# Bell inequalities as a tool to probe quantum chaos

Albert Aloy<EMAIL_ADDRESS>Institute for Quantum Optics and Quantum
Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna,
Austria  Vienna Center for Quantum Science and Technology (VCQ), Faculty of
Physics, University of Vienna, Vienna, Austria  Guillem Müller-Rigat  ICFO-
Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, Castelldefels (Barcelona) 08860, Spain.  Maciej Lewenstein  ICFO-
Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, Castelldefels (Barcelona) 08860, Spain.  ICREA, Pg. Lluís Companys
23, 08010 Barcelona, Spain.  Jordi Tura  $\langle aQa^{L}\rangle$ Applied
Quantum Algorithms, Universiteit Leiden  Instituut-Lorentz, Universiteit
Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands  Matteo Fadel
<EMAIL_ADDRESS>Department of Physics, ETH Zürich, 8093 Zürich,
Switzerland

###### Abstract

We introduce a permutationally invariant multipartite Bell inequality for
many-body three-level systems and use it to explore a possible connection
between the presence of nonlocality and quantum chaos. For this, we interpret
the Bell operator resulting from fixing the measurement settings as a system
Hamiltonian and explore the energy level spacing distribution for different
measurement choices and SU(3) irreducible representations. Interestingly, we
find that in every irreducible representation displaying nonlocality, the
measurement settings yielding maximal violation result in a Bell operator
whose energy level spacing distribution signals the absence of quantum chaos.
This situation changes when the measurement settings slightly depart from the
optimal ones, as we generally observe a level spacing distribution
characteristic of chaotic systems. Our work suggests an intriguing connection
between Bell nonlocality, quantum chaos and random matrix theory. These
subjects belonged to the beloved areas of the late Fritz Haake, and we
dedicate this paper to his memory.

Nonlocality is a quantum resource that enables tasks impossible by classical
means, a main example being the violation of Bell inequalities [1].
Considerable theoretical and experimental progress has been recently made in
the detection and characterization of nonlocality, also in many-body
scenarios. However, the implications of nonlocal correlations in relation to
macroscopic physical properties and observables of multipartite systems remain
unclear.

For systems composed of many spin-$1/2$ particles, several methods and
inequalities have been derived for detecting nonlocality and Bell correlations
in a practical and scalable way [2]. This made possible to unveil interesting
connections between the onset of nonlocality and phase transitions [3],
quantum-critical points [4], and metrological advantage [5]. On the other
hand, Bell inequalities for many multi-level systems (such as spin-$S$ with
$S\geq 1$) have been studied in only a few works [6, 7]. In this case, it is
natural to consider Bell scenarios with $m\geq 2$ inputs and $d\geq 3$
(typically $d=2S+1$) outputs per party. Notoriously, the discovery of useful
inequalities in such scenarios becomes extremely hard as $m$ and $d$ increase,
because enumerating all tight inequalities scales as
${\mathcal{O}}\left((d^{mn})^{\lfloor D/2\rfloor}\right)$, where
$D\in{\mathcal{O}}\left((dm)^{n}\right)$ and $n$ is the number of parties [8].

To mitigate such a prohibitive scaling, a successful approach to facilitate
nonlocality detection in the large $n$ regime relies on focusing on few-body
correlations only, as well as on exploiting symmetries already present in
physical systems of interest. Examples include Bell inequalities involving
one- and two-body observables with translational [9, 10] and permutational
invariance [11, 12, 13, 14, 15], which can make the complexity of the Bell
scenario scale polynomially in $n$. This enabled the experimental detection of
Bell correlations in ensembles of $n\approx 500$ [16] and $n\approx 5\cdot
10^{5}$ [17] spin-1/2 particles. To date, however, no experiment reported the
detection of Bell correlations in ensembles of particles with larger spin.

Systems of spin-1 or, more generally, of three-level particles play an
important role in nuclear shell models [18, 19, 20], ultracold atomic
ensembles [21, 22, 23, 24] and in solid-state physics [25, 26, 27]. This
motivated the development of ideas to simulate qudit Hamiltonians using
trapped ions, superconducting circuits and ultracold atoms. Moreover, it was
noted that such SU(3) models provide a minimal example of autonomous chaotic
dynamics, as opposed to SU(2) models where quantum chaos is typically observed
by inducing time-dependent interactions or periodic driving [28, 29].

While classical chaos is associated to an exponentially fast growing distance
between phase-space trajectories, quantum chaos is often associated to
spectral properties of the system’s Hamiltonian. The celebrated Bohigas-
Giannoni-Schmit (BGS) conjecture [30] suggests that the spectrum statistics of
quantum systems whose classical limit show chaotic behaviour are expected to
be characterized by correlated energy levels similar to the ones encountered
in Random Matrix Theory [31, 32]. A Poisson distribution of energy-level
spacing is a characteristic indicator of a system that is integrable in the
classical limit, while a Wigner-Dyson distribution of energy-level spacing is
an indicator of chaotic dynamics [32].

In this work, we investigate the connection between quantum chaos and the
presence of nonlocality. To this end, we introduce a three-outcome
permutationally invariant Bell inequality (PIBI) from which we derive a Bell
operator for multipartite three-level systems. We show that, depending on the
chosen SU(3) representation, the Bell operator shows energy level spacing
distributions characteristic of an integrable (resp. chaotic) model and
generically the presence (resp. absence) of nonlocality. Our work sheds light
on a possible connection between quantum correlations, the dynamical
properties of a quantum system and random matrix theory.

A three-level PIBI and its corresponding Bell operator.– For our analysis we
introduce the multipartite two-setting three-outcome PIBI

$\displaystyle
B=(\mathcal{P}_{0|0}+\mathcal{P}_{0|1}+\mathcal{P}_{1|0}+\mathcal{P}_{1|1})+$
(1)
$\displaystyle(\mathcal{P}_{00|00}+\mathcal{P}_{00|11}+\mathcal{P}_{11|00}+\mathcal{P}_{11|11})-2(\mathcal{P}_{01|01}+\mathcal{P}_{01|10})\geq
0\;,$

where $\mathcal{P}_{a|x}=\sum_{i\in[n]}p_{i}(a|x)$ represents the collective
one-body conditional probability of obtaining outcome $a$ for measurement
setting $x$. Analogously, $\mathcal{P}_{ab|xy}=\sum_{i\neq
j\in[n]}p_{ij}(ab|xy)$ represents the collective two-body conditional
probability. In the Supplementary Material [33] we prove that Eq. (1) is
indeed a Bell Inequality by showing that it has classical bound $\beta_{c}=0$.
Therefore, observing $B<0$ signals nonlocal correlations.

To show that Bell inequality (1) can be violated by quantum mechanics, one has
to find appropriate quantum states and measurements. To this end, we associate
each measurement setting to a self-adjoint operator, such that Eq. (1) can be
written as a Bell operator $\mathcal{B}(\bm{\theta})$, where $\bm{\theta}$ is
a vector parametrizing a specific measurement setting. The maximum quantum
violation of the inequality can then be related to
$\min\limits_{\bm{\theta}}\lambda_{\text{min}}(\mathcal{B}(\bm{\theta}))$,
where $\lambda_{\text{min}}$ indicates the minimum eigenvalue. To carry out
this minimization over $\bm{\theta}$ for qutrits, we need a way to parametrize
measurements. Contrary to the two-level case, i.e. SU(2), where any local
measurement can be efficiently parametrized by two angles, for higher-
dimensional systems the parametrization is somewhat more involved [34].

For this work, we adopt the following continuous parametrization of unitary
SU(3) operators. We start by choosing a set of unitaries
$\\{U_{0},\ldots,U_{m-1}\\}$, all with the same spectrum ordered by complex
phase argument based on $d$-roots of unity $\\{1,\zeta,\ldots,\zeta^{d-1}\\}$,
where $\zeta=e^{i2\pi/d}$. Taking $d=3$, we consider the Heisenberg-Weyl
operators $X$ and $Z$, which act on the computational basis as
$X\ket{\alpha}=\ket{\alpha+1\mod d}$ and
$Z\ket{\alpha}=\zeta^{\alpha}\ket{\alpha}$ [35]. These satisfy
$X^{d}=Z^{d}=\mathbb{I}$. In order to parameterize any unitary of
$\mathrm{SU}(3)$, we take a set of generators $g_{\ell}$ such that
$V_{\ell}=e^{ig_{\ell}}$ are the elements of the set
$\\{X,Z,X^{2},XZ,ZX,XZ^{2},X^{2}Z,Z^{2}X,X^{2}Z^{2}\\}$. Note that this choice
is sufficient for $g_{\ell}$ to generate the Lie algebra
$\bm{\mathfrak{su}}(3)$.

For simplicity, here we consider that the measurement outcomes are
$\zeta^{i}$, as a generalization of the $\pm 1$ outcomes one typically
associates with the $d=2$ case with Pauli matrices. Therefore, we are going to
parameterize measurement operators as unitaries with spectrum $\zeta^{i}$.
Note that the fact that these are non-Hermitian is irrelevant, as the only
relevant information needed is the basis in which these unitaries diagonalize
[36]. Such basis will then be used to create the rank-1 projectors used to
construct $\mathcal{B}$. To this end, let us define

$g(\bm{\theta}):=g_{0}+\sum\limits_{\ell=1}^{M}\theta_{\ell}(g_{\ell}-g_{0}),$
(2)

where $\bm{\theta}\in\mathbb{R}^{M}$, so that we have the desired unitary
parametrization

$U(\bm{\theta}):=e^{{i\mkern 1.0mu}g(\bm{\theta})}De^{-{i\mkern
1.0mu}g(\bm{\theta})},$ (3)

where $D$ is a diagonal matrix containing eigenvalues $\zeta^{i}$. Concluding,
instead of directly combining different unitaries $U$ to obtain new ones, we
have moved the parametrization to the Hermitian matrices $g_{\ell}$ which form
a $\mathbb{R}$-vector space. Therefore, by varying $g_{\ell}$ using the
parametrization in Eq. 2 we have obtained an interpolating function
$U(\bm{\theta})$ in Eq. 3, which allows us to implement typical numerical see-
saw and gradient descend optimization techniques.

