# Bell Nonlocality and the Reality of Quantum Wavefunction

Anandamay Das Bhowmik  Physics and Applied Mathematics Unit, Indian
Statistical Institute, 203 BT Road, Kolkata, India.  Preeti Parashar  Physics
and Applied Mathematics Unit, Indian Statistical Institute, 203 BT Road,
Kolkata, India.  Manik Banik  School of Physics, IISER Thiruvananthapuram,
Vithura, Kerala 695551, India.

###### Abstract

Status of quantum wavefunction is one of the most debated issues in quantum
foundations – whether it corresponds directly to the reality or just
represents knowledge or information about some aspect of reality. In this
letter we propose a $\psi$-ontology theorem that excludes a class of
ontological explanations where quantum wavefunction is treated as mere
information. Our result, unlike the acclaimed Pusey-Barrett-Rudolph’s theorem,
does not presume the absence of holistic ontological properties for product
quantum preparations. At the core of our derivation we utilize the seminal no-
go result by John S. Bell that rules out any local realistic world view for
quantum theory. We show that the observed phenomenon of quantum nonlocality
cannot be incorporated in a class of $\psi$-epistemic models. Using the well
known Clauser-Horne-Shimony-Holt inequality we obtain a threshold bound on the
degree of epistemicity above which the ontological models are not compatible
with quantum statistics.

##  I Introduction

What does a quantum wavefunction stand for? Does it represent the state of
reality of the physical system ($\psi$-ontic doctrine) or it merely does
provide information about the system ($\psi$-epistemic doctrine)? This
question is at the core of quantum foundational debate ever since the advent
of the theory Bacciagaluppi09 . A mathematically precise formulation of this
question, within a broad class of realist approaches to quantum theory, can be
made in the ontological model framework of Harrigan & Spekkens Harrigan09
(see also Spekkens05 ; Harrigan07 ). Epistemicity, in this framework, is
defined (as well as quantified) through the amount of overlap between
probability distributions over the ontic states resulting from different
quantum preparations.

In a fascinating development, Pusey, Barrett, and Rudolph (PBR) have shown
that $\psi$-epistemic interpretation contradicts prediction of quantum theory
in any model where independently prepared systems have independent physical
states (named as preparation independence (PI) assumption) Pusey12 . This
result drew attention of quantum foundations community and within few days
several researchers reported similar theorems Hardy13 ; Colbeck12 ; Colbeck17
; Patra13 , commonly called as $\psi$-ontology theorems Leifer14 , derived
under different assumptions. Subsequently, several criticisms were raised
regarding the assumptions used in those $\psi$-ontology theorems
Schlosshauer12 ; Emerson13 ; Ballentine14 ; Schlosshauer14  (see also Leifer14
). In particular, the authors in Schlosshauer14  have shown that the physical
rationale for composition principles such as PI overreaches and thus places
the no-go theorem put forward by PBR into jeopardy.

Interestingly, Owen Maroney came up with a new kind of $\psi$-ontology theorem
that uses no compositional assumption and rules out a class of ontological
models with certain degree of epistemicity Maroney12 ; Maroney12(1) .
Subsequently, several other results were obtained excluding $\psi$-epistemic
models with increasingly lower degree of epistemicity and consequently
imposing higher degree of onticity on quantum wavefunction Leifer13 ;
Barrett14 ; Leifer14(1) ; Branciard14 ; Ringbauer15 . However, all these
theorems apply to Hilbert spaces of dimension strictly greater than two.

In this work we derive a new $\psi$-ontology theorem that excludes the
maximally $\psi$-epistemic model as well as non-maximal ones with certain
degree of epistemicity. Importantly our theorem does not assume the ontic
composition principles of PI. In fact it does not presuppose that ontic state
space for quantum product preparations should be Cartesian product of their
individual ontic state spaces only. In other words, we consider that the two
or more quantum systems prepared even in product state can possess holistic
ontic features accessible only through global measurements, which broadens the
scope of our result over PBR’s theorem. Furthermore, unlike the
$\psi$-ontology theorems reported in Refs.Maroney12 ; Maroney12(1) ; Leifer13
; Barrett14 ; Leifer14(1) ; Branciard14 ; Ringbauer15 , the present theorem
applies to qubit Hilbert space too. Quite importantly, our result demonstrates
an interesting connection between degree of epistemicity and Bell nonlocality
Bell64 ; Bell66 ; Mermin93 ; Brunner14 . While it was already recognized that
$\psi$-complete and $\psi$-ontic models for quantum theory are inconsistent
with the concept of locality Bacciagaluppi09 ; Harrigan09 , our result
establishes the fact in the reverse direction. It shows that the phenomenon of
Bell nonlocality prefers $\psi$-ontic interpretation for quantum wavefunction
as $\psi$-epistemicity models having epistemicity above a threshold degree
cannot incorporate the observed quantum nonlocality.

##  II Framework

We first recall the ontological model framework as developed in Harrigan09
(see also Harrigan07 ; Spekkens05 ; Leifer14 ; Ballentine14 ). Such a model
consists of a space $\Lambda$ of ontic states that completely identify the
possible physical properties of a system. A quantum system prepared in the
state $\ket{\psi}$ is associated with a probability distribution
$\mu(\lambda|\psi)$ over $\Lambda$, where each realization of the preparation
$\ket{\psi}$ results in an ontic state $\lambda\in\Lambda$ sampled with
probability measure $\mu(\lambda|\psi)$. The probability distribution of the
resulting ontic states is called the epistemic state associated with
$\ket{\psi}$ and
$\mbox{Supp}[\mu(\lambda|\ket{\psi})]\equiv\Lambda_{\psi}:=\\{\lambda\in\Lambda\leavevmode\nobreak\
|\leavevmode\nobreak\ \mu(\lambda|\psi)>0\\}$ is called the ontic support of
$\ket{\psi}$. For all $\ket{\psi}$, we must have
$\int_{\Lambda_{\psi}}d\lambda\mu(\lambda|\psi)=1$. Whenever an observable $M$
is measured on a system, the possible outcomes are the eigenvalues $\phi_{k}$
with the associated eigenvectors $\ket{\phi_{k}}$, i.e.,
$M=\sum_{k}\phi_{k}\ket{\phi_{k}}\bra{\phi_{k}}$. Here we restrict our
attention to rank-1 projective measurements only, although quantum theory
allows more general measurement process described by positive operator valued
measure (POVM). Given a system in the ontic state $\lambda$, the probability
of obtaining the $k^{th}$ outcome is given by an response function
$\xi(\phi_{k}|\lambda,M)\in[0,1]$. A generic ontological model keeps open the
possibility for this outcome response to be contextual Spekkens05 ; whenever
measurement context is not important we will denote the response function
simply as $\xi(\phi_{k}|\lambda)$. An outcome deterministic (OD) ontological
model demands $\xi(\phi_{k}|\lambda,M)\in\\{0,1\\}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ k,\lambda,M$. Denoting
$\mbox{Supp}[\xi(\psi|\lambda)]:=\\{\lambda\in\Lambda\leavevmode\nobreak\
|\leavevmode\nobreak\ \xi(\psi|\lambda)>0\\}$ and
$\mbox{Core}[\xi(\psi|\lambda)]:=\\{\lambda\in\Lambda\leavevmode\nobreak\
|\leavevmode\nobreak\ \xi(\psi|\lambda)=1\\}$, the following set inclusion
relations are immediate:

$\Lambda_{\psi}\subseteq\mbox{Core}[\xi(\psi|\lambda)]\subseteq\mbox{Supp}[\xi(\psi|\lambda)].$

An OD model satisfies
$\mbox{Core}[\xi(\psi|\lambda)]=\mbox{Supp}[\xi(\psi|\lambda)]$, whereas a
model with $\Lambda_{\psi}=\mbox{Core}[\xi(\psi|\lambda)]$ is termed as
reciprocal Ballentine14 . Interestingly, the author in Ballentine14  have also
established that

$Maximally\leavevmode\nobreak\ \psi-epistemic\leavevmode\nobreak\
\Leftrightarrow\leavevmode\nobreak\ OD\leavevmode\nobreak\
\wedge\leavevmode\nobreak\ Reciprocal.$ (1)

An operational transformation procedures $T$ at ontological level corresponds
to a transition matrix $\Gamma_{T}(\lambda,\tilde{\lambda})$ denoting the
probability density for a transition from the ontic state $\lambda$ to the
ontic state $\tilde{\lambda}$. In prepare and measure scenario, the
reproducibility of the Born rule at operational level demands
$\int_{\Lambda}d\lambda\xi(\phi_{k}|\lambda,M)\mu(\lambda|\psi)=|\langle\phi_{k}|\psi\rangle|^{2}:=\mbox{Pr}(\phi_{k}|\psi)$.

##  III Degree of epistemicity

In a maximally $\psi$-epistemic model the quantum overlap
$|\langle\psi|\phi\rangle|^{2}$ for any two state vectors $\ket{\psi}$ and
$\ket{\phi}$ is completely accounted for by the overlap between the
corresponding epistemic distributions $\mu(\lambda|\psi)$ and
$\mu(\lambda|\phi)$, i.e.,
$\int_{\Lambda_{\phi}}\mu(\lambda|\psi)d\lambda=|\langle\psi|\phi\rangle|^{2}$
Maroney12 . Maroney’s theorem Maroney12  and the subsequent results
Maroney12(1) ; Leifer13 ; Barrett14 ; Leifer14(1) ; Branciard14 ; Ringbauer15
exclude maximally $\psi$-epistemic model and a class of ontological models
with increasingly lower degree of epistemicity. To quantify the degree of
epistemicity of an ontological model, please note
that,$\int_{\Lambda_{\phi}}d\lambda\mu(\lambda|\psi)=\int_{\Lambda_{\phi}}d\lambda\xi(\phi|\lambda)\mu(\lambda|\psi)\leq\int_{\Lambda}d\lambda\xi(\phi|\lambda)\mu(\lambda|\psi)=|\langle\phi|\psi\rangle|^{2}$.
The first equality is due to the fact that
$\Lambda_{\phi}\subseteq\mbox{Core}[\xi(\phi|\lambda)]$. One can express the
above inequality as a equality of the following form

$\int_{\Lambda_{\phi}}d\lambda\mu(\lambda|\psi)=\Omega(\phi,\psi)\leavevmode\nobreak\
|\langle\phi|\psi\rangle|^{2},$ (2)

where $\Omega(\phi,\psi)$ captures degree of epistemicity of a model and for
any pair of non-orthogonal quantum states $\Omega(\phi,\psi)\in[0,1]$. For a
maximally $\psi$-ontic model $\Omega(\phi,\psi)=0$ for all non-orthogonal
pairs of state, while $\Omega(\phi,\psi)=1$ for all such pairs in a maximally
$\psi$-ontic model (see Maroney12  for more elaboration). Larger the departure
of $\Omega(\phi,\psi)$ from its maximum possible value more the model is
$\psi$-ontic and consequently less $\psi$-epistemic it is.

