# Classicality concept test on neutral pseudoscalar meson qubits with Wigner
inequalities

A. Yu. Efimova<EMAIL_ADDRESS>University of Geneva, Switzerland  N. V.
Nikitin<EMAIL_ADDRESS>Physical Faculty, M.V. Lomonosov Moscow
State University, Russia  Skobeltsyn Institute of Nuclear Physics, Lomonosov
Moscow State University, Russia  National Research Centre "Kurchatov
Institute", Institute for Theoretical and Experimental Physics, Russia  Moscow
Center for Advanced Studies, Russia

###### Abstract

In this study, we introduce the concept of Classicality and derive Wigner
inequalities that depend on two instants, with a potential extension to three
instants. We explore the experimental feasibility of testing the violations of
these inequalities in both pure and mixed flavor states of $K^{0}-$, $D^{0}$-,
and $B_{s}$\- meson pairs. Using the Werner noise model, we demonstrate that
violations of time-dependent Wigner inequalities can be detected even when
background processes constitute up to $50\%$ of the system.

##  I Introduction

During the last century, quantum mechanics has shown many useful applications
in various fields, ranging from condensed matter to quantum cryptography.
However, throughout its development, many attempts have been made to fit it
into the classical paradigm. The reason is that a final interpretation of
quantum mechanics has not yet been found, and many postulates seem unexplained
or at least counterintuitive, such as the collapse of the wavefunction or
nonlocality. Therefore, theories such as hidden variables Einstein:1935rr  or
the de Broglie-Bohm pilot waves PhysRev.85.166 ; PhysRev.85.180 ;
Frauchiger_2018  were created.

To align quantum mechanics with the classical paradigm, a set of strict
criteria for a physical object in the classical world should be determined.
These criteria must then be expressed in mathematical language.

The first set of such criteria was presented in the work of A. Einstein, B.
Podolsky, and N. Rosen Einstein:1935rr . Years later, this set of criteria has
been termed "Local Realism" (LR). Through this concept, multiple types of
inequalities have been derived, such as Bell inequalities bell , Clauser-
Horne-Shimony-Holt (CHSH) Clauser:1969ny ; muynck1986  inequalities, and time-
independent (or static) Wigner inequalities wigner . Local Realism is based on
three principles:

1) Classical Realism (CR): All properties of any physical system exist
jointly, and independently of any observer, even if these properties cannot be
measured by any classical measuring device.

In this context, a "classical measuring device" refers to an apparatus that
receives classical input, and, after interacting with a system (which may be
either quantum or classical), yields a purely classical output. Causality is
crucial in this framework, signifying that a physical process, which occurred
before the measurement, and required time to unfold, is the cause of the
obtained measurement values. This causal link can be challenged by quantum
nonlocality. Thus, this leads us to the next postulate.

2) Locality: If there are two classical measurement devices $D_{a}$, and
$D_{b}$ that are spacelike separated in 4D spacetime throughout the entire
measurement process, then the measurements of $D_{a}$ do not influence the
outcomes at $D_{b}$, and vice versa. Locality is linked to the principle that
information cannot be transferred faster than the speed of light eberhard ;
Herbert1982 ; Wootters1982 ; DIEKS198227 ; 2020MUPB…75..541N .

3) Freedom of choice: The state of the system cannot influence the
experimenter’s subsequent choice of actions in the experiment with $100\%$
probability.

Here it is important to emphasize “with $100\%$ probability” ’’ i.e. it is
clear that experimenter planes his following experiments based on the former
data, however, we would better leave this question to the philosophers.

In the language of mathematics, LR can be formulated in terms of joint
probabilities, which arise naturally from the CR principle. Since in the
quantum paradigm it is not always possible to correctly define joint
probabilities, this opens the possibility to demonstrate that the quantum
paradigm cannot be reduced to a classical one. Locality can be expressed in
the language of the No-signaling condition (NSC) Cirelson:1980 ; PRboxes:1984
. Furthermore, the freedom of choice of the experimenter allows treating such
probabilities as independent. Section IV will examine in more detail how the
LR concept is related to NSC, and why, upon strict examination, this concept
must be supplemented with another principle.

Much more often than the CR principle or the experimenter’s freedom of choice,
the literature discusses issues specifically related to the principle of
locality. It is often believed that the violation of Bell’s inequalities
within the quantum paradigm is solely due to the nonlocality of entangled
composite quantum systems RevModPhys.81.1727 ; RevModPhys.86.419 . The quantum
correlations that arise as a result of such entanglement should formally
contradict both Hans Reichenbach’s classical principle of common cause
PhysRevX.7.031021 , and N.N. Bogolyubov’s principle of correlation weakening.
These issues are discussed in more detail in review articles, and collections
RevModPhys.86.419 ; Rosset:2014tsa ; QuantumUnSpeakables ;
QuantumUnSpeakables-II , where numerous references to original works can also
be found.

This work is organized as follows. The "Introduction" section introduces the
concept of Local Realism and justifies its main principles, which are crucial
for formulating the Classicality concept afterwards. Section II presents the
derivation of the time-independent Wigner inequalities, discusses the problem
of noise admixture, and its impact on the violation of the time-independent
Wigner inequalities. In Section III, we explore another concept, Macroscopic
Realism, and discuss its advantages that may also apply to the Classicality
concept. Section IV provides a detailed examination of the NSC and introduces
the Enhanced NSC, followed by the formulation of the Classicality concept.
Based on this concept, time-dependent Wigner inequalities are derived in
Section V. These inequalities are analyzed for both pure and mixed states in
Sections VI and VII, respectively, where their benefits for particle physics
are also explained. The Conclusion briefly highlights the main results of the
article.

##  II Time independent Wigner inequalities

The idea of using kaons, and later neutral pseudoscalar mesons, for testing
the Bell inequalities is not new. Initially, this concept was proposed by F.
Uchiyama Uchiyama:1996va  and a bit later developed by R. Bertlmann and B. C.
Hiesmayr Bertlmann:2001sk . These inequalities are based on the concept of
Local Realism. Later, it was demonstrated in Anna:2020ofp  that implementing
the proposed test in a real experiment is almost impossible. Let us briefly
explain the arguments.

Assume Local Realism applies to a system consisting of two correlated
subsystems, ‘‘I’’ and ‘‘II.’’ Each subsystem possesses a set of dichotomous
observables $a^{i}$, $b^{i}$, and $c^{i}$, where $i=\\{I,II\\}$. Observables
with only two possible outcomes allow us to consider qubit systems, which are
crucial for Bell inequalities Clauser:1969ny . Wigner inequalities can also be
derived using qubits; for instance, the observable $a^{I}$ might have two
distinct values, $a_{+}^{I}$ and $a_{-}^{I}$, and similarly for $b^{i}$ and
$c^{i}$. It is interesting to note that for Wigner inequalities, it is not
necessary for the values of the observables to be exactly $+1$ or $-1$, as was
the case with CHSH inequalities. This flexibility offers significant
advantages in the realm of particle physics, which will be discussed later.

Neutral kaons can also be considered as qubits. It is natural to consider the
following bases: the flavour basis, which can be described in terms of the
quarks inside the kaon and antikaon. It is important to note that neutral
kaons are not truly neutral particles; $K^{0}$ differs from $\bar{K}^{0}$ in
that the quarks are as follows:

$\left|\,K^{0}\,\right>=\left|\,d\bar{s}\,\right>,\qquad\left|\,\bar{K}^{0}\,\right>=\left|\,\bar{d}s\,\right>.$
(1)

However, these states are not eigenstates of the charge parity operator
$\hat{C}\hat{P}$. Therefore, one can consider the following basis:

$\left|\,K^{0}_{1}\,\right>=\frac{\left|\,K_{0}\,\right>+\left|\,\bar{K}^{0}\,\right>}{\sqrt{2}},\qquad\left|\,K^{0}_{2}\,\right>=\frac{\left|\,K^{0}\,\right>-\left|\,\bar{K}^{0}\,\right>}{\sqrt{2}}.$
(2)

$K_{1}^{0}$, and $K_{2}^{0}$ are chosen to satisfy the eigenvalues equation:

$\hat{C}\hat{P}\left|\,K_{1}^{0}\backslash
K^{0}_{2}\,\right>=\pm\left|\,K_{1}^{0}\backslash K^{0}_{2}\,\right>.$ (3)

Finally, it was observed in Christenson:1964CP  that CP eigenstates are not
the same as eigenstates of the Hamiltonian, it was observed that CP symmetry
can be violated. The third basis, which consists of states with specified
masses and lifetimes, can be considered

$\begin{split}\left|\,K^{0}_{L}\,\right>&=\frac{1}{\sqrt{1+|\varepsilon|^{2}}}(\varepsilon\left|\,K^{0}_{1}\,\right>+\left|\,K^{0}_{2}\,\right>)=p\left|\,K^{0}\,\right>-q\left|\,\bar{K}^{0}\,\right>,\\\
\left|\,K^{0}_{S}\,\right>&=\frac{1}{\sqrt{1+|\varepsilon|^{2}}}(\varepsilon\left|\,K^{0}_{2}\,\right>+\left|\,K^{0}_{1}\,\right>)=p\left|\,K^{0}\,\right>+q\left|\,\bar{K}^{0}\,\right>,\end{split}$
(4)

where $\varepsilon$ is CP-violation parameter. This parameter was measured
experimentally PDG22 :

$\begin{split}|\varepsilon|&=(2.228\pm 0.011)\times 10^{-3},\\\
Re(\varepsilon)&=(1.596\pm 0.013)\times 10^{-3}.\end{split}$ (5)

Time independent Wigner inequalities can be written in a general form as:

$w(a_{+}^{II},b_{+}^{I}|D_{a}^{II},D_{b}^{I})\leq
w(c_{+}^{II},b_{+}^{I}|D_{c}^{II},D_{b}^{I})+w(a_{+}^{II},c_{+}^{I}|D_{a}^{II},D_{c}^{I}),$
(6)

where $a_{\pm}$, $b_{\pm},$, and $c_{\pm}$ are dichotomous variables, and
$D_{a}$, $D_{b}$, and $D_{c}$ are macro measuring devices. The higher index
indicates the particle number. $a$, $b$, and $c$ can be replaced by different
combinations of the states of the three bases which were described above.

