# Clustering of dark matter in the cosmic web as a probe of massive neutrinos

Mohadese Khoshtinat1, Mohammad Ansarifard2, Farbod Hassani3, Shant Baghram1,4

1 Department of Physics, Sharif University of Technology, P. O. Box
11155-9161, Tehran, Iran
2 School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
P.O. Box 19395-5531, Tehran, Iran
3 Institute of Theoretical Astrophysics, Universitetet i Oslo, 0315 Oslo,
Norway
4 Research Center for High Energy Physics, Department of Physics, Sharif
University of Technology, P.O.Box 11155-9161, Tehran, Iran
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The large-scale structure of the universe is distributed in a cosmic web.
Studying the distribution and clustering of dark matter particles and halos
may open up a new horizon for studying the physics of the dark universe. In
this work, we investigate the nearest neighbour statistics and spherical
contact function in cosmological models with massive neutrinos. For this task,
we use the relativistic N-body code, gevolution and study particle snapshots
at three different redshifts. In each snapshot, we find the halos and evaluate
the letter functions for them. We show that a generic behaviour can be found
in the nearest neighbour, $G(r)$, and spherical contact functions, $F(r)$,
which makes these statistics promising tools to constrain the total neutrino
mass.

###### keywords:

Large Scale Structures, the total neutrino mass, 1-point statistics, the
spatial point patterns

††pubyear: 2023††pagerange: Clustering of dark matter in the cosmic web as a
probe of massive neutrinos–B

##  1 Introduction

The data sets from ongoing (Aghamousa et al., 2016) and upcoming (Amendola et
al., 2018) Large Scale Structure (LSS) surveys not only let tighter
constraints on the cosmological parameters but make it possible to address
some fundamental questions in physics and cosmology- e.g., the equation-of-
state of dark energy, the characteristics of dark matter, and the total
neutrino mass. With a wealth of raw data, the task is to extract as much
possible information in a computationally efficient way from these data sets.
Due to the presence of non-linearities, especially in the late-time Universe,
this will be a challenging task. On one hand, the Linear Perturbation Theory
(LPT) still holds on large scales (Dodelson, 2003), where the density contrast
field can also be considered a Gaussian random field. So, the traditional
2-point correlation function contains all the information and is the most
suitable statistic for extracting the information from the LSS data (Romanello
et al., 2023). On the contrary, at small scales (Cooray & Sheth, 2002), the
nonlinear gravitational evolution of matter leads to a non-Gaussian field as a
result of mode couplings. In this situation, information diffuses into the
higher-order correlation functions, and the 2-point correlation function does
not contain all the information. So it fails to grasp the complete content of
the data set. The direct way to capture leaked information from the second
moment is to compute higher-order correlation functions, e.g., bispectrum
(Sefusatti et al., 2016). As going to deep non-linear regimes, it is not
straightforward that the information is stored in low moments of the
correlation function. It means that the whole Probability Distribution
Function (PDF) incorporates the non-Gaussian effects. One approach to
circumvent these difficulties is to consider the 2-point correlation function
(or its Fourier counterpart, the power spectrum) equipped with extra
information - e.g., the marked power spectrum (Stoyan, 1984) \- or by
calculating it in different environments like voids, nodes, sheets and
filaments in real space (Bonnaire et al., 2022), or in redshift space
(Bonnaire et al., 2023). Parallel to these efforts, many non-linear semi-
analytical models have been introduced, that use the relation between linear
scale matter distribution statistics and non-linear observables such as the
number density of dark matter halos. Peak theory (Bardeen et al., 1986; Sheth
& Tormen, 1999) and the excursion set theory (Bond et al., 1991; Sheth et al.,
2001; Zentner, 2007; Nikakhtar & Baghram, 2017; Nikakhtar et al., 2018) are
two non-linear structure formation models, which use the relation between
linear and non-linear scales to probe the distribution of matter over a wide
range of scales.
Another strategy is to search for other statistical measures sensitive to all
higher-order correlation functions that are computed for one point in space.
To mention a few - the PDF of the matter density field smoothed at some radius
(Uhlemann et al., 2020; Boyle et al., 2021; Gough & Uhlemann, 2022), k-nearest
neighbours Cumulative Distribution Functions (CDF) or peaked-CDF (Banerjee &
Abel, 2020; Banerjee et al., 2022; Yuan et al., 2023), and the pioneering work
of (White, 1979) using the void probability function to study the clustering
of the tracers, e.g., halos or galaxies. Fard et al. (2022) employed the
nearest neighbour CDF, or $G(r)$ and the spherical contact CDF, or $F(r)$
within the context of the standard model of cosmology based on cosmological
constant $\Lambda$ and cold dark matter (CDM) known as $\Lambda$CDM model
using the SMDPL simulations (Moliné et al., 2023) with different mass cuts and
at redshifts $z=0,0.5,1$. They demonstrated that these statistical measures
could differentiate between samples constructed by mass cuts at various
redshifts.

The subsequent query is if these statistics (i.e., $G$ and $F$) can
distinguish between different extensions of the $\Lambda$CDM.
In this paper, we compute these summary statistics for the base $\Lambda$CDM
cosmology and cosmologies with massive neutrinos, $\nu\Lambda$CDM. After a
brief review of the impact of massive neutrino on the late time cosmology and
the statistics used in this work in Section 2, we describe the simulations and
the data preparation in Section 3. The results of the analysis are depicted in
Section 4. Finally, besides wrapping up, we point out the future direction in
Section 5.

##  2 Theoretical background

In this section, we present the theoretical background needed for this work.
In the first subsection, we review the idea of one-point statistics, and in
the second subsection, we address the effect of massive neutrinos on large-
scale structures.

###  2.1 One point statistics

In the spatial point analysis, we use the cumulative distribution functions of
the Nearest Neighbour Distances (NNDs) such as $F(R)$ and $G(R)$ to reveal the
clustering feature of different samples. By comparing these functions to those
calculated for a Poisson point process know as Complete Spatial Randomness
(CSR) with the same average intensity (roughly, the number of points per
volume), we can effectively identify and analyse spatial patterns.

In our case, the nearest neighbour distances measure the separation between
two events from the data set (here, the position of halos in the halo
catalogue) or the distance between an event and a reference point (in our
analysis, the reference points are randomly generated points in the simulation
box with the same intensity as the halo catalogue). The nearest neighbour CDF,
$G(r)$ indicates the proportion of the halos shown by $i$ having their first
nearest neighbour on a sphere of radius $r_{i}\leq r$ around them. On the
other hand, the spherical contact CDF, $F(r)$ shows the fraction of the
reference points indicated by $j$ with an event point (position of a halo)
within the distance ${{r}}_{j}\leq r$. The presence of clustering leads to
higher values for $G(r)$ and $F(r)$ in comparison to CSR (Hand, 2008).
Moreover, Fard et al. (2022) argued that $G(r)$ is advantageous for studying
the high-density regions and small scales while $F(r)$ comes in handy to
investigate the larger scales.
An interpretable combination of these two functions is the $J-$function:

$\\\ J(r)\equiv\frac{1-G(r)}{1-F(r)},$ (1)

with the value one for CSR while in the presence of clustering, this function
is less than unity.
The letter functions (i.e., $G$, $F$, and $J$) are the statistics we use in
this work. In the context of cosmology, whenever dealing with new statistics,
two natural questions always arise: how much information these statistics
contain and what is the connection between these statistics and the classic
n-point correlation functions. To address these, White (1979) introduced
conditional correlation functions $\Xi_{i}$,

$\displaystyle\Xi_{i}(\textbf{r}_{1},...,\textbf{r}_{i};V)=$
$\displaystyle\sum_{j=0}^{\infty}\dfrac{(-n)^{j}}{j!}$ (2)
$\displaystyle\int...\int\xi_{i+j}(\textbf{r}_{1},...,\textbf{r}_{i+j})\,dV_{i+1}...dV_{i+j},$

where the integrals are over an empty volume of interest $V$, except for the
points $r_{i}$ and $\xi_{i}$ are the n-point correlation functions. Note that
by definition $\xi_{0}=0$ and $\xi_{1}=1$. White (1979) showed the probability
of finding an empty sphere of radius $r$ around a point, i.e., the void
probability - $P_{0}(V)$, is related to $\Xi_{0}$ through
$P_{0}(V)=\exp\left[\Xi_{0}(V(r))\right]$, which is just the complement of
$F(r)$. The probability of a halo having its first nearest neighbour on the
surface or inside of a sphere of volume $V$ is the complement of no other halo
at this distance, which reads
$G(r)=1-\Xi_{1}(\textbf{r}_{o};V(r))\exp\left[\Xi_{0}(V(r))\right]$.
These relations, along with the equation 2, manifest the dependence of these
statistics on not only the 2-point correlation function but on all the higher-
order correlations. Accordingly, it is reasonable to use these letter
functions to investigate cosmological models, especially in non-linear regime.