##  IV Ontic composition

The discussions so far consider ontological models for a single system only.
More involving situations arise for composite systems. Consider System-$A$ and
System-$B$ with ontic state space $\Lambda^{A}$ and $\Lambda^{B}$,
respectively. For a composite quantum system prepared in a state $\rho_{AB}$,
a naive classical thinking suggests the joint ontic state $\lambda_{joint}$ to
be in $\Lambda_{A}\times\Lambda_{B}$, i.e.,
$\Lambda_{\rho_{AB}}\subseteq\Lambda^{A}\times\Lambda^{B}$. However, lesson
from the seminal Bell’s theorem indicates towards a more intricate description
of the ontic state for composite system. Violation of any local realistic
inequality by some joint quantum preparation $\rho_{AB}$ necessitates some
‘nonlocal’ ontic state space $\Lambda^{NL}$. To say more precisely, this
nonlocal variable captures the essence of nonlocal correlation in the sense of
Bell Bell64 ; Bell66 . At this point it should be reminded that these nonlocal
correlations are perfectly compatible with no-signaling principle that
prohibits instantaneous transfer of information. Therefore, we can say,
$\Lambda_{\rho_{AB}}\subseteq(\Lambda^{A}\times\Lambda^{B})\cup\Lambda^{NL}$,
where $\Lambda^{NL}$ accounts for Bell type local realistic inequality
violation. Importantly, all product states are Bell local (same is the case
for separable state, and local entangled states Werner89 ; Barrett02 ;
Bowles15 ). But, does it assert that such a product quantum preparation should
not contain any holistic ontic feature, i.e their ontic support ought to be a
subset of $\Lambda^{A}\times\Lambda^{B}$? PBR in their derivation have
considered such an assumption. In fact their PI assumption is even restrictive
as it explicitly spells
$\mu(\lambda_{A},\lambda_{B}|\psi_{A}\otimes\psi_{B})=\mu(\lambda_{A}|\psi_{A})\mu(\lambda_{B}|\psi_{B})$
Pusey12 . However, such an assumption is conservative as it considers (local)
reality of the product measurements only. It is possible that a composite
system even prepared in product state contains properties that are accessible
through global measurements only, e.g., Bell basis measurement. More
dramatically, the phenomena of ‘nonlocality without entanglement’ indeed
indicates towards such a situation even without involving any entanglement in
the measurement basis Bennett99 ; Halder19 ; Bhattacharya20 . Thus for a
bipartite product (also for separable and local entangled) quantum state
$\rho_{AB}$ the ontic support should be considered as
$\Lambda_{\rho_{AB}}\subseteq\Lambda^{A}\times\Lambda^{B}\times\Lambda^{G}$,
where $\Lambda^{G}$ carries strictly relational holistic information about the
two systems and remains hidden under local measurements, i.e. presumed not to
be accessible by local measurements. Here we make a clear distinction between
$\Lambda^{G}$ and $\Lambda^{NL}$. Both are non-classical features of
ontological state space for bipartite quantum system. But they correspond to
two completely distinct operational non-classicality of quantum theory.
$\Lambda^{NL}$ contains variables that are nonlocal strictly in the sense of
Bell inequality violation, which is revealed through local measurements
performed on spatially separated subsystems of the composite system - no
global measurement is involved or required in this case. This is clearly not
the case for ontic elements belonging to $\Lambda^{G}$, that accounts for the
global properties corresponding to non-classical joint measurements allowed in
quantum theory due to its richer bipartite effect space structure containing
entangled effects in addition to product effects. These global measurements
with entangled effects correspond to holistic properties or observables of the
composite system whose outcomes can never be realised by locally measuring the
individual subsystems. At this point one may consider the possibility that the
product quantum states have ontic support within $\Lambda^{NL}$ in such a way
the ‘nonlocal’ effect gets averaged out in the operational statistics.
However, our following proposition (see Supple  for the proof) puts a no-go on
such assertion.

######  Proposition 1.

In a maximally $\psi$-epistemic model (more generally in any outcome
deterministic ontological model) quantum product preparations do not possess
any nonlocal ontic state.

At this point we leave this question open for further investigation weather
the proposition can be extended for any $\psi$-epistemic model. However, for
other $\psi$-epistemic models (apart from those considered in Proposition 1)
nonexistence of nonlocal ontic state in the support of quantum product
preparation can be justified from the principle of Occam’s razor, which
researchers in quantum foundation have applied in different context
Ionicioiu11 . For an entangled state $\rho_{AB}$ exhibiting Bell nonlocality
the ontic description will be
$\Lambda_{\rho_{AB}}\subseteq(\Lambda^{A}\times\Lambda^{B}\times\Lambda^{G})\cup\Lambda^{NL}$
with $\Lambda_{\rho_{AB}}\cap\Lambda^{NL}\neq\emptyset$. The Cartesian product
structure assumed above is not necessary for our argument but it provides a
simple way to present the idea.

##  V A new $\psi$-ontology theorem

We are now in a position to prove our $\psi$-ontology theorem. To this aim we
consider a quantum copying machine $\mathbb{M}$ that perfectly copies the
states $\ket{0}$ and $\ket{1}$. The pioneering ‘no-cloning’ theorem does not
prohibit existence of such a machine as the coping states are mutually
orthogonal Wootters82 . The action of $\mathbb{M}$ can be described by a
unitary evolution $U_{\mathbb{M}}$ satisfying
$U_{\mathbb{M}}\ket{i}\ket{r}=\ket{i}\ket{i}$, where $\ket{r}$ is some fixed
reference state and $i\in\\{0,1\\}$. The machine has two input ports, one fed
with particles $R$ prepared in some reference state $\ket{r}$ and the other
fed with system $S$ prepared in state $\ket{0}$ or $\ket{1}$.

The action of this copying machine is worth analyzing in the ontological
picture. Whenever $S$ is prepared in the state $\ket{i}$, a composite ontic
state $\lambda_{joint}=(\lambda^{S},\lambda^{R},\lambda^{G})$ is fed into the
machine, where
$\lambda^{S}\in\Lambda_{i}\subseteq\Lambda^{S},\leavevmode\nobreak\
\lambda^{R}\in\Lambda_{r}\subseteq\Lambda^{R}$, and
$\lambda^{G}\in\Lambda^{G}$, i.e.
$\Lambda_{ir}:=\\{\mu(\lambda_{joint}|\ket{ir})>0\\}\subseteq\Lambda^{S}\times\Lambda^{R}\times\Lambda^{G}$.
After the successful completion of cloning, the machine yields the outcome
$\ket{ii}$, i.e., it yields an ontic state belonging in
$\Lambda_{ii}\subseteq\Lambda^{S}\times\Lambda^{R}\times\Lambda^{G}$. Let us
now feed the above copying machine with the system state prepared in
$\ket{+}=(\ket{0}+\ket{1})/\sqrt{2}$. At the ontological level the machine
receives some ontic state
$\lambda_{joint}\in\Lambda^{S}\times\Lambda^{R}\times\Lambda^{G}$ sampled with
probability distribution
$\mu(\lambda_{joint}|\ket{+r})=\mu(\lambda^{S},\lambda^{R},\lambda^{G}|\ket{+r})$.
Let us consider a maximally $\psi$-epistemic model underlying quantum theory.
This will imply
$\Lambda_{+}=(\Lambda_{0}\cap\Lambda_{+})\cup(\Lambda_{1}\cap\Lambda_{+})$ and
$\Lambda_{+r}=(\Lambda_{0r}\cap\Lambda_{+r})\cup(\Lambda_{1r}\cap\Lambda_{+r})\subseteq\Lambda_{0r}\cup\Lambda_{1r}$.
Since the ontic state obtained after the machine’s action only depends on the
input ontic state fed into the machine, therefore
$\mbox{Supp}[\mu(\lambda_{joint}|U_{\mathbb{M}}[\ket{+r}])]\subseteq\mbox{Supp}[\mu(\lambda_{joint}|U_{\mathbb{M}}[\ket{0r}])]\cup\mbox{Supp}[\mu(\lambda_{joint}|U_{\mathbb{M}}[\ket{+r}])]\equiv\Lambda_{00}\cup\Lambda_{11}$.
Since a product preparation can’t have any ‘nonlocal’ ontic reality, i.e.
$\Lambda_{00}\cap\Lambda^{NL}=\emptyset=\Lambda_{11}\cap\Lambda^{NL}$ (see
Proposition 1), therefore it, in turn, implies
$\mbox{Supp}[\mu(\lambda_{joint}|U_{\mathbb{M}}[\ket{+r}])]\cap\Lambda^{NL}=\emptyset$.

But the above conclusion is in direct contradiction with predictions of
quantum theory. Due to linearity of the machine’s action one will obtain the
output $\ket{\phi^{+}}=(\ket{0}\ket{0}+\ket{1}\ket{1})/\sqrt{2}$ whenever the
machine $\mathbb{M}$ is supposed to copy the state $\ket{+}$. Being the
maximally entangled state it exhibits maximum violation of Clauser-Horne-
Shimony-Holt (CHSH) inequality Clauser69  for suitable measurement choice on
its local parts. Hence the entire ontic support of the output pairs cannot be
contained within $\Lambda^{S}\times\Lambda^{R}\times\Lambda^{G}$; in other
words $\Lambda_{\phi^{+}}\cap\Lambda^{NL}\neq\emptyset$. At this point it is
important to note that, like PBR we have neither considered ontic state space
for a product preparation to be Cartesian product of their individual state
space nor we presume the strong assumption of ‘preparation independence’. We
have not also invoked the assumption of ‘Local independence’, i.e. $\int
d\lambda^{G}\mu(\lambda^{S},\lambda^{R},\lambda^{G}|\ket{ir})=\mu(\lambda^{S}|i)\mu(\lambda^{R}|r)$
as considered in Emerson13  and the assumption of ‘Ontic indifference’ used in
Hardy13 .

A similar proof runs if we feed any state
$\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$ into the machine $\mathbb{M}$ instead
of the state $\ket{+}$, where $\mathbb{C}\ni\alpha,\beta\neq 0$ and
$|\alpha|^{2}+|\beta|^{2}=1$. In a maximally $\psi$-epistemic theory the ontic
support $\Lambda_{\psi}$ entirely gets shared between $\Lambda_{0}$ and
$\Lambda_{1}$. Whereas for $\ket{+}$ these two shares are equal, for
$\ket{\psi}$ the shares are proportional to the quantum overlaps. Similar
argument in this case implies
$\mbox{Supp}[\mu(\lambda_{joint}|U_{\mathbb{M}}[\ket{\psi
r}])]\cap\Lambda^{NL}=\emptyset$. But this is again in contradiction with
quantum prediction as the machine’s action on $\ket{\psi r}$ yields the state
$\alpha\ket{00}+\beta\ket{11}$ which is known to be Bell nonlocal Gisin91 .

Our result thus establishes that maximally $\psi$-epistemic model cannot
account for the Bell nonlocal correlations. In the ‘orthodox’ interpretation
of quantum theory, the wavefunction $\psi$ alone provides the complete
description of reality which itself can be considered as $\psi$-complete
ontological model. Researchers have already acknowledged that Einstein had
shown incompatibility of $\psi$-complete model with locality through a simple
argument at the Solvay conference Bacciagaluppi09 . In Einstein’s own words
Schilpp70 :

… One arrives at very implausible theoretical conceptions, if one attempts to
maintain the thesis that the statistical quantum theory is in principle
capable of producing a complete description of an individual physical system
….

Extending this result, the authors in Harrigan09  have shown that

Any $\psi$-ontic ontological model that reproduces the quantum statistics
(QSTAT) violates locality.

In brief, these two results together assert "$\psi$-complete and/or
$\psi$-ontic $\Rightarrow$ $\neg$ locality". Our theorem can be viewed as a
converse of this claim as it shows that "Nonlocality (in the sense of Bell)
$\Rightarrow$ $\neg$ maximal $\psi$-epistemicity". Here we recall the result
of Ref. Leifer13 , where it has been shown that “maximally $\psi$-epistemic
$\Rightarrow$ Kochen-Specker noncontextual". Therefore Kochen-Specker
contextuality Kochen67  excludes maximally $\psi$-epistemic model for quantum
systems with Hilbert space dimension strictly greater that two. Similarly, our
theorem shows that the Bell nonlocality rules out maximally $\psi$-epistemic
ontological model. Manifestly, the scope of our theorem is broader than that
in Leifer13  as it excludes maximally $\psi$-epistemic model even for a qubit
system.