In the derivation of these inequalities the fact that particles $I$ and $II$
exhibit an anticorrelation property in between their observables, i.e. if
particle $I$ has the observable $a^{I}$ with a value of $a_{+}^{I}$, then for
the second particle, this observable will have the value $a_{-}^{II}$.

This property can be achieved if these two particles are in the
$\left|\,\Psi^{\pm}\,\right>$ Bell state. In this work we will consider
$\left|\,\Psi^{-}\,\right>$ state:

$\left|\,\Psi^{-}\,\right>=\dfrac{1}{\sqrt{2}}\left(\left|\,K^{II}\,\right>\otimes\left|\,\bar{K}^{I}\,\right>+\left|\,\bar{K}^{II}\,\right>\otimes\left|\,K^{I}\,\right>\right).$
(7)

We consider this state because it can be experimentally obtained from the
$\phi(1020)$\- meson decay, there $K^{0}$, and $\bar{K}^{0}$ produced exactly
in $\left|\,\Psi^{-}\,\right>$ flavor state.

After calculating the probabilities in (6) for $\left|\,\Psi^{-}\,\right>$
state one can obtain two possible inequalities in terms of the CP-violation
parameter $\varepsilon$ Uchiyama:1996va :

$\begin{split}|Re(\varepsilon)|&\geq-1,\\\
|Re(\varepsilon)|&\geq-|\varepsilon|^{2},\\\ \end{split}$ (8)

It is very obvious that only the last inequality from (8) can be highly
violated if the modulus takes a negative value. However, it is always
important to take possible noise into account because it exists in any
experiment. One of the simplest models of noise is Werner state, where in
addition of the density matrix of the considered state the admixture of the
identity matrix which models the noise is added. The density matrix of the
Werner state which is formed from the pure $\left|\,\Psi^{-}\,\right>$ Bell
state can be written as:

$\rho^{(W)}_{\Psi^{-}}=x\left|\,\Psi^{-}\,\right>\left<\,\Psi^{-}\,\right|+\dfrac{1-x}{4}I_{4\times
4},$ (9)

where $I_{4\times 4}$ is the identity four by four matrix, and $x$ is the
purity parameter which vary from $0$ to $1$, and indicate the fraction of the
pure Bell state in the Werner state. The higher the value of $x$, the closer
the Werner state is to the pure $\left|\,\Psi^{-}\,\right>$ state.

Inequalities which are analogues of the inequalities (8) for the Werner state
do not depend on $\varepsilon$ only but on $x$. The last inequality in (8) is
transformed to:

$x(1+2Re(\varepsilon)-|\varepsilon|^{2})\leq 1+|\varepsilon|^{2}.$ (10)

From equation (10), one can find the minimum value of the purity parameter x
at which it becomes possible to experimentally detect violations of the time-
independent Wigner inequalities.

$x\leq\dfrac{1+|\varepsilon|^{2}}{1+2Re(\varepsilon)-|\varepsilon|^{2}}.$ (11)

From (11) it is easy to show that the violations can be observed only for $x$
close to 1 such that only less than $0.1\%$ of noise is allowed, which is
impossible in current experiments. Therefore, further modifications of this
type of inequality are needed.

##  III A Novel Approach to Testing Quantum Paradigms

Before formulating the new concept, it is necessary to emphasize that Local
Realism (LR) is a stationary concept where time evolution does not exist.
Therefore, the original Bell and CHSH inequalities bell ; Clauser:1969ny ,
Wigner inequality wigner , and Mermin paradox Mermin1990  do not depend on
time.

Another attempt to build inequalities emphasizing the differences of a quantum
system was made in Leggett:1985zz ; PhysRevA.100.062314 ; PhysRevA.91.032117 ;
Formaggio:2016cuh , where the authors, A. J. Leggett and A. Garg, employed a
different concept, namely Macroscopic Realism (MR). The systems they
considered consist of only one observable but measured at different instants
PhysRevA.100.062314 . MR encompasses the following three principles:

1) Macroscopic realism per se: If a physical system can exist in multiple
macroscopically distinct states, then at any given instant, the system is in
only one of those states. This principle is analogous to the Classical Realism
principle from LR and is closely related to $\psi$-epistemism Harrigan2010 .

2) Noninvasive Measurement (NIM). This principle states that it is possible to
measure the system with arbitrary accuracy while having a minimal effect on
the subsequent system dynamics. It is evident that this principle contradicts
the projective measurement technique. For example, the von Neumann formula for
conditional probabilities is not applicable. This principle necessitates the
use of weak measurements, which are not feasible in high-energy physics.

3) Induction: The result of the current measurement cannot with certainty
influence the experimenter’s behavior or their choice of manipulations on the
system in the future. This principle is similar to the Freedom of choice
principle from LR, and it also permits the use of statistical methods for data
processing, which are designed to be applicable to both quantum and classical
approaches.

It is possible to reformulate MR in mathematical terms. MR per se allows us to
assume that each observable $a(t)$ at any given instant $t_{i}$ can have a
unique value from the spectrum: $a(t_{i})=a_{i}$. This value can either remain
the same or change subsequently. This principle enables the use of joint
probabilities of the form:

$\displaystyle 0\leq
w_{ji}(a_{j},\,\ldots,\,a_{k},\,\ldots,\,a_{i}\,|\,t_{j},\,\ldots,\,t_{k},\,\ldots\,t_{i})\leq
1,$ (12)

where at $t_{k}\neq\,\\{t_{i},\,t_{j}\\}$ the observable $a(t)$ is not
measured but it is measured at $t_{i}$, and $t_{j}$.

From the principle of Noninvasive Measurement, the ’’No-signaling in time’’
(’’NSIT’’) follows:

$\displaystyle\sum\limits_{a_{k}}w_{ji}(a_{j},\,\ldots,\,a_{k},\,\ldots,\,a_{i}\,|\,t_{j},\,\ldots,\,t_{k},\,\ldots,\,t_{i})\,=\,w_{ji}(a_{j},\,\ldots,\,a_{i}\,|\,t_{j},\,\ldots,\,t_{i}),$
(13)

meaning that the probability of measuring $a_{i}$, and $a_{j}$ values from the
spectrum of $a(t)$ does not depend of whether there were measurements of
$a(t)$ at $t_{k}\,\neq\,\\{t_{i},\,t_{j}\\}$.

Considering these two different concepts: LR, and MR, and the inequalities
based on them it is natural to try to form inequalities which consider
multiple observables that could be measured at different instants. For this
purpose, a new concept with a set of criteria that fulfills the requirements
of a multivariable system with time evolution should be constructed. In
addition, we require the projective measurement description to be applicable
for testing these inequalities because these measurements are commonly used in
high-energy physics. We will call these inequalities ’’Time-dependent Wigner
inequalities for $n$ instants’’. In the current work we consider $n=2$,
however, higher $n$ are also possible and were considered by authors.

##  IV Classicality

For the the new concept proposed in the previous section we have chosen the
name ’’Classicality’’ (CL). It should contain both static and dynamical
features of the classical paradigm. Additionally, the goal of this article is
to apply this concept to elementary particle physics and the accelerator
experiments. This area is of great importance because it deals with massive
ultra-relativistic particles which have not been considered yet in the field
of testings of the quantum paradigm. Such processes are usually described by
quantum field theory (QFT), and do not involve closed systems with the fixed
number of particles at least due to the production of the virtual particle -
anti-particle pairs from vacuum. Therefore, Classicality should not be static
as it was in LR, however, it should contain a principle similar to Locality
which is absent in MR.

In the intersection of high-energy physics and nonlocality, attention has
mostly been paid to the CHSH inequalities and time-independent Wigner
inequalities Privitera:1992xc ; Uchiyama:1996va ; Bramon:1998nz ;
Bertlmann:2001sk ; Baranov2008 ; Nikitin:2009sr ; PhysRevLett.127.161801 ;
Barr:2022 ; EPJC82Severi ; EPJC82Saavedra ; PhysRevD.107.016012 ;
PhysRevD.107.093002 . Leggett-Garg inequalities cannot be tested in high-
energy physics due to the absence of weak measurements, requiring instead
projective measurements. An attempt to build a new concepts has already been
made, and this concept was called Realism PhysRevA.100.062314 ;
PhysRevA.95.052103 ; Formaggio:2016cuh ; Pusey:2011de . However, the authors
believe that ’Classicality’ or ’CL’ is a more suitable name for the concept.
Up to now, there has been no well-formulated set of criteria for CL.