###  2.2 Massive neutrinos in cosmology

The detection of the atmospheric neutrino oscillation indicates that at least
two of three neutrino flavors of the standard model of particle physics have
mass. The latest data release from KATRIN experiment (Aker et al., 2022)
suggests that the upper limit for the total neutrino mass would be around
$\sum m_{\nu}\lesssim 0.8\ $eV ($90\%$ C. L.). For cosmology, this
experimental fact means neutrinos belong to the matter sector rather than the
radiation in late times. Consequently, the neutrino energy density at redshift
zero reads,

$\\\ \omega_{\nu}=\frac{\sum m_{\nu}}{94.13\rm eV}\ ,$ (3)

with the neutrino mass fraction $f_{\nu}={\omega_{\nu}}/{\omega_{m}}$, where
$\omega_{m}=\Omega_{m}h^{2}$. Hence, cosmological data sets can constrain the
total neutrino mass through the constraints on the $\omega_{\nu}$
independently from the laboratory kinematics constraints. This amount would be
around $\sum m_{\nu}\lesssim 0.12\ $eV (Aghanim et al., 2020), with severe
restrictions on the nature of the dark energy to be near the cosmological
constant (Hannestad, 2005)111 If the dark energy equation-of-state and its
derivative change concurrently, then the upper limit increases by a factor of
three.. Unlike cold dark matter particles, neutrinos fall out of equilibrium
as relativistic particles. Due to the universe’s expansion, neutrinos lose
energy and eventually become non-relativistic. This process introduces a non-
relativistic length scale $\lambda_{nr}$ as an ultimate scale beyond which
neutrinos act like cold dark matter. Below this maximum scale, neutrinos
stream freely and, due to their thermal velocities, can not cluster
effectively under the influence of gravity. Accordingly, for a given redshift
and neutrino species, the free-streaming wave number is defined as Agarwal &
Feldman (2011):

$\\\ k_{f_{s}}(a)\approx 0.063\,h{\rm Mpc}^{-1}\,\frac{m_{\nu}}{0.1{\rm
eV}}\frac{a^{2}H(a)}{H_{0}}\ ,$ (4)

with $m_{\nu}$ is the mass of one neutrino specie, $a$ scale factor, $H(a)$
the Hubble parameter, and $H_{0}=100h$ (km $\rm{Mpc}^{-1}$ $\rm s^{-1}$) the
Hubble constant. Moreover, free-streaming of the neutrinos smooth the matter
density fluctuation field below $\lambda_{fs}\sim k_{f_{s}}^{-1}$. This
process leads to less clustering in the universe and suppression of structures
compared to the $\Lambda$CDM universe. LoVerde (2014) shows that the presence
of massive neutrinos reduces the abundance of massive halos, hosting galaxy
clusters, by slowing down the collapse time. Non-linear clustering of massive
neutrinos counter acts this delay slightly, but the outcome is the reduced
number of massive halos by $1\%$ for the total neutrino mass below $0.5\ $eV.
Yet, there is a claim that the free-streaming of light relics induces a cut-
off on the minimum mass of halos Schneider et al. (2013). The lighter the
relic particles are, the cut-off in the halo mass gets smaller.
One of the main cosmological observations to probe the effect of massive
neutrinos is the matter power spectrum both in linear and non-linear regimes
(Hannestad et al., 2020). To study the non-linear regime, one often uses the
halo model approach. In this context, Hannestad et al. (2020) shows the
suppression in power starts at $k\simeq 0.01\,\rm{h/Mpc}$, decreases up to
$k\simeq 1\,\rm{h/Mpc}$, and continues in larger wavenumbers as a plateau.
Hannestad et al. (2020) refer to the former feature as a "slide" and the
latter as a "spoon" feature and state the spoon feature as a generic behaviour
detected in different simulations. As an example, Rossi (2017) used the
simulation on the cosmic web and used the Lyman alpha observations to study
the effect of neutrinos on small scales. Also, there are studies on the effect
of massive neutrinos on void statistics and distribution (Massara et al.,
2015), also on the bias parameter (Hassani et al., 2022).
In this work, instead of using the power spectrum, we use the 1-point
statistics to investigate the distribution of matter in the presence of
massive neutrinos. We will show the probable relation of some specific scales,
introduced in the two frameworks (power spectrum and 1-point statistics).

##  3 Simulation and sample preparation

In the following subsections, we discuss the simulation used and the data
preparation process.

###  3.1 Simulation

Figure 1: The nearest-neighbour distance PDF for $\Lambda$CDM model (top) and
$\sum m_{\nu}=0.3\ $ eV (bottom). The black dotted lines represent PDF in
comoving space, the red dash-dotted lines show it after removing the
intersecting halos, the blue dashed lines depict the PDF in the redshift
space, and the solid yellow lines signify it after omitting the remaining
intersecting halos in the redshift space. The x-axis represents the distance
in real/redshift space, depending on whether the analysis is performed in
either space.

To study the effect of neutrino mass on the statistics introduced in Section
2, we use gevolution code (Adamek et al., 2016a, b)
222https://github.com/gevolution-code . Gevolution is a relativistic N-body
code based on the weak field expansion of the general relativity. In this
code, one can treat massive neutrinos as separate relativistic species or by
using their linear perturbations from the transfer functions obtained from
CLASS (Blas et al., 2011). Adamek et al. (2017) showed that the results of the
two approaches are consistent. In this work we use the halo catalogues of the
simulations presented in Adamek et al. (2017). Specifically, we use three
simulations considering the total neutrino mass of $\sum\ m_{\nu}=0.06,\
0.20,\ 0.30\ $ eV plus one for the vanilla $\Lambda$CDM cosmology. The
fiducial cosmology parameters are given in Table 1. For the cosmologies with
massive neutrinos, we set the density parameters of the cold dark matter
according to the following relation,
$\omega_{c}\equiv\Omega_{c}h^{2}=0.12038-\omega_{\nu}$, to keep the total
matter density ($\omega_{\nu}+\omega_{b}+\omega_{c}$) unchanged. In the
previous relation $\omega_{\nu}$ is computed using Eq. 3 and
$\omega_{c}=0.12038$ is the value we consider for the cold dark matter density
in the $\Lambda$CDM scenario. These simulations contain $4096^{3}$ particles
and grids in a cube of the comoving box size
$L=2048~{}{\text{Mpc}}/{\text{h}}$ initialised at redshift $z=100$. Using
these high resolution simulations we study the letter functions at $z=0$.
Moreover, to study the redshift evolution of the 1-point statistics, we run
lower resolution simulations, where we consider $L=300$ Mpc/h and $N_{\rm
grids}=N_{\rm pcl}=1024^{3}$. We output snapshots at three redshifts
$z=0,0.5,1$. The results for these simulations are presented in Appendix A. It
is worth mentioning that we construct the halo catalogues for each snapshot
using ROCKSTAR halo finder (Behroozi et al., 2012) at each redshift.

$\omega_{c}$ | $\omega_{b}$ | $h$ | $n_{s}$ | $T_{\rm CMB}$ | $A_{s}$
---|---|---|---|---|---
$0.12038$ | $0.022032$ | $0.67556$ | $0.9619$ | $2.7255$ | $2.215\times 10^{-9}$
Table 1: The cosmology parameters of standard model where
$\omega_{c}=\Omega_{c}h^{2}$, $\omega_{b}=\Omega_{b}h^{2}$ are baryonic and
dark matter densities normalises to critical density. $h=H_{0}/100$, $n_{s}$,
$T_{\rm CMB}$, $A_{s}$ are respectively the reduced Hubble parameter, spectral
index, the CMB temperature, and the amplitude amplitude of scalar
perturbations.

###  3.2 Data Preparation

Data preparation follows these steps. First, we remove all sub-halos from the
halo catalogue. Then we eliminate the halos which do not meet the virial
condition: " $\text{T}\ /\ \text{U}\ <2$", where T and U stand for kinetic and
potential energies, respectively. As we use a specific mass cut for sample
preparation, we omit the sub-halos to control the effect of substructures. In
observational proposal this means that we use a volume limited sample for our
analysis. The halos which do not meet the virial condition are considered as
ones which are not gravitationally bound objects (Bett et al., 2007). At this
stage, if we compute the nearest neighbour distance and plot its Probability
Distribution Function (PDF)—referred to as NND-PDF—against the comoving
radius, $r$, we observe a bimodal behaviour in the PDF. This bimodality
appears for all simulations regardless of neutrino mass, i.e., dashed black
line in the top and bottom panel of figure 1, for $\Lambda$CDM and $\sum
m_{\nu}=0.3$ eV respectively. Since our analysis depends on the behaviour of
probability functions, it is critical to understand whether it is something
physical or just an artefact of the resolution of the simulation or the halo
finder used. Our study suggests that most halos that contribute to the first
mode are intersecting halos—denoting that these halos overlap with each other
or share common spatial volume. To illustrate this, we compare the distance
between the neighbouring halos with the sum of their virial radii and we find
that some of the neighbouring halos intersect. If we omit the intersecting
halos from the sample and then compute the NND, we end up with the red dash-
dotted line in figure 1. As we discuss in Appendix B, we observe that if we do
the same computations in the redshift space, where we use the redshift space
distances instead of the comoving ones, we obtain a smooth PDF with one peak,
and most of the contribution from intersecting halos introduced by the method
of halo finder will disappear which is presented with the solid yellow line in
figure 1. To do this, we use the plane-parallel approximation and reposition
all the halos along one Cartesian axis of the simulation box using the
relation between the comoving positions, $\textbf{x}_{c}$, and the one in the
redshift space $\textbf{x}_{r}$

$\textbf{x}_{r}=\textbf{x}_{c}+\bigl{(}\frac{\textbf{v}\ \cdot\
\hat{\textbf{n}}}{aH}\bigr{)}\ \hat{\textbf{n}}$ (5)

with $\hat{\textbf{n}}$ being the unit vector along the line of sight, $a$
scale factor, and $H(z)$ the Hubble parameter at redshift $z$. We consider two
catalogues in redshift space: one that takes into account all halos and
another that is filtered to eliminate intersecting halos. Eventually, the halo
catalogues in the redshift space represent a smooth NND-PDF with one peak, the
solid yellow and dashed blue lines in figure 1. Hereafter, we do all the
computations for the catalogues in the redshift space unless we state it
directly. The results are almost independent of the specific direction we
choose in simulation because of the statistical isotropy of the sample. Note
that in the upcoming results the distances are in redshift space obtained from
equation 5.

