# Multipartite quantum correlation, spatially anisotropic coupling, and finite
temperature effects in a triangular Ising system with tunable interactions

Jun Ren1 Fang-Man Liu1 Yan-Chao Li2 Li-Hang Ren 1 Z. D. Wang3<EMAIL_ADDRESS>Yan-Kui Bai1,3<EMAIL_ADDRESS>1 College of Physics and Hebei Key Laboratory
of Photophysics Research and Application, Hebei Normal University,
Shijiazhuang, Hebei 050024, People’s Republic of China
2 College of Materials Science and Opto-Electronic Technology, University of
Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
3 Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Department of
Physics, and HK Institute of Quantum Science & Technology, The University of
Hong Kong, Pokfulam Road, Hong Kong, China

###### Abstract

We investigate multipartite quantum correlation (MQC), spatially anisotropic
coupling, and finite temperature effects in a triangular Ising system with
tunable interactions using the exact diagonalization method. We demonstrate
that spatially anisotropic coupling serves as an effective means to modulate
MQC in the antiferromagnetic ground state, which is achievable with current
experimental technologies. Moreover, we explore the interplay between MQC and
spatially anisotropic coupling in the Ising system at finite temperatures. Our
findings reveal a three-way trade-off relationship among high MQC, robust
thermal stability, and anisotropic strength in the triangular Ising system
with antiferromagnetic interactions, though the MQC in the ferromagnetic case
is quite susceptible to temperature changes. These insights contribute to our
understanding of ground state properties and MQC modulation in quantum many-
body systems.

###### pacs:

03.67.Mn, 03.65.Ud, 64.70.Tg

##  I Introduction

Quantum correlation in the ground state of many-body systems is a kind of
important physical resource for quantum information processing amico08rmp ;
horod09rmp ; modi12rmp ; bera18rpp . Frustrated spin systems, in particular,
have garnered significant attention due to their highly degenerate ground
states, which can result in multipartite quantum correlations (MQCs)
sach99book ; chiara18rpp . The most basic form of spin frustration can be
observed in a triangular lattice with three spins and antiferromagnetic
interactions binder86rmp , where the spins cannot organize into the preferred
antiparallel pattern, resulting in a degenerate manifold of ground states due
to the competition between interactions and lattice geometry. The ground state
of many-body spin systems can be in different phases at zero temperature
sach99book , and the MQC serves as an effective tool for characterizing the
properties of ground state in different quantum phases weitc05pra ;
olivei06pra ; olivei06prl ; facchi10njp ; mon10pra ; giampa13pra ; sunzy14pra
; hofm14prb ; Bayat17prl ; Yamasaki18pra ; haldar20prb . Recent theoretical
and experimental research has demonstrated that spin lattices with spatially
anisotropic triangular configurations possess unique phases and potential
applications in quantum computing Yoshida15np ; Zhang16prl ; Ito16prb ;
Keles18prl ; Tala20prapp .

Quantitative characterization of MQC is a fundamental problem in quantum
information processing. Entanglement monogamy in multipartite systems implies
that quantum entanglement cannot be freely shared among different subsystems
bennett96pra  and the quantitative monogamous relations can be used to
construct the measures or indicators for genuine MQC coffman00pra ;
osborne06prl ; christandl04jmp ; fan07pra ; byw07pra ; kimjs09pra ; bai06pra ;
byw09pra ; kimjs10jpa ; cornelio10pra ; bxw14prl ; songbai16pra ; ren21npj ;
bai22pra . Using the tripartite quantum correlation based on the monogamy
relation of negativity fan07pra ; vidal02pra , Rama Koteswara Rao _et al_
explored the ground state of quantum transverse Ising model in a fixed
triangular configuration through a nuclear magnetic resonance (NMR) system
rkrao13pra . They found that, unlike the bipartite quantum correlation,
multipartite quantum correlation can differentiate between frustrated and non-
frustrated regimes in the ground state of a three-spin system, with higher MQC
values in the non-frustrated regime compared to the frustrated one rkrao13pra
. In addition, it is worth noting that a system’s ground state property is
highly dependent on its anisotropy, such as anisotropic Heisenberg chains
wang01pla ; kamta02prl ; zhou03pra ; zhang05pra  and spatially anisotropic
triangular lattices yunoki06prb ; eckardt10epl ; hauke13prb . Consequently, it
is of interest to study the ground state properties in triangular Ising
systems with tunable spatially anisotropic couplings and investigate a more
general relationship between MQC and anisotropy in both ferromagnetic and
antiferromagnetic ground states of Ising systems, particularly in systems
beyond the three-spin case.

On the other hand, it is also essential to further analyze the MQC in tunable
Ising systems at finite temperatures, as MQC plays a vital role in quantum
information processing cory98prl ; knill98prl ; rauss01prl ; datta08prl ;
lanyon08prl ; ren21arxiv ; monz09prl ; lanyon09natp . Over the past two
decades, significant efforts have been made to study thermal quantum
entanglement or quantum correlation in spin systems, as highlighted in the
review papers amico08rmp ; bera18rpp . In this context, we aim to investigate
the intrinsic relationship between MQC and spatially anisotropic strength in
tunable Ising systems at finite temperatures and explore an effective and
feasible method for MQC modulation using current experimental technologies
kim10nat ; Struck11sci ; wang18natnano ; niskanen07sci ; Ryazanov01prb ;
Grosz11prb ; Kosior13pra ; Seo13prl . It remains an open question whether
there exists an inherent connection between high MQC, strong thermal
robustness, and spatially anisotropic strength, even in the simplest three-
spin case.

In this paper, we investigate the multipartite quantum correlations (MQC),
spatially anisotropic coupling, and finite temperature effects in a triangular
Ising system with tunable interactions up to 15 spins, using the exact
diagonalization method. The paper is organized as follows: In Section II, we
first introduce the relevant background knowledge, including the tunable
triangular Ising system and the MQC measure. We then examine the properties of
MQC as the spatial anisotropy changes in both ferromagnetic and
antiferromagnetic ground states. In Section III, we analyze the effects of
finite temperature on MQC and explore the inherent relationship among the
three key elements, namely, high MQC, its thermal robustness, and the
anisotropic strength in the antiferromagnetic case. We also explain why MQC in
the ferromagnetic case is highly susceptible to temperature. Finally, in
Section IV, we discuss the experimental possibilities for MQC modulation via
spatially anisotropic coupling in a system of cold atoms trapped in an optical
lattice and provide a brief conclusion.

##  II The MQC property in the ground state of a tunable triangular Ising
system

###  II.1 The tunable triangular Ising system and the MQC characterization

In this work, we address the two-dimensional triangular Ising lattices as
shown in Fig. 1, for which the Hamiltonian with only nearest-neighbor
couplings has the form

$H=\sum_{\left\langle
i,j\right\rangle}J_{ij}S_{i}^{x}S_{j}^{x}+h\sum_{i}S_{i}^{z},$ (1)

where $J_{ij}$ is the coupling strength between the $i$th and the $j$th spins,
$h$ means the strength of the transverse field, $S_{i}^{x}$ and $S_{i}^{z}$
are spin-1/2 operators with the subscript being the spin located at the vertex
$i$ in the two-dimensional triangular lattices. As shown in the figure, the
nearest-neighbor coupling $J_{ij}$s in different directions are indicated by
blue, red and green line segments, for which their magnitudes can be expressed
as $\omega J$, $\eta J$ and $J$, respectively, with $\omega$ and $\eta$ being
two nonnegative tunable parameters. It is noted that the triangular structure
with fixed couplings in the NMR system rkrao13pra  is covered by the
Hamiltonian. When $J>0$, the system possesses antiferromagnetic interactions
and may result in frustration, while the interactions are ferromagnetic in the
case of $J<0$ and the system has no frustration. The eigenstates and
eigenenergies for the Ising system can be solved by an exact diagonalization
approach. Hereafter we set the parameter $h=1$ for simplicity.

Figure 1: (Color online) The schematic diagram of two-dimensional triangular
Ising lattices with the tunable couplings (blue lines and red lines) via
spatially anisotropic parameters $\omega$ and $\eta$.

Entanglement negativity vidal02pra ; plenio05prl ; zyc98pra  is an important
entanglement measure, which is computable for bipartite mixed states and can
be defined as vidal02pra

$N(\rho_{AB})=\|\rho_{AB}^{T_{A}}\|_{1}-1,$ (2)

where $\|\cdot\|_{1}$ denotes the trace norm and equals the sum of the moduli
of eigenvalues for the partial transposed matrix $\rho_{AB}^{T_{A}}$.
Entanglement monogamy can be utilized to constitute the measure of
multipartite quantum correlation horod09rmp . Ou and Fan proved the monogamy
property of squared negativities in pure multiqubit states, and show genuine
tripartite quantum correlation in a three-qubit pure state
$|\psi\rangle_{ABC}$ can be characterized by the residual correlation
$\pi_{3}^{A|BC}=N_{A|BC}^{2}-N_{AB}^{2}-N_{AC}^{2}$ fan07pra  in which
$N_{A|BC}$ is the bipartite negativity in the partition $A|BC$ and $N_{Aj}$
quantifies the two-qubit entanglement with $j=\\{B,C\\}$.

For the ground state of the tunable triangular Ising system with $N$ spins,
the MQC can be characterized by the residual entanglement of the squared
negativities. Here, we make use of a modified multipartite quantum correlation
measure in an $N$-qubit state $|\Psi\rangle$ via the monogamous property of
squared negativities

$T_{N}(A_{i})=[N_{A_{i}|\bar{A}_{i}}^{2}-\sum_{j\neq
i}N_{A_{i}|A_{j}}^{2}]^{\frac{1}{2}},$ (3)

where $N_{A_{i}|\bar{A}_{i}}$ is the bipartite negativity between qubit
$A_{i}$ and the other subsystems $\bar{A}_{i}$, $N_{A_{i}|A_{j}}$ is the two-
qubit negativity between the $i$th and $j$th qubits, and the letter $A_{i}$ in
the parenthesis for the MQC $T_{N}(A_{i})$ means that the qubit $A_{i}$ is the
central qubit in the $N$-partite correlation.

Figure 2: (Color online) The schematic diagrams of the tunable triangular
Ising lattices, two-spin quantum correlation $N_{12}$, bipartite multi-spin
correlation $N_{2|\bar{2}}$, and the MQC $T_{N}$ as the functions of the
spatially anisotropic parameter $\eta$ and interaction strength $J$ in the
ground state. (a) Three-spin case: from left to right, the four panels are
schematic diagram for three-spin lattice with one tunable interaction, two-
spin quantum correlation $N_{12}$, bipartite three-spin correlation
$N_{2|\bar{2}}$, the MQC $T_{3}(A_{2})$, respectively. (b) Six-spin case: the
schematic diagram for the tunable lattices with six spins, and three
correlation measures $N_{12}$, $N_{2|\bar{2}}$ and $T_{6}(A_{2})$. (c) Ten-
spin case: schematic diagram for the ten-spin lattices, and three quantum
correlations $N_{12}$,$N_{2|\bar{2}}$ and $T_{10}(A_{2})$, respectively.

###  II.2 The MQC property and spatial anisotropy in the ground states of
tunable triangular Ising system

We first consider the triangular Ising model with one tunable coupling
parameter $\eta$ by setting the other parameter $\omega=1$, which means that
the nearest-neighbor couplings with the magnitude $J_{ij}=\eta J$ in
horizontal direction is tunable, as shown in Fig. 2. In experiments, the
tunable triangular configuration with two equally coupled legs has been
realized in ion traps kim10nat  and optical lattices Struck11sci .

In Fig. 2(a), we first analyze the three-spin case, where the first panel is
the schematic diagram with the tunable coupling $\eta J$ (red line) and the
central qubit with a dotted circle is the spin number $2$. The ground state
belongs to the class of tetrahedral states Carteret00jpa ; Brun01pla  and can
be used to perform the controlled teleportation Li14pra . In the second panel
of Fig. 2(a), the negativity $N_{12}$ is plotted as a function of the tunable
parameter $\eta$ and the interaction strength $J$. As shown in the panel, the
value of $N_{12}$ is small, where $N_{12}$ decreases along with the increase
of $\eta$ for a given $J$ in the ferromagnetic regime and the negativity is
almost zero when $\eta>1$ in antiferromagnetic regime. The third panel of Fig.
2(a) shows the bipartite negativity $N_{2|\bar{2}}$ as the function of $\eta$
and $J$, which quantifies the bipartite quantum correlation between qubit $2$
and other qubits $\bar{2}$. As shown in the panel, $N_{2|\bar{2}}$ in
ferromagnetic regime has the higher values and is insensitive to the
anisotropic parameter $\eta$ for larger $J$, while $N_{2|\bar{2}}$ in
antiferromagnetic regime is sensitive to the anisotropy and has lower values
for the configuration near the isotropic lattices. In the last panel of Fig.
2(a), the tripartite quantum correlation
$T_{3}(A_{2})=[N_{2|13}^{2}-N_{12}^{2}-N_{23}^{2}]^{1/2}$ is plotted, where
the MQC in the ferromagnetic ground state (away from the critical point $J=0$)
has a relatively large value and is insensitive to the spatially anisotropic
parameter $\eta$, but the correlation in the antiferromagnetic case is quite
different and decreases along with the parameter $\eta$ increases. By
comparing $N_{2|\bar{2}}$ with $T_{3}$, we obtain that the bipartite
correlation $N_{2|\bar{2}}$ in ferromagnetic regime is dominated by MQC
$T_{3}$ and the case of antiferromagnetic regime with small $\eta$ is similar,
while $N_{2|\bar{2}}$ in antiferromagnetic regime with large $\eta$ is mainly
composed of two qubit negativities. According to Fig. 2(a), the MQC $T_{3}$
can be effectively modulated from zero to almost the maximum via tuning the
anisotropy parameter $\eta$.

Figure 3: (Color online) The schematic diagram for the relationship between
the spatially anisotropic coupling and quantum correlation structure in the
ground state of ten spins far away from the critical point, where the left
part is the ferromagnetic ground state being insensitive to the anisotropic
parameter $\eta$ and the right part is the antiferromagnetic ground state
being sensitive to the anisotropy.

In Figs. 2(b) and 2(c), we further analyze correlation properties in the
tunable triangular Ising lattices with six and ten spins, respectively, where
the couplings (red lines) in horizontal direction are tunable via the
spatially anisotropic parameter $\eta$ and the central qubit is still the spin
number 2 as shown in the first panels of the two figures. In the second panel
of Fig. 2(b), the two-spin entanglement $N_{12}$ is plotted as a function of
the parameters $\eta$ and $J$, where both ferromagnetic and antiferromagnetic
regimes have small values of two-qubit quantum correlation. In the third panel
of Fig. 2(b), we plot the bipartite entanglement $N_{2|\bar{2}}$ for the
lattices of six spins, which is similar to the three-spin case and
$N_{2|\bar{2}}$ in the ferromagnetic regime is insensitive to $\eta$ for
larger $J$ and the bipartite multi-spin correlation has a lower value for the
configuration near isotropic lattices. The MQC $T_{6}(A_{2})$ in the ground
state of the triangular lattices is plotted in the last panel of Fig. 2(b),
where the anisotropic parameter $\eta$ has little influence on the MQC in the
ferromagnetic ground state (away from the critical point) but can modulate the
MQC in the antiferromagnetic region. The case for the lattices with ten spins
was shown in Fig. 2(c), where the correlations $N_{12}$, $N_{2|\bar{2}}$, and
$T_{10}(A_{2})$ have the similar behaviors along with the changes of the
parameters $\eta$ and $J$ as those of six-spin case and the spatially
anisotropic parameters $\eta$ can effectively modulate the MQC $T_{10}$ in
antiferromagnetic ground state. In addition, along with the increase of spin
numbers, the high quantum correlation areas of $N_{2|\bar{2}}$ and $T_{N}$ are
enlarged.

The influence of the spatially anisotropic parameter $\eta$ on MQC $T_{N}$ is
quite different in the ferromagnetic and antiferromagnetic ground states as
shown in Fig. 2. Here, we analyze the reason and consider the regions of
interaction strength $J$ far away from the critical point. As an example, we
study the triangular lattices with ten spins. In the ferromagnetic regime,
regardless of the anisotropic strength of $\eta$, the ground state of the
system always tends to the state of all the spins being spin-up or spin-down
at the same time. In the case of $J\rightarrow-\infty$, the ground state is
just the ferromagnetic state (GHZ-type)
$|\Psi\rangle=(|0\rangle^{\otimes{10}}+|1\rangle^{\otimes{10}})/\sqrt{2}$ with
the maximally multipartite quantum correlation as shown in the left part of
Fig. 3. Therefore, the anisotropic parameter $\eta$ does not change the
property of the ferromagnetic ground state and has little influence on the MQC
$T_{10}$ when the interaction strength $J$ in the nonfrustration regime has a
larger value as exhibited in the last panel of Fig. 2(c). However, the MQC
property of $T_{10}$ in the antiferromagnetic ground state is sensitive to the
spatially anisotropic parameter $\eta$ as shown in the right part of Fig. 3.
When the parameter $\eta\rightarrow 0$ and the coupling strength
$J\rightarrow\infty$, the interactions in horizontal direction disappear and
there is no frustration in the lattices. As shown in the figure, the ground
state is a kind of GHZ state
$|\Phi\rangle=(|011000111\rangle+|100111000\rangle)/\sqrt{2}$, for which the
MQC has the maximum. When $\eta=1$ with a larger $J$, the antiferromagnetic
ground state has an isotropic structure and tends to the equal probability
superposition of a lot of degenerate states. In this case, the ground state
has the large frustration since one-third nearest-neighbor spin pairs cannot
order in the favored antiparallel pattern and its MQC is nonzero but not
maximal. Along with the increases of the parameter $\eta$, the anisotropy will
affect the competition among these degenerate states, and thus change the MQC
in the ground state. In the case of $\eta\rightarrow\infty$, the
antiferromagnetic ground state becomes a tensor product state, which can be
written as
$|\Psi^{\prime}\rangle=(|1\rangle+|0\rangle)\otimes(|10\rangle+|01\rangle)\otimes(|101\rangle+|010\rangle)\otimes(|1010\rangle+|0101\rangle)/4$,
which results in the MQC $T_{10}(A_{2})$ being zero. It should be noted that
our analysis on the relation between the anisotropy of Ising lattices and the
MQC in the ground state is not limited to the case of ten spins, and the
qualitative results still hold for the case of more spins.

According to the numerical results in Fig. 2 and our qualitative analysis
shown in Fig. 3, we can obtain that, regardless of the number of spins, the
anisotropy parameter $\eta$ has little influence on the MQC $T_{N}$ in the
ground state with ferromagnetic interaction away from the critical point, but
for the antiferromagnetic case ($J>0$), the anisotropic parameter $\eta$ can
greatly change the ground state property and its MQC $T_{N}$. In particular,
when $\eta$ tends to zero with $J\rightarrow\infty$, the ground state tends to
the GHZ state which has the maximal MQC. When $\eta\rightarrow\infty$ with
large $J$, the ground state tends to a tensor product state and its MQC is
zero. In the case of $\eta$ close to 1, the ground state has large frustration
for which the MQC has a medium value. Moreover, although the modulation effect
of the MQC has a little difference along with the increase of spin numbers,
the MQC in the antiferromagnetic ground state (with a proper value of $J$) can
be continuously adjusted on a scale near $(0,1)$.

In order to illustrate our results, we further plot the MQC $T_{N}(A_{2})$ as
a function of parameter $J$ in the ground state of Ising lattices with the
anisotropic parameter $\eta=0.2,1$, and $2.5$, where the numbers of spins are
chosen to be $3,6,10$, and $15$ respectively. As shown in Fig. 4, the
anisotropy has little influence on the high values of the MQCs in the
ferromagnetic regime ($J<0$), while the values of the MQCs in the
antiferromagnetic regime ($J>0$) decrease along with the increase of $\eta$.
For example, the value of the MQC $T_{15}$ with $J=6$ is modulated from almost
the maximum $1$ to the lower value $0.15$ when the observer changes the
anisotropic parameter $\eta$ from $0.2$ to $2.5$. In addition, the MQC
$T_{N}(A_{2})$ in both two regimes increases along with the increase of the
number of lattice spins.

Figure 4: (Color online) The MQC $T_{N}(A_{2})$ as a function of parameter $J$
in the ground state of Ising lattices with the anisotropic parameter
$\eta=0.2,1$, and $2.5$, where the numbers of spins are chosen to be $3,6,10$,
and $15$ respectively.

We have studied the relation between the MQC property and spatial anisotropy
in the triangular Ising lattices with one asymmetric interaction. In practical
quantum systems, there also exist completely spatially anisotropic coupling
systems, such as the dipole-dipole interaction Heisenberg model Keles18prl ,
the tunable artificial spin ice structure Tala20prapp , and the
superconducting qubit architecture Grosz11prb , and so on. In the next
section, we will analyze the completely spatially anisotropic triangular Ising
system and explore the thermal robustness of the MQC.

##  III The MQC and its thermal robustness in the spatially anisotropic Ising
system

Characterization of thermal quantum resources is an important issue in quantum
information processing, and considerable efforts have been made to study the
thermal bipartite entanglements or quantum correlations in spin systems where
the effects of magnetic field and temperature are analyzed, see, for example,
Refs. wang01pla ; kamta02prl ; zhou03pra ; zhang05pra ; arnesen01prl ;
wang01pra ; wang02pra ; sun05njp ; abliz06pra ; ww06pra ; hide09prl ;
park17prl ; Shim08prb  and the review papers amico08rmp ; bera18rpp . Since
multipartite quantum correlation is a kind of important physical resource
vedral04njp ; Nakata09pra ; Campbell13njp ; sun19pra , it is necessary to
investigate the property of the MQC at finite temperatures. Here, we will
focus on the inherent relation among the MQC, its thermal robustness and
spatial anisotropy in the tunable Ising system with the triangular
configuration.