Before formulating the criteria, it would be useful to consider two important
conditions in advance. First, as mentioned earlier, the Locality principle
from LR introduces the No-signaling condition, which can be described as
follows: let us again assume that there are two subsystems, I and II, and we
consider an observable $a$ with the spectrum $a_{i}$ that can be measured by
the measuring device $D_{a}$ for subsystem I, and an observable $b$ with the
spectrum $b_{j}$ that can be measured by the measuring device $D_{b}$ for
subsystem II. Then, the result of the measurement of subsystem I should not
depend on the state of the measuring device $D_{b}$, whatever it measured on
subsystem II. Mathematically, this can be expressed as a sum over all possible
values of the spectrum of observable $b$ of the joint probability for both
subsystems, followed by the absence of dependence on the measuring device
$D_{b}$ in the probability of finding a certain value from the spectrum of
observable $a$ for subsystem I only:

$\displaystyle\sum\limits_{j}w\left(a_{i}^{(I)},\,b_{j}^{(II)}\,|\,D_{a}^{(I)},\,D_{b}^{(II)}\right)\,=\,w\left(a_{i}^{(I)}\,|\,D_{a}^{(I)}\right).$
(14)

Relation (14) is called No-signaling condition (NSC) Cirelson:1980 ;
PRboxes:1984 . Obviously the same relation is true for the subsystem ’’$II$’’:

$\displaystyle\sum\limits_{i}w\left(a_{i}^{(I)},\,b_{j}^{(II)}\,|\,D_{a}^{(I)},\,D_{b}^{(II)}\right)$
$\displaystyle=$ $\displaystyle w\left(b_{j}^{(II)}\,|\,D_{b}^{(II)}\right).$
(15)

Without the NSC this relation would be summed by the rules of the Kholmogorov
probability theory, and the result would differ from the one we have just
obtained:

$\sum\limits_{j}w\left(a_{i}^{(I)},\,b_{j}^{(II)}\,|\,D_{a}^{(I)},\,D_{b}^{(II)}\right)\,=\,w\left(a_{i}^{(I)}\,|\,D_{a}^{(I)},\,D_{b}^{(II)}\right).$
(16)

The latest equation allows for faster-than-light transitions between devices
$D_{a}$ and $D_{b}$, which is unphysical.

Let us now consider a more complicate case, and assume that for the subsystem
I there is another observable $c$ with the spectrum $c_{k}$ which can be
measured by the measuring device $D_{c}$. It is still easy to write two NSCs:

$\displaystyle\sum\limits_{j}w\left(a_{i}^{(I)},\,c_{k}^{(I)},\,b_{j}^{(II)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right)\,=\,w\left(a_{i}^{(I)},\,c_{k}^{(I)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)}\right);$
(17)
$\displaystyle\sum\limits_{i,\,\,k}w\left(a_{i}^{(I)},\,c_{k}^{(I)},\,b_{j}^{(II)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right)\,=\,w\left(b_{j}^{(II)}\,|\,D_{b}^{(II)}\right).$
(18)

We face some difficulties when trying to explore the following probability:

$\sum\limits_{k}w\left(a_{i}^{(I)},\,c_{k}^{(I)},\,b_{j}^{(II)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right).$

Using the Classical Realism principle, we can use the following expression for
the conditional probability:

$\displaystyle
w\left(a_{i}^{(I)},\,c_{k}^{(I)},\,b_{j}^{(II)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right)\,=$
$\displaystyle=$ $\displaystyle
w\left(a_{i}^{(I)},\,b_{j}^{(II)}\,|\,\,c_{k}^{(I)},\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right)\times
w\left(c_{k}^{(I)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right).$

Assume that the second term in (IV) fulfills the Locality principle, then

$w\left(c_{k}^{(I)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right)\,=\,w\left(c_{k}^{(I)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)}\right),$

i.e., the probability of measuring of a certain value from the spectrum of
$c^{(I)}$ does not depend on the state of the measuring device $D_{B}^{(II)}$.
Then

$\displaystyle
w\left(a_{i}^{(I)},\,c_{k}^{(I)},\,b_{j}^{(II)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right)\,=$
$\displaystyle=$ $\displaystyle
w\left(a_{i}^{(I)},\,b_{j}^{(II)}\,|\,\,c_{k}^{(I)},\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right)\times
w\left(c_{k}^{(I)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)}\right).$

However, it is still not forbidden by LR that the probability of measuring the
spectrum of the observable $c^{(I)}$ can depend on the measuring device
$D_{a}^{(I)}$.

It is shown from (IV) that LR concept is not full. It is needed to add an
additional principle which would be an static analogue for NIM. We will call
this principle as ’’Principle of the Classical Measurement’’. It can be
formulated in the following way: If a system has two observables, $a$ and $c$,
which can be either jointly measurable or not, then the final result of the
measurement of the spectra of these two observables does not depend on the
measuring order.

Let us assume that the classical measuring devices $D_{a}$ и $D_{c}$, which
measure observables $a$ and $c$, are quite close to each other, or at least
are not separated by a space-like interval. It means that one cannot apply the
NSC to these observables. Mathematically, the new criteria can be written as
Enhanced NSC or ENSC:

$\displaystyle\sum\limits_{k}w\left(a_{i}^{(I)},\,c_{k}^{(I)},\,b_{j}^{(II)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right)\,=\,w\left(a_{i}^{(I)},\,b_{j}^{(II)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right).$
(21)
$\displaystyle\sum\limits_{i}w\left(a_{i}^{(I)},\,b_{j}^{(II)}\,|\,D_{a}^{(I)},\,D_{c}^{(I)},\,D_{b}^{(II)}\right)\,=\,w\left(b_{j}^{(II)}\,|\,D_{b}^{(II)}\right).$
(22)

From (21) follows that the measurement of one of the variables of the
subsystem $I$ does not destroy the state of the subsystem $I$, and allows an
experimenter to measure the second observable afterwards. Relations (21), and
(22), do not contradict (18).

Here we can finish the remark, and come back to the new concept formulation.

In the current work we propose the following set of the Classicality concept
criteria, which are quite close to the criteria in LR, and MR but includes the
necessary changes justified above:

1) Ontism: At any instant $t_{i}$ the physical system is in just one of the
possible ’’ontic states’’ Pusey:2011de ; Harrigan2010 .

Any ontic state of the system can be fully described by a unique set of values
of the spectrum of all the observables corresponding to the system. However,
we do not assume that all these observables can be simultaneously measurable
by any measuring device. Thus, it is possible to use probabilities for the
existence of each ontic state at any instant $t$:

$\displaystyle 0\leq
w(a_{\alpha}(t),\,b_{\beta}(t)\,\ldots\,|\,D_{a},\,D_{b},\,\ldots)\leq 1.$
(23)

2) Epistemism: The experimenter always deals only with the "epistemic state"
of the physical system or the state that was observed by him or her
Pusey:2011de ; Harrigan2010 . Epistemic states differ from each other by the
values of the observables which can be jointly measured at $t_{i}$.

3) Consistency: At any instant $t_{i}$ for the values of the spectra of the
observable of the system in the ontic state the formulas (17), (18) NSC, and
(21), (22) ENSC are valid.

This principle is needed to derive probabilities of the epistemic state at
$t_{i}$ from the ontic state probabilities by summing over all observables
which are not simultaneously measurable. Ontism assumes that probabilities of
obtaining different values of the spectrum of a given observable are
probabilities of incompatible events, and these can be summed using the
formulas: (17), (18), (21), and (22).

4) Classicality of the measurement: For the spectrum of any observable of the
system in the ontic state, the NSIT (13) principle is fulfilled.

5) Independence: The experimenter has free will, and the planning and analysis
of an experiment are not $100\%$ dependent on the results of previous
experiments. This condition allows for the statistical independence of
different experiments.

In the framework of the classical paradigm, ontic states and epistemic states
of the physical systems usually coincide. For instance, it is always true for
classical mechanics. However, in statistical physics, they can be different.
One can consider an ideal gas in a vessel of a given volume; in this case, the
ontic state is the set of coordinates and velocities of all the molecules of
the gas, while the epistemic state is described in terms of temperature and
gas pressure.

In quantum physics, there are different opinions about ontic states and
epistemic states, and their differences. However, it does not affect the
implementation of this concept in the time-dependent Wigner inequalities in
the current article. Although the authors believe that Classicality can also
be useful in the future, for example, for issues of the applicability of
classical and quantum statistical methods.

In the framework of Classicality, the general probability for the spectrum of
observable $a$ evolving between $t_{0}$, and $t>t_{0}$ can be written as:

$\displaystyle
w\Big{(}a_{\epsilon}(t),\,b_{\beta}(t_{0})\,\ldots\,|\,D_{a},\,D_{b},\,\ldots\Big{)}\,=$
$\displaystyle=$
$\displaystyle\sum\limits_{\alpha}w\Big{(}a_{\alpha}(t_{0})\to
a_{\epsilon}(t)\Big{)}\,w\Big{(}a_{\alpha}(t_{0}),\,b_{\beta}(t_{0})\,\ldots\,|\,D_{a},\,D_{b},\,\ldots\Big{)},$

i.e. the observalbe $a$ evolves independently of other observables of the
system. This formula is crucial for the derivation of the time-dependent
Wigner inequalities in the following section.

##  V Time-dependent Wigner inequalities for two different measurement times

Let us assume that Classicality works for the system consisted from the two
subsystems ’’I’’, and ’’II’’, these systems can be correlated. Each of these
subsystems has a set of dichotomous observables $a^{(i)}$, $b^{(i)}$, and
$c^{(i)}$, where $i=\\{I,\,II\\}$. Observables with only two possible values
allow us to talk about qubit systems which are important for Bell inequalities
Clauser:1969ny . Wigner inequalities also can be obtained using qubits, one
can assume that the observable $a^{I}$ has two different possible values
$a_{+}^{I}$, and $a_{-}^{I}$.