So far we have shown that Bell nonlocality prohibits ontological model that is
maximally $\psi$-epistemic. Naturally the question arises what about the non-
maximally $\psi$-epistemic models [see Eq. (2)]? Interestingly, we will now
show that a broader class of ontological models having certain degree of
epistemicity can also be excluded from a similar reasoning.

Whenever the state $\ket{+}$ is fed into the machine copying the state
$\ket{0}$ and $\ket{1}$ perfectly, the resulting state being maximally
entangled exhibits Bell nonlocality. Thus part of the ontic support
$\Lambda_{+}$ must lie outside $\Lambda_{0}\cup\Lambda_{1}$, which accounts
for the observed CHSH violation and consequently this will imposes bound on
degree of epistemicity. For suitable choices of measurements maximally
entangled state can exhibits CHSH violation up-to $2\sqrt{2}$, i.e.
$\bra{\phi^{+}}\mathbb{CHSH}\ket{\phi^{+}}=2\sqrt{2}$; $\mathbb{CHSH}$ denotes
the CHSH expression/operator. Quantum reproducibility condition, therefore,
demands

$\displaystyle\int_{\Lambda_{+}}d\lambda^{S}\int_{\Lambda^{R}}\int_{\Lambda^{G}}d\lambda^{R}d\lambda^{G}\mu(\lambda_{joint}|\ket{+r})\mathbb{CHSH}_{\mathbb{M}[\lambda_{joint}]}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
=\bra{\phi^{+}}\mathbb{CHSH}\ket{\phi^{+}}=2\sqrt{2}.$

Here $\lambda_{joint}\equiv(\lambda^{S},\lambda^{R},\lambda^{G})$ is the input
ontic state and $\mathbb{M}[\lambda_{joint}]$ is the output ontic state after
the machine’s action. The domain of integration for the variable $\lambda^{S}$
can be divided into three disjoint parts, i.e.
$\int_{\Lambda_{+}}d\lambda^{S}=\int_{\Lambda_{0}\cap\Lambda_{+}}d\lambda^{S}+\int_{\Lambda_{1}\cap\Lambda_{+}}d\lambda^{S}+\int_{\Lambda_{+}\setminus(\Lambda_{0}\cup\Lambda_{1})}d\lambda^{S}$.

Note that whenever $\lambda^{S}\in\Lambda_{i}\cap\Lambda_{+}$, the joint ontic
state $\lambda_{joint}=(\lambda^{S},\lambda^{R},\lambda^{G})$ belonging in
$\Lambda_{+r}$ as well as in $\Lambda_{ir}$ can not lead to the observed
nonlocality since $\mathbb{M}[\lambda_{joint}]\in\Lambda_{ii}$. However, there
may exist $(\lambda^{S},\lambda^{\prime R},\lambda^{\prime G})\in\Lambda_{+r}$
s.t. $(\lambda^{S},\lambda^{\prime R},\lambda^{\prime G})\notin\Lambda_{ir}$,
where $\lambda^{S}\in\Lambda_{i}\cap\Lambda_{+}$ for some $i\in\\{0,1\\}$. In
such a case the machine can distinct the input preparation $\ket{+}$ or
$\ket{i}$ by accessing $\lambda^{\prime R}$ and/or $\lambda^{\prime G}$ and
the observed nonlocal behaviour can be well explained. At this point we assume
that

$\displaystyle\forall\leavevmode\nobreak\ \lambda^{S}$
$\displaystyle\in\Lambda_{i}\cap\Lambda_{+},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
(\lambda^{S},\lambda^{\prime R},\lambda^{\prime G})\in\Lambda_{+r}$
$\displaystyle\implies(\lambda^{S},\lambda^{\prime R},\lambda^{\prime
G})\in\Lambda_{ir}.$ (3)

Importantly, this is a strictly weaker assumption than the ontic composition
principle of PI used by PBR. To argue this first note that, for
$\lambda^{S}\in\Lambda_{+}\cap\Lambda_{i}$ the Bayes’ rule allows us to write
$\mu(\lambda^{S},\lambda^{R},\lambda^{G}|\ket{+r})=\mu(\lambda^{S}|\ket{+r})\mu(\lambda^{R}|\lambda^{S},\ket{+r})\mu(\lambda^{G}|\lambda^{S},\lambda^{R},\ket{+r})$.
We can also write
$\mu(\lambda^{S},\lambda^{R},\lambda^{G}|\ket{ir})=\mu(\lambda^{S}|\ket{ir})\mu(\lambda^{R}|\lambda^{S},\ket{ir})$
$\mu(\lambda^{G}|\lambda^{S},\lambda^{R},\ket{ir})$. Clearly, the assumptions

$\displaystyle\mu(\lambda^{R}|\lambda^{S},\ket{+r})$
$\displaystyle=\mu(\lambda^{R}|\lambda^{S},\ket{ir}),$ (4a)
$\displaystyle\mu(\lambda^{G}|\lambda^{S},\lambda^{R},\ket{+r}$
$\displaystyle=\mu(\lambda^{G}|\lambda^{S},\lambda^{R},\ket{ir}))$ (4b)

will imply Eq.(11). Unlike the PI assumption, Eq.(12) do not prohibit
correlation at the ontic level for two operationally independent preparations
and hence they are weaker assumptions Self . Furthermore, conditions in
Eq.(12) suffice our purpose, but they are not at all the necessary
requirements. The following weaker conditions

$\displaystyle\mu(\lambda^{R}|\lambda^{S},\ket{ir}$
$\displaystyle=0\implies\mu(\lambda^{R}|\lambda^{S},\ket{+r})=0,\leavevmode\nobreak\
\&$ (5a) $\displaystyle\mu(\lambda^{G}|\lambda^{S},\lambda^{R}\ket{ir}$
$\displaystyle=0\implies\mu(\lambda^{G}|\lambda^{S},\lambda^{R}\ket{+r})=0,$
(5b)

serve good for our purpose. Therefore
$\Lambda_{+}\setminus(\Lambda_{0}\cup\Lambda_{1})$ should have non-zero
measure in order to explain the observed nonlocality.

The area of the domain $\Lambda_{+}\setminus(\Lambda_{0}\cup\Lambda_{1})$
required to explain the observed quantum nonlocal imposes bound on the degree
of epistemicity of the underlying ontological model. Assuming
$\Omega(+,0)=\Omega(+,1):=\Omega$ it turns out that $\Omega\leq 2-\sqrt{2}$
Supple . Therefore a threshold amount of onticity is required in the
ontological model to incorporate the observed CHSH violation in quantum
theory. Taking more general consideration of a cloning machine
$\mathbb{M}_{\phi}$ that perfectly copies the state $\ket{\phi}$ and
$\ket{\phi^{\perp}}$, one can obtain the following general bound
$|\alpha|^{2}\Omega(\phi,\psi)+|\beta|^{2}\Omega(\phi^{\perp},\psi)\leq
2-\sqrt{1+4|\alpha|^{2}|\beta|^{2}}$, where
$\ket{\psi}=\alpha\ket{\phi}+\beta\ket{\phi^{\perp}}$ and $\alpha\neq 0,1$
Supple .

##  VI Discussions

The PBR theorem initiated a surge in research interest regarding the reality
of quantum wavefunction Hardy13 ; Colbeck12 ; Colbeck17 ; Patra13 ; Leifer14 ;
Lewis12 ; Montina12 ; Bub12 ; Banik14 ; Jennings15 ; Bandyopadhyay17 . Between
the two competitive views – $\psi$-ontic vs. $\psi$-epistemic – it favours the
former. To this claim it uses an assumption, called PI, regarding the ontology
of a composite system. Establishing such a powerful doctrine about the reality
of quantum state, the theorem goes through detailed scrutiny and the PI
assumption gained several criticism Schlosshauer12 ; Emerson13 ; Ballentine14
; Schlosshauer14 . In particular the criticism in Ref. Schlosshauer14  is
quite severe. The authors there consider ontic composition assumption that is
weaker than the assumption of ‘preparation independence’ and reject vast class
of deterministic hidden-variables theories, including those consistent on
their targeted domain. This result puts challenge on the compositional aspect
of the real/ontic states one might wish to assume through preparation
independence while modeling a tensor-product quantum state. It therefore
motivates renewed aspiration to establish $\psi$-ontic nature of quantum
wavefunction from more rational assumption or using no such assumption at all.
At this point our theorem starts contributing. Our result established that if
we take maximally $\psi$-epistemic doctrine then quantum nonlocality cannot be
explained in such a model. Importantly, unlike PBR, we do not consider any
compositional assumption. In fact, we consider that ontic state space for
product preparation can be more general than Cartesian product of their
individual ontic states as they can possess composite ontic properties. In
this regard, the toy model of Spekkens Spekkens07  is worth mentioning. The
model is maximally $\psi$-epistemic by construction. Though it reproduces
number of phenomena as observed in quantum theory, it is a Bell local model.
Our theorem establishes a general result in this direction as it shows that no
maximally $\psi$-epistemic model can incorporate the nonlocal behavior of
quantum theory. Furthermore, we show that the observed phenomenon of quantum
nonlocality excludes not only the maximally $\psi$-epistemic models, but it
also imposes bound on the degree of epistemicity of the underlying models.
Extent of our theorem is also broader than the $\psi$-ontology theorems in
Maroney12 ; Maroney12(1) ; Leifer13 ; Barrett14 ; Leifer14(1) ; Branciard14 ;
Ringbauer15  as these results apply to quantum systems with dimensions greater
than two but remain silent for qubit system. Here it is worth mentioning the
theorem proved by Aaronson et. al. Aaronson13 . There the authors have
considered only those psi-epistemic models in which every pair of non-
orthogonal states have ontic overlap of nonzero measure. However, our theorem
does not limit itself to such pairwise $\psi$-epistemic models i.e. given any
two non-orthogonal states, the theorem dictates an upper bound on their ontic
overlap without assuming anything about the degree of ontic overlap between
all other pairs of non-orthogonal states. Our result opens up new research
possibilities. In our work we have considered the CHSH inequality. It would be
interesting to study whether more stringent restriction(s) on the underlying
ontological models can be obtained from other classes of local realistic
inequalities. It may also be interesting to study what new kind of restriction
genuine quantum nonlocality would impose on the nature of quantum
wavefunction.

###### Acknowledgements.

ADB and MB acknowledge fruitful discussions with Guruprasad Kar, Some Sankar
Bhattacharya, and Tamal Guha. MB would like to thank Michael J. W. Hall for
useful suggestions (through private communication). MB acknowledges research
grant through INSPIRE-faculty fellowship from the Department of Science and
Technology, Government of India.

##  Appendix A Proof of Proposition 1

Before going to the main part of the proof, let us discuss first what
conclusion one can make regarding the evolution of the underlying ontic states
when some measurement is performed on some operational quantum preparation.

Single system: Consider that a qubit is prepared in the state $\ket{0}$, the
$+1$ eigenstate of $\sigma_{z}$. At the ontological level the system is
prepared in some ontic state
$\lambda\in\Lambda_{0}:=\\{\lambda\leavevmode\nobreak\ |\leavevmode\nobreak\
\mu(\lambda|\ket{0}>0\\}\subset\Lambda$. Suppose $\sigma_{x}$ measurement is
performed on system.