##  4 Results

As mentioned earlier, to study the impact of neutrino mass, we use point
process analysis tools to investigate the clustering of the halo catalogues.
We specifically use the the so-called letter functions, i.e., $G(r)-$ the
nearest neighbour distance cumulative distribution function, $F(r)-$the
spherical contact cumulative distribution function, and $J-$function as
defined in equation 1. The main task is to compare the letter functions for
the base $\Lambda$CDM and cosmologies with massive neutrinos ($\sum
m_{\nu}=0.06,0.2,0.3$ eV). To do this, we compute the letter functions for
each of the halo catalogues in different cosmologies. Then, we plot the
subtraction of these functions for the catalogues with massive neutrinos and
the $\Lambda$CDM, i.e., $\Delta_{Y}=Y_{\nu\Lambda{\rm CDM}}-Y_{\Lambda{\rm
CDM}}$, where Y is a member of the set $\\{G,F,J\\}$.
As G and F functions are CDFs, we subtract them in each radius for comparing
two distinct cosmological models. This method is widely used in Banerjee &
Abel (2020); Uhlemann et al. (2020). We also compare the J-function in the
same way to be consistent. The errors, for each simulation and for each $Y$,
are determined utilising the Jack-Knife method (Wolter & Wolter, 2003). For
this task, we divide the simulation box into $n^{3}$ sub-boxes and
subsequently omit each sub-box at the time, calculate the desired quantity,
and at the end, perform averaging over the $n^{3}-1$ quantities, and then
using the mean as the data points and the standard deviation as the errors.
The total error bars on the $\Delta_{Y}$ are calculated by the sum of the
uncorrelated errors of the $Y_{\Lambda{\rm CDM}}$ and $Y_{\nu\Lambda{\rm
CDM}}$. The error bars are smaller than the size of the points.

### Letter function’s dependence on the total neutrino mass

Figure 2: The $\Delta_{Y}$ versus radius, blue circles related to $\sum
m_{\nu}=0.06$ eV, green squares to $\sum m_{\nu}=0.2$ eV and red triangles to
$\sum m_{\nu}=0.3$ eV. The top panel is $\Delta_{G}$, the middle $\Delta_{F}$,
and the bottom $\Delta_{J}$.

In figure 2, we plot $\Delta_{Y}$ versus distances in the redshift space at
redshift $z=0$. The blue circles, green squares, and red triangles are
representative of the catalogue with the total neutrino mass of 0.06, 0.2, and
0.3 eV, respectively.
The top panel depicts $\Delta_{G}$, which manifests a generic behaviour,
starting at zero, meeting a mild maximum value at around $\sim 2$ Mpc$h^{-1}$,
then decreasing and hitting its minimum value around $\sim 7$ Mpc$h^{-1}$, and
finally reaching zero again. The depth of the $\Delta_{G}$ is highly sensitive
to the sum of the neutrino’s mass. As the total mass increases, the
$\Delta_{G}$ deepens more. On the contrary, the peak of $\Delta_{G}$ is
affected slightly by the change in the total mass. To interpret the behaviour,
recall that $G(r)$ is the probability of having the first nearest neighbour at
distances less than $r$ for the halos in the catalogue. Wherever the
$\Delta_{G}$ takes a positive value, it implies finding the immediate
neighbour is more probable in cosmology with massive neutrino than
$\Lambda$CDM cosmology. This behaviour can be understood due to relation of
the biased power spectrum in two cosmological models (Hassani et al., 2022).
The specific shape of a excess in smaller scales and a deficit on larger
scales is related to the characteristics of one-halo and two-halo terms in the
matter power spectrum. The negative value for $\Delta_{G}$ translates to
larger distances between neighbouring halos in neutrino cosmologies.
Accordingly, the shape of the $\Delta_{G}$ suggests that a small portion of
halos in cosmology with massive neutrinos have their nearest neighbour at
smaller distances than their counterpart in the base $\Lambda$CDM, as for a
large number of halos, the immediate neighbour is at further distances.
In the middle panel, $\Delta_{F}$ also reveals a generic behaviour that starts
at zero, decreases, reaches a minimum at around $\sim 10$ Mpc$h^{-1}$, and
then increases and goes to zero. Yet the total neutrino mass causes the change
in the depth of the minimum value. As the sum of neutrinos’ mass increases,
the minimum value decreases. $F(r)$ probes the clustering of the point process
by measuring the distances between a set of randomly generated points and the
halos from the catalogue. The shortfall of the probability for the neutrino
cosmologies from their $\Lambda$CDM counterpart suggests that a randomly
located sphere with a radius around $\sim 10$ Mpc$h^{-1}$ in a $\nu\Lambda$CDM
is less populated than one placed in the $\Lambda$CDM universe. It is worth
mentioning that in general, the $F(r)$ is expected to probe larger scales than
$G(r)$. This happens because in $G(r)$ both points are chosen from the halo
samples while in $F(r)$ one of the points is chosen randomly.
In the bottom panel of figure 2, we plot the $\Delta_{J}$. The $J$-function
encapsulates the information in the $G$ and $F$ functions as defined in
equation 1. The deviation of this function from unity toward lesser values
represents clustering in the point pattern. The positive value for
$\Delta_{J}$ means that halos in $\nu\Lambda$CDM cosmologies are less
clustered than those halos in the $\Lambda$CDM one, and vice versa. The
generic behaviour of $\Delta_{J}$, inherited from $G$ and $F$ functions, once
more suggests a slightly excessive clustering at a smaller scale and a fall-
off of clustering on intermediate scales in cosmology with neutrinos to the
base model. Note that the extremum on the $J$ and $G$ happens on the same
scale, while the extremum on the $F$ is at a larger scale. This is because of
the shape of the function of $J$ and the small correction of $F$ in the
denominator causes this effect. An interesting point is that, in the matter
power spectra, $\Lambda$CDM is always larger than massive neutrinos as
neutrinos cannot cluster as effectively as CDM particles. But we see a maximum
in $\Delta G$ which seems to hint at a clustering at certain scales. But this
is regarding the halos, not CDM particles. Although, the maximum is less
significant in comparison to the minimum of the $\Delta_{G}$, however, it
could be a specific fingerprint of the massive neutrinos. This fingerprint can
be used to distinguish the massive neutrino from other types of the modified
cold dark matter scenario, e.g., Kousha et al. (2023).
These results propose that the notable signature of cosmology with massive
neutrinos is in the dip of the $G$ and $F$ plots in the scales of $\sim 7-10$
Mpc $h^{-1}$. The dip of the $G$ and $F$ is consistent with the less power in
cosmology with massive neutrinos in the aforementioned scales. In order to
propose an observational probe, we have to translate this distribution of dark
matter halos to the galaxy distribution. This translation is very complicated
due to the halo occupation distribution and the bias parameter.

### Mass Cut

In this subsection, at $z=0$, we investigate the dependency of mass cut on our
results. The mass cut specifies the minimum mass for which we consider halos
to be equal to or larger than it. By including a mass cut in our halo
catalogues, we study various halo types that have undergone distinct dynamics.
In figure 3, we plot the $\Delta F$, $\Delta G$ and $\Delta J$ function versus
the radius in the redshift space. The colour bar diagrams show the mass cut of
these functions. By increasing the mass-cut we observe that the minima in the
$\Delta_{Y}$ where $Y=\\{F,G\\}$ deepen and occur at larger scales. This
behaviour is very similar to the behaviour of matter spectrum for different
halo mass functions in small scales (Hannestad et al., 2020). Higher mass cuts
eliminate low-mass dark matter halos, which has the consequence of finding the
nearest neighbours at larger distances. Among all letter functions the
J-function shows the largest deviation from unity for large mass-cuts.

Figure 3: The $\Delta_{Y}$ versus radius in the redshift space, for $\sum
m_{\nu}=0.3$ eV is plotted. The colour-bar shows the minimum mass of the
sample.

##  5 Conclusions and Remarks

Massive neutrinos affect the LSS mainly in non-linear regimes. Accordingly,
the one point statistics which is a probe of the LSS in non-linear scales will
be useful as a complementary method to traditional power spectrum
calculations. In the data preparation, we follow the same procedure for the
massive neutrino and $\Lambda$CDM cosmologies. We start the simulation from
the same seed number and consider similar spatial resolution and halo finding
processes. This approach helps in mitigating discrepancies arising from
simulations’ randomness, numerics, or differences in spatial resolution.
Consequently, the differences between the letter functions in the two
cosmologies become more physical, minimising susceptibility to artificial
effects introduced by the simulation. In addition to the halo catalogue
obtained from the halo finder, we prepare a halo catalogue in the redshift
space. In this process, we transform the position of dark matter halos to the
one in the redshift space observed by a distant observer and then we eliminate
the halos that are in separation less than their total virial radius. In
Appendix B we show how this correction eliminates the artificial effect of
halo halo-finding process.
We find the difference of the letter functions, namely G, F, and J functions,
between the $\nu\Lambda$CDM and the base $\Lambda$CDM cosmologies. The
specific shape of an excess in small scales and a deficit in larger scales is
probably related to the characteristic scales appearing in the matter power
spectrum. In the context of the Halo model in smaller scales the one-halo term
will be the dominant one and the two-halo terms of matter power spectrum
affect the larger scales. The investigation of this relation is out of the
scope of this work and is the subject of further study.
The redshift evolution of the letter functions is discussed in the Appendix A.
We show that the difference between the two models occurs on almost the same
scale with redshift evolution and the difference is more significant in higher
redshifts. This suggests that high-$z$ surveys will be more convenient to
distinguish the neutrino cosmology from $\Lambda$CDM using the letter
functions.
The depth of the $\Delta_{G}$ and $\Delta_{F}$ is highly sensitive to the sum
of neutrino masses; As the total neutrino mass increases, the $\Delta_{G}$
deepens more. On the contrary, the peak of $\Delta_{G}$ is affected slightly
by the change in the total neutrino mass.
The extension of this work would be to investigate mock catalogues and study
the distribution of galaxies in the light of 1-point statistics in both
neutrino cosmology and standard model.

## Acknowledgments

We would like to thank Negin Khosravaninezhad for many fruitful discussions.
We are very grateful to Julian Adamek and Martin Kunz for giving us access to
the halo catalogues of high resolution neutrino simulations that are presented
in (Adamek et al., 2017). This work is supported by the University of Oslo
computational facilities and a grant from the Swiss National Supercomputing
Centre (CSCS) under project ID s1051. FH is supported by the Overseas Research
Fellowship from the Research Council of Norway and the UNINETT Sigma2 - the
National Infrastructure for High Performance Computing and Data Storage in
Norway. SB is partially supported by Abdus Salam International Center of
Theoretical Physics (ICTP) under the junior associateship scheme. MA is
partially supported by Iran’s National Elites Foundation (INEF) under the
Chamran plan for young researchers.