For spin $s^{I}=1/2$ qubits, one might consider single or double projection
for simplicity. to the three non-coplanar axes $\vec{a}$, $\vec{b}$, and
$\vec{c}$ as three dichotomous variables. For CHSH inequalities, only double
spin-$\frac{1}{2}$ projections should be considered having values $\pm 1$;
this is not necessary for Wigner inequalities. For the neutral pseudoscalar
mesons $M$ instead of the ’’derections’’ in space the following states bases
could be considered: 1. states with a certain flavour ($M$, and $\bar{M}$), 2.
states with a certain $CP$–parity ($M_{1}$, and $M_{2}$), and 3. states with a
certain mass, and life time ($M_{L}$, and $M_{H}$):

$\displaystyle\left|\,M_{1}^{(i)}\,\right>=\frac{1}{\sqrt{2}}\,\Big{(}\left|\,M^{(i)}\,\right>+\left|\,\bar{M}^{(i)}\,\right>\Big{)},\quad\left|\,M_{2}^{(i)}\,\right>=\frac{1}{\sqrt{2}}\,\Big{(}\left|\,M^{(i)}\,\right>-\left|\,\bar{M}^{(i)}\,\right>\Big{)};$
$\displaystyle\left|\,M_{L}^{(i)}\,\right>=p\,\left|\,M^{(i)}\,\right>+q\,\left|\,\bar{M}^{(i)}\,\right>,\quad\left|\,M_{H}^{(i)}\,\right>=p\,\left|\,M^{(i)}\,\right>-q\,\left|\,\bar{M}^{(i)}\,\right>,$

where
$\hat{C}\,\hat{P}\,\left|\,M_{1}^{(i)}\,\right>\,=\,+\,\left|\,M_{1}^{(i)}\,\right>$
и
$\hat{C}\,\hat{P}\,\left|\,M_{2}^{(i)}\,\right>\,=\,-\,\left|\,M_{2}^{(i)}\,\right>$.

There are generalized states that were used for kaons described in Section II.

Let us consider two different moments of time: $t_{0}$ – initial time аnd
$t>t_{0}$. Assume that the spectrum of the variables $a^{(i)}$, and $b^{(i)}$
of each subsystem ’’I’’ or ’’II’’ are measured at $t$, and that at initial
time $t_{0}$ the spectrum of the observavbles $a^{(i)}$, $b^{(i)}$, and
$c^{(i)}$ of the subsystems ’’I’’, and ’’II’’ are correlated. For example, for
the variable $a^{(i)}$ this correlation can be written as:

$\displaystyle\left\\{a_{\pm}^{I}(t_{0}),\,D^{II}_{a}\right\\}\,\to\,\left\\{a_{\mp}^{II}(t_{0}),\,D^{II}_{a}\right\\}.$
(25)

Such behavior can be demonstrated in accelerator experiments. For example, in
the decays $\phi(1020)\to K^{0}\bar{K}^{0}$ or $\Upsilon(4S)\to
B^{0}\bar{B}^{0}$, a pair of pseudoscalar mesons is produced in the singlet
Bell state in flavor.

$\displaystyle\left|\,\Psi^{-}\,\right>=\frac{1}{\sqrt{2}}\Big{(}\left|\,M^{I}\,\right>\otimes\left|\,{\bar{M}}^{II}\,\right>-\left|\,{\bar{M}}^{I}\,\right>\otimes\left|\,M^{II}\,\right>\Big{)}.$
(26)

In the state $\left|\,\Psi^{-}\,\right>$ there is not only anticorrelation
(25) in flavour of pseudoscalar mesons which can be seen directly from the
equation (26), but also anticorrelation in $CP$–parity states, and in states
$\left|\,M_{L,\,H}\,\right>$.

At $t_{0}$ in Classicality due to Сonsistency principle, and the correlation
condition (25) for the Wigner inequalities are as follows::

$\displaystyle
w\left(a_{+}^{II},b_{+}^{I},t_{0}\right)\,\leq\,w\left(c_{+}^{II},b_{+}^{I},t_{0}\right)\,+\,w\left(a_{+}^{II},c_{+}^{I},t_{0}\right);$
(27) $\displaystyle
w\left(a_{+}^{II},b_{-}^{I},t_{0}\right)\,\leq\,w\left(c_{+}^{II},b_{-}^{I},t_{0}\right)\,+\,w\left(a_{+}^{II},c_{-}^{I},t_{0}\right);$
(28) $\displaystyle
w\left(a_{-}^{II},b_{+}^{I},t_{0}\right)\,\leq\,w\left(c_{+}^{II},b_{+}^{I},t_{0}\right)\,+\,w\left(a_{-}^{II},c_{+}^{I},t_{0}\right);$
(29) $\displaystyle
w\left(a_{-}^{II},b_{-}^{I},t_{0}\right)\,\leq\,w\left(c_{+}^{II},b_{-}^{I},t_{0}\right)\,+\,w\left(a_{-}^{II},c_{+}^{I},t_{0}\right).$
(30)

For brevity, we use the following simplification::

$w\left(a_{+}^{II},\,b_{+}^{I},t_{0}\right)\,=\,w\left(a_{+}^{II}(t_{0}),\,b_{+}^{I}(t_{0})\,|\,D^{II}_{a},\,D^{II}_{c},\,D^{I}_{b}\right),$

and so on.

Wigner inequalities have already been derived in several different ways
starting from the original article wigner ; however, in this work, we aim to
demonstrate the importance of the ENSC principle. Let us derive the equation
(27). Applying the epistemism principle (21), one can write that

$\displaystyle
w\left(a_{+}^{II}(t_{0}),\,b_{+}^{I}(t_{0})\,|\,D^{II}_{a},\,D^{II}_{c},\,D^{I}_{b}\right)=\sum\limits_{k=\\{-,\,+\\}}w\left(a_{+}^{II}(t_{0}),\,c_{k}^{II}(t_{0}),\,b_{+}^{I}(t_{0})\,|\,D^{II}_{a},\,D^{II}_{c},\,D^{I}_{b}\right).$
(31)

Then

$\displaystyle
w\left(c_{+}^{II}(t_{0}),\,b_{+}^{I}(t_{0})\,|\,D^{II}_{a},\,D^{II}_{c},\,D^{I}_{b}\right)=\sum\limits_{i=\\{-,\,+\\}}w\left(a_{i}^{II}(t_{0}),\,c_{+}^{II}(t_{0}),\,b_{+}^{I}(t_{0})\,|\,D^{II}_{a},\,D^{II}_{c},\,D^{I}_{b}\right).$
(32)

The second term on the right-hand side (32) is the same as the first term on
the right-hand side (31). The first term on the right-hand side (32) is
greater than zero due to ontism and (23). Finally:

$\displaystyle
w\left(a_{+}^{II}(t_{0}),\,c_{+}^{I}(t_{0})\,|\,D^{II}_{a},\,D^{I}_{b},\,D^{I}_{c}\right)=\sum\limits_{j=\\{-,\,+\\}}w\left(a_{+}^{II}(t_{0}),\,b_{j}^{I}(t_{0}),\,c_{+}^{I}(t_{0}),|\,D^{II}_{a},\,D^{I}_{b},\,D^{I}_{c}\right).$
(33)

The second term in (33) can be written as follows, due to anticorrelation
property (25):

$w\left(a_{+}^{II}(t_{0}),\,b_{+}^{I}(t_{0}),\,c_{+}^{I}(t_{0}),|\,D^{II}_{a},\,D^{I}_{b},\,D^{I}_{c}\right)=w\left(a_{+}^{II}(t_{0}),\,c_{-}^{II}(t_{0}),\,b_{+}^{I}(t_{0})\,|\,D^{II}_{a},\,D^{II}_{c},\,D^{I}_{b}\right).$

Then it equals the first term in (31). The first term in (33) due to Ontism is
non-negative. Then one obtains inequality (27). Inequalities (28) – (30) can
be obtained in the same way.

Calssicality allows us to connect probability
$w\left(a_{+}^{II}(t),\,b_{+}^{I}(t)\right)$ with other four probabilities
$w\left(a_{+}^{II},b_{+}^{I},t_{0}\right)$,
$w\left(a_{+}^{II},b_{-}^{I},t_{0}\right)$,
$w\left(a_{-}^{II},b_{+}^{I},t_{0}\right)$ и
$w\left(a_{-}^{II},b_{-}^{I},t_{0}\right)$, which satisfy the Wigner
inequalities (27) – (30). This allows us to write the equation dependent on
two instants, $t_{0}$, and $t$.