In such a scenario, what can we say about the post-measurement ontic state
(say) $\lambda^{\prime}$? Can we give a deterministic evolution from the pre-
measurement ontic state $\lambda$ to the post-measurement ontic state
$\lambda^{\prime}$? Answer of this question is NO. However, we can make some
conclusion regarding $\lambda^{\prime}$. Given that on the pre-measurement
ontic state $\lambda$ the measurement $\sigma_{x}$ is performed, we can say
that post-measurement ontic state $\lambda^{\prime}$ must belong to
$(\Lambda_{+}\cup\Lambda_{-})$ i.e.
$\Lambda_{0}\ni\lambda\xrightarrow{\sigma_{x}}\lambda^{\prime}\in(\Lambda_{+}\cup\Lambda_{-})$.

Suppose that on the quantum preparation $\ket{0}$ the $\sigma_{z}$ measurement
is performed instead of the $\sigma_{x}$ measurement. Clearly, the operational
state remains identical as a measurement does not disturb its eigenkets. In
this case, can we make the conclusion that the post-measurement ontic state
$\lambda^{\prime}$ is same as the pre-measurement ontic state $\lambda$? Again
the answer is NO. However, we can make the conclusion that $\lambda^{\prime}$
must belong to the support $\Lambda_{0}$. In other words, a measurement that
does not disturb the operational preparation can disturb the ontic
preparation. Such a disturbance is indeterministic in general but the final
ontic state remains within the support on the operational preparation, i.e.
$\Lambda_{0}\ni\lambda\xrightarrow{\sigma_{z}}\lambda^{\prime}\in\Lambda_{0}$.

Therefore, in general we can say a measurement, disturbing or non-disturbing
on an operational preparation, should not assume any deterministic state-
update rule for the underlying ontic states. The only thing that can be
specified is the set of possible post-measurement ontic states for a given
pre-measurement ontic state.

Bipartite system: Consider a two-qubit bipartite system $\mathcal{S}_{AB}$. As
discussed in the manuscript, the ontic state space of such a system has three
parts: $\Lambda_{A}\times\Lambda_{B}$ corresponding local reality,
$\Lambda^{G}$ carrying strictly relational information about the two systems
and remains hidden under local measurements, and $\Lambda^{NL}$ reality
resulting to the Bell nonlocal feature. Since in the following we will
consider local measurements on the bipartite system, without loss of any
generality we only consider the joint ontic states
$\lambda_{joint}\in(\Lambda_{A}\times\Lambda_{B})\cup\Lambda^{NL}$. Any
$\lambda_{joint}\in(\Lambda_{A}\times\Lambda_{B})$ can be thought as
$\lambda_{joint}\equiv(\lambda_{A},\lambda_{B})$, where
$\lambda_{A}\in\Lambda^{A}$ and $\lambda_{B}\in\Lambda^{B}$. On the other
hand, a $\lambda_{joint}\in\Lambda^{NL}$ must violates at least one of the
conditions of outcome independence (OI) and parameter independence.

Outcome independence:

$\displaystyle
p(a|b,x,y,\lambda_{joint})=p(a|x,y,\lambda_{joint}),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ a,b,x,y;$ (6a) $\displaystyle
p(b|a,x,y,\lambda_{joint})=p(b|x,y,\lambda_{joint}),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ a,b,x,y.$ (6b)

Here, $a,b$ respectively denote Alice’s and Bob’s outcome for their
respectively local measurements $x$ and $y$.

Parameter independence:

$\displaystyle
p(a|x,y,\lambda_{joint})=p(a|x,\lambda_{joint}),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ a,x,y;$ (7a) $\displaystyle
p(b|x,y,\lambda_{joint})=p(b|y,\lambda_{joint}),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ b,x,y.$ (7b)

In a OI model, a nonlocal ontic state $\lambda_{joint}\in\Lambda^{NL}$ must
violates the condition Parameter independence which can happen in three ways:
(i) Violation of Eq.(7a) for some choice of $a,x,y$. Such a $\lambda_{joint}$
we will represent as $\lambda_{joint}\equiv(\lambda_{A\leftarrow
B},\lambda_{B})$, as Bob’s measurement choice effects Alice’s marginal outcome
probabilities.
(ii) Violation of Eq.(7b);
$\lambda_{joint}\equiv(\lambda_{A},\lambda_{{A\rightarrow B}})$.
(iii) Violation of Eq.(7a) and Eq.(7b);
$\lambda_{joint}\equiv\lambda_{{A\leftrightarrow B}}$.

For a better clarification, specific examples of these four types of
$\lambda_{joint}$’s are discussed below.
Case (1):
$\lambda_{joint}\equiv(\lambda_{A},\lambda_{B})\in\Lambda_{A}\times\lambda_{B}$.

$\leavevmode\nobreak\ \leavevmode\nobreak\ \\#\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $xy(\downarrow)/ab(\rightarrow)$ | $+1,+1$ | $+1,-1$ | $-1,+1$ | $-1,-1$
---|---|---|---|---|---
$1$ | $\sigma_{x}\otimes\sigma_{x}$ | $1$ | $0$ | $0$ | $0$
$2$ | $\sigma_{x}\otimes\sigma_{y}$ | $0$ | $1$ | $0$ | $0$
$3$ | $\sigma_{y}\otimes\sigma_{x}$ | $0$ | $0$ | $1$ | $0$
$4$ | $\sigma_{y}\otimes\sigma_{y}$ | $0$ | $0$ | $0$ | $1$
Table 1: Note that $\lambda_{joint}$ satisfies the condition of Parameter
Independence. None of Alice’s outcome depends on the measurement context(s)
chosen by Bob, and vice versa. Therefore, observables of both A and B possess
local reality (independent of any remote context chosen by other observer).

Such a $\lambda_{joint}$ can be described by Cartesian product of ontic states
$\lambda_{A}\in\Lambda_{A}$ and $\lambda_{B}\in\Lambda_{B}$ corresponding the
following sub-tables.

| $\leavevmode\nobreak\ \leavevmode\nobreak\ \\#\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $x(\downarrow)/a(\rightarrow)$ | $+1$ | $-1$
---|---|---|---
$1$ | $\sigma_{x}$ | $1$ | $0$
$2$ | $\sigma_{y}$ | $0$ | $1$
| $\leavevmode\nobreak\ \leavevmode\nobreak\ \\#\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $y(\downarrow)/b(\rightarrow)$ | $+1$ | $-1$
---|---|---|---
$1$ | $\sigma_{x}$ | $1$ | $0$
$2$ | $\sigma_{y}$ | $0$ | $1$

Case (2-i): $\lambda_{joint}\equiv(\lambda_{A\leftarrow
B},\lambda_{B})\in\Lambda^{NL}$.

$\leavevmode\nobreak\ \leavevmode\nobreak\ \\#\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $xy(\downarrow)/ab(\rightarrow)$ | $+1,+1$ | $+1,-1$ | $-1,+1$ | $-1,-1$
---|---|---|---|---|---
$1$ | $\sigma_{x}\otimes\sigma_{x}$ | $1$ | $0$ | $0$ | $0$
$2$ | $\sigma_{x}\otimes\sigma_{y}$ | $1$ | $0$ | $0$ | $0$
$3$ | $\sigma_{y}\otimes\sigma_{x}$ | $1$ | $0$ | $0$ | $0$
$4$ | $\sigma_{y}\otimes\sigma_{y}$ | $0$ | $0$ | $1$ | $0$
Table 2: Note that Alice’s outcome for $\sigma_{y}$ measurement depends on the
Bob’s measurement choice(s) (compare $3^{rd}$ and $4^{th}$ rows). However,
Bob’s outcomes are independent of measurement contexts chosen by Alice.

In this case, observables of A system do not possess any local reality, i.e.
it cannot be described by an ontic state belonging to its local ontic space
$\Lambda^{A}$. However, observables of B system take values independent of
measurement contexts chosen by Alice, hence they possess local ontic reality,
i.e. B has its local ontic state $\lambda_{B}\in\Lambda^{B}$.

| $\leavevmode\nobreak\ \leavevmode\nobreak\ \\#\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $y(\downarrow)/b(\rightarrow)$ | $+1$ | $-1$
---|---|---|---
$1$ | $\sigma_{x}$ | $1$ | $0$
$2$ | $\sigma_{y}$ | $1$ | $0$

Case(2-ii): $\lambda_{joint}\equiv(\lambda_{A},\lambda_{A\rightarrow
B})\in\Lambda^{NL}$.

$\leavevmode\nobreak\ \leavevmode\nobreak\ \\#\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $xy(\downarrow)/ab(\rightarrow)$ | $+1,+1$ | $+1,-1$ | $-1,+1$ | $-1,-1$
---|---|---|---|---|---
$1$ | $\sigma_{x}\otimes\sigma_{x}$ | $1$ | $0$ | $0$ | $0$
$2$ | $\sigma_{x}\otimes\sigma_{y}$ | $1$ | $0$ | $0$ | $0$
$3$ | $\sigma_{y}\otimes\sigma_{x}$ | $1$ | $0$ | $0$ | $0$
$4$ | $\sigma_{y}\otimes\sigma_{y}$ | $0$ | $1$ | $0$ | $0$
Table 3: Note that $\sigma_{y}$’s value for Bob depends on the measurement
context chosen by Alice (compare $2^{nd}$ and $4^{th}$ rows). Importantly,
Alice’s outcomes do not depend on Bob’s measurement choice(s).

In this case outcome of the observables on A system do not depend on Bob’s
measurement choice. Therefore, though B does not possess local reality ( i.e.
context independent reality) the system A is in a local ontic state
$\lambda^{A}\in\Lambda^{A}$.

| $\leavevmode\nobreak\ \leavevmode\nobreak\ \\#\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $x(\downarrow)/a(\rightarrow)$ | $+1$ | $-1$
---|---|---|---
$1$ | $\sigma_{x}$ | $1$ | $0$
$2$ | $\sigma_{y}$ | $1$ | $0$

Case(2-iii): $\lambda_{joint}\equiv\lambda_{A\leftrightarrow B}$.

$\leavevmode\nobreak\ \leavevmode\nobreak\ \\#\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $xy(\downarrow)/ab(\rightarrow)$ | $+1,+1$ | $+1,-1$ | $-1,+1$ | $-1,-1$
---|---|---|---|---|---
$1$ | $\sigma_{x}\otimes\sigma_{x}$ | $1$ | $0$ | $0$ | $0$
$2$ | $\sigma_{x}\otimes\sigma_{y}$ | $1$ | $0$ | $0$ | $0$
$3$ | $\sigma_{y}\otimes\sigma_{x}$ | $0$ | $1$ | $0$ | $0$
$4$ | $\sigma_{y}\otimes\sigma_{y}$ | $0$ | $0$ | $0$ | $1$
Table 4: Parameter independence is violated both from A to B and B to A.
Alice’s $\sigma_{y}$ measurement takes different outcome depending on Bob’s
measurement context. Similarly, Bob’s $\sigma_{x}$ outcome depends Alice’s
measurement context. Therefore, none of Alice’s and Bob’s observables possess
local reality.

We are now in a position to prove the Proposition stated in the main text.

###### Proof.

Consider the product quantum preparation $\ket{0}_{A}\ket{0}_{B}$. Contrary to
Proposition 1, assume that there exists some nonlocal joint ontic state
$\lambda_{joint}\in\Lambda^{NL}$ lying in the support $\Lambda_{00}$. Since,
we are considering the ontological model to be maximally $\psi$-epistemic,
therefore the model must be outcome deterministic and reciprocal Ballentine14
. Thus the nonlocal $\lambda_{AB}$ must violate at least one of the
assumptions of parameter independence and outcome independence (OI) Jarrett84
; Shimony84 . Since outcome deterministic models satisfy OI Hall11 , therefore
$\lambda_{AB}$ must violate parameter independence. Note that, nonlocal
correlations have a classical like explanation if we assume that the agent
sacrifice their free choice/ measurement independence Hall10 ; Barrett11 ;
Banik12 ; Bub12 ; Banik13 . However, here we are considering that the agents
enjoy their free choice.

