# Tripartite GHZ Entanglement in Monitored Random Clifford Circuits

Guanglei Xu  Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  Yu-Xiang Zhang<EMAIL_ADDRESS>Institute of Physics, Chinese
Academy of Sciences, Beijing 100190, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Hefei
National Laboratory, Hefei, 230088, China

(August 28, 2024)

###### Abstract

Multipartite quantum entanglement of a manybody is not well understood. Here
we numerically study the amount of tripartite Greenberger-Horne-Zeilinger
(GHZ) states that can be extracted from the state generated by random Clifford
circuits with probabilistic single-qubit projective measurements. We find a
GHZ-entangled phase where this amount is finite and a GHZ-trivial phase where
no tripartite entanglement is available. The transition between them is either
measurement-induced, at $p_{c}\approx 0.16$, or partition-induced when a party
contains more than half of the qubits. We find that the GHZ entanglement can
be enhanced by measurements in certain regimes, which could be understood from
the perspective of quantum Internet. Effects of the measurements to the growth
of GHZ entanglement are also studied.

There are countless inequivalent ways in which particles of a manybody system
can be entangled. However, only bipartite quantum entanglement is relatively
well-understood Horodecki  _et al._ (2009); Gühne and Tóth (2009); Walter  _et
al._ (2016). Bipartite entanglement omits all the complexities within the same
party, resulting in a vastly coarse-grained description. Despite that,
nontrivial features have already emerged at this level, particularly a
transition between phases with area-law and volume-law bipartite entanglement
Amico  _et al._ (2008); Eisert  _et al._ (2010); Laflorencie (2016); Abanin
_et al._ (2019). Recently, this transition has also been observed in random
circuits where qubits arranged in a one-dimensional (1D) layout undergo random
two-qubit unitaries intercepted with probabilistic single-qubit projective
measurements Li  _et al._ (2018); Chan  _et al._ (2019); Skinner  _et al._
(2019); Szyniszewski  _et al._ (2019); Gullans and Huse (2020a). Entanglement
transition in this system has been mapped to a classical statistical mechanics
model of Potts-like spins Bao  _et al._ (2020); Jian  _et al._ (2020); Zhou
and Nahum (2019). The growth of bipartite entanglement entropy in the volume-
law phase can be understood from entanglement domain walls Nahum  _et al._
(2017, 2018) and directed polymer in a random environment Li  _et al._ (2023).
Monitored 1D random circuits have become a high-profile platform for studying
the interplay between information-theoretical concepts and quantum manybody
dynamics. Recent works have also extended the systems to circuits with long-
range gates Nahum  _et al._ (2021); Block  _et al._ (2022); Sharma  _et al._
(2022), multi-qubit measurements Lavasani  _et al._ (2021a, b), other
dimensions Turkeshi  _et al._ (2020); Sierant  _et al._ (2022); Liu  _et al._
(2022); Feng  _et al._ (2023); Jian  _et al._ (2021) spacetime duality
Ippoliti and Khemani (2021); Lu and Grover (2021); Ippoliti  _et al._ (2022),
error correction properties Choi  _et al._ (2020); Gullans  _et al._ (2021);
Fan  _et al._ (2021); Li and Fisher (2021), noisy circuits Weinstein  _et al._
(2022); Liu  _et al._ (2023, 2024) and experiments Hoke  _et al._ (2023), etc.

Figure 1: (a) Schematic of the setup: Random Clifford gates implemented upon
neighboring qubits are arranged in a brickwork configuration. Each layer of
the circuits consists of two levels of random unitaries and after each level,
projective Z-measurements are applied independently to every qubit with
probability $p$. (b) Illustration of the tripartite decomposition of a
stabilizer state into Bell pairs $\ket{\psi}$ and the GHZ state
$\ket{\text{GHZ}}$ omitting isolated single qubits.

Considering these substantial discoveries and the significant gap between the
coarse-grained bipartite descriptions and the finest multipartite structures,
we are motivated to advance to tripartite entanglement, for which only a few
results are available Zou  _et al._ (2021); Nezami and Walter (2020) in the
context of manybody systems. As illustrated in Fig. 1(a), we accommodate
ourselves by restricting the random two-qubit gates to the _Clifford_ group.
The problem is simplified for two reasons. Firstly, Clifford circuits can be
simulated efficiently Gottesman (1998); Aaronson and Gottesman (2004); Anders
and Briegel (2006). Secondly, states generalized by Clifford circuits, the
_stabilizer states_ , has a clear structure of tripartite entanglement.
Namely, up to unitaries localized in each party, a stabilizer state
$\ket{\Psi}_{ABC}$ is equivalent to a collection of isolated single-qubit
states, two-qubit Bell states $\ket{\psi}=(\ket{00}+\ket{11})/\sqrt{2}$ and
three-qubit GHZ states $\ket{\text{GHZ}}=(\ket{000}+\ket{111})/\sqrt{2}$
Bravyi  _et al._ (2006), cf. Fig. 1(b). Neglecting the isolated qubits that
are irrelevant to entanglement, the equivalence reads:

$\displaystyle\ket{\Psi}_{ABC}\sim$ $\displaystyle\ket{\psi}_{AB}^{\otimes
n_{AB}}\otimes\ket{\psi}_{BC}^{\otimes n_{BC}}\otimes\ket{\psi}_{AC}^{\otimes
n_{AC}}$ (1) $\displaystyle\otimes\ket{\text{GHZ}}^{\otimes g}_{ABC}.$

The index $g$, which counts the entanglement of the GHZ type Fattal  _et al._
(2004), will be the focus of this Letter. It should _not_ be confused with the
notion of genuinely multipartite entanglement Horodecki  _et al._ (2009);
Avakian  _et al._ (2024), which is referred to states cannot be prepared by a
mixture of product states in the form of $\rho_{A}\otimes\rho_{BC}$,
$\rho_{B}\otimes\rho_{AC}$ and $\rho_{C}\otimes\rho_{AB}$. This notion does
not exclude the Bell states contained in Eq. (1).

