# Revisiting kink-like parametrization and constraints using OHD/Pantheon/BAO
samples

Simran Arora 0000-0003-0326-8945<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

We reexamine the kink-like parameterization of the deceleration parameter to
derive constraints on the transition redshift from cosmic deceleration to
acceleration. This is achieved using observational Hubble data, Type Ia
supernovae Pantheon samples, and Baryon acoustic oscillations/cosmic microwave
background. In this parametrization, the value of the initial $q$ parameter is
$q_{i}$, the final value is $q_{f}$, the present value is denoted by $q_{0}$,
and the transition duration is given by $\alpha$. We perform our calculations
using the Monte Carlo method, utilizing the emcee package. Under the
assumption of a flat geometry, we constrain the range of possible values for
three scenarios: when $q_{f}$ is unrestricted, when $q_{f}$ is equal to $-1$,
and when $\alpha$ is equal to $1/3$. This is done assuming that $q_{i}=1/2$.
Here, we achieved that the OHD data fixes the free parameters as in the flat
$\Lambda$CDM for unrestricted $q_{f}$. In addition, if we fix $q_{f}=-1$, the
model behaves well as the $\Lambda$CDM for the combined dataset. The
individual supernova data is causing tension in our determination when
contrasted to the $\Lambda$CDM model. We also acquired the current value of
the deceleration parameter, which is consistent with the latest results from
the Planck Collaboration that assume the $\Lambda$CDM model. Furthermore, we
observe a deviation from the standard $\Lambda$CDM model in the current model
based on the evolution of $j(z)$, and it is evident that the universe
transitions from deceleration to acceleration and will eventually reach the
$\Lambda$CDM model in the near future.

Keywords: kink-like parametrization, deceleration-acceleration, transition,
observational constraints

###### Contents

  1. I Introduction
  2. II The $q(z)$ parametrization
  3. III Observational Data Analysis
    1. III.1 Observational Hubble Data
    2. III.2 Pantheon SN Ia Sample
    3. III.3 Baryon Acoustic Oscillations/Cosmic Microwave Background
    4. III.4 Statistical evaluation
  4. IV Results and Cosmological parameters
    1. IV.1 When $q_{f}$ is free
    2. IV.2 When $q_{f}$ is $-1$
    3. IV.3 When $\alpha=1/3$
  5. V Conclusion

##  I Introduction

The standard cosmological model is challenged by significant physical
inconsistency caused by the presence of mysterious dark energy and dark
matter, the characteristics of which are currently the subject of extensive
theoretical discussion. The most significant conundrum in the standard
cosmological model is the possibility that 75% of the universe content
consists of dark energy, which is responsible for the accelerated expansion of
the universe. In contrast, dark matter accounts for approximately 20% of the
universe content and is an unknown component that is crucial for the formation
of structures. Gaining an in-depth understanding of both these components
poses a major challenge within the standard cosmological model, also known as
the $\Lambda$CDM paradigm [1].
While $\Lambda$CDM cosmology aligns with most observational data, it
encounters significant challenges, notably fine-tuning [2]. According to
observations, the magnitude of $\Lambda$ is very low, sufficient to drive this
current accelerating phase. Whereas the theoretically predicted value of
$\Lambda$, estimated from the quantum theory of fields, is enormously higher,
and consequently, there is a huge discrepancy of the order of $10^{120}$
between the observed and theoretical estimations of $\Lambda$. Secondly,
$\Lambda$ and matter magnitudes are unexpectedly comparable, leading to a
coincidence problem [3]. Consequently, these challenges have stimulated the
quest for alternative physical explanations for explaining the dynamics of the
universe, resulting in plenty of alternative approaches [4]. The present
issues highlight the necessity for additional frameworks that include the
$\Lambda$CDM model [5, 6]. Supernovae Ia (SNe Ia) observations [7, 8] provided
the first evidence that the universe is accelerating, which was further
supported by data from cosmic microwave background (CMB) radiation [9, 10],
baryonic acoustic oscillations [29, 11], and the observed Hubble measurements
[12, 22].
A further approach to understanding cosmic acceleration is to examine
kinematic variables such as the Hubble parameter $H$, the deceleration
parameter $q$, or the jerk parameter $j$, which are derived from the
derivatives of the scale factor. The kinematic technique is helpful since it
does not require any model-specific assumptions, such as the composition of
the universe. A metric theory of gravity describes it, and it is assumed that
the universe is homogeneous and isotropic at cosmological scales. Numerous
attempts in the literature have been made to constrain the current values of
$H$, $q$, and $j$ by parametrizing $q$ or $j$ [39, 13, 14]. Already, there are
some relevant investigations in this direction. Several works in the
literature estimated cosmological parameters independently of energy content,
and some authors used parametrization in these estimates [15, 16, 17]. Riess
et al. [18] used a simple linear redshift parameterization of $q$, i.e.,
$q(z)=q_{0}+q_{1}\,z$, to measure a transition from an early decelerating to a
current accelerating phase. However, at high redshift, this model is
unreliable. In a recent study, Xu et al. [19] considered the linear first-
order expansion of $q$, i.e., $q(z)=q_{0}+\frac{q_{1}\,z}{1+z}$, constant jerk
parameter, and third-order growth of luminosity distance. Using a $q(z)$
parametrization, in contrast to a parametrization for the equation of state,
offers the advantage of broader generality. In this scenario, we are not
limited to the principles of general relativity, and the assumptions made
about dark matter and energy are kept to a minimum. We simply need to adopt a
metric theory of gravity.
We aim to find a parametrization that accurately represents several
cosmological models across a broad range of redshift values. In the same vein,
the major purpose of this study is to investigate several simple kinematic
models for cosmic expansion based on a kink-like parameterization for $q(z)$
[20]. There are some special cases described by our parametrization, such as a
flat standard cosmological model with a constant dark energy equation of state
($\omega$CDM), $\Lambda$CDM, the flat DGP braneworld model, the flat
quartessence Chaplygin model, modified Polytropic Cardassian model [37].
Our primary focus lies in the transition from cosmic deceleration to
acceleration. We are interested in the redshift of this transition, its
duration, and the current value of the deceleration parameter. To constrain
the parameters of our model, we utilize Observational Hubble data (OHD),
Supernovae Pantheon samples, and Baryon Acoustic Oscillations/Cosmic Microwave
Background (BAO/CMB). We additionally constrain the value of $H_{0}$ by
utilizing datasets rather than assuming uniform or Gaussian priors, as done in
previous studies. However, our work is more general and different from similar
works in different ways. Firstly, we do not assume the $H_{0}$ as a prior but
rather allow our model to behave more generally through observational data.
Secondly, in this work, we go one step further by studying the evolution of
the equation of state and jerk parameters for a general deceleration
parameter. Lastly, we employ the latest Pantheon samples here.
The framework of this study is as follows: Section II provides an overview of
the background cosmology and introduces our kink-like parametrization along
with its associated features. Section III concisely explains the observational
data used in our analysis. This section presents constraints on the model
parameters determined by the data sets. It also addresses the potential model
selection using the Akaike and Bayesian criteria. The cosmological model’s
kinematic parameters, including the deceleration parameter, equation of state
parameter, and jerk parameter, are discussed in Section IV. Section V contains
a concise overview and remarks on the obtained outcomes.

##  II The $q(z)$ parametrization

A spatially homogeneous and isotropic universe is described by the Friedmann-
Lemaître-Robertson-Walker (FLRW) metric

$ds^{2}=-c^{2}dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\phi^{2}\right],$ (1)

where $a(t)$ represents the scale factor and $k=0,\pm 1$ depicts the
curvature. In this study, our primary focus is on a flat cosmological model.
The Hubble parameter is defined as $H=\frac{\dot{a}}{a}$, where $\dot{a}$
denotes derivative with respect to the cosmic time. It is possible to rewrite
all cosmological parameters as functions of redshift $z$, using
$\frac{a_{0}}{a}=\frac{1}{1+z}$ ($a(0)=a_{0}$).
A dimensionless measure of the cosmic acceleration is determined by the
deceleration parameter, given by

$q=-\frac{\ddot{a}}{aH^{2}}.$ (2)

Cosmological measurements suggest that the universe is currently experiencing
an era of accelerated expansion, i.e., $q<0$. Nevertheless, this acceleration
must have commenced in the near past and is not an enduring characteristic of
evolution. This transition from a decelerated to an accelerated phase of
expansion is characterized by a change in the signature of $q$, which occurs
at some specific $z_{t}$, known as the deceleration-acceleration transition
redshift.
Here, we adopt a flat cosmological model whose deceleration parameter can be
described by the given equation after the dominance of radiation [20].

$q(z)=q_{f}+\frac{\left(q_{i}-q_{f}\right)}{1-\frac{q_{i}}{q_{f}}\left(\frac{1+z_{t}}{1+z}\right)^{1/\alpha}}.$
(3)

The parameter $z_{t}$ represents the redshift at which the transition occurs
from cosmic deceleration to acceleration, whereas $\alpha$ denotes the width
of the transition and is related to the jerk at the transition by
$\frac{1}{\alpha}=\left(\frac{1}{q_{i}}-\frac{1}{q_{f}}\right)j(z_{t})$.
We can rewrite the Hubble parameter using $q$ as

$H=H_{0}\,exp\int_{0}^{z}\frac{1+q(y)}{1+y}dy,$ (4)

which further leads to

$H(z)=H_{0}(1+z)^{(1+q_{i})}\left[\frac{q_{i}\left(\frac{1+z_{t}}{1+z}\right)^{1/\alpha}-q_{f}}{q_{i}\left(1+z_{t}\right)^{1/\alpha}-q_{f}}\right]^{\alpha(q_{i}-q_{f})}.$
(5)

The final value of the deceleration parameter $q_{f}=q|_{z=-1}$ can be linked
to the current value $q_{0}=q|_{z=0}$ by

$q_{f}=\frac{q_{i}\left(1+z_{t}\right)^{1/\alpha}}{1-\frac{q_{i}}{q_{0}}\left[1-(1+z_{t})^{1/\alpha}\right]}.$
(6)

As previously stated, our objective with the $q(z)$ parametrization is to
describe the transition from a decelerated phase to an accelerated phase.
Therefore, given that $q_{i}>0$ and $q_{f}<0$, it is straightforward to show
that the parameter $\alpha$ is constrained to the range
$0<\alpha<\frac{ln(1+z_{t})}{ln\left(1-\frac{q_{0}}{q_{i}}\right)}$.