The thermal state under the equilibrium at a finite temperature $T$ can be
written as sach99book  $\rho(T)=\frac{1}{Z}\mbox{exp}(-\beta H)$ in which
$Z=\mbox{tr}[\mbox{exp}(-\beta H)]$ is the partition function with $H$ being
the system Hamiltonian and $\beta=1/k_{B}T$ (for simplicity, the Boltzmann
constant is set to be $k_{B}=1$). In the basis of eigenvectors, the thermal
state of the tunable Ising system can be further expressed as

$\rho(T)=\frac{1}{Z}\sum_{i}e^{-E_{i}/T}|e_{i}\rangle\langle e_{i}|,$ (4)

where $E_{i}$s are energy eigenvalues of the Hamiltonian given in Eq. (1) and
the coefficient $e^{-E_{i}/T}$ determines the proportion of the eigenstate
$|e_{i}\rangle$ in the mixed state.

Figure 5: (Color online) The MQC and its thermal robustness in the spatially
anisotropic triangular lattices. (a)-(c) The anisotropic lattices with spin
numbers $N=3,6,10$ and the MQC $T_{N}$ as the function of anisotropic
parameters $\omega$ and $\eta$ with the coupling strength $J=6$ at the
temperatures $T=0,0.05$, and $0.1$, respectively. (d), (e) The three-way
trade-off relations among the MQC, its thermal robustness and the anisotropic
strength in the triangular Ising system with antiferromagnetic interactions.

The triangular Ising lattices which we consider in this section are completely
spatially anisotropic, and the nearest-neighbor couplings are $J_{ij}=\omega
J$, $\eta J$ and $J$, respectively, in which the asymmetric couplings are
tunable via the parameters $\omega$ and $\eta$. In order to analyze the finite
temperature effect on the MQC, we choose the coupling strength $J=6$ away from
the critical point and consider three typical temperatures $T=0,0.05$, and
$0.1$, respectively. In Fig. 5, we plot the MQC $T_{N}$ as a function of the
anisotropic parameters $\omega$ and $\eta$ in the antiferromagnetic regime at
different temperatures, where the spin numbers of the Ising lattices are $3$,
$6$ and $10$ with the schematic diagrams shown in the left panels of Figs.
5(a)-(c). For the three-spin case, the MQC $T_{3}(A_{1})$ at zero temperature
is plotted as a function of the parameters $\omega$ and $\eta$ in the second
panel of Fig. 5(a), where the lattice structure with larger anisotropy (orange
and red regions) has the high value of the MQC, the structure with two smaller
equal couplings (blue regions) has the low value of $T_{3}$, and the structure
close to the isotropy (green central region) has the medium value. Along with
the number of spins increases ($N=6$ and $N=10$), the area of high MQC (red
regions) enlarges for the strong spatially anisotropic lattices, the low MQC
(green and yellow regions) still corresponds to the structures with two
smaller equal couplings, and the MQC for the structure close to isotropic
lattices (central regions) increases as shown in the second panels in Figs.
5(b) and 5(c). At the finite temperature $T=0.05$, the MQC $T_{N}$ in the
structure with larger anisotropy decreases rapidly but the MQC decreases
slowly in the structure with weak spatial anisotropy (two smaller equal
couplings and approximate isotropic couplings) as shown in the third panels of
Figs. 5(a)-5(c). Moreover, along with the spin number increases, the MQC
$T_{N}$ is more sensitive to the anisotropy and the area of strong thermal
robustness is reduced. The case for the temperature $T=0.1$ is similar, where
the MQC for the Ising lattices with weak spatial anisotropy is thermally
robust and the area of nonzero MQC further decreases along the temperature as
shown in the fourth panels of Figs. 5(a)-5(c). However, the thermal robustness
of two-spin quantum correlations is quite different, and, in Appendix A, we
analyze $\sum_{i}N_{5i}$ in the lattices of ten spins at finite temperature
for which the configurations near the isotropic case have zero correlation.

According to the above analysis, we can obtain that there exists a three-way
trade-off relation among high MQC, strong thermal robustness and the
anisotropic strength in the triangular Ising system with antiferromagnetic
interactions. The stronger the anisotropy of interactions, the higher the
value of the MQC $T_{N}$, but the weaker the thermal robustness of the MQC.
Conversely, the weaker the anisotropy of interactions, the lower the value of
the MQC $T_{N}$, but the stronger the thermal robustness of the MQC. In Figs.
5(d) and 5(e), the schematic diagrams for three-way trade-off relations among
the anisotropy, the MQC and its thermal robustness are plotted. Therefore, the
anisotropy of the tunable Ising lattices is an important quantity to modulate
the MQC and its thermal robustness. In particular, when both the relatively
high MQC and strong thermal robustness are required in a certain task of
quantum information processing, one can tune the spatially anisotropic
interactions of triangular Ising system to a proper configuration.

Unlike the case of antiferromagnetic regime, the spatial anisotropy has little
influence on the high MQC $T_{N}$ in the ground state of ferromagnetic case,
since the ground state tends to the GHZ state as shown in the previous
analysis of Fig. 3. However, the MQC $T_{N}$ in the ferromagnetic regime is
very fragile at finite temperature ren21arxiv . This property comes from the
structure of thermal state in Eq. (5), which is closely related to the system
eigenvalues and eigenstates. The mixed probability of component
$|e_{i}\rangle$ is proportional to the value $e^{-E_{i}}$, and thus the
eigenvector with the negative energy eigenvalue is dominant, while the one
with positive eigenvalue has a small weight in the thermal state for a given
lower temperature. As an example, we consider the case of three spins and plot
the eigenvalue $E_{i}$ as a function of the coupling strength $J$ for the
isotropic coupling $\eta=\omega=1$ in Fig. 6(a) and the anisotropic coupling
$\eta=0.5$ and $\omega=1.5$ in Fig. 6(b). As shown in the figures, when $J<0$
in the ferromagnetic regime, the energy eigenvalues $E_{0}$ (purple solid
line) and $E_{1}$ (green solid line) are negative and other eigenvalues are
positive (black dotted lines), which leads to the thermal state being mainly
comprised of the eigenvectors $|e_{0}\rangle$ and $|e_{1}\rangle$. Along with
the increase of temperature, the ferromagnetic thermal state tends to be the
mixture of the two eigenvectors with the equal probabilities due to the small
gap between the ground state and the first excited state [see the inset of
Fig. 6(a)], which results in a very low MQC and exhibit the weaker thermal
robustness. In the isotropic case of antiferromagnetic thermal state ($J>0$),
the ground state $|e_{0}\rangle$ is dominant in a certain temperature range
and the MQC has the stronger thermal robustness, because the eigenenergies
have a relatively large gap between $E_{0}$ and $E_{1}$ as shown in the inset
of Fig. 6(a). But the anisotropic case of antiferromagnetic thermal state
($J>0$) is different, the MQC is fragile due to the small gap between $E_{0}$
and $E_{1}$ as shown in the inset of Fig. 6(b). Similar conclusions can be
obtained when the spin number increases. For the case of ten spins, we further
plotted the first ten lowest eigenenergies as the functions of coupling
parameter $J$ in Figs. 6(c) and 6(d), which correspond to the isotropic and
anisotropic lattices, respectively. For the ferromagnetic thermal state
($J<0$) away from the critical point, the gap between $E_{0}$ and $E_{1}$ is
very small and the anisotropy has little influence on the two lowest
eigenenergies, which gives rise to the fragile MQC at finite temperatures. For
the antiferromagnetic case ($J>0$), the anisotropy changes the structure of
spectrum as shown in the Figs. 6(c) and 6(d), where the gaps between $E_{0}$
and $E_{1}$ have the smaller values in comparison with that of the isotropic
case, and then the MQC in the thermal state for the isotropic lattices has the
stronger thermal robustness. We further calculate the eigenenergies for the
case of 15 spins and obtain the similar conclusion. Moreover, the numerical
results show that the MQC $T_{15}$ in the antiferromagnetic regime, its
thermal robustness, and the spatial anisotropy of the Ising system still obey
the three-way trade-off relations in Figs. 5(d) and 5(e). At the same time,
the MQC $T_{15}$ in the ferromagnetic regime is very fragile at finite
temperatures. In Appendix B, we plot the finite temperature effect of MQC
$T_{N}$ for some typical configurations of the triangular lattices with the
number of spins being $N=3,6,10$, and $15$, respectively.

Figure 6: (Color online) The energy spectrum for the isotropic and anisotropic
Ising systems with the spin number $N=3$ and $N=10$ for which only the first
ten lowest eigenenergies are plotted. (a) The isotropic case for $N=3$ with
$\omega=\eta=1$. (b) The anisotropic case for $N=3$ with
$\eta=0.5,\omega=1.5$. (c) The isotropic case for $N=10$ with $\omega=\eta=1$.
(d) The anisotropic case for $N=10$ with $\eta=0.5,\omega=1.5$. The insets in
the four panels represent the energy gaps between the ground state and the
first excited state.

##  IV discussion and conclusion

Since multipartite quantum correlation is an important resource in quantum
information processing, it is desirable to study the MQC properties in the
triangular Ising system with the tunable interactions. Moreover, in
experiments, the tunable spatially anisotropic triangular structure can be
realized in various physical platforms, such as trapped ions kim10nat , cold
atoms Struck11sci , artificial-spin-ice heterosystem wang18natnano ,
superconducting systems niskanen07sci , and so on. In this paper, we find that
unlike the lattices with fixed couplings rkrao13pra , the MQC $T_{N}$ in the
antiferromagnetic ground state is sensitive to the spatial anisotropy of the
lattices as shown in Fig. 2.

The tuning of spatially anisotropic couplings will provide us an effective
strategy for the MQC modulation in the antiferromagnetic ground state. Here,
we discuss an experimental possibility for the MQC modulation via the
spatially anisotropic coupling in the system of cold atoms. In Ref.
Struck11sci , the authors utilized the motional degrees of freedom of atoms
trapped in a triangular optical lattice to simulate frustrated magnetism. The
key technology in their experiment is the independent tuning of the nearest-
neighbor couplings $J$ and $J^{\prime}$ by introducing a fast oscillation of
the lattice eckardt10epl . The atoms at each site $i$ of the lattice have a
local phase $\theta_{i}$, which can be regarded as a classical vector spin
$\textbf{S}_{i}=[\mbox{cos}(\theta_{i}),\mbox{sin}(\theta_{i})]$. The ground
state of this system corresponds to the minimum of the energy Struck11sci

$E(\\{\theta_{i}\\})=-\sum_{\langle
i,j\rangle}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j},$ (5)

where the sum extends over all pairs with the nearest-neighbor coupling, and
the coefficient $J_{ij}$ has the meaning of coupling strength between the
$i$th and $j$th sites. In the experimental system, the sign and magnitude of
$J_{ij}$ can be tuned independently via an elliptical shaking of the lattice,
which provides a tunable anisotropic parameter $\eta=J/J^{\prime}$ with the
same range of $(0,3)$ as that of our analysis in Fig. 2. Moreover, the recent
progress on the tunable couplings chen14prl ; Guo19prl ; li20prapp  and the
breakthrough in superconducting circuits Arute19nat ; gong21sci ; wu21prl ;
zhang22nat ; feng22prapp  can make dozens of qubit be arranged on two-
dimensional lattices with tunable nearest-neighbor couplings, which makes it
possible to construct an anisotropic triangular structure.