The index $g$ is typically small for stabilizer states Smith and Leung (2006)
and stabilizer tensor networks Nezami and Walter (2020), see also an
estimation in sp . However, having few in numbers does not mean lacking of
features. We find that the monitored random Clifford circuits exhibit a GHZ-
entangled phase where $g$ is finite and a GHZ-trivial phase where $g$
vanishes. The transition between them can be either measurement-induced, at
the measurement probability $p\approx 0.16$ which is close to (if not
identical to) the critical point of the bipartite entanglement transition Li
_et al._ (2018); Skinner  _et al._ (2019) and the purification phase
transition Gullans and Huse (2020b), or _partition-induced_ , which might be
unique to multipartite entanglement. Throughout this Letter, we consider the
geometry shown in Fig. 1(b), where each party is simply connected. The open
boundary condition is assumed unless stated otherwise. We also assume that A
and C have the same number of qubits. Thus, the way of partition is fixed by
the ratio $N_{B}/N$. We find that the GHZ-entangled phase appears only when
$N_{B}/N<1/2$. Indeed, partition itself does not alter the underlying many-
body state. It indicates an unexplored fundamental limit to the extent of
multipartite entanglement established by local interactions.

Additionally, we also find that GHZ entanglement can be _enhanced_ by
measurements, contrasting with the negative impact on bipartite entanglement.
The growth of $g$ is studied with a comparison to the long-range bipartite
entanglement.

## Preliminaries.

As shown in Fig. 1(a), we investigate the dynamics of 1D array of $N$ qubits
governed by random Clifford gates arranged in a brickwork configuration,
intercepted by single-qubit measurement of Pauli $Z$ operator with a
probability of $p$. In our simulation, each circuit comprises $N$ layers and
each layer consists of two levels of random gates. The specific realization of
all random Clifford gates and the outcomes of the probabilistic measurements
determine a trajectory of state evolution. Our numerical results of $\langle
g\rangle$ are obtained by averaging 400 trajectories.

We initialize the qubits in $\ket{0}$, an eigenstate of the Pauli Z operator
($Z\ket{0}=\ket{0}$). The initial state is a stabilizer state with the
stabilizers $\\{Z_{1},Z_{2},\cdots Z_{N}\\}$ Lavasani  _et al._ (2021a).
Tracking the state evolution is equivalent to updating the set of stabilizers
Gottesman (1998). By the definition of Clifford gates, the stabilizers are
always a product of the identity operator $\mathbb{I}$ and the standard Pauli
operators $X,Y,Z$. A stabilizer is said to act trivially on some qubit if the
corresponding factor is $\mathbb{I}$. To compute $g$, one needs to identify a
space (denoted by $\mathcal{S}$) generated by all $N$ stabilizers, following
the rule that if $s_{1}$ and $s_{2}$ are both in $\mathcal{S}$, then
$s_{1}s_{2}$ also belongs to $\mathcal{S}$. The space $\mathcal{S}$ has three
subspaces denoted by $S_{\hat{\alpha}}$ for $\alpha\in\\{A,B,C\\}$, where
$S_{\hat{\alpha}}$ contains all elements of $\mathcal{S}$ that act trivially
on Party-$\alpha$. Then, it has been proven that Fattal  _et al._ (2004);
Bravyi  _et al._ (2006)

$g=N-\mathrm{dim}\big{(}\cup_{\alpha}S_{\hat{\alpha}}\big{)}.$ (2)

We simulate the circuits using QISKIT software packages to track the $N\times
2N$ tableau to the binary representations of the stabilizers Nielsen and
Chuang (2010). All indices of Eq. (1) are obtained accordingly sp .

Figure 2: Measurement-induced and partition-induced entanglement transition.
(a,b) $\langle g\rangle$ as a function of $p$ for the partition that
$N_{B}/N=1/3$ in (a) and $8/15$ in (b). The inset of (a) demonstrates the data
collapse result, where the horizontal axis label is
$x_{p}\equiv(p-p_{c})L^{1/\nu}$ and $p_{c}\approx 0.16$. (c) $\langle
g\rangle$ as a function of $p$ and $N_{B}/N$ for $N=240$. (d) $\langle
g\rangle$ as a function of $N_{B}/N$ for $N=240$ and a series of $p$. (e,f)
$\langle g\rangle$ as a function of $N_{B}/N$ for $p=0$ in (e) and 0.08 in
(f). The corresponding data collapse results are shown in the insets of (e,f),
where the horizontal axis label is $x_{\rm N_{\rm
B}}\equiv(n_{B}-n^{cr}_{B})L^{1/\mu}$ where $n_{B}\equiv N_{B}/N$ and
$n^{cr}_{B}\approx 0.5$.

## Phase transitions.

We start with an equal partition, $N_{A}{=}N_{B}{=}N_{C}{=}N/3$, and calculate
the average value $\langle g\rangle$ for $N=120{\sim}240$ (in increments of
30). The results are plotted as a function of $p$ in Fig. 2(a). It shows that
$\langle g\rangle$ is a finite value, insensitive to $N$, but changes with $p$
abruptly in the interval $0.1<p<0.2$. The larger $N$ is, the steeper the
decline of $\langle g\rangle$. Thus, we anticipate a _measurement_ -induced
continuous transition of $\langle g\rangle$. In the inset of Fig. 2(a), we
attempt a data collapse with the scaling form $\langle
g\rangle=F[(p-p_{c})N^{1/\nu}]$ with $F$ being an arbitrary function and $\nu$
a critical exponent, and find that $p_{c}\approx 0.159\pm 0.002$ and
$\nu\approx 1.36$. We refer to the phase for $p<p_{c}$ as _GHZ-entangled_ and
the phase for $p>p_{c}$ as _GHZ-trivial_. The critical point $p_{c}$ aligns
with those of the measurement-induced bipartite entanglement transition Li
_et al._ (2018); Skinner  _et al._ (2019) and the purification transition
Gullans and Huse (2020b). It is worth noting that $\nu$ may not be universal,
though we have not reached a conclusive dependence on $N_{B}/N$.

Perhaps surprisingly, we find that the critical mutual information displayed
in Fig. 7 of an early work on the bipartite entanglement transition Li  _et
al._ (2019), is actually contributed by the tripartite GHZ entanglement sp .
The geometry used there is different: The periodic boundary condition is
applied and one of the three party is not simply connected.

The dependence of $\langle g\rangle$ on $p$ changes significantly when we
extend the size of B from $N/3$ to $8/15N$, as shown in Fig. 2(b). Instead of
the plateau observed in the GHZ-entangled phase, we see $\langle g\rangle$
grows with $p$ in a wide region of $p<0.15$, demonstrating an enhancement
induced by measurements. The striking difference between Figs. 2(a) and (b)
prompts us to scan $\langle g\rangle$ over both $p$ and $N_{B}/N$ in Fig.
2(c), for the case of $N=240$. Values of $g$ are represented by color mapping.
The zigzags of the contours are due to statistical errors. Nevertheless, the
plot demonstrates a clear plateau of $\langle g\rangle>1$. Moreover, dense
color changes, indicating “cliffs” of $\langle g\rangle$, are observed near
the contours of $\langle g\rangle=0.4$ and $0.7$. These two contours are
adjacent to the vertical line of $p_{c}\approx 0.16$ and the horizontal line
of $N_{B}/N=1/2$. The former coordinates the measurement-induced transition,
suggesting that the latter may imply another.

To learn Fig. 2(c) better, we analyze its vertical sections by plotting
$\langle g\rangle$ as a function of $N_{B}/N$ for given values of $p$ in Fig.
2(d). It manifests a congruence of the red curves ($p<0.16$) near
$N_{B}/N=1/2$. Then, we collect the curves of $\langle g\rangle$ for $p=0$ and
$p=0.08\approx p_{c}/2$ across five different values of $N$ in Figs. 2(e,f).
Both figures exhibit curve crossings similar to those in Fig. 2(a). We attempt
data collapse with the scaling form $\langle
g\rangle=G[(N_{B}/N-1/2)N^{1/\mu}]$, where $G$ an arbitrary function, and show
the result in the insets of Figs. 2(e, f). The critical indices are found to
be $\mu_{0}=1.011$ and $\mu_{0.08}=1.356$. Thus, we believe there is a phase
transition at $N_{B}/N=1/2$. This transition is _partition-induced_ , as
particularly evidenced in Fig. 2(e), where measurement is irrelevant. From the
above observations, we infer that the GHZ-entangled phase is confined within
the rectangle defined by $p\leq p_{c}\approx 0.16$ and $N_{B}/N\leq 1/2$.

The partition-induced phase transition suggests that the GHZ entanglement
transition may not be described by a conformal field theory (CFT), as the
bipartite entanglement transition with long-range gates Block  _et al._
(2022). The inconsistency with CFT is corroborated by other evidences. The
decomposition (1) implies that $g=I_{A\mathrel{\mathop{\mathchar
58\relax}}B}-2N_{A,B}$ Sang  _et al._ (2021) where
$I_{A\mathrel{\mathop{\mathchar 58\relax}}B}$ is mutual information between A
and B, and $N_{A,B}=\log_{2}||\rho_{AB}^{T_{A}}||_{1}$ is the entanglement
negativity Vidal and Werner (2002) (with $\rho_{AB}^{T_{A}}$ being the
partially transposed reduced density matrix). Meanwhile, for state (1)
$N_{A,B}$ is equivalent to the _computable cross norm negativity_ Chen and Wu
(2003) $\mathcal{E}_{A,B}=\log_{2}||R_{AB}||_{1}$ where $R_{AB}$ “realigns”
the elements of the density matrix:
$\innerproduct{a,a^{\prime}|R_{AB}|b,b^{\prime}}{a,a^{\prime}|R_{AB}|b,b^{\prime}}=\innerproduct{a,b|\rho_{AB}|a^{\prime},b^{\prime}}{a,b|\rho_{AB}|a^{\prime},b^{\prime}}$.
Both types of negativity, when A and B are adjacent parties, have been
evaluated in CFT. However, they do not agree:
$N_{A,B}=\frac{c}{4}\log_{2}\frac{l_{a}l_{b}}{l_{a}+l_{b}}+\text{const.}$
Calabrese  _et al._ (2012) whereas
$\mathcal{E}_{A,B}=\frac{c}{8}[2\log_{2}(l_{a}l_{b})-3\log_{2}(l_{a}+l_{b})]+\text{const.}$
Yin and Liu (2023), where $c$ is the central charge and $l_{i}=(N/\pi)\sin(\pi
N_{i}/N)$ with $i=A,B$. Furthermore, Ref. Zou  _et al._ (2021) proposed a
tripartite entanglement measure $g(A{\mathrel{\mathop{\mathchar
58\relax}}}B)=2E_{P}(A{\mathrel{\mathop{\mathchar
58\relax}}}B)-I_{A\mathrel{\mathop{\mathchar 58\relax}}B}$ where $E_{P}$ is
the entanglement of purification Terhal  _et al._ (2002). It can be proved
that $g(A{\mathrel{\mathop{\mathchar 58\relax}}}B)$ equals $g$ when applied to
stabilizer states (1). However, $g(A{\mathrel{\mathop{\mathchar 58\relax}}}B)$
in a CFT is found insensitive to partition, which is incompatible with the
strong dependence on partition observed here.

Figure 3: (a) The random circuit can be viewed as a quantum Internet linked in
a 1D configuration, where the two-qubit gates are realized by teleportation-
based methods. All the operations, unitary or measurements, are implemented
locally in each node. (b) Teleported CNOT gate. The circuit needs one photonic
Bell pair, local CNOT gates between the atomic and photonic qubits, and local
Z or X unitary controlled by the outcome of the remote measurements.

## A quantum Internet perspective.

Evidences of $\langle g\rangle$ being enhanced by single-qubit measurements
can be seen from Figs. 2(b-d). Some of them are finite-size effect within the
GHZ-trivial phase. For the GHZ-tangled phase, it is observed from Figs. 2(e,f)
that the plateau displayed in Fig. 2(f), $p=0.08$, is higher than that of Fig.
2(e) where $p=0$, by about 0.05. To understand why measurements is not
necessarily detrimental, let us map the qubit array into a chain Internet
where each qubit, which will be called _atomic_ qubit, is located on a node,
and a photonic source distributes Bell pairs to the neighboring nodes, as
illustrated in Fig. 3(a). The gates between atomic qubits can be realized
using the photonic Bell pairs in the manner of quantum teleportation Bennett
_et al._ (1993), as exemplified by the CNOT gate Chou  _et al._ (2018) in Fig.
3(b). Therein, outcomes of the measurements on the photonic qubits must be
shared, allowing the other node to control local Z and X operations. However,
here communication is unnecessary since local Z and X do not affect the
sampling of Clifford gates.

Generally, two photonic Bell pairs are sufficient to implement any two-qubit
gate Eisert  _et al._ (2000); Collins  _et al._ (2001). Thus, simulating a
random circuit of T layers is equivalent to distributing 2T Bell pairs for
every two neighbors, followed by sequential unitaries and measurements
performed locally within each node. We package all the local implementations
specified by a trajectory into an operator $\mathcal{M}_{i}$ acting on $2T$
photonic and one atomic qubits. Thus, the final state is expressed as:

$\ket{\Psi}_{T}=\bigotimes_{i=1}^{N}\mathcal{M}_{i}\circ\ket{\Psi_{0}}$ (3)

where the resource state contains only Bell type short-range entanglement

$\ket{\Psi_{0}}=\ket{0}_{1}\ket{\psi}_{1,2}^{\otimes
2T}\ket{0}_{2}\ket{\psi}_{2,3}^{\otimes 2T}\cdots\ket{\psi}_{N-1,N}^{\otimes
2T}\ket{0}_{N}.$ (4)

Now, let us abstract the nature of $\mathcal{M}_{i}$ and treat it as a general
operator. If $\mathcal{M}_{i}$ is unitary, no GHZ-entanglement can be obtained
from $\ket{\Psi}_{T}$, because local unitaries do not change anything about
entanglement. In this sense, the GHZ-entangled phase obtained without
measurement ($p=0$) effectively arises from measuring the photonic qubits.
Thus, it is reasonable that $g$ benefits from further measuring some atomic
qubits ($p>0$). This argument indicates that the multipartite entanglement is
converted from the fewer-party entanglement. Such theory is currently under
development de Jong  _et al._ (2024).

Figure 4: Entanglement growth in the GHZ-entangled phase (equal partition
$N_{B}=N/3$). (a) Values of $\langle g\rangle$ (upper panel, scattering dots
represent the raw data of $\langle g\rangle$ while the smooth curves are
obtained by fitting the data) and $n_{AC}$ (lower panel) as a function of
circuit layer $t$. The inserts show the speed of growth. (b) The distribution
of the first layer $t_{0}$ when $\langle g\rangle$ (upper panel) and $n_{AC}$
becomes nonzero.

## GHZ entanglement growth.

At last, we study the growth of $\langle g\rangle$ for the equal partition
$N_{A}=N_{B}=N_{C}$. We plot $\langle g\rangle$ as a function of the circuit
layer denoted by $t$ (for $p\in[0,0.16]$) in the upper panel of Fig. 4(a). To
illuminate the difference between GHZ entanglement with long-range but
bipartite Bell entanglement, we plot parallel results of $n_{AC}$ in the lower
panel of Fig. 4(a). This comparison is intriguing because previous studies of
Clifford circuits rely on the _clipped gauge_ of the stabilizers Nahum  _et
al._ (2017). The clipped gauge fails to distinguish the GHZ-entanglement from
the Bell-entanglement between A and C, because it records only the first and
the last qubit where a stabilizer acts nontrivially, regardless of the
supports on B.

The plots show that $\langle g\rangle$ saturates earlier than $n_{AC}$, and
there are regimes where constant speed of growth is well-defined, see the
insets of Fig. 4(a). The speed of growth of bipartite entanglement is
described by the Kardar-Parisi-Zhang equation Nahum  _et al._ (2017), while
that of the GHZ-entanglement is open. $\langle g\rangle$ grows with a lower
speed than $n_{AC}$, and slows down when $p$ is larger. To learn the effect of
measurements better, we consider $p=0,0.02,0.1$ and plot the distribution of
the first moment $t_{0}$ when $g$ and $n_{AC}$ transit to non-zero values in
the upper and lower panels of Fig. 4(b), respectively. It shows that a larger
$p$ yields smaller $t_{0}$ while flattening the distribution. In addition,
$\min t_{0}$ of GHZ is visibly smaller, meaning that tripartite GHZ
entanglement dominates the A-C bipartite entanglement at early dynamics.

## Conclusions and Discussions.

To summarize, we numerically studied the tripartite GHZ entanglement of the
monitored random Clifford circuits and found a GHZ-entangled phase and a GHZ-
trivial phase. The transition between them can be measurement-induced at
$p=0.16$ or partition-induced at $N_{B}/N=1/2$. The partition-induced
transition is a new discovery of the multipartite entanglement. We found that
GHZ entanglement can be enhanced by measurements. The growth of GHZ
entanglement is compared with the growth of long-range bipartite entanglement.

There are questions remain open. Firstly, imagine a phase diagram of $\langle
g\rangle$ akin to Fig. 2(c). Our data is not sufficient to resolve weather the
line of $p=0.16$ and $N_{B}/N>1/2$ is critical or not. It might be because
Fig. 2(d) indicates an unobserved feature: curves of $\langle g\rangle$
appears concave when $p<0.16$ (red) and convex when $p>0.16$ (blue), while
near $p_{c}$ (dashed line) the curve is roughly linear, giving a chance of
nonzero $\langle g\rangle$ for $N_{B}/N>1/2$ in the large $N$ limit.