In most cosmological scenarios, the formation of large-scale structures
necessitates that the universe, during its early stages, undergoes a period
dominated by matter, where $q=\frac{1}{2}$. Given that our parametrization of
$q(z)$ is specifically intended to reflect the evolution of the universe
starting from a phase dominated by matter and experiencing deceleration, we
set $q_{i}=\frac{1}{2}$. It is important to note that other flat cosmological
models explored in the literature can be seen as specific instances of kink-
like parametrization, such as $\omega$CDM, $\Lambda$CDM, etc. It is seen that
while our parametrization is true at high redshift, it is not applicable
during the radiation-dominated era when $q=1$.
We aim to provide a sensible approach for determining constraints on the
unknown parameters $(H_{0},q_{0},z_{t},\alpha)$.

##  III Observational Data Analysis

Within this section, we employ several combinations of datasets, such as the
Observational Hubble data (OHD) data and Type Ia Supernova (SN) distance
modulus data, to reconstruct the cosmic deceleration parameter $q$ as a
function of the redshift $z$. In addition, we perform a reconstruction of $q$
using the combined $OHD$, $SN$, and $BAO/CMB$ datasets. To explore the
parameter space, we will be using the Markov Chain Monte Carlo (MCMC)
methodology and Python package emcee [21].

###  III.1 Observational Hubble Data

Observational Hubble Data is a widely accepted and significant method for
examining the expansion history of the universe beyond GR and $\Lambda$CDM.
The OHD sample mainly derives from the differential age of galaxies method,
often known as DAG [22]. The Hubble rate is usually obtained from the
following formula

$H(z)=\frac{-1}{1+z}\frac{dz}{dt}.$ (7)

In this work, we mainly utilize the OHD points obtained from the Cosmic
Chronometers (CC), which refer to the massive and passively evolving galaxies.
By employing CC, one can get the rate of change $dz/dt$ by calculating $\Delta
z/\Delta t$, where $\Delta z$ is the redshift separation in the galaxies
sample. This value can be established with precision and accuracy through
precise spectroscopy. However, obtaining $\Delta z$ is considerably more
complex and requires conventional clocks. To accomplish this, we might employ
old, massive, stellar populations that are passively evolving and exist across
a broad spectrum of redshifts. To determine the priors and likelihood
functions (which are necessary), we take into account the compilation
consisting of 31 points. To constrain our model, we introduce the chi-square
function

$\chi^{2}_{OHD}=\sum_{i=1}^{N_{H}}\bigg{[}\frac{H^{\mathrm{th}}_{i}(\Theta,z_{i})-H^{\mathrm{obs}}_{i}(z_{i})}{\sigma_{H(z_{i})}}\bigg{]}^{2},$
(8)

where $\Theta$ is a parameter space, $N_{H}$ denotes the number of data
points, $z_{i}$ represent the redshift at which $H(z_{i})$ has been measured,
$H_{obs}$ and $H_{th}$ are the measured and predicted values of $H(z)$,
respectively, and $\sigma_{H(z_{i})}$ is the standard deviation of the
$i_{th}$ point. The likelihood function that we will be using for MCMC
sampling has its usual exponential form

$\mathcal{L}=\exp(-\chi^{2}/2).$ (9)

###  III.2 Pantheon SN Ia Sample

We have also opted to work with the Pantheon sample, which represents one of
the most extensive collections of SNeIa. This dataset comprises 279 Type Ia
supernovae from the PANSTARRS DR1 (PS1) Medium Deep Survey. The SNeIa has
redshift values ranging from 0.03 to 0.68. Additionally, distance estimates of
SNeIa from the Sloan Digital Sky Survey, SuperNovae Legacy Survey (SNLS), and
other low-z and Hubble Space Telescope Samples were combined with PS1 data.
Consequently, this combined dataset represents a considerable collection of
SNeIa, a total of 1048 supernovae with redshift values ranging from 0.01 to
2.3 [23, 24]. The corresponding chi-square function reads

$\chi^{2}_{SN}=\sum^{N_{SN}}_{i,j=1}\frac{\Delta\mu_{i}}{\sigma_{\mu(z_{i})}},\quad\Delta\mu_{i}=\mu^{th}(\Theta)-\mu_{i}^{obs}.$
(10)

The distance moduli theoretically can be defined as

$\mu^{th}=5\log_{10}D_{L}(z)+\mu_{0},\quad\mu_{0}=5\log_{10}\frac{H_{0}^{-1}}{\mathrm{Mpc}}+25.$
(11)

where

$D_{L}(z)=\frac{c(1+z)}{H_{0}}S_{K}\bigg{(}H_{0}\int^{z}_{0}\frac{d\overline{z}}{H(\overline{z})}\bigg{)}.$
(12)

The function $S_{K}(x)$ is given by

$S_{K}(x)=\begin{cases}\sinh(x\sqrt{\Omega_{K}})/\Omega_{K},\quad~{}\Omega_{K}>0\\\
x,\quad\quad\quad\quad\quad\quad\quad\quad~{}~{}\Omega_{K}=0\\\
\sin(x\sqrt{|\Omega_{K}|})/|\Omega_{K}|,~{}~{}\Omega_{K}<0\end{cases}\,.$ (13)

Since we have a spatially flat universe, therefore $\Omega_{K}=0$.
The nuisance parameters in the Tripp formula [25] $\mu=m_{B}-M_{B}+\alpha
x_{1}-\beta c+\Delta_{M}+\Delta_{B}$ were recovered using the novel method
known as BEAMS with Bias Correction (BBC) [26]. As a result, the observed
distance modulus is now equal to the difference between the corrected apparent
magnitude $M_{B}$ and the absolute magnitude $m_{B}$
$\left(\mu=m_{B}-M_{B}\right)$. Hence, one can define the chi-square function
as follows [27]

$\chi^{2}_{\mathrm{SN}}=\Delta\bm{\mu}^{T}\mathbf{C}^{-1}\Delta\bm{\mu}.$ (14)

The total uncertainty matrix of distance modulus is given by

$\mathbf{C}=\mathbf{D}_{\mathrm{stat}}+\mathbf{C}_{\mathrm{sys}}$ (15)

It is assumed that the diagonal matrix of statistical uncertainties has the
following form $\mathbf{D}_{\mathrm{stat},ii}=\sigma^{2}_{\mu(z_{i})}$.
Besides, systematic uncertainties are derived using the Bias Corrections (BBC)
method, presented in [23].

$\mathbf{C}_{ij,\mathrm{sys}}=\sum^{K}_{k=1}\bigg{(}\frac{\partial\mu^{obs}_{i}}{\partial
S_{k}}\bigg{)}\bigg{(}\frac{\partial\mu^{obs}_{j}}{\partial
S_{k}}\bigg{)}\sigma^{2}_{S_{k}}.$ (16)

Indexes $\\{i,j\\}$ denote the redshift bins for distance modulus, $S_{k}$
indicates the magnitude of systematic error, and $\sigma_{S_{k}}$ is its
standard deviation uncertainty.

###  III.3 Baryon Acoustic Oscillations/Cosmic Microwave Background

Finally, we constrain our model using Baryon Acoustic Oscillations (BAOs).
BAOs appear early in the evolution of the universe. The so-called sound
horizon $r_{s}$, which is visible at the photon decoupling epoch with redshift
$z_{*}$, defines the characteristic scale of the BAO

$r_{s}=\frac{c}{\sqrt{3}}\int^{\frac{1}{1+z_{*}}}_{0}\frac{da}{a^{2}H\sqrt{1+(3\Omega_{b0}/4\Omega_{\gamma
0})a}}.$ (17)

In this case, the present baryon mass density is denoted by $\Omega_{b0}$, and
$\Omega_{\gamma 0}$ is the photon mass density at present. Here, $z_{*}$ is
the redshift of photon decoupling, and we use $z_{*}=1091$. To obtain the
BAO/CMB constraints, we use the “acoustic scale”
$l_{A}=\pi\frac{d_{A}(z_{*})}{r_{s}(z_{*})}$, where
$d_{A}(z)=\int^{z}_{0}\frac{dz^{\prime}}{H(z^{\prime})}$ is the angular
diameter distance in the comoving coordinates. Here, $D_{V}(z_{BAO})$ is the
dilation scale

$D_{V}(z)=\left[\frac{d_{A}(z)^{2}cz}{H(z)}\right]^{1/3}.$ (18)

Finally, we obtain the BAO/CMB constraints
$\left(\frac{d_{A}(z_{*})}{D_{V}(z_{BAO})}\right)$ [28]. This dataset was
gathered from these references [31, 29, 30].