Let us start with $w\left(a_{+}^{II},b_{+}^{I},t_{0}\right)$. If the state
$\\{a_{+}^{II}(t_{0}),b_{+}^{I}(t_{0})\\}$ transitions to
$\\{a_{+}^{II}(t),b_{+}^{I}(t)\\}$, it is necessary to determine the
probability of the transitions $a_{+}^{II}(t_{0})\to a_{+}^{II}(t)$, and
$b_{+}^{I}(t_{0})\to b_{+}^{I}(t)$. Using Classicality and (IV), one can show
that the total probability for $\\{a_{+}^{II}(t),b_{+}^{I}(t)\\}$ is:

$w\left(a_{+}^{II}(t_{0})\to
a_{+}^{II}(t_{2})\right)\,w\left(b_{+}^{I}(t_{0})\to
b_{+}^{I}(t_{1})\right)\,w\left(a_{+}^{II},\,b_{+}^{I},\,t_{0}\right),$

i.e. the transition probability form the state
$\\{a_{+}^{II}(t_{0}),b_{+}^{I}(t_{0})\\}$ to the state
$\\{a_{+}^{II}(t),b_{+}^{I}(t)\\}$ can be written as a product of
probabilities of two transitions $a_{+}^{II}(t_{0})\to a_{+}^{II}(t)$, and
$b_{+}^{I}(t_{0})\to b_{+}^{I}(t)$. For the remaining three combinations –
$\\{a_{+}^{II}(t_{0}),b_{-}^{I}(t_{0})\\}$,
$\\{a_{-}^{II}(t_{0}),b_{+}^{I}(t_{0})\\}$ and
$\\{a_{-}^{II}(t_{0}),b_{-}^{I}(t_{0})\\}$ – one can write the probabilities
in a similar way. Then by Classicality the total probability
$\\{a_{+}^{II}(t),b_{+}^{I}(t)\\}$ can be written as:

$\displaystyle\qquad\qquad\qquad w\left(a_{+}^{II}(t),\,b_{+}^{I}(t)\right)=$
$\displaystyle=$ $\displaystyle w\left(a_{+}^{II}(t_{0})\to
a_{+}^{II}(t)\right)\,w\left(b_{+}^{I}(t_{0})\to
b_{+}^{I}(t)\right)\,w\left(a_{+}^{II},b_{+}^{I},t_{0}\right)\,+$
$\displaystyle+$ $\displaystyle w\left(a_{-}^{II}(t_{0})\to
a_{+}^{II}(t)\right)\,w\left(b_{+}^{I}(t_{0})\to
b_{+}^{I}(t)\right)\,w\left(a_{-}^{II},b_{+}^{I},t_{0}\right)\,+$
$\displaystyle+$ $\displaystyle w\left(a_{+}^{II}(t_{0})\to
a_{+}^{II}(t)\right)\,w\left(b_{-}^{I}(t_{0})\to
b_{+}^{I}(t)\right)\,w\left(a_{+}^{II},b_{-}^{I},t_{0}\right)\,+$
$\displaystyle+$ $\displaystyle w\left(a_{-}^{II}(t_{0})\to
a_{+}^{II}(t)\right)\,w\left(b_{-}^{I}(t_{0})\to
b_{+}^{I}(t)\right)\,w\left(a_{-}^{II},b_{-}^{I},t_{0}\right).$

Incorporating inequalities (27) – (30) into (V), one obtains:

$\displaystyle\qquad\qquad\quad
w\left(a_{+}^{II}(t),\,b_{+}^{I}(t)\right)\leq$ $\displaystyle\leq$
$\displaystyle w\left(a_{+}^{II}(t_{0})\to
a_{+}^{II}(t)\right)w\left(b_{+}^{I}(t_{0})\to b_{+}^{I}(t)\right)\cdot$
$\displaystyle\cdot$
$\displaystyle\left[w\left(c_{+}^{II},b_{+}^{I},t_{0}\right)+w\left(a_{+}^{II},c_{+}^{I},t_{0}\right)\right]\,+$
$\displaystyle+$ $\displaystyle w\left(a_{-}^{II}(t_{0})\to
a_{+}^{II}(t)\right)w\left(b_{+}^{I}(t_{0})\to b_{+}^{I}(t)\right)\cdot$
$\displaystyle\cdot$
$\displaystyle\left[w\left(c_{+}^{II},b_{+}^{I},t_{0}\right)+w\left(a_{-}^{II},c_{+}^{I},t_{0}\right)\right]\,+$
$\displaystyle+$ $\displaystyle w\left(a_{+}^{II}(t_{0})\to
a_{+}^{II}(t)\right)w\left(b_{-}^{I}(t_{0})\to b_{+}^{I}(t)\right)\cdot$
$\displaystyle\cdot$
$\displaystyle\left[w\left(c_{+}^{II},b_{-}^{I},t_{0}\right)+w\left(a_{+}^{II},c_{+}^{I},t_{0}\right)\right]\,+$
$\displaystyle+$ $\displaystyle w\left(a_{-}^{II}(t_{0})\to
a_{+}^{II}(t)\right)w\left(b_{-}^{I}(t_{0})\to b_{+}^{I}(t)\right)\cdot$
$\displaystyle\cdot$
$\displaystyle\left[w\left(c_{+}^{II},b_{-}^{I},t_{0}\right)+w\left(a_{-}^{II},c_{+}^{I},t_{0}\right)\right].$

Equation (V) can be rewritten in a more convenient form:

$\displaystyle\qquad\qquad\quad
w\left(a_{+}^{II}(t),\,b_{+}^{I}(t)\right)\leq$ $\displaystyle\leq$
$\displaystyle w\left(a_{+}^{II},c_{+}^{I},t_{0}\right)\cdot
w\left(a_{+}^{II}(t_{0})\to a_{+}^{II}(t)\right)\cdot$ $\displaystyle\cdot$
$\displaystyle\left[w\left(b_{+}^{I}(t_{0})\to
b_{+}^{I}(t)\right)+w\left(b_{-}^{I}(t_{0})\to b_{+}^{I}(t)\right)\right]+$
$\displaystyle+$ $\displaystyle w\left(a_{-}^{II},c_{+}^{I},t_{0}\right)\cdot
w\left(a_{-}^{II}(t_{0})\to a_{+}^{II}(t)\right)\cdot$ $\displaystyle\cdot$
$\displaystyle\left[w\left(b_{+}^{I}(t_{0})\to
b_{+}^{I}(t)\right)+w\left(b_{-}^{I}(t_{0})\to b_{+}^{I}(t)\right)\right]+$
$\displaystyle+$ $\displaystyle w\left(c_{+}^{II},b_{+}^{I},t_{0}\right)\cdot
w\left(b_{+}^{I}(t_{0})\to b_{+}^{I}(t)\right)\cdot$ $\displaystyle\cdot$
$\displaystyle\left[w\left(a_{+}^{II}(t_{0})\to
a_{+}^{II}(t)\right)+w\left(a_{-}^{II}(t_{0})\to a_{+}^{II}(t)\right)\right]+$
$\displaystyle+$ $\displaystyle w\left(c_{+}^{II},b_{-}^{I},t_{0}\right)\cdot
w\left(b_{-}^{I}(t_{0})\to b_{+}^{I}(t)\right)\cdot$ $\displaystyle\cdot$
$\displaystyle\left[w\left(a_{+}^{II}(t_{0})\to
a_{+}^{II}(t)\right)+w\left(a_{-}^{II}(t_{0})\to a_{+}^{II}(t)\right)\right].$

Inequality (V) is referred to as the time-dependent Wigner inequality for
three moments in time. This inequality was previously published in the work
Nikitin:2014yaa  without a derivation, and without references to the concept
of Classicality. The authors of Nikitin:2014yaa  deemed it obvious that
obtaining inequality (V) required only the concept of Local Realism and
’common sense’. Over time, the authors’ views evolved. It became apparent that
to demonstrate the Wigner inequalities (27) – (30) and to accurately define
the state space Nikitin:2014yaa ; PhysRevA.100.062314 , it was necessary to
enhance the concept of Local Realism with the principle of Classical
Measurement (which should not be confused with the principle of Classicality
of measurement, included in the concept of Classicality). Subsequently, it
became evident that even this was insufficient to prove (V). The Realism
Hypothesis Nikitin:2014yaa ; Nikitin:2015bca ; PhysRevA.95.052103 ;
PhysRevA.100.062314  emerged, which in this work has evolved into a more
formal concept of Classicality.

It is worth noting that to derive BCHSH inequalities Clauser:1969ny  under the
assumption of hidden parameters, the original form of the concept of Local
Realism is entirely sufficient. However, if one abandons the hidden
parameters, as done in work muynck1986 , and relies only on the existence of
joint probabilities of spectra of all observables characterizing the given
physical system, then to correctly derive the BCHSH inequalities, a principle
similar to the one discussed in this article, the principle of Classical
Measurement, is required.

One can also consider more measurement times $t_{2}\neq t_{1}\neq t_{0}$,
which were considered in the article PhysRevA.95.052103  again in the
assumption of the sufficiency of the Local Realism. The analysis of this type
of inequalities is similar to the one presented below.

##  VI Violations of the Time-Dependent Wigner Inequalities for Pure Systems

Let us briefly describe the results from Nikitin:2015bca . Consider the
inequalities (V) for the pair of meson $M$, and antimeson $\bar{M}$, which
were produced at $t_{0}=0$ in a pure (26) Bell state. We consider $K^{0}$,
$D^{0}$ и $B^{0}_{s}$–mesons. Let us consider flavor states, $CP$–parity, and
energy /lifetime states as dichotomous variables. Then make all possible
connections between these states, and the variables $a_{\pm}^{(i)}$,
$b_{\pm}^{(i)}$, and $c_{\pm}^{(i)}$ from the derived above Wigner
inequalities. These correspondences are represented in the Table 1.

If the quantum paradigm can still be described by classical physics, which we
used to derive Wigner inequalities, then (V) is never violated, as can be
shown, for example, in the systems of neutral pseudoscalar mesons. Let us show
that this is not true using probabilities from Nikitin:2014yaa ;
Nikitin:2015bca ; PhysRevA.95.052103 , and represent the inequality (V) as a
set of inequalities of the type:

$\displaystyle R_{N}\left(t,\,t_{0},\,\ldots\right)\,\geq\,1,$ (37)

where $N=\\{1,\,\ldots\,8\\}$ is the number of each inequality , which
correspond to $N$ in the Table 1, and the number of the function
$R_{N}(\ldots)$ from Nikitin:2015bca .

Obviously, if we consider $R_{N}(t=t_{0}=0,\,\ldots)$, we revert to the time-
independent Wigner inequalities. On the other hand, it is also possible to
increase the number of instants when the measurements are performed, however,
in the current article we would stop on $t$, and $t_{0}$.