For each measurement setting $\bm{\theta}$, via the inverse Fourier transform,
we obtain the corresponding projectors $P_{j,k}=(\zeta^{0\cdot
j}U_{k}^{3}+\zeta^{1\cdot j}U_{k}^{2}+\zeta^{2\cdot j}U_{k})/3$, where
$j\in\\{0,1,2\\}$ labels the measurement outcomes and $k\in\\{0,1\\}$ the
measurement settings. In principle, the two measurement settings
$\bm{\theta}_{k}^{(i)}$ can be chosen differently for every party
$i\in\\{1,\ldots,n\\}$, but this would result in a too demanding optimization
task. Therefore, we restrict ourselves to the same two measurement settings
for every party, i.e. $\bm{\theta}_{k}^{(i)}=\bm{\theta}_{k}$, as numerical
evidence supports the intuition that this choice is sufficient for
permutationally invariant Bell inequalities [37].

By virtue of Schur-Weyl duality, PI Bell operators block-diagonalize in a
symmetry-adapted basis in blocks of polynomial size [3]. These are indexed by
the irreducible representations of the permutation group. Therefore, we can
further restrict our search for the optimal measurements to each different
blocks separately. We outline the details of this restricted optimization in
the supplementary material [33].

Figure 1: Nearest-neighbour spacing distributions (NNSD) for PIBI (1) with
$n=25$ and optimal measurement settings. The irreps $(p,q)=(21,2)$ (in blue)
and $(9,8)$ (in red) have been chosen for illustrative purposes, which in the
symmetric subspace have 825 and 855 eigenvalues, respectively. Solid lines are
fits using the Brody distribution Eq. (5), which interpolates between Poisson
($\omega\sim 0$, typical of an integrable system) and Wigner-Dyson
distributions ($\omega\sim 1$, typical of a chaotic system).

SU(3) and quantum chaos.– The Hilbert space of a three-level particle is
spanned by the basis vectors $\ket{0}=(1,0,0)^{T}$, $\ket{1}=(0,1,0)^{T}$ and
$\ket{2}=(0,0,1)^{T}$. The transition from state $\ket{\beta}$ to
$\ket{\alpha}$ is then described by one of the nine ladder operators
$\tau_{\alpha\beta}=\ket{\alpha}\bra{\beta}$, for
$\alpha,\beta\in\\{0,1,2\\}$. For a system of $n$ three-level particles we
introduce the collective operators
$S_{\alpha\beta}=\sum_{i}\tau_{\alpha\beta}^{(i)}$, where
$\tau_{\alpha\beta}^{(i)}$ denotes the operator $\tau_{\alpha\beta}$ applied
on the $i$th particle. Note that we have $S_{00}+S_{11}+S_{22}=n\mathbb{I}$.
Following the convention used in particle physics, we define the third
component of the isospin $T_{3}=(S_{00}-S_{11})/2$ and the hypercharge
$Y=(S_{00}+S_{11}-2S_{22})/2$. An irreducible representation of SU(3) (irrep
for short) is uniquely characterized by its highest-weight vector $\ket{\mu}$,
which is defined as the common eigenstate of $T_{3}$ an $Y$ with eigenvalues

$T_{3}\ket{\mu}=\dfrac{p}{2}\ket{\mu}\;,\quad
Y\ket{\mu}=\dfrac{p+2q}{3}\ket{\mu}\;,$ (4)

and it is annihilated by $S_{01}\ket{\mu}=S_{12}\ket{\mu}=S_{02}\ket{\mu}=0$.
Analogously to the spin number $0\leq J\leq n/2$ used to label SU(2) irreps,
we use here the pair of non-negative integers $(p,q)$ to label SU(3) irreps.
The dimension of one such representation is then $(1+p)(1+q)(2+p+q)/2$.

SU(3) Hamiltonians, such as the one describing paradigmatic spin-1 models
[18], have been investigated extensively in the context of quantum chaos [32].
This is often associated to a chaotic dynamics in the classical limit, namely
when taking the dimension of the irreducible representation to infinity [38,
39]. A commonly used indicator of quantum chaos consists in analyzing the
nearest-neighbour energy level spacing distribution (NNSD) of the unfolded
energy spectrum [32, 33]. If the NNSD follows a Poisson distribution, then the
NNSD illustrates so-called “level attraction”, that in general signals an
integrable system in the classical limit. If, on the contrary, the NNSD
follows a Wigner-Dyson distribution, then the NNSD illustrates so-called
“level repulsion”, which signals a chaotic system in the classical limit
(equivalently described by random matrix theory). For this reason, it is
convenient to fit the NNSD computed numerically with the so-called Brody
distribution [40, 41], which gives a function interpolating between these two
limit cases. For random matrices sampled from the Gaussian orthogonal
ensemble, the Brody distribution reads

$P(s,\omega)=A(\omega+1)s^{\omega}\exp\left(-As^{\omega+1}\right)\;,$ (5)

where $A=\Gamma^{\omega+1}\left(\frac{\omega+2}{\omega+1}\right)$ and
$0\leq\omega\leq 1$ is the Brody parameter characterizing the interpolation
between Poisson ($\omega\sim 0$) and Wigner-Dyson ($\omega\sim 1$)
distributions. See Fig. 1 for an example.

Figure 2: Maximal quantum violation of PIBI (1) when restricted to the $(p,q)$
irrep of SU(3) for $n=25$. The classical bound is $\beta_{c}=0$ and,
consequently, $\braket{B}<0$ certifies nonlocality. We use $r=p/(p+q)$ as a
means to quantify permutational invariance of each irrep, with $r=1$ being the
fully symmetric case. The pairs $(p,q)$ in blue correspond to Poisson NNSD
statistics (signalled by $\omega\sim 0$, integrable), while the pairs in red
correspond to Wigner-Dyson NNSD statistics (signalled by $\omega>0$, chaotic).
Dashed lines connects points with same $p+2q$ and $p-q$. We observe the
largest amount of violation for to the totally symmetric sector, as expected
from PI of the Bell operator [37, 3].

PIBI irreducible representations and energy spacing distribution.– The PIBI
Eq. (1) we have introduced is capable of detecting nonlocality in low energy
states of SU(3) Hamiltonians [42] that are relevant in the context of quantum
chaos [43, 39, 18]. In particular, every Bell operator can itself be
interpreted as an Hamiltonian whose low-energy states can show nonlocal
correlations [44, 3]. This allows us to explore a possible connection between
the detection of Bell nonlocality and quantum chaos. For this investigation,
we proceed in the following way. First, given PIBI (1) and a number of parties
$n$, we choose an irrep $(p,q)$. Next, we express in this SU(3) sector the
Bell operator associated to our PIBI and maximize its quantum violation by
optimizing over the measurement settings as explained above. Finally, we
compute the NNSD of the resulting Bell operator and fit it to the Brody
distribution Eq. (5) to extract the parameter $\omega$, see e.g. Fig. 1.
Repeating this approach for all inequivalent irreps $(p,q)$ gives us a
relation between quantum violations and Brody parameters $\omega$. These
results are summarized in Fig. 2 for $n=25$, where we represent the quantum
violations as a function of $r=p/(p+q)\in[0,1]$. This parameter $r$ can be
understood as a quantifier for the degree of symmetry of each irrep, with
$r=1$ representing the fully symmetric case. Interestingly, different values
of $r$ can be associated to different classical limits where chaos may or may
not manifest [39], a feature which is absent in the SU(2) case. In Fig. 2 we
indicate whether the NNSD associated to the specific irrep is characteristic
of an integrable (blue, $\omega\sim 0$) or chaotic (orange, $\omega>10^{-3}$)
system.

From our analysis we note that, when restricting to the optimal measurements,
irreps that exhibit nonlocality generically display a Poisson NNSD
distribution. Conversely, irreps where no detection of nonlocality is observed
generically display a Wigner-Dyson NNSD distribution. We observe this
behaviour for different number of parties $n$, starting from $n=8$ (when the
PIBI starts detecting nonlocality) up to $n=32$ (beyond which a numerical
analysis becomes computationally expensive). Based on this observation we
conjecture that, for the PIBI Eq. (1) with optimal measurements, irreps
$(p,q)$ exhibiting nonlocality have a NNSD with $\omega\sim 0$ as $n$ goes to
infinity, which signals integrability in the classical limit. Figure 2 shows
cases of irreps apparently contradicting this conjecture (the orange points
below zero). However, we attribute them to finite size effects, since for
$n=25$ the scarce number of energy levels results in coarse-grained NNSDs.

Remarkably, we note the emergence of interesting patterns when plotting the
maximal quantum violation for each irrep as a function of $r$. As an example,
we highlight with thick dashed lines in Fig. 2 the connection between points
with same $p+2q$, meaning a constant value of the hypercarge $Y$ Eq. (4). The
emergent structure suggests that there may be analytical expressions relating
the maximal quantum violation of the PIBI with the degree of permutation
symmetry in the specific subsector being probed. Exploring this potential
connection is however beyond current scopes and is left for future work.

Let us emphasize that the results presented in this section are based on
optimizing the measurement settings entering the PIBI. The measurement choice
plays a crucial role, since it defines the Bell operator and, therefore, its
quantum violation and NNSD. After having observed that when restricting to the
measurements leading to maximal violation we generically obtain a Bell
operator with Poisson NNSD, we proceed with exploring what happens for more
general measurement choices.

Random measurement settings.– To investigate the generic case, we study here
the connection between the violation of PIBI Eq. (1) and the NNSD in the case
of pairs of random measurement settings. For every irrep, we have generated
more than $10^{3}$ random projectors obtained by appropriately sampling
matrices from SU(3) and then computed the NNSD of the resulting Bell operator.
We show in Fig. 3 the histogram of the fitted Brody parameters for the
illustrative case $(p,q)=(25,0)$ and $10^{4}$ samples.

