all the bipartite entangled states
shared with $\mathcal{A}_{1}$ are disentangled with all the particles in
$\mathcal{A}_{1}$, but re-entangled with particles in $\mathcal{B}$ if
$s_{i;in}+2t_{i}\geq s_{i;out}$. This is the basic fact of entanglement
swapping BBC . Hence, if $s_{i;in}+2t_{i}\geq s_{i;out}$, there is a CPTP
mapping
$\mathcal{E}_{\mathcal{A}}=SWAP_{\mathcal{A}}\circ\mathcal{E}_{\mathcal{A}_{1}}$
satisfying

$\displaystyle\mathcal{E}_{\mathcal{A}}:\rho_{A_{1}\cdots
A_{n}}\mapsto|0\rangle_{A_{1}}\langle
0|\otimes\hat{\rho}_{A_{2}\cdots{}A_{n}}$ (29)

where $\rho_{A_{1}\cdots A_{n}}$ denotes the total state of $\mathcal{N}_{q}$,
$\hat{\rho}_{A_{2}\cdots{}A_{n}}$ denotes a proper state of subnetwork
consisting of $\textsf{A}_{2},\cdots,\textsf{A}_{n}$, and
$SWAP_{\mathcal{A}}=\otimes_{A_{i}\in\mathcal{A}_{1}}SWAP(A_{i}B_{i})$. This
means that there is a local $k$-partite mapping for transforming
$\mathcal{N}_{q}$ into a disconnected network. Hence, $\mathcal{N}_{q}$ is
$k$-connection biseparable state. This has completed the proof. $\Box$

Figure 6: Schematic quantum network. A general quantum network with subnetwork
consisting of all parties in $\mathcal{A}$. Here, there is a chain subnetwork
shown in red lines.

Lemma 2. An $n$-partite $c$-connected quantum network $\mathcal{N}_{q}$ is
$k$-connection biseparable state with $2k\geq c+1$, where the
$c$-connectedness means that there are $c$ numbers of different chain
subnetworks connecting each pair $(\textsf{A}_{i},\textsf{A}_{j})$ of
$\mathcal{N}_{q}$.

In Lemma 2, a chain subnetwork consists of
$\textsf{A}_{i_{1}},\cdots,\textsf{A}_{i_{s}}$ such that each adjacent pair
share one entanglement.

Proof of Lemma 2. For any $n$-partite $c$-connected quantum network
$\mathcal{N}_{q}$, there is one party ($\textsf{A}_{1}$ for example) who
shares $c$ bipartite entangled states with others. Moreover, for any $c$ and
$k$ there exists a $k$-partite subnetwork $\mathcal{N}_{q}^{(\mathcal{A})}$
with $\textsf{A}_{1}\in\mathcal{A}$ such that all the parties (except for
$\textsf{A}_{1}$) in $\mathcal{A}$ have at least one chain subnetwork, as
shown red lines in Fig.6. This implies that

$\displaystyle s_{1;in}+2t_{1}\geq 2k-2$ (30)

where an $m$-partite chain network has $2m-2$ particles. From Lemma 1 and
$s_{1;out}=c-1$ we have completed the proof. $\Box$

Input: Finite-size network $\mathcal{N}_{q}$

Output: $k$, satisfying that $\mathcal{N}_{q}$ is at most $k$-connection
entanglement

  * (i)

Find the connectedness degree $\ell_{i}$ for any each party $\textsf{A}_{i}$
with $i=1,\cdots,n$.

  * (ii)

Rearrange all parties with decreasing order into
$\textsf{A}_{1},\cdots,\textsf{A}_{n}$ (for simplicity).

  * (iii)

Find $J$ such that $\ell_{j}=\min\\{\ell_{i}\\}$ with $j\in J$. Let $m=|J|$.

  * (iv)

For $s=1:m$

  * (v)

For $t=1:\lfloor\frac{\ell_{j}+1}{2}\rfloor$

  * (a) Let $\mathcal{A}_{1}=\\{\textsf{A}_{j}\\}$ and $\overline{\mathcal{A}}_{t}=\\{\textsf{A}_{1},\cdots,\textsf{A}_{n}\\}-\mathcal{A}_{t}$.