## Data Availability

The halo catalogue data, along with data for all plots derived from our
theoretical models, can be made available upon request.

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##  Appendix A Redshift evolution of 1-point statistics

To investigate the evolution of the letter functions (F, G, and J) in
redshift, we run three additional simulations with a lower resolution
including the total neutrino mass of $\sum m_{\nu}=0,\ 0.3,\ 0.6\ $ eV using
the gevolution code. In these simulations, we use the degenerate approximation
for the mass of neutrino species, i.e., one massless and the two others having
equal mass. Moreover, in gevolution, we consider the effect of massive
neutrinos through their linear transfer functions provided by CLASS
333https://github.com/lesgourg/class_public (Blas et al., 2011).

The fiducial cosmology parameters are consistent with Aghanim et al. (2020)
given in Table 2. For the cosmologies with massive neutrinos, we set the
density parameters of the dark matter according to the following relation,
$\omega_{c}=0.1200-\sum m_{\nu}/94.13\ $eV, to keep the total matter density
unchanged. In these simulations, we consider the same number of particles and
grid points $N_{\rm pcl}=N_{\rm grid}=1024^{3}$ in a cube of the comoving
length $L=300$ Mpc $h^{-1}$ from $z_{\rm ini}=100$ to $z=0$. We output three
snapshots at redshifts 0, 0.5, and 1. We construct the halo catalogues for
each snapshot using the ROCKSTAR halo finder (Behroozi et al., 2012). The
preparation of the datasets follows the steps discussed in Section 3 with an
additional condition on the minimum mass of halos considered in all the
snapshots. This mass reads $M_{min}=5\times 10^{12}M_{\odot}\ h^{-1}$, and the
idea behind it is closely connected to constructing a volume-limited galaxy
catalogue.
In figure (4), we plot the same letter functions as in figure 2, for redshifts
$z=1,0.5,0$. We note that the local maximum on small scales disappears in
higher redshifts in the $G(r)$. However, the difference between the two models
that are introduced in larger scales becomes more significant in higher
redshifts. This means that surveys operating at higher redshifts are prominent
to distinguish the neutrino cosmology models from the standard model. Our
results are in the direction of the argument that high-$z$ surveys are more
important than lower-$z$ surveys for neutrino mass detection (Takada et al.,
2006).

$\omega_{c}$ | $\omega_{b}$ | $h$ | $n_{s}$ | $\sigma_{8}$ | $ln(10^{10}A_{s})$
---|---|---|---|---|---
$0.12000$ | $0.02237$ | $0.6736$ | $0.9649$ | $0.8111$ | $3.044$
Table 2: The cosmology parameters of standard model where
$\omega_{c}=\Omega_{c}h^{2}$, $\omega_{b}=\Omega_{b}h^{2}$ are baryonic and
dark matter densities normalises to critical density. $h=H_{0}/100$, $n_{s}$,
$\sigma_{8}$ and $A_{s}$ are respectively the reduced Hubble parameter,
spectral index, variance of perturbations in $8$ Mpc and the amplitude of
scalar perturbations. Figure 4: We plot $\Delta_{Y}$ versus radius, where
$Y=G,F,J$ for different redshifts $z=0,0.5,1$ and two total neutrino masses
$\sum m_{\nu}=0.3,0.6$ eV.

##  Appendix B Data preparation and simulation characteristics

As mentioned in Section 2, we observe two peaks in the NND-PDF of the halo
catalogues from the gevelotion. By comparing the distance between two
neighbours with the sum of their virial radii, we find that the intersecting
halos contribute the most to the first peak. We see two possible roots for the
situation. First, gevolution is a fixed-mesh simulation which results in a
fixed spatial resolution and as a result the simulation might not be able to
resolve the small scales properly. On the other hand, the ROCKSTAR performs
the halo identification procedure in a 6+1 dimensional phase space. In this
procedure, the velocity of the particles alters the spatial closeness, which
is the main criterion used by the other halo-finder algorithms, such as
Friend-of-Friend (FoF). To find out the extent of each factor on the peaks, we
compare the NND-PDF within different settings as follows:

  * •

The effect of Force Resolution (FR) for one simulation: The force resolution
which is a parameter in the halo finder ROCKSTAR, is related to the spatial
resolution of the simulation. In our case, we consider
$FR\equiv(3/4\pi)^{1/3}(Ln^{-1})$ where $L$ is the box size and $n^{3}$ is the
number of grids. In figure 5, we plot the NND-PDF for different FR values for
one snapshot from gevolution. It is evident from the plot that the first peak
is ever-present, but it becomes less dominant as the FR increases.

  * •

The comparison of two different particle mesh schemes: We compare two halo
catalogues, both prepared by ROCKSTAR, but from two distinct simulations,
namely SMDPL (Moliné et al., 2023), and gevolution. To ensure the
comparability of the two catalogues, we apply a mass cut and we compute the
NND-PDF, which are shown in figure 6. The dash-dotted blue and solid yellow
lines correspond to the SMDPL simulation, while the dotted black and red
dashed lines represent the gevolution. In the case of the SMDPL, the higher
spatial resolution of the simulation results in smaller initial distances
between neighbouring halos, but it seems that the behaviour of the PDFs at
large values is similar. Interestingly, the first peak of the NND-PDF also
appears for the SMDPL when the mass cut is $10^{13}M_{\odot}$.

  * •

The comparison between two halo finder: The SMDPL simulation provides
different halo catalogues, each prepared with a different algorithm. Figure 7
shows the NND-PDFs for the ROCKSTAR (RoS in the figure) and the FoF catalogues
for two different mass cuts. The FoF algorithm does the halo identification in
the 3 dimensional coordinate space, and particle velocities do not play a role
in the process. The figure shows that the PDFs for the ROCKSTAR catalogues
have broader widths, and the peaks happen at smaller distances than for the
FoF catalogues. In the next part, we argue that the transformation to the
redshift space will clean the effect of the halo finder algorithm in the halo
identification process.

  * •

The impact of the transformation from coordinate space to the redshift space
on the halo catalogues of the SMDPL simulation: We use the ROCKSTAR and FoF
halo catalogues from the SMDPL both with the mass cut of $10^{10}M_{\odot}$
and transform the position of the halos in both catalogues from the coordinate
space to the redshift space using equation 5 and using the parallel plane
approximation. Figure 8 shows the result in both the coordinate and the
redshift spaces. Surprisingly, the two NND-PDF are alike in the redshift
space, the dash-dotted red and solid yellow lines. In figure 8, we plot the
NND-PDF versus scale in SMDPL simulation, where we applied the redshift-space-
distortion correction. An important point to indicate is that after this
correction the distribution of halos is similar in both halo-finding methods.
Accordingly, as we mentioned in the data preparation Section we use the RSD
correction for the sample we work on in this paper.

Figure 5: The effect of the Force Resolution (FR) parameter on the peaks of
the NND-PDF for one simulation. The blue line in both the upper and lower
panels corresponds to the actual force resolution. In the upper panel, the red
dotted line and the dashed yellow line represent FR, which are half and two-
thirds of the actual FR, respectively. In the lower panel, the red dotted line
and the dashed yellow line represent the FR, which are two times and 1.5 times
the actual FR, respectively. We calculate PDFs in the coordinate space. Figure
6: The NND-PDF for gevolution and SMDPL in the coordinate space. Figure 7:
The NND-PDFs for SMDPL simulation and two different halo finder algorithms
versus distance in the coordinate space. The yellow solid and dashed red lines
correspond to catalogues prepared by the ROCKSTAR (RoS) and Friend of Friend
(FoF) when the minimum mass is set to $10^{11}M_{\odot}$. The blue dash-dotted
and dotted black lines relate to the RoS and FoF catalogues when the minimum
mass is set to $10^{12}M_{\odot}$.  Figure 8: The effect of the RSD
transformation on the NND-PDF in SMDPL simulation for two halo finder
algorithms. The dotted black and dashed blue lines correspond to the real
space distances for the RoS and FoF algorithms, respectively. The dot-dashed
red and solid yellow lines represent the NND-PDF after the RSD transformation
of the distances for the RoS and FoF algorithms, respectively.


We present in this paper that the CHSH game admits one and only one optimal state and so remove all ambiguity of representations.
More precisely, we use the well-known universal description of quantum commuting correlations as state space on the universal algebra for two player games,
and so allows us to unambigiously compare quantum strategies as states an this common algebra. As such we find that the CHSH game leaves a single optimal state on this commong algebra.

In turn passing to any non-minimal Stinespring dilation for this unique optimal state is the only source of ambiguity (including self-testing):More precisely, any state on some operator algebra may be uniquely broken up into its minimal Stinespring dilation as an honest representation for the operator algebra followed by its vector state.
Any other Stinespring dilation however arises simply as an extension of the minimal Stinespring dilation (i.e. as an embedding of the minimal Hilbert space into some random ambient one).
As such this manifests the only source of ambiguity appearing in most (but not all!) traditional self-testing results such as for the CHSH game as well as in plenty of similar examples.

We then further demonstrate the simplicity of our arguments on the Mermin–Peres magic square and magic pentagram game.

Most importantly however, we present this article as an illustration of operator algebraic techniques on optimal states and their quotients,and we further pick up the results of the current article in another following one (currently under preparation) to derive a first robust  result in the quantum commuting model.

§ STINESPRING DILATION

The Stinespring dilation for states on operator algebras will play the major role of the current article —
more precisely it provides a convenient approach for verifying uniqueness of optimal states.
As such we begin with an introduction for convenience of the reader new to operator algebras.

Meanwhile we moreover explain the issue of randomly chosen non-minimal Stinespring dilations (in random ambient spaces) which do not reflect any of the properties of the original state. As such this accounts for all of the ambiguity arising in most (but not all!) self-testing results in literature.