Figure 3: Histogram of Brody parameters $\omega$ resulting from fitting the
NNSD of the Bell operator constructed from the PIBI (1) using random
projectors. Here $n=25$ and $(p,q)=(25,0)$, i.e. the lowest point in Fig. 2,
but other irreps show a similar behaviour despite having a lower fraction of
points exhibiting nonlocality.

Generically, we observe that the NNSD now approaches a Wigner-Dyson
distribution, independently of the quantum violation. Interestingly, however,
note that when sampling over random projectors we obtain a high percentage of
nonlocality detection. We attribute this behaviour to the restriction of using
the same measurement settings for all parties. In an even more general
scenario, where each party would be free to set different random measurements,
we would expect the departure from Poisson NNSDs to be even more accentuated.
We thus conclude that generically observing a Poisson NNSD when setting the
optimal measurements for each irrep is a remarkable property, since we see
that typically the Bell observable displays level-repulsion.

Volume of Poisson-like behavior around the optimal point.– To quantify the
robustness of the connection between Poisson NNSD and measurement optimality,
we proceed as follows. For a given $n$ and irrep, we start from the Bell
operator corresponding to the optimal measurements and then gradually deviate
from it by smoothly perturbing the measurements until the NNSD of the new Bell
operator stops being Poissonian. This allows us to estimate a region of
measurements leading do Poisson NNSD, whose volume we observe shrinking as $n$
increases.

Figure 4: Estimated volume of the region of measurement settings resulting in
a Poisson NNSD plotted as a function of the number of parties $n$. We observe
that the volume decreases as the number of parties $n$ increases. The inset
for $n=25$ shows that only a small percentage of the randomly generated
observables fall inside the estimated Poissonian region (see the text for
details).

For this analysis, we use the parametrization Eq. 3 to generate a random
direction $\bm{\theta}^{\text{rand}}\in\mathbb{R}^{9}$ with
$||\bm{\theta}^{\text{rand}}||\ll 1$, that we iteratively add to the optimal
measurements $\bm{\theta}^{\text{opt}}$ as a perturbation. At each step, we
check the new Bell operator and its NNSD. After a sufficient number of
iterations $\alpha\in\mathbb{N}$, we observe the NNSD transition from being
Poisson ($\omega\sim 0$) to Wigner-Dyson ($\omega>0$). The point at which this
transition occurs defines the measurement settings
$\tilde{\bm{\theta}}=\bm{\theta}^{\text{opt}}+\alpha\bm{\theta}^{\text{rand}}$.
By repeating this procedure for several random directions
$\bm{\theta}^{\text{rand}}_{j}$ we obtain a set of points in the measurement
settings space, centered around the optimal setting, which samples the
boundary of the region where the NNSD remains Poissonian. After having
collected enough boundary points $\tilde{\bm{\theta}}$, we use the
corresponding projectors to obtain observables $A(\tilde{\bm{\theta}})$ and
use the Frobenius distance to compute the average radius $R^{\text{avg}}$ of
the region with respect to the optimal settings. That is, we compute
$R^{\text{avg}}=\sum_{j=1}^{s}R_{j}/s$, where $s$ is the number of estimated
boundary points we obtained and
$R_{j}=\mathrm{Tr}\left[\left(A(\bm{\theta}^{\text{opt}})-A(\tilde{\bm{\theta}}_{j})\right)\left(A(\bm{\theta}^{\text{opt}})-A(\tilde{\bm{\theta}}_{j})\right)^{\dagger}\right]^{1/2}$.
Therefore, we identify $R^{\text{avg}}$ as the radius of a ball approximating
the Poissonian region. Finally, we use a Monte Carlo method to estimate the
volume of this region: we generate random observables $A(\bm{\theta})$ and
estimate their membership in the Poissonian region. Concretely, we check
whether their radius $R$ is smaller or larger than the average radius of the
region $R^{\text{avg}}$. To ensure that the random observables have been
generated uniformly, we construct them from random unitaries
$U\in\mathbb{C}^{3\times 3}$ sampled from the Haar measure.

In Fig. 4, for each shown value of $n$ we have estimated the Poissonian
regions using $10^{3}$ random directions and $10^{5}$ random unitaries $U$. We
observe that the volumes associated to Poisson NNSD become smaller as the
number of parties increases. Our numerical study suggests that this volume
tends to zero in the asymptotic limit. Therefore, this further supports our
conjecture that the Poisson NNSD observed around the point maximally violating
the Bell inequality (1) is indeed a special case. This observation applies to
the maximal violation of each irrep displaying nonlocality.

Conclusions and outlook.– We have investigated the connection between Bell
nonlocality and quantum chaos. To this end, we introduced a novel two-input
three-outcome multipartite Bell inequality and computed for qutrits the Bell
operators resulting from different choices of measurement settings. We have
characterized the NNSD of these operators and found that, generically,
measurements yielding maximal violation give Poisson statistics, while non-
optimal measurements give Wigner-Dyson statistics.

Our results lead us to conjecture that, in the thermodynamic limit, only the
optimal measurement settings lead to a Bell operator with NNSD characteristic
of an integrable system. Even a slight perturbation from these settings,
although it would not alter the amount of Bell violation observed, quickly
results in an operator characteristic of a chaotic system.

Our work represents a first step towards using Bell inequalities as a
diagnostic tool to probe quantum chaos. The structure present in the points of
maximal violation suggest that the absence of chaos can allow for elegant
characterizations of the quantum state and measurements, which is an essential
ingredient for self-testing of quantum many-body systems [45]. In future
works, it would be interesting to further explore how our findings manifest in
more general Bell scenarios.

Acknowledgments.– AA acknowledges support from the Austrian Science Fund (FWF)
(projects P 33730-N and 10.55776/PAT2839723) and by the ESQ Discovery
programme (Erwin Schrödinger Center for Quantum Science & Technology), hosted
by the Austrian Academy of Sciences (ÖAW).

GM and ML acknowledge support from: European Research Council AdG NOQIA;
MCIN/AEI (PGC2018-0910.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, Plan National STAMEENA PID2022-139099NB, I00, project
funded by MCIN/AEI/10.13039/501100011033 and by the “European Union
NextGenerationEU/PRTR” (PRTR-C17.I1), FPI); QUANTERA MAQS PCI2019-111828-2);
QUANTERA DYNAMITE PCI2022-132919, QuantERA II Programme co-funded by European
Union’s Horizon 2020 program under Grant Agreement No 101017733); Ministry for
Digital Transformation and of Civil Service of the Spanish Government through
the QUANTUM ENIA project call - Quantum Spain project, and by the European
Union through the Recovery, Transformation and Resilience Plan -
NextGenerationEU within the framework of the Digital Spain 2026 Agenda;
Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social
Fund FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2023-3-0024); Funded by the
European Union. Views and opinions expressed are however those of the
author(s) only and do not necessarily reflect those of the European Union,
European Commission, European Climate, Infrastructure and Environment
Executive Agency (CINEA), or any other granting authority. Neither the
European Union nor any granting authority can be held responsible for them
(HORIZON-CL4-2022-QUANTUM-02-SGA PASQuanS2.1, 101113690, EU Horizon 2020 FET-
OPEN OPTOlogic, Grant No 899794), EU Horizon Europe Program (This project has
received funding from the European Union’s Horizon Europe research and
innovation program under grant agreement No 101080086 NeQSTGrant Agreement
101080086 — NeQST); ICFO Internal “QuantumGaudi” project; European Union’s
Horizon 2020 program under the Marie Sklodowska-Curie grant agreement No
847648; “La Caixa” Junior Leaders fellowships, La Caixa” Foundation (ID
100010434): CF/BQ/PR23/11980043.

JT has received support from the European Union’s Horizon Europe program
through the ERC StG FINE-TEA-SQUAD (Grant No. 101040729). JT also acknowledges
support from the Quantum Delta NL program. This publication is part of the
‘Quantum Inspire – the Dutch Quantum Computer in the Cloud’ project (with
project number [NWA.1292.19.194]) of the NWA research program ‘Research on
Routes by Consortia (ORC)’, which is funded by the Netherlands Organization
for Scientific Research (NWO).

MF was supported by the Swiss National Science Foundation Ambizione Grant No.
208886, and The Branco Weiss Fellowship – Society in Science, administered by
the ETH Zürich.

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##  I Supplemental material for “Bell inequalities as a tool to probe quantum
chaos”

###  I.1 Classical bound for the Bell inequality (1)

Here we provide a proof showing that the Bell inequality $B$ introduced in the
main text has classical bound $\beta_{c}=0$ for any number of parties $n$.
That is, we want to show that

$\displaystyle B$ $\displaystyle=$
$\displaystyle(\mathcal{P}_{0|0}+\mathcal{P}_{0|1}+\mathcal{P}_{1|0}+\mathcal{P}_{1|1})+(\mathcal{P}_{00|00}+\mathcal{P}_{00|11}+\mathcal{P}_{11|00}+\mathcal{P}_{11|11})-2(\mathcal{P}_{01|01}+\mathcal{P}_{01|10})\geq
0.$ (6)

First, in order to account for all local hidden variables, we want to come up
with a parametrization to describe the conditional probabilities in terms of
Local Deterministic Strategies (LDS). Because our Bell inequality is
permutationally invariant, many Bell inequality coefficients take the same
values at many LDSs, which leads to redundancies. Hence, instead of
considering all the $d^{mn}$ possibilities of the general case ($3^{2n}$ for
our case with $3$ outcomes an $2$ measurements), we propose the following
(much more efficient) parametrization: Suppose that at each run all the
outcomes for each possible measurement and party are predetermined. Then, let
$c_{a,a^{\prime}}$ be the total number of parties that have predetermined the
pair of outcomes $a,a^{\prime}\in\\{0,1,2\\}$ for the two possible measurement
settings $x\in\\{0,1\\}$ respectively. It follows by definition that
$c_{a,a^{\prime}}\geq 0$ and $\sum_{a,a^{\prime}}c_{a,a^{\prime}}=n$.
Therefore, following this parametrization, the symmetrized one-body
conditional probabilities $\mathcal{P}_{a|x}$ under a given LDS can be
expressed as:

$\displaystyle\mathcal{P}_{a|x}:=\sum\limits_{i=1}^{n}p_{i}(a|x)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
LDS}}}}{{=}}}\left\\{\begin{array}[]{cc}c_{a,0}+c_{a,1}+c_{a,2}&\textrm{for
}x=0\\\ c_{0,a}+c_{1,a}+c_{2,a}&\textrm{for }x=1\end{array}\right..$ (9)

On the other hand, the symmetric two-body conditional probabilities
$\mathcal{P}_{ab|xy}$ factorize under a given LDS as:

$\begin{split}\mathcal{P}_{ab|xy}:=&\sum\limits_{i\neq
j}p_{ij}(ab|xy)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
LDS}}}}{{=}}}\sum\limits_{i\neq j}p_{i}(a|x)p_{j}(b|y)\\\
=&\underbrace{\sum\limits_{i\neq
j}p_{i}(a|x)p_{j}(b|y)+\sum\limits_{i}p_{i}(a|x)p_{i}(b|y)}_{\mathcal{P}_{a|x}\cdot\mathcal{P}_{b|y}}-\underbrace{\sum\limits_{i}p_{i}(a|x)p_{i}(b|y)}_{:=\mathcal{Q}_{ab|xy}}\;,\end{split}$
(10)

where we have defined

$\displaystyle\mathcal{Q}_{ab,xy}:=\left\\{\begin{array}[]{l}\mathcal{P}_{a|x}\\\
0\\\ c_{a,b}\\\ c_{b,a}\end{array}\right.\begin{array}[]{l}\text{if
}a=b,x=y\\\ \text{if }a\neq b,x=y\\\ \text{if }x=0,y=1\\\ \text{if
}x=1,y=0\end{array}\;.$ (19)

Note that one can neglect one of the outcomes without loss of generality by
means of the NS principle,

$P(a_{1},\ldots,\hat{a}_{i},\ldots,a_{n}|x_{1},\ldots,\hat{x}_{i},\ldots,x_{n})\equiv\sum\limits_{a_{i}\in\\{0,1,2\\}}P(a_{1},\ldots,a_{n}|x_{1},\ldots,x_{n})\;,$
(20)

where $\hat{\cdot}$ denotes the absence of that coordinate and the $\equiv$
symbol means that the LHS of Eq. 20 is well-defined; i.e., it does not depend
on the value of $x_{i}$. Hence, for instance we can take Eqs. 9, 10 and 19
with $a,b\in\\{0,1\\}$. Notice also that it is straightforward to generalize
our parametrization to any number of outcomes $d$.

Finally, in Tab. 1 we express the one-body terms and the factorized two-body
terms as a function of the quantities $c_{a,a^{\prime}}$. Therefore, all the
possible local-realist correlations in a $(n,2,3)$ Bell-type experiments can
be described in terms of the relations in Table 1 and shared randomness.
Moreover, the local polytope for an $(n,2,3)$ permutationally invariant Bell
scenario characterized by one- and two-body correlators is formed by the
convex hull of all the configurations satisfying the relations in Table 1.

$\mathcal{P}_{0|0}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}c_{0,0}+c_{0,1}+c_{0,2}$ | $\mathcal{P}_{00|00}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{0|0}^{2}-\mathcal{P}_{0|0}$ | $\mathcal{P}_{00|01}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{0|0}\mathcal{P}_{0|1}-c_{0,0}$ | $\mathcal{P}_{00|10}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{00|01}$ | $\mathcal{P}_{00|11}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{0|1}^{2}-\mathcal{P}_{0|1}$
---|---|---|---|---
$\mathcal{P}_{1|0}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}c_{1,0}+c_{1,1}+c_{1,2}$ | $\mathcal{P}_{01|00}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{0|0}\mathcal{P}_{1|0}$ | $\mathcal{P}_{01|01}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{0|0}\mathcal{P}_{1|1}-c_{0,1}$ | $\mathcal{P}_{01|10}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{10|01}$ | $\mathcal{P}_{01|11}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{0|1}\mathcal{P}_{1|1}$
$\mathcal{P}_{0|1}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}c_{0,0}+c_{1,0}+c_{2,0}$ | $\mathcal{P}_{10|00}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{01|00}$ | $\mathcal{P}_{10|01}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{1|0}\mathcal{P}_{0|1}-c_{1,0}$ | $\mathcal{P}_{10|10}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{01|01}$ | $\mathcal{P}_{10|11}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{01|11}$
$\mathcal{P}_{1|1}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}c_{0,1}+c_{1,1}+c_{2,1}$ | $\mathcal{P}_{11|00}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{1|0}^{2}-\mathcal{P}_{1|0}$ | $\mathcal{P}_{11|01}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{1|0}\mathcal{P}_{1|1}-c_{1,1}$ | $\mathcal{P}_{11|10}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{11|01}$ | $\mathcal{P}_{11|11}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny LDS}}}}{{=}}}\mathcal{P}_{1|1}^{2}-\mathcal{P}_{1|1}$
Table 1: Resulting one- and two-body conditional probabilities under an LDS.

Now that we have a parametrization to incorporate the LDSs into the
conditional probabilities, we are ready to substitute the corresponding
conditional probabilities in the Bell inequality. After rearranging the terms,
one ends up with the following polynomial:

$\displaystyle B$ $\displaystyle=$
$\displaystyle(c_{00}+c_{02})^{2}+(c_{00}+c_{20})^{2}+(c_{11}+c_{12})^{2}+(c_{11}+c_{21})^{2}$
(21)
$\displaystyle+(c_{00}-c_{12})^{2}+(c_{00}-c_{21})^{2}+(c_{11}-c_{02})^{2}+(c_{11}-c_{20})^{2}$
$\displaystyle+2(c_{10}+c_{01})-2(c_{00}+c_{11})^{2}-(c_{12}+c_{20})^{2}-(c_{02}+c_{21})^{2},$

where $c_{i,j}\geq 0$ for all $i,j\in\\{0,1,2\\}$ and they fulfill the
constraint $\sum\limits_{0\leq i,j<3}c_{ij}=n$ with $n$ the total number of
parties. Notice that the term $c_{22}$ does not appear in the expression, thus
we can set any $0\leq c_{22}\leq n$ without contributing in the classical
bound. Thus it is trivial to see that there exists at least one strategy
leading to $B=0$, i.e. setting $c_{22}=n$. Consequently, we just have to prove
that $B$ cannot take negative values.

#### Proof that $B\geq 0$:

We are interested in the minimal value that (21) can achieve. Since the terms
$2(c_{10}+c_{01})$ will always add a positive or zero contribution, we can set
them to $c_{10}=c_{01}=0$ without loss of generality to find the minimal value
of $B$. Therefore we simplify the problem to look at the minimal value of:

$\displaystyle\tilde{B}$ $\displaystyle=$
$\displaystyle(c_{00}+c_{02})^{2}+(c_{00}+c_{20})^{2}+(c_{11}+c_{12})^{2}+(c_{11}+c_{21})^{2}$
(22)
$\displaystyle+(c_{00}-c_{12})^{2}+(c_{00}-c_{21})^{2}+(c_{11}-c_{02})^{2}+(c_{11}-c_{20})^{2}$
$\displaystyle-2(c_{00}+c_{11})^{2}-(c_{12}+c_{20})^{2}-(c_{02}+c_{21})^{2}.$

After expanding and rearranging the terms we reach the following equivalent
polynomial:

$\displaystyle\tilde{B}$ $\displaystyle=$ $\displaystyle
2\bigg{[}c_{00}^{2}+c_{11}^{2}+\frac{c_{02}^{2}+c_{20}^{2}}{2}+\frac{c_{12}^{2}+c_{21}^{2}}{2}$
(23)
$\displaystyle+c_{00}(c_{02}+c_{20})-c_{11}(c_{02}+c_{20})+c_{11}(c_{12}+c_{21})-c_{00}(c_{12}+c_{21})$
$\displaystyle-c_{02}c_{21}-c_{12}c_{20}-2c_{00}c_{11}\bigg{]}.$

Then, the condition for (23) to take negative values corresponds to the
following inequality

$c_{11}(c_{02}+c_{20})+c_{00}(c_{12}+c_{21})+c_{02}c_{21}+c_{12}c_{20}+2c_{00}c_{11}>c_{00}^{2}+c_{11}^{2}+\frac{c_{02}^{2}+c_{20}^{2}}{2}+\frac{c_{12}^{2}+c_{21}^{2}}{2}+c_{00}(c_{02}+c_{20})+c_{11}(c_{12}+c_{21})\;,$
(24)

which can be rearranged as:

$(c_{00}-c_{11})(c_{12}+c_{21}-c_{02}-c_{20})>(c_{00}-c_{11})^{2}+\frac{(c_{02}-c_{21})^{2}}{2}+\frac{(c_{12}-c_{20})^{2}}{2}\;.$
(25)

Our goal is to find that such condition leads to a contradiction for all cases
in order to show that $I$ cannot take a negative value.

First it is convenient to define de variables
$x:=c_{00}-c_{11},y:=c_{12}-c_{20},z:=c_{02}-c_{21}$, so that the condition
gets expressed as:

$x(y-z)-\left(x^{2}+\frac{y^{2}}{2}+\frac{z^{2}}{2}\right)>0.$ (26)

Take now $f(x,y,z)=x(y-z)-(x^{2}+\frac{y^{2}}{2}+\frac{z^{2}}{2})$ in order to
find its critical points $\nabla f(x,y,z)=(-2x+y-z,x-y,-z-x)$, $\nabla
f(x^{*},y^{*},z^{*})=0\Rightarrow x^{*}=y^{*},z^{*}=-x^{*}$, where
$f(x^{*},y^{*},z^{*})=0$. Next, by looking at its Hessian matrix
$\bm{H}\left(f(x,y,z)\right)$, where
$\left(\bm{H}\left(f(x,y,z)\right)\right)_{ij}=\frac{\partial^{2}f}{\partial
x_{i}\partial x_{j}}$, one sees that the resulting Hessian matrix has
eigenvalues $\\{-3,-1,0\\}$ and therefore it is negative semidefinite. Thus,
the critical point corresponds to the maximum.

