# Persistency Of Non $n$-Local Correlations In Noisy Linear Networks

Kaushiki Mukherjee<EMAIL_ADDRESS>Department of Mathematics,
Government Girls’ General Degree College, Ekbalpore, Kolkata-700023, India.
Indranil Chakrabarty<EMAIL_ADDRESS>Center for Quantum
Science and Technology, International Institute of Information Technology,
Gachibowli, Hyderabad 500 032, Telangana, India.
Center for Security Theory and Algorithmic Research, International Institute
of Information Technology, Gachibowli, Hyderabad 500 032, Telangana, India.
Ganesh Mylavarapu<EMAIL_ADDRESS>Center for Quantum
Science and Technology, International Institute of Information Technology,
Gachibowli, Hyderabad 500 032, Telangana , India.

###### Abstract

Linear $n$-local networks are compatible with quantum repeaters based
entanglement distribution protocols. Different sources of imperfections such
as error in entanglement generation, communication over noisy quantum channels
and imperfections in measurements result in decay of quantumness across such
networks. From practical perspectives it becomes imperative to analyze non
classicality of quantum network correlations in presence of different types of
noise. Present discussion provides a formal characterization of non $n$-local
feature of quantum correlations in noisy network scenario. In this context,
persistency of non $n$-locality has been introduced. Such a notion helps in
analyzing decay of non $n$-local feature of network correlations with
increasing length of the linear network in presence of one or more causes of
imperfections.

##  I Introduction

Formulation of the Einstein-Podolsky-Rosen (EPR) paradox[1] points out
inexplicability of quantum predictions in terms of only local hidden variable
models. Such an impossibility in turn gives rise to the notion of
nonlocality[2]. Quantum nonlocality serves as a resource in multifaceted
practical tasks[4, 5, 6, 7, 8, 9, 10]. Over the past few years study of
nonlocality has been extended beyond paradigm of standard Bell scenario.
Manifestation of nonlocal network correlations has been a recent trend of
analysis in the field of quantum information theory[11].

Unlike standard Bell-CHSH scenario, any measurement scenario compatible with
network topology involves multiple distant sources. Each of the sources
distributes physical systems to a subset of distant observers. In case all the
sources in the entire network are independent of each other($n$-local
assumption), non $n$-local correlations may emerge under suitable measurement
contexts.[11]. The simplest of this type of networks, commonly known as
bilocal network(see Fig.1 for $n$$=$$2$) was first introduced in [12] followed
by a vivid analysis in [13]. Keeping pace with utility of quantum networks in
various information processing tasks[15, 16, 17, 18, 19], study of $n$-local
networks has witnessed multi directional development[20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33].

Assumption of source independence adds new physical insights in analyzing non
classical behavior of quantum network correlations. For example, consider a
$(n+1)$-partite entanglement swapping network(see Fig.1) involving $n$
independent sources $\mathcal{S}_{i}(i$$=$$1,2,..,n)$ arranged in a linear
fashion. Each source distributes a two qubit entangled state between a pair of
parties(detailed discussion in II.3). All the parties are thus not receiving
qubits from a single source. Hence, unlike standard Bell scenario, initially
they do not share any common past. Moreover some of the parties perform a
single measurement. This leads to another striking difference with standard
Bell experiment where each party must randomly and independently choose from a
collection of two or more inputs[10]$.\,$ $n$-local assumption thus reduces
requirements for exploiting non classicality in quantum networks[14, 12, 13].

Quantum repeaters form building blocks of any network meant for distributing
entanglement between distant observers across a large length of quantum
channel[15]. Now entanglement swapping forms the basis of designing quantum
repeater networks. So any such network structure can be considered as a
$n$-local network[11]. In ideal scenario, under suitable measurement contexts,
nonlocality in terms of non $n$-locality is thus generated in the network.
However in practical situations various factors of difficulties such as
imperfection in entanglement generation, communication over noisy quantum
channels and many others hinder distribution of entanglement over the entire
length of the chain. Consequently, unlike that in idealistic scenario,
simulation of non $n$-local correlations in the entire network structure
becomes impossible. At this junction it becomes imperative to explore for how
long such non classical behavior can be observed. To facilitate the discussion
we have introduced the concept of persistency in this context.

In literature, idea of persistency has been used to characterize different
types of multipartite quantum correlations[34, 35, 36]. Starting from a given
$m$ partite state $\rho$(say) exhibiting some form of quantum correlation
$\mathcal{C}$(say), number of parties is gradually decreased so as to find the
minimum number of parties $m^{{}^{\prime}}$(say) such that none of the
possible $m^{{}^{\prime}}$ partite reduced state exhibits $\mathcal{C}.$
$m^{{}^{\prime}}$ is usually referred to as persistency of $\rho$ with respect
to the specified quantum correlation($\mathcal{C}$). For current discussion,
we have introduced concept of persistency on a different note. Here it will be
used for exploiting sustainability of non $n$-local feature varying with
length of a network in presence of different types of noise.

We have analyzed generation of non $n$-local correlations in presence of
various sources of imperfection. For our purpose we have considered $n$-local
linear[21] networks. There may be error in entanglement generation at the
sources. Distribution of qubits may then occur over noisy channels. Also the
observers may be using local imperfect measurement devices at their end. We
have considered all such potential sources of errors. For rest of our work,
$n$-local networks affected by at least one such type of imperfection are
referred to as noisy $n$-local networks. To characterize non $n$-local
correlations in such networks we put forward the notion of persistency of non
$n$-locality.