Consider a state on a unital operator algebra

\[
\phi:A\to \COMPLEX:\TAB\phi\geq0,\TAB\phi(1)=1.
\]

From this we may define an inner product pairing

\[
A\times A\to \COMPLEX:\TAB \braket{x|y}:= \phi(x^*y)
\]

which defines a Hilbert space. We formally denote the Hilbert space in braket notation such that the inner product conveniently reads as composition,

\[
H=\mathquote{A\ket\phi}:\TAB  \phi(x^*y) = \bra\phi x^*y \ket\phi.
\]

This allows us to later split such expressions to our desire.
[This can be made rigorous in the setting of Hilbert modules.]
Based on the Hilbert space we obtain an induced representation by left multiplication

\[
A\to B\left(\overline{A\ket\phi}\right): \TAB a\big(x\ket\phi\big) := ax\ket\phi
\]

together with an induced vector state by $\ket\phi=1\ket\phi$:

\[
B\left(\overline{A\ket\phi}\right)\to\COMPLEX:\TAB T\mapsto \bra\phi T\ket\phi.
\]

Both combined however retrieve the original state $\bra\phi a\ket\phi = \phi(a)$.

As such our state allows for a dilation to the vector state:

\begin{gather*}
\begin{tikzcd}
A \arrow[rr, "\phi(\BLANK)"]\drar &[-0.5cm]&[-0.5cm] \COMPLEX: \\
& B\left(~\overline{A\ket\phi}=\mathrm{minimal~space}~\right)
\urar[swap]{\bra\phi\BLANK\ket\phi}
\end{tikzcd}
\end{gather*}

This defines the (so-called) minimal Stinespring dilation in the following sense:
Each dilation arises simply as an embedding of this minimal dilation into some random ambient space. More precisely, consider another dilation say

\[
\begin{tikzcd}
A \arrow[rr, "\phi(\BLANK)"]\drar &[-0.5cm]&[-0.5cm] \COMPLEX. \\
& B\left(~H=\mathrm{some~space}~\right) \urar[swap]{\bra{\phi'}\BLANK\ket{\phi'}}
\end{tikzcd}
\]

Then the space defines an ambient space for the minimal one above:

More precisely, the embedding is formally given by the isometry

\[
A\ket\phi\subset A\ket{\phi'}\subset H:\TAB x\ket\phi= x\ket{\phi'},\TAB \ket\phi=\ket{\phi'}.
\]

Indeed one easily verifies using the above dilation

\[
\bra\phi x^*y\ket\phi=\phi(x^*y)=\bra{\phi'}x^*y\ket{\phi'}.
\]

Using the orthogonal decomposition we may invoke matrix formalism

\begin{gather*}
\MATRIX{ B\left(\overline{A\ket\phi}\right) & B\left(\OTHER\to\overline{A\ket\phi}\right) \\ B\left(\overline{A\ket\phi}\to\OTHER\right) & B\Big(\OTHER\Big) }
\end{gather*}

and so the representation into the random ambient dilation reads:

\begin{gather*}
A\to B\Big(A\ket\phi\oplus\OTHER\Big):\TAB a\mapsto \SMATRIX{a&*\\{*}&*},\\[2\jot]
B\Big(A\ket\phi\oplus\OTHER\Big)\to\COMPLEX:\TAB \SMATRIX{a&*\\{*}&*}\mapsto\SMATRIX{\bra\phi&0}\SMATRIX{a&*\\{*}&*}\SMATRIX{\ket\phi\\0}.
\end{gather*}

As such we have found that each dilation simply arises as an embedding into some random ambient space, and the description in matrix formalism illustrates how these ambient spaces do not reflect any properties of the original state.

On the other hand, the Stinespring dilation defines also the minimal quotient to which the original state descends. More precisely, the minimal quotient is given by the kernel (see <cit.> and <cit.>)

\begin{gather*}
\begin{tikzcd}
0\rar & I(\phi)\rar & A\rar & B\left(\overline{A\ket\phi}\right):
\end{tikzcd}\\
\begin{tikzcd}
A\rar & A/I(\phi)\subset B\left(\overline{A\ket\phi}\right)\rar{\phi(\BLANK)} & \COMPLEX
\end{tikzcd}
\end{gather*}

while any other quotient lies in between:

\begin{gather*}
I\ILEQ A:\TAB I\subset\ker(\phi)\implies I\subset I(\phi):\\[2\jot]
\begin{tikzcd}
A\rar & A/I\rar &
\rar{\phi(\BLANK)} & \COMPLEX.
\end{tikzcd}
\end{gather*}

Summarizing, every state decomposes uniquely as a representation onto its minimal quotient followed by its dilation as vector state
(a good mnemonic here is the analogous decomposition of morphisms into epi plus mono):

Aquotient     A/I(ϕ)
⟨ϕ||ϕ⟩     :ϕ=∘

In turn, one may use the unique decomposition for comparison of states as we will illustrate in this article for the CHSH and the Mermin–Peres games.

Finally let us make another remark regarding the canonical decomposition:
Suppose one chooses some factorization via some unital completely positive map (in short ucp-map) instead of an honest representation (which accounts for a choice of POVM instead of some honest PVM):

B( S=small space ) [swap]⟨ϕ||ϕ⟩

Using a minimal Stinespring dilation for the ucp-map we obtain an embedding into another ambient space

A[equal]     B(H=ambient space)S⊂H[bend left]⟨ϕ||ϕ⟩

A[swap]     B(S=small space) [swap]⟨ϕ||ϕ⟩     .

In other words, this may be understood as a mistaken decomposition:

\[
\phi = (\mono\circ\mono)\circ \epi  = \mono \circ (\mono\circ\epi) = \bra\phi\BLANK\ket\phi\circ \ucp.
\]

Moreover, our previous discussion reveals this also as an ambient space for the original state, and as such reveals our small space as an incompatible cut:

\begin{gather*}
\begin{tikzpicture}[scale=2/3]
\draw (-5.5,-3.5) rectangle (5.5,3.5);
\draw[fill=black!10] (0,0) ellipse [x radius=3, y radius=2,rotate=0];
\draw[dashed] (0,-3.5) -- (2,3.5);
\node at (0,4) {$H=\mathrm{small~space}\oplus\mathrm{remainder}$};
\node at (0,0) {$\overline{A\ket\phi}$};
\node at (2,0) {$\ket\phi$};
\node at (3,-2.5) {$S=\mathrm{small~space}$};
\node at (-3,-2.5) {$\mathrm{remainder}$};
\end{tikzpicture}
\end{gather*}

That is any choice of POVM (instead of an honest PVM) simply arises as the minimal Stinespring dilation embedded in some ambient space and then cut incompatibly onto some small portion.
As such this accounts for an additional, much worse source of ambiguity for self-testing.

This concludes our introduction and discussion on the Stinespring dilation from an operator algebraic perspective.
With this at hand, we may now proceed to nonlocal games and their optimal states — and their uniqueness.

§ THE TWO-PLAYER ALGEBRA

We continue with an introduction on the two-player algebra, which provides the universal setting for describing quantum correlations all as states on this common algebra — and so allows us to compare them on a common ground.

It is well-known that the universal -algebra generated by a single projection valued measure agrees with the full group -algebra

\[
\CSTAR(\INTEGERS/A)=\CSTAR\Big(e_1,\ldots,e_A\in\proj\Big|e_1+\ldots +e_A=1\Big)
\]

and as such we obtain as the universal -algebra per player for a given number of questions and answers the free product of groups

\[
\CSTAR(X=\inputs|A=\outputs):=\CSTAR\Big(\INTEGERS/A*\ldots*\INTEGERS/A:\text{$X$-many}\Big).
\]

Finally a commuting pair of families of observables may be gathered algebraically by the maximal tensor product of their operator algebras and as such the algebra for two-player games has the universal description (with corresponding question and answer sets)

\[
\CSTAR(\twoplayer):=\CSTAR(\Alice)\tensor\CSTAR(\Bob)=\CSTAR(X|A)\tensor\CSTAR(Y|B)
\]

and any quantum commuting strategy arises simply as a state

\[
\phi:\CSTAR(\twoplayer)\to\COMPLEX:\TAB p(ab|xy)=\phi\Big(e(a|x)\tensor e(b|y)\Big).
\]

Indeed, applying any Stinespring dilation (see section <ref>) one obtains the classical description of correlations as a commuting representation of projection valued measures together with a vector state,

\[
\hspace{-1cm}\begin{tikzcd}[column sep = small]
\CSTAR(\Alice)\tensor\CSTAR(\Bob) \arrow[rr, "\phi(\BLANK)\phantom{++++}"]\drar[swap]{\pi=\alpha\tensor\beta} &[1/2cm]&[1cm] \COMPLEX. \\
& B\Big(~\overline{\CSTAR(\twoplayer)\ket\phi}\subset H~\Big) \urar[swap]{\bra\phi\BLANK\ket\phi}
\end{tikzcd}
\]

This classical description however has the disadvantage that a Stinespring dilation introduces a level of ambiguity due to the chosen representation, which is basically the cause for the usual self-testing results.

Instead the description in terms of states on the two-player algebra allows now for a direct comparison of such on a common universal algebra, and so with this at hand we may now approach uniqueness of optimal states as follows.

§ CHSH GAME: SINGLE OPTIMAL STATE

We proof in this section that the CHSH game has one and only one optimal state (compared to merely unique optimal correlation).
For this we begin with a quick review on the CHSH game phrased in the language of operator algebras.

Recall from [4] that the CHSH game is a binary-input binary-output game, and as such the two-player algebra
is generated by a pair of  unitaries for every player (compare section <ref>):

\begin{gather*}
\CSTAR(\twoplayer) = \CSTAR(\Alice)\tensor\CSTAR(\Bob):\\
\CSTAR(\Alice)=\CSTAR(u^2=1=v^2)=\CSTAR(\Bob).
\end{gather*}

Note that the CHSH game is an XOR-game as introduced in [5].As such we may equivalently describe the CHSH game by its bias polynomial

\[
\win = \{a\oplus b =xy\}:\TAB \CHSH := u \tensor u + u \tensor v + v\tensor u - v\tensor v
\]

and an optimal state for the corresponding nonlocal game translates to maximizing the bias (see appendix <ref> for a summary on XOR games):

\[
\phi:\CSTAR(\twoplayer)\to\COMPLEX:\TAB \phi(\CHSH)=\|\CHSH\|.
\]

This concludes our quick review on the CHSH game. With this at hand, we may now proof the desired uniqueness of the optimal state for the CHSH game.