We have looked into the property of the MQC $T_{N}$ and its thermal robustness
in the anisotropic Ising systems at finite temperatures. It is revealed that
there exists a three-way trade-off relation among the high MQC, strong thermal
robustness and the anisotropic strength in the triangular Ising system with
antiferromagnetic interactions as shown in Fig. 5. In particular, one can
obtain the relatively high MQC and strong thermal robustness at the same time
via tuning the spatially anisotropic interactions in the antiferromagnetic
Ising lattices. The trade-off relation originates from the structure of the
first two lowest eigenenergies as shown in Fig. 6, which has close relation
with the anisotropy and gives rise to different kinds of thermal states. The
qualitative analysis still holds for the lattices of more spins.
Experimentally, the tunable triangular lattices can be utilized as a basic
building unit for constructing complex multipartite quantum networks
wehner18sci ; zhong21nat  and served as a potential platform for detecting
quantum nonlocality in many-body systems luo18prl ; renou19prl ; tejada21prl .

In conclusion, we have investigated the multipartite quantum correlations
(MQCs), spatially anisotropic coupling, and finite temperature effects in
triangular Ising systems with tunable interactions. In the ground states of
the Ising lattices, the influence of anisotropy on MQC varies, allowing us to
modulate MQC in the antiferromagnetic ground state using anisotropic strength.
This modulation is feasible with current experimental technologies. Moreover,
we have uncovered a three-way trade-off relationship between MQC, thermal
robustness, and anisotropic coupling in the tunable Ising system with
antiferromagnetic couplings, while MQC in the ferromagnetic case is highly
susceptible to finite temperatures. Our findings offer valuable insights into
ground state properties and MQC modulation in quantum many-body systems.

###### Acknowledgements.

This work was supported by the NSF-China (Grant Nos. 11575051, 11904078,
12074376 and 12105074), Hebei NSF (Grant Nos. A2021205020, A2019205263 and
A2019205266), and the CRF of Hong Kong (C6009-20G). JR was also funded by the
project of China Postdoctoral Science Foundation (Grant No. 2020M670683).

##  Appendix A finite temperature effect of two-spin negativities in the
tunable lattices of ten spins

Figure 7: (Color online) The two-spin negativities $\sum_{i}N_{5i}$ as a
function of parameters $\omega$ and $\eta$ and its thermal robustness in the
spatially anisotropic triangular lattices, where the temperature is chosen to
be $T=0,0.05$ and $0.1$, respectively.

According to the analysis in Sec. III of main text, we know that the MQC in
the isotropic triangular Ising structure has the strongest thermal robustness
and there exists a three-way trade-off relation among the MQC, its thermal
robustness and the spatial anisotropy. However, the behavior of two-spin
negativities is still not clear. Here, we consider the tunable Ising lattices
of ten spins and analyze the relation among two-spin negativities
$\sum_{i}N_{5i}$, its thermal robustness and the spatial anisotropy. In Fig.
7, we plot the negativities $\sum_{i}N_{5i}$ as a function of anisotropic
parameters $\omega$ and $\eta$ at the temperature $T=0,0.05$, and $0.1$,
respectively.

In comparison with the MQC in Fig. 5(c), the behavior of negativities
$\sum_{i}N_{5i}$ is quite different and, in particular, $\sum_{i}N_{5i}$ has
almost zero values for the configurations close to isotropic lattices. At zero
temperature, the higher negativities exist in the configuration close to weak
spatial anisotropic lattices (_i.e._ , the lattices with two smaller equal
couplings) as shown in Fig. 7(a). Along with the increase of temperature, the
two-spin correlations have the strong thermal robustness for the
configurations of weak spatial anisotropy as shown in Figs. 7(b) and 7(c).
Moreover, we further study the two-spin negativities $\sum_{i}N_{5i}$ in the
tunable Ising lattice of $15$ spins and can obtain the similar results.

##  Appendix B finite temperature effect of the MQC for some typical
configurations

Figure 8: (Color online) Multipartite quantum correlation $T_{N}$ as a
function of temperature $T$ with different parameters of the tunable
interactions in ferromagnetic regime (blue dashed line) and antiferromagnetic
(green dash-dotted line, black solid line, and red dotted line) regimes.
Panels (a)-(d) correspond to the case of spin numbers $N=3,6,10$, and $15$,
respectively. The insets in panels (c) and (d) highlight the MQC in the
ferromagnetic regime and that in the antiferromagnetic regime with the strong
spatial anisotropy.

In order to further illustrate the relation between the MQC $T_{N}$ and the
anisotropic configuration of the lattices at finite temperatures, we plot
$T_{N}$ as a function of the temperature $T$ for some typical configurations
with the spin numbers $N=3,6,10$, and $15$, respectively. As shown in Fig. 8,
for the ferromagnetic regime ($J=-6$), we consider the isotropic lattices with
the parameters $\eta=\omega=1$ (blue dashed line), and for the
antiferromagnetic regime ($J=6$), we choose three typical configurations,
_i.e._ , the strong spatial anisotropy ($\eta=0.5$, $\omega=1.5$, green dash-
dotted line), the weak spatial anisotropy ($\eta=0.9$, $\omega=1$, green black
solid line), and the isotropy case ($\eta=\omega=1$, red dotted line).

In Fig. 8(a), the three-spin case is plotted, where the MQC $T_{3}$ in the
ferromagnetic regime (blue dashed line) decays fastest and the one with
isotropic structure (red dotted line) in the antiferromagnetic regime has the
strongest thermal robustness. In addition, the strong anisotropic case (green
dash-dotted line) has the higher initial $T_{3}$ than that of the weak
anisotropic one (black solid line) but decays much faster. Along with the
number of spins increases, we find that the decay of $T_{N}$ in the
ferromagnetic regime (blue dashed line) speeds up as shown in Figs. 8(b)-(d),
and the case for the strong anisotropic structure (green dash-dotted line) is
similar, for which the reason can be explained by the analysis on structure of
energy spectrum in Fig. 6 of main text. Moreover, the MQC $T_{N}$ for the weak
anisotropy (black solid line) has higher initial correlation than that of the
isotropic case (red dotted line) but decays a little faster along with
temperature. The numerical results for the finite temperature effects of
$T_{N}$ with $N=3,6,10,15$ coincide with our previous theoretical analysis in
Sec. III of main text.