Consider a $\lambda_{joint}\equiv\lambda_{A\leftrightarrow B}$ [Type (2-iii)
nonlocal]. Such a $\lambda_{joint}$ requires two different tables of
assignments: one for when Alice measures first (which would be [$\bf
Type(2-ii)$ non-local]), and a different one when Bob measures first (which
would be [$\bf Type(2-i)$] local). But for simplicity, we consider a fixed
order of their measurements : Alice measures her part of system before Bob
performs any measurement on his part , so that a single table of assignments
is sufficient. A typical example of such
$\lambda_{joint}\equiv\lambda_{A\leftrightarrow B}\in\Lambda_{00}$ is the
following :

$\leavevmode\nobreak\ \leavevmode\nobreak\ \\#\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $xy(\downarrow)/ab(\rightarrow)$ | $+1,+1$ | $+1,-1$ | $-1,+1$ | $-1,-1$
---|---|---|---|---|---
$1$ | $\sigma_{z}\otimes\sigma_{z}$ | $1$ | $0$ | $0$ | $0$
$2$ | $\sigma_{z}\otimes\sigma_{x}$ | $1$ | $0$ | $0$ | $0$
$3$ | $\sigma_{x}\otimes\sigma_{z}$ | $1$ | $0$ | $0$ | $0$
$4$ | $\sigma_{x}\otimes\sigma_{x}$ | $1$ | $0$ | $0$ | $0$
$5$ | $\sigma_{x}\otimes\sigma_{y}$ | $1$ | $0$ | $0$ | $0$
$6$ | $\sigma_{y}\otimes\sigma_{x}$ | $0$ | $1$ | $0$ | $0$
$7$ | $\sigma_{y}\otimes\sigma_{y}$ | $1$ | $0$ | $0$ | $0$
$8$ | $\sigma_{z}\otimes\sigma_{y}$ | $1$ | $0$ | $0$ | $0$
$9$ | $\sigma_{y}\otimes\sigma_{z}$ | $0$ | $1$ | $0$ | $0$
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
Table 5: $\lambda_{joint}\equiv\lambda_{A\leftrightarrow B}$.

Note that in accordance with our chosen order of measurements, the table of
assignment for $\lambda_{joint}\equiv\lambda_{A\leftrightarrow B}$ is of
[$\bf{Type(2-ii)}$ nonlocal]. Here, we do not provide any table of
[$\bf{Type(2-i)}$ nonlocal] associated with
$\lambda_{joint}\equiv\lambda_{A\leftrightarrow B}$ that considers the case
when Alice and Bob measures in the reverse temporal order (i.e. Bob first,
Alice second) since it is straightforward to construct such a table.

On this pre-measurement joint ontic state let Alice first measures
$\sigma_{z}$ on her system. This will cause all observable at Bob’s end to
take values in accordance with the chosen context of $\sigma_{z}$ at Alice’s
side. Therefore, post-measurement $\lambda^{\prime}_{joint}$ will be of Type
(2-ii) nonlocal, i.e. $(\lambda^{\prime}_{A\leftarrow
B},\lambda^{\prime}_{B})$; where $\lambda^{\prime}_{B}$ is described as the
the following table:

$\leavevmode\nobreak\ \leavevmode\nobreak\ \\#\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $y(\downarrow)/b(\rightarrow)$ | $+1$ | $-1$
---|---|---|---
$1$ | $\sigma_{z}$ | $1$ | $0$
$2$ | $\sigma_{x}$ | $1$ | $0$
$3$ | $\sigma_{y}$ | $1$ | $0$
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
Table 6:

The measurement $\sigma_{z}$ on A system thus leads to the following ontic
transformation:

$\displaystyle\lambda^{joint}$ $\displaystyle\equiv\lambda_{A\leftrightarrow
B}\xrightarrow[]{\sigma_{z}\otimes\mathbb{I}}\lambda^{\prime}_{joint}$
$\displaystyle\equiv(\lambda^{\prime}_{A\leftarrow
B},\lambda^{\prime}_{B})\in\mathbf{Type(2-ii)}.$ (8)

Following reasoning clarifies such a post-measurement evolution of the ontic
state. Before Alice’s measurement, observables of system B possess no local
reality, which is reflected in the pre-measurement ontic state
$\lambda_{A\leftrightarrow B}$. For instance, Bob’s $\sigma_{x}$ observable
does not have context independent reality in the state
$\lambda_{A\leftrightarrow B}$ (see Table 5). In other words, one cannot
assign value to $\sigma_{x}$ of B independent of Alice’s choice of measurement
(From the $2^{nd}$ and $6^{th}$ row of Table 5 it is evident.) But, as soon as
Alice chooses one particular measurement context Bob’s observables take the
corresponding values in accordance with the Alice’s chosen measurement
context. To see that consider the $2^{nd}$ row of pre-measurement state (Table
5) which asserts: “If Alice measures $\sigma_{z}$ on A, Bob will obtain +1 if
he measures $\sigma_{x}$ on B". When Alice already performed the measurement
$\sigma_{z}$, Bob’s $\sigma_{x}$ observable must take the value $+1$ in the
post-measurement state $\lambda^{\prime}_{joint}$, otherwise proposition of
the $2^{nd}$ row of pre-measurement state gets violated. If Alice had measured
$\sigma_{y}$ instead of $\sigma_{z}$, Bob’s $\sigma_{x}$ observable would have
taken the value $-1$, otherwise proposition of the $6^{th}$ row of pre-
measurement state would have been violated. Since Alice measures $\sigma_{z}$,
not $\sigma_{y}$, the possibility of Bob’s $\sigma_{x}$ value to be $-1$ is no
more there in the post measurement state $\lambda^{\prime}_{joint}$. Therefore
Alice’s choice of a particular context assigns Bob’s observable context
independent local reality. Similarly, after Alice’s $\sigma_{z}$ measurement,
$8^{th}$ row demands B should have $\sigma_{y}=1$ in the post-measurement
joint ontic state $\lambda^{\prime}_{joint}$.

In general, in the post-measurement joint ontic state
$\lambda^{\prime}_{joint}$ , $\sigma_{n}$ of B will take the value obeying the
condition imposed by the row corresponding to $\sigma_{z}\otimes\sigma_{n}$ of
pre-measurement joint ontic state $\lambda_{A\leftrightarrow B}$. Thus all the
observables of B take some fixed particular value, therefore attain local
reality, as soon as Alice chooses her measurement context. Therefore, in the
post-measurement state, though Parameter Independence can be violated from B
to A there is no such violation from A to B and hence in
$\lambda^{\prime}_{joint}$, B is in the ontic state
$\lambda^{\prime}_{B}\in\Lambda_{B}$ (given in Table 6).

Please note that we can only claim that the post measurement ontic state
$\lambda_{joint}\equiv(\lambda^{\prime}_{A\leftarrow
B},\lambda^{\prime}_{B})\in\mathbf{Type(2-ii)}$, but which particular
$\lambda^{\prime}_{joint}$ it is that is not specified and for our argument it
is not required. Suppose Bob now performs $\sigma_{z}$ measurement on the
state $\lambda^{\prime}_{joint}$. A similar reasoning as above will imply the
final ontic state
$\lambda^{\prime\prime}_{joint}\equiv(\lambda^{\prime\prime}_{A},\lambda^{\prime\prime}_{B})\in\mathbf{Type(1)}$.
Thus we have,

$\displaystyle\mathbf{Type(2-iii)}\ni\lambda_{A\leftrightarrow
B}\xrightarrow[]{\sigma_{z}\otimes\mathbb{I}}(\lambda^{\prime}_{A\leftarrow
B},\lambda^{\prime}_{B})\in\mathbf{Type(2-ii)}$
$\displaystyle\xrightarrow[]{\mathbb{I}\otimes\sigma_{z}}(\lambda^{\prime\prime}_{A},\lambda^{\prime\prime}_{B})\in\mathbf{Type(1)}.$

Here, in the second step of ontic evolution in equation $(4)$, we have made
the following assumption regarding ontic evolution of a certain class of joint
ontic states.

In any joint ontic state $\lambda_{joint}^{AB}$, if there exists no parameter
dependence from $A\rightarrow B$ (or from $B\rightarrow A$), local measurement
on B ( or A) cannot generate parameter dependence from $A\rightarrow B$ (or
$B\rightarrow A$ ) in the post-measurement state $\lambda^{\prime
AB}_{joint}$. Due to this, in the second step of ontic evolution in $(4)$, the
following has not been the case : $(\lambda^{\prime}_{A\leftarrow
B},\lambda^{\prime}_{B})\in\mathbf{Type(2-ii)}\xrightarrow[]{\mathbb{I}\otimes\sigma_{z}}(\lambda^{\prime\prime}_{A},\lambda^{\prime\prime}_{A\rightarrow
B})\in\mathbf{Type(2-i)}.$ More precisely, as there was no parameter
dependence from $A\rightarrow B$ in $(\lambda^{\prime}_{A\leftarrow
B},\lambda^{\prime}_{B})$, measuring $\sigma_{z}$ on B cannot generate
parameter dependence from $A\rightarrow B$ in the post-measurement state.

As the local joint ontic states [ i.e.
$\lambda_{joint}\equiv(\lambda_{A},\lambda_{B})\in\bf{Type(1)}$ ] do not
possess parameter dependence in either direction ( i.e. neither $A\rightarrow
B$ nor $A\leftarrow B$), the assumption immediately leads to the following:

$\displaystyle(\lambda_{A},\lambda_{B})\in\mathbf{Type(1)}\xrightarrow[]{\sigma_{z}\otimes\mathbb{I}}(\lambda^{\prime}_{A},\lambda^{\prime}_{B})\in\mathbf{Type(1)}$
$\displaystyle\xrightarrow[]{\mathbb{I}\otimes\sigma_{z}}(\lambda^{\prime\prime}_{A},\lambda^{\prime\prime}_{B})\in\mathbf{Type(1)}.$

Joint ontic states of system AB which are local remain local under local
measurements done on subsystems A and/or B. [Note : as we have fixed a
specific temporal order in which Alice measures first and Bob measures second,
$\lambda_{A\leftrightarrow B}$ $\in\mathbf{Type(2-iii)}$ is essentially of
$\mathbf{Type(2-ii)}$ in which no parameter dependence exists from
$B\rightarrow A$. Therefore, we could have as well used the assumption in the
very first step of ontic evolution in $(4)$ to obtain
$\mathbf{Type(2-iii)}\ni\lambda_{A\leftrightarrow
B}\xrightarrow[]{\sigma_{z}\otimes\mathbb{I}}(\lambda^{\prime}_{A},\lambda^{\prime}_{B})\in\mathbf{Type(1)}$].

Therefore from eq $(4)$, for any pre-measurement
$\lambda_{joint}\in\Lambda^{NL}$ that were assumed to lie within the ontic
support of $\ket{00}$, the post-measurement joint ontic state after Alice and
Bob performing their local measurements will be
$\lambda_{joint}\equiv(\lambda^{\prime\prime}_{A},\lambda^{\prime\prime}_{B})\in\Lambda_{A}\times\Lambda_{B}$.
Therefore the post-measurement quantum state cannot contain any
$\lambda_{joint}\in\Lambda^{NL}$ in its ontic support. But since the post
measurement quantum state remains unchanged, i.e. it is again
$\ket{0}_{A}\ket{0}_{B}$; this implies ontic support of
$\ket{0}_{A}\ket{0}_{B}$ cannot contain any $\lambda_{joint}\in\Lambda^{NL}$,
which is in contradiction with our initial assumption.