Firstly, we have derived the non $n$-locality detection criterion for noisy
networks. That criterion is further used to develop the concept of
persistency. First concept of persistency has been introduced in presence of
single noise factor at a time. Then the notion has been generalized for more
practical situations when the network is affected by two or more noise factors
simultaneously.

Rest of the work is organized as follows: Some basic preliminaries are
discussed in Sec.II. Characterization of noisy $n$-local linear is given in
Sec.III. Persistency of non $n$-local correlations is studied in Sec.IV
followed by some concluding remarks in Sec.V.

##  II Preliminaries

We first proceed to discuss some basic pre-requisites to be used in
forthcoming sections.

###  II.1 Density Matrix Representation Of Arbitrary Two Qubit State

Let $\varrho$ denote an arbitrary two qubit state. Density matrix of $\varrho$
in terms of Bloch parameters is given by:

$\small{\varrho}=\small{\frac{1}{4}(\mathbb{I}_{2}\times\mathbb{I}_{2}+\vec{a}.\vec{\sigma}\otimes\mathbb{I}_{2}+\mathbb{I}_{2}\otimes\vec{b}.\vec{\sigma}+\sum_{j_{1},j_{2}=1}^{3}w_{j_{1}j_{2}}\sigma_{j_{1}}\otimes\sigma_{j_{2}})},$
(1)

where $\vec{\sigma}$$=$$(\sigma_{1},\sigma_{2},\sigma_{3}),$ $\sigma_{j_{k}}$
denote Pauli operators along three mutually perpendicular directions
($j_{k}$$=$$1,2,3$). $\vec{a}$$=$$(x_{1},x_{2},x_{3})$ and
$\vec{b}$$=$$(y_{1},y_{2},y_{3})$ denote local bloch vectors
($\vec{a},\vec{b}$$\in$$\mathbb{R}^{3}$) corresponding to party $\mathcal{A}$
and $\mathcal{B}$ respectively with $|\vec{a}|,|\vec{b}|$$\leq$$1$ and
$(w_{i,j})_{3\times 3}$ denotes correlation tensor $\mathcal{W}$(real). Matrix
elements $w_{j_{1}j_{2}}$ are given by
$w_{j_{1}j_{2}}$$=$$\textmd{Tr}[\rho\,\sigma_{j_{1}}\otimes\sigma_{j_{2}}].$
$\mathcal{W}$ can be diagonalized by subjection it to suitable local unitary
operations[39, 40]. Simplified expression is then given by:

$\small{\varrho}^{{}^{\prime}}=\small{\frac{1}{4}(\mathbb{I}_{2}\times\mathbb{I}_{2}+\vec{\mathfrak{a}}.\vec{\sigma}\otimes\mathbb{I}_{2}+\mathbb{I}_{2}\otimes\vec{\mathfrak{b}}.\vec{\sigma}+\sum_{j=1}^{3}t_{jj}\sigma_{j}\otimes\sigma_{j})},$
(2)

$T$$=$$\textmd{diag}(t_{11},t_{22},t_{33})$ denote the correlation matrix in
Eq.(2) where $t_{11},t_{22},t_{33}$ are the eigen values of
$\sqrt{\mathcal{W}^{T}\mathcal{W}},$ i.e., singular values of $\mathcal{W}.$

###  II.2 $n$-local Linear Networks

Consider a network with $n$ sources
$\mathcal{S}_{1},\mathcal{S}_{2},...\mathcal{S}_{n}$ and $n+1$ parties
$\mathcal{A}_{1},\mathcal{A}_{2},...,\mathcal{A}_{n+1}$ arranged in a linear
pattern(see Fig.1). $\forall i$$=$$1,2,...,n,$ source $\mathcal{S}_{i}$
independently distributes physical systems(characterized by $\lambda_{i}$) to
$\mathcal{A}_{i}$ and $\mathcal{A}_{i+1}.$ For each of $i$$=$$2,3,..,n,$
$\mathcal{A}_{i}$ receives two particles and is referred to as central party.
Each of other two parties $\mathcal{A}_{1}$ and $\mathcal{A}_{n+1}$ receives
one particle and is referred to as extreme party. $\mathcal{S}_{i}$ is
characterized by variable $\lambda_{i}.$ As sources are independent, joint
distribution of $\lambda_{1},...,\lambda_{n}$ is factorizable:

$\rho(\lambda_{1},...\lambda_{n})=\Pi_{i=1}^{n}\rho_{i}(\lambda_{i})$ (3)

where $\forall i,\,\rho_{i}$ denotes the normalized distribution of
$\lambda_{i}.$ Eq.(3) represents $n$-local constraint.
$\forall i$$=$$2,3,...n-1$ party $\mathcal{A}_{i}$ performs single measurement
$y_{i}$ on joint state of two subsystems received from $\mathcal{S}_{i-1}$ and
$\mathcal{S}_{i}.$ Each of $\mathcal{A}_{1}$ and $\mathcal{A}_{n+1}$ selects
from a collection of two dichotomous inputs. $n+1$ partite network
correlations are local if:

$\displaystyle\small{p(o_{1},\vec{\mathfrak{o}}_{2},...,\vec{\mathfrak{o}}_{n},o_{n+1}|y_{1},y_{n+1})}=$
$\displaystyle\int_{\Lambda_{1}}\int_{\Lambda_{2}}...\int_{\Lambda_{n}}d\lambda_{1}d\lambda_{2}...d\lