# Molecular Nanomagnet
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ as Resource
for Quantum Entanglement, Coherence, and Spin Squeezing

Azadeh Ghannadan  0009-0005-1448-0919 Saeed’s Quantum Information Group, P.O.
Box 19395-0560, Tehran, Iran  Hamid Arian Zad  0000-0002-1348-1777
Corresponding author<EMAIL_ADDRESS>Department of Theoretical
Physics and Astrophysics, Faculty of Science, P. J. Sǎfárik University, Park
Angelinum 9, 041 54 Košice, Slovak Republic  Saeed Haddadi
0000-0002-1596-0763 Faculty of Physics, Semnan University, P.O. Box 35195-363,
Semnan, Iran  Saeed’s Quantum Information Group, P.O. Box 19395-0560, Tehran,
Iran  Jozef Strečka  0000-0003-1667-6841 Department of Theoretical Physics and
Astrophysics, Faculty of Science, P. J. Sǎfárik University, Park Angelinum 9,
041 54 Košice, Slovak Republic  Zhirayr Adamyan  Laboratory of Theoretical
Physics, Yerevan State University, Alex Manoogian 1, 0025 Yerevan, Armenia
CANDLE, Synchrotron Research Institute, 31 Acharyan Str., 0040 Yerevan,
Armenia  Vadim Ohanyan  0000-0002-7810-7321 Laboratory of Theoretical Physics,
Yerevan State University, Alex Manoogian 1, 0025 Yerevan, Armenia  CANDLE,
Synchrotron Research Institute, 31 Acharyan Str., 0040 Yerevan, Armenia

###### Abstract

We investigate key quantum characteristics of the mixed spin-(1/2,1,1/2)
Heisenberg trimer under the influence of an external magnetic field.
Specifically, we analyze the distributions of bipartite and tripartite
entanglement quantified through the respective negativities, the $l_{1}$-norm
of coherence, and spin squeezing with the help of rigorous analytical and
numerical methods. Our findings suggest that the heterotrinuclear molecular
nanomagnet
$[\\{\text{Cu}^{\text{II}}\text{L}\\}_{2}\text{Ni}^{\text{II}}(\text{H}_{2}\text{O})_{2}](\text{ClO}_{4})_{2}3\text{H}_{2}\text{O}$,
which represents an experimental realization of the mixed spin-(1/2,1,1/2)
Heisenberg trimer, exhibits a significant bipartite entanglement between
$\text{Cu}^{\text{II}}$ and $\text{Ni}^{\text{II}}$ magnetic ions along with
robust tripartite entanglement among all three constituent magnetic ions. The
significant bipartite and tripartite entanglement persists even at relatively
high temperatures up to $37\,\text{K}$ and magnetic fields up to
$50\,\text{T}$, whereby the coherence is maintained even at elevated
temperatures. In addition, we investigate the spin squeezing parameter within
thermal states of the spin-(1/2,1,1/2) Heisenberg trimer. Our exact results
reveal optimal conditions for achieving the highest degree of the spin
squeezing, which are achieved at zero magnetic field around $T\approx
30\,\text{K}$.

##  I Introduction

Entanglement [1], a cornerstone of quantum mechanics, plays a crucial role in
the study of molecular magnets [2, 3], driving advancements in quantum
information and condensed matter physics [4, 5, 6]. In this context,
Heisenberg spin clusters serve as exemplary systems for exploring fundamental
quantum properties, such as entanglement [7, 8, 9, 10], quantum coherence [11,
12, 13], and spin squeezing [14, 15, 16, 17, 18, 19]. These properties not
only illuminate the intricate behavior of quantum systems but also pave the
way for practical applications in quantum information [19, 20, 21, 22, 23],
magnetic resonance [24] and improving the sensor sensitivity [25, 26, 27]. Our
investigation is particularly inspired by the experimental developments in
molecular magnets, which provide a tangible platform for testing theoretical
predictions. By studying the quantum characteristics of Heisenberg spin
clusters, we aim to deepen our understanding of these phenomena and their
implications for future quantum technologies [5, 1].

Different states coexist simultaneously through superposition in quantum
systems, which is known as quantum coherence [28]. Many quantum technologies
including cryptography and computing depend on this feature. It is well known
that real systems can exhibit interference effects when they are coherent,
which makes it easier to accurately control and manipulate their behavior.
However, coherence is brittle and easily disrupted by interactions with the
surroundings, leading to decoherence [29, 30]. As such maintaining coherence
is a significant obstacle to the development of useful quantum technologies.
Generally speaking, the quantum coherence of real systems is affected by
various factors, including environmental noise and general decoherence
mechanisms [31, 32]. Indeed, the environmental noise caused by some factors
such as thermal fluctuations can disrupt the fragile quantum superpositions
required for coherence. Thus, studying quantum coherence is crucial for
exploring the quantum characteristics of these systems under decoherence.

According to the hierarchy of quantum resources [33], quantum coherence and
entanglement are different but related phenomena (quantum coherence
$\supseteq$ entanglement), and they serve distinct purposes in quantum
information processing. Quantum coherence, as mentioned earlier, refers to the
ability of a quantum system to exist in a superposition of states, meaning
that it can be in multiple states simultaneously. This property is vital for
some interesting tasks like quantum communication and quantum computation,
where controlling and preserving coherence is crucial for performing complex
calculations or transmitting information securely [34, 35, 36]. Entanglement,
on the other hand, is a special kind of correlation that exists between the
states of two or more quantum systems, where the states of each system are
intrinsically linked, even when they are physically separated. In some cases,
depending on the specific application, examining quantum coherence may be
preferred over entanglement. Therefore, the reason to choose quantum coherence
over entanglement (or vice versa) depends on the requirements of the
considered quantum protocol or program, as well as specific constraints and
resources.

Theoretical advancements over the past two decades have deepened our
understanding of spin squeezing [14, 37, 38, 39, 40], from its creation by
interactions in optical lattices [41, 42, 43] and Rydberg atoms [44] to its
application in programmable quantum sensors [25, 26, 27]. A coherent spin
state is defined as a minimum-uncertainty state where all components of the
collective spin operator of an ensemble of spinful particles have the same
variance. Spin squeezing refers to the redistribution of quantum fluctuations
in such a state, resulting in the variance of one spin component being smaller
than that of the other components. This phenomenon allows for measurements
that surpass the shot-noise limit, facilitating high-precision measurements,
and the generation of many-body entangled and many-body Bell correlated states
[38, 39].

The study and experimental use of spin squeezing in spin models have garnered
significant interest in recent years [45]. In quantum metrology [46, 47, 48],
spin squeezing has been employed to improve atomic precision measurements
beyond the standard quantum limit [49, 50], which is defined as the maximum
phase sensitivity achievable with separable states. This advancement has
important implications for various technologies, including Ramsey spectroscopy
[51, 52, 53], quantum interferometer [27, 54] and atomic clocks [55, 56]. By
enhancing the accuracy of measurements, spin squeezing contributes to the
development of more precise measurement devices.

In the present work, we investigate the mixed-spin (1/2,1,1/2) Heisenberg
trimer, which can be experimentally realized by the heterotrinuclear molecular
nanomagnet
$[\\{\text{Cu}^{\text{II}}\text{L}\\}_{2}\text{Ni}^{\text{II}}(\text{H}_{2}\text{O})_{2}](\text{ClO}_{4})_{2}\cdot
3\text{H}_{2}\text{O}$ hereafter referred to as the
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ molecular
complex. The magnetic properties of the
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ complex
characterized by a linear structure and a rather weak single-ion anisotropy
were studied in Ref. [57], where the strength of the exchange coupling between
CuII and NiII ions were reported along with the single-ion anisotropy of the
NiII ions.

In an earlier work [58], the magnetic properties of the
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ complex were
studied without considering single-ion anisotropy. Later, Hari _et al._ [59]
synthesized and characterized a
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ complex with
nonzero $\text{Cu}\cdots\text{Cu}$ exchange coupling, which was found to be
much smaller than the $\text{Cu}\cdots\text{Ni}$ exchange couplings. They then
investigated its magnetic properties. Comparing the results obtained for the
magnetic behavior of the two structural models—one with and one without
$\text{Cu}\cdots\text{Cu}$ exchange coupling—it was observed that the
magnetization and magnetic susceptibility behavior did not change
significantly. In a very recent theoretical work [60], the ground-state phase
diagram, bipartite entanglement, and magnetization properties of a trimeric
model similar to that reported in Ref. [59], but with different Landé
$g$-factors of CuII and NiII ions, have been investigated in detail.

Motivated by the results reported in Ref. [57], we selected the
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ complex
without $\text{Cu}\cdots\text{Cu}$ exchange interaction to investigate the
effects of the applied magnetic field and temperature on its quantum
properties. This study provides a comprehensive examination of mixed
spin-(1/2,1,1/2) Heisenberg trimer model with nonzero single-ion anisotropy in
the presence of an external magnetic field, offering novel quantum insights
that have not been previously explored in this context.

The structure of the paper is as follows: In Sec. II, we present the model and
derive its partition function from the exact solution. Section III outlines
the key formulas and theoretical approaches used in our analysis of the global
density matrix of the system, as well as two reduced density matrices for a
nonzero value of the single-ion anisotropy of the spin integer particle. This
section also explores the tripartite entanglement negativity and two bipartite
negativities with respect to the temperature and magnetic field. In Sec. IV we
study the quantum coherence of the system and compare it with the entanglement
negativities. The results, including numerical plots illustrating the thermal
evolution of the spin squeezing parameter, are discussed in Sec. V. Finally,
Sec. VII provides a summary and outlook.

##  II  Model

Let us consider the mixed spin-(1/2,1,1/2) Heisenberg trimeric cluster, which
is experimentally motivated by the heterotrinuclear
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ complex. The
model can be described by the Hamiltonian:

$\displaystyle\\!\hat{H}\\!\\!=\\!\\!J\big{(}\hat{{\bm{s}}}_{\text{a}}\\!\cdot\\!\hat{{\bm{S}}}_{\text{b}}+\hat{{\bm{S}}}_{\text{b}}\\!\cdot\\!\hat{{\bm{s}}}_{\text{c}}\big{)}\\!\\!+\\!\\!D\big{(}\hat{{S}}_{\text{b}}^{z}\big{)}^{2}\\!\\!\\!-\\!g\mu_{\text{B}}B\big{(}\hat{{s}}_{\text{a}}^{z}+\hat{{S}}_{\text{b}}^{z}+\hat{{s}}_{\text{c}}^{z}\big{)}.$

Here, the first term of the Hamiltonian (II) expresses the nearest-neighbor
exchange interactions between the spin operators $\hat{{\bm{s}}}_{\text{a}}$
and $\hat{{\bm{s}}}_{\text{c}}$ assigned to Cu atoms and the spin operator
$\hat{{\bm{S}}}_{\text{b}}$ assigned to the Ni atom respectively, with the
antiferromagnetic coupling constant $J>0$. The second term refers to the
single-ion anisotropy of the Ni atom with constant $D$. Finally, there is a
Zeeman term, where $\mu_{\text{B}}$ is the Bohr magneton, $g$ is the Landé
$g$-factor and $B$ is the static magnetic field applied along the $z$-axis.
The Hamiltonian (II) can be solved analytically through the exact
diagonalization method and a full set of eigenvalues and eigenvectors can be
obtained. The twelve energy eigenvalues and their corresponding eigenvectors,
written in standard bases of Hilbert space $\rvert\phi\rangle=\rvert
s_{\text{a}}^{z},S_{\text{b}}^{z},s_{\text{c}}^{z}\rangle$ are given in Table
1 and the probability amplitudes of the eigenvectors are given in Appendix A.
Next, the Gibbs free energy of the system can be calculated using the
following expression:

$\displaystyle\mathcal{G}=-k_{\text{B}}T~{}\ln Z,$ (2)

where $Z$ is the partition function of the system that can be explicitly
obtained from the sum of the exponentials of the eigenenergies as:

$\begin{split}&Z=\sum_{i=1}^{12}\exp(-\beta E_{i})=1+2~{}\cosh(\beta
h)\exp(-\beta D)\\\
&+\exp\left(-\\!\beta(D-J)\right)\\!+\\!2~{}\cosh\left(2\beta
h\right)\exp\left(-\beta(D+J)\right)\\\ &+4~{}\cosh\left(\beta
h\right)\exp\left(-\frac{\beta
D}{2}\right)\cosh\left(\frac{\beta}{2}\sqrt{D^{2}+4J^{2}}\right)\\\
&+2~{}\cosh\left(\frac{\beta}{2}\sqrt{(D-J)^{2}+8J^{2}}\right)\exp\left(-\frac{\beta}{2}(D-J)\right).\end{split}$
(3)

Here, $\beta=1/(k_{\text{B}}T)$, $k_{\text{B}}$ is being the Boltzmann
constant and $T$ is the absolute temperature. For convenience, we introduced a
new parameter $h=g\mu_{\text{B}}B$ representing the Zeeman term related to the
applied magnetic field. Using the Gibbs free energy (2), we can derive other
quantities such as magnetization:

$M=-\frac{\partial\mathcal{G}}{\partial B}.$ (4)

##  III The density matrices and entanglement measures

Having accomplished the eigenvalue problem of the mixed spin-(1/2,1,1/2)
Heisenberg trimer and having the partition function in hand, one can step
forward to evaluation of the density matrix which allows the calculation of
bipartite and tripartite entanglement via a measure of negativity. As the
first step, the overall density operator is defined by the formula:

$\displaystyle\hat{\rho}=\frac{1}{Z}\exp(-\beta\hat{H})=\frac{1}{Z}\sum_{i=1}^{12}\exp(-\beta
E_{i})\rvert\psi_{i}\rangle\langle\psi_{i}\lvert,$ (5)

and the density matrix corresponding to the density operator in the standard
basis $\rvert\phi\rangle$ is written.

Then, to evaluate the bipartite entanglement between any spin pair, one should
follow the calculation of the reduced density operator related to the spin
pair by tracing out the degrees of freedom of the remaining spin of the mixed
spin-(1/2,1,1/2) Heisenberg cluster. Although the reduced density operators
$\hat{\varrho}_{\text{ab}}$ and $\hat{\varrho}_{\text{bc}}$ are not
equivalent, the results of the negativity (as well as coherence) are
identical, namely $N_{\text{ab}}=N_{\text{bc}}$. Therefore, hereafter the
bipartite entanglement of parties $s_{\text{a}}$ and $S_{\text{b}}$ will be
considered that is given by:

$\hat{\varrho}_{\text{ab}}\\!\\!=\\!\text{Tr}_{s_{\text{c}}}\hat{\rho}\\!=\\!\\!\frac{1}{Z}\\!\sum_{i=1}^{12}\\!\sum_{s_{\text{c}}^{z}=\pm
1/2}\\!\\!\\!\\!\\!\text{exp}(-\\!\beta E_{i})\langle
s_{\text{c}}^{z}\rvert\psi_{i}\rangle\langle\psi_{i}\lvert
s_{\text{c}}^{z}\rangle.$ (6)

In a similar way, the reduced density operator relevant to $s_{\text{a}}$ and
$s_{\text{c}}$ is calculated by tracing out the degrees of freedom of
$S_{\text{b}}$:

$\hat{\varrho}_{\text{ac}}\\!\\!=\\!\\!\text{Tr}_{S_{\text{b}}}\hat{\rho}\\!=\\!\\!\frac{1}{Z}\\!\sum_{i=1}^{12}\\!\sum_{S_{\text{b}}^{z}=0,\pm
1}\\!\\!\\!\\!\\!\text{exp}(-\\!\beta
E_{i})\langle\\!S_{\text{b}}^{z}\rvert\psi_{i}\rangle\langle\psi_{i}\lvert
S_{\text{b}}^{z}\rangle.$ (7)

The next step is to write the matrix form of the reduced density operators and
partially transpose them with respect to one of the spins of their basis. The
elements of the partially transposed density matrices $\varrho_{\text{ab}}$
and $\varrho_{\text{ac}}$ are written as follows respectively:

$\begin{split}&\langle
s_{\text{a}}^{z_{{}^{\prime}}},S_{\text{b}}^{z_{{}^{\prime}}}\lvert\varrho_{\text{ab}}\rvert
s_{\text{a}}^{z},S_{\text{b}}^{z}\rangle^{\text{T}_{\text{a}}}\\!=\\!\langle
s_{\text{a}}^{z},S_{\text{b}}^{z_{{}^{\prime}}}\lvert\varrho_{\text{ab}}\rvert
s_{\text{a}}^{z_{{}^{\prime}}},S_{\text{b}}^{z}\rangle,\\\ &\langle
s_{\text{a}}^{z_{{}^{\prime}}},s_{\text{c}}^{z_{{}^{\prime}}}\lvert\varrho_{\text{ac}}\rvert
s_{\text{a}}^{z},s_{\text{c}}^{z}\rangle^{\text{T}_{\text{c}}}\\!=\\!\langle
s_{\text{a}}^{z_{{}^{\prime}}},s_{\text{c}}^{z}\lvert\varrho_{\text{ac}}\rvert
s_{\text{a}}^{z},s_{\text{c}}^{z_{{}^{\prime}}}\rangle.\end{split}$ (8)

Ultimately, the bipartite negativities ${N}_{\text{ab}}$ and ${N}_{\text{ac}}$
measuring the magnitude of entanglement between the spin pairs
$s_{\text{a}}$-$S_{\text{b}}$ and $s_{\text{a}}$-$s_{\text{c}}$ are calculated
according to the formula by Vidal and Werner [61] that originates from the
Peres-Horodecki separability criterion [62]:

$\displaystyle{N}_{\text{ab}}=\sum_{i=1}^{6}\frac{|\lambda_{i}|-\lambda_{i}}{2},\hskip
14.22636pt{N}_{\text{ac}}=\sum_{i=1}^{4}\frac{|\lambda_{i}|-\lambda_{i}}{2},$
(9)

where $\lambda_{i}$ are the eigenvalues of the respective partially transposed
density matrices. Both of the negativities $N_{\text{ab}}$ and
${N}_{\text{ac}}$ are calculated analytically and the possible negative
eigenvalues that contribute to $N_{\text{ab}}$ and ${N}_{\text{ac}}$ are
presented in Appendix B.

The tripartite entanglement of the mixed spin-(1/2,1,1/2) Heisenberg cluster
is quantified through the tripartite negativity, which is calculated from the
three different partial transposes of the overall density matrix. The first
partial transpose of the overall density matrix with respect to the first spin
of the basis $\rho^{\text{T}_{\text{a}}}$ is performed by fixing the quantum
numbers $S_{\text{b}}^{z}$ and $s_{\text{c}}^{z}$ in the basis $\rvert
s_{\text{a}}^{z},S_{\text{b}}^{z},s_{\text{c}}^{z}\rangle$ and interchanging
quantum number $s_{\text{a}}^{z}$. The same pattern is repeated to obtain the
partial transpose of the overall density matrix with respect to the second and
third spins of the basis:

$\begin{split}\langle
s_{\text{a}}^{z_{{}^{\prime}}},S_{\text{b}}^{z_{{}^{\prime}}},s_{\text{c}}^{z_{{}^{\prime}}}\lvert\rho\rvert
s_{\text{a}}^{z},S_{\text{b}}^{z},s_{\text{c}}^{z}\rangle^{\text{T}_{\text{a}}}\\!=\\!\langle
s_{\text{a}}^{z},S_{\text{b}}^{z_{{}^{\prime}}},s_{\text{c}}^{z_{{}^{\prime}}}\lvert\rho\rvert
s_{\text{a}}^{z_{{}^{\prime}}},S_{\text{b}}^{z},s_{\text{c}}^{z}\rangle.\\\
\langle
s_{\text{a}}^{z_{{}^{\prime}}},S_{\text{b}}^{z_{{}^{\prime}}},s_{\text{c}}^{z_{{}^{\prime}}}\lvert\rho\rvert
s_{\text{a}}^{z},S_{\text{b}}^{z_{{}^{\prime}}},s_{\text{c}}^{z}\rangle^{\text{T}_{\text{b}}}\\!=\\!\langle
s_{\text{a}}^{z_{{}^{\prime}}},S_{\text{b}}^{z},s_{\text{c}}^{z_{{}^{\prime}}}\lvert\rho\rvert
s_{\text{a}}^{z},S_{\text{b}}^{z_{{}^{\prime}}},s_{\text{c}}^{z}\rangle,\\\
\langle
s_{\text{a}}^{z_{{}^{\prime}}},S_{\text{b}}^{z_{{}^{\prime}}},s_{\text{c}}^{z_{{}^{\prime}}}\lvert\rho\rvert
s_{\text{a}}^{z},S_{\text{b}}^{z},s_{\text{c}}^{z}\rangle^{\text{T}_{\text{c}}}\\!=\\!\langle
s_{\text{a}}^{z_{{}^{\prime}}},S_{\text{b}}^{z_{{}^{\prime}}},s_{\text{c}}^{z}\lvert\rho\rvert
s_{\text{a}}^{z},S_{\text{b}}^{z},s_{\text{c}}^{z_{{}^{\prime}}}\rangle.\end{split}$
(10)

The bipartite negativities that are calculated from the partially transposed
density matrices (10) are respectively denoted by ${N}_{\text{a-bc}}$,
${N}_{\text{b-ac}}$ and ${N}_{\text{c-ab}}$ such that
${N}_{\text{I-JK}}=\sum_{i=1}^{12}\frac{|\lambda_{i}|-\lambda_{i}}{2}$ with
$\text{I}=\text{a,b,c}$ and $\text{JK}=\text{bc},\text{ac},\text{ab}$.
Consequently, the tripartite negativity of the mixed spin-(1/2,1,1/2)
Heisenberg trimer can be calculated as the geometric mean of the bipartite
negativities [63]:

${N}_{\text{abc}}=\sqrt[3]{{N}_{\text{a-bc}}~{}{N}_{\text{b-ac}}~{}{N}_{\text{c-ab}}}~{}.$
(11)

The matrix forms of overall density matrix, reduced density matrices, and
their partially transposed density matrices are presented in Appendix A.

Now, let us begin with a detailed analysis of the ground-state behavior of the
mixed spin-(1/2,1,1/2) Heisenberg trimer in presence of a magnetic field. To
proceed to the results, all interaction terms involved in the Hamiltonian (II)
will be normalized with respect to the antiferromagnetic coupling constant
$J>0$, which will henceforth serve as an energy unit. The ground-state phase
diagram of the mixed spin-(1/2,1,1/2) Heisenberg trimer is presented in Fig.
1(a) in the parameter space $D/J$ vs. $g\mu_{\text{B}}B/J$. It turns out that
there are three different ground states unambiguously given by the
eigenvectors $\rvert\psi_{9}\rangle$, $\rvert\psi_{5}\rangle$ and
$\rvert\psi_{11}\rangle$ specifically quoted in Appendix A. The inspection of
energy eigenvalues reveals the following phase boundaries between three
available ground states $\rvert\psi_{9}\rangle$, $\rvert\psi_{5}\rangle$ and
$\rvert\psi_{11}\rangle$:

$\begin{split}&\rvert\psi_{5}\rangle-\rvert\psi_{9}\rangle:h\\!=\\!\frac{J}{2}\\!+\\!\frac{1}{2}\left(\sqrt{(D\\!-\\!J)^{2}\\!+\\!8J^{2}}\\!-\\!\sqrt{D^{2}\\!+\\!4J^{2}}\right)\\\
&\rvert\psi_{11}\rangle-\rvert\psi_{5}\rangle:h=J+\frac{D}{2}+\frac{1}{2}\sqrt{D^{2}+4J^{2}}\end{split}$
(12)

Figure 1: (a) The ground-state phase diagram of the mixed spin-(1/2,1,1/2)
Heisenberg trimer in the parameter space $D/J$ vs. $g\mu_{\text{B}}B/J$; (b)
The zero-temperature magnetization normalized with respect to the saturation
value ($M_{\text{s}}=2g\mu_{\text{B}}$) as a function of the magnetic field
for three different values of the single-ion anisotropy $D/J$; (c) The zero-
temperature density plot of the tripartite negativity ${N}_{\text{abc}}$ in
the $g\mu_{\text{B}}B/J-D/J$ plane; (d) The tripartite negativity
$N_{\text{abc}}$ as a function of the magnetic field for three different
values of $D/J$ at zero temperature.

In Fig. 1(b), the zero-temperature magnetization curves displaying zero, one-
half and saturation plateaus are illustrated at three different cuts of the
relevant ground-state phase diagram. At low magnetic fields, the singlet
ground state $\rvert\psi_{9}\rangle$ having character of a quantum
superposition of four basis states $\rvert\downarrow 1\downarrow\rangle$,
$\rvert\uparrow-1\uparrow\rangle$, $\rvert\uparrow 0\downarrow\rangle$, and
$\rvert\downarrow 0\uparrow\rangle$, with zero total spin moment
$S_{\text{tot}}^{z}=0$ is favored. This ground state corresponds to a zero
magnetization plateau. At moderate magnetic fields, one contrarily encounters
the triplet ground state $\rvert\psi_{5}\rangle$ with character of a quantum
superposition of three basis states $\rvert\uparrow 1\downarrow\rangle$,
$\rvert\downarrow 1\uparrow\rangle$, $\rvert\uparrow 0\uparrow\rangle$ with
the total spin moment $S_{\text{tot}}^{z}=1$.

Figure 2: (a) The tripartite negativity $N_{\text{abc}}$, (b) the bipartite
negativity $N_{\text{ab}}$, and (c) the bipartite negativity $N_{\text{ac}}$
of the mixed spin-(1/2,1,1/2) Heisenberg trimer in the
$g\mu_{\text{B}}B/J-k_{\text{B}}T/J$ plane by assuming $D/J=0.05$; (d)
Temperature dependencies of the $N_{\text{abc}}$, (e) $N_{\text{ab}}$ and (f)
$N_{\text{ac}}$ for a few selected values of the magnetic field and
$D/J=0.05$.

The ground state $\rvert\psi_{5}\rangle$ thus corresponds to an intermediate
one-half magnetization plateau being stable in a rather wide region of the
magnetic fields. Finally, one detects the fully saturated ground state
$\rvert\psi_{11}\rangle=\rvert\uparrow 1\uparrow\rangle$, which is reached at
high enough magnetic fields where all three spins are fully aligned with the
magnetic-field direction.

Fig. 1(c) displays the density plot of the tripartite negativity
$N_{\text{abc}}$ for the mixed spin-(1/2,1,1/2) Heisenberg trimer in the same
parameter region as used in Fig. 1(a). As expected, the tripartite negativity
is highest in the ground state $\rvert\psi_{9}\rangle$ reaching a maximum
value of $N_{\text{abc}}=0.625$ at zero single-ion anisotropy. The singlet
ground state $\rvert\psi_{9}\rangle$ thus exhibits the most pronounced
collective quantum features. In contrast, the collective quantum features are
somewhat reduced within another quantum ground state $\rvert\psi_{5}\rangle$
due to partial spin alignment towards the magnetic field realized in this one-
half plateau state. Consequently, the tripartite negativity is substantially
lower in the triplet ground state $\rvert\psi_{5}\rangle$ in comparison with
the singlet ground state $\rvert\psi_{9}\rangle$. Finally, the tripartite
negativity becomes zero in the fully polarized ground state
$\rvert\psi_{11}\rangle$. For better insight, the magnetic field variations of
the tripartite negativity are shown in Fig. 1(d) for three different cuts of
the relevant density plot. A prominent feature of single-ion anisotropy is the
reduction of tripartite negativity, whereby this effect is much more
pronounced for the triplet ground state $\rvert\psi_{5}\rangle$ compared to
the singlet ground state $\rvert\psi_{9}\rangle$. It is also worth mentioning
that there is a substantial drop in tripartite negativity at the phase
boundary between $\rvert\psi_{9}\rangle$ and $\rvert\psi_{5}\rangle$, which
relates to existence of a mixed state at the relevant phase boundary that
reduces tripartite entanglement.

To provide a clear understanding of the thermal behavior of the mixed
spin-(1/2,1,1/2) Heisenberg trimer, the density plot of tripartite as well as
bipartite negativities are depicted in Fig. 2(a)-(c) for the particular case
with the fixed value of single-ion anisotropy $D/J=0.05$. It is quite clear
that the magnetic-field-driven variations of the tripartite negativity
$N_{\text{abc}}$ at sufficiently low temperatures are in a perfect agreement
with the abrupt changes of the tripartite negativity detected at zero
temperature around the magnetic fields $g\mu_{\text{B}}B/J\approx 1$ and $2$
[cf. Fig. 2(c)-(d)]. We note that, the tripartite negativity ranges from 0 to
1, where for our case its maximum is 0.625. It can be inferred from Figs. 2(b)
and 2(c) that the bipartite negativities ${N}_{\text{ab}}$ and
${N}_{\text{ac}}$ form at low enough temperatures plateaus quite similarly as
the tripartite negativity $N_{\text{abc}}$ does. The tripartite negativity
$N_{\text{abc}}$ and the bipartite negativity $N_{\text{ab}}$ display their
highest achievable values, $N_{\text{abc}}=0.625$ and $N_{\text{ab}}=1/3$, in
a low-field region inherent to the singlet ground state
$\rvert\psi_{9}\rangle$, whereas the maximum of bipartite negativity
${N}_{\text{ac}}\approx 0.0974$ can be found at moderate magnetic fields
supporting the triplet ground state $\rvert\psi_{5}\rangle$. Generally, the
maximum values of both bipartite negativities are restricted to between 0 and
0.5. Absence of a direct exchange interaction between the corner spins
$s_{\text{a}}$ and $s_{\text{c}}$ generally results in a lower value of the
bipartite entanglement ${N}_{\text{ac}}$ compared to the ${N}_{\text{ab}}$ one
assigned to directly interacting central and corner spins. A gradual vanishing
of tripartite and bipartite negativities is observed along the temperature
axis in all three density plots shown in upper panel.

The bottom panel in Fig. 2 demonstrates typical temperature dependencies of
the tripartite and bipartite negativities $N_{\text{abc}}$, ${N}_{\text{ab}}$
and ${N}_{\text{ac}}$ for a few selected values of the magnetic field. A
common property of all temperature dependencies of tripartite and bipartite
negativities in Fig. 2(d)-(f) is the gradual thermally-induced decline in the
tripartite and bipartite entanglement. In Fig. 2(d) the tripartite negativity
starts from its highest value $N_{\text{abc}}=0.625$ achieved at zero magnetic
field and vanishes at $k_{\text{B}}T/J\approx 1.15$. As the magnetic field
strengthens, the initial value of the tripartite negativity gradually
decreases, while the threshold temperature at which it becomes zero remains
unchanged. Contrary to this, the negativity ${N}_{\text{ac}}$ measuring the
bipartite entanglement between the corner spins is zero at low magnetic
fields, but starts from the initial value ${N}_{\text{ac}}\approx 0.0974$ at
moderate magnetic fields $g\mu_{\text{B}}B/J\gtrsim 1$ and is rapidly
suppressed upon increasing temperature terminating at the threshold
temperature $k_{\text{B}}T/J\approx 0.4$ [see Fig. 2(e)]. The bipartite
negativity ${N}_{\text{ab}}$ follows trends similar to those of the tripartite
negativity $N_{\text{abc}}$ when reaching a maximum initial value
${N}_{\text{ab}}=1/3$ and vanishing at the same threshold temperature
$k_{\text{B}}T/J\approx 1.1$. A less pronounced reentrance in the tripartite
and bipartite entanglement can be found at magnetic fields slightly exceeding
the saturation value $g\mu_{\text{B}}B/J\gtrsim 2.5$ due to thermal
excitations towards higher-energy levels. Besides, the peculiar changes in
tripartite and bipartite entanglement observable around the magnetic fields
$g\mu_{\text{B}}B/J\approx 1$ and $2$ can be again attributed to the relevant
magnetic-field-driven phase transitions.

##  IV Quantum coherence

There are several ways to evaluate quantum coherence [31]. Among these, the
$l_{1}$-norm of quantum coherence, which is widely used in quantum physics,
has the following definition:

$C=\sum_{i\neq j}\mid\rho_{i,j}\mid.$ (13)

According to Eq. (13), the $l_{1}$-norm of coherence can be expressed as the
sum of absolute values of the off-diagonal elements corresponding to the
selected basis [32].

To find more details regarding the quantum resources in the system under
consideration, we have plotted the $l_{1}$-norm of quantum coherence for the
tripartite ($C_{\text{abc}}$) and bipartite ($C_{\text{ab}}$ and
$C_{\text{ac}}$) states in Fig. 3. As can be seen in Fig. 2, the tripartite
and bipartite entanglement disappears at a certain threshold temperature,
however, the quantum coherence of tripartite and bipartite states captured by
$l_{1}$-norm of coherence (13) remains nonzero even at higher temperatures as
illustrated in Fig. 3(a)-(c) for a few selected values of the magnetic field
and $D/J=0.05$. This result is expected because increasing temperature can
reduce quantum correlations, such as entanglement, due to rising role of
thermal fluctuations in the system. The quantum coherence can nevertheless
remain a significant resource compared to entanglement based on the hierarchy
of quantum resources [33] as the amount of entanglement is bounded by the
quantum coherence [64]. It is worth noting that the qualitative behavior of
quantum coherence is similar to that of entanglement, apart from some minor
differences [compare Figs. 2(d)-(f) and 3(a)-(c)]. Moreover, despite the
quantitative differences between tripartite and bipartite quantum coherences,
their qualitative behavior is often the same, except for the bipartite
coherence $C_{\text{ac}}$ at low temperatures.

Fig. 3(d)-(f) shows the thermal coherence of the mixed spin-(1/2,1,1/2)
Heisenberg trimer as a function of the magnetic field $g\mu_{\text{B}}B/J$ for
a few selected values of temperature and one fixed value of the single-ion
anisotropy $D/J=0.05$. Regardless of temperature and single-ion anisotropy
effects, one can see that the tripartite and bipartite quantum coherences
decrease upon strengthening of the magnetic field. The only exception to this
rule is the $l_{1}$-norm of the bipartite coherence $C_{\text{ac}}$, which
displays a peculiar transient strengthening due to the magnetic field. In
general, the suppression of thermal coherence by the magnetic field can be
related to the Zeeman effect, which causes splitting of energy levels into
sublevels (i.e. splitting of spin multiplets) and energetically favors the
sublevels with the highest value of $z$-component of the total spin. As a
result, the qualitative behavior of tripartite coherence [Fig. 3(d)] at zero
and low temperatures is quite similar to the tripartite entanglement and
magnetization [see blue lines in Figs. 1(b) and 1(d)]. The inclusion of a
relatively small single-ion anisotropy may further cause zero-field splitting
of energy levels and hence, it may additionally enhance the effect of external
magnetic field. Altogether, the combined effect of magnetic field and single-
ion anisotropy results in a more rapid loss of quantum coherence of the mixed
spin-(1/2,1,1/2) Heisenberg trimer under thermal conditions. While the
tripartite coherence and tripartite entanglement disappear at absolute zero
temperature at the saturation field, it can be observed from Figs. 2 and 3
that increasing temperature can delay this disappearance for larger magnetic
fields.

Figure 3: (a)-(c) The temperature dependencies of the quantum coherence
(tripartite and bipartite) of the mixed spin-(1/2,1,1/2) Heisenberg trimer for
a few selected values of the magnetic field and one fixed value of the single-
ion anisotropy $D/J=0.05$, (d)-(f) the quantum coherence as a function of the
magnetic field for the fixed value of the single-ion anisotropy $D/J=0.05$ and
four selected temperatures.

##  V Spin squeezing

A complementary investigation of quantum properties of the mixed
spin-(1/2,1,1/2) Heisenberg trimer involves the spin squeezing, which can be
quantified through the spin squeezing parameter whose definition is based on
the uncertainty of spin measurements along different spatial directions. This
measure quantifies the degree to which quantum correlations within a multi-
particle spin state reduce the variance of a specific spin component [38]:

$\xi^{2}=\frac{2(\Delta\mathcal{\hat{\bm{J}}}_{\vec{n}_{\perp}})^{2}}{\lvert\langle\mathcal{\hat{\bm{J}}}\rangle\rvert},$
(14)

where
$\mathcal{\hat{J}}=\hat{\bm{s}}_{\text{a}}+\hat{\bm{S}}_{\text{b}}+\hat{\bm{s}}_{\text{c}}$
is the sum of spin operators. For our mixed spin model, we consider the
maximum total number of spins as $L^{\prime}=4$ where
$\mid\\!\langle\mathcal{\hat{J}}\rangle\\!\mid=L^{\prime}/2$. This originates
from the effective four spin-1/2 system by assuming a spin-1 particle as two
interacting spin-1/2 particles. Keeping this assumption in mind, and
considering the mean spin along the $z$-direction, $\vec{n}_{\perp}$ will
align in the $x$-$y$ plane:
$\mathcal{\hat{J}}_{\vec{n}_{\perp}}=\cos\theta\mathcal{\hat{J}}_{x}+\sin\theta\mathcal{\hat{J}}_{y}$,
with the variance

$\begin{split}(\Delta\mathcal{\hat{J}}_{\vec{n}_{\perp}})^{2}=&\langle\mathcal{\hat{J}}_{\vec{n}_{\perp}}^{2}\rangle-\langle\mathcal{\hat{J}}_{\vec{n}_{\perp}}\rangle^{2}\\\
=&\\!\langle\mathcal{\hat{J}}_{x}^{2}+\mathcal{\hat{J}}_{y}^{2}\rangle\\!+\cos(2\theta)\langle\mathcal{\hat{J}}_{x}^{2}\\!-\\!\mathcal{\hat{J}}_{y}^{2}\rangle\\!\\\
&+\sin(2\theta)\langle\mathcal{\hat{J}}_{x}\mathcal{\hat{J}}_{y}\\!+\\!\mathcal{\hat{J}}_{y}\mathcal{\hat{J}}_{x}\rangle.\end{split}$
(15)

Consequently, the spin squeezing parameter is calculated by minimizing the
variance with respect to $\theta$. Note that
$\langle\mathcal{\hat{J}}_{\alpha}\rangle=\langle\mathcal{\hat{J}}_{\alpha}\mathcal{\hat{J}}_{z}\rangle=\langle\mathcal{\hat{J}}_{z}\mathcal{\hat{J}}_{\alpha}\rangle=0,~{}\alpha\in\\{x,y\\}$.
After a straightforward manipulation the spin squeezing parameter becomes
[38]:

$\begin{split}\xi^{2}=&\dfrac{2}{L^{\prime}}\left[\langle\mathcal{\hat{J}}_{x}^{2}+\mathcal{\hat{J}}_{y}^{2}\rangle-\sqrt{\langle\mathcal{\hat{J}}_{x}^{2}-\mathcal{\hat{J}}_{y}^{2}\rangle^{2}+\langle[\mathcal{\hat{J}}_{x},\mathcal{\hat{J}}_{y}]_{+}\rangle^{2}}~{}\right],\end{split}$
(16)

where
$[\mathcal{\hat{J}}_{x},\mathcal{\hat{J}}_{y}]_{+}=\mathcal{\hat{J}}_{x}\mathcal{\hat{J}}_{y}+\mathcal{\hat{J}}_{y}\mathcal{\hat{J}}_{x}$
is the anticommutator for operators $\mathcal{\hat{J}}_{x}$ and
$\mathcal{\hat{J}}_{y}$. To bring insight into the spin squeezing of the mixed
spin-(1/2,1,1/2) Heisenberg trimeric cluster, the spin squeezing parameter was
calculated according to Eq. (16) in a thermal state $\hat{\rho}$ as a function
of the magnetic field $g\mu_{\text{B}}B/J$ at a few selected temperatures and
a fixed value of the single-ion anisotropy $D/J=0.05$. Fig. 4(a) bears
evidence that the ground state of the mixed spin-(1/2,1,1/2) Heisenberg trimer
can be either a squeezed, unsqueezed or a coherent state. The triplet ground
state emerging at moderate magnetic fields $g\mu_{\text{B}}B/J>1$ belongs to a
coherent state with $\xi^{2}=1$, whereas the singlet ground state emerging at
lower fields $g\mu_{\text{B}}B/J<1$ features a spin squeezing $\xi^{2}<1$ (see
black curve for $k_{\text{B}}T/J=0.002$). It is also evidence that close to
the critical magnetic field $g\mu_{\text{B}}B/J=1$, the state of the system
becomes unsqueezed ($\xi^{2}>1$). With increase of the temperature, the state
of the system remains squeezed for higher magnetic fields. The inset of Fig.
4(a) actually verifies the behavior of $\xi^{2}$ as a function of the magnetic
field at sufficiently low temperature $k_{\text{B}}T/J=0.1$. It is evident
from the inset that the system is indeed in a squeezed state $\xi^{2}<1$ at
low enough magnetic fields $g\mu_{\text{B}}B/J\lesssim 1$, while it is
evolving to an unsqueezed state $\xi^{2}=1$ at higher magnetic fields
$g\mu_{\text{B}}B/J\gtrsim 1$. With a further increase of the magnetic field,
the state of the system becomes coherent. This illustrates how the applied
magnetic field can significantly influence the nature of the state of a
quantum system at low temperature.

In Fig. 4(b), we illustrate temperature variations of the spin squeezing
parameter $\xi^{2}$ for the mixed spin-(1/2,1,1/2) Heisenberg trimer at six
different values of the magnetic field. As the magnetic field increases, the
temperature at which the largest spin squeezing is achieved (i.e. the smallest
$\xi^{2}$) shifts to higher values. This indicates that lower magnetic fields
stabilize the squeezed state, which consequently allows a more significant
spin squeezing at higher temperatures. Interestingly, for the critical
magnetic field $g\mu_{\text{B}}B/J=1$ (blue solid line in Fig. 4(b)), the
behavior of the squeezing parameter is quite different. It reaches its maximum
at finite low temperatures and then gradually decreases with increasing
temperature. From this figure, it can be seen that for fields below the
critical value $g\mu_{\text{B}}B/J=1$, the squeezing parameter shows a global
minimum, while for higher fields $g\mu_{\text{B}}B/J>1$, it exhibits a smooth
decrease upon heating. Eventually, this parameter tends to the value
$\xi^{2}=5/6$ at high temperatures that is the indication of the thermal
steady-squeezed state.

Figure 4(c) displays a density plot of the spin squeezing parameter $\xi^{2}$
across $g\mu_{\text{B}}B/J-k_{\text{B}}T/J$ plane. The color scale at the top
indicates values of the spin squeezing parameter $\xi^{2}$ ranging from 0.508
to 1.5. At lower magnetic fields than the critical value
$g\mu_{\text{B}}B/J=1$, the system can achieve a global minimum
$\xi^{2}_{\text{min}}$. The line of yellow circles illustrates positions of
this global minimum $\xi^{2}_{\text{min}}$ in the
$g\mu_{\text{B}}B/J-k_{\text{B}}T/J$ plane. At low magnetic fields,
$\xi^{2}_{\text{min}}$ is found at lower temperatures indicating significant
spin squeezing in this regime. As the magnetic field increases, the position
of $\xi^{2}_{\text{min}}$ shifts towards higher temperatures. This shift
suggests that stronger magnetic fields are needed to maintain the global
minimum of spin squeezing at higher temperatures. This behavior indicates the
intricate interplay between thermal fluctuations and the applied magnetic
field in the degree of the squeezing parameter. To summarize, it has been
verified that the mixed spin-(1/2,1,1/2) Heisenberg trimer can be driven from
a coherent state to a spin squeezed state and hence, it has potential
application in quantum information processing and quantum sensing where the
enhancement of spin squeezing is crucial.

Figure 4: (a) The spin squeezing parameter (16) of the mixed spin-(1/2,1,1/2)
Heisenberg trimer as a function of the magnetic field at several different
temperatures. The inset shows a detail for the parameter space at low enough
temperature $k_{\text{B}}T/J=0.1$. (b) Temperature variations of the spin
squeezing parameter $\xi^{2}$ for several values of the magnetic field. (c)
The density plot the squeezing parameter in the
$g\mu_{\text{B}}B/J-k_{\text{B}}T/J$ plane. The line marked with yellow
circles indicates the position of global minimum squeezing
$\xi^{2}_{\text{min}}$ in the $g\mu_{\text{B}}B/J-k_{\text{B}}T/J$ plane. The
light region indicates that the state od the system becomes unsqueezed where
the squeezing parameter reaches its maximum 1.5 close to the critical magnetic
field $g\mu_{\text{B}}B/J=1$ at low temperature. In all panels, a fixed value
of the single-ion anisotropy $D/J=0.05$ was assumed.

Figure 5: The Husimi $Q$-function calculated for a thermal state of the mixed
spin-(1/2,1,1/2) Heisenberg trimer characterized through the density operator
$\rho$ (5) at a fixed value of temperature $k_{\text{B}}T/J=0.1$, the single-
ion anisotropy $D/J=0.05$ and a few selected values of the magnetic field: (a)
$g\mu_{\text{B}}B/J=0.0$, (b) $g\mu_{\text{B}}B/J=0.5$, (c)
$g\mu_{\text{B}}B/J=1.0$, (d) $g\mu_{\text{B}}B/J=1.5$, (e)
$g\mu_{\text{B}}B/J=2.0$, (f) $g\mu_{\text{B}}B/J=2.5$.

Next, let us examine the impact of the applied magnetic field on a spin
squeezing state within the mixed spin-(1/2,1,1/2) Heisenberg trimer by
utilizing a quasi-probability distribution based on the Husimi $Q$-function.
The Husimi $Q$-function is mathematically defined as:

$Q(\theta,\phi)=\frac{1}{\pi}\langle\alpha|\hat{\rho}|\alpha\rangle,$ (17)

where $\hat{\rho}$ (5) represents the density operator of the system. The
vector $|\alpha\rangle$ corresponds to a spin coherent state unambiguously
given by spherical coordinates $\theta$ and $\phi$ on the Bloch sphere
combining these two spherical angles into a single complex parameter:

$\alpha=\sin\theta\cos\phi+{\rm i}\sin\theta\sin\phi.$ (18)

The spin coherent state $|\alpha\rangle$ is the state obtained by rotating the
highest-weight $\hat{S}^{z}_{\text{T}}$ eigenstate $|\uparrow
1\uparrow\rangle$ counterclockwise by angle $\theta$ about the $y$-axis, and
then by angle $\phi$ about the $z$-axis:
$|\alpha\rangle=e^{i\phi\hat{\mathcal{J}}_{z}}e^{i\theta\hat{\mathcal{J}}_{y}}|\uparrow
1\uparrow\rangle$. The coherent state $|\alpha\rangle$ can be also introduced
in terms of the angular momentum raising and lowering operators
$\hat{\mathcal{J}}_{\pm}=\hat{\mathcal{J}}_{x}\pm i\hat{\mathcal{J}}_{y}$ as
follows:

$|\alpha\rangle=\exp\left[\frac{\theta}{2}\left(\hat{\mathcal{J}}_{-}e^{i\phi}-\hat{\mathcal{J}}_{+}e^{-i\phi}\right)\right]|\uparrow
1\uparrow\rangle.$ (19)

We have implemented QuTiP toolbox [65, 66] for calculating the Husimi
$Q$-function for the mixed spin-(1/2,1,1/2) Heisenberg trimer at a fixed value
of temperature $k_{\text{B}}T/J=0.1$, the single-ion anisotropy $D/J=0.05$ and
a few selected values of the magnetic field, which provides additional insight
into the squeezing effect particularly illuminated in the inset of Fig. 4(a).
In the squeezed region $g\mu_{\text{B}}B/J<1$ with $\xi^{2}<1$, this operation
produces a squeezed state that is stretched both along $x$\- and
$y$-directions as illustrated in Figs. 5(a) and (b). In a vicinity of the
transition field $g\mu_{\text{B}}B/J\approx 1$, the squeezing property starts
to fade as exemplified by Fig. 5(c). Fig. 5(d)-(f) illustrates the Husimi
$Q$-function for the unsqueezed parameter region $g\mu_{\text{B}}B/J>1$, where
the spin squeezed state counterproductively diminishes. Therefore, the state
is sufficiently squeezed only if a strength of the magnetic field is lower
than the transition field $g\mu_{\text{B}}B_{\text{c}}/J\approx 1$. For
instance, an unsqueezed kicked top displayed in Fig. 5(f) showcases a coherent
state with a Gaussian distribution of quantum fluctuations, which results from
a quasi-probability distribution with a circular cross-section whose
probability density is highest at the center and gradually diminishes
outwards.

##  VI Quantum features of molecular nanomagnet
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$

In this section, we present a theoretical prediction for the degree of
tripartite entanglement, coherence, and spin squeezing parameter of the
heterotrinuclear molecular nanomagnet
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$, which
represents an experimental realization of the mixed spin-(1/2,1,1/2)
Heisenberg trimer. It has been verified in Ref. [57] that magnetic properties
of the molecular compound
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ can be
accurately modeled by the mixed spin-(1/2,1,1/2) Heisenberg trimer with an
isotropic exchange-coupling constant $J=22.8\,\text{cm}^{-1}$, a small nonzero
single-ion anisotropy $D=0.05\,\text{cm}^{-1}$, and an average gyromagnetic
ratio $g=2.227$ for $\text{Cu}^{2+}$ and $\text{Ni}^{2+}$ magnetic ions [57].
We will use this set of parameters to make theoretical predictions for the
tripartite negativity, coherence, and spin squeezing parameter of the
heterotrinuclear molecular complex
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$.

Fig. 6(a) illustrates temperature dependencies of the tripartite negativity
$N_{\text{abc}}$ of the mixed spin-(1/2,1,1/2) Heisenberg trimer adjusted to a
theoretical modeling of the molecular nanomagnet
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ for several
values of the magnetic field. At zero magnetic field, the tripartite
negativity gradually decreases upon increasing temperature when starting from
a zero-temperature asymptotic limit $N_{\text{abc}}=5/8$ and vanishing
approximately at the threshold temperature $T\approx 37\,K$. The same general
trends can be also detected at magnetic fields below critical value
$B=J$(black line marked with circles) favoring the singlet ground state
$|\psi_{9}\rangle$. In the range of magnetic fields between special points
$B=J$ and $B=2J$(blue line marked with hexagons), the tripartite negativity
starts from a lower local maximum of around $N_{\text{abc}}\approx 0.454$. The
weaker value of the tripartite entanglement bears a close relation with
presence of the triplet ground state $|\psi_{5}\rangle$, but the respective
threshold temperature remains unaffected by this change. A notable non-
monotonous thermal dependence of the tripartite negativity is finally observed
when the magnetic field slightly exceeds the saturation value $B\approx 2J$.
Under such conditions, the tripartite negativity $N_{\text{abc}}$ exhibits a
reentrant behavior when it first becomes nonzero at a lower threshold
temperature, then rises steadily to a relatively low local maximum, which is
successively followed by a gradual monotonous decline until it completely
disappears at a higher threshold temperature (see the particular case for
$B=50$ T).

Figure 6: Temperature variations of three basic quantum characteristics of the
mixed spin-(1/2,1,1/2) Heisenberg trimer with an exchange-coupling constant
$J=22.8\,\text{cm}^{-1}$, a single-ion anisotropy $D=0.05\,\text{cm}^{-1}$,
and gyromagnetic ratio $g=2.227$ adjusted according to Ref. [57] to a
theoretical modeling of the molecular nanomagnet
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ at a few
different magnetic field strengths: (a) the tripartite negativity
$N_{\text{abc}}$, (b) the quantum coherence $C_{\text{abc}}$, (c) the spin
squeezing parameter $\xi^{2}$. The labels for the magnetic field is given in
the panel (a).