## References

  * (1) L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80, 517 (2008).
  * (2) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  * (3) K. Modi, A. Brodutch, H. Cable,T. Paterek, and V. Vedral, The classical-quantum boundary for correlations: Discord and related measures, Rev. Mod. Phys. 84, 1655 (2012).
  * (4) A. Bera, T. Das, D. Sadhukhan, S. Singha Roy, A. Sen(De) and U. Sen, Quantum discord and its allies: a review of recent progress, Rep. Prog. Phys. 81, 024001 (2018).
  * (5) S. Sachdev, Quantum phase transitions (Cambridge University Press, Cambridge, UK, 1999).
  * (6) G. De Chiara and A. Sanpera, Genuine quantum correlations in quantum many-body systems: a review of recent progress, Rep. Prog. Phys. 81, 074002 (2018).
  * (7) K. Binder and A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys. 58, 801 (1986).
  * (8) T. C. Wei, D. Das, S. Mukhopadyay, S. Vishveshwara, and P. M. Goadbart, Global entanglement and quantum criticality in spin chains, Phys. Rev. A 71, 060305(R) (2005).
  * (9) T. R. de Oliveira, G. Rigolin, and M. C. de Oliveira, Global entanglement and quantum criticality in spin chains, Phys. Rev. A 73, 010305(R) (2006).
  * (10) T. R. de Oliveira, G. Rigolin, M. C. de Oliveira, and E. Miranda, Multipartite entanglement signature of quantum phase transitions, Phys. Rev. Lett. 97, 170401 (2006).
  * (11) P. Facchi, G. Florio, U. Marzolino, G. Parisi, and S. Pascazio, Multipartite entanglement and frustration, New J. Phys. 12, 025015 (2010).
  * (12) A. Montakhab and A. Asadian, Multipartite entanglement and quantum phase transitions in the one-, two-, and three-dimensional transverse-field Ising model, Phys. Rev. A 82, 062313 (2010).
  * (13) S. M. Giampaolo and B. C. Hiesmayr, Genuine multipartite entanglement in the XY model, Phys. Rev. A 88, 052305 (2013).
  * (14) Z.-Y. Sun, Y.-Y. Wu, J. Xu, H.-L. Huang, B.-F. Zhan, B. Wang, and C. B. Duan, Characterization of quantum phase transition in the XY model with multipartite correlations and Bell-type inequalities, Phys. Rev. A 89, 022101 (2014).
  * (15) M. Hofmann, A. Osterloh, and O. Gühne, Scaling of genuine multiparticle entanglement close to a quantum phase transition, Phys. Rev. B 89, 134101 (2014).
  * (16) A. Bayat, Scaling of Tripartite Entanglement at Impurity Quantum Phase Transitions, Phys. Rev. Lett. 118, 036102 (2017).
  * (17) H. Yamasaki, A. Pirker, M. Murao, W. Dür, and B. Kraus, Multipartite entanglement outperforming bipartite entanglement under limited quantum system sizes, Phys. Rev. A 98, 052313 (2018).
  * (18) S. Haldar, S. Roy, T. Chanda, A. Sen(De), and U. Sen, Multipartite entanglement at dynamical quantum phase transitions with nonuniformly spaced criticalities, Phys. Rev. B 101, 224304 (2020).
  * (19) Y. Yoshida, H. Ito, M. Maesato, Y. Shimizu, H. Hayama, T. Hiramatsu, Y. Nakamura, H. Kishida, T. Koretsune, C. Hotta, and G. Saito, Spin-disordered quantum phases in a quasi-one-dimensional triangular lattice, Nat. Phys. 11, 679 (2015).
  * (20) X.-F. Zhang, S. Hu, A. Pelster, and S. Eggert, Quantum Domain Walls Induce Incommensurate Supersolid Phase on the Anisotropic Triangular Lattice, Phys. Rev. Lett. 117, 193201 (2016).
  * (21) H. Ito, T. Asai, Y. Shimizu, H. Hayama, Y. Yoshida, and G. Saito, Pressure-induced superconductivity in the antiferromagnet $\rm\kappa$-$\rm(ET)_{2}CF_{3}SO_{3}$ with quasi-one-dimensional triangular spin lattice, Phys. Rev. B 94, 020503(R) (2016).
  * (22) A. Keles and E. Zhao, Absence of Long-Range Order in a Triangular Spin System with Dipolar Interactions, Phys. Rev. Lett. 120, 187202 (2018).
  * (23) A. Talapatra, N. Singh, and A. O. Adeyeye, Magnetic Tunability of Permalloy Artificial Spin Ice Structures, Phys. Rev. Appl. 13, 014034 (2020).
  * (24) C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53, 2046 (1996).
  * (25) V. Coffman, J. Kundu, and W. K. Wootters, Distributed entanglement, Phys. Rev. A 61, 052306 (2000).
  * (26) T. J. Osborne and F. Verstraete, General monogamy inequality for bipartite qubit entanglement, Phys. Rev. Lett. 96, 220503 (2006).
  * (27) M. Christandl and A. Winter, Squashed entanglement: An additive entanglement measure, J. Math. Phys. 45, 829 (2004).
  * (28) Y.-C. Ou and H. Fan, Monogamy inequality in terms of negativity for three-qubit states, Phys. Rev. A 75, 062308 (2007).
  * (29) Y.-K. Bai, D. Yang, and Z. D. Wang, Multipartite quantum correlation and entanglement in four-qubit pure states, Phys. Rev. A 76, 022336 (2007).
  * (30) J. S. Kim, A. Das, and B. C. Sanders, Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity, Phys. Rev. A 79, 012329 (2009).
  * (31) Y.-K. Bai, S.-S. Li, H.-Z. Zheng, and Z. D. Wang, Detecting a set of entanglement measures in an unknown tripartite quantum state by local operations and classical communication, Phys. Rev. A 74, 022305 (2006).
  * (32) Y.-K. Bai, M.-Y. Ye, and Z. D. Wang, Entanglement monogamy and entanglement evolution in multipartite systems, Phys. Rev. A 80, 044301 (2009).
  * (33) J. S. Kim and B. C. Sanders, Monogamy of multi-qubit entanglement using Rényi entropy, J. Phys. A: Math. Theor. 43, 445305 (2010).
  * (34) M. F. Cornelio and M. C. de oliveira, Strong superadditivity and monogamy of the Rényi measure of entanglement, Phys. Rev. A 81, 032332 (2010).
  * (35) Y.-K. Bai, Y.-F. Xu, and Z. D. Wang, General monogamy relation for the entanglement of formation in multiqubit systems, Phys. Rev. Lett. 113, 100503 (2014).
  * (36) W. Song, Y.-K. Bai, M. Yang, M. Yang, and Z.-L. Cao, General monogamy relation of multiqubit systems in terms of squared Rényi-$\alpha$ entanglement, Phys. Rev. A 93, 022306 (2016).
  * (37) L.-H. Ren, M. Gao, J. Ren, Z. D. Wang, and Y.-K. Bai, Resource conversion between operational coherence and multipartite entanglement in many-body systems, New J. Phys. 23, 043053 (2021).
  * (38) L.-L. Su, J. Ren, Z. D. Wang, and Y.-K. Bai, Long-range multipartite quantum correlations and factorization in a one-dimensional spin-1/2 XY chain, Phys. Rev. A 106, 042427 (2022).
  * (39) G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  * (40) K. Rama Koteswara Rao, H. Katiyar, T. S. Mahesh, A. Sen(De), U. Sen, and A. Kumar, Multipartite quantum correlations reveal frustration in a quantum Ising spin system, Phys. Rev. A 88, 022312 (2013).
  * (41) X. Wang, Effects of anisotropy on thermal entanglement, Phys. Lett. A 281, 101 (2001).
  * (42) G. L. Kamta and A. F. Starace, Anisotropy and Magnetic Field Effects on the Entanglement of a Two Qubit Heisenberg XY Chain, Phys. Rev. Lett. 88, 107901 (2002).
  * (43) L. Zhou, H. S. Song, Y. Q. Guo, and C. Li, Enhanced thermal entanglement in an anisotropic Heisenberg XYZ chain, Phys. Rev. A 68, 024301 (2003).
  * (44) G. F. Zhang and S. S. Li, Thermal entanglement in a two-qubit Heisenberg XXZ spin chain under an inhomogeneous magnetic field, Phys. Rev. A 72, 034302 (2005).
  * (45) S. Yunoki and S. Sorella, Two spin liquid phases in the spatially anisotropic triangular Heisenberg model, Phys. Rev. B 74, 014408 (2006).
  * (46) A. Eckardt, P. Hauke, P. Soltan-Panahi, C. Becker, K. Sengstrock, and M. Lewenstein, Frustrated quantum antiferromagnetism with ultracold bosons in a triangular Lattice, Europhys. Lett. 89, 10010 (2010).s
  * (47) P. Hauke, Quantum disorder in the spatially completely anisotropic triangular lattice, Phys. Rev. B 87, 014415 (2013).
  * (48) D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, Experimental Quantum Error Correction, Phys. Rev. Lett. 81, 2152 (1998).
  * (49) E. Knill and R. Laflamme, Power of One Bit of Quantum Computation, Phys. Rev. Lett. 81, 5672 (1998).
  * (50) R. Raussendorf and H. Briegel, A One-way Quantum Computer, Phys. Rev. Lett. 86, 5188 (2001).
  * (51) A. Datta, A. Shaji, and C. M. Caves, Quantum Discord and the Power of One Qubit, Phys. Rev. Lett. 100, 200502 (2008).
  * (52) B. P. Lanyon, M. Barbieri, M. P. Almeida, and A. G. White, Experimental Quantum Computing Without Entanglement, Phys. Rev. Lett. 101, 200501 (2008).
  * (53) J. Ren, F-M. Liu, L.-H. Ren, Z. D. Wang, Y.-K. Bai, arXiv 2109.03576v2
  * (54) T. Monz, K. Kim, W. Hansel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, Realization of the Quantum Toffoli Gate with Trapped Ions, Phys. Rev. Lett. 102, 040501 (2009).
  * (55) B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, Simplifying quantum logic using higher-dimensional Hilbert spaces, Nat. Phys. 5, 134 (2009).
  * (56) K. Kim, M. S. Chang, S. Korenblit, R. Islam, E. E. Edwards, J. K. Freericks, G. D. Lin, L. M. Duan, and C. Monroe, Quantum simulation of frustrated Ising spins with trapped ions, Nature 465, 590 (2010).
  * (57) J. Struck, C. Olschlager, R. Le Targat, P. Soltan-Panahi, A. Eckardt, M. Lewenstein, P. Windpassinger, and K. Sengstock, Quantum simulation of frustrated classical magnetism in triangular optical lattices, Science 333, 996 (2011).
  * (58) Y.-L. Wang, X. Ma, J. Xu, Z.-L. Xiao, A. Snezhko, R. Divan, L. E. Ocola, J. E. Pearson, B. Janko, and W.-K. Kwok, Switchable geometric frustration in an artificial-spin-ice uperconductor heterosystem, Nat. Nanotechnol. 13, 560 (2018).
  * (59) A. O. Niskanen, K. Harrabi, F. Yoshihara, Y. Nakamura, S. Lloyd, and J. S. Tsai, Quantum coherent tunable coupling of superconducting qubits, Science 316, 723 (2007).
  * (60) V. V. Ryazanov, V. A. Oboznov, A. V. Veretennikov, and A. Y. Rusanov, Intrinsically frustrated superconducting array of superconductor-ferromagnet-superconductor junctions, Phys. Rev. B 65, 020501(R) (2001).
  * (61) P. Groszkowski, A. G. Fowler, F. Motzoi, and F. K. Wilhelm, Tunable coupling between three qubits as a building block for a superconducting quantum computer, Phys. Rev. B 84, 144516 (2011).
  * (62) A. Kosior and K. Sacha, Simulation of frustrated classical XY models with ultracold atoms in three-dimensional triangular optical lattices, Phys. Rev. A 87, 023602 (2013).
  * (63) M. Seo, H. K. Choi, S.-Y. Lee, N. Kim, Y. Chung, H.-S. Sim, V. Umansky, and D. Mahalu, Charge Frustration in a Triangular Triple Quantum Dot, Phys. Rev. Lett. 110, 046803 (2013).
  * (64) M. B. Plenio, Logarithmic Negativity: A Full Entanglement Monotone That is not Convex, Phys. Rev. Lett. 95, 090503 (2005).
  * (65) K. Życzkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Volume of the set of separable states, Phys. Rev. A 58, 883 (1998).
  * (66) H. A. Carteret and A. Sudbery, Local symmetry properties of pure three-qubit states, J. Phys. A. Math. Gen. 33, 4981 (2000).
  * (67) T. A. Brun and O. Cohen, Parametrization and distillability of three-qubit entanglement, Phys. Lett. A 281, 88 (2001).
  * (68) X. Li and S. Ghose, Control power in perfect controlled teleportation via partially entangled channels, Phys. Rev. A 90, 052305 (2014).
  * (69) M. C. Arnesen, S. Bose, and V. Vedral, Natural Thermal and Magnetic Entanglement in the 1D Heisenberg Model, Phys. Rev. Lett. 87, 017901 (2001).
  * (70) X. Wang, Entanglement in the quantum Heisenberg XY model, Phys. Rev. A 64, 012313 (2001).
  * (71) X. Wang, Thermal and ground-state entanglement in Heisenberg XX qubit rings, Phys. Rev. A 66, 034302 (2002).
  * (72) Z. Sun, X. Wang and Y.-Q. Li, Entanglement in dimerized and frustrated spin-one Heisenberg chains, New. J. Phys. 7, 83 (2005).
  * (73) A. Abliz, H. J. Gao, X. C. Xie, Y. S. Wu, and W. M. Liu, Entanglement control in an anisotropic two-qubit Heisenberg XYZ model with external magnetic fields, Phys. Rev. A 74, 052105 (2006).
  * (74) X. Wang and Z. D. Wang, Thermal entanglement in ferrimagnetic chains, Phys. Rev. A 73, 064302 (2006).
  * (75) J. Hide, W. Son, and V. Vedral, Enhancing the Detection of Natural Thermal Entanglement with Disorder, Phys. Rev. Lett. 102, 100503 (2009).
  * (76) Y. Park, J. Shim, S.-S. B. Lee, H.-S. Sim, Nonlocal Entanglement of 1D Thermal States Induced by Fermion Exchange Statistics, Phys. Rev. Lett. 119, 210501 (2017).
  * (77) J. Shim, H.-S. Sim, and S.-S. B. Lee, Numerical renormalization group method for entanglement negativity at finite temperature, Phys. Rev. B 97, 155123 (2018).
  * (78) V. Vedral, Mean-field approximations and multipartite thermal correlations, New J. Phys. 6, 22 (2004).
  * (79) Y. Nakata, D. Markham, and M. Murao, Thermal robustness of multipartite entanglement of the one-dimensional spin-1/2 XY model, Phys. Rev. A 79, 042313 (2009).
  * (80) S. Campbell, L. Mazzola, G. De Chiara, T. J. G. Apollaro, F. Plastina, T. Busch, and M. Paternostro, Global quantum correlations in finite-size spin chains, New J. Phys. 15, 043033 (2013).
  * (81) Z.-Y. Sun, M. Wang, Y.-Y. Wu, and B. Guo, Multipartite nonlocality and boundary conditions in one-dimensional spin chains, Phys. Rev. A 99, 042323 (2019).
  * (82) Y. Guo, R. M. Kroeze, V. D. Vaidya, J. Keeling, and B. L. Lev, Sign-Changing Photon-Mediated Atom Interactions in Multimode Cavity Quantum Electrodynamics, Phys. Rev. Lett. 122, 193601 (2019).
  * (83) Y. Chen, C. Neill, P. Roushan, et al., Qubit Architecture with High Coherence and Fast Tunable Coupling, Phys. Rev. Lett. 113, 220502 (2014).
  * (84) X. Li, T. Cai, H. Yan, Z. Wang, X. Pan, Y. Ma, W. Cai, J. Han, Z. Hua, X. Han, Y. Wu, H. Zhang, H. Wang, Y. Song, L. Duan, and L. Sun, Tunable Coupler for Realizing a Controlled-Phase Gate with Dynamically Decoupled Regime in a Superconducting Circuit, Phys. Rev. Appl. 14, 024070 (2020).
  * (85) F. Arute et al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * (86) M. Gong et al., Quantum walks on a programmable two-dimensional 62-qubit superconducting processor, Science 372, 948 (2021).
  * (87) Y. Wu et al., Strong Quantum Computational Advantage Using a Superconducting Quantum Processor, Phys. Rev. Lett. 127, 180501 (2021).
  * (88) X. Zhang et al., Digital quantum simulation of Floquet symmetry-protected topological phases, Nature 607, 468 (2022).
  * (89) W. Feng, G.-Q. Zhang, Q.-P. Su, J.-X. Zhang, and C.-P. Yang, Generation of Greenberger-Horne-Zeilinger States on Two-Dimensional Superconducting-Qubit Lattices via Parallel Multiqubit-Gate Operations, Phys. Rev. Appl. 18, 064036 (2022).
  * (90) S. Wehner, D. Elkouss, R. Hanson, Quantum internet: A vision for the road ahead, Science 362, 303 (2018).
  * (91) Y. Zhong, H.-S. Chang, A. Bienfait, E. Dumur, M.-H. Chou, C. R. Conner, J. Grebel, R. G. Povey, H. Yan, D. I. Schuster, and A. N. Cleland, Deterministic Multi-Qubit Entanglement in a Quantum Network, Nature 590, 571 (2021).
  * (92) M.-X. Luo, Computationally Efficient Nonlinear Bell Inequalities for Quantum Networks, Phys. Rev. Lett. 120, 140402 (2018).
  * (93) M. O. Renou, E. Bäumer, S. Boreiri, N. Brunner, N. Gisin, and S. Beigi, Genuine Quantum Nonlocality in the Triangle Network, Phys. Rev. Lett. 123, 140401 (2019).
  * (94) P. Contreras-Tejada, C. Palazuelos, and J. I. de Vicente, Genuine Multipartite Nonlocality Is Intrinsic to Quantum Networks, Phys. Rev. Lett. 126, 040501 (2021).