We conclude that (26) leads to a contradiction for all values of $c_{ij}$ and,
consequently, $I$ cannot take negative values. Finally, since the argument is
independent of $n$ and we have seen that $I=0$ is a valid local deterministic
strategy, it follows that the classical bound is $\beta_{c}=0$ for all $n$.

$\square$

###  I.2 Measurement optimization restricted to some irreducible
representation of SU$(3)$

In the task of optimizing the measurement settings for a specific irrep, one
encounters the problem of representing the symmetrized version of a one-body
observable $A$ in that specific irrep. Normally, general formulas for an irrep
$(p,q)$ exist only for specific matrices, so one has to do the appropriate
change of coordinates to represent an arbitrary $A$ in the irrep $(p,q)$. Here
we outline this method. Following the notation of Ref. [46], we start by
fixing the eight basis matrices with the identity element
$\\{\mathbb{I},T_{+},T_{-},T^{3},V_{+},V_{-},U_{+},U_{-},U^{3}\\}^{(p,q)}$ of
the finite dimensional $(p,q)-$irrep of SU$(3)$. In Ref. [46] one can find a
possible representation of these for any $(p,q)$. Then, define $G$ as the Gram
matrix formed from this basis for the local irrep $(p,q)=(1,0)$; i.e.,
$G_{i,j}=\braket{v_{i},v_{j}}$, for
$v_{i}\in\\{\mathbb{I},T_{+},T_{-},T^{3},V_{+},V_{-},U_{+},U_{-},U^{3}\\}^{(1,0)}$.
Next, define a column-vector
$\bm{b}=(\braket{v_{1},A},\braket{v_{2},A},\ldots,\braket{v_{9},A})^{T}$
consisting of the inner products of this basis for some one-party observable
$A$ parametrized as described in the text. Finally, solve the system of linear
equations $G\bm{x}=\bm{b}$ in order to find $\bm{x}$. We are now ready to
represent an observable $A$ in any chosen irrep $(p,q)$ through the following
expression

$A^{(p,q)}=nx_{1}\mathbb{I}^{(p,q)}+\sum\limits_{i=2}^{9}x_{i}v_{i}^{(p,q)}\;,$
(27)

where the first term has been taken out of the sum to go together with the
extra identity elements that need to be added in order to guarantee the proper
normalization.

Now, if we choose the one-party observable $A$ to be a one-body projective
unitary qudit operator (for example, $A=\mathcal{P}_{0|0}$), we can use the
approach introduced in the main text, which allows us to perform typical
optimization techniques restricted to the chosen irrep $(p,q)$ subsector.
Finally, the two-body terms take the form

$\mathcal{P}_{ab|xy}^{(p,q)}=\mathcal{P}_{a|x}^{(p,q)}\mathcal{P}_{b|y}^{(p,q)}-(\mathcal{P}_{a|x}\mathcal{P}_{b|y})^{(p,q)}\;.$
(28)

###  I.3 Unfolding the energy spectrum

To compare the nearest neighbour energy level space distribution of different
operators, it is desirable to normalize it appropriately. The procedure we
follow takes the name of spectrum unfolding [32] and it consists of the
following steps.

First, the energy spectrum is sorted so that the energy levels
$\\{x_{i}\\}_{i}$ are in ascending order $\\{x_{1}\leq x_{2}\leq\ldots\leq
x_{k}\\}$. Then, we compute the cumulative distribution function $I_{x}(E)$
counting the number of energy levels up to energy $E$. This is a discrete
function, which it is convenient to interpolate with a continuous polynomial
function $\tilde{I}_{x}(E)$. The goal is now to rescale the sequence
$\\{x_{i}\\}_{i}$ into a sequence $\\{y_{i}\\}_{i}$, such that its cumulative
function $\tilde{I}_{y}(E)$ is a straight line. This is achieved by inverting
the function $\tilde{I}_{x}(E)$ and using it to rescale the sequence
$\\{x_{i}\\}_{i}$. This spectrum unfolding ensures that the local density of
states of the renormalized levels $\\{y_{i}\\}_{i}$ is unity. From the latter
sequence we compute the nearest-neighbour energy level spacing as
$s_{i}=y_{i}-y_{i-1}$, which are used to obtain the NNSD.


2023 III Workshop Astronomy for Inclusion

# Sound training platform applied to astronomy

N. M. M. Bertaina Lucero  11affiliation: Instituto de Bioingeniería,
Universidad de Mendoza, Ciudad de Mendoza, Argentina. 22affiliation: Consejo
Nacional de Investigaciones Científicas y Técnicas - CONICET. J. Casado
11affiliationmark:  B. García  22affiliationmark:  33affiliation: Universidad
Tecnológica Nacional, Facultad Regional Mendoza, Ciudad de Mendoza, Argentina.
and G. Cayo33affiliationmark:

###### Abstract

The convergence between astronomy and data sonification represents a
significant advancement in the approach and analysis of cosmic information. By
surpassing the visual exclusivity in data analysis in astronomy, innovative
projects have developed software that goes beyond visual representation,
transforming data into auditory and tactile displays. However, it has been
evidenced that this novel technique requires specialized training,
particularly for audio format data. This work describes the initial
development of a platform aimed at providing training for data analysis in
astronomy through sonification. The integration of these tools in astronomical
education and research opens new horizons, facilitating a more inclusive and
multisensory participation in the exploration of space science.

La convergencia entre la astronomía y la sonorización de datos marca un avance
significativo en la forma en que se aborda y analiza la información del
cosmos. Superando la exclusividad visual en el análisis de datos en
astronomía, proyectos innovadores han desarrollado software que va más allá de
la representación visual, transformando datos en despliegues sonoros y
táctiles. Sin embargo, se ha evidenciado que esta novedosa técnica requiere un
entrenamiento especializado, particularmente para datos en formato de audio.
En este trabajo se describe el desarrollo inicial de una plataforma que tiene
como objetivo principal contener entrenamientos para el análisis de datos en
astronomía a través de la sonorización. Lograr la integración de estas
herramientas en la educación y la investigación astronómica abre nuevos
horizontes, facilitando una participación más inclusiva y multisensorial en la
exploración de las ciencias del espacio.

Sonorización de datos Entrenamientos Accesibilidad Multisensorialidad

##  0.1 Introduction

When discussing astronomy, the mind often associates it with numbers, images
of celestial bodies, and data in various formats, but it is always presented
in tables for visual inspection. This predominance of visual expressions in
astronomy can be challenging for people with visual impairments. However,
there are projects aimed at promoting inclusion in this field. These projects
have led to the development of software that goes beyond visual representation
of data, transforming it into sound and even tactile information. Some of the
most relevant projects to date with active support include: HighCharts
Sonification Studio (Kondak et al., 2017), sonoUno (Casado et al., 2019) and
StarSound (Foran et al., 2022). These projects offer a user interface that
makes them more accessible to people without programming knowledge.

On the other hand, some packages allow for the sonification of data without a
user interface, such as Strauss (Harrison et al., 2021),
Astronify111https://astronify.readthedocs.io/en/latest/ and Soni-py (Patton
and Levesque, 2021). In addition to being used for the analysis of
astronomical data, all of these tools can be applied to sonify data of various
types, allowing for scientific research and development in a wide range of
areas.

With the use of this innovative approach to data interpretation in recent
years, the need for specialized training has become evident so that users can
approach, understand, and analyze data through sonification (Lopez and Tello,
2024). Although the sense of hearing is used from the moment a person is born,
the study and subsequent use of a new technique always requires training that
indicates exactly what is being represented and how it should be interpreted.

Most of the programs available for auditory training with visual support have
focused on areas related to language and its comprehension. Among the
diversity of these, could be found some such as that of Hitchcock et al.
(2023), which summarizes the training with visual-auditory biofeedback that is
used to improve speech in individuals with difficulties detecting these
sounds. However, in the search for more innovative applications, some
proposals sonify numerical data to explore the relationship between sound
information and the understanding of visual representations, determining
whether the sonification of graphics can improve understanding, especially in
the educational field, and the identification of features in signals, over
noise, in the case of scientific research Bonebright et al. (2001).

Following this line of research, the early development of a web platform for
training in sonification of different types of astronomical data will be
described. These programs can not only help understand sound perception but
also offer the opportunity to integrate this novel tool into data analysis
that would ultimately allow more people to engage in science, harnessing
multiple senses simultaneously to explore nature.

##  0.2 Preevious works

In 2022, Bertaina Lucero (2023) conducted a series of trainings using the
Psychopy tool (Peirce et al., 2022), which is primarily useful for conducting
experiments in behavioral sciences such as neuroscience, linguistics, and
psychology, among others. In the first proposed training, specific spectral
data such as emission and absorption lines were used, which were displayed
both graphically and sonorously so that the participant could classify whether
what they heard and saw was one of the two mentioned signals (Fig.1). In
addition to a block of visual and auditory display, another block with
auditory display only was included to test whether visual support should be
included at the beginning of the training (Lopez and Tello, 2024).

Figure 1: A. Image and sound display in the training designed with Psychopy.
B. Question and options are displayed for the user to answer using the Arrow
Key block of the keyboard.

In the second stage of development, a training workshop was conducted over two
sessions of 2 hours each. Different blocks or work units were programmed for
this workshop, with the sonification of various types of data, such as noise
produced in the detector of the Gravitational Wave Observatory (EGO)
(Glitches), particles from the Large Hadron Collider (LHC), and cosmic muons
used for muongraphy. Those data were extracted from Casado’s doctoral
thesis(Casado, 2023). These tools and techniques can be applied to various
fields, including astronomy, and can help improve data interpretation and
analysis skills.