Fig. 6(b) displays typical temperature dependencies of the quantum coherence
$C_{\text{abc}}$ of the mixed spin-(1/2,1,1/2) Heisenberg trimer by adapting
the same set of parameters as used in Fig. 6(a) for a theoretical modeling of
the tripartite entanglement of the molecular complex
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$. At low
temperatures, the coherence $C_{\text{abc}}$ exhibits a quite analogous
behavior to the tripartite negativity including three different zero-
temperature asymptotic values assigned to the singlet ground state
$|\psi_{9}\rangle$, the triplet ground state $|\psi_{5}\rangle$, and the fully
polarized ground state $|\psi_{11}\rangle$, respectively. In contrast to the
tripartite negativity, the coherence $C_{\text{abc}}$ does not completely
vanish at the threshold temperature but it persists of being nonzero in a
high-temperature regime due to a more gradual thermally-induced decay. This
observation bears evidence that the molecular nanomagnet
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ retains a
certain degree of quantum coherence even at elevated temperatures, which are
well above the threshold temperature where the quantum entanglement is
completely lost.

Last but not least, Fig. 6(c) illustrates typical temperature variations of
the spin squeezing parameter $\xi^{2}$ calculated from Eq. (16) specifically
for the mixed spin-(1/2,1,1/2) Heisenberg trimer resembling the molecular
complex $\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$. The
spin squeezing parameter $\xi^{2}$ allows to distinguish a coherent spin state
from a squeezed one. For nonzero magnetic field with excluding critical point
$B=J$ (black line marked with circles), at low enough temperatures, the spin
squeezing parameter approaches the value $\xi^{2}=1$ indicating a perfect
coherent state. With raising the temperature, this parameter starts to
increase and reaches a round maximum then gradually decreases until surpasses
the coherent limit. At zero magnetic field (blue line), the spin squeezing
parameter $\xi^{2}$ drops down upon increasing temperature to a global minimum
observed around $T\approx 30\,\text{K}$. This global minimum becomes gradually
flattens and shifts towards higher temperatures as the magnetic field
strengthens. After reaching the global minimum, the squeezing parameter
$\xi^{2}$ rises steadily with temperature that suggests a reduction in quantum
correlations and entanglement. The increase in the spin squeezing parameter
$\xi^{2}$ continues up to a certain temperature, beyond which it tends to
$\xi^{2}\approx 5/6$ where a the state of the system is steady-state spin
squeezed. As discussed in Fig. 4(b), for the special case $B=J$we see that the
parameter $\xi^{2}$ is in its maximum value 1.5 at $T\approx 0$. Then with
increase of the temperature, it decreases and surpasses the coherent limit
near the temperature $T\approx 20\,\text{K}$ and reaches constant value
$\xi^{2}\approx 0.75$ at sufficiently high temperature.

The observed behavior of the spin squeezing parameter can be related to the
previously discussed tripartite entanglement and quantum coherence. At low
temperatures, the initial drop in the the spin squeezing parameter $\xi^{2}$
corresponds to the presence of tripartite entanglement with still sizable
quantum correlations at the global minimum. On the other hand, the
nonmonotonic behavior of the tripartite negativity $N_{\text{abc}}$ observed
at sufficiently high magnetic fields $B\gtrsim 2J$ T, where the negativity
first increases before ultimately declining, may also be reflected in a
gradual decrease of the spin squeezing parameter $\xi^{2}$ without reaching a
global minimum under similar conditions (see black dotted lines in Fig. 6(a)
and 6(c) for $B=50\,\text{T}$). Although the spin squeezing parameter
$\xi^{2}$ increases with temperature in a high-temperature regime it never
approaches the maximal value $\xi^{2}=1$ what signifies a loss of quantum
coherence due to the thermal decoherence.

##  VII Conclusions

In the present paper, we have comprehensively investigated basic quantum
characteristics of the mixed spin-(1/2,1,1/2) Heisenberg trimer in a presence
of the external magnetic field, which was designed for a theoretical modeling
of molecular nanomagnets such as the heterotrinuclear coordination compound
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$. By
employing rigorous analytical and numerical calculations, we have examined
distributions of bipartite and tripartite entanglement, the $l_{1}$-norm of
quantum coherence, and the spin squeezing parameter under the influence of an
external magnetic field. The key finding stemming from the present study is
that the mixed spin-(1/2,1,1/2) Heisenberg trimer may display significant
bipartite and tripartite entanglement up to relatively high temperatures and
magnetic fields, whereby the quantum coherence is maintained even at elevated
temperatures compared to entanglement. This indicates that the investigated
quantum spin system may retain a high degree of the quantum coherence despite
strong thermal fluctuations. As far as the spin squeezing is concerned, the
temperature at which the largest spin squeezing is achieved shifts to higher
values upon strengthening of the magnetic field. This result implies that
smaller magnetic fields stabilize the squeezed state and allow more
significant spin squeezing at lower temperatures. The intricate interplay
between quantum fluctuations and thermal effects is crucial for optimizing
conditions for achieving maximal spin squeezing, which is vital for
applications in quantum information processing and quantum sensing.

Exact results for quantum features of the mixed spin-(1/2,1,1/2) Heisenberg
trimer has been also adapted for a theoretical modeling of the
heterotrinuclear molecular nanomagnet
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$. It has been
verified that the molecular complex
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ exhibits a
significant bipartite entanglement between
$\text{Cu}^{\text{II}}$-$\text{Ni}^{\text{II}}$ magnetic ions and a
significant tripartite entanglement between
$\text{Cu}^{\text{II}}$-$\text{Ni}^{\text{II}}$-$\text{Cu}^{\text{II}}$
magnetic ions, which persist up to relatively high temperatures $37\,\text{K}$
and magnetic fields $50\,\text{T}$. In addition, it has been theoretically
predicted that molecular nanomagnet
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ displays a
high degree of the quantum coherence persisting even at elevated temperatures
in spite of strong thermal fluctuations. Last but not least, our theoretical
results for the molecular compound
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ also reveal
optimal conditions for achieving the highest degree of the spin squeezing,
which are achieved at zero magnetic field around $T\approx 20\,\text{K}$
corresponding to the global minimum of the spin squeezing parameter. To
summarize, our analysis underscores the importance of the magnetic field and
temperature in modulating the bipartite and tripartite entanglement, quantum
coherence, and spin squeezing in molecular nanomagnet
$\text{Cu}^{\text{II}}\text{Ni}^{\text{II}}\text{Cu}^{\text{II}}$ and related
group of magnetic materials, which pave the way for advanced applications in
quantum technologies

## Acknowledgements

A.G. acknowledges Saeed’s Quantum Information Group (SSQIG) for accommodating
hospitality during the course of this project. H.A.Z. acknowledges the
financial support provided under the postdoctoral fellowship program of P. J.
Šafárik University in Košice, Slovakia. S.H. was supported by Semnan
University under Contract No. 21270. J.S. acknowledges financial support by
the grant of Slovak Research and Development Agency provided under the
contract No. APVV-22-0172 and by the grant of The Ministry of Education,
Research, Development and Youth of the Slovak Republic under the contract No.
VEGA 1/0695/23. V.O and Zh. A. acknowledge partial financial support form
ANSEF (Grant No. PS-condmatth-2884) and from CS RA MESCS (Grants No.
21AG-1C047, 21AG-1C006 and 23AA-1C032).

##  Appendix A Eigenvectors and eigenenergies

Eigenvectors of the mixed spin-(1/2,1,1/2) Heisenberg trimer given by the
Hamiltonian II are defined in Table 1, whereby the respective probability
amplitudes entering into the relevant eigenvectors are specifically given in
Table 2.

Table 1:  Eigenvectors, total quantum spin number $S_{\text{T}}^{z}$, and eigenenergies of the mixed spin-(1/2,1,1/2) Heisenberg trimer. Eigenvectors | $S_{\text{T}}^{z}$ | Eigenenergies
---|---|---
$\rvert\psi_{1}\rangle$ = $\frac{1}{\sqrt{2}}(\rvert\downarrow 0\uparrow\rangle-\rvert\uparrow 0\downarrow\rangle$) | 0 | $E_{1}=0$
$\rvert\psi_{2}\rangle$ = $\frac{1}{\sqrt{2}}(\rvert\downarrow 1\downarrow\rangle-\rvert\uparrow-1\uparrow\rangle$) | 0 | $E_{2}=D-J$
$\rvert\psi_{3}\rangle$ = $\frac{1}{\sqrt{2}}(\rvert\downarrow 1\uparrow\rangle-\rvert\uparrow 1\downarrow\rangle$) | 1 | $E_{3}=D-h$
$\rvert\psi_{4}\rangle=\frac{1}{\sqrt{2}}(\rvert\downarrow-1\uparrow\rangle-\rvert\uparrow-1\downarrow\rangle$) | -1 | $E_{4}=D+h$
$\rvert\psi_{5}\rangle=a(\rvert\uparrow 1\downarrow\rangle+\rvert\downarrow 1\uparrow\rangle)+b\rvert\uparrow 0\uparrow\rangle$ | 1 | $E_{5}=\frac{D}{2}-h-\frac{1}{2}\sqrt{D^{2}+4J^{2}}$
$\rvert\psi_{6}\rangle=c(\rvert\uparrow 1\downarrow\rangle+\rvert\downarrow 1\uparrow\rangle)+d\rvert\uparrow 0\uparrow\rangle$ | 1 | $E_{6}=\frac{D}{2}-h+\frac{1}{2}\sqrt{D^{2}+4J^{2}}$
$\rvert\psi_{7}\rangle=a(\rvert\uparrow-1\downarrow\rangle+\rvert\downarrow-1\uparrow\rangle)+b\rvert\downarrow 0\downarrow\rangle$ | -1 | $E_{7}=\frac{D}{2}+h-\frac{1}{2}\sqrt{D^{2}+4J^{2}}$
$\rvert\psi_{8}\rangle=c(\rvert\uparrow-1\downarrow\rangle+\rvert\downarrow-1\uparrow\rangle)+d\rvert\downarrow 0\downarrow\rangle$ | -1 | $E_{8}=\frac{D}{2}+h+\frac{1}{2}\sqrt{D^{2}+4J^{2}}$
$\rvert\psi_{9}\rangle=e(\rvert\uparrow 0\downarrow\rangle+\rvert\downarrow 0\uparrow\rangle)+f(\rvert\uparrow-1\uparrow\rangle+\rvert\downarrow 1\downarrow\rangle)$ | 0 | $E_{9}=\frac{D-J}{2}-\sqrt{(\frac{D-J}{2})^{2}+2J^{2}}$
$\rvert\psi_{10}\rangle=g(\rvert\uparrow 0\downarrow\rangle+\rvert\downarrow 0\uparrow\rangle)+h(\rvert\uparrow-1\uparrow\rangle+\rvert\downarrow 1\downarrow\rangle)$ | 0 | $E_{10}=\frac{D-J}{2}+\sqrt{(\frac{D-J}{2})^{2}+2J^{2}}$
$\rvert\psi_{11}\rangle=\rvert\uparrow 1\uparrow\rangle$ | 2 | $E_{11}=D+J-2h$
$\rvert\psi_{12}\rangle=\rvert\downarrow-1\downarrow\rangle$ | -2 | $E_{12}=D+J+2h$

with the corresponding probability amplitudes given in Table 2.