# On Perceived Emotion in Expressive Piano Performance:
Further Experimental Evidence for the Relevance of Mid-level Perceptual
Features

###### Abstract

Despite recent advances in audio content-based music emotion recognition, a
question that remains to be explored is whether an algorithm can reliably
discern emotional or expressive qualities between different performances of
the same piece. In the present work, we analyze several sets of features on
their effectiveness in predicting arousal and valence of six different
performances (by six famous pianists) of Bach’s Well-Tempered Clavier Book 1.
These features include low-level acoustic features, score-based features,
features extracted using a pre-trained emotion model, and Mid-level perceptual
features. We compare their predictive power by evaluating them on several
experiments designed to test performance-wise or piece-wise variations of
emotion. We find that Mid-level features show significant contribution in
performance-wise variation of both arousal and valence – even better than the
pre-trained emotion model. Our findings add to the evidence of Mid-level
perceptual features being an important representation of musical attributes
for several tasks – specifically, in this case, for capturing the expressive
aspects of music that manifest as perceived emotion of a musical performance.

##  1 Introduction

A musical performance, particularly in the Western music tradition, is not
merely a literal acoustic rendering of a notated piece or composition. Rather,
the piece is transformed by the performer’s own expressive performance
choices, relating to such dimensions as the choice of tempo, expressive tempo
and timing variations, dynamics, articulation, and so on. The emotional effect
of a performance on a listener can be a consequence both of the composition
itself, with its musical properties and structures, and of the performance,
the way the piece was played. In fact, it has been convincingly demonstrated
[1, 2] that performers are capable of communicating, with high accuracy,
intended emotional qualities by their playing.

The analysis of emotion in music recordings has a long history in Music
Information Retrieval, with many works addressing content-based emotion
regression and classification typically using low-level or hand-crafted audio
and musical features [3, 4, 5, 6] or using deep learning based methods [7, 8,
9]. However, there has been little research on the more subtle problem of
identifying emotional aspects that are due to the actual performance, and even
less on models that can automatically recognize this from audio recordings. On
the latter problem – the one to be addressed in this paper – the most directly
relevant prior work we are aware of is [10], where 324 6-second audio snippets
of different genres (classical, jazz, blues, metal, etc.) were annotated in
terms of perceived emotion (valence and arousal), and various regressors were
trained to predict these two dimensions from a set of standard audio features.
The regression models were then used to predict valence-arousal trajectories
over 5 different recordings of 4 Chopin pieces, but no ground truth in terms
of human emotion annotations was collected. The relevance of the model
predictions was evaluated only indirectly, by comparing similarity scores
between predicted profiles with overall performance similarity ratings by
three human listeners, which showed some non-negligible correlations.

In a recent focused study [11], Battcock & Schutz (referred to as “B&S”
henceforth) investigate how three specific score-based cues (Mode, Pitch
Height, and Attack Rate111Actually, attack rate as computed by B&S is also
informed by the average tempo of the performance; thus, it is not strictly a
score-only feature.) work together to convey emotion in J.S.Bach’s preludes
and fugues collected in his Well-tempered Clavier (WTC). They used recordings
of the complete WTC Book 1 (48 pieces) of one famous pianist (Friedrich Gulda)
as stimuli for human listeners to rate each performance on perceived arousal
and valence. Their findings suggest that within this set of performances,
arousal is significantly correlated with attack rate and valence is affected
by both the attack rate and the mode. However, that study was based on only
one set of performances, making it impossible to decide whether the human
emotion ratings used as ground truth really reflect aspects of the
compositions themselves, or whether they were also (or even predominantly)
affected by the specific (and, in some cases, rather unconventional) way in
Friedrich Gulda plays the pieces – that is, whether the emotion ratings
reflect piece or performance aspects.

The purpose of the present paper is to try to disentangle the possible
contributions and roles of different features in capturing
composer-(piece-)specific and performer-(recording-)specific aspects. To this
end, we collected human ratings of perceived valence and arousal in six
complete sets of recordings of WTC Book 1, and then performed a systematic
study with feature sets derived from various levels of musical abstraction,
including some extracted by pre-trained deep neural networks.

##  2 Data Collection

###  2.1 Pieces and Recordings

J.S.Bach’s Well-tempered Clavier (WTC) is ideally suited for systematic and
controlled studies of this kind, as it comprises a stylistically coherent set
of keyboard pieces from a particular period, evenly distributed over all keys
and major/minor modes, with a pair of two pieces (a prelude, followed by a
fugue) in each of the 24 possible keys, for a total of 48 pieces. Each piece
has its own distinctive musical character, and despite being written in a
rather strict style and not meant to be played in ‘romantic’ ways, the music
offers pianists (or pianists take) lots of liberties in ornamentation, but
also overall performance parameters (e.g., tempo and articulation). For
example, there are pieces in our set of recordings that one pianist takes more
than twice (!) as fast as another.

For a broad set of diverse performances, we selected six recordings of the
complete WTC Book 1, by six famous and highly respected pianists, all of whom
can be considered Bach specialists to various degrees. The recordings are
listed in Table 1.

Pianist | Recording | Year
---|---|---
Glenn Gould | Sony 88725412692 | 1962-1965
Friedrich Gulda | MPS 0300650MSW | 1972
Angela Hewitt | Hyperion 44291/4 | 1997-1999
Sviatoslav Richter | RCA 82876623152 | 1970
András Schiff | ECM 4764827 | 2011
Rosalyn Tureck | DG 4633052 | 1952-1953
Table 1: Pianists and recordings.

###  2.2 Emotion Annotations and Pre-processing

In accordance with B&S, we will only use the first 8 bars of each recording
for the annotation process and our experiments. These were cut out manually.
The participants of our annotation exercise were students of a course at a
university, without a specifically musical background. Each participant heard
a subset of the recordings (all 48 pieces as played by one pianist) and was
asked to rate the valence on a scale of -5 to +5 (11 levels) and the arousal
on a scale of 0 to 100 (increments of 10; a total of 11 levels). They could
listen to a recording as many times as they liked. Each recording was rated by
29 participants. In total, we collected 8,352 valence-arousal annotation
pairs.

For the purposes of this paper, we take the mean arousal and mean valence
ratings for each recording, and these values serve as our ground-truth values
for all following experiments. The distributions (over the 6 performances) of
these mean ratings for each piece are summarised in Figure 1.

Figure 1: Distribution of emotion ratings across pianists for every piece.

##  3 Feature Sets

In this section, we briefly describe the four feature sets that we use to
model arousal and valence.

###  3.1 Low-level Features

These consist of hand-crafted musical features (such as onset rate, tempo,
pitch salience) as well as generic audio descriptors (such as spectral
centroid, loudness). Taken together, they reflect several musical
characteristics such as tone colour, dynamics, and rhythm. A brief description
of all low-level features that we use is given in Table 2. We use Essentia
[12] and Librosa [13] for extracting these. The audio is sampled at 44.1kHz
and the spectra computed (when required) with a frame size of 1024 samples and
a hop size of 512 samples. Each feature is aggregated over the entire duration
of an audio clip by computing the mean and standard deviation over all the
frames of the clip (a ‘clip’ being an 8 bar initial segment from a recording).