It is not hard to see that one arrives at similar contradiction for every
product quantum preparation. Furthermore, the arguments also holds in any
outcome deterministic ontological model instead of the maximally
$\psi$-epistemic one only. This completes our proof. ∎

##  Appendix B General bound on degree of epistemicity

Consider a quantum copying machine $\mathbb{M}_{\phi}$ that perfectly copies
the state $\ket{\phi}$ and $\ket{\phi^{\perp}}$. While the machine is fed with
system $S$ prepared in state $\ket{\phi}$ and $\ket{\phi^{\perp}}$ and the
reference system $R$ prepared in some fixed state $\ket{r}$, its action at the
ontological level is similar as discussed in the manuscript. If we fed
$\mathbb{M}_{\phi}$ with $\ket{\psi}_{S}\ket{r}_{R}$, with
$\ket{\psi}=\alpha\ket{\phi}+\beta\ket{\phi^{\perp}}\leavevmode\nobreak\
(\alpha\neq 0,1)$ then linearity of quantum theory results in the output state
$\ket{\Theta}_{SR}=\alpha\ket{\phi}_{S}\ket{\phi}_{R}+\beta\ket{{\phi^{\perp}}}_{S}\ket{\phi^{\perp}}_{R}$.
For suitable choices of measurements this state can exhibits CHSH inequality
violation up-to $2\sqrt{1+4|\alpha|^{2}|\beta|^{2}}$ Gisin91 . Assuming
maximally $\psi$-epistemicity an argument similar to the one presented in the
manuscript implies
$\mbox{Supp}[\mu(\lambda_{joint}|U_{\mathbb{M}_{\phi}}[\ket{\psi
r}])]\cap\Lambda^{NL}=\emptyset$ and hence the observed nonlocality cannot be
explained. To incorporate this nonlocality we need to depart from
$\psi$-maximal epistemicity. Now quantum reproducibility condition for the
observed nonlocality demands,

$\displaystyle\int_{\Lambda^{S}}d\lambda^{S}\int_{\Lambda^{R}}\int_{\Lambda^{G}}d\lambda^{R}d\lambda^{G}\mu(\lambda_{joint}|\ket{\psi
r})\mathbb{CHSH}_{\mathbb{M}_{\phi}[\lambda_{joint}]}$
$\displaystyle=\bra{\Theta_{SR}}\mathbb{CHSH}\ket{\Theta_{SR}}=2\sqrt{1+4|\alpha|^{2}|\beta|^{2}}.$
(9)

where, $\lambda_{joint}=(\lambda^{S},\lambda^{R},\lambda^{G})$ and
$\mathbb{M}_{\phi}[\lambda_{joint}]$ is the evolved ontic state after the
machine action, and $\mathbb{CHSH}$ denotes the CHSH expression. As argued in
the manuscript $\mathbb{M}_{\phi}[\lambda_{joint}]\notin\Lambda^{NL}$ whenever
$\lambda_{joint}\in(\Lambda_{\psi}\cap\Lambda_{\phi})\cup(\Lambda_{\psi}\cap\Lambda_{\phi^{\perp}})$.
Let us, therefore, consider that the ontic region
$\Lambda_{\psi}\setminus(\Lambda_{\phi}\cup\Lambda_{\phi^{\perp}})$ has
nonzero measure, which will be the case for non-maximally $\psi$-epistemic
models. Thus, the domain of integration for $\lambda^{S}$, in the left hand
side of Eq. (9) can be divided into three disjoint regions
$\Lambda_{\psi}\cap\Lambda_{\phi}$, $\Lambda_{\psi}\cap\Lambda_{\phi^{\perp}}$
and $\Lambda_{\psi}\setminus(\Lambda_{\phi}\cup\Lambda_{\phi^{\perp}})$ [see
Fig. 1]. Thus we have

$\displaystyle\int_{\Lambda_{\psi}\cap\Lambda_{\phi}}d\lambda^{S}\int
d\lambda^{R}d\lambda^{G}\mu(\lambda^{S},\lambda^{R},\lambda^{G}|\ket{\psi
r})\mathbb{CHSH}_{\mathbb{M}_{\phi}[\lambda_{joint}]}$
$\displaystyle=\int_{\Lambda_{\psi}\cap\Lambda_{\phi}}d\lambda^{S}\int
d\lambda^{R}d\lambda^{G}\mu(\lambda^{S},\lambda^{R},\lambda^{G}|\ket{\psi
r})\times 2$
$\displaystyle=2\int_{\Lambda_{\phi}}d\lambda^{S}\mu(\lambda^{S}|\psi)=2\Omega(\phi,\psi)|\langle\phi|{\psi}\rangle|^{2}$
$\displaystyle=2|\alpha|^{2}\Omega(\phi,\psi).$ (10)

Figure 1: (Color on-line) Ontic support
$\Lambda_{\chi}\equiv\mbox{Supp}[\mu(\lambda|\chi)]$ for
$\ket{\chi}\in\left\\{\ket{\phi},\ket{\phi^{\perp}},\ket{\psi}=\alpha\ket{\phi}+\beta\ket{\phi^{\perp}}\right\\}$.
Along $x$-axis ontic states for the system $S$ are shown and along $y$-axis
probability distribution on these ontic states for different quantum
preparations are plotted. In case of a non-maximally $\psi$-epistemic model,
$\Lambda_{\psi}\cap(\Lambda_{\phi}\cup\Lambda_{\phi^{\perp}})^{\mathrm{C}}\neq\emptyset$.
Area of the sky-blue shaded region is given by
$\int_{\Lambda_{\phi}}d\lambda^{S}\mu(\lambda^{S}|\psi)$ and area of the pink
shaded (black dotted) region is
$\int_{\Lambda_{\phi^{\perp}}}d\lambda^{S}\mu(\lambda^{S}|\psi)$.

We have consider that a local ontic states yields the maximum possible value
$2$ for the CHSH expression, and used the fact that $\int
d\lambda^{R}d\lambda^{G}\mu(\lambda^{S},\lambda^{R},\lambda^{G}|\ket{\psi
r})=\mu(\lambda^{S}|\psi r)=\mu(\lambda^{S}|\psi)$. Similar reasoning yields

$\displaystyle\int_{\Lambda_{\psi}\cap\Lambda_{\phi^{\perp}}}d\lambda^{S}\int
d\lambda^{R}d\lambda^{G}\mu(\lambda^{S},\lambda^{R},\lambda^{G}|\ket{\psi
r})\mathbb{CHSH}_{\mathbb{M}_{\phi}[\lambda_{joint}]}$
$\displaystyle=2|\beta|^{2}\Omega(\phi^{\perp},\psi).$ (11)

Whenever,
$\lambda_{joint}\in\Lambda_{\psi}\setminus(\Lambda_{\phi}\cup\Lambda_{\phi^{\perp}})$,
the evolved ontic state $\mathbb{M}_{\phi}[\lambda_{joint}]$ may lie in
$\Lambda^{NL}$ and consequently contributes to the observed quantum
nonlocality. Assuming that all such $\lambda_{joint}$’s yield the maximum
possible CHSH value (i.e. $4$) we obtain

$\displaystyle\int_{\Lambda_{\psi}\setminus(\Lambda_{\phi}\cup\Lambda_{\phi^{\perp}})}d\lambda^{S}\int
d\lambda^{R}d\lambda^{G}\mu(\lambda^{S},\lambda^{R},\lambda^{G}|\ket{\psi
r})\mathbb{CHSH}_{\mathbb{M}_{\phi}[\lambda_{joint}]}$
$\displaystyle\leavevmode\nobreak\
=\int_{\Lambda_{\psi}\setminus(\Lambda_{\phi}\cup\Lambda_{\phi^{\perp}})}d\lambda^{S}\int
d\lambda^{R}d\lambda^{G}\mu(\lambda^{S},\lambda^{R},\lambda^{G}|\ket{\psi
r})\times 4$
$\displaystyle=4\left[\int_{\Lambda_{\psi}}d\lambda^{S}\mu(\lambda^{S}|\psi)-\int_{\Lambda_{\phi}}d\lambda^{S}\mu(\lambda^{S}|\psi)-\int_{\Lambda_{\phi^{\perp}}}d\lambda^{S}\mu(\lambda^{S}|\psi)\right]$
$\displaystyle=4\left[1-|\alpha|^{2}\Omega(\phi,\psi)-|\beta|^{2}\Omega(\phi^{\perp},\psi)\right].$
(12)

Eqs. (9), (10), (11), & (12) all-together therefore imply

$\displaystyle 2\sqrt{1+4|\alpha|^{2}|\beta|^{2}}$ $\displaystyle\leq
2\leavevmode\nobreak\ |\alpha|^{2}\Omega(\phi,\psi)+2\leavevmode\nobreak\
|\beta|^{2}\Omega(\phi^{\perp},\psi)\leavevmode\nobreak\ $
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+4\leavevmode\nobreak\
\left[1-|\alpha|^{2}\Omega(\phi,\psi)-|\beta|^{2}\Omega(\phi^{\perp},\psi)\right].$
(13)

Here, instead of equality, we put inequality as some local $\lambda$’s can
yield CHSH value less than $2$ and some nonlocal $\lambda$’s can yield CHSH
value less than $4$. Simplifying the above expression we obtain

$|\alpha|^{2}\Omega(\phi,\psi)+|\beta|^{2}\Omega(\phi^{\perp},\psi)\leq
2-\sqrt{1+4|\alpha|^{2}|\beta|^{2}}.$ (14)

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# Short-Term Trajectory Prediction for Full-Immersive Multiuser Virtual
Reality with Redirected Walking

Filip Lemic12, Jakob Struye2, Jeroen Famaey2  1NaNoNetworking Center in
Catalunya (N3Cat), Polytechnic University of Catalonia, Spain  2Internet
Technology and Data Science Lab (IDLab), Universiteit Antwerpen - imec,
Belgium
Email: {filip.lemic, jakob.struye<EMAIL_ADDRESS>
###### Abstract

Full-immersive multiuser Virtual Reality (VR) envisions supporting
unconstrained mobility of the users in the virtual worlds, while at the same
time constraining their physical movements inside VR setups through redirected
walking. For enabling delivery of high data rate video content in real-time,
the supporting wireless networks will leverage highly directional
communication links that will “track” the users for maintaining the Line-of-
Sight (LoS) connectivity. Recurrent Neural Networks and in particular Long
Short-Term Memory (LSTM) networks have historically presented themselves as a
suitable candidate for near-term movement trajectory prediction for natural
human mobility, and have also recently been shown as applicable in predicting
VR users’ mobility under the constraints of redirected walking. In this work,
we extend these initial findings by showing that Gated Recurrent Unit (GRU)
networks, another candidate from the RNN family, generally outperform the
traditionally utilized LSTMs. Second, we show that context from a virtual
world can enhance the accuracy of the prediction if used as an additional
input feature in comparison to the more traditional utilization of solely the
historical physical movements of the VR users. Finally, we show that the
prediction system trained on a static number of coexisting VR users be scaled
to a multi-user system without significant accuracy degradation.