Consequently, to perform the MCMC sampling, we need to define the chi-square
function for our BAO/CMB dataset

$\chi^{2}_{BAO/CMB}=X^{T}C^{-1}X,$ (19)

Where $X$ and $C^{-1}$ are of the form [30]

$X=\left(\begin{array}[]{c}\frac{d_{A}(z_{*})}{D_{V}(0.106)}-30.95\\\
\frac{d_{A}(z_{*})}{D_{V}(0.2)}-17.55\\\
\frac{d_{A}(z_{*})}{D_{V}(0.35)}-10.11\\\
\frac{d_{A}(z_{*})}{D_{V}(0.44)}-8.44\\\
\frac{d_{A}(z_{*})}{D_{V}(0.6)}-6.69\\\
\frac{d_{A}(z_{*})}{D_{V}(0.73)}-5.45\end{array}\right),\quad
C^{-1}=\begin{pmatrix}0.48435&-0.101383&-0.164945&-0.0305703&-0.097874&-0.106738\\\
-0.101383&3.2882&-2.45497&-0.0787898&-0.252254&-0.2751\\\
-0.164945&-2.454987&9.55916&-0.128187&-0.410404&-0.447574\\\
-0.0305703&-0.0787898&-0.128187&2.78728&-2.75632&1.16437\\\
-0.097874&-0.252254&-0.410404&-2.75632&14.9245&-7.32441\\\
-0.106738&-0.2751&-0.447574&1.16437&-7.32441&14.5022\end{pmatrix}.$ (20)

Additionally, we minimized $\chi^{2}_{OHD}+\chi^{2}_{SN}+\chi^{2}_{BAO}$ to
carry out the joint analysis from the combined $OHD+SN+BAO$. The results are
numerically derived from MCMC using OHD, Pantheon, BAO, and joint datasets,
which are presented in Table 1. Furthermore, the $1-\sigma$ and $2-\sigma$
likelihood contours for the possible subsets of parameter space are presented
for different cases in the next section. Here are the priors that we consider
for free parameters

$\begin{gathered}H_{0}\in[60,80],\quad q_{0}\in[-1,0],\\\
z_{t}\in[0,1],\quad\alpha\in[0,0.8].\\\ \end{gathered}$ (21)

These priors are extremely relaxed. Therefore, there is little to no bias.

###  III.4 Statistical evaluation

To assess the effectiveness of our MCMC study, a statistical evaluation has to
be conducted utilizing the Akaike Information Criterion (AIC) and Bayesian
Information Criterion (BIC). The AIC can be expressed as follows
$\mathrm{AIC}=\chi^{2}_{\mathrm{min}}+2d$ [32], where $d$ is the number of
free parameters in a considered model. To compare our results with the
standard $\Lambda$CDM model, we use the AIC difference between our model and
the standard cosmology
$\Delta\mathrm{AIC}=|\mathrm{AIC}_{\Lambda\mathrm{CDM}}-\mathrm{AIC}_{\mathrm{Model}}|$.
Here, if $\Delta\mathrm{AIC}<2$, there is strong evidence in favor of our
model, while for $4<\Delta\mathrm{AIC}\leq 7$, there is little evidence in
favor of the model of our consideration. Finally, for the case with
$\Delta\mathrm{AIC}>10$, there is practically no evidence in favor of our
model [33].
In addition, BIC is defined by $\mathrm{BIC}=\chi^{2}_{\mathrm{min}}+d\ln N$.
Here, $N$ is the number of data points used for MCMC. For BIC, if
$\Delta\mathrm{BIC}<2$, there is no strong evidence against a chosen model
that deviates from $\Lambda$CDM, if $2\leq\Delta\mathrm{BIC}<6$, there is
evidence against the model and finally for $\Delta\mathrm{BIC}>6$ there is
strong evidence against our model. We therefore store the
$\chi^{2}_{\mathrm{min}}$/AIC/BIC data for our model in the Table 1.
We see that, for $q_{f}$ free case, $\Delta\rm AIC=1.9$ for $OHD$, $2.4$ for
$SN$, and $8.0$ for $OHD+SN+BAO$. In this case, the model strongly favors
$OHD$ data and is slightly favorable for $SN$. Whereas, in case of $q_{f}=-1$,
$\Delta\rm AIC=0.464$ for $OHD$, $1.1$ for $SN$, and $4.6$ for $OHD+SN+BAO$.
The model strongly favors all the combinations of datasets. According to BIC
analysis, the present model for the case of $q_{f}=-1$ is consistent and
favorable.

##  IV Results and Cosmological parameters

###  IV.1 When $q_{f}$ is free

Let us examine the scenario when we set $q_{f}$ as a free parameter and
establish $q_{i}=1/2$. It can assume any value. Instead of working with
$q_{f}$, we use $q_{0}$. In general, three model parameters need to be
constrained in this case, namely $(H_{0},q_{0},z_{t},\alpha)$. It is
noteworthy that we explicitly use the restriction on $\alpha$, as mentioned
before.

Figure 1: The 1-dimensional and 2-dimensional marginalized posterior
distributions in the parameter space $(H_{0},q_{0},z_{t},\alpha)$ for the case
when $q_{f}$ is free.

Figure 2: The 1-dimensional and 2-dimensional marginalized posterior
distributions in the parameter space $(H_{0},z_{t},\alpha)$ for the case when
$q_{f}=-1$.

In FIG. 2, we display 2D marginalized confidence level regions for parameter
space $(H_{0},q_{0},z_{t},\alpha)$ using $OHD$, $SN$ and $OHD+SN+BAO$.
Finally, we plot the deceleration parameter $q(z)$, the equation of state
parameter $\omega(z)$, and the jerk parameter $j(z)$ for all these considered
data in FIG. LABEL:fig2. The behavior of $q(z)$ shows a signature flip and
$z_{t}$ indicates the epoch when the expansion of the universe went from a
decelerating to an accelerating phase in the recent past. There is an
uninspired phase of the universe when $z\rightarrow-1$ since $q(z)$ diverges
more from the de-sitter phase for the combined dataset. Another choice for
$q_{f}$ could also be made to ensure the model’s consistency in the distant
future. Consequently, an additional analysis is required. The plot also
displays the evolution of the equation of state parameter with respect to $z$.
It reveals that the $\omega$ is positive at higher redshifts and remains
greater than $-1$ (quintessence phase) until the present time for all the
datasets but crosses a phantom divide for the far future. The present value of
$\omega$ is $-0.72\pm 0.1$, $-0.78^{+0.09}_{-0.07}$, and
$-0.76^{+0.08}_{-0.07}$ for $OHD$, $SN$, and $OHD+SN+BAO$, respectively.
However, the function $j(z)=q(2q+1)+(1+z)\frac{dq}{dz}$ demonstrates the
deviation of $j$ from the flat $\Lambda$CDM model $(j=1)$ in the best-fit
model. The jerk parameter is often used to discriminate against various dark
energy models. Observations indicate that, in this model, the current value of
$j$ exceeds 1 for all the datasets ($j_{0}=1.3$, $1.71$, $1.7$ for $OHD$,
$SN$, $OHD+SN+BAO$, respectively) [36]. Therefore, based on the existing model
(where $j\neq 1$ and $q_{0}<0$), it is evident that the dynamic dark energy
model considered is the most probable explanation for the current acceleration
and needs attention. But in this case, the $j(z)$ is not well constrained as
it may be connected to the appearance of abrupt future singularities [34, 35].

###  IV.2 When $q_{f}$ is $-1$

Let us consider the case in which, besides fixing $q_{i}=1/2$, the final value
of the deceleration parameter is fixed to the value $q_{f}=-1$, such that
asymptotically in the future, i.e., $z=-1$, the model approaches to a de-
sitter phase. Models such as $\Lambda$CDM, DGP, quintessence Chaplygin gas,
etc., belong to this class of models [37]. By fixing $q_{f}$, we are left with
only three free parameters, namely $(H_{0},z_{t},\alpha)$. We present the
$68\%$ and $95\%$ confidence level for this parameter space in FIG. 2. The
present values of $q$ for $OHD$, $SN$, and $OHD+SN+BAO$ are obtained
explicitly using equation (6), as mentioned in Table 1.
We plot the deceleration parameter $q(z)$, equation of state parameter
$\omega(z)$, and jerk parameter $j(z)$ for all these considered data in FIG.
LABEL:fig4. The behavior of $q(z)$ shows a signature flip and $z_{t}$
indicates the epoch when the expansion of the universe went from a
decelerating to an accelerating phase in the recent past and approaches a de-
sitter phase as expected. The present values of $\omega$ are obtained as
$-0.74^{+0.03}_{-0.06}$, $-0.81^{+0.02}_{-0.04}$, $-0.75^{+0.02}_{-0.03}$ for
$OHD$, $SN$, $OHD+SN+BAO$, respectively. It reveals that the $\omega$ is
positive at higher redshifts and remains greater than -1 (quintessence phase).
It does not cross a phantom divide for the far future in this case.
For this mode, the jerk parameter $j(z)$ is found to be evolving, which
indicates a tendency of deviation of the universe from the standard
$\Lambda$CDM model. Observations indicate that, in this model, the current
value of $j$ exceeds 1 for all the datasets ($j_{0}=1.3$, $1.51$, $1.2$ for
$OHD$, $SN$, $OHD+SN+BAO$, respectively). Therefore, based on the existing
model (where $j\neq 1$ ), it is evident that the present model seems to
converge to 1, i.e., the standard $\Lambda$CDM in the near future. This would
indicate that the higher value of $H_{0}$ leads to a lower jerk constraint and
decreasing $z_{t}$. The significant deviations from the $\Lambda$CDM
predictions suggest a preference for using a parametrization of the dynamical
parameter for understanding the dark sector of the universe. It should be
noted that the present values of $q_{0}$, $H_{0}$, $z_{t}$ are consistent with
the values obtained by many authors [38, 39, 40] (also shown in figures
LABEL:fig8 and 9).

###  IV.3 When $\alpha=1/3$

We also considered the case of $\Lambda$CDM, when $\alpha$ is also fixed, i.e.
$\alpha=1/3$. The parameter space here reduces to $(H_{0},z_{t})$. We have
obtained $q_{0}$ from the same equation (6), mentioned in Table 1. We present
the $68\%$ and $95\%$ confidence levels for this parameter space in FIG. 5. It
is clear that the $\Lambda$CDM is in good agreement with the data and used for
comparing with the two previous cases. Here, the BAO measurements are giving
us tighter constraints on the parameters. The behavior of $q(z)$ shows a
signature flip and $z_{t}$ indicates the epoch when the expansion of the
universe went from a decelerating to an accelerating phase in the recent past
and approaches a de-sitter phase as expected. In this case, the jerk parameter
is $j=1$ for the best-fit values, consistent with the observations.