The CHSH game has one and only one optimal state.

More precisely, consider the bias polynomial for the CHSH game

\[
\CHSH=u\tensor u+ u\tensor v+v\tensor u -v\tensor v
\]

and suppose a state realises the maximal bias (see proposition <ref>)

\[
\phi:\CSTAR(\twoplayer)\to\COMPLEX:\TAB\phi(\CHSH)=\|\CHSH\|.
\]

Then the state necessarily factors over the matrix algebra

()() [swap] [rr, ""name=phi]    [-0.5cm]   [0.5cm]

M(2)M(2) [from=phi, phantom, "ϕ()++"]

given by the canonical quotient (see proposition <ref>)
[column sep=1cm]

Note that the collection of words in generators reduces, by the anticommutation in the quotient, to the finite list of words (up to linear span)

\[
\words(u,v)=\{1,u,v,uv=-vu,uvu=-v,\ldots\}\implies \{1,u,v,uv\}
\]

whence every state is already linearly determined on those:

\[
\phi:M(2)\tensor M(2)\to\COMPLEX:\TAB \phi\Big(\{1,u,v,uv\}\tensor\{1,u,v,uv\}\Big)=\ ?
\]

With this observation in mind, the optimal state is uniquely and entirely determined (including all higher and mixed moments) by the correlation table

\[\def\arraystretch{1.3}\begin{array}{c|cccc}
\phantom{+}\phi(\ldots)=\phantom{+} & \BLANK\tensor1 & \BLANK\tensor u & \BLANK\tensor v & \BLANK\tensor uv \\[\jot]
\hline
1\tensor\BLANK  & 1 & 0 &  0 & 0 \\
u\tensor\BLANK  & 0      & \phantom{+}\tfrac{1}{\sqrt2}\phantom{+} &  \phantom{+}\tfrac{1}{\sqrt2}\phantom{+} & 0 \\
v\tensor\BLANK  & 0      & \phantom{+}\tfrac{1}{\sqrt2}\phantom{+} & -\tfrac{1}{\sqrt2}\phantom{+} & 0 \\
uv\tensor\BLANK & 0      & 0 &  0 & -1
\end{array}\]

from which there is no ambiguity left anymore! (Recall section <ref>.)

In short, there is one and only one optimal state for the CHSH game.

Before we begin with the proof,
let us note how our notion of uniqueness on states compares with the traditional uniqueness on correlations:

The entire correlation table for the optimal state above entails all quantum commuting correlations,
since these are given by the state space restricted on two-moments of the form

\[
\begin{array}{ccccccc}
\phi(1\tensor 1), & \phi(1\tensor u), & \phi(1\tensor v), &
\cancel{\phi(1\tensor uv)}, & \cancel{\phi(1\tensor vu)}, & \cancel{\phi(1\tensor uvu)} & \cancel{\ldots} \\
\phi(u\tensor 1), & \phi(u\tensor u), & \phi(u\tensor v), &
\cancel{\phi(u\tensor uv)}, & \cancel{\phi(u\tensor vu)}, & \cancel{\phi(u\tensor uvu)} & \cancel{\ldots} \\
\phi(v\tensor 1), & \phi(v\tensor u), & \phi(v\tensor v), &
\cancel{\phi(v\tensor uv)}, & \cancel{\phi(v\tensor vu)}, & \cancel{\phi(v\tensor uvu)} & \cancel{\ldots} \\
\cancel{\phi(uv\tensor 1)}, & \cancel{\phi(uv\tensor u)}, & \cancel{\phi(uv\tensor v)}, & \cancel{\ldots} \\
\end{array}
\]

These values are equivalently determined by the traditional correlation table

\[
p\Big(a,b=0,1\Big|x,y=0,1\Big)=\phi\Big(E(a|x)\tensor E(b|y)\Big)
\]

given by the corresponding spectral projections

\[
u = E(0|0) - E(1|0),\TAB v = E(0|1) - E(1|1)
\]

which reduces to the familiar values for the CHSH game

\begin{align*}
p\Big(a\oplus b=xy\Big)     = \frac{1+1/\sqrt{2}}{4}, \TAB
p\Big(a\oplus b\neq xy\Big) = \frac{1-1/\sqrt{2}}{4}.
\end{align*}

In particular, the traditional uniqueness of correlations — given for example by self-testing — excludes all higher and mixed moments of order-2 unitaries.

Our approach follows the original one by Landau from [12] and its sequel [13], which we however extend to uniqueness of states using some additional arguments from operator algebras.
We begin with the following observation: Suppose a state is optimal for the bias polynomial, then it is necessarily also optimal for the squared bias

\[
\phi(\CHSH)=\|\CHSH\|\implies \phi\left(\CHSH^2\right)=\|\CHSH\|^2
\]

which indeed follows by Cauchy–Schwarz:
[One may either use the bound $\ket\phi\bra\phi\leq \|\phi\|^2$ together with $a\leq b\implies c^*ac\leq c^*bc$,or equivalently the general Cauchy–Schwarz for ucp-maps $\phi(a)^*\phi(a)\leq\phi(a^*a)$ as originally proven by Choi in [3].
For a neat proof of the latter see also <cit.>.]

\begin{align*}
\|\CHSH\|^2 &= \bra\phi\CHSH^*\ket\phi\bra\phi\CHSH\ket\phi \\
\end{align*}

In our case the squared bias however reduces to

\[
\CHSH^2 = 4 - [u,v]\tensor [u,v]:\TAB \|\CHSH\|^2=8
\]

and so an optimal state necessarily satisfies

\[
\phi\left(\CHSH^2\right)=8\implies \phi\Big([u,v]\tensor[u,v]\Big)=-4.
\]

We note that this defines only a necessary condition.

Arguing once more by Cauchy–Schwarz we obtain as above

\[
\phi\Big([u,v]^*\tensor [u,v]^*[u,v]\tensor[u,v]\Big)=16
\]

and this once more is only a necessary but not sufficient condition.

Noting that $u^2=1=v^2$, this expression reads written out

\begin{align*}
16 &= \phi\Big(( 2-uvuv-vuvu)\tensor (2-uvuv-vuvu)\Big) \\
&= \phi(2\tensor 2) -\phi(2\tensor uvuv) - \ldots + \phi(vuvu\tensor vuvu).
\end{align*}

Note that each summand involves a unitary:

\[
1\tensor1,\TAB 1\tensor uvuv,\TAB \ldots,\TAB  vuvu\tensor vuvu.
\]

Thus by a simple counting argument this is only possible if the state achieves the value plus/minus one with correct sign on each of the unitaries.
In particular

\[
\phi(uvuv\tensor1)  = -1 = \phi(1\tensor uvuv).
\]

Within the Stinespring dilation we may however cut such relations into

\[
\braket{x|y}:=\bra\phi uv\tensor1| uv\tensor1\ket\phi =\phi(uvuv\tensor1)=-1
\]

with both pieces of norm-one

\begin{align*}
\braket{x|x}=\bra\phi uv\tensor1|vu\tensor1\ket\phi = 1\\
\braket{y|y}=\bra\phi vu\tensor1|uv\tensor1\ket\phi=1
\end{align*}

and similarly for the second tensor factor $\phi(1\tensor uvuv)$.

That is however only possible if these are colinear with

\[
uv\tensor1\ket\phi=- vu\tensor1\ket\phi,\TAB 1\tensor uv\ket\phi=- 1\tensor vu\ket\phi
% \footnote{Use the familiar equivalence for $\braket{e|e}=1=\braket{f|f}$: $\braket{e|f}=1$ if and only if $\ket e=\ket f$.}
\]

and so we have the anticommutation

\[
\{u,v\}\tensor 1\ket\phi= 0=1\tensor \{u,v\}\ket\phi.
\]

Finally note that these anticommutators lie in the center

\[
\{u,v\}\tensor1,1\tensor\{u,v\}\in\CENTER\CSTAR(\twoplayer)
\]

as one easily verifies on generators:

\[
u\{u,v\} = v + uvu = \{u,v\}u,\TAB v\{u,v\} = vuv+v = \{u,v\}v.
\]

The anticommutator therefore also vanishes on the entire subspace

\begin{gather*}
H := \overline{\CSTAR(\twoplayer)\ket\phi} = \overline{\CSTAR(\Alice)\tensor\CSTAR(\Bob)\ket\phi}:\\[2\jot]
\{u,v\}\tensor1\ \CSTAR(\Alice)\tensor\CSTAR(\Bob)\ket\phi = 0\\
1\tensor\{u,v\}\ \CSTAR(\Alice)\tensor\CSTAR(\Bob)\ket\phi = 0.
\end{gather*}

that is within the minimal Stinespring dilation (see section <ref>)

\[
\CSTAR(\twoplayer)\to B(H=\min):\TAB \{u,v\}\tensor1=0=1\tensor \{u,v\}.
\]

In other words, the state factors as desired via (see proposition <ref>)

\[
M(2)\tensor M(2) = \CSTAR\big(u^2=1=v^2\big|\{u,v\}=0\big)\tensor\CSTAR\big(u^2=1=v^2\big|\{u,v\}=0\big)
\]

which gives us the diagram for the optimal state:
[column sep=small]
()() [swap] [rrr]    [-0.5cm]   [0.5cm]   [1cm] .