##  I Introduction

Virtual Reality (VR) continues to revolutionize our digital perceptions and
interactions [1]. It also promises novel applications with utility in
healthcare, tourism, education, entertainment, and occupational safety [2, 3],
to name a few. VR setups and contents are being rapidly improved, with primary
efforts targeting the enhancements in the immersiveness of VR experiences.
Toward this end, the research is mainly focused on enhancing the quality of
video content that is being presented to the VR users [4], as well as on
“cutting the wire” and enabling truly wireless delivery of the video content
[5, 6]. An additional important objective is to enable multiuser experiences,
in which the users are able to collaborate in a way that the action of one
user in the virtual world is delivered as a virtual content to other,
potentially collocated users [7], and consequently affects their virtual
experiences.

In the near future, VR systems will support multiple fully immersed and
coexisting VR users with constraint-free mobility in the virtual worlds. Such
setups will be supported through high frequency wireless communication
networks primary operating in the millimeter Wave (mmWave) (i.e., 30-300 GHz)
frequency band [8]. For supporting the delivery of high-quality video content
to the mobile VR users in real-time, the underlying wireless communication
will have to be highly directional on both transmit and receive sides [9].
These directional mmWave beams are simultaneously expected to “track” the
users’ movements for continuously maintaining Line-of-Sight (LoS) connectivity
for maximizing the quality of video delivery. At the same time, redirected
walking is envisioned to be utilized for avoiding collisions between the
collocated users, and between the users and the boundaries of the constrained
VR environments [7]. This will enable immersion of the VR users in the virtual
worlds by allowing them to roam freely, while simultaneously and ideally
imperceivably redirecting them in the physical setups for collision avoidance.

For supporting continuous LoS maintenance with each of the VR users, the
directional mmWave beams will have to be transmitted in such a way that they
also provide coverage with high-quality wireless signals at near-future
locations of the VR users. This in turn motivates the need for VR users’
short-term movement trajectory prediction in full-immersive multiuser VR
setups. Short-term movement trajectory prediction for a natural human walk is
a well-established topic in the research community, which yielded RNNs in
general and LSTM networks as their particular RNN instance suitable for this
task (e.g., [10, 11]). However, it is intuitive that neither imperceivable nor
perceivable re-steers of the VR users resemble a natural human walk,
suggesting the need for evaluating the suitability of RNNs for predicting VR
users’ mobility under the constraints stemming from redirecting walking. This
topic has received substantially less attention in the research community.
Nonetheless, an early work from Nescher and Kunz [12] has shown that such
precision is possible with reasonable accuracy, while [13, 14] demonstrated
that RNNs, specifically LSTMs, can be utilized for this purpose as they
feature encouraging prediction accuracy. All three approaches base the
prediction of the near-future trajectories on the historical movement
trajectories of the VR users.

In this work, we extend these initial findings along several dimensions.
First, we consider GRUs in addition to LSTMs for the near-future movement
trajectory prediction. We consider GRUs due to their faster training and
execution in comparison to LSTMs, which would conceptually make them a more
suitable candidate for such prediction in case they feature comparable
prediction accuracies. Second, we consider information from the virtual world
(i.e., the VR users’ movement trajectory in the virtual world) as an
additional input feature for the two considered approaches, which is in
contrast to the existing works from the literature that base the prediction
solely on historical movement trajectories in a physical space. Finally, we
evaluate the effects of a varying number of coexisting VR users (e.g., the
users dynamically entering and terminating immersive experiences) on the
prediction performance of the considered approaches trained with the data from
only a two-user system.

Our results demonstrate that GRU outperforms LSTM not only in terms of the
execution times, but also in the prediction accuracy, suggesting their utility
in the considered scenario. Moreover, we show that the information from the
virtual experience, in our case the virtual movement trajectory, provides
additional useful context and, therefore, improves the accuracy of prediction
for both LSTM and GRU-based approaches. Finally, our results show that a
predictor trained on data from two-user systems only can be scaled to a multi-
user system without significant accuracy degradation, which has a potential
for enabling dynamic immersion in and termination of virtual experiences in
multiuser full-immersive VR setups.

##  II System Overview

###  II-A Considered Scenario

We consider a full-immersive multiuser VR scenario as depicted in Figure 1.
Specifically, we consider a constrained physical environment in which the
full-immersive VR setup is deployed. The environment is constrained in terms
of its physical sizes in order to create a boundary inside of which it is safe
to engage in virtual experiences. As such, it is assumed that there are no
obstacles inside the deployment environment, which would represent a tripping
hazard if not properly accounted for. In other words, the only factors to be
considered as potential collision perils from the perspective of a VR user are
the environmental boundaries and other users, as depicted in the figure.

In the scenario, multiple VR users can be collocated in the same deployment
environment. The VR content is streamed to the users using highly directional
mmWave communication. Specifically, highly directional beams are transmitted
by an Access Point (AP) in such a way that they follow the movements of the VR
users. This is done in order to continuously maintain LoS connectivity with
each of the users, resulting in a maximized link quality and consequently
enhancing the VR users’ Quality of Experience (QoE) [15]. To do so, we
envision the VR users’ Head-Mounted Devices reporting their physical locations
to the AP, which is then utilized for supporting both redirected walking and
beamsteering. The assumption of the availability of physical locations of the
VR users is well-established in the existing literature (e.g., [9]) and the
contemporary VR headsets such as Oculus Quest 2 and Vive Cosmos already
support its generation and provisioning. It is also worth emphasizing that
these VR headsets are able to support the generation and provisioning of VR
users’ location information without the support of external devices such as
cameras or localization anchors.

In terms of transmitter-side beamsteering, the AP is envisioned to utilize the
current and near future locations of a VR user for forming a beam in a way
that it covers both current and near future locations of the user, enabling
LoS maintenance during a current time instance, as well as in the near future
instances. Simultaneously, redirected walking is utilized for steering of the
VR users in a way that guarantees collision avoidance between the collocated
users, as well as between the users and the environmental boundaries. The aim
of redirected walking is to enable immersion of the VR users (i.e., allowing
them to roam freely in potentially unconstrained virtual worlds), while
simultaneously and (ideally) imperceivably redirecting them in the physical
space.

Note that on the receiver side, beamforming is envisioned to adapt in real-
time to the VR user’s head rotations, relying on the HMD’s built-in sensors
providing accurate orientation estimates. The receiver-side beamforming and
beamsteering is considered as out of scope of this work. Interested readers
can find more details on this aspect of the system in [6], where we present
coVRage, a receiver beamforming solution in which, based on past and current
head orientations, the HMD predicts how the Angle of Arrival (AoA) from the AP
will change in the near future, and covers this AoA trajectory with a
dynamically shaped beam.

Note that we consider as out of scope the potential interruptions of the LoS
connectivity between the AP and the users due to the obstructions caused by
the other users. An intuitive solution to the issue can be based on an AP
handover if it is predicted that one user will block the LoS path of another
one, again motivating the need for short-term prediction of the VR users’
movements. Alternative solutions based on Intelligent Reflective Surfacess
have also been presented [9].

Figure 1: Considered scenario

###  II-B Short-Term Trajectory Prediction

(a) RNN structure

(b) LSTM neuron

(c) GRU neuron

(d) Legend

Figure 2: Considered types of recurrent neural networks

Based on the above discussion, short-term prediction of the VR users’ movement
trajectory is envisioned to be utilized for both redirected walking and
transmitter-side beam steering, while also being potentially beneficial for
LoS obstruction avoidance. It is intuitive that the accuracy of such
prediction will affect the performance of other parts of the envisioned
system, suggesting that the optimization of this accuracy will be beneficial
from the perspective of the VR’s QoE.

As mentioned before, RNNs are arguably the most suitable technological
candidates for predicting the trajectory of human movements, even in
redirected walking scenarios. An RNN is a class of artificial neural networks
which contains multiple neurons of the same type, each passing a message to a
succeeding neuron, as depicted in Figure 2(a). This allows the RNNs to exhibit
temporally dynamic behavior, making them suitable for tasks such as
handwriting or speech recognition and time-series prediction [16]. In this
work, we consider two of the most promising types of RNNs for such prediction,
namely LSTM and GRU. Single neurons of both types of networks are depicted in
Figure 2, together with the indications of their main building blocks.

In LSTM, a _sigmoid_ layer called the forget gate is utilized for deciding on
the retainment of the previous cell state. Moreover, a _sigmoid_ layer called
the input gate jointly with a _tanh_ layer are utilized for updating the cell
state. The output of the cell is then based on our cell state filtered through
a sigmoid layer for deciding the part of the cell state to be outputted,
normalized through a _tanh_ layer, as depicted in Figure 2(b). GRU neuron is
comparable with the one from LSTM, with the main differences stemming from the
combined forget and input gates into a one called update gate and merged cell
and hidden states, as depicted in Figure 2(c).

The aforementioned related works targeting short-term trajectory prediction
for full-immersive VR with redirected walking generally utilize LSTMs, while
here we also consider GRU networks due to their faster learning and execution
times. Moreover, in the literature the near future trajectory predictions are
predominantly based on previous locations and historical movements patters of
the users, taking their inspiration from works targeting the prediction of
walking trajectories assuming natural human walk. However, in full-immersive
VR setups with redirected walking the VR users’ movement is not natural, as
they are continuously being re-steered by the redirected walking for collision
avoidance. To do so, redirected walking will deliver VR content to the users
in such a way that the users are drawn toward physical locations where there
is no collision hazard, while has two important implications. First, as the
movement trajectories of VR users do not resemble a natural walk, the
approaches developed for predicting near-future trajectories based on
historical locations assuming naturally walking users are unoptimized for
full-immersive VR with redirected walking. Second, the redirections that are
envisioned to occur in the virtual world in the near future to avoid
collisions are the defining feature of the mobility of VR users in the
physical setups.

Our intuition is that the redirected walking-related inputs from the virtual
worlds are a valuable source of information for optimizing the near-future
movement trajectory prediction for full-immersive VR users. To model such
information, we define a notion of a virtual location of the VR user, i.e.,
its location in the virtual space. We then utilize the stream of historical
virtual locations of the VR users as an input feature to the LSTM and GRU
networks, in addition to the stream of their historical physical locations.
Based on the type of utilized input information, we distinguish the “baseline”
(i.e., LSTM/GRU-B) and “virtual” (i.e., LSTM/GRU-V) versions of the utilized
RNN approaches. The former is inspired by the existing literature, while the
latter additionally accounts for the specificities of the VR users’ mobility
by utilizing the users’ virtual coordinates. Also note that, given that the
decisions stemming from redirected walking, and consequently also the virtual
coordinates of the users, are assumed to be known one step ahead in time
compared to the physical coordinates of the users, as shown in Figure 3. We
consider this to be a natural assumption, given that the virtual coordinates
for the next time instance are in redirected walking derived using the
physical coordinates from a current time instance.

Figure 3: Input features of the considered RNN approaches

##  III Evaluation

Figure 4: Simulator overview [17]

###  III-A Evaluation Setup

To simulate the users’ mobility in multiuser VR setups with redirected
walking, we utilize the simulator from [17]. The core idea of the simulator is
to provide a mapping between design decisions about the users’ mobility in the
virtual words, and their effects on the physical mobility in constrained VR
environments. Specifically, given a set of VR users with their virtual
movement trajectories, the outline of the physical environment, and a
redirected walking algorithm for avoiding physical collisions, the simulator
is able to derive the physical movements of the users, as indicated in Figure
4. Based on the derived physical movements, the simulator is able to output a
set of performance metrics characterizing the number of perceivable resets and
the distances between such resets for each user. We have extended the
simulator to support VR users’ near future movement trajectory predictions and
have implemented the four considered prediction approaches (i.e., LSTM-B,
LSTM-V, GRU-B, GRU-V). As the performance metric for characterizing the
accuracy of prediction, we have defined the Squared Error (SE) between the
predicted and true near future physical locations of the VR users.