Figure 5: The 1-dimensional and 2-dimensional marginalized posterior
distributions in the parameter space $(H_{0},z_{t})$ for the case when
$\alpha=1/3$.

In FIG. 6, we have shown the evolution of the Hubble parameter $H(z)$ as a
function of redshift $z$ compared with 31 $H(z)$ measurements. We have plotted
the data points for $H(z)$ with $1\sigma$ error bars. We have observed that
our model is well consistent with the $H(z)$ data against redshift for free
$q_{f}$, $q_{f}=-1$, $\alpha=1/3$. For a convenient display, we also show
$H(z)/(1+z)$ in FIG. 6.

Figure 6: Figure shows the evolution of Hubble parameter $H(z)$ and $H(z)/(1+z)$ as a function of $z$. We have plotted 31 $H(z)$ measurements with its $1\sigma$ error bars for all the cases considered. Model | Datasets | $H_{0}$ | $q_{0}$ | $z_{t}$ | $\alpha$ | $\chi_{\rm min}^{2}$ | $\rm AIC$ | $\rm BIC$
---|---|---|---|---|---|---|---|---
| OHD | $68.90\pm 0.24$ | $-0.59^{+0.19}_{-0.15}$ | $0.606^{+0.081}_{-0.13}$ | $0.33^{+0.13}_{-0.19}$ | $16.128$ | $24.128$ | $29.864$
$q_{f}$ is free | SN | $68.91\pm 0.25$ | $-0.68^{+0.14}_{-0.11}$ | $0.548^{+0.069}_{-0.13}$ | $0.27^{+0.11}_{-0.22}$ | $1043.385$ | $1051.385$ | $1071.203$
| OHD+SN+BAO | $68.93^{+0.23}_{-0.29}$ | $-0.65^{+0.13}_{-0.11}$ | $0.627^{+0.062}_{-0.085}$ | $0.38^{+0.12}_{-0.16}$ | $1062.084$ | $1086.084$ | $1110.805$
Model | Datasets | $H_{0}$ | $q_{0}$ | $z_{t}$ | $\alpha$ | $\chi_{\rm min}^{2}$ | $\rm AIC$ | $\rm BIC$
| OHD | $69.19\pm 0.10$ | $-0.619^{+0.05}_{-0.10}$ | $0.544^{+0.053}_{-0.063}$ | $0.245^{+0.051}_{-0.065}$ | $16.657$ | $22.657$ | $26.959$
$q_{f}=-1$ | SN | $69.196\pm 0.098$ | $-718^{+0.04}_{-0.06}$ | $0.510^{+0.063}_{-0.074}$ | $0.191^{+0.038}_{-0.044}$ | $1044.081$ | $1050.081$ | $1064.945$
| OHD+SN+BAO | $69.199\pm 0.099$ | $-0.629^{+0.03}_{-0.04}$ | $0.626\pm 0.041$ | $0.269^{+0.034}_{-0.038}$ | $1064.676$ | $1082.676$ | $1101.217$
Model | Datasets | $H_{0}$ | $q_{0}$ | $z_{t}$ | $\alpha$ | $\chi_{\rm min}^{2}$ | $\rm AIC$ | $\rm BIC$
| OHD | $68.14\pm 0.11$ | $-0.522^{+0.037}_{-0.034}$ | $0.623\pm 0.060$ | $1/3$ | $18.193$ | $22.193$ | $25.061$
$\alpha$ fixed | SN | $68.13\pm 0.10$ | $-0.545^{+0.027}_{-0.028}$ | $0.663\pm 0.049$ | $1/3$ | $1044.981$ | $1048.981$ | $1058.89$
| OHD+SN+BAO | $68.129\pm 0.099$ | $-0.54^{+0.022}_{-0.023}$ | $0.654\pm 0.039$ | $1/3$ | $1066.039$ | $1078.039$ | $1090.4$
Table 1: Summary of the best-fit values of model parameters and statistical
analysis using OHD, SN, and OHD+SN+BAO, including the 68% and 95% confidence
levels. Figure 9: The plots represents the present values of $H_{0}$ with
their corresponding error bars obtained from several works. The green, blue,
and black lines represent the best-fit estimated values obtained for the
current model in all three scenarios (free $q_{f}$, $q_{f}=-1$, $\alpha=1/3$),
with the related $1\sigma$ error shown as shaded portions [40].

##  V Conclusion

In this work, we reexamined a kink-like parametrization for the deceleration
parameter to investigate the transition from cosmic deceleration to
acceleration in a spatially flat FLRW universe. The functional form of $q(z)$
involves $q_{f}$, $q_{i}$, $q_{0}$, $z_{t}$, $\alpha$, and $H_{0}$ as free
parameters. Given that our parametrization specifically intends to explain the
cosmic evolution that begins with a matter-dominated decelerated phase, we set
the value of $q_{i}=1/2$. However, as new data pour in and new techniques
evolve, revisiting the nature of $q$ with newer datasets is imperative. So,
here, we focussed on recent OHD, SN (Pantheon samples), and BAO/CMB
observations and carried out a Markov chain Monte Carlo (MCMC) analysis of the
model, which allowed us to determine the optimal values of the model
parameters.
We first consider the case in which we do not impose a fixed value for
$q_{f}$. So, we constrained four parameters namely
$(H_{0},q_{0},z_{t},\alpha)$. The comparison of the observational data on the
OHD shows a good agreement with the $\Lambda$CDM model. The AIC analysis also
substantiates the presence of a little tension for $OHD+SN+BAO$ between the
current model for this scenario and the $\Lambda$CDM predictions. However, to
definitively address this matter, additional observational evidence covering a
more comprehensive range of redshifts is required. There is an uninspired
phase of the universe when $z\rightarrow-1$ since $q(z)$ diverges more from
the de-sitter phase for the combined dataset. Another choice for $q_{f}$ could
also be made to ensure the model’s consistency in the distant future. The
evolution of the equation of state parameter reveals that the $\omega$ is
positive at higher redshifts and remains greater than $-1$ (quintessence
phase) until the present time but crosses a phantom divide for the far future.
The present value of $\omega$ is $-0.72$, $-0.78$, and $-0.76$ for $OHD$,
$SN$, and $OHD+SN+BAO$, respectively. The current value of $j$ exceeds $1$ for
all the datasets ($j_{0}=1.3$, $1.71$, $1.7$ for $OHD$, $SN$, $OHD+SN+BAO$,
respectively). But in this case, the $j(z)$ is not well constrained as it may
be connected to the appearance of abrupt future singularities.

We also examined the scenario where the model exhibits the final de-Sitter
phase, characterized by $q_{f}=-1$. The Pantheon samples are the primary
source for improvements compared to prior works. It is anticipated that these
samples lead to a substantial enhancement in the constraints, owing to their
higher statistics. By incorporating these datasets, we analyze the influence
of $H_{0}$ on the transition $z_{t}$. We find that the cosmological
deceleration-acceleration transition depends on the value of $H_{0}$ but is
otherwise mildly dependent on other cosmological parameters. The Hubble
parameter is not so far from the current bounds, agreeing with the Scolnic
2018 results [23]. Here, the jerk parameter is well constrained as it
converges to $j_{0}=1$ at late times.
We also considered the case of $\Lambda$CDM, when $\alpha$ is also fixed, i.e.
$\alpha=1/3$. The parameter space here reduces to $(H_{0},z_{t})$. The values
of cosmological parameters, including the jerk parameter $j=1$, are consistent
with the observations considered.
Our results have shown that the present model may represent an interesting
alternative to dark energy for the case of $q_{f}=-1$ to tighten the
constraints on the parameters for $OHD$, $Pantheon$, and $BAO$. Furthermore,
it is worth noting the potential of future surveys such as DESI, LSST, SH0ES,
and BINGO observations in effectively constraining the range of these model
parameters.

## Data Availability Statement

There are no new data associated with this article.

## Acknowledgments

SA & PKS thank IUCAA, Pune, India, for the hospitality where part of the work
was carried out. PKS acknowledges the Science and Engineering Research Board,
Department of Science and Technology, Government of India, for financial
support to carry out Research Project No.: CRG/2022/001847.

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# Relational Quantum Mechanics with Cross-Perspective Links Postulate: an
Internally Inconsistent Scheme

Marcin Markiewicz  International Centre for Theory of Quantum Technologies
(ICTQT), University of Gdansk, 80-308 Gdansk, Poland  Marek Żukowski
International Centre for Theory of Quantum Technologies (ICTQT), University of
Gdansk, 80-308 Gdansk, Poland

(August 28, 2024)

###### Abstract

We discuss the status of relative facts – the central concept of Relational
Quantum Mechanics (RQM) – in the context of the new amendment to RQM called
cross-perspective links postulate. The new axiom states that by a proper
measurement one learns the value of the relative outcome/fact earlier obtained
by another observer-system. We discuss a Wigner-Friend-type scenario in which,
without cross-perspective links postulate, relative facts have no predictive
or causal power, whereas including cross-perspective links makes them
effectively hidden variables, which causally determine outcomes of specific
measurements. However, cross-perspective links axiom invalidates the other
axiom of RQM, the one which states that in a Wigner-Friend scenario, RQM
assigns an entangled state to the Friend and System after the unitary
transformation of their interaction, despite the appearance of the relative
fact for the Friend. This quantum mechanical state according to RQM properly
describes the situation for Wigner. From this we show that RQM with cross-
perspective links axiom is an internally inconsistent hidden variable theory
and therefore cannot be treated as an interpretation of quantum mechanics in
any sense.

##  I Introduction

In this work we study consequences of a modification of Relational Quantum
Mechanics (RQM) introduced by Adlam and Rovelli [1]. The new axiom introduced
therein is called cross-perspective links and verbatim reads:

  * •

“cross-perspective links:” In a scenario where some observer Alice measures a
variable $V$ of a system $S$, then provided that Alice does not undergo any
interactions which destroy the information about $V$ stored in Alice’s
physical variables, if Bob subsequently measures the physical variable
representing Alice’s information about the variable $V$, then Bob’s
measurement result will match Alice’s measurement result. 111The cross-
perspective-links axiom is in the preprint version of [1] numbered 4.1, as it
is to replace axiom RQM-4. We will be referring to this axiom throughout the
paper either as cross-perspective links or RQM-4.1

Most importantly it replaces according to the Authors the following axiom:

  * •

RQM-4. Relativity of comparisons: it is meaningless to compare the accounts
relative to any two systems except by invoking a third system relative to
which the comparison is made.

As RQM is claimed to be an interpretation of quantum mechanics, all rules of
quantum mechanics must be applicable within it. It must give predictions which
are in tune with the quantum ones. The additional notions introduced in RQM,
with respect to quantum formalism, can interpret the nature of quantum
predictions, but cannot change them. Also any theory has to be internally
consistent, cannot lead to contradictions. But a theory which is an
interpretation of another one must be consistent with the theory it is
intended to interpret. Otherwise it is a different theory, with testable
predictive discrepancies.

Based on the above we shall assume that quantum mechanical terms, which are
used by the followers of RQM, mean what they mean in quantum mechanics,
provided that a caveat is not expressed by the followers of RQM stressing that
this or that must be understood in an RQM way.

With this respect we see no difference between quantum understanding of the
notion of "state" of a system and the one in RQM. It is a tool for prediction
of future results based on earlier facts. 222In [9] one can read: ”What is
then a “quantum state”? In RQM, it is a bookkeeping of known facts, and a tool
for predicting the probability of unknown facts, on the basis of the available
knowledge. Since it summarises knowledge about relative facts, the quantum
state of a system (and a fortiori its density matrix) does not pertain solely
to the system. It pertains also to the other system involved in the
interactions that gave rise to the facts considered known. Hence it is always
a relative state.” This might seem to be an expression of the most ”orthodox”
approach to the quantum state, as a mathematical tool for encoding predictions
for the future possible measurements, based on a preparation procedure, which
effectively is also a measurement (or was tested in earlier measurements to
give the states in question). Still, as we shall see the phrase ”it summarizes
knowledge of relative facts” is going against the orthodox approach in which
no notion of ”relative facts” as defined in RQM exist. Using the RQM parlance
[9] ”orthodox” quantum mechanics uses only ”stable facts”. Still, another
statement in [9] seems to corroborate with quantum mechanics. Namely ”A moment
of reflection shows that any quantum state used in real laboratories, where
scientists use quantum mechanics concretely, is always a relative state. Even
a radical believer in a universal quantum state would concede that the $\psi$
that physicists use in their laboratories to describe a quantum system is not
the hypothetical universal wave function: it is the relative state, in the
sense of Everett, that describes the properties of the system, relative to the
apparata it is interacting with.” However, in ”orthodox” quantum mechanics
apparata are always assumed to be macroscopic. The predictions are
probabilistic in quantum mechanics and in RQM.

The main difference openly spelled out by followers of RQM is that it
introduces the notion of "relative facts", which are produced during a
(unitary) interaction of two quantum systems. Such relative facts/outcomes do
not appear in the "orthodox" 333There is just one quantum mechanics, based on
facts occurring in preparation and measurement apparata. We use the term
”orthodox” to inform about this our standpoint, in the wording of those who
are not satisfied with quantum mechanics as it is. Note that ”orthodox”
quantum mechanics has not been falsified in any experiment, and since 1925 is
successfully used for making predictions. quantum theory. The following
quotation might be best in explaining the new notion:

  * •

[9] "For relative facts, every interaction can be seen as a “Copenhagen
measurement”, but only for the systems involved. Any physical system can play
the role of the “Copenhagen observer”, but only for the facts defined with
respect to itself. From this perspective, RQM is nothing else than a minimal
extension of the textbook Copenhagen interpretation, based on the realisation
that any physical system can play the role of the “observer” and any
interaction can play the role of a “measurement”: this is not in contradiction
with the permanence of interference through interactions because the
“measured” values are only relative to the interacting systems themselves and
do not affect other physical systems."

One should comment here that the word "realisation" should, we think, be
replaced here by "RQM assertion". We also cannot agree, that the extension is
"minimal". Still, this quotation defines the RQM standpoint.

###  I.1 The basis of our analysis

Our analysis is interpretation neutral. Interpretations usually involve a
specific understanding of the notion of the quantum state. For us a quantum
state is a theory-specific description of a statistical ensemble of
equivalently prepared systems, which allows for statistical (probabilistic)
predictions of future measurements, via the Born rule. 444In [8] we read in
page 13 ”In a strict sense, quantum theory is a set of rules allowing the
computation of probabilities for the outcomes of tests which follow specified
preparations.”, and in pages 25-26, one can find: ”Note that the word “state”
does not refer to the photon by itself, but to an entire experimental setup
involving macroscopic instruments. […] The essence of quantum theory is to
provide a mathematical representation of states (that is, of preparation
procedures), together with rules for computing the probabilities of the
various outcomes of any test.”  The state describes an individual system only
as a member of such an ensemble. The theory itself is "a set of rules for
calculating probabilities for macroscopic detection events, upon taking into
account any previous experimental information", [5]. Or if you like, one can
use E. P. Wigner’s statement (1961) “the wave function is only a suitable
language for describing the body of knowledge - gained by observations - which
is relevant for predicting the future behaviour of the system" [11]. Note that
all internally consistent interpretations (not modifications) of quantum
mechanics agree with the above. They only add some other properties to the
quantum state and/or variables to the theory (hidden causes, hidden ontic
states, hidden many worlds, etc.), or to individual members of the ensemble
(systems), without any modification of the calculational rules of quantum
theory (based on the statistical ensemble approach).

###  I.2 The codification of RQM, and its change

In [1] (pre-cross-perspectives) RQM assumptions are listed in the form of the
following set of axioms. We quote verbatim from [1]:

  * •

RQM-1. Relative facts: Events, or facts, can happen relative to any physical
system.

  * •

RQM-2. No hidden variables: unitary quantum mechanics is complete.

  * •

RQM-3. Relations are intrinsic: the relation between any two systems $A$ and
$B$ is independent of anything that happens outside these systems’
perspectives.

  * •

RQM-4. Relativity of comparisons: it is meaningless to compare the accounts
relative to any two systems except by invoking a third system relative to
which the comparison is made.

  * •

RQM-5. Measurement: an interaction between two systems results in a
correlation within the interactions between these two systems and a third one;
that is, with respect to a third system $W$, the interaction between the two
systems $S$ and $F$ is described by a unitary evolution that potentially
entangles the quantum states of $S$ and $F$.

  * •

RQM-6. Internally consistent descriptions: In a scenario where $F$ measures
$S$, and $W$ also measures $S$ in the same basis, and $W$ then interacts with
$F$ to ‘check the reading’ of a pointer variable (i.e., by measuring $F$ in
the appropriate ‘pointer basis’), the two values found are in agreement.

As RQM is to be an interpretation of quantum mechanics let us discuss quantum
mechanical meaning of these postulates.

  * •

RQM-1. A new notion of event or fact is introduced. In quantum mechanics
events, or facts, happen only in the preparation stage of an experiment
(selection of systems so that they are in a specific initial pure or mixed
state) and the final measurement (they are the results), and in both cases
they are macroscopically accessible. In between we have an evolution, which
might be controlled during the experiment or not.

  * •

RQM-2. An unquestionable definition of hidden variables in quantum mechanics
can be put as follows: Hidden variables are any additional parameters or
variables which are introduced to the description and are not present in the
quantum formalism. Their aim is to causally influence the results of final
measurements. The causal influence does not have to obey relativistic
constraints (witness: Bohmian Mechanics).

Concerning unitary quantum mechanics, by this usually people mean quantum
mechanics without the collapse postulate. Yes, it is complete provided one
uses it correctly. E.g. the unitary description of proper quantum measurements
with macroscopically accessible results must involve decoherence, see e.g.
[17], or in relation to the matters discussed here [14]. The decoherence is
due to the macroscopic nature of measuring devices which produce "outcomes".
Decoherence is the way to understand the emergence of classicality of
macroscopic objects from the point of view of unitary description of the
dynamics of the atomic structures that constitute them [16]. This is a modern
version of Bohr’s insistence on the classical nature of our observation of
measurement events in quantum experiments [2].

One can also use a simplified description involving states of memories
encoding the results. The states of the memories are not allowed to be in a
superposition. In short measurement devices in quantum mechanics and their
memories are on the macroscopic side of Heisenberg’s Cut.

  * •

RQM-3. This seems to be also applicable to the facts of point 1.

  * •

RQM-4. A purely RQM postulate, having no relation with formalism of quantum
mechanics, because it involves relative facts, see RQM-1.

  * •

RQM-5. The first sentence of the axiom is most probably a typo (as after $S$
and $F$ interaction there is no reason for the state of $W$ to be correlated
with them). We shall interpret it in the following way (consistent with
earlier writings of authors supporting RQM ideas555See the following statement
by Rovelli in [9]: ”[…]in the Wigner’s friend scenario, the friend interacts
with a system and a fact is realised with respect to the friend. But this fact
is not realised with respect to Wigner, who was not involved in the
interaction, and the probability for facts with respect to Wigner (realised in
interactions with Wigner) still includes interference effects. A fact with
respect to the friend does not count as a fact with respect to Wigner. With
respect to Wigner, it only corresponds to the establishment of an
“entanglement”, namely the expectation of a correlation, between the friend
and the system.”):

An interaction between two systems $S$ and $F$ is quantum mechanically
described by a unitary evolution and in RQM results in an outcome which is a
relative fact for these (it is uncertain in RQM whether for both or one of
them, but the symbol $F$ suggests the Friend in Wigner’s Friend scenario,
therefore there seems to be a preference in RQM to think that the fact occurs
for $F$ only). With respect to a third system $W$ (this seemingly suggests
Wigner), the only consequence of the (unitary) interaction between the two
systems $S$ and $F$ is that $S$ and $F$ are entangled after this process.

This can be put in a form of quantum formulas, which give the proper
(according to RQM) state of $S\otimes F$, "with respect to $W$". Namely after
$S$-$F$ interaction producing a fact/outcome for $F$ only, denoted here as
$r_{F}$, related with her RQM measurement of an observable
$\hat{O}=\sum_{l=1}^{d}r_{l}\outerproduct{l}{l}_{S}$, the state that $W$
faces666Important: the state he faces, not the state he thinks that he faces,
or ascribes, or must ascribe, or whatever. A quantum state gives proper
predictions, no matter whether the observer knows or not the state. Knowledge
of a state may influence the action of the observer, but not the measurement
results.  is as follows:

$\ket{{for-W}}_{S\otimes
F}=\frac{1}{\sqrt{2}}\big{(}\sum_{l=1}^{d}c_{l}\ket{l}_{S}\ket{l}_{F}\big{)}.$
(1)

Note that one of $r_{l}$’s must be $r_{F}$, but this is not (more, cannot be)
encoded in the state $\ket{{for-W}}_{S\otimes F}.$

This state is the basis of any quantum prediction for all possible later
measurements by W, because such is the role of "states" in quantum theory. We
spell this out, and add an obvious remark: all predictions for W’s
measurements are based only on $\ket{{for-W}}_{S\otimes F}$.

  * •

RQM-6. What is the difference between this postulate and the cross-perspective
links one? In RQM-6 $W$ does not have to measure $S$.

Also, in RQM-6 it is not stated whether the measurement of $S$ by $W$ is
simultaneous with the RQM measurement by $F$. $W$ seems to be Wigner in the
Wigner-Friend scenario, and this suggests that he measures later. Definitely
$W$ cannot act before $F$’s measurement of $S$: if $W$ measures first, then
effectively $F$ and $W$ switch their roles. Thus, if axiom RQM-6 is not to be
in a conflict with axiom RQM-5, and consequently with (1), one must assume
that it applies to simultaneous measurements of $S$ by $F$ and $W$. This is
because, if $W$ acts later than $F$ a conflict with (1) arises. State
$\ket{{for-W}}_{S\otimes F}$ is assumed to properly describe the system
$S\otimes F$ before $W$’s action and it does not single out $F$’s relative
result/outcome/fact, denoted here by $r_{F}$ . Therefore, there is no reason
whatsoever for $W$ to get result $r_{W}=r_{F}$. In other words in RQM a
unitary interaction of $F$ with $S$ produces a result $r_{F}$, but no collapse
to

$\ket{r_{F}}_{S}\ket{l=r_{F}}_{F}$ (2)

occurs, as the superposition $\ket{{for-W}}_{S\otimes F}$ according to RQM
properly describes for $W$ the post-RQM measurement state of $S\otimes F$.
777The collapse postulate can be replaced here a by description following
quantum theory of measurement based on decoherence [17]. Still, please note
that the collapse postulate has not been shown wrong in any experiment, and is
an effective way to describe the post-measurement situation (selection of a
sub-ensemble of systems which gave a specific measurement result).

##  II No Cross-perspective links: Relative facts vs EPR-type correlations

In this section Alice, or $A$, is the Friend. Bob, denoted by $B$ initially
plays the role of Wigner. They share a maximally entangled state of two qubits
(say, polarizations of two separate photons):

$\ket{ENT_{max}}_{S}=\sum_{l=0,1}c_{l}\ket{l}_{1}\ket{l}_{2}.$ (3)

The state above is in its Schmidt form, thus $c_{l}$ are real. In order to
remove the ambiguity of the Schmidt basis we assume that $c_{1}\neq c_{2}$,
with $c_{1}c_{2}\neq 0$, which guarantees that it is unique.

The labs of Friend and Wigner are separate but do not have to be far away from
each other. Two photons reach the labs, one photon per lab (not necessarily at
the same time!). Despite the fact that we have two photons, their
polarizations form one system $S=S_{1}+S_{2},$ or more formally
$S=S_{1}\otimes S_{2}$. Measurement by Alice on her photon $S_{1}$ constitutes
a measurement of the whole system $S$ which is degenerate in eigenvalues.
Alice makes an RQM measurement on photon 1, and therefore on $S$. The RQM
measurement is assumed to be in the Schmidt basis for photon 1, and we assume
that this fact is known to Bob, in full analogy to the original Wigner-Friend
scenario [11] in which Friend’s measurement basis is known to Wigner.

Thus, according to RQM Alice obtains a relative outcome, denoted here as
${r_{a}}$. This is one of the possible (eigen-)values of an observable
$\hat{A}=\sum_{l=1}^{2}r_{l}\outerproduct{l}{l}_{1}$, where
$\outerproduct{l}{l}_{1}$ denotes a projector onto $\ket{l}_{1}$. Note that
the eigenbasis of the observable is the basis for particle $1$, in the Schmidt
decomposition of state $\ket{ENT_{max}}_{S}$. What is the state of $S$ after
Alice receives her outcome is undefined in the axioms of RQM. However RQM is
definitive when postulating what is for Wigner the state of $S$ and $A$ after
their interaction. Namely according to RQM (see Eq. (1) and discussion
around), Wigner faces an $S+A$ state described by:

$\ket{for-B}_{S\otimes A}=\sum_{l=0,1}c_{l}\ket{l}_{1}\ket{l}_{2}\ket{l}_{A}.$
(4)

Most importantly, according to RQM, this situation is such no matter what
was/is the value of Alice’s relative outcome $r_{a}$. Wigner-Bob has no access
to $r_{a}$ without an additional interaction with Alice-Friend or $S$.

Wigner-Bob receives photon 2, for our argument this is relativistically
objectively after Alice’s RQM measurement. He makes a quantum mechanical
measurement or an RQM measurement of his photon $S_{2}$. Since the quantum
state Bob faces, $\ket{for-B}_{S\otimes A}$, contains maximal and complete
information for predicting measurements results, according to quantum
mechanics, the prediction is that his outcomes $r_{b}$ would appear with
probabilities $P(r_{b}=r_{l})=c_{l}^{2}$. Most importantly, conditioning on
$r_{a}$ does not change this distribution, since Alice’s RQM measurement is
not imprinted into the state Bob faces, $\ket{for-B}_{S\otimes A}$. Thus, RQM
relative values of Alice and Bob are not EPR correlated. This is in stark
contrast with a situation, in which Alice performs a proper quantum mechanical
measurement, and not the RQM one. Assume that she gets a result $r_{a_{QM}}$.
Then according to quantum mechanics, if Bob does not have an access to the
result, he faces a mixed state:

$\rho_{S\otimes A}^{post-
meas}=\sum_{l=0,1}c_{l}^{2}\ket{l}\bra{l}_{1}\otimes\ket{l}\bra{l}_{2}\otimes\ket{l}\bra{l}_{A}.$
(5)

However a communication of a message from Alice disclosing $r_{a_{QM}}$
changes Bob’s probability assignment from uniform to deterministic. More, as
Bob in the Wigner-Friend scenario is aware that Alice is to do her
measurements, he additionally knows that even without her message, his result
would match hers, $r_{b}=r_{a_{QM}}.$ This is because the probabilistic
mixture reflected in $\rho_{S\otimes A}^{post-meas}$ is only due to him not
knowing $r_{a_{QM}}$.

But if we treat $A\otimes B$ as one system making an RQM measurement in the
product basis $\ket{l}_{1}\ket{k}_{2}$ of the two qubits constituting $S$,
they get RQM values always correlated $r_{a}=r_{b}$, i.e. they are constrained
to $l=k$. The order of performing measurements by $A$ and $B$ is here totally
irrelevant.

Notice that the freedom of defining of a system (which is perfectly allowed in
quantum mechanics), as the $A\otimes B$ interacting with $S=S_{1}\otimes
S_{2}$ or $A$ acting on $S_{1}$ and separately $B$ acting on $S_{2}$, changes
the RQM prediction. In contrast, Quantum mechanics gives an unambiguous
$r_{a}=r_{b}$ in both cases.

Other remarks:

  * •

If Alice’s relative outcome $r_{a}$ has no conditional consequences for Bob,
as above, then it is metaphysical, by which we mean that it does not have any
predictive or causal consequences, since conditioning on its value does not
affect Bob’s probability assignment in a situation in which Alice’s and Bob’s
outcomes are expected to be correlated. In other words existence or non-
existence of Alice’s outcome, or its actual for Alice value, has no influence
on Bob’s observation aimed at repeating her RQM measurement. 888The entangled
state of $S+A$ that Bob predicts/faces after the measurement by $A$ is a basis
of prediction for results of his possible actions (not his predictions,
simply: predictions). It is not a state of his mind with no consequences for
his possible measurements. Obviously, if he errs in his description of the
state, the outcomes of his actions will generally not be following predictions
based on his erroneous belief. In the Wigner-Friend gedanken experiment one
should not assume that Bob is stupid.

  * •

If we assume cross-perspective links, then Alice’s result is a hidden variable
for Bob, which complements the description of his situation. Namely he faces
$\ket{for-B}_{S\otimes A}$ and the hidden unknown to him $r_{a}$. It is
$r_{a}$ which causally (deterministically !) implies that his result is to be
$r_{b}=r_{a}$. Otherwise there is no reason for $r_{b}=r_{a}$ to occur with
probability 1, as nothing like this is a consequence of the state faced by
Bob, $\ket{for-B}_{S\otimes A}$.

##  III RQM with Cross-perspective links: a hidden variable theory

Here we discuss the consequences of the postulate quoted as cross-perspective
links [item 4.1 of ref. [1]], using the wording of [1]. Below we show the
incompatibility of the new postulate (which incorporates relative facts) with
"Wigner-Friend" postulate RQM-5 of RQM and quantum mechanics.

Counterargument: Let us consider a scenario in which Bob measures "the
physical variable representing Alice’s information about the variable $V$" of
a system $S$. In order to indicate the contradiction which arises when one
applies the cross-perspective links axiom to this situation let us go step by
step through the entire process. The initial state of the system $S$ expressed
in eigenbasis of the observable $V$ may be put as:

$\ket{system}_{S}=\sum_{j}c_{j}\ket{V_{j}}_{S},$ (6)

where kets $\ket{V_{j}}_{S}$ represent eigenstates of the variable $V$. RQM
states that from Bob’s perspective Alice gets entangled with $S$, and these
eigenstates enter the Schmidt decomposition of the resulting entangled state.
Namely Bob faces the following state of the system $S$ and Alice’s physical
variables:

$\ket{for-B}_{S\otimes APV}=\sum_{j}c_{j}\ket{V_{j}}_{S}\ket{APV_{j}}_{APV},$
(7)

where $\ket{APV_{j}}_{APV}$ represent orthogonal states of Alice, associated
with different results of the RQM measurement ($APV$ stands in the notation
for “Alice’s physical variables"). Here $j=1,2$, but there is no problem to
generalize the situation beyond qubits/polarization and have $j=1,2.,...,d$.

Importantly, we shall make a principal assumption: within RQM Bob is not
stupid. By this we mean that the state vector $\ket{for-B}$ is a correct
according to RQM Bob’s description of the state of the full system $S\otimes
APV$, after the interaction between $S$ and $APV$ producing an inaccessible
for Bob RQM relative outcome. This is a correct description of the situation
he faces, not his perhaps wrong assumption. If Bob is replaced by somebody
else, say Cecile, then the new agent would face a similar situation, described
quantum mechanically in an identical way.999Note that in quantum mechanics
such a unitary interaction involving only $S$ and $APV$ does not produce any
result. In the quantum measurement theory this is just a pre-measurement, see
e.g. [10].

Only one of the kets $\ket{APV_{j}}_{APV}$ is related with Alice’s relative
fact (outcome) say $V_{j=r_{a}}$, namely $\ket{APV_{r_{a}}}_{APV}$, but this
is not a relative fact for Bob, as he faces the situation described, according
to RQM by (7). He must take an action to learn more. Assume that he interacts
with Alice in such a way that it does not affect the information stored in her
physical variables which are related with her RQM measurement of $S$,
concerning variable $V$. This means that the interaction must have the
property that the evolution leaves states related with possible different
results of her measurement intact,
$\ket{APV_{j}}_{APV}\rightarrow\ket{APV_{j}}_{APV}$. As according to RQM
during an RQM measurement a unitary evolution entangles Bob with the measured
object (in this case Alice), this must lead to the total state of all involved
subsystems in the form:

$\sum_{j}c_{j}\ket{V_{j}}_{S}\ket{APV_{j}}_{APV}\ket{j}_{B},$ (8)

and then Bob gets a result/outcome, denoted here as $r_{b}$, as a relative RQM
fact or for Wigner-Bob we have a stable fact (RQM with cross-perspective links
assumes that these two methods would lead to the same result fixing $j$). The
interaction leading to the state (8) is a Bob-Alice’s physical variables
entangling process:

$\ket{APV_{j}}_{APV}\ket{initial}_{B}\rightarrow\ket{APV_{j}}_{APV}\ket{j}_{B},$
(9)

which leaves Alice unperturbed, that is
$\ket{APV_{j}}_{APV}\rightarrow\ket{APV_{j}}_{APV}$. Under such an interaction
the reduced density matrix of Alice does not change, it is still the one
implied by $\ket{for-B}_{S\otimes APV}$. The interaction (9) changes only the
state of Bob. Applying Born rule to the state (8) we get a non-zero
probability of Bob having a wrong result:

$P(r_{b}\neq r_{a})=\sum_{j\neq r_{a}}|c_{j}|^{2}\neq 0.$ (10)

However, the cross-perspective links axiom assures that in a situation
precisely described above "Bob’s measurement result will match Alice’s
measurement result", that is we should expect $P(r_{b}\neq r_{a})=0$. We see
that the cross-perspective links axiom leads to a prediction at odds with the
quantum mechanical one based on state $\ket{for-B}_{S\otimes APV}$, which in
turn is defined by axiom RQM-5. An interpretation of quantum mechanics based
on RQM with amendment 4.1 is therefore internally contradictory.

The whole discussion shows that cross-perspective links axiom cannot be merged
with quantum mechanical formalism, therefore it can be interpreted only as an
introduction of hidden variables to quantum mechanics. Namely in the described
scenario, relative fact of Alice is a hidden variable for Bob, as it causally
forces outcome of his measurement to be consistent with Alice’s. This is not
supported by the quantum formalism, if one considers (9) as the proper
description of the situation, and contradicts also the axiom RQM-2 on non-
existence of hidden variables. Note that relative fact is a specific type of a
hidden variable. In contrast to the typical hidden variables discussed in the
context of Bell-type scenarios, or in Bohmian mechanics, it is not produced
during the preparation of initial state of system(s), but during the RQM
measurement.

All that raises further questions, the most important is the following one.
Any physical interaction happens in a finite time interval ("immediate"
quantum gates are non-physical idealizations). Therefore one should ask at
which stage of the entangling interaction the relative fact is already
produced.

##  IV Application of cross-perspective links postulate leads to a
contradiction in the case of GHZ correlations

In [6] an inconsistency of the notion of relative facts (outcomes) is proved,
provided one assumes the following two principles. 1-LMZ: RQM is an
interpretation of quantum mechanics, and therefore especially the Born rule
holds in it, and 2-LMZ: “if an interpretation of quantum theory introduces
some conceptualization of outcomes of a measurement, then probabilities of
these outcomes must follow the quantum predictions as given by the Born rule.”
In the next subsection we present a modification of the proof from [6] which
instead of 2-LMZ uses cross-perspective links axiom. Let us however start from
a short summary of the original argument presented in [6].

The gedanken-experiment setup for showing the contradiction [6] consists of
three qubits treated as a single compound system $S$ prepared initially in a
GHZ state of its constituents, and later measured in a relativistcally
objective temporally separated sequence by two "observers" $A$ and $B$ in such
a way that the situation as a whole is an operational scheme of the so-called
extended Wigner-Friend scenario, introduced in [4]. Namely, the first
“observer” $A$ performs an RQM measurement on the 8-dimensional system $S$, in
such a way that we have a unitary interaction composed of three commuting
unitary sub-transformations, each acting on a different qubit degree of
freedom of the compound system $S$, which is tensor-factorized into such,
$S=\otimes_{l=1}^{3}S_{l}$. Such a compound transformation, as well as its
factors, form a pre-measurement in quantum mechanics (for a discussion of this
concept see e.g. [15]). However, in RQM it is assumed that they also lead to a
“relative outcome”. In the discussed case the outcome can take one of eight
possible non-degenerate values. Further on, observer $B$ (who can be thought
of as just another system, or as a Wigner, the final observer, from the
Wigner-Friend scenario of the extended form by Deutsch) performs a measurement
on joint system consisting of qubits and the observer $A$ in a basis, which is
effectively complementary [14] to the one chosen by $A$. Since both observer
$A$ performs the RQM measurement on three qubits, and $B$ performs a quantum
mechanical measurement on effectively an 8-dimensional system, they obtain
outcomes which can be represented as two three-tuples of numbers, or three-
valued sequences $\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3}$ and
$\mathcal{B}_{1},\mathcal{B}_{2},\mathcal{B}_{3}$ respectively. We use the
Bell-GHZ convention that $\mathcal{A}_{l}$ and $\mathcal{B}_{l}$ have values
$\pm 1$. A contradiction is then shown, which can be reduced to a statement
that there is no assignment of real-valued outcomes to these numbers
$\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3},$
$\mathcal{B}_{1},\mathcal{B}_{2},\mathcal{B}_{3}$ if they are to satisfy
constraints predicted by quantum mechanics.

###  IV.1 Our argument rephrased

The system $S$ is composed of three qubits, which are all RQM measured by $A$.
$S$ is prepared in a GHZ state:

$\displaystyle\ket{GHZ}_{S}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\otimes_{m=1}^{3}\ket{+1^{(1)}}_{S_{m}}+\otimes_{m=1}^{3}\ket{-1^{(1)}}_{S_{m}}\right).$
(11)

A measurement of observables of a tensor factorizable form, like
$X=X_{1}\otimes X_{2}\otimes X_{3}$ can be treated as a single measurement.
This is consistent with quantum mechanics, see the Appendix.

We have a sequential measurement. First Alice makes her RQM measurement on
each of qubits in the GHZ state, in a basis of the Mermin’s argument. The
unitary operation related with this RQM measurement (for each qubit subsystem)
is given by

$\displaystyle\hat{U}_{m}^{SA}\left(\ket{l^{(3)}}_{S_{m}}\ket{\textrm{initial}}_{A_{m}}\right)$
$\displaystyle=$ $\displaystyle\ket{l^{(3)}}_{S_{m}}\ket{l^{(3)}}_{A_{m}}$
(12) $\displaystyle\equiv$ $\displaystyle\ket{l^{(3)}}_{SA_{m}},$

where

$\displaystyle\ket{\pm
1^{(3)}}_{S_{m}}=\frac{1}{\sqrt{2}}\left(\ket{+1^{(1)}}_{S_{m}}\pm
i\ket{-1^{(1)}}_{S_{m}}\right),$ (13)

and subscript $A_{m}$ denotes Alice’s physical variables coupled via the
interaction to qubit $S_{m}$. We use here the usual "Bell" convention for
eigenvalues of the associated observable: their value is $\pm 1$ for the
respective eigenstate. Alice gets her relative values
$\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3}$. Each equal to $1$ or $-1$.
Next Bob makes his effectively complementary measurements of the compound
system $S\otimes A$ in an orthonormal basis which contains tensor product
vectors of the form $\otimes_{m=1}^{3}\ket{\pm 1^{(2)}}_{SA_{m}}$, where

$\displaystyle\ket{\pm
1^{(2)}}_{SA_{m}}=\frac{1}{\sqrt{2}}\left(\ket{+1^{(3)}}_{SA_{m}}\pm
i\ket{-1^{(3)}}_{SA_{m}}\right).$

As the state after the first unitary transformation
$\otimes_{m=1}^{3}\hat{U}_{m}^{SA}$ is expressible in terms of states
$\otimes_{m=1}^{3}\ket{\pm 1^{(2)}}_{SA_{m}},$ one does not have to define the
other 56 elements of the basis. One can simplify the discussion by assuming
that $B$ is Wigner, and therefore makes a real quantum mechanical measurement
from the very beginning. The outcomes of such a measurement are accessible to
any information storing devices in his lab, including brains, as they are
stable and amplified to the macroscopic level.

Following the algebra in [6] one can see that Bob-Wigner’s results satisfy the
following relation:

$\mathcal{B}_{1}^{W}\mathcal{B}_{2}^{W}\mathcal{B}_{3}^{W}=1.$ (15)

We have added the subscript $W$ to stress that these are "stable outcomes" in
RQM parlance. However the relative results of Alice are potentially readable
in the counterfactual situation of Wigner performing on them the procedure
postulated in the cross-perspective links axiom RQM-4.1. If he decides to
apply RQM-4.1 to read out $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$ they must
follow relation with $\mathcal{B}_{1}^{W}$ of the following form:

$\mathcal{B}_{1}^{W}\mathcal{A}_{2}\mathcal{A}_{3}=-1.$ (16)

In the alternative counterfactual cases (different pairs of Alice’s results
checked by procedure RQM-4.1) one obtains the following constraints:

$\displaystyle\mathcal{A}_{1}\mathcal{B}_{2}^{W}\mathcal{A}_{3}=-1,$ (17)
$\displaystyle\mathcal{A}_{1}\mathcal{A}_{2}\mathcal{B}_{3}^{W}=-1.$ (18)

Wigner knows that he is taking part in a Friend-type experiment, thus is aware
of the "agreed" unitary action of $A$ leading to her relative result which can
be expressed in the form of three values
$\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3}$, see the Appendix. He also
knows the initial state of $S$.

Therefore each of $B$’s qubit-convention values $\mathcal{B}_{i}^{W}$ must
take the value of the negative product of $A$’s relative outcomes:
$\mathcal{B}_{i}^{W}=-\mathcal{A}_{j}\mathcal{A}_{k}$, with $i,j,k$ of
different values. Thus, he does not have to perform these counterfactual
measurements at all. The products of values of Alice are encoded in his
results, as $\mathcal{B}_{i}^{W}=-\mathcal{A}_{j}\mathcal{A}_{k}$. As he sees
$\mathcal{B}^{W}_{1}\mathcal{B}^{W}_{2}\mathcal{B}^{W}_{3}=1$, he concludes
that $\mathcal{A}_{i}$ cannot exist because a trivial algebra gives:

$\mathcal{A}_{1}\mathcal{A}_{2}\mathcal{A}_{3}=\pm i,$

while their values are supposed to be real, $\mathcal{A}_{i}=\pm 1$.

##  V Conclusions concerning contradictions between axioms

One can introduce the following requirement: a new notion can be introduced in
physics if one can show that it has a predictive power (or causal influence on
events).

Relative facts without cross-perspective links axiom do not have the required
above property of physical notions. They are metaphysical only, whereas the
cross-perspective axiom makes them effectively hidden variables, as they
determine, from Bob-Wigner perspective, the result of a possible measurement
in a specific context. This context is a measurement which is described in
cross-perspective links postulate, or in the axiom RQM-6 (if one assumes that
$W$ acts objectively after $F$). Note that this does not have to be the
context of the actual measurement by Bob.

###  V.1 Contradictions listed

  * •

The relative facts do not exist in quantum description. They are an RQM notion
(axiom RQM-1). They have a predictive power by cross-perspective links axiom
RQM-4.1, therefore they are hidden variables (axiom RQM-3 cannot hold).

  * •

Axioms RQM-5 and cross-perspective links RQM-4.1 are contradictory.

Note that the replacement of axiom RQM-4 by RQM-4.1 makes a defence of RQM
much more difficult, as such defence is usually based on axiom RQM-4. Axioms
RQM-4.1 and RQM-6 corroborate for the case of measurement by $W$ occurring
objectively after the RQM measurement by $F$, and it is difficult to
understand what for the replacement of RQM-4 by RQM-4.1 is done.

## Acknowledgements

MM and MZ are supported by Foundation for Polish Science (FNP), IRAP project
ICTQT, contract no. 2018/MAB/5, co-financed by EU Smart Growth Operational
Programme. We cordially thank Jay Lawrence for months of intensive
discussions.

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##  Appendix A Observables of compound quantum systems

In [3], the authors reject our inconsistency proof. The main argument on which
this refutation is based on is a statement that the quantum mechanical
constraint on the product of outcomes:

$\mathcal{B}_{1}\mathcal{B}_{2}\mathcal{B}_{3}=1,$ (19)

does not necessarily hold in RQM, since [[3], p. 5]:
One cannot however simply define an observer $B=B_{1}\otimes B_{2}\otimes
B_{3}$ relative to which constraint (19) holds, if there is no interaction
involving those three systems after their measurements take place.

Let us comment on this statement. The discussion here is an expansion of the
one in [7]. First of all in quantum mechanics there is no absolute definition
of a single or compound system. A tensor product nature of a Hilbert space,
like e.g. $\mathcal{H}=\mathcal{H}^{\prime}\otimes\mathcal{H}^{\prime\prime}$,
does not necessarily imply, that (i) the system under consideration is
bipartite. Moreover, (ii) sometimes it is not physically possible to separate
parts corresponding to two Hilbert spaces, as is the case of so-called virtual
tensor products [12, 13]. In the latter paper one finds a description of a
situation, in which a subspace of a multipartite quantum system decomposes
into two subspaces, which do not correspond to a physical partition of the
system under consideration, nevertheless one can implement measurements on
them which mutually commute. To sum up, quantum mechanics allows us to define
a single measurement of local factorizable observables like $X=X_{1}\otimes
X_{2}\otimes X_{3}$, and each $X_{l}$ acts in a two dimensional Hilbert space
$\mathcal{H}_{l}^{(2)}$. The spaces constitute the full Hilbert space for the
problem via a tensor product
$\mathcal{H}^{(8)}=\otimes_{l=1}^{3}\mathcal{H}^{(2)}_{l}$. Hilbert spaces
$\mathcal{H}^{(2)}_{l}$ do not have to describe states of separated qubits.

Further, we can even change encoding of outcomes, or in other words ascribe
other eigenvalues, which overtly are associated with the states of the
compound system $S$. This is a general feature, often omitted in textbook
presentations. It is most evident when one considers observables which are
unrelated with classical dynamical variables, e.g. related to detection events
in arrays of detectors. In our context this can be illustrated in the
following way. Let three dichotomic observables $X_{i}$, where $i=1,2,3,$ have
eigenvalues $x_{i}=\pm 1$, and let each pertain to a different qubit degree of
freedom of a compound system of three qubits. That is, we assume that our
eight-dimensional Hilbert space is tensor-factorized into three two-
dimensional spaces
$\mathcal{H}^{(8)}=\mathcal{H}_{1}^{(2)}\otimes\mathcal{H}_{2}^{(2)}\otimes\mathcal{H}_{3}^{(2)}$.
Each of the factor observables are of the form $X_{1}=X_{1}^{(2)}\otimes
I_{2}^{(2)}\otimes I_{3}^{(2)}$, and similarly are defined other $X_{i}$’s.
Finally, observables with subscript $i$ act only in $\mathcal{H}_{i}^{(2)}$,
and $I_{i}^{(2)}$ is an identity. Let us assign new eigenvalues to the
outcomes using the following mapping: $+1\rightarrow 0,-1\rightarrow 1$. The
new values are bits. In this way each three-qubit result $(x_{1},x_{2},x_{3})$
can be transformed to bit values, say $b_{n}$ such that the following relation
holds: $x_{n}=(-1)^{b_{n}}$. The compound result $(x_{1},x_{2},x_{3})$ gets
transformed to:

$(x_{1},x_{2},x_{3})\rightarrow(b_{1},b_{2},b_{3}).$

Next we can use a one-to-one map of the bit sequences to integers $v$ from 0
to 7 written down in the binary system, which can be put as
$(b_{1},b_{2},b_{3})\rightarrow b_{3}b_{2}b_{1}(binary).$ Note that the
(decimal) value of the integers reads
$v(b_{1},b_{2},b_{3})=\sum_{i=1}^{3}2^{i-1}b_{i}$.

Since RQM aims at being an interpretation, not a modification of quantum
mechanics, it must allow for treating measurement of observable $X$ as a
single measurement, also in its redefined version with the new eigenvalues
$v$. Note that a rejection of such a possibility implies that a definition of
an RQM “observer” system should demand that it can be only a prime-dimensional
one.