M(2)M(2)    B(H=min)[swap]⟨ϕ||ϕ⟩

This finishes the first part of the proposition and so we may freely restrict our attention from now on to the optimal state already acting on the quotient

()()   [-4] M(2)M(2) ϕ()    [1cm]
and so also the bias polynomial for the CHSH game

\[
\CHSH= u\tensor(u+v) + v\tensor(u-v)\in\mathrm{M}(2)\tensor\mathrm{M}(2).
\]

From here the rest of the theorem basically follows from traditional results on the CHSH game.
We however present a new more algebraic approach below.But at first we untwist the CHSH polynomial. For this we note that the combined expressions (in brackets above) are easily seen to anticommute

\[
\{u+v,u-v\} = 1 - \{u,v\}+\{u,v\} - 1=0.
\]

Due to the anticommutation of our generators,
these define also unitaries of order-2 (up to normalization)

\[
(u \pm v)^2= u^2 \pm (uv + vu) + v^2 = 1 \pm \{u,v\} +1 = 2.
\]

In other words, these define just another abstract pair of Pauli matrices and so they give rise to the automorphism (see proposition <ref>)

\[
M(2)\to M(2):\TAB u\mapsto \frac{u+v}{\sqrt2},\TAB v\mapsto\frac{u-v}{\sqrt2}
\]

which is at the same time its own inverse.
Using this automorphism we obtain the untwisted bias polynomial (basically in the new basis)

\[
\CHSH/\sqrt2 = u\tensor u + v\tensor v.
\]

From here we give a new algebraic approach by verifying the trace property from which we deduce the final uniqueness,
and from which we may moreover easily read off the values for the correlation table:
Suppose our state maximises the untwisted bias polynomial and so also the unitary summands

\[
\phi(u\tensor u+v\tensor v)=2\implies \phi(u\tensor u)=1=\phi(v\tensor v).
\]

From this it follows within the Stinespring dilation (similar as above):

\begin{align*}
u\tensor 1\ket\phi = 1\tensor u\ket\phi,\TAB
v\tensor 1\ket\phi = 1\tensor v\ket\phi.
\end{align*}

Indeed one easily verifies (and similarly for the other generator)

\[
\bra\phi(u\tensor1-1\tensor u)^*(u\tensor1-1\tensor u)\ket\phi=\bra\phi2(1\tensor1-u\tensor u)\ket\phi=0.
\]

The entire trace property follows now by iteration on words of generators

\begin{align*}
(u_1\ldots u_n u)\tensor1\ket\phi &= (u_1\ldots u_n)\tensor u\ket\phi= \ldots = 1\tensor (u u_n\ldots u_1)\ket\phi
\end{align*}

and we note that the order gets flipped (which is a cause for the transpose).Full matrix algebras however carry a unique tracial state and so we obtain as desired the unique optimal state for the CHSH game:

\[
\phi\Big(u_1\ldots u_m\tensor v_1\ldots v_n\Big) = \tau\Big(u_1\ldots u_mv_n\ldots v_1\Big).
\]

The particular values for the correlation table may now be easily derived from this relation: Indeed we obtain for example

\[
\phi(u\tensor v)=\tau\Big(u\cdot\frac{u+v}{\sqrt2}\Big)=\frac{1}{\sqrt2}\tau(1) + \frac{1}{\sqrt2}\tau(uv) = \frac{1}{\sqrt2}.
\]

This concludes the proof whence the optimal state is unique.

§ MERMIN–PERES GAMES: SINGLE PERFECT STATE

We continue in this section with another example of nonlocal games:the Mermin–Peres magic square and magic pentagram games.
Both admit a unique winning state for rather obvious reasons (see the theorem below).In order to understand this uniqueness argument however, we will need a more thorough introduction first to linear constraint system games and their game algebra (and we note that the following arguments also apply to nonlinear constraint system games with any complex variables).

A linear constraint system game is a nonlocal game given by a system of linear equations (which we formulate in multiplicative form) such as

\begin{gather*}
u,v,\ldots\in\exp\{2\pi i~\INTEGERS/2\}:\TAB uvw=1,\TAB vz=-1,\TAB\ldots
\end{gather*}

The players are asked an equation each and so

\[
\]

and each player is asked to fill out the variables of their equation say

\[
x=\left(vz=-1\right)\TAB\implies\TAB a=\left(v=-1,z=1\right).
\]

As a first rule the players are asked to answer in a synchronous way:

\[
x=\mathrm{some~equation}=y\implies a=b.
\]

As such a winning state (being synchronous) defines a tracial state on the algebra for a single player (see <cit.>).
More precisely, consider the two-player algebra (from section <ref>) for identical algebras per player

\begin{gather*}
\CSTAR(\twoplayer)=\CSTAR(\player)\tensor\CSTAR(\player)
\end{gather*}

and consider a winning state

\[
\phi:\CSTAR(\player)\tensor\CSTAR(\player)\to\COMPLEX:\TAB\phi\Big(E(\losing)\Big)=0.
\]

Then the synchronicity rule enforces on the Stinespring dilation:

\[
\phi\Big(E(a\neq b|x=y)\Big)=0\implies E(a|x)\tensor1\ket\phi =
1\tensor E(a|x)\ket\phi.
\]

By iteration such states enjoy the special property (note the flip in order)

\begin{gather*}
\phi\Big(E(a_1|x_1)\ldots E(a_m|x_m)\tensor E(b_1|y_1)\ldots E(b_n|y_n)\Big) \\
= \phi\Big(E(a_1|x_1)\ldots E(a_m|x_m)E(b_n|y_n)\ldots E(b_1|y_1)\tensor1\Big)
\end{gather*}

and similar for flipping words in other ways from left to right.

Thus such states define a tracial state on either algebra per player

\begin{align*}
\phi(\BLANK\tensor1)&:\CSTAR(\player)\tensor1\to\COMPLEX\\
\phi(1\tensor\BLANK)&:1\tensor\CSTAR(\player)\to\COMPLEX
\end{align*}

which both allow us to recover the original state from above relation.

So we may from now on restrict our attention to a tracial state

\[
\tau:\CSTAR(\player)\to\COMPLEX:\TAB\tau(ab)=\tau(ba).
\]

For tracial states however every additional rule enforces the corresponding  of losing pairs to vanish in the minimal Stinespring dilation (see [15] and [9]). As such we are left to determine the game algebra for our synchronous game

\begin{gather}\label{GAME-ALGEBRA}
\CSTAR(\mathrm{sync~game}) := \CSTAR\Big(\player\Big|E(\losing)=0\Big)
\end{gather}

where we denote for shorthand (similar as in appendix <ref>)

\[
E(ab|xy):= E(a|x)E(b|y):\TAB E(S)=\{E(s)|s\in S\}
\]

and any tracial state on the quotient defines a winning correlation:

\begin{gather}\label{SYNC=TRACES}
\tau:\CSTAR(\mathrm{sync~game})\to\COMPLEX:\TAB p(ab|xy)=\tau\Big(E(a|x)E(b|y)\Big).
\end{gather}

With this at hand we now get to the first particular rule for linear constraint system games.
For this we first note that for each equation as input we obtain a projection valued measure whence commuting projections for each variable. For the equation $x=(uvw=1)$ we obtain for example the triple

\[
\]

which we retrieve from the projection valued measure as a disjoint union like

\[
E(v=-1|x):=\sum E\Big(u\in\mathquote{\INTEGERS/2},v=-1,w\in\mathquote{\INTEGERS/2}\Big|x\Big)
\]

and we recover the projection valued measure simply as intersection

\[
E(u,v,w|x) = E(u|x) E(v|x) E(w|x).
\]

This is all easiest remembered as a Venn diagram (for commuting projections):

(-1,1) ellipse [x radius=2.5, y radius=2,rotate=0];
(3/2,2) ellipse [x radius=2.5, y radius=2,rotate=20];
(3/2,0) ellipse [x radius=2.5, y radius=2,rotate=-30];
(-5,-3) rectangle (6,5);
at (-2,1) $u=1$;
at (-3,3.5) $u=-1$;
at (2,3) $v=1$;
at (4.5,4) $v=-1$;
at (2,-1) $w=1$;
at (4.5,-2) $w=-1$;
at (-2,-2.3) $E(u=1,v=-1,w=1)$;
[->] (-2,-2) to[bend right=30] (-0.2,-0.5);

The first particular rule for linear constraint system games asks now the players to answer consistently for shared variables, for example

\[
x=(uvw=1),\TAB y=(vz=-1)\TAB\implies\TAB\mathquote{a(v)=b(v)}
\]

which boils down to the equality for our projections (inside the game algebra).

\[
E\Big(v\Big|x=(uvw=1)\Big) = E\Big(v\Big|y=(vz=-1)\Big).
\]

Indeed this follows very quickly from (think in terms of Venn-diagrams!)

\[
E(v=1|x)E(v=-1|y) =0= E(v=-1|x)E(v=1|y).
\]

That is the projections are all independent of the chosen equation (cf. [6]).

With this at hand we may now get to the second particular rule for linear constraint system games.
For this we first pass to their generating unitaries

\[
U:= E(u=1)-E(u=-1),\TAB V:= E(v=1)-E(v=-1),\TAB\ldots
\]

We now ask these unitaries to satisfy each equation, for example

\[
f(u,v,w)= uvw = 1\TAB\implies\TAB f(U,V,W)=UVW=1.
\]

Note that the unitaries appearing in a common equation arose as commuting projections. As such they (isomorphically) generate the abelian subalgebra

\[
\CONTINUOUS(\spectrum U)\tensor\CONTINUOUS(\spectrum V)\tensor\CONTINUOUS(\spectrum W) \subset\CSTAR(\player)
% \CONTINUOUS\Big(\{\pm1\}\times\{\pm1\}\times\{\pm1\}\Big) =
\]

and we may have instead also asked for any continuous function such as

\[
g(U,V,W)=\cos(UV^3) + W^2-1=0.
\]

The generated subalgebra however agrees with the isomorphic copy

\[
\CSTAR(g)=\CONTINUOUS(\supp g)E(\supp g)\subset\CSTAR(\player)
\]

and so modding out (the ideal generated) by either side results in the same quotient,
that is the requirement above reduces to its support projection:

\[
g(U,V,W)=0\TAB\iff\TAB E(\supp g)=0.
\]

That is we obtain for our example (read backwards)

\[
E(u,v,w)=E(u)E(v)E(w)=0,\TAB \forall uvw\neq1 \TAB\iff\TAB UVW=1.
\]

This is nothing but our second particular rule for linear constraint system games:
The players need to respond with variables satisfying their equation like

\[
x = (uvw=1)\TAB\implies\TAB a=(u,v,w=\pm1):\TAB a(u)a(v)a(w)=1.
\]

Thus the game algebra (<ref>) for linear constraint systems reads (see also [6]):

\begin{align*}
&=\CSTAR\Big(U^2=1,V^2=1,\ldots\Big| UVW=1: [U,V]=0,\ldots; VZ=-1:\ldots\Big).
\end{align*}

While allowing also for continuous functions, note that there was also nothing special about unitaries of . Instead all the above also holds perfectly fine for variables with values in any (possibly different) list of complex numbers

\[
u\in\{\alpha_1,\ldots,\alpha_u\},\TAB v\in\{\beta_1,\ldots,\beta_v\},\TAB\ldots
\]

upon replacing the generators with

\[
U=\alpha_1 E(u=\alpha_1) + \ldots + \alpha_u E(u=\alpha_u).
\]

This concludes the introduction to linear constraint system games.