  * (b) Let $\mathcal{A}_{t+1}=\mathcal{A}_{t}\cup\\{\textsf{A}_{u}\\}$, where $\textsf{A}_{u}$ has shared the most bipartite entangled states with parties in $\mathcal{A}_{t}$ compared with other parties in $\overline{\mathcal{A}}_{t}$, and $\textsf{A}_{u}\in\overline{\mathcal{A}}_{t}$.

  * (c) Evaluate $s_{j;in}$ and $t_{j}$.

  * (d) If $s_{j;in}+2t_{j}<s_{j;out}$

  * $t\to t+1$

  * Otherwise

  * Return $v_{s}=\min\\{\ell_{j},t\\}$

  * $s\to s+1$

  * (vi)

Output $k\leq\min\\{v_{1},\cdots,v_{m}\\}$

Algorithm 1 Verifying any $n$-partite quantum network $\mathcal{N}_{q}$
consisting of bipartite entanglement

It is general difficult to find the largest $k$ for any entangled quantum
network $\mathcal{N}_{q}$ such that the total state of $\mathcal{N}_{q}$ is
$k$-CGE. Here, from Lemmas 1 and 2, we present an efficient method for
obtaining the upper bound of $k$, as shown in Algorithm 1.

For each party $\textsf{A}_{j}$ with the minimal connectedness degree
$\ell_{j}$, from Lemma 1 if $s_{j;in}+2t_{j}\geq s_{j;out}$ for some $t$ there
is a CPTP mapping to disentangle all the particles shared by $\textsf{A}_{j}$.
Hence, the total state of $\mathcal{N}_{q}$ is $t$-connection biseparable. The
time complexity of the step (i) is at most $O(n^{2})$. The time complexity of
the step (iii) is at most $O(n)$. For a given party $\textsf{A}_{j}$ with
$j\in J$, the time complexity of the step (b) is at most $O(n^{2})$. Hence,
the total time complexity is at most $O(n^{4})$.

Figure 7: Schematic quantum networks. (a) Chain network. (b) Star network. (c)
Cyclic network. (d) Complete connected network. (e) Planar network (or cluster
states). (d) Cubic network (or cluster states). Here, each line with two dots
denotes one bipartite entanglement. $\mathcal{A}$ denotes the set consisting
of all parties who cooperate to disentangle the party $\textsf{A}_{1}$. The
red lines denote the bipartite entanglements shared by $\textsf{A}_{1}$ with
others out of $\mathcal{A}$. The green lines denote the bipartite
entanglements shared parties in $\mathcal{A}$.

Some examples are shown in Fig.7. For the chain network in Fig.7(a), the party
$\textsf{A}_{1}$ shares one bipartite entanglement (red line) with other
parties out of $\mathcal{A}$, and shares one bipartite entanglement (green
line) with $\textsf{A}_{2}$. From Lemma 1 the chain network in Fig.7(a) is
2-connection biseparable, where the party $\textsf{A}_{1}$ can be disentangled
with other parties out of $\mathcal{A}$ by using joint operation of
$\textsf{A}_{1}$ and $\textsf{A}_{2}$. Moreover, it is genuinely multipartite
entanglement Luo2021 ; Sv , that is, any local operation cannot disentangle
one party. Thus the chain network in Fig.7(a) is 1-CGE. Similar result holds
for the star network in Fig.7(b) and cyclic network in Fig.7(c). For a
complete connected network in Fig.7(d), the party $\textsf{A}_{1}$ shares two
bipartite entanglements (red lines) with other parties out of $\mathcal{A}$.
There are three bipartite entanglements (green lines) shared by parties in
$\mathcal{A}$. From Lemma 1, it is 3-connection biseparable while any two
parties cannot jointly disentangle one party. Hence, this network is 2-CGE,
where the party $\textsf{A}_{1}$ can be disentangled with other parties out of
$\mathcal{A}$ by using joint operation of the parties
$\textsf{A}_{1},\textsf{A}_{2}$ and $\textsf{A}_{3}$. In general, we can prove
an $n$-partite complete connected network is $k_{\max}$-CGE with
$k_{\max}=\lfloor\frac{n}{2}\rfloor$. Similarly, from Lemma 2 the planar
network in Fig.7(e) is 1-CGE while the cubic network is 2-CGE.