Table 2:  Probability amplitudes of the eigenvectors of the mixed spin-(1/2,1,1/2) Heisenberg trimer. $a=\frac{\frac{D}{2}-\sqrt{J^{2}+\left(\frac{D}{2}\right)^{2}}}{\sqrt{2J^{2}+2\left(\frac{D}{2}-\sqrt{J^{2}+\left(\frac{D}{2}\right)^{2}}\right)^{2}}}$ | $b=\frac{J}{\sqrt{J^{2}+\left(\frac{D}{2}-\sqrt{J^{2}+(\frac{D}{2})^{2}}\right)^{2}}}$
---|---
$c=\frac{\frac{D}{2}+\sqrt{J^{2}+\left(\frac{D}{2}\right)^{2}}}{\sqrt{2J^{2}+2\left(\frac{D}{2}+\sqrt{J^{2}+\left(\frac{D}{2}\right)^{2}}\right)^{2}}}$ | $d=\frac{J}{\sqrt{J^{2}+\left(\frac{D}{2}+\sqrt{J^{2}+(\frac{D}{2})^{2}}\right)^{2}}}$
$e=\frac{J}{\sqrt{2J^{2}+\left(\frac{D-J}{2}-\sqrt{2J^{2}+(\frac{D-J}{2})^{2}}\right)^{2}}}$ | $f=\frac{\frac{D-J}{2}-\sqrt{2J^{2}+\left(\frac{D-J}{2}\right)^{2}}}{\sqrt{4J^{2}+2\left(\frac{D-J}{2}-\sqrt{2J^{2}+(\frac{D-J}{2})^{2}}\right)^{2}}}$
$g=\frac{J}{\sqrt{2J^{2}+\left(\frac{D-J}{2}+\sqrt{2J^{2}+(\frac{D-J}{2})^{2}}\right)^{2}}}$ | $h=\frac{\frac{D-J}{2}+\sqrt{2J^{2}+\left(\frac{D-J}{2}\right)^{2}}}{\sqrt{4J^{2}+2\left(\frac{D-J}{2}+\sqrt{2J^{2}+(\frac{D-J}{2})^{2}}\right)^{2}}}$

##  Appendix B The total, partially transposed and reduced density matrices

The total density matrix $\rho$ of the mixed spin-(1/2,1,1/2) Heisenberg
trimer reads:

$\displaystyle\rho=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}\rho_{1,1}&0&0&0&0&0&0&0&0&0&0&0\\\
0&\rho_{2,2}&\rho_{2,3}&0&0&0&\rho_{2,7}&0&0&0&0&0\\\
0&\rho_{2,3}&\rho_{3,3}&0&0&0&\rho_{3,7}&0&0&0&0&0\\\
0&0&0&\rho_{4,4}&\rho_{4,5}&0&0&\rho_{4,8}&\rho_{4,9}&0&0&0\\\
0&0&0&\rho_{4,5}&\rho_{5,5}&0&0&\rho_{5,8}&\rho_{5,9}&0&0&0\\\
0&0&0&0&0&\rho_{6,6}&0&0&0&\rho_{6,10}&\rho_{6,11}&0\\\
0&\rho_{2,7}&\rho_{3,7}&0&0&0&\rho_{7,7}&0&0&0&0&0\\\
0&0&0&\rho_{4,8}&\rho_{5,8}&0&0&\rho_{8,8}&\rho_{8,9}&0&0&0\\\
0&0&0&\rho_{4,9}&\rho_{5,9}&0&0&\rho_{8,9}&\rho_{9,9}&0&0&0\\\
0&0&0&0&0&\rho_{6,10}&0&0&0&\rho_{10,10}&\rho_{10,11}&0\\\
0&0&0&0&0&\rho_{6,11}&0&0&0&\rho_{10,11}&\rho_{11,11}&0\\\
0&0&0&0&0&0&0&0&0&0&0&\rho_{12,12}\end{pmatrix}$

The partially transposed density matrices with respect to each spin
$s_{\text{a}}$, $S_{\text{b}}$ and $s_{\text{c}}$ are denoted by
$\rho^{\text{T}_{\text{a}}}$, $\rho^{\text{T}_{\text{b}}}$ and
$\rho^{\text{T}_{\text{c}}}$, given by:

$\displaystyle\rho^{\text{T}_{\text{a}}}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}\rho_{1,1}&0&0&0&0&0&0&\rho_{2,7}&\rho_{3,7}&0&0&0\\\
0&\rho_{2,2}&\rho_{2,3}&0&0&0&0&0&0&\rho_{4,8}&\rho_{5,8}&0\\\
0&\rho_{2,3}&\rho_{3,3}&0&0&0&0&0&0&\rho_{4,9}&\rho_{5,9}&0\\\
0&0&0&\rho_{4,4}&\rho_{4,5}&0&0&0&0&0&0&\rho_{6,10}\\\
0&0&0&\rho_{4,5}&\rho_{5,5}&0&0&0&0&0&0&\rho_{6,11}\\\
0&0&0&0&0&\rho_{6,6}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&\rho_{7,7}&0&0&0&0&0\\\
\rho_{2,7}&0&0&0&0&0&0&\rho_{8,8}&\rho_{8,9}&0&0&0\\\
\rho_{3,7}&0&0&0&0&0&0&\rho_{8,9}&\rho_{9,9}&0&0&0\\\
0&\rho_{4,8}&\rho_{4,9}&0&0&0&0&0&0&\rho_{10,10}&\rho_{10,11}&0\\\
0&\rho_{5,8}&\rho_{5,9}&0&0&0&0&0&0&\rho_{10,11}&\rho_{11,11}&0\\\
0&0&0&\rho_{6,10}&\rho_{6,11}&0&0&0&0&0&0&\rho_{12,12}\end{pmatrix}$

$\displaystyle\rho^{\text{T}_{\text{b}}}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}\rho_{1,1}&0&0&\rho_{2,3}&0&0&0&0&\rho_{3,7}&0&0&\rho_{5,8}\\\
0&\rho_{2,2}&0&0&0&0&\rho_{2,7}&0&0&\rho_{4,8}&0&0\\\
0&0&\rho_{3,3}&0&0&\rho_{4,5}&0&0&0&0&\rho_{5,9}&0\\\
\rho_{2,3}&0&0&\rho_{4,4}&0&0&0&0&\rho_{4,9}&0&0&\rho_{6,10}\\\
0&0&0&0&\rho_{5,5}&0&0&0&0&0&0&0\\\
0&0&\rho_{4,5}&0&0&\rho_{6,6}&0&0&0&0&\rho_{6,11}&0\\\
0&\rho_{2,7}&0&0&0&0&\rho_{7,7}&0&0&\rho_{8,9}&0&0\\\
0&0&0&0&0&0&0&\rho_{8,8}&0&0&0&0\\\
\rho_{3,7}&0&0&\rho_{4,9}&0&0&0&0&\rho_{9,9}&0&0&\rho_{10,11}\\\
0&\rho_{4,8}&0&0&0&0&\rho_{8,9}&0&0&\rho_{10,10}&0&0\\\
0&0&\rho_{5,9}&0&0&\rho_{6,11}&0&0&0&0&\rho_{11,11}&0\\\
\rho_{5,8}&0&0&\rho_{6,10}&0&0&0&0&\rho_{10,11}&0&0&\rho_{12,12}\end{pmatrix}$

$\displaystyle\rho^{\text{T}_{\text{c}}}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}\rho_{1,1}&0&0&\rho_{2,3}&0&0&0&\rho_{2,7}&0&0&0&0\\\
0&\rho_{2,2}&0&0&0&0&0&0&0&0&0&0\\\
0&0&\rho_{3,3}&0&0&\rho_{4,5}&\rho_{37}&0&0&\rho_{4,9}&0&0\\\
\rho_{2,3}&0&0&\rho_{4,4}&0&0&0&\rho_{4,8}&0&0&0&0\\\
0&0&0&0&\rho_{5,5}&0&0&0&\rho_{5,9}&0&0&\rho_{6,11}\\\
0&0&\rho_{4,5}&0&0&\rho_{6,6}&\rho_{5,8}&0&0&\rho_{6,10}&0&0\\\
0&0&\rho_{3,7}&0&0&\rho_{5,8}&\rho_{7,7}&0&0&\rho_{8,9}&0&0\\\
\rho_{2,7}&0&0&\rho_{4,8}&0&0&0&\rho_{8,8}&0&0&0&0\\\
0&0&0&0&\rho_{5,9}&0&0&0&\rho_{9,9}&0&0&\rho_{10,11}\\\
0&0&\rho_{4,9}&0&0&\rho_{6,10}&\rho_{8,9}&0&0&\rho_{10,10}&0&0\\\
0&0&0&0&0&0&0&0&0&0&\rho_{11,11}&0\\\
0&0&0&0&\rho_{6,11}&0&0&0&\rho_{10,11}&0&0&\rho_{12,12}\end{pmatrix}$

The reduced density matrix $\varrho_{\text{ab}}$ is given by:

$\displaystyle\varrho_{\text{ab}}\\!=\\!\\!\begin{pmatrix}\rho_{1,1}+\rho_{2,2}&0&0&0&0&0\\\
0&\rho_{3,3}+\rho_{4,4}&0&\rho_{2,3}+\rho_{4,5}&0&0\\\
0&0&\rho_{5,5}+\rho_{6,6}&0&\rho_{4,5}+\rho_{6,10}&0\\\
0&\rho_{2,3}+\rho_{4,5}&0&\rho_{7,7}+\rho_{8,8}&0&0\\\
0&0&\rho_{4,5}+\rho_{6,10}&0&\rho_{9,9}+\rho_{10,10}&0\\\
0&0&0&0&0&\rho_{11,11}+\rho_{12,12}\end{pmatrix}.$