With this at hand one may now very easily settle uniqueness of winning states:

The game algebra (<ref>) for the Mermin–Peres magic square and magic pentagram game read respectively

\begin{gather*}
\CSTAR(\magicsquare) = \mathrm{M}_2\tensor\mathrm{M}_2\\
\CSTAR(\magicpenta) = \mathrm{M}_2\tensor\mathrm{M}_2\tensor\mathrm{M}_2
\end{gather*}

and as such both admit one and only one winning state given by the unique tracial state on either matrix algebra:

\begin{gather*}
\end{gather*}

As for the CHSH game, this entails all quantum commuting strategies.

The algebra for the Mermin–Peres magic square is generated by

North [thick,dash][thick,dashed, no head]

West [thick,dash][thick,dash]
Center [thick,dash][thick,dashed,no head]
East [thick,dash]

South [thick,dash]     v1

where we have commuting rows and columns automatically built in as tensor product while using abstract copies of order two unitaries:

\[
u^2=1=v^2:\TAB u\tensor1,\TAB v\tensor1,\TAB 1\tensor u,\TAB 1\tensor v.
\]

Also all the boxes on outer outer edges are already determined:

\[
\boxed{\text{North}} = u\tensor u,\TAB \boxed{\text{West}} = u\tensor v,\TAB \boxed{\text{East}}= v\tensor u,\TAB \boxed{\text{South}} = v\tensor v.
\]

Using the center box one now easily derives their anticommutation:

\begin{align*}
\boxed{\text{West}}\cdot\boxed{\text{East}} = \boxed{\text{Center}} = -\boxed{\text{North}}\cdot\boxed{\text{South}} &\implies 1\tensor\{u,v\}=0,\\[2\jot]
\boxed{\text{West}}\cdot\boxed{\text{East}} = \boxed{\text{Center}} = -\boxed{\text{South}}\cdot\boxed{\text{North}} &\implies \{u,v\}\tensor1=0.
\end{align*}

But the universal algebra generated by a pair of anticommuting  unitaries is nothing but the abstract Pauli algebra from proposition <ref> and so we found

\[
\CSTAR(\magicsquare) = \mathrm{M}_2\tensor\mathrm{M}_2.
\]

Similarly one argues for the magic pentagram and the proof is complete.

§ ACKNOWLEDGEMENTS

The author would like to thank his supervisor Søren Eilers for his kind support and encouragements,
as well as Connor Paddock for suggesting the quite valuable survey article [18].
Moreover, the author acknowledges the support under the Marie–Curie Doctoral Fellowship No. 801199.

§ PAULI ALGEBRA

The algebra of $2x2$-matrices (equivalently the algebra generated by its system of matrix units) has the equivalent description as the universal algebra generated by anticommuting  unitaries

\[
\]

which identifies the generating order-2 unitaries as Pauli matrices

\[
u=\SMATRIX{1\\&-1}=Z,\TAB v=\SMATRIX{&1\\1}=X
\]

or any other (possibly rotated) pair of anticommuting Pauli matrices.

We thus suggestively denote this as the abstract algebra of Pauli matrices.

Note that the system of matrix units is equivalently generated by the partial isometry with the orthogonality relation for its range and source:

\[
\]

The algebra generated by such a partial isometry is now easily seen to be equivalent to the algebra generated by anticommuting  unitaries.
Indeed with the suggested identification from the proposition one has

\[
\]

with inverse translation given by

\begin{gather*}
\end{gather*}

and one easily verifies the orthogonality relation for the range and source,

as well as the anticommutation relation, respectively.

§ OVERVIEW ON XOR-GAMES

We provide an overview of XOR-games for convenience of the reader, and we do so from an abstract, operator algebraic perspective.
Meanwhile, the author would like to acknowledge that the considerations of the current section arose as joint work with Azin Shahiri from a forthcoming work on the systematic (operator algebraic and representation-theoretic) classification of optimal states for the tilted CHSH games from [14] and further [1] with longterm goal for a better understanding of the I3322-inequality from [8] and [2].

An XOR-game is a nonlocal game in which the players answer with bits and such that the winning condition depends only on the parity of their answers:

\[
A=\{0,1\}=B:\TAB\win=\{(a\oplus b|xy)|\ldots\}\subset \{0,1\}\times X\times Y.
\]

The most prominent example is of course the CHSH game as discussed in the current article.
To give another less obvious example however one may also consider the graph 2-coloring game, whose winning condition may be equivalently formulated as

\begin{gather*}
x\sim y\implies a\oplus b = 1,\\
x = y \implies a\oplus b = 0.
\end{gather*}

For convenience we consider bits from now on as $\pm1$-valued:

\[
A=\{1,-1\}=B:\TAB a\oplus b=0,1\TAB\rightsquigarrow\TAB ab=\pm1.
\]

The point of such games is that they allow to verify the winning probability based on the bias instead, at which we take a closer look now.
Recall for this the two-player algebra with generators (see section <ref>):

\begin{gather*}
\CSTAR(\twoplayer)=\CSTAR(X|A=2)\tensor\CSTAR(Y|B=2):\\[\jot]
u(x)= E(a=0|x)- E(a=1|x).
\end{gather*}

Now for nonlocal games in general the winning probability may be described by the operator norm for the game polynomial (which we take unnormalized)

\[
\game := \sum E(\win):\TAB\winningprob_{qc}(\game)=\frac{\|\game\|}{|X\times Y|}
\]

and where we used the shorthand notation

\[
E(ab|xy):= E(a|x)\tensor E(b|y):\TAB E(S)=\{E(s)|s\in S\}.
\]

For XOR-games we may now introduce the bias polynomial (also unnormalized)

\[
\bias := \sum_{xy}\beta(x,y)u(x)\tensor u(y)
\]

with bias function given by the sum over the possible winning parities

\[
\beta(x,y):=\sum\Big\{\parity=\pm1\Big|(\parity|xy)\in\win\Big\}=1,0,-1.
\]

For the 2-coloring example above this reads for example

\[
\beta(x\sim y)=-1,\TAB\beta(x=y)=1,\TAB\beta(x\nsim y, x\neq y)=0.
\]

The point is now that the bias polynomial satisfies

\[
2\game = |\win| + \bias.
\]

Recall that we take here the winning set as $\win\subset \{0,1\}\times X\times Y$.

Since the game polynomial is positive we obtain for the spectrum

\[
0\leq\sigma(\game)\leq\|\game\|:\TAB \max\sigma(\game)=\|\game\|
\]

and by the relation above

\begin{gather*}
2\max\spectrum(\game)=\max\spectrum(\bias) + |\win|.
\end{gather*}

We may therefore relate the winning probability to the maximal upper bound for the bias.
We remark that the above procedure — that is passing from the norm to the spectrum — may be seen as a linearization process since

\[
\|a+1\|\neq\|a\|+1\TAB\rightsquigarrow\TAB\spectrum(a+1)=\spectrum(a)+1,\TAB \spectrum(\lambda a)=\lambda\spectrum(a).
\]

The goal is now to undo the linearization process — that is passing from the spectrum for the bias polynomial back to its norm.
We achieve this with the following observation:

Bias polynomials for XOR-games have symmetric spectrum,
from which the maximal bias agrees with the operator norm:

\[
\spectrum(\bias)=-\spectrum(\bias):\TAB \max\spectrum(\bias)=\|\bias\|.
\]

As a consequence for any XOR-game

\[
\]

whence the winning probability relates to the problem (see [7]):

Can you compute the operator norm for the bias polynomial?

The automorphism which swaps the measurement outcomes for Alice and leaves the measurements for Bob untouched (or vice versa)

\begin{gather*}
\Phi:\CSTAR(X|2)\tensor\CSTAR(Y|2)\to\CSTAR(X|2)\tensor\CSTAR(Y|2):\\
u(x)\tensor1\mapsto - u(x)\tensor1,
\TAB
1\tensor u(y)\mapsto 1\tensor u(y)
\end{gather*}

sends any bias polynomial to its negative

\[
\Phi(\bias) = -\sum \beta(xy)u(x)\tensor u(y) = -\bias.
\]

The spectrum however is invariant under automorphism whence

\[
\spectrum(\bias)=\spectrum(\Phi\bias)=\spectrum(-\bias)=-\spectrum(\bias).
\]

The remaining statements follow now from the symmetic spectrum.

One needs to stay cautious since polynomials (which do not strictly arise as bias polynomial) generally fail to have symmetric spectrum:for example the I3322 inequality from [8] and as reinvented in [2] might even have distinct values for the minimal and maximal bias.

We finish this overview with an interesting negative result from [5],
and for which we note that it easily extends to include also quantum commuting strategies simply by using a Stinespring dilation as discussed in section <ref>:

Nonlocal games with binary outputs are never pseudo–telepathy games — neither for quantum commuting strategies.

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