The deployment environment is defined as a square with size of 7.5 m. In the
training of the considered approaches, we assume two VR users coexist in the
environment, fully immersed in a VR experience. A redirected walking algorithm
is then utilized to map the users’ movements in the virtual world to the
corresponding movements in the physical deployment environment. We utilize the
redirected walking algorithms from Bachmann  _et al._ [7]. Specifically, the
Artificial Potential Field Redirected Walking (APF-RDW) algorithm for
imperceivable redirected walking in multiuser full-immersive VR applications,
and the Artificial Potential Field Reseting (APF-R) algorithm to re-orient the
VR user towards the safest available area in case there is a need for
perceivable resets to avoid collisions with other users or environmental
boundaries. Example virtual and physical mobility patterns of the users are
also depicted in Figure 4, while the other simulator parameters are summarized
in Table I. The full implementation of the simulator is available as an open
source tool, while the reproducibility of the results delivered in the
following sections is enabled through open data access111Both available at:
https://bitbucket.org/filip_lemic/pm4vr/src/master/.

###  III-B Hyperparameter Tuning

(a) LSTM activation

(b) GRU activation

(c) LSTM optimizer

(d) GRU optimizer

(e) Number of LSTM neurons

(f) Number of GRU neurons

Figure 5: Selected hyperparameter tuning results for the considered RNNs TABLE I: Simulation parameters Parameter |  Value
---|---
Number of points per trajectory |  10 [points/sec]
Duration of simulation |  3600 [sec]
Redirected walking |
Exponential falloff due to other users |  1.4
Arc radius or redirected walking |  7.5 [m]
Maximum rotational rate |  15 [°]
Velocity threshold |  0.1 [m/s]
TABLE II: Hyperparameter values used in RNNs’ training Parameter | Initial | LSTM-B | LSTM-V | GRU-B | GRU-V
---|---|---|---|---|---
Activation | _softplus_ | _relu_ | _relu_ | _softsign_ | _softmax_
Optimizer | _sgd_ | _nadam_ | _adam_ | _nadam_ | _nadam_
# neurons | 20 | 80 | 80 | 40 | 80
# batch/epochs | 20/10 | 20/40 | 60/50 | 80/30 | 80/30

In this step of the evaluation, our goal is to determine which hyperparameters
have a considerable effect on the performance of the considered RNN
approaches. We do that with the aim of reducing the hyperparameters’
optimization space, i.e., the hyperparameter tuning would be needed only for
influential hyperparameters. We define influential hyperparameters as the ones
whose change affects the average prediction error by more than 10%,
specifically the activation function, optimizer, number of neurons in the
hidden LSTM/GRU layer, and number of batches and epochs. For the influential
hyperparameters, our consequent goal is to determine the
hyperparameterizations that yield optimal performance for a given approach for
the near-future movement trajectory prediction in full-immersive multiuser VR
setups with redirected walking. In our optimization approach, we tune the
influential hyperparameters individually, while keeping the ones not already
tuned at their initial values (cf., Table II). Hence, we are able to derive a
set of locally optimal hyperparameters for each of the considered approaches,
as shown in the table. We follow this procedure, in contrast to performing an
extensive multihyperparameter search (which would result in a globally optimal
hyperparameters) because of an extremely high computational complexity and
time overhead of the alternative. A selected set of hyperparameter tuning
results is depicted in Figure 5 for the LSTM-B and GRU-B approaches, while the
other hyperparameters and considered approaches (i.e., LSTM-V, GRU-V) are left
out for brevity. Note that in the figure we depict only the hyperparameter
values with “reasonable” prediction accuracies, while the ones with
comparatively larger prediction errors are left out for clarity. The
hyperparameter value that maximizes the prediction accuracy is then considered
as the optimal one. For example, the optimal activation functions of the
LSTM-B and GRU-B approaches are respectively _relu_ and _softsign_ , as they
yielded the smallest SE of the prediction, as shown in the figure. The
resulting set of hyperparameters for the four considered approaches is given
in Table II.

###  III-C Evaluation Results

We present the performance results achieved by the four considered prediction
approaches, with their hyperparameters optimized following the above-discussed
procedure. Figures 6 and 7 depict the SEs observed by different versions of
the LSTM and GRU-based prediction approaches, respectively. For each approach,
we depict the baseline performance (i.e., GRU/LSTM-B), as well as the
performance of their virtual counterparts (i.e., LSTM/GRU-V), both with
locally-optimal hyperparameters. Moreover, we also depict two intermediary
approaches. In LSTM/GRU-I1, the hyperparameters of a predictor have not been
changed in comparison to the baseline approach to isolate solely the effect of
utilizing virtual trajectory as a feature for prediction. In LSTM/GRU-I2, only
the number of neurons in the hidden layer and the number of epochs have been
doubled in comparison to the baseline approach. This has been done with an
intuition of better handling the complexity of utilizing virtual coordinates
as an additional input feature compared to LSTM/GRU-I1.

As visible in the figure, for both LSTM- and GRU-based approaches, the
versions utilizing the virtual coordinates generally outperform the baselines
based on utilizing solely historical knowledge about the physical movement
trajectories of the VR users. Additionally, there are also benefits of
hyperparameter tuning when utilizing virtual coordinates as an additional
input feature. For example, the average per-user SE of the prediction is
deceased from roughly 0.0003 m2 in LSTM-B to less than 0.0001 m2 in LSTM-V,
representing a threefold increase in the prediction accuracy. Moreover, a
roughly fivefold improvement is observed for the GRU-based prediction when
utilizing virtual coordinates of the VR users in comparison to the more
traditional baseline. We argue that these results demonstrate that certain
contextual information from the virtual world, in this case the virtual
coordinates of the VR users, can be utilized for improving the performance of
the RNN-based near-term movement trajectory prediction approaches in the
physical VR setups. We believe that this conclusion can be generalized and the
fact that we have demonstrated the benefits for two types of prediction
approaches makes an argument for supporting this intuition.

Figure 6: SEs achieved by different versions of the LSTM approach Figure 7:
SEs achieved by different versions of the GRU approach Figure 8: Training time
of different approaches

The training duration of the four considered approaches is depicted in Figure
8. Note that the execution times have been captured using the simulator
running on a contemporary MacBook Air laptop and as such their absolute values
are indicative solely to that platform. Nonetheless, several interesting
trends can be observed. The figure depicts the execution times of different
approaches in cases when the short-term trajectory prediction is enabled and
vice-versa. Intuitively, where the trajectory prediction is not enabled, the
simulation times of both “baseline” and “virtual”approaches are comparable, as
shown in the figure. This is an expected behaviour as the prediction
approaches are in this case neither trained nor involved in the prediction,
but it nonetheless makes an argument about the correctness of the
implementation.

In case the short-term trajectory prediction is enabled, one can first observe
that the GRU-based approaches generally features lower simulation times than
the LSTM-based alternatives. This is a well-known behaviour demonstrating one
of the primarily benefits of GRU over LSTMs, namely their faster execution and
training. Finally and perhaps counterintuitively, the “virtual” version of the
LSTM-based approach features higher simulation time than its ’baseline’
counterpart, while that is not the case for GRU-flavored prediction methods.
We argue this discrepancy comes from the fact that an increase in the number
of input features of the “virtual” version of both approaches is not the
primary factor in their execution times, despite the fact that it is certainly
to an extent contributive to the increase in the execution times. On the
contrary, we argue that the execution times are to a greater extent affected
by the hyperparameterization of the approaches, with the most significant
contributions stemming from the selected number of batches, epochs, and
neurons. In case of LSTM, the comparatively higher number of sample per batch
in LSTM-V allows for greater parallelization of the training process, which
arguably has a larger effect on the execution time than the combined effect of
a slight increase in the number of training epochs and the increase in the
input feature dimensionality compared to LSTM-B, hence the execution time of
LSTM-V is lower than for LSTM-B. In case of GRU, an increase in the number of
neurons in the hidden GRU layer and in the dimensionality of the input
features jointly affect the increase in the execution time of GRU-V compared
to GRU-B. In conclusion, we argue the increase in the dimensionality of the
input features when considering virtual coordinates has only a slight negative
effect on the training and execution times of the considered approaches, while
a more substantial affect can be expected from hyperparameter tuning.

(a) LSTM-V

(b) GRU-V

Figure 9: Multiuser performance results for LSTM-V and GRU-V

Finally, the extended simulator enables straightforward evaluation of the
performance of redirected walking and short-term trajectory prediction for a
varying number of coexisting VR users. We utilize this feature to evaluate if
the considered approaches can, once optimally trained for a two-user system,
be used for such prediction for a varying number of coexisting users. Figure 9
depicts the output of the simulator (i.e., a set of performance metrics) for
the two optimally-parameterized prediction approaches for different numbers of
coexisting VR users. The graphs in the first row depict the number of
perceivable resets experienced by each VR user for a varying number of the VR
users. Similarly, the graphs in the second row depict the distances the users
are able to traverse between two consecutive perceivable resets. As visible
from the figure, the numbers of perceivable resets and the distances between
them are highly comparable for the two utilized approaches. This is an
expected behaviour as the same redirected walking approach, which has an
affect on the number of perceivable rests and their respective distances, has
been utilized for both shot-term trajectory prediction approaches. Moreover
and as visible in the figure, the number of such resets increases (and,
therefore, the corresponding distances between consecutive resets decrease)
with an increase in the number of users, which is again an expected behaviour
as reported in [17]. We depict these two metrics solely to demonstrate that
there are significant differences in the performance of the prediction system
as a function of a varying number of VR users.

The above suggests that datasets used for training of short-term trajectory
prediction are not comparable for a varying number of users. Nonetheless, the
prediction performance for both of the utilized approaches is highly
comparable across a varying number of coexisting VR users, as indicated int he
bottom graphs of the figure depicting the SEs of predictions. This suggests
that both systems could, without significant prediction performance
degradation, be used in scenarios in which the users dynamically immerse in or
terminate shared VR experiences. In addition, one can observe that the GRU-V
approach generally outperforms LSTM-V, with average reduction in the observed
SEs of around 50%, suggesting their utility for the movement trajectory
prediction in full-immersive multiuser VR setups with redirected walking.

##  IV Conclusion

We have shown that Long Short-Term Memory (LSTM) and Gated Recurrent Unit
(GRU)-based Recurrent Neural Networks (RNNs) are promising candidates for the
near-future movement trajectory prediction in multiuser full-immersive Virtual
Reality (VR) with redirected walking. We have also demonstrated the benefits
of utilizing virtual context such as the movement trajectory in the virtual
world as an input feature for such prediction, as well as shown that that the
prediction system, once trained on assuming on a static number of users, can
maintain the prediction accuracy once the number of users in the system
changes, without the need for re-training. The main direction of our ongoing
and future work is and will be focused on experimentally confirming our
insights with realistic traces.

## Acknowledgments

Filip Lemic was supported by the EU H2020 Marie Skłodowska-Curie project
”Scalable Localization-enabled In-body Terahertz Nanonetwork” (nr. 893760).

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  *[VR]: Virtual Reality
  *[LoS]: Line-of-Sight
  *[LSTM]: Long Short-Term Memory
  *[GRU]: Gated Recurrent Unit
  *[RNNs]: Recurrent Neural Network
  *[RNN]: Recurrent Neural Network
  *[GRUs]: Gated Recurrent Unit
  *[LSTMs]: Long Short-Term Memory
  *[AP]: Access Point
  *[QoE]: Quality of Experience
  *[AoA]: Angle of Arrival
  *[SE]: Squared Error
  *[APF-RDW]: Artificial Potential Field Redirected Walking
  *[APF-R]: Artificial Potential Field Reseting
  *[SEs]: Squared Error