The mentioned workshop can be considered a focus group that allowed for a
first approach to the use of the tool and to collect user feedback. This is
how the need for training on a dedicated website arose, as Psychopy is a
desktop program that offers a website for executing the experiments but has
the disadvantage of being paid. In addition, during these meetings,
suggestions for improvement were obtained from users, which were considered
for the original, open, and free web deployment.

##  0.3 Methods and Tools

Taking into account the existing tools for developing a website, ranging from
designing the visible part for the user (front-end) to the part that works
with the server (back-end) and programming the database for collecting user
responses, it is proposed to use the Django web development
framewor2222005-2023 Django Software Foundation and individual contributors.
Available at: https://www.djangoproject.com/. It is a high-level open-source
web framework with active support to date. It offers extensive documentation,
which is up-to-date with the most current version to date (v5.0). It allows
the creation of complex websites, programming most of its parts in Python,
standing out for its effective Model-View-Tempered architecture. Each of these
components is used to handle different parts of a website. On the one hand,
the models generate structures that represent different tables in the database
used and help in its management and manipulation. The views will be in charge
of handling the interactions of the models and templates that the user sees.
Finally, templates are HTML-type files that represent what will be shown to
the user. As for the database, Postgresql was used 3331996-2023 The PostgreSQL
Global Development Group. Available at: https://www.postgresql.org/. Finally,
some functionalities of Bootstrap 5 framework were integrated 444Bootstrap
team. Available at: https://getbootstrap.com/, which includes design and
utilities in a few lines of code that make the development of a clean and
organized front-end more fluid, both at the code and user interface level.
Additionally, it integrates classes in the tags that allow for responsive
design (adapts to different screen sizes).

##  0.4 Development

The website was completely implemented using Django, which allowed the
programming of user views using HTML and CSS files, as well as the management
of the database exclusively with the Python language. Concerning the database,
among the various management systems available, PostgreSQL was chosen due to
its easy integration with the Django project, providing an effective
connection between the developed application and the programmed database. In
addition to this, a database made with PostgreSQL stands out for its ability
to manage large volumes of data, which is crucial for the scalability of the
site in the future. The database not only provides support for the management
of the data received when the user finishes the different training sessions
but also helps to manage all the files that will be displayed, both visually
and by sonification, as well as the user information for the start session
(Fig. 2).

Figure 2: Basic diagram of the design of the database, represented and used in
Django models.

The entire database is controlled through an administration view that helps
obtain the necessary statistics to adjust the different training programs that
are developed. From these same statistics, it is possible to have a report on
the general progress of the users, asking them to adjust the programs as
required. In addition, with the data obtained from the number of correct
answers given, the user can be kept informed of their progress. In this same
database, responses to small surveys conducted with users will be stored,
which will contribute to the continuous improvement of training programs.

Regarding the front-end construction of the site, attention was paid to the
appropriate use of tags and an effective distribution of all its sections.
This was mainly guided by the goal of making the site accessible to as many
users as possible. Although the process is subject to changes with the
changing versions of the languages used in its construction, it is aimed to
achieve the highest accessibility percentage possible, although it cannot be
guaranteed at a hundred percent.

The platform designed for training is intended to be connected to the sonoUno
web, providing a better understanding of sonified data. It is planned to track
each user’s progress, classifying them into three categories: beginner,
intermediate, and advanced (Fig.3). Depending on the category in which a user
is, they will have access to a series of training adapted to their level. As
users correctly complete the training, they will be promoted to a higher
category, unlocking new training with increasing levels of difficulty.

Figure 3: Training page, with the list corresponding to the beginner level.

Each training is organized into blocks, which are further divided into four
displays arranged in a form. Each of these latter consists of presenting an
image, audio, and a question with a series of options for the user to answer,
maintaining a paginated overall view (see Fig.4). This structure helps to
prevent the user from having to scroll down to interact with the next display,
making the experience more comfortable and avoiding adding additional
cognitive load.

Regarding the images and sounds used in the development of the training, these
were obtained from previously selected real astronomical data using sonoUno.
The only exception to this characteristic is found in the initial training
intended for all users, which consists of presenting simple mathematical
functions (Fig.4), serving as an introduction to the training methodology.

It is important to clarify about sonified data, that it is not mandatory to
exclusively use sonoUno. Any of the sonification programs mentioned earlier
can be used. This makes the platform versatile for training users in data
sonification in general, regardless of the software used for this purpose.

Figure 4: Example of deployment of mathematical functions training: beginner
level.

The final website is in the development stage. For it to move to the
production stage, some accessibility tests must be completed, such as testing
the deployment with screen readers. Ensuring accessibility and inclusion is
considered a fundamental pillar of this development.

##  0.5 Conclusion

Sonification of different types of data represents an advance in the way of
approaching research, as well as can create the path towards greater inclusion
in science. This is why the introduction of different software that translates
data into sound, visual, and tactile displays not only challenges
accessibility barriers but also offers a new perspective for exploring and
analyzing information.

The development of specific training programs is essential to unlocking the
full potential of this innovative approach. In addition to this, a specific
educational approach is required to be able to interpret the subtleties found
in this new display of information, which mainly involves discriminating a
signal from noise that may be natural or linked to instrumental problems. This
task is not trivial; it requires an exhaustive study, which allows topics such
as those linked to perception to be included in the results.

On this path to providing an optimal training platform, it is important to
choose a framework that allows a comprehensive approach, from the user
interface to database management, as allowed by Django. Although the website
is still in development, its main goal is to provide adaptive training that
allows an innovative educational experience for the interpretation of sonified
data.

## References

  * Bertaina Lucero (2023) Bertaina Lucero, N. M. M. 2023, in: ArXiv, abs/2305.06943.
  * Bonebright et al. (2001) Bonebright, T. L., Nees, M. A., Connerley, T. T., & McCain, G. R. 2001, in: Testing the effectiveness of sonified graphs for education: A programmatic research project. (Proceedings of the 2001 International Conference on Auditory Display)
  * Casado (2023) Casado, J. 2023, in: https://arxiv.org/pdf/2305.05635.pdf
  * Casado et al. (2019) Casado, J., De La Vega, G., Díaz-Merced, W., Gandhi, P., & García, B. 2019 in: SonoUno: a user-centred approach to sonification. Proceedings of the International Astronomical Union. 2019;15(S367):120-123. doi:10.1017/S174392132100079X
  * De la Vega et al. (2023) De la Vega, G., Dominguez, L., Casado, J., & García, B. 2023, in: SonoUno web: an innovative user centred web interface. (10.48550/arXiv.2302.00081)
  * Foran et al. (2022) Foran, G., Cooke, J., & Hannam, J. 2022, Revista Mexicana de Astronomía y Astrofísica Serie de Conferencias, 54, 1-8.
  * Harrison et al. (2021) Harrison, C., Trayford, J., Harrison, L., & Bonne, N. 2021, in: Audio universe tour of the solar system: using sound to make the universe more accessible. Astronomy and Geophysics, 63(2):2.38–2.40.
  * Hitchcock et al. (2023) Hitchcock, E. R., Ochs, L. C., Swartz, M. T., Leece, M. C., Preston, J. L., & McAllister, T. 2023, in: Tutorial: Using Visual–Acoustic Biofeedback for Speech Sound Training. American journal of speech-language pathology, 32(1), 18-36.
  * Kondak et al. (2017) Kondak, Z., Liang, T. A., and Tomlinson, B., & Walker, B. N. 2017, in: Web sonification sandbox-an easy-to-use web application for sonifying data and equations. In Proceedings of 3rd Web Audio Conference, London.
  * Patton and Levesque (2021) Patton, L., & Levesque, E. 2021, in GitHub (soni-py: A pitch-based data sonification package.) Department of Astronomy, University of Washington, Seattle; Harvard-Smithsonian Center for Astrophysics, Cambridge.
  * Lopez and Tello (2024) Lopez, N. M., & Tello, E. 2024 in: Advances in Bioengineering and Clinical Engineering. Proceedings of the XXIII Argentinian Congress of Bioengineering (SABI 2022) and the XII Clinical Engineering Conference.
  * Peirce et al. (2022) Peirce, J. W., Hirst, R. J., & MacAskill, M. R. 2022. Building Experiments in PsychoPy. 2nd Edn London: Sage.


# States of finite effect algebras

G. Bińczak1 1 Faculty of Mathematics and Information Sciences
Warsaw University of Technology
00-662 Warsaw, Poland  ,  J. Kaleta2 2 Department of Applied Mathematics
Warsaw University of Life Sciences
02-787 Warsaw, Poland  and  A.Zembrzuski3 3 Department of Informatics, Faculty
of Applied Informatics and Mathematics, Warsaw University of Life Sciences
02-787 Warsaw, Poland  1<EMAIL_ADDRESS><EMAIL_ADDRESS>3
<EMAIL_ADDRESS>
###### Abstract.

In this paper we characterize finite effect algebras which have a state. We
construct two matrices $A$ and $B$ assigned to a finite effect algebra $E$ and
show that if $E$ has a state then rank$A=$ rank$B$.

###### Key words and phrases:

effect algebras, state of effect algebra

###### 2010 Mathematics Subject Classification:

81P10, 81P15

##  1\. Introduction

Effect algebras have been introduced by Foulis and Bennet in 1994 (see [5])
for the study of foundations of quantum mechanics (see [4]). Independtly,
Chovanec and Kôpka introduced an essentially equivalent structure called
$D$-poset (see [9]). Another equivalent structure was introduced by Giuntini
and Greuling in [6]).

The most important example of an effect algebra is $(E(H),0,I,\oplus)$, where
$H$ is a Hilbert space and $E(H)$ consists of all self-adjoint operators $A$
on $H$ such that $0\leq A\leq I$. For $A,B\in E(H)$, $A\oplus B$ is defined if
and only if $A+B\leq I$ and then $A\oplus B=A+B$. Elements of $E(H)$ are
called effects and they play an important role in the theory of quantum
measurements ([2],[3]).

A quantum effect may be treated as two-valued (it means $0$ or $1$) quantum
measurement that may be unsharp (fuzzy). If there exist some pairs of effects
$a,b$ which posses an orthosum $a\oplus b$ then this orthosum correspond to a
parallel measurement of two effects.