Dissonance |  Total harmonic dissonance computed from pairwise dissonance of all spectral peaks.
---|---
Dynamic Complexity |  The average absolute deviation from the global loudness level estimate in dB.
Loudness |  Mean loudness of the signal computed from the signal amplitude.
Onset Rate |  Number of onsets (note beginnings or transients) per second.
Pitch Salience |  A measure of tone sensation, computed from the harmonic content of the signal.
Spectral Centroid |  The weighted mean frequency in the signal, with frequency magnitudes as the weights.
Spectral Flatness |  A measure to quantify how much noise-like a sound is, as opposed to being tone-like.
Spectral Bandwidth |  The second order bandwidth of the spectrum.
Spectral Rolloff |  The frequency under which 85% of the total energy of the spectrum is contained.
Spectral Complexity |  The number of peaks in the input spectrum.
Tempo (BPM) |  Tempo estimate from audio in beats per minute.

Table 2: Low-level Features

###  3.2 Score Features

The following set of features was computed directly from the musical score
(i.e., sheet music) of the pieces instead of the audio files. The unit of
score time, “beat”, is defined by the time signature of the piece (e.g., 4/4
means that there are 4 beats of duration 1 quarter in a bar). The score
information and the audio files were linked using automatic score-to-
performance alignment. Table 3 describes the score features in detail.

Inter Onset Interval |  The time interval between consecutive notes per beat.
---|---
Duration |  Two features describing the empirical mean and standard deviation of the notated duration per beat in the snippet.
Onset Density |  The number of note onsets per beat. A chord constitutes a single onset.
Pitch Density |  The number of unique notes per beat.
Mode |  Binary feature denoting major/minor modality, computed using the Krumhansl-Schmuckler key finding algorithm [14] (to reflect the fact that the dominant key over the segment may be different from the given key signature).
Key Strength |  This feature represents how much does the tonality by the "Mode" feature fit the snippet.

Table 3: Score Features

###  3.3 Mid-level Features

Mid-level features, described in [15], are perceptual musical features that
are intuitively understandable to the average listener. They seem well-suited
to bridge the “semantic gap" between low-level audio features and high-level
descriptors such as emotion and have been shown to be useful in explainable
music emotion recognition [16]. We learn these features from the Mid-level
Dataset [15] using a receptive-field regularised residual neural network (RF-
ResNet) model [17]. Since we intend to use this model to extract features from
solo piano recordings (a genre that is not covered by the original training
data), we use a domain-adaptive training approach as described in [18]. We use
an input audio length of 30 seconds, padded or cropped as required. As these
features cannot be strictly defined, Table 4 lists a rough description adapted
from the questions in [15] that were shown to the annotators of the dataset to
help them rate the audio clips.

Melodiousness |  How singable is this music?
---|---
Articulation |  Overall impression of articulation in terms of staccato or legato playing style. Higher means more staccato.
Rhythmic Stability |  How easy is it to march-along with the music?
Rhythmic Complexity |  How difficult is it to follow the music by tapping? Rhythmic layers and different meters correlate with higher complexity.
Dissonance |  Noisier timbre or presence of dissonant intervals (tritones, seconds, etc.)
Tonal Stability |  How clear or apparent the tonic and key are.
Minorness |  Relates to the perceived tonality. More “minor-sounding" music will have higher minorness.

Table 4: Mid-level Features

###  3.4 DEAMResNet Emotion Features

To compare the mid-level features with another deep neural network based
feature extractor, we train a model with the same architecture (RF-ResNet) and
training strategy on the DEAM dataset [19] to predict arousal and valence from
spectrogram inputs. Since this model is trained to predict arousal and
valence, it is expected to learn representations suitable for this task. As
with the mid-level model, we perform domain adaptation while training this
model also.

Features are extracted from the penultimate layer of the model, which gives us
512 features. Since these are too many features to use for our dataset
containing only 288 data points, we perform dimensionality reduction using PCA
(Principal Component Analysis), to obtain 9 components explaining at least 98%
of the variance. These 9 features are named as pca_x with x being the
principal component number.

##  4 Feature Evaluation Experiments

In this section, we evaluate the four feature sets. The aim of this section is
to answer the following questions:

  1. 1.

How well can each feature set fit the arousal and valence ratings? How do
these feature sets compare to the ones used by B&S? (Sections 4.1 and 4.2)

  2. 2.

In each feature set, which features are the most important? (Section 4.3)

  3. 3.

Which feature set best explains variation of arousal and valence between
pieces? (Section 4.4)

  4. 4.

Which feature set best explains variation of arousal and valence between
different performances of the same piece? (Section 4.5)

We use ordinary least squares fitting on the dataset in question and calculate
the regression metrics. The metrics we report are adjusted coefficient of
determination ($\tilde{R}^{2}$), root mean squared error between true and
predicted values (RMSE), and Pearson’s correlation coefficient between true
and predicted values (Corr).

###  4.1 Evaluation on B&S Data

As a starting point, we take the data used by B&S in Experiment 3 of their
paper – Gulda’s performances rated on valence and arousal We perform
regression with our feature sets and compare with the values obtained by B&S
using their features Attack Rate, Pitch Height, and Mode. The results are
summarised in Table 5.

| Arousal | Valence
---|---|---
$\tilde{R}^{2}$ | RMSE | Corr | $\tilde{R}^{2}$ | RMSE | Corr
Mid-level | 0.84 | 0.36 | 0.93 | 0.79 | 0.42 | 0.91
DEAMResNet | 0.91 | 0.27 | 0.96 | 0.69 | 0.50 | 0.86
Low-level | 0.86 | 0.29 | 0.96 | 0.67 | 0.45 | 0.89
Score | 0.31 | 0.74 | 0.67 | 0.61 | 0.55 | 0.83
B&S (exp 3) | 0.48 | - | - | 0.75 | - | -

Table 5: Regression on Gulda data from B&S [11].

We can see that all three audio-based features perform considerably well for
both arousal and valence to motivate further analysis.

###  4.2 Evaluation on Our Dataset

Next, we perform regression on our complete dataset (comprising of 288 unique
recordings – 48 pieces $\times$ 6 pianists). The results summary can be seen
in Table 6(a). Here again, we observe that while DEAMResNet Emotion features
perform best on arousal and Score features perform best on valence, Mid-level
features show a balanced performance across both the emotion dimensions.

To evaluate generalizability, we perform cross-validation with three different
kinds of splits – piece-wise (all 6 performances of a piece are test samples
in a fold, for a total of 48 folds), pianist-wise (all 48 pieces of a pianist
are test samples in a fold, for a total of 6 folds), and leave-one-out (one
recording is the test sample per fold, for a total of 288 folds). This is
summarized in Table 6(b).

| Arousal | Valence
---|---|---
Feature Set | $\tilde{R}^{2}$ | RMSE | Corr | $\tilde{R}^{2}$ | RMSE | Corr
Mid-level | 0.68 | 0.56 | 0.83 | 0.63 | 0.60 | 0.80
DEAMResNet | 0.70 | 0.54 | 0.84 | 0.42 | 0.72 | 0.69
Low-level | 0.62 | 0.59 | 0.81 | 0.41 | 0.74 | 0.67
Score | 0.41 | 0.75 | 0.65 | 0.75 | 0.49 | 0.87

(a) Regression metrics with our data

| Piece-wise | Pianist-wise | LOO
---|---|---|---
Feature Set | A | V | A | V | A | V
Mid-level | 0.68 | 0.63 | 0.68 | 0.64 | 0.69 | 0.65
DEAMResNet | 0.67 | 0.37 | 0.61 | 0.41 | 0.68 | 0.43
Low-level | 0.54 | 0.20 |  $-$0.11 |  $-$0.05 | 0.57 | 0.30
Score | 0.08 | 0.67 | 0.39 | 0.75 | 0.37 | 0.74

(b) $\tilde{R}^{2}$ for different cross-validation splits. A: Arousal, V:
Valence, LOO: Leave-One-Out

Table 6: Evaluation results on our data

We see that Mid-level features show good generalization for arousal and are
robust to different kinds of splits. They also show balanced performance
between arousal and valence for all splits. The good performance of the Score
features on the valence dimension (V), here and in the previous experiment, is
mostly due to the Mode feature; there is a substantial correlation in the
annotations between major/minor mode and positive/negative valence.

Figure 2: Feature importance for audio features using T-statistic. Only
features with p<0.05 are shown.

###  4.3 Feature Importance within Feature Sets

We use the absolute value of the t-statistic of a feature as the importance
measure. T-statistic is defined as the estimated weight scaled with its
standard error. We focus on the audio-based feature sets here, as in most
realistic applications scenarios, the score information will not be available
(and, being constant across different performances, will not be able to
distinguish performance aspects). We perform a regression using all audio-
based features (numbering 39 in total) and compare the t-values in Figure 2.

We see that the top-4 and top-2 features in arousal and valence, respectively,
are Mid-level features. These features also make obvious musical sense –
modality is often correlated with valence (positive or negative emotional
quality), and tempo, rhythm, and articulation with arousal (intensity or
energy of the emotion).

###  4.4 Explaining Piece-wise Variation

We observe from the annotation data (see Figure 1) that the distribution of
emotion ratings of each piece is distinct (here, the 6 performances for each
piece form the “distribution” of the piece). In essence, the mean value of
arousal or valence depends on the piece in question. Therefore, to take into
account this factor of variation, we use linear mixed models [20] to model
arousal and valence.

In this linear mixed effect model, the piece id is considered as a “random
effect” intercept, which models part of the residual remaining unexplained by
the features we are evaluating (“fixed effects”). A feature set that models
piece-wise variation better than another set would naturally have a lesser
residual variation to be explained by the random effect. We therefore look at
which feature set has the least fraction of residual variance explained by the
random effect of piece id, defined as:

$\textrm{E}_{\textrm{random}}=\frac{\textrm{Var}_{\textrm{random}}}{\textrm{Var}_{\textrm{random}}+\textrm{Var}_{\textrm{residual}}}$
(1)

where $\textrm{Var}_{\textrm{random}}$ is the variance of the random effect
intercept and $\textrm{Var}_{\textrm{residual}}$ is the variance of the
residual that remains after mixed effects modeling.

Feature Set | Arousal | Valence
---|---|---
Mid-level | 0.50 | 0.86
DEAMResNet | 0.47 | 0.89
Low-level | 0.66 | 0.90
Score | 0.63 | 0.68
Table 7: Fraction of residual variance explained by the random effect of
“piece id”.

We see from Table 7 that the DEAMResNet emotion features best explain piece-
wise variation in arousal, followed closely by Mid-level features. For
valence, the performance of all three audio-based features are close, with
Mid-level features performing the best, however, score features outperform
them with a large margin. This is again due to the relationship between mode
and valence, and mode covarying tightly with the piece ids.

###  4.5 Explaining Performance-wise Variation

Evaluation of performance-wise variation modelling cannot be done with the
mixed effects approach as in the previous section because the means (of
arousal or valence across all pieces) are nearly identical for each pianist.