If at least for one possible value of $N$ there is such $t$ that
$R_{N}\left(t,\,t_{0},\,\ldots\right)\,<\,1$, then inequality (V) is violated,
and as a consequence quantum world cannot be described by the Classicality
concrept representing the classicla approach.

For $K^{0}$-, and $D^{0}$\- mesons the following functions give violations of
(37): $R_{5,6}(\ldots)$. For $B^{0}_{s}$-mesons they are $R_{7,8}(\ldots)$.
Let us assume that $\frac{q}{p}=e^{i\,\zeta}$. Then
$R_{5}(\ldots)\,\equiv\,R_{6}(\ldots)$, and
$R_{7}(\ldots)\,\equiv\,R_{8}(\ldots)$. Here we consider $\zeta$ as they were
presented in Nikitin:2015bca , namely, $\zeta_{K}=-0.18^{\circ}$,
$\zeta_{D}=-10^{\circ}$ и $\zeta_{B_{s}}=185^{\circ}$.

(a) $K^{0}$-mesons

(b) $D^{0}$\- mesons

(c) $B_{s}$-mesons

Figure 1: Plots of the $R_{N}$ function for various mesons, initially in the
pure flavor state as defined in Equation 26, with $c\,t$ values measured in
millimeters. The phase ratios are specific to each meson type, indicated by
$q/p$. A dashed line on each plot represents where the function
${R}_{N}(t,t_{0}=0,\ldots)$ equals 1, with regions below this line indicating
strong violations of the Wigner inequalities as defined in Equation 37. (a)
For $K^{0}$ mesons, the function ${R}_{5,6}(t,t_{0}=0,\ldots)$ is plotted,
with $q/p=e^{i\,\zeta_{K^{0}}}$. (b) For $D^{0}$ mesons, the function
${R}_{5,6}(t,t_{0}=0,\ldots)$ is plotted, with $q/p=e^{i\,\zeta_{D^{0}}}$. (c)
For $B_{s}$ mesons, the function ${R}_{7,8}(t,t_{0}=0,\ldots)$ is plotted,
with $q/p=e^{i\,\zeta_{B_{s}}}$.

On the Fig.1(a) one can see the dependence of the function
$R_{5,6}(t,\,t_{0}=0,\,\ldots)$ from $ct$ measured in $mm$., where $c$–is the
speed of light. The inequality (37) is violated when
$R_{5,6}\left(t,\,t_{0}=0,\,\ldots\right)$ is below $1$. For convenience,
there is a dashes line at the level of $1$ on the plot.

It is interesting to mention that in the Fig.1(a), if one considers
$t=t_{0}=0$ it is possible to see that it is slightly below $1$, which is
exactly the result obtained in Section (II). In this case, the function
$R_{6}(t,\,t_{0}=0,\,\ldots)$ can be written as:

$R_{6}(t,\,t_{0}=0,\,\ldots)=1+|\varepsilon|^{2}-Re(\varepsilon)\approx
0.998\,\leq 1.$ (38)

This means that for the function $R_{6}(t,\,t_{0}=0,\,\ldots)$, inequality
(37) is only slightly violated, even for the pure $\left|\,\Psi^{-}\,\right>$
state, which is difficult to measure experimentally. However, the time-
dependent Wigner inequalities (27) can be violated more significantly. But
this violation is within the measurement error of the function
$R_{6}\left(t,\,t_{0}=0,\,\ldots\right)$, and as it was shown in the section
(II) disappears with admixture of noise Uchiyama:1996va ; Bertlmann:2001sk ;
Anna:2020ofp . If $ct>0$, then the inequality (37) is violated much stronger,
this result is in agreement with the work Nikitin:2015bca .

The analysys of the violation of the inequalities (37) for $D^{0}$, and
$B_{s}$ mesons is similar to that for $K^{0}$ mesons. The function
$R_{6}\left(t,\,t_{0}=0,\,\ldots\right)$ dependence on $ct$ for $D^{0}$\-
mesons, and $R_{8}\left(t,\,t_{0}=0,\,\ldots\right)$ for $B_{s}$-mesons are
represented on the Fig. 1(b), and Fig. 1(c) respectively.

In this section we showed that if a pair of neutral pseudoscalar mesons was
produced in the pure entangled state (26) at $t_{0}=0$, then there is a big
area of violation during the certain segment on the axis $ct$ further in time
where Wigner inequalities (V) are violated. Authors believe that the violation
is strong enough to be measured experimentally.

##  VII Violation of time-dependent WIgner inequalities for the Werner state

As discussed in Section II, there is noise in any experiment, for example, in
$B$-factories, where $B_{s}{\bar{B}_{s}}$ pairs are produced in a mixture of
the pure state (26)due to electron-positron collisions during the decay of
$\Upsilon(5S)\to B_{s}\bar{B_{s}}$, and from background processes, which can
arise fro example in the chain of decays $\Upsilon(5S)\to\left(B^{*}_{s}\to
B_{s}\gamma\right)\,\bar{B_{s}}$. In the hadron expariments (for example LHC)
a fraction of noise is usually much higher than on $B$-factories. Same is true
for the $K^{0}$– и $D^{0}$–mesons production on the $\phi$-factories, and
(super) С-tau factories respectively. As it was done for the time-independent
Wigner inequalities we consider the Werner state noise model and use the
following density matrix again: werner1989 :

$\displaystyle\hat{\rho}^{(W)}_{\Psi^{-}}(x)\,=\,x\,\left|\,\Psi^{-}\,\right>\left<\,\Psi^{-}\,\right|\,+\,\frac{1}{4}(1-x)\,I_{4x4},$
(39)

where $I_{4\times 4}$ is the identity $4\times 4$ matrix. The purity parameter
is $0\leq x\leq 1$. Therefore, for the Werner state, we should consider not
only $c\,t$ but also $x$ as a parameter. It is possible to express all Wigner
inequalities for the Werner state in the form:

$\displaystyle\tilde{R}_{N}\left(t,\,t_{0}=0,\,x,\,\ldots\right)\,\geq\,1.$
(40)

The number $N$ of the functions $\tilde{R}_{N}(\ldots)$ fully correlates with
the corresponding numbers of $R_{N}(\ldots)$ that were considered in the
previous Section VI. It is clear that

$\displaystyle\tilde{R}_{N}\left(t,\,t_{0}=0,\,x=1,\,\ldots\right)\,=\,R_{N}\left(t,\,t_{0}=0,\,\ldots\right),$
(41)

i.e., $R_{N}(\ldots)$ is a special case of $\tilde{R}_{N}(\ldots)$.

If for any number $N$ there are values of $t$ and $x$ such that
$\tilde{R}_{N}(\ldots)\,<\,1$, then inequalities (40) for noisy systems are
violated, and consequently, inequalities (V) are also violated what means that
is impossible to describe quantum phenomena using classical approach because
Classicality is not correct for quantum systems. It is intuitive trying to
find violations of the inequalities(40) for the same $N$, for which
inequalities (37) were violated.

The function $\tilde{R}_{N}(\ldots)$’s behavior depending on $t$ and $x$ is
illustrated in Figures 2, where it can be seen that as $x$ decreases, the
range of $t$ where inequalities are violated also decreases; however, this
decrease is not as rapid as it was for time-independent Wigner inequalities.
It is possible to observe violations experimentally for noise levels higher
than $30\%$, and for kaons, this ratio can reach even $50\%$. We can discuss
certain values of $x$ only in relation to a specific time $t$, optimally
chosen for each meson type. For less convenient measuring instants, the noise
ratio is notably worse.

(a) $K^{0}$-mesons

(b) $D^{0}$\- mesons

(c) $B_{s}$-mesons

Figure 2: Plots of the function $\tilde{R}_{N}(t,t_{0}=0,x,\ldots)$ for
various mesons, initially in a mixed flavor state as defined in Equation 39,
with values of $c\,t$ measured in millimeters, where $c$ is the speed of
light. Each meson type is characterized by a specific phase ratio $q/p$. A
contour line indicates where $\tilde{R}_{5,6}(t,t_{0}=0,x,\ldots)=1.0$,
corresponding to the reference in Figure 1. (a) Function
$\tilde{R}_{5,6}(t,t_{0}=0,x,\ldots)$ or $K^{0}$ mesons, with
$q/p=e^{i\,\zeta_{K}}$. (b) Function $\tilde{R}_{5,6}(t,t_{0}=0,x,\ldots)$ for
$D^{0}$ mesons, with $q/p=e^{i\,\zeta_{D}}$. (c) Function
$\tilde{R}_{7,8}(t,t_{0}=0,x,\ldots)$ for $B_{s}$ mesons, with
$q/p=e^{i\,\zeta_{B_{s}}}$.

Fig. 2(a), 2(b) depict the functions $\tilde{R}_{5,6}(t,x)(\ldots)$ for
$K^{0}$ and $D^{0}$ mesons over a range of ct values from $0$ to $400\,mm.$
and from $0$ to $40\,mm$., respectively. For $B_{s}$ mesons on the Fig. 2(c)
the $\tilde{R}_{7,8}(t,x)(\ldots)$ functions are plotted in the range of $ct$
values from $0$ to $25\,mm$. The color graphs illustrate the values of
$\tilde{R}_{5,6}(t,x)(\ldots)$ and $\tilde{R}_{7,8}(t,x)(\ldots)$ which
decrease as the purity parameter $x$ decreases from $1.0$ to $0.0$. There is a
line at $\tilde{R}_{N}(t,x)(\ldots)=1$ to better indicate the violation range.
The dark area above the curve highlights regions where Wigner inequalities are
violated, showing a significant area of violation.