The partially transposed reduced density matrix
$\varrho_{\text{ab}}^{\text{T}_{\text{b}}}$ is as follows:

$\displaystyle\varrho_{\text{ab}}^{\text{T}_{\text{b}}}=\begin{pmatrix}\rho_{1,1}+\rho_{2,2}&0&0&0&\rho_{2,3}+\rho_{4,5}&0\\\
0&\rho_{3,3}+\rho_{4,4}&0&0&0&\rho_{4,5}+\rho_{6,10}\\\
0&0&\rho_{5,5}+\rho_{6,6}&0&0&0\\\ 0&0&0&\rho_{7,7}+\rho_{8,8}&0&0\\\
\rho_{2,3}+\rho_{4,5}&0&0&0&\rho_{9,9}+\rho_{10,10}&0\\\
0&\rho_{4,5}+\rho_{6,10}&0&0&0&\rho_{11,11}+\rho_{12,12}\end{pmatrix}.$

The reduced density matrix $\varrho_{\text{ab}}$ is given by:

$\displaystyle\varrho_{\text{ab}}\\!=\\!\\!\begin{pmatrix}\rho_{1,1}+\rho_{7,7}&0&0&0&0&0\\\
0&\rho_{2,2}+\rho_{8,8}&\rho_{2,3}+\rho_{4,5}&0&0&0\\\
0&\rho_{2,3}+\rho_{8,9}&\rho_{3,3}+\rho_{9,9}&0&0&0\\\
0&0&0&\rho_{4,4}+\rho_{10,10}&\rho_{4,5}+\rho_{10,11}&0\\\
0&0&0&\rho_{4,5}+\rho_{10,11}&\rho_{5,5}+\rho_{11,11}&0\\\
0&0&0&0&0&\rho_{6,6}+\rho_{12,12}\end{pmatrix}.$

The partially transposed reduced density matrix
$\varrho_{\text{ab}}^{\text{T}_{\text{b}}}$ is as follows:

$\displaystyle\varrho_{\text{ab}}^{\text{T}_{\text{b}}}=\begin{pmatrix}\rho_{1,1}+\rho_{7,7}&0&0&\rho_{2,3}+\rho_{8,9}&0&0\\\
0&\rho_{2,2}+\rho_{8,8}&0&0&0&0\\\
0&0&\rho_{3,3}+\rho_{9,9}&0&0&\rho_{4,5}+\rho_{10,11}\\\
\rho_{2,3}+\rho_{8,9}&0&0&\rho_{4,4}+\rho_{10,10}&0&0\\\
0&0&0&0&\rho_{5,5}+\rho_{11,11}&0\\\
0&0&\rho_{4,5}+\rho_{10,11}&0&0&\rho_{6,6}+\rho_{12,12}\end{pmatrix}.$

The reduced density matrix $\varrho_{\text{ac}}$ would be given by the
following formula:

$\displaystyle\varrho_{\text{ac}}=\begin{pmatrix}\rho_{1,1}+\rho_{3,3}+\rho_{5,5}&0&0&0\\\
0&\rho_{2,2}+\rho_{4,4}+\rho_{6,6}&\rho_{2,7}+\rho_{4,9}+\rho_{6,11}&0\\\
0&\rho_{2,7}+\rho_{4,9}+\rho_{6,11}&\rho_{7,7}+\rho_{9,9}+\rho_{11,11}&0\\\
0&0&0&\rho_{8,8}+\rho_{10,10}+\rho_{12,12}\end{pmatrix}.$

The partially transposed density matrix
$\varrho_{\text{ac}}^{\text{T}_{\text{c}}}$ takes the form:

$\displaystyle\varrho_{\text{ac}}^{\text{T}_{\text{a}}}=\begin{pmatrix}\rho_{1,1}+\rho_{3,3}+\rho_{5,5}&0&0&\rho_{2,7}+\rho_{4,9}+\rho_{6,11}\\\
0&\rho_{2,2}+\rho_{4,4}+\rho_{6,6}&0&0\\\
0&0&\rho_{7,7}+\rho_{9,9}+\rho_{11,11}&0\\\
\rho_{2,7}+\rho_{4,9}+\rho_{6,11}&0&0&\rho_{8,8}+\rho_{10,10}+\rho_{12,12}\end{pmatrix}.$

The elements of the density matrix are presented below where $Z$ is the
partition function given in formula (3):

$\displaystyle\rho_{1,1}$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\exp\left(-\beta{E_{11}}\right)$
$\displaystyle\rho_{2,2}$ $\displaystyle=$
$\displaystyle\rho_{7,7}=\frac{1}{Z}\left[\frac{1}{2}\exp\left(-\beta
E_{3}\right)+a^{2}\exp\left(-\beta E_{5}\right)+c^{2}\exp\left(-\beta
E_{6}\right)\right]$ $\displaystyle\rho_{2,3}$ $\displaystyle=$
$\displaystyle\rho_{3,7}=\frac{1}{Z}\left[ab\exp\left(-\beta
E_{5}\right)+cd\exp\left(-\beta E_{6}\right)\right]$ $\displaystyle\rho_{2,7}$
$\displaystyle=$ $\displaystyle\frac{1}{Z}\left[-\frac{1}{2}\exp\left(-\beta
E_{3}\right)+a^{2}\exp\left(-\beta E_{5}\right)+c^{2}\exp\left(-\beta
E_{6}\right)\right]$ $\displaystyle\rho_{3,3}$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\left[b^{2}\exp\left(-\beta
E_{5}\right)+d^{2}\exp\left(-\beta E_{6}\right)\right]$
$\displaystyle\rho_{4,4}$ $\displaystyle=$
$\displaystyle\rho_{99}=\frac{1}{Z}\left[\frac{1}{2}\exp\left(-\beta
E_{1}\right)+e^{2}\exp\left(-\beta E_{9}\right)+g^{2}\exp\left(-\beta
E_{10}\right)\right]$ $\displaystyle\rho_{4,5}$ $\displaystyle=$
$\displaystyle\rho_{4,8}=\rho_{5,9}=\rho_{8,9}=\frac{1}{Z}\left[ef\exp\left(-\beta
E_{9}\right)+gh\exp\left(-\beta E_{10}\right)\right]$
$\displaystyle\rho_{4,9}$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\left[-\frac{1}{2}\exp\left(-\beta
E_{1}\right)+e^{2}\exp\left(-\beta E_{9}\right)+g^{2}\exp\left(-\beta
E_{10}\right)\right]$ $\displaystyle\rho_{5,5}$ $\displaystyle=$
$\displaystyle\rho_{88}=\frac{1}{Z}\left[\frac{1}{2}\exp\left(-\beta
E_{2}\right)+f^{2}\exp\left(-\beta E_{9}\right)+h^{2}\exp\left(-\beta
E_{10}\right)\right]$ $\displaystyle\rho_{5,8}$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\left[-\frac{1}{2}\exp\left(-\beta
E_{2}\right)+f^{2}\exp\left(-\beta E_{9}\right)+h^{2}\exp\left(-\beta
E_{10}\right)\right]$ $\displaystyle\rho_{6,6}$ $\displaystyle=$
$\displaystyle\rho_{11,11}=\frac{1}{Z}\left[\frac{1}{2}\exp\left(-\beta
E_{4}\right)+a^{2}\exp\left(-\beta E_{7}\right)+c^{2}\exp\left(-\beta
E_{8}\right)\right]$ $\displaystyle\rho_{6,10}$ $\displaystyle=$
$\displaystyle\rho_{10,11}=\frac{1}{Z}\left[ab\exp\left(-\beta
E_{7}\right)+cd\exp\left(-\beta E_{8}\right)\right]$
$\displaystyle\rho_{6,11}$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\left[-\frac{1}{2}\exp\left(-\beta
E_{4}\right)+a^{2}\exp\left(-\beta E_{7}\right)+c^{2}\exp\left(-\beta
E_{8}\right)\right]$ $\displaystyle\rho_{10,10}$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\left[b^{2}\exp\left(-\beta
E_{7}\right)+d^{2}\exp\left(-\beta E_{8}\right)\right]$
$\displaystyle\rho_{12,12}$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\exp\left(-\beta E_{12}\right)$

All eigenvalues of the partially transposed reduced density matrix
$\varrho_{\text{ab}}^{\text{T}_{\text{b}}}$ that constitute bipartite
negativity $N_{\text{ab}}$ (according to the formula) are as follows:

$\displaystyle\lambda_{1}=\rho_{2,2}+\rho_{8,8},\qquad\lambda_{2}=\rho_{5,5}+\rho_{11,11},$

$\displaystyle\lambda_{3;4}=\frac{1}{2}\left(\rho_{1,1}+\rho_{7,7}+\rho_{4,4}+\rho_{10,10}\pm\sqrt{(\rho_{1,1}+\rho_{7,7}-\rho_{4,4}-\rho_{10,10})^{2}+4(\rho_{2,3}+\rho_{8,9})^{2}}\right),$

and

$\displaystyle\lambda_{5;6}=\frac{1}{2}\left(\rho_{3,3}+\rho_{9,9}+\rho_{6,6}+\rho_{12,12}\pm\sqrt{(\rho_{3,3}+\rho_{9,9}-\rho_{6,6}-\rho_{12,12})^{2}+4(\rho_{10,11}+\rho_{4,5})^{2}}\right).$

Besides, the eigenvalue of the partially transposed reduced density matrix
$\varrho_{\text{ac}}^{\text{T}_{\text{c}}}$ that constitutes bipartite
negativity $N_{\text{ac}}$ would be expressed by:

$\displaystyle\lambda_{1}=\rho_{2,2}+\rho_{4,4}+\rho_{6,6},\qquad\lambda_{2}=\rho_{7,7}+\rho_{9,9}+\rho_{11,11},$

and

$\displaystyle\lambda_{3;4}=\frac{1}{2}\left(\rho_{1,1}+\rho_{3,3}+\rho_{5,5}+\rho_{8,8}+\rho_{10,10}+\rho_{12,12}\pm\sqrt{(\rho_{1,1}+\rho_{3,3}+\rho_{5,5}-\rho_{8,8}-\rho_{10,10}-\rho_{12,12})^{2}+4(\rho_{2,7}+\rho_{4,9}+\rho_{6,11})^{2}}\right)$

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