Secondly, the partition-induced transition indicates that the relative size
matters. Beside that, it is yet to be explored that how the multipartite
entanglement depends on the topology of the partitions. For example, let us
consider Kitaev’s toric code Kitaev (2003) and divide all the qubits into five
parties (GHZ state extracted from local junction point Bravyi  _et al._ (2006)
is avoided because each stabilizer of the toric code is supported by four
qubits). For five-party GHZ entanglement, Eq. (2) still applies Bravyi  _et
al._ (2006); Fattal  _et al._ (2004). Then $g\neq 0$ only if there is a logic
operator that cannot be cleaned Bravyi and Terhal (2009) from any party so
that it does not belong to $\cup_{\alpha}S_{\hat{\alpha}}$, which requires
every party to be topologically nontrivial.

###### Acknowledgements.

Thanks to Zi-Xiang Li and Run-Ze Chi for useful discussions. Y.-X. Zhang
acknowledges the financial support from National Natural Science Foundation of
China (Grant No. 12375024), Innovation Program for Quantum Science and
Technology (Grant No. 2023ZD0301100 and No. 2- 6), CAS Project for Young
Scientists in Basic Research (YSBR-100).

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# On a variant of Schu’s lemma

C. Zălinescu Octav Mayer Institute of Mathematics, Iaşi Branch of Romanian
Academy, Iaşi, Romania, email<EMAIL_ADDRESS>
###### Abstract

In this note we show that a false statement obtained by replacing a “lim” by
“limsup” for a sequence in Lemma 1.3 of the paper [J. Schu: Weak and strong
convergence to fixed points of asymptotically nonexpansive mappings, Bull.
Austral. Math. Soc. 43 (1991), 153–159] is used in many papers. We do not know
which is the first paper that used that false assertion, but the first one
that we identified is from 2005 and the last one is from 2024.

In J. Schu’s paper [1], the following result is stated without proof, where
[8] is our reference [2]:

“LEMMA 1.3. (compare [8, p.484]) Let $(E,\left\|\cdot\right\|)$ be a uniformly
convex Banach space, $0<b<c<1$, $a\geq 0$, $(t_{n})\subset[b,c]^{\mathbb{N}}$
and $(x_{n}),(y_{n})\in E^{\mathbb{N}}$ such that
$\limsup\left\|x_{n}\right\|\leq a$, $\limsup\left\|y_{n}\right\|\leq a$ and
$\lim\left\|t_{n}x_{n}+(1-t_{n})y_{n}\right\|=a$. Then
$\lim\left\|x_{n}-y_{n}\right\|=0$.”

Taking $t_{n}:=t\in(0,1)$ for $n\in\mathbb{N}^{\ast}$
$(:=\\{1,2,...,n,...\\})$, [1, Lemma 1.3] reduces to Problem 1.10(c) from [2,
p. 484]. Observe also that the version of [1, Lemma 1.3] for $[b,c]$ replaced
by $[\varepsilon,1-\varepsilon]$ with $\varepsilon\in(0,1)$ is attributed to
“(Zeidler (1986))” in [3, Lemma 2.1]. Moreover, the following text can be
found in [4]:

“In 1991, Sahu [21] established an important property of UCBS, which can be
stated as follows.

Lemma 2.2. [21] Assume that $\Omega$ is a UCBS and $\\{z_{\ell}\\}$ be a
sequence in $(0,1)$ for all $\ell\geq 1$. If $\\{a_{\ell}\\}$ and
$\\{b_{\ell}\\}$ are in $\Omega$ such that
$\limsup_{\ell\rightarrow\infty}\left\|a_{\ell}\right\|\leq m$,
$\limsup_{\ell\rightarrow\infty}\left\|b_{\ell}\right\|\leq m$ and
$\lim_{\ell\rightarrow\infty}\left\|(1-z_{\ell})a_{\ell}+z_{\ell}b_{\ell}\right\|=m$
for some $m\geq 0$. Then
$\lim_{\ell\rightarrow\infty}\left\|a_{\ell}-b_{\ell}\right\|=0$.”

In fact the reference [21] from the preceding quoted text is nothing else than
[1]. Observe that the previous version of [1, Lemma 1.3] is false as the next
example shows.

###### Example 1

Let $(X,\left\|\cdot\right\|)$ be a non trivial normed vector space, $x\in
X\setminus\\{0\\}$, $x_{n}:=x$ and $y_{n}:=-x$ for $n\in\mathbb{N}^{\ast};$
moreover, consider $\\{s_{n}\\}\subset(0,1)$ with $s_{n}\rightarrow 0$,
$t_{n}:=s_{n}$ for $n$ odd and $t_{n}:=1-s_{n}$ for $n$ even. Then
$\left\|x_{n}\right\|=\left\|y_{n}\right\|=\left\|x\right\|$, and so
$\limsup_{n\rightarrow\infty}\left\|x_{n}\right\|=\limsup_{n\rightarrow\infty}\left\|y_{n}\right\|=\left\|x\right\|>0$;
moreover, $t_{n}x_{n}+(1-t_{n})y_{n}=(2s_{n}-1)x$ for $n$ odd,
$t_{n}x_{n}+(1-t_{n})y_{n}=(1-2s_{n})x$ for $n$ even, and so
$\lim_{n\rightarrow\infty}\left\|t_{n}x_{n}+(1-t_{n})y_{n}\right\|=\lim_{n\rightarrow\infty}\left|1-2s_{n}\right|\cdot\left\|x\right\|=\left\|x\right\|.$
It is obvious that
$\lim_{n\rightarrow\infty}\left\|x_{n}-y_{n}\right\|=2\left\|x\right\|\neq 0$.

Notice that [4, Lemma 2.2] is used in the proofs of [4, Lemmas 3.3 & 3.4]
which, at their turn, are used in the proofs of [4, Lemma 3.5 & Theorem 5.1]
and [4, Lemma 3.5 & Theorems 3.9, 5.1], respectively; [4, Lemma 3.5] is used
in the proof of [4, Theorem 3.9].

In the paper [5] by W. Laowang and B. Panyanak one finds the text:

“The following result is a characterization of uniformly convex hyperbolic
spaces which is an analog of Lemma 1.3 of Schu [25]. It can be applied to a
CAT$(0)$ space as well.