In this paper we assign to each finite effect algebra a matrix $A$ such that
$E$ has a state if and only if rank $A$=rank $(A|1)$ and matrix equation
$AX=\left[\begin{array}[]{c}1\\\ \vdots\\\ 1\end{array}\right]$ has a solution
in the form $X=\left[\begin{array}[]{c}x_{1}\\\ \vdots\\\
x_{n}\end{array}\right]$ such that $x_{i}\geq 0$ for $1\leq i\leq n$.

Let us start with the following definition of an effect algebra.

######  Definition 1.1.

In [5] an effect algebra is defined to be an algebraic system $(E,0,1,\oplus)$
consisting of a set $E$, two special elements $0,1\in E$ called the zero and
the unit, and a partially defined binary operation $\oplus$ on $E$ that
satisfies the following conditions for all $p,q,r\in E$:

  1. (1)

[Commutative Law] If $p\oplus q$ is defined, then $q\oplus p$ is defined and
$p\oplus q=q\oplus p$.

  2. (2)

[Associative Law] If $q\oplus r$ is defined and $p\oplus(q\oplus r)$ is
defined, then $p\oplus q$ is defined, $(p\oplus q)\oplus r$ is defined, and
$p\oplus(q\oplus r)=(p\oplus q)\oplus r$.

  3. (3)

[Orthosupplementation Law] For every $p\in E$ there exists a unique $q\in E$
such that $p\oplus q$ is defined and $p\oplus q=1$.

  4. (4)

[Zero-unit Law] If $1\oplus p$ is defined, then $p=0$.

For simplicity , we often refer to $E$, rather than to $(E,0,1,\oplus)$, as
being an effect algebra.

######  Definition 1.2.

For effect algebras $E_{1},E_{2}$ a mapping $\phi\colon E_{1}\to E_{2}$ is
said to be an isomorphism if $\phi$ is a bijection, $a\perp
b\iff\phi(a)\perp\phi(b)$, $\phi(1)=1$ and $\phi(a\oplus
b)=\phi(a)\oplus\phi(b)$.

If $p,q\in E$, we say that $p$ and $q$ are orthogonal and write $p\perp q$ iff
$p\oplus q$ is defined in $E$. If $p,q\in E$ and $p\oplus q=1$, we call $q$
the orthosupplement of $p$ and write $p^{\prime}=q$.

It is shown in [5] that the relation $\leq$ defined for $p,q\in E$ by $p\leq
q$ iff $\exists r\in E$ with $p\oplus r=q$ is a partial order on $E$ and
$0\leq p\leq 1$ holds for all $p\in E$. It is also shown that the mapping
$p\mapsto p^{\prime}$ is an order-reversing involution and that $q\perp p$ iff
$q\leq p^{\prime}$. Furtheremore, $E$ satisfies the following cancellation
law: If $p\oplus q\leq r\oplus q$, then $p\leq r$.

For $n\in{\mathbb{N}}$ and $x\in E$ let $nx=x\oplus x\oplus\ldots\oplus x$
($n$-times). We write ${\rm ord}(x)=n\in{\mathbb{N}}$ if $n$ is the greatest
integer such that $nx$ exists in $E$.

An atom of an effect algebra $E$ is a minimal element of $E\setminus\\{0\\}$.
An effect algebra $E$ is atomic if for every non-zero element $x\in E$ there
exists an atom $a\in E$ such that $a\leq x$.

######  Definition 1.3.

A state on an effect algebra $E$ is a map $s\colon
E\to[0,1]\subseteq{\mathbb{R}}$ such that $s(1)=1$ and if $a\oplus b$ is
defined, then $s(a)+s(b)\leq 1$ and $s(a\oplus b)=s(a)+s(b)$.

######  Proposition 1.4.

[10, Proposition 3.1] Let $E$ be an orthocomplete atomic effect algebra. Then
for every $x\in R$, there is a set $\\{a_{i}|i\in I\\}$ of mutually different
atoms in $E$ and a set $\\{k_{i}|i\in I\\}$ of positive integers such that
$x=\bigoplus\\{k_{i}a_{i}|i\in I\\}$.

Let $E$ be a finite effect algebra. Then $A$ is orthocomplete and atomic. If
$E$ has atoms $a_{1},\ldots a_{m}$ then by 1.4 for every $x\in E$ there exist
non-negative integers $k_{1},\ldots,k_{m}$ such that
$x=\bigoplus\limits_{i=1}^{m}k_{i}a_{i}$. By $M_{n\times n}({\mathbb{N}})$ we
denote the set of all $n\times m$ matrices which quantities are natural
numbers (including $0$).

######  Definition 1.5.

Let

$A=\left[\begin{array}[]{ccc}y_{11}&\ldots&y_{1m}\\\ \vdots&&\vdots\\\
y_{n1}&\ldots&y_{nm}\end{array}\right]\in M_{n\times m}({\mathbb{N}})$

then denote

$\left[\begin{array}[]{cccc}y_{11}&\ldots&y_{1m}&1\\\ \vdots&&\vdots&\vdots\\\
y_{n1}&\ldots&y_{nm}&1\end{array}\right]$

by $(A|1)$.

######  Definition 1.6.

Let $E$ be a finite effect algebra with atoms $a_{1},\ldots,a_{m}$. Then
define
$Seq(E)=\\{(x_{1},\ldots,x_{m})\in{\mathbb{N}}^{m}\colon\bigoplus\limits_{t=1}^{m}x_{t}a_{t}=1\\}$.
Let $n=card(Seq(E))$ and

$\begin{array}[]{l}M(E)=\left\\{\left[\begin{array}[]{ccc}y_{11}&\ldots&y_{1m}\\\
\vdots&&\vdots\\\ y_{n1}&\ldots&y_{nm}\end{array}\right]\right.\in M_{n\times
m}({\mathbb{N}})\colon\mathop{\forall}\limits_{1\leq i<j\leq
n}\quad\mathop{\exists}\limits_{1\leq t\leq m}y_{it}\not=y_{jt},\\\
\\{(y_{i1},\ldots,y_{im})\in{\mathbb{N}}^{m}\colon 1\leq i\leq
n\\}=Seq(E)\Bigg{\\}}.\end{array}$

It turns out that rows in $A\in M(E)$ are all elements of $Seq(E)$. Moreover
if $A,B\in M(E)$ then $B$ can be obtained from $A$ by permuting its rows.

In order to prove the main Theorem, we need use the following two lemmas:

######  Lemma 1.7.

Let $E$ be a finite effect algebra with atoms $a_{1},\ldots,a_{m}$ and

$A=\left[\begin{array}[]{ccc}y_{11}&\ldots&y_{1m}\\\ \vdots&&\vdots\\\
y_{n1}&\ldots&y_{nm}\end{array}\right]\in M(E).$

If $x=\bigoplus\limits_{t=1}^{m}k_{t}a_{t}\in E$ then there exists $1\leq
i\leq n$ such that $k_{t}\leq y_{it}$ for $1\leq t\leq m$.

###### Proof.

Let $E$ be a finite effect algebra with atoms $a_{1},\ldots,a_{m}$ and

$A=\left[\begin{array}[]{ccc}y_{11}&\ldots&y_{1m}\\\ \vdots&&\vdots\\\
y_{n1}&\ldots&y_{nm}\end{array}\right]\in M(E).$

Let $x=\bigoplus\limits_{t=1}^{m}k_{t}a_{t}\in E$ and
$x^{\prime}=\bigoplus\limits_{t=1}^{m}l_{t}a_{t}$ then
$\bigoplus\limits_{t=1}^{m}(k_{t}+l_{t})a_{t}=x\oplus x^{\prime}=1$ so
$(k_{1}+l_{1},\ldots,k_{m}+l_{m})\in Seq(E)$ and there exists $1\leq i\leq k$
such that $(k_{1}+l_{1},\ldots,k_{m}+l_{m})=(y_{i1},\ldots,y_{im})$. Hence
$k_{t}\leq k_{t}+l_{t}=y_{it}$ for $1\leq t\leq m$. ∎

######  Lemma 1.8.

Let $E$ be a finite effect algebra with atoms $a_{1},\ldots,a_{m}$ and

$A=\left[\begin{array}[]{ccc}y_{11}&\ldots&y_{1m}\\\ \vdots&&\vdots\\\
y_{n1}&\ldots&y_{nm}\end{array}\right]\in M(E).$

If
$x=\bigoplus\limits_{t=1}^{m}k_{t}a_{t}=\bigoplus\limits_{t=1}^{m}l_{t}a_{t}\in
E$ then there exists $1\leq p,q\leq k$ such that $k_{t}\leq y_{pt}$,
$l_{t}\leq y_{qt}$ and $y_{pt}-k_{t}=y_{qt}-l_{t}$ for $1\leq t\leq m$.

###### Proof.

Let $E$ be a finite effect algebra with atoms $a_{1},\ldots,a_{m}$ and

$A=\left[\begin{array}[]{ccc}y_{11}&\ldots&y_{1m}\\\ \vdots&&\vdots\\\
y_{n1}&\ldots&y_{nm}\end{array}\right]\in M(E).$

Let
$x=\bigoplus\limits_{t=1}^{m}k_{t}a_{t}=\bigoplus\limits_{t=1}^{m}l_{t}a_{t}\in
E$. Let $z=x^{\prime}=\bigoplus\limits_{t=1}^{m}r_{t}a_{t}$ then
$\bigoplus\limits_{t=1}^{m}(k_{t}+r_{t})a_{t}=x\oplus x^{\prime}=1$ and
$\bigoplus\limits_{t=1}^{m}(l_{t}+r_{t})a_{t}=x\oplus x^{\prime}=1$ so
$(k_{1}+r_{1},\ldots,k_{m}+r_{m})\in Seq(E)$ and
$(l_{1}+r_{1},\ldots,l_{m}+r_{m})\in Seq(E)$ and there exist $1\leq p,q\leq n$
such that $(k_{1}+r_{1},\ldots,k_{m}+r_{m})=(y_{p1},\ldots,y_{pm})$ and
$(l_{1}+r_{1},\ldots,l_{m}+r_{m})=(y_{q1},\ldots,y_{qm})$. Hence $k_{t}\leq
k_{t}+r_{t}=y_{pt}$, $l_{t}\leq l_{t}+r_{t}=y_{qt}$ and
$y_{pt}-k_{t}=r_{t}=y_{qt}-l_{t}$ for $1\leq t\leq m$. ∎

The following theorem (the main theorem) gives the necessary and sufficient
conditions enabling algebra to have a state.