The purity parameter for the meson factories is usually around $x\geq 0.5$ х.
It is possible to increase $x$ by the selection criteria. For the hadronic
experiments initially $x\to 0$, however, again using selection criteria it is
possible to form a set of events where the purity parameter is much higher.
However, this set is always open for the detection loophole eberhard1993 , and
contextual loophole baere1984-1 ; baere1984-2 . In addition, for any real
experiment it is necessary to exam the locality loophole Storz:2023jjx .
Moreover, it is obvious that the real noise model are more complicated than a
Werner state, therefore, it would require more detailed investigation of the
range of parameters where time-dependent Wigner inequalities can be violated
in each particular experiment.

##  VIII Conclusion

In this research, we have established several key findings with implications
for the field of quantum mechanics, particularly in the formulation and
testing of Wigner inequalities. Firstly, it has been demonstrated that for the
testing of time-independent Wigner inequalities, the noise level in
experiments described by Werner model should be less than $1\%$, which makes
testing these inequalities in high-energy physics impossible and necessitates
a new type of inequality.

It has been shown that to correctly derive the time-independent Wigner
inequality, it is necessary to supplement the concept of Local Realism with
the principle of Classical Measurement.

We have defined the concept of Classicality, which encompasses principles such
as Ontism, Epistemism, Consistency, Classical Measurement, and Independence.
The introduction of each principle has been justified, reinforcing
Classicality as a robust framework for testing the quantum paradigm.

Based on the Classicality framework, we have successfully derived time-
dependent Wigner inequalities for two instants, which allow us to explore
inequalities with multiple observables and multiple instants.

Our investigations also explored the potential violations of these
inequalities in systems of neutral pseudoscalar mesons— $K^{0}$, $D^{0}$ and
$B_{s}$ —particularly when an $M\bar{M}$ pair is initially produced either in
a pure flavor state or a mixed Werner state. This study demonstrates that
time-dependent Wigner inequalities are more noise robust than time-independent
ones and can be violated even with a noise contribution as high as $50\%$
within the Werner approximation. This finding is pivotal as it illustrates the
robustness of Wigner inequalities against significant experimental noise,
thereby guiding future experimental designs and the interpretation of outcomes
in environments with substantial noise.

This study also set the stage for future experimental investigations that
might further challenge or validate the foundational principles of quantum
mechanics through high-energy physics, where in principle some relativistic
effects can be incorporated into the described testing system.

###### Acknowledgements.

Previous works Nikitin:2014yaa ; Nikitin:2015bca ; PhysRevA.95.052103 ;
PhysRevA.100.062314 , and the article Nikitin:2009sr  about this topic were
created, and written together with our friend, and colleague K.S. Toms. This
is the first work written without him. We would like to dedicate this article
to the blessed memory of K.S. Toms.

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Table 1:  Table of the correspondence between the spectrum of dichotomous
variables $a^{(i)}$, $b^{(i)}$, $c^{(i)}$, from the inequality (V), and
physical characteristics $M/\bar{M}$, $M_{1}/M_{2}$, and $M_{L}/M_{H}$ of the
system of the neutral pseudoscalar mesons. Number $N$ in the first column is
the same as $N$, in Nikitin:2014yaa .
N | Correspondence of the spectum elements
---|---
1 |  $a_{+}\to M_{1}$; $b_{+}\to\bar{M}$; $c_{+}\to M_{H}$; $a_{-}\to M_{2}$; $b_{-}\to M$; $c_{-}\to M_{L}$
2 |  $a_{+}\to M_{1}$; $b_{+}\to M$; $c_{+}\to M_{H}$; $a_{-}\to M_{2}$; $b_{-}\to\bar{M}$; $c_{-}\to M_{L}$
3 |  $a_{+}\to M_{2}$; $b_{+}\to\bar{M}$; $c_{+}\to M_{H}$; $a_{-}\to M_{1}$; $b_{-}\to M$; $c_{-}\to M_{L}$
4 |  $a_{+}\to M_{2}$; $b_{+}\to M$; $c_{+}\to M_{H}$; $a_{-}\to M_{1}$; $b_{-}\to\bar{M}$; $c_{-}\to M_{L}$
5 |  $a_{+}\to M_{1}$; $b_{+}\to\bar{M}$; $c_{+}\to M_{L}$; $a_{-}\to M_{2}$; $b_{-}\to M$; $c_{-}\to M_{H}$
6 |  $a_{+}\to M_{1}$; $b_{+}\to M$; $c_{+}\to M_{L}$; $a_{-}\to M_{2}$; $b_{-}\to\bar{M}$; $c_{-}\to M_{H}$
7 |  $a_{+}\to M_{2}$; $b_{+}\to\bar{M}$; $c_{+}\to M_{L}$; $a_{-}\to M_{1}$; $b_{-}\to M$; $c_{-}\to M_{H}$
8 |  $a_{+}\to M_{2}$; $b_{+}\to M$; $c_{+}\to M_{L}$; $a_{-}\to M_{1}$; $b_{-}\to\bar{M}$; $c_{-}\to M_{H}$


# SSUP-HRI: Social Signaling in Urban Public
Human-Robot Interaction dataset

Fanjun Bu  Cornell University, Cornell TechNew YorkUSA<EMAIL_ADDRESS>and
Wendy Ju  Cornell University, Cornell TechNew YorkUSA<EMAIL_ADDRESS>
###### Abstract.

This paper introduces our dataset featuring human-robot interactions (HRI) in
urban public environments.111Dataset information is available at
https://github.com/FAR-Lab/SSUP-HRI. This dataset is rich with social signals
that we believe can be modeled to help understand naturalistic human-robot
interaction. Our dataset currently comprises approximately 15 hours of video
footage recorded from the robots’ perspectives, within which we annotated a
total of 274 observable interactions featuring a wide range of naturalistic
human-robot interactions. The data was collected by two mobile trash barrel
robots deployed in Astor Place, New York City, over the course of a week. We
invite the HRI community to access and utilize our dataset. To the best of our
knowledge, this is the first dataset showcasing robot deployments in a
complete public, non-controlled setting involving urban residents.

Dataset, human-robot interaction, wizard-of-oz, public interaction, field
experiment, urban spaces, behavioral elicitation, social robots

††copyright: rightsretained††conference: Workshop on Social Signal Modelling;
March 11, 2024; Boulder, CO, USA††booktitle: SS4HRI: Workshop on Social Signal
Modelling, March 11, 2024, Boulder, CO, USA††isbn: ††ccs: Human-centered
computing Collaborative and social computing systems and tools††ccs: Human-
centered computing Empirical studies in interaction design††ccs: Human-
centered computing Empirical studies in collaborative and social computing
Figure 1. Our dataset features hundreds of human-robot interactions in public
spaces from the robots’ perspective, which showcase rich social signals in a
variety of contexts.

##  1\. Introduction

While mobile robots are not yet a common feature in public spaces, the
emergence of autonomous agents such as self-driving vehicles and unmanned food
delivery robots is on the horizon. It will be important to understand how
everyday people who encounter these robots might interact with these robots;
as part of this, the social signals that they perform in and around the robots
will be important for the robots to pick up on and respond to. It is also
important to understand how people perceive the movement of the robot itself
as a form of social signaling. People inherently attribute social meaning to
movement, even of the most basic of shapes (Heider and Simmel, 1944; Smith and
Gasser, 2005). Robots, even those not designed for verbal or social
interaction, need to be adept in understanding the signals people make at
them, and making sure they respond appropriately in kind. This understanding
is crucial to ensure they navigate public spaces safely and respectfully,
avoiding causing inconvenience or offense to passersby. Moreover, it is
imperative for mobile robots to align their movements with the existing
dynamics, norms, and expectations of the public spaces they inhabit.

In this paper, we present a unique dataset that captures interactions between
people and two mobile trash barrel robots in a bustling public plaza in New
York City. These robots are limited to motion as their sole mode of
interaction, and are controlled in a Wizard-Of-Oz (WoZ) method to make sure
their movements align with human expectations in the situated context. Our
dataset will contribute to the social science understanding of public human-
robot interactions and assist in the modeling of social signaling in human-
robot interactions in the wild.

##  2\. Related Works

###  2.1. HRI in Public Spaces

Previous research on human-robot interactions in semi-public settings like
hotels and hospitals has provided specific insights related to particular
sites and tasks (Pan et al., 2013, 2015; Díaz-Boladeras et al., 2015; Onyeulo
and Gandhi, 2020; Anvari and Wurdemann, 2020). Broader studies have
concentrated on modeling and predicting human movement to ensure robots can
navigate without causing obstructions (Taylor et al., 2020; Adeli et al.,
2021; Coscia et al., 2018; Alahi et al., 2017). Furthermore, significant work
in social contexts has delved into proxemic behaviors, developing models to
determine the ideal distance between robots and humans during interactions to
maintain comfort (Mumm and Mutlu, 2011; Henkel et al., 2012b, a). However,
models for cooperative interaction in public also need to account for the
social factors that influence whether people want to interact with a robot or
not, such as Koile et al. (2003)’s work on ”activity zones” in homes, which
highlights spatial influences on interactions. Public interaction is similar,
but far more complex, with subtle factors like bench arrangement and sunlight
direction significantly influencing behavior patterns and social dynamics
(Whyte, 1980).