Lemma 2.9. Let $(X,d,W)$ be a uniformly convex hyperbolic space with modulus
of convexity $\eta$, and let $x\in X$. Suppose that $\eta$ increases with $r$
(for a fixed $\varepsilon$) and suppose that $\\{t_{n}\\}$ is a sequence in
$[b,c]$ for some $b,c\in(0,1)$, and $\\{x_{n}\\}$, $\\{y_{n}\\}$ are sequences
in $X$ such that $\limsup_{n\rightarrow\infty}d(x_{n},x)\leq r$,
$\limsup_{n\rightarrow\infty}d(y_{n},x)\leq r$, and
$\lim_{n\rightarrow\infty}d((1-t_{n})x_{n})\oplus t_{n}y_{n},x)=r$ for some
$r\geq 0$. Then $\lim_{n\rightarrow\infty}d(x_{n},y_{n})=0$. (2.17)”

Observe that [5, Lemma 2.9] is announced in [6, Lemma 2.7].

Changing few words in the above text one obtains [7, Lemma 4.5] and its
preamble; the same remark is valid for the proofs of [5, Lemma 2.9] and [7,
Lemma 4.5].

Lemma 2.9 from [5] is stated (and proved) for $X$ a CAT$(0)$ space in Lemma
3.2 of [8] under the supplementary condition $0<b(1-c)\leq\tfrac{1}{2}$;
observe that this condition is superfluous. First notice that $b,c\in(0,1)$
implies that $0<b(1-c)$, while the fact that (there exists) $t_{n}\in[b,c]$
implies that $b\leq c$; hence $0<b\leq c<1$. Because ($0<$)
$b^{2}+(b-1)^{2}=2b^{2}-2b+1$ and ($c\geq$) $b>0$, one gets ($c\geq$)
$b>1-1/(2b)$, whence $1/(2b)>1-c$, and so $b(1-c)<\tfrac{1}{2}$. The
supplementary condition $0<b(1-c)\leq\tfrac{1}{2}$ is present in quite many
papers from our bibliography. Notice that the version for CAT(0) spaces of [5,
Lemma 2.9] (with the same conditions on $b,c\in(0,1)$ as above) is stated and
proved in [9, Lemma 3.2].

Taking $t_{n}:=1/2$ for $n\in\mathbb{N}^{\ast}$, [5, Lemma 2.9] reduces to
[10, Lemma 2.2].

Note that the formulation of [11, Lemma 2.5] is (practically) the same as that
of [5, Lemma 2.9] but one provides _another proof_ ; in that proof one finds
the next _false assertion_ : “Since
$\limsup_{n\rightarrow\infty}d(x_{n},x)\leq r$ and
$\limsup_{n\rightarrow\infty}d(y_{n},x)\leq r$ we have: (i) $d(x_{n},x)\leq
r+\frac{1}{n}$; (ii) $d(y_{n},x)\leq r+\frac{1}{n}$ for each $n\geq 1$”.

Lemma 2.9 from [5] is reobtained as Lemma 2.5 in [12] using [12, Lemma 2.3].

The proof of [5, Lemma 2.9] seems to be an adaptation to this context of that
for Problem 10.1 (c) in [2]; possibly, J. Schu did a similar adaptation of the
proof of [2, Problem 10.1 (c)] for his Lemma 1.3.

The next version of [1, Lemma 1.3] for modular function spaces is from [13]
(see also [14, Lemma 4.2]):

“Lemma 3.2. Let $\rho\in\Re$ be (UUC1) and let $\\{t_{k}\\}\subset(0,1)$ be
bounded away from $0$ and $1$. If there exists $R>0$ such that
$\limsup_{n\rightarrow\infty}\rho(f_{n})\leq R$,
$\limsup_{n\rightarrow\infty}\rho(g_{n})\leq R$,
$\lim_{n\rightarrow\infty}\rho(tf_{n}+(1-t_{n})g_{n})=R$, then
$\lim_{n\rightarrow\infty}\rho(f_{n}-g_{n})=0.$”

An _alternative proof for[1, Lemma 1.3]_ is the following:

Because $\limsup\left\|x_{n}\right\|\leq a$ and
$\limsup\left\|y_{n}\right\|\leq a$, there exists $R>0$ such that
$\left\|x_{n}\right\|,\left\|y_{n}\right\|\leq R$ for every
$n\in\mathbb{N}^{*}$. Because $(E,\left\|\cdot\right\|)$ is uniformly convex,
$\left\|\cdot\right\|^{2}$ is uniformly convex on $B:=\\{x\in
E\mid\left\|x\right\|\leq R\\}$ by [15, Theorem 4.1 (ii)], and so there exists
a strictly increasing function $\rho:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$
with $\rho(0)=0$ such that

$\left\|\lambda
x+(1-\lambda)y\right\|^{2}\leq\lambda\left\|x\right\|^{2}+(1-\lambda)\left\|y\right\|^{2}-\lambda(1-\lambda)\rho\left(\left\|x-y\right\|\right)\quad\forall
x,y\in B,\ \lambda\in[0,1].$

Setting $\gamma:=\min\\{b,1-c\\}$ $(>0)$ one has that
$\lambda(1-\lambda)\geq\gamma^{2}$, and so

$\gamma^{2}\rho\left(\left\|x_{n}-y_{n}\right\|\right)\leq
t_{n}\left\|x_{n}\right\|^{2}+(1-t_{n})\left\|y_{n}\right\|^{2}-\left\|t_{n}x_{n}+(1-t_{n})y_{n}\right\|^{2}\quad\forall
n\in\mathbb{N}.$ (1)

Assume that the conclusion does not hold. Then there exist a strictly
increasing sequence $(n_{k})\subset\mathbb{N}^{\ast}$ and $\varepsilon>0$ such
that $t_{n_{k}}\rightarrow t\in[b,c]$ and
$\left\|x_{n_{k}}-y_{n_{k}}\right\|\geq\varepsilon$ for
$k\in\mathbb{N}^{\ast}$; replacing, if necessary, $n$ by $n_{k}$ in (1) we
assume that $n_{k}=k$ for every $k\in\mathbb{N}^{\ast}$, and so (1) holds.
Because $\rho\left(\left\|x_{n}-y_{n}\right\|\right)\geq\rho(\varepsilon)$,
passing to $\limsup$ in (1), one gets the contradiction
$\gamma^{2}\rho(\varepsilon)\leq ta^{2}+(1-t)a^{2}-a^{2}=0$. $\square$