######  Theorem 1.9.

Let $E$ be a finite effect algebra with atoms $a_{1},\ldots,a_{m}$. Let $A\in
M(E)$ and $B=(A|1)$. Then $E$ has a state if and only if rank $A=$ rank $B$
and there exist $s_{1},\ldots,s_{m}\in[0,+\infty)$ such that

$A\cdot\left[\begin{array}[]{c}s_{1}\\\ \vdots\\\
s_{m}\end{array}\right]=\left[\begin{array}[]{c}1\\\ \vdots\\\
1\end{array}\right].$

###### Proof.

Let

$A=\left[\begin{array}[]{ccc}y_{11}&\ldots&y_{1m}\\\ \vdots&&\vdots\\\
y_{k1}&\ldots&y_{km}\end{array}\right]\in M(E).$

  1. $\Leftarrow$

Suppose that there exist $s_{1},\ldots,s_{m}\in[0,+\infty)$ such that

$A\cdot\left[\begin{array}[]{c}s_{1}\\\ \vdots\\\
s_{m}\end{array}\right]=\left[\begin{array}[]{c}1\\\ \vdots\\\
1\end{array}\right].$

Then $\sum\limits_{t=1}^{m}y_{it}s_{t}=1$ for $1\leq i\leq k$.

Let $h\colon E\to{\mathbb{R}}$ be a map such that
$h(\bigoplus\limits_{t=1}^{m}x_{t}a_{t})=\sum\limits_{t=1}^{m}x_{t}s_{t}$. By
1.4 $h$ is defined for all $x\in E$.

First we show that $h$ is well-defined.

Let $x\in E$ and
$x=\bigoplus\limits_{t=1}^{m}k_{t}a_{t}=\bigoplus\limits_{t=1}^{m}l_{t}a_{t}$
by 1.8 there exist $1\leq p,q\leq k$ such that $k_{t}\leq y_{pt}$, $l_{t}\leq
y_{qt}$ and $y_{pt}-k_{t}=y_{qt}-l_{t}$ for $1\leq t\leq m$. Hence
$\sum\limits_{t=1}^{m}k_{t}s_{t}-\sum\limits_{t=1}^{m}l_{t}s_{t}=\sum\limits_{t=1}^{m}(k_{t}-l_{t})s_{t}=\sum\limits_{t=1}^{m}(y_{pt}-y_{qt})s_{t}=\sum\limits_{t=1}^{m}y_{pt}s_{t}-\sum\limits_{t=1}^{m}y_{qt}s_{t}=1-1=0$.
Thus $\sum\limits_{t=1}^{m}k_{t}s_{t}=\sum\limits_{t=1}^{m}l_{t}s_{t}$ so
$h(\bigoplus\limits_{t=1}^{m}k_{t}a_{t})=h(\bigoplus\limits_{t=1}^{m}l_{t}a_{t})$.

Now we show that $h\colon E\to[0,1]$. If
$x=\bigoplus\limits_{t=1}^{m}k_{t}a_{t}$ then there exists $1\leq i\leq n$
such that $k_{t}\leq y_{it}$ for $1\leq t\leq m$ by 1.7. Hence

$0\leq\sum_{t=1}^{m}k_{t}s_{t}=h(\bigoplus_{t=1}^{m}k_{t}a_{t})\leq\sum_{t=1}^{m}y_{it}s_{t}=1,$

that is why $0\leq h(x)\leq 1$.

Since $1=\bigoplus\limits_{t=1}^{m}y_{1t}a_{t}$ so
$h(1)=\sum\limits_{t=1}^{m}y_{1t}s_{t}=1$.

Let $x_{1},x_{2}\in E$, $x_{1}=\bigoplus\limits_{t=1}^{m}k_{t}a_{t}$ and
$x_{2}=\bigoplus\limits_{t=1}^{m}l_{t}a_{t}$. Suppose $x_{1}\oplus x_{2}$ is
defined then

$x_{1}\oplus
x_{2}=\bigoplus_{t=1}^{m}k_{t}a_{t}\oplus\bigoplus_{t=1}^{m}l_{t}a_{t}=\bigoplus_{t=1}^{m}(k_{t}+l_{t})a_{t}$

so by 1.7 there exists $1\leq i\leq n$ such that $k_{t}+l_{t}\leq y_{it}$ for
$1\leq t\leq m$. Thus
$h(x_{1})+h(x_{2})=\sum\limits_{t=1}^{m}k_{t}s_{t}+\sum\limits_{t=1}^{m}l_{t}s_{t}=\sum\limits_{t=1}^{m}(k_{t}+l_{t})s_{t}\leq\sum\limits_{t=1}^{m}y_{it}s_{t}=1$.

Moreover $h(x_{1}\oplus
x_{2})=h(\bigoplus\limits_{t=1}^{m}(k_{t}+l_{t})a_{t})=\sum\limits_{t=1}^{m}(k_{t}+l_{t})s_{t}=\sum\limits_{t=1}^{m}k_{t}s_{t}+\sum\limits_{t=1}^{m}l_{t}s_{t}=h(x_{1})+h(x_{2})$.
Hence $h$ is a state.

  2. $\Rightarrow$

Suppose that $E$ has a state $h\colon E\to[0,1]$. Let $s_{t}=h(a_{t})\geq 0$
for $1\leq t\leq m$. We know that $\bigoplus\limits_{t=1}^{m}y_{it}a_{t}=1$
for $1\leq i\leq n$ since $A\in M(E)$. Hence
$1=h(1)=h(\bigoplus\limits_{t=1}^{m}y_{it}a_{t})=\sum\limits_{t=1}^{m}y_{it}h(a_{t})=\sum\limits_{t=1}^{m}y_{it}s_{t}$.
Thus

$A\cdot\left[\begin{array}[]{c}s_{1}\\\ \vdots\\\
s_{m}\end{array}\right]=\left[\begin{array}[]{c}1\\\ \vdots\\\
1\end{array}\right].$

By Rouché-Capelli Theorem rank $A=$ rank $B$, where $B=(A|1)$

∎

In [7] Greechie gives an example of finite effect algebra that has no states.
This effect algebra $E$ has 12 atoms $\\{a_{1},\ldots,a_{12}\\}$ such that
$a_{1}\oplus a_{2}\oplus a_{3}\oplus a_{4}=1$, $a_{5}\oplus a_{6}\oplus
a_{7}\oplus a_{8}=1$, $a_{9}\oplus a_{10}\oplus a_{11}\oplus a_{12}=1$,
$a_{1}\oplus a_{5}\oplus a_{9}=1$, $a_{2}\oplus a_{6}\oplus a_{10}=1$,
$a_{3}\oplus a_{7}\oplus a_{1}1=1$, $a_{4}\oplus a_{8}\oplus a_{12}=1$. Let

$A=\left[\begin{array}[]{cccccccccccc}1&1&1&1&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&1&1&1&0&0&0&0\\\ 0&0&0&0&0&0&0&0&1&1&1&1\\\
1&0&0&0&1&0&0&0&1&0&0&0\\\ 0&1&0&0&0&1&0&0&0&1&0&0\\\
0&0&1&0&0&0&1&0&0&0&1&0\\\ 0&0&0&1&0&0&0&1&0&0&0&1\end{array}\right]$

then $A\in M(E)$ and the echelon form of $(A|1)$ is (using Maxima)

$\left[\begin{array}[]{ccccccccccccc}1&0&0&0&1&0&0&0&1&0&0&0&1\\\
0&1&0&0&0&1&0&0&0&1&0&0&1\\\ 0&0&1&0&0&0&1&0&0&0&1&0&1\\\
0&0&0&1&0&0&0&1&0&0&0&1&1\\\ 0&0&0&0&1&1&1&1&0&0&0&0&1\\\
0&0&0&0&0&0&0&0&1&1&1&1&1\\\ 0&0&0&0&0&0&0&0&0&0&0&0&1\end{array}\right].$

So rank$A=6$ and rank$(A|1)=7$. Thus $E$ has no state by 1.9.

We found (using Matlab) smaller example of finite effect algebra $E_{4}$ that
has no states. This effect algebra $E_{4}$ has 3 atoms
$\\{a_{1},a_{2},a_{3}\\}$ such that $a_{1}\oplus a_{2}\oplus a_{3}=1$,
$4a_{1}=4a_{2}=4a_{3}=1$. Let

$B=\left[\begin{array}[]{ccc}1&1&1\\\ 4&0&0\\\ 0&4&0\\\
0&0&4\end{array}\right]$

then $B\in M(E_{4})$ and rank$B=3$ and rank$(B|1)=4$ thus $E_{4}$ has no state
by 1.9.

The effect algebra $E_{4}$ is represented by the following Hasse diagram:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3a_{3}=a_{1}\oplus
a_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a_{1}\oplus
a_{3}=3a_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a_{2}\oplus
a_{3}=3a_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2a_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2a_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2a_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

It remains to check if in Theorem 1.9 the condition that there exist
$s_{1},\ldots,s_{m}\in[0,+\infty)$ such that

$A\cdot\left[\begin{array}[]{c}s_{1}\\\ \vdots\\\
s_{m}\end{array}\right]=\left[\begin{array}[]{c}1\\\ \vdots\\\
1\end{array}\right].$

can be omitted.

Problem. Prove or disprove: if $E$ is a finite effect algebra with atoms
$a_{1},\ldots,a_{m}$, $A\in M(E)$ and $B=(A|1)$ then $E$ has a state if and
only if rank $A=$ rank $B$.

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