Therefore, we look at one piece at a time and compute the fraction of variance
unexplained (FVU) and Pearsons’s correlation coefficient (Corr) between
predicted and true values across performances for each such test piece. This
is done as leave-one-piece-out cross-validation, and aggregated by taking the
means. The p-values of the correlation coefficients are counted and we report
the percentage of pieces for which $p<0.1$. With only 6 performances per
piece, a significance level of $p<0.05$ is obtained for only a handful of
pieces. Since score-features based predictions are exactly equal for all
performances of a piece, these metrics are not meaningful, and hence the Score
feature set is not included here.

| Arousal | Valence
---|---|---
Feature Set | FVU | Corr (p<0.1) | FVU | Corr (p<0.1)
Mid-level | 0.31 |  0.58 (47.9%) | 0.36 | 0.42 (27.0%)
DEAMResNet | 0.32 | 0.54 (43.8%) | 0.61 |  0.47 (37.5%)
Low-level | 0.43 | 0.56 (54.2%) | 0.75 | 0.38 (22.9%)

Table 8: Evaluation metrics for performance-wise variation. FVU: Fraction of
Variance Unexplained. Corr: Pearson’s correlation coefficient.

Again, Mid-level features come out at the top in most measures. To illustrate
the modelling of performance-wise variation, we select a few example pieces
that have a high variation of emotion between performances and plot them
together with the predicted values using mid-level features in Figure 3. The
predicted emotion dimensions follow the ratings closely, even for performances
that deviate from the average (e.g. the arousals of Gulda’s performance of
Prelude in A major and Tureck’s performance of Fugue in E minor.)

Figure 3: Some example pieces with high emotion variability between
performances which are modeled particularly well using mid-level features.

##  5 Probing Further

We now describe two additional experiments designed to further probe the
predictive power of our feature sets.

###  5.1 Predicting Emotion of Outlier Performances

Figure 4 shows two examples of pieces where one performance has a vastly
different emotional character than the others – in the first example, Gould
even produces a negative valence effect (mostly through tempo and
articulation) in the E-flat major prelude, which the others play in a much
more flowing fashion. A challenge for any model would thus be to predict the
emotion of such idiosyncratic performances, not having seen them during
training.

Figure 4: Two examples of outlier performances: Prelude # 7 in Db major,
outlier is Gould (left); Prelude #2 in C minor (Gulda, right).

We therefore create a test set by picking out the outlier performance for each
piece in arousal-valence space using the elliptic envelope method [21]. This
gives us a split of 240 training and 48 test samples (the outliers). We train
a linear regression model using each of our feature sets and report the
performance on the outlier test set in Figure 5. We see again that Mid-level
features outperform the others, for both emotion dimensions. We take this as
another piece of evidence for the ability of the mid-level features to capture
performance-specific aspects. The surprisingly good performance of score
features for valence can be attributed to the fact that for most pieces, the
outlier points are separated mostly in the arousal dimension – the spread of
valence is rather small (though not always: see the Gould case in Fig. 4) –
and the score feature “mode” is an important predictor of valence (see earlier
sections).

Figure 5: $\tilde{R}^{2}$ scores on outlier performances. The outliers were
selected using elliptic envelope on the rated arousal-valence space. Out of 48
pieces, the number of times each pianist was an outlier are Gould: 13, Gulda:
10, Tureck: 10, Schiff: 5, Hewitt: 5, Richter: 5.

###  5.2 Predicting Discrete Emotions

Finally, we evaluate how the feature sets perform in a discrete emotion
classification task, which might be relevant in a music recommendation
setting, for instance. The emotion ratings are converted to classes simply by
reducing them to quadrants in the arousal-valence space. In the literature,
these are often associated with the basic emotion labels happy, relaxed, sad,
and angry (in clockwise fashion, starting at upper right). We then train
logistic regression models using our feature sets and report the leave-one-out
cross-validation accuracy in Figure 6. We observe that in this task, all
feature sets perform more-or-less equally well, again with a slight advantage
for the Mid-level features. Note that the random choice baseline accuracy is
0.25. An emotion classification model based on mid-level perceptual features
might be attractive for performance-emotion-aware music recommendation, being
able to offer the mid-level concepts not only as explanations but also as
‘handles’ to search for performances with certain characteristics.

Figure 6: Discrete emotion classification.

##  6 Conclusion

In this work, we evaluated four feature sets – mid-level perceptual features,
pre-trained emotion features, low-level audio features, and score-based
features on their ability to model and predict emotion in terms of arousal and
valence. Specific focus was given on the three audio-based features and their
modelling power over performance-wise variation of emotion. Mid-level features
emerge as the most robust and important among these.

The search for good features to model music emotion is a worthwhile objective
since emotional effect is a very fundamental human response to music. Features
that provide a better handle on content-based emotion recognition can have a
significant impact on applications such as search and recommendation.
Modelling emotion is also becoming increasingly relevant in generative music,
allowing possibilities such as expressivity- or emotion-based snippet
continuation and emotion-aware human-computer collaborative music.

From the experiments presented in this paper, it is clear that deep-learning-
based feature extractors are strong competition to the audio features
typically used for emotion recognition [3]. Here the importance of Mid-level
features is even more pronounced – in addition to being able to model both
arousal and valence well under different conditions, they also provide
intuitive musical meaning to each feature, and have been previously used as
the basis for explainable emotion recognition in [16].

##  7 Acknowledgements

This work is supported by the European Research Council (ERC) under the EU’s
Horizon 2020 research & innovation programme, grant agreement No. 670035 (“Con
Espressione”), and the Federal State of Upper Austria (LIT AI Lab). The
authors would like to thank Carlos Cancino Chacón for helpful discussions and
calculating score-performance alignments and score features, and Jan Schlüter
and Rainer Kelz for data collection and preparation. Thanks to Aimee Battcock,
Cameron Anderson, and Michael Schutz for sharing their annotation data.

## References

  * [1] A. Gabrielsson and P. N. Juslin, “Emotional expression in music performance: Between the performer’s intention and the listener’s experience,” _Psychology of Music_ , vol. 24, no. 1, pp. 68–91, 1996.
  * [2] J. Akkermans, R. Schapiro, D. Müllensiefen, K. Jakubowski, D. Shanahan, D. Baker, V. Busch, K. Lothwesen, P. Elvers, T. Fischinger _et al._ , “Decoding emotions in expressive music performances: A multi-lab replication and extension study,” _Cognition and Emotion_ , vol. 33, no. 6, pp. 1099–1118, 2019.
  * [3] R. Panda, R. M. Malheiro, and R. P. Paiva, “Audio features for music emotion recognition: a survey,” _IEEE Transactions on Affective Computing_ , pp. 1–1, 2020.
  * [4] M. Soleymani, A. Aljanaki, Y.-H. Yang, M. N. Caro, F. Eyben, K. Markov, B. W. Schuller, R. Veltkamp, F. Weninger, and F. Wiering, “Emotional analysis of music: A comparison of methods,” in _Proceedings of the 22nd ACM international conference on Multimedia_ , 2014, pp. 1161–1164.
  * [5] Y. E. Kim, E. M. Schmidt, R. Migneco, B. G. Morton, P. Richardson, J. Scott, J. A. Speck, and D. Turnbull, “Music emotion recognition: A state of the art review,” in _Proceedings of the 11th International Society for Music Information Retrieval Conference, ISMIR 2010_ , vol. 86, 2010, pp. 937–952.
  * [6] Y.-H. Yang and H. H. Chen, “Machine recognition of music emotion: A review,” _ACM Transactions on Intelligent Systems and Technology (TIST)_ , vol. 3, no. 3, pp. 1–30, 2012.
  * [7] R. Orjesek, R. Jarina, M. Chmulik, and M. Kuba, “DNN based music emotion recognition from raw audio signal,” in _29th International Conference Radioelektronika 2019 (RADIOELEKTRONIKA)_. IEEE, 2019, pp. 1–4.
  * [8] M. B. Er and I. B. Aydilek, “Music emotion recognition by using chroma spectrogram and deep visual features,” _International Journal of Computational Intelligence Systems_ , vol. 12, no. 2, pp. 1622–1634, 2019.
  * [9] N. HE and S. Ferguson, “Multi-view neural networks for raw audio-based music emotion recognition,” in _2020 IEEE International Symposium on Multimedia (ISM)_. IEEE, 2020, pp. 168–172.
  * [10] J. Grekow, “Musical performance analysis in terms of emotions it evokes,” _Journal of Intelligent Information Systems_ , vol. 51, no. 2, pp. 415–437, 2018.
  * [11] A. Battcock and M. Schutz, “Acoustically expressing affect,” _Music Perception: An Interdisciplinary Journal_ , vol. 37, no. 1, pp. 66–91, 2019.
  * [12] D. Bogdanov, N. Wack, E. Gómez Gutiérrez, S. Gulati, H. Boyer, O. Mayor _et al._ , “Essentia: An audio analysis library for music information retrieval.” International Society for Music Information Retrieval (ISMIR), 2013, pp. 493–498.
  * [13] B. McFee, C. Raffel, D. Liang, D. P. Ellis, M. McVicar, E. Battenberg, and O. Nieto, “librosa: Audio and music signal analysis in python,” in _Proceedings of the 14th python in science conference_ , vol. 8, 2015, pp. 18–25.
  * [14] C. L. Krumhansl, _Cognitive foundations of musical pitch_. Oxford University Press, 2001.
  * [15] A. Aljanaki and M. Soleymani, “A Data-driven Approach to Mid-level Perceptual Musical Feature Modeling,” in _Proceedings of the 19th International Society for Music Information Retrieval Conference, ISMIR 2018, Paris, France_ , 2018, pp. 615–621.
  * [16] S. Chowdhury, A. Vall, V. Haunschmid, and G. Widmer, “Towards Explainable Music Emotion Recognition: The Route via Mid-level Features,” in _Proceedings of the 20th International Society for Music Information Retrieval Conference, ISMIR 2019_ , 2019.
  * [17] K. Koutini, H. Eghbal-Zadeh, M. Dorfer, and G. Widmer, “The Receptive Field as a Regularizer in Deep Convolutional Neural Networks for Acoustic Scene Classification,” in _2019 27th European signal processing conference (EUSIPCO)_. IEEE, 2019, pp. 1–5.
  * [18] S. Chowdhury and G. Widmer, “Towards explaining expressive qualities in piano recordings: Transfer of explanatory features via acoustic domain adaptation,” in _ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2021, pp. 561–565.
  * [19] A. Aljanaki, Y.-H. Yang, and M. Soleymani, “Developing a Benchmark for Emotional Analysis of Music,” _PloS one_ , vol. 12, no. 3, p. e0173392, 2017\.
  * [20] J. Pinheiro, D. Bates, S. DebRoy, D. Sarkar, and R Core Team, _nlme: Linear and Nonlinear Mixed Effects Models_ , 2021, r package version 3.1-152. [Online]. Available: https://CRAN.R-project.org/package=nlme
  * [21] P. J. Rousseeuw and K. V. Driessen, “A fast algorithm for the minimum covariance determinant estimator,” _Technometrics_ , vol. 41, no. 3, pp. 212–223, 1999.