###  2.2. Datasets for HRI

Datasets focusing on physical interactions between humans and robots are
scarce because collecting real-time robotics data is inefficient.
Consequently, many datasets are centered around human-human interaction data,
with the aim for researchers to use the data to create algorithms applicable
to robots (Kshirsagar et al., 2023; Carfì et al., 2019; Ondras et al., 2023).
Joo et al. (2019) released a dataset of nonverbal signals among people in a
triadic social interaction scenario, with the hope of teaching machines
nonverbal communication skills. In efforts to incorporate robots into data
collection, the Wizard-of-Oz method is often employed, wherein robots are
remotely controlled by unseen researchers to simulate authentic behaviors
(Horvitz et al., 1998). For example, Tian et al. (2023) collected a human-
robot collaboration dataset featuring handovers with a researcher
teleoperating the robot. The MHHRI dataset features triadic interactions
between two participants and a robot to study personality and engagement
(Celiktutan et al., 2017). The robot’s dialogue and expressions are also
managed through Wizard-of-Oz. Given the complex and context-sensitive nature
of human-robot interactions, the Wizard-of-Oz approach is frequently utilized
as a reliable method for capturing genuine interaction data.

Different from the aforementioned datasets, which were all collected in
controlled lab settings, our dataset features social signaling in a
naturalistic environment without explicitly engineered tasks. Here, the
signals from passersby are unprompted. Though the range of activities and
factors in the dataset are noisier, models built from the data might be
anticipated to be more ecologically valid.

##  3\. Data Collection

In this section, we detailed our study deployment and data collection process.
All deployments took place in a public space in New York City and complied
with local regulations. Our study was approved by Cornell Tech’s IRB protocol
#IRB014571.

###  3.1. Robot Deployments

We deployed two trash barrel robots, one for recycling, and the other for
landfill, in Astor Place, New York City. The robots roved the public square
and collected trash from people passing by or sitting at the chairs and tables
set out in the area. We operated the robots for 2 hours in the early
afternoons for 5 days when weather permitted. The hours coincided with high
traffic and high occupancy moments in the day.

Methodologically, we used a semi-supervised Wizard-of-Oz protocol (Horvitz et
al., 1998; Zamfirescu-Pereira et al., 2021). Trained researchers piloted the
robot, with the goal of engaging the users as a service robot would. The
Wizard-of-Oz method allows us to have the robots respond to social signals in
ways that the users were likely to expect; the decisions of the robot
operators were as much part of the study as the reactions of the public. For
safety reasons, the operators always maintained a line of sight to the robots
they were controlling; however, they obscured their role by hiding the
controllers or wearing sunglasses so it was less obvious where they were
looking. Following people’s interactions with the robot, we had another
researcher approach to interview willing individuals for their feedback and
thoughts on the interaction.

During the deployment, the wizards were encouraged to keep two robots
physically close to each other. They were also encouraged to communicate with
each other and coordinate the robots’ motions. In rare cases where one robot
got stuck, the other robot was expected to ”bump” the robot in need, instead
of direct human intervention.

###  3.2. Collection Site

To make sure we were collecting naturalistic social signals, we looked for
public spaces that were intensively used by residents daily, where the use
case of having a mobile robot in the space seemed plausible. Other site
selection criteria included foot traffic, availability of public seating,
business around the plaza, and ease of research activity operation.

This deployment eventually took place in Astor Place, a public plaza located
in the heart of Manhattan, New York City. This public plaza was a trapezium-
shaped traffic island with multiple tables and chairs laid out for people to
socialize. Originally we intended the robots to be robot chairs and tables,
but in our meetings with the business improvement district that operates the
plaza, it was determined that robot trash barrels, like those previously
deployed by Yang et al. at Stanford (Yang et al., 2015), would fit in better
and gain more interaction. The arrangements of the seating were not fixed day
to day. As a result, the wizards sat at different locations for each
deployment.

Figure 2. A bird-eye view of our deployment space.

###  3.3. Recording Apparatus

Each trash barrel robot is a simple mobile platform. Each robot primarily
interacts with passersby through its movement. This minimalism helps focus our
study and subsequent model on presence and movement, without the confounds
that other sounds or visual elements would bring. On the other hand, such
design choice also constrains the range of people’s signaling towards the
robots, which also largely depends on motion and gesture instead of spoken
instructions.

To capture the social signals around them, each trash barrel robot is equipped
with an Insta360 One R
camera222https://www.insta360.com/product/insta360-oner_twin-edition with
built-in microphones and an external IMU to record the surrounding and robot
motions. The robots are powered by recycled hoverboards; the robust metal
chassis provides the structural integrity needed to navigate outdoor urban
terrain. Computationally, each robot currently runs on a Raspberry Pi 4
running the Ubuntu 20.04 server and Robot Operating System (ROS 1 Noetic)
(Quigley et al., 2009).

Figure 3. Exploded diagram of robot.

###  3.4. Data Annotation

To isolate the social signals in the dataset, two researchers from our
research group annotated the dataset and labeled every explicit or implicit
interaction moment. Between two of the researchers, one of them was involved
in the data collection process, where he was on-site for each deployment. The
other researcher was not involved during the collection process. The two
researchers labeled the dataset individually and then consolidated a
spreadsheet of all interaction moments. The spreadsheet will also be publicly
available. Based on our interests, our annotation focused on explicit
interactions between plaza users and trash barrel robots. For every
interaction, each annotator labeled time, which robot was involved in the
interaction, and a summary of the scenario.

##  4\. Dataset Overview

Overall, we observed 274 interactions between the plaza users and trash barrel
robots, ranging from normal disposal of trash to people helping the robot to
get unstuck. Each interaction consists of unique exchanges of social signals
between the users and the robots.

Within the 274 observed interactions, 118 interactions feature people and
trash barrel robots negotiated to dispose of trash, for which we provide more
detailed annotations focusing on behavioral signaling. Among trash disposal
interactions, people interacted with both robots equally (48 interactions with
the recycling robot, 54 interactions with the landfill robot, and 16
interactions with both). Over half (N = 68) of the interactions feature
negotiations between a single user and the robots, while party sizes range
from 1 to 5 people. Even though people use different signals to initiate and
terminate interactions with the robots, we are able to categorize them into
the categories shown in table 1. Lastly, 18 individuals or groups try to have
a conversation with the robot during the interaction.

Initiation | Signal Type | Count
---|---|---
Passive: Robot approached | 23
Waving without objects | 7
Waving/holding objects | 47
Walking towards robots | 37
Standing up | 4
Termination | Walking away | 47
Verbal | 8
Turning around | 4
Sitting down | 6
Hand Gesture | 16
Passive: Robots drove away | 11
Looking away | 25
Table 1. Different signals people use to initiate and terminate interactions.

##  5\. Use Cases

We anticipate our dataset to significantly impact social science
understandings of human-robot interaction in public spaces and aid in
advancing social intelligence in machine learning models. Methodologically, we
believe our dataset enables bridging qualitative and quantitative practices in
the field of human-robot interaction. Pragmatically, our dataset can help
establish guidelines to improve the quality of interaction provided by social
robots in public.

###  5.1. Qualitative Analysis

Ethnographic researchers may find our video clips valuable and can discover
insights generalizable to other applications. Our dataset features
naturalistic exchanges of social signals in the context of social navigation
and service robotics. Furthermore, interview transcripts can be used for
thematic analysis to understand the mental models and thought processes people
go through when encountering robots in uncontrolled environments and deciding
ways of signaling. Furthermore, the narration and reflection of the encounter
depict how users interpret social signals presented by the robots. Our dataset
touches upon topics including, but not limited to, people’s sensemaking
processes, assumptions, expectations, interaction sequences, and cultural
influences on human-robot interaction.

Figure 4. LLM generated captions with inputs from our field study.

###  5.2. Quantitative Analysis

Our dataset also opens room for researchers to perform quantitative analysis
to provide numerical support for qualitative analysis. From the data,
statistics on activities such as people’s initiation and termination actions,
count party sizes and physical formation, characterize stances (sitting,
standing, or walking), and interaction duration are easily observable.
Additionally, researchers can profile the robots’ actions, based on metrics
like approach angles, speed, and distance covered. Our dataset also offers a
new playground to analyze existing theoretical models, such as the social
force model and F-Formation (Kendon, 1990; Cristani et al., 2011; Helbing and
Molnar, 1995).

###  5.3. Machine Learning Test Bed

While our dataset volume is not yet large enough to train a machine learning
algorithm from scratch, it provides a test bed for computer scientists to test
out their algorithms with realistic real-world data. It is possible both to
pick out the social signals generated by passersby who want a robot to
approach or stay away and also to model what motions and signals the robots
are giving off that people are responding to. Furthermore, given the recent
progress in the development of foundational models, we are optimistic that our
dataset will play a role in establishing a foundation model for interactive
mobile robots navigating social environments in urban public spaces.

##  6\. Future Direction

This project marks the initial phase of establishing a dataset focused on
social interactions between humans and mobile robots within urban
environments. Our objective is to extend the deployment of these robots to
numerous public plazas, with the ultimate aim of covering all five boroughs of
New York City. We intend to expand and enrich our dataset as we continually
gather new data.

##  7\. Conclusion

We present SSUP-HRI, a dataset that features human-robot interaction in public
social spaces. The dataset is exploratory in nature, where both the users and
the wizards controlling the robots are trying to decipher the social signals
presented by each other. Since the data were collected in unconstrained public
space, we observed parties of various sizes negotiated with the robots for
various purposes under different contexts. Therefore, we believe our dataset
will be beneficial to both researchers and practitioners in the understanding
of social signals in the context of human-robot interaction in urban spaces.

###### Acknowledgements.

We thank the Village Alliance for permission to run the study at Astor Place.
This work was supported by research funding from Tata Consulting Services.
This study would not have been possible without the wizards behind the scenes:
Rei Lee, Saki Suzuki, Ricardo Gonzalez, Stacey Li, Maria Teresa Parreira, and
Jorge Pardo Gaytan.

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