In [16], one states a version of [1, Lemma 1.3] with a weaker hypothesis:

“Lemma 2.1 (Schu [12]). Let $E$ be a uniformly convex Banach space and
$\\{\alpha_{n}\\}$ a sequence in $[\varepsilon,1-\varepsilon]$ for some
$\varepsilon\in(0,1)$. Suppose $\\{x_{n}\\}$ and $\\{y_{n}\\}$ are sequences
in $E$ such that ${\limsup_{n\rightarrow\infty}}$$\left\|x_{n}\right\|\leq r$,
${\limsup_{n\rightarrow\infty}}$$\left\|y_{n}\right\|\leq r$ and
${\limsup_{n\rightarrow\infty}}$$\left\|t_{n}x_{n}+(1-t_{n})y_{n}\right\|=r$
hold for some $r\geq 0$. Then $\lim\left\|x_{n}-y_{n}\right\|=0$.”

Notice that no proof is provided for [16, Lemma 2.1]. However, in the book
[17] one finds the following result for which one provides a proof:

“Theorem 2.3.13 Let $X$ be a uniformly convex Banach space and let
$\\{t_{n}\\}$ be a sequence of real numbers in $(0,1)$ bounded away from $0$
and $1$. Let $\\{x_{n}\\}$ and $\\{y_{n}\\}$ be two sequences in $X$ such that

${\limsup_{n\rightarrow\infty}\left\|x_{n}\right\|\leq a,\ \
{\limsup_{n\rightarrow\infty}}\left\|y_{n}\right\|\leq a}\text{\ \ {and}}{\
~{}\limsup_{n\rightarrow\infty}\left\|t_{n}x_{n}+(1-t_{n})y_{n}\right\|=a}$

for some $a\geq 0$. Then $\lim\left\|x_{n}-y_{n}\right\|=0$.”

In fact [16, Lemma 2.1] (as well as [17, Theorem 2.3.13]) is false as the next
example shows.

###### Example 2

Let $(X,\left\|\cdot\right\|)$ be a non trivial normed vector space, $x\in X$
with $\left\|x\right\|=1$ $(=:r)$, $x_{n}:=x,$ $y_{2n}:=0,$ $y_{2n+1}:=x,$
$t_{n}:=\lambda\in(0,1)$ for $n\in\mathbb{N}^{\ast}$. Then
$\limsup_{n\rightarrow\infty}\left\|x_{n}\right\|=1,$
$\limsup_{n\rightarrow\infty}\left\|y_{n}\right\|=1,$
$t_{n}x_{n}+(1-t_{n})y_{n}=\lambda x$ for even $n$,
$t_{n}x_{n}+(1-t_{n})y_{n}=x$ for odd $n$, and so
$\limsup_{n\rightarrow\infty}\left\|t_{n}x_{n}+(1-t_{n})y_{n}\right\|=\limsup_{n\rightarrow\infty}1=1.$
It is obvious that $\lim_{n\rightarrow\infty}\left\|x_{n}-y_{n}\right\|$ does
not exist, and so it is different from $0$.

Observe that [16, Lemma 2.1] is used in the proof of [16, Theorem 3.3], which,
at its turn, is used in the proof of [16, Theorem 3.5] (which “extends some
known results”), and so these results could be false.

In what concerns [17, Theorem 2.3.13], this is used in the proofs of
Proposition 2.4.5(i) (used, at its turn in the proofs of Proposition 3.1.13
$\rightarrow$ 3.1.14 $\rightarrow$ 3.1.15, Theorem 6.4.7 $\rightarrow$ Theorem
6.4.13 $\rightarrow$ Corollary 6.4.14, Theorem 6.4.16 $\rightarrow$ Theorem
6.7.4 and Theorem 6.5.2).

The assertion from [16, Lemma 2.1] (with $E$ replaced by $X$) in which
“$[\varepsilon,1-\varepsilon]$ for some $\varepsilon\in(0,1)$” is often
replaced by $[b,c]$ with $b,c\in(0,1)$, possibly with the supplementary
condition(s) $b<c$ (and $0<b(1-c)\leq\tfrac{1}{2}$), is used in the references
mentioned later on.

Observe that the following assertions are equivalent to [16, Lemma 2.1], and
generally one gives as reference [1, Lemma 1.3]; the exceptions will be
mentioned explicitly. Of course, no proofs are provided, but they are used in
the proofs of other results:

[18, Lemma 2.3], [19, Lemma 2.4], [20, Lemma 2.1], [21, Lemma 5], [22, Lemma
2.1], [23, Lemma 2.1] (attributed to [24]), [25, Lemma 2.1], [26, Lemma 2.2],
[27, Lemma 2.2], [28, Lemma 2.3], [29, Lemma 2.2] (attributed to [30]), [31,
Lemma 2.6], [32, Lemma 2.1], [33, Lemma 2], [34, Lemma 1.3], [35, Lemma 2.9],
[36, Lemma 3.2], [37, Lemma 1.4], [38, Lemma 2.2], [39, Lemma 12], [40, Lemma
4.2], [41, Lemma 2.4], [42, Lemma 2.4], [43, Lemma 2.1], [44, Lemma 2.10],
[45, Lemma 2.2] (attributed to [2]), [46, Lemma 2.3], [47, Lemma 2.2], [48,
Lemma 2.1], [49, Lemma 2], [50, Lemma 2.4], [51, Lemma 2.4], [52, Lemma 2.6],
[53, Lemma 3], [54, Lemma 1.4], [55, Lemma 2.5], [56, Lemma 2.1], [57, Lemma
1.4], [58, Lemma 1], [59, Lemma 2.6], [60, Lemma 5], [61, Lemma 3], [62, Lemma
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Lemma 4], [73, Lemma 2.2], [74, Lemma 3], [75, Lemma 2.16], [76, Lemma 2.14],
[77, Lemma 3], [78, Lemma 2.10], [79, Lemma 5], [80, Lemma 4], [81, Lemma
2.4], [82, Lemma 2.3], [83, Lemma 1] (with $\rho_{n}\in(0,1)$), [84, Lemma 3],
[85, Lemma 2.3], [86, Lemma 2] (attributed to [41]), [87, Lemma 6], [88, Lemma
12], [89, Lemma 2], [90, Lemma 1], [91, Lemma 2.6], [92, Lemma 2.6], [93,
Lemma 2.2], [94, Lemma 2.10], [95, Lemma 1.4] (with $t_{n}\in[0,1]$, and
attributed to [17]), [96, Lemma 2.1], [97, Lemma 2.2], [98, Lemma 3], [99,
Lemma 1], [100, Lemma 1], [101, Lemma 2.3], [102, Theorem 2.6] (attributed to
[2]), [103, Lemma 2.3], [104, Lemma 2.], [105, Lemma 5], [106, Lemma 5], [107,
Lemma 4], [108, Lemma 2.12], [109, Lemma 2.1], [110, Lemma 1.3], [111, Lemma
1], [112, Lemma 2.4], [113, Lemma 2.3], [114, Lemma 2.3], [115, Lemma 1],
[116, Lemma 3],

It is worth observing that for [43, Lemma 2.1], [94, Lemma 2.10], [108, Lemma
2.12] and [109, Lemma 2.1] one cites [1] even if among the co-authors of these
papers there is one of the co-authors of the book [17].

Related to [5, Lemma 2.9] (recalled on the first page), one finds the next
statement in [117] (Received 6 November 2014):

“Lemma 2.2 ([6, Lemma 2.9]). Let $(M,d)$ be a CAT$(0)$ space and let $x\in M$.
Suppose $\\{t_{n}\\}$ is a sequence in $[b,c]$ for some $b,c\in(0,1)$ and
$\\{x_{n}\\},$ $\\{y_{n}\\}$ are sequences in $M$ such that
$\limsup_{n\rightarrow\infty}d(x_{n},x)\leq r$,
$\limsup_{n\rightarrow\infty}d(y_{n},x)\leq r$ and
$\limsup_{n\rightarrow\infty}d((1-t_{n}x_{n})\oplus t_{n}y_{n},x)=r$ for some
$r\geq 0$. Then $\lim_{n\rightarrow\infty}d(x_{n},y_{n})=0$.”

Of course, there is not a proof for this “stronger” version of [5, Lemma 2.9]
in [117].

Because any Hilbert space is CAT$(0)$, Example 2 shows that [117, Lemma 2.2]
is false, and so [117, Lemmas 4.4, 4.9 & Theorem 4.10] could be false, too.

Observe that the following “results” are equivalent to [117, Lemma 2.2] and
they are used in the proofs of other results; for each paper we add the
version of [5, Lemma 2.9] which is cited:

[118, Lemma 2.3] ([7, Lemma 4.5]), [119, Lemma 2.2] ([11, Lemma 2.5]), [120,
Lemma 2.5] ([10, Lemma 2.2]), [121, Lemma 1.12] ([7, Lemma 4.5]), [122, Lemma
1.11] ([11, Lemma 2.5] via [9, Lemma 1.3]), [123, Lemma 2] ([7, Lemma 4.5]),
[124, Lemma 2.11] ([5, Lemma 2.9]), [125, Lemma 2.5] ([7, Lemma 4.5]), [126,
Lemma 2.2] ([7, Lemma 4.5]), [127, Lemma 2.7] ([5, Lemma 2.9]), [128, Lemma
2.6] ([11, Lemma 2.5] via [122, Lemma 1.11]), [129, Lemma 2.11] ([11, Lemma
2.5]), [130, Lemma 2.5] ([6, Lemma 2.7]), [131, Lemma 5] ([5, Lemma 2.9]),
[132, Lemma 2.2] ([7, Lemma 4.5]), [133, Lemma 2.1] ([11, Lemma 2.5]), [134,
Lemma 3] ([5, Lemma 2.9]), [135, Lemma 2.10] ([5, Lemma 2.9]), [136, Lemma 12]
([1, Lemma 1.3]), [137, Lemma 2.10] ([6, Lemma 2.7]), [138, Lemma 2.8] ([11,
Lemma 2.5]), [139, Lemma 5] ([1, Lemma 1.3]).

Related to [13, Lemma 3.2] (recalled before the alternative proof of [1, Lemma
1.3]), one finds the next statement in [140]:

“Lemma 1. Suppose that $\rho$ satisfies property (UUC1) and let
$\\{\alpha_{l}\\}\subset[a,b]$, where $0<a<b<1$. If there exists a positive
real number $r$ such that
$\limsup_{l\rightarrow\infty}\rho(\alpha_{l}x_{l}+(1-\alpha_{l})y_{l})=r$,
$\limsup_{l\rightarrow\infty}\rho(x_{l})\leq r$ and
$\limsup_{l\rightarrow\infty}\rho(y_{l})\leq r$, then
$\lim_{l\rightarrow\infty}\rho(x_{l}-y_{l})=0$ ([8], cf. [5]).”

In [140, Lemma 1], [8] and [5] are our references [141] and [13],
respectively; notice that [141, Lemma 2.7] is nothing else than [13, Lemma
3.2]; of course, there is no proof for the above “stronger” version of [13,
Lemma 3.2] in [140]. Moreover, [140, Lemma 1] is used in the proof of [140,
Theorem 1], which, at its turn, is used in the proof of [140, Corollary 1].

Observe that the following “results” are equivalent to [140, Lemma 1] and they
are used in the proofs of other results; for each paper we add the versions of
[13, Lemma 3.2] which are cited:

[142, Lemma 2.6] is attributed to [14] & [13] via [143] (even if in [143]
there is not a result equivalent to [140, Lemma 1]), while [144, Lemma 1] is
attributed to [14].

A last remark: We did not check the correctness of the results from the works
mentioned in the bibliography; moreover, the list of those works using
equivalent versions of [16, Lemma 2.1], [117, Lemma 2.2] or [140, Lemma 1] is
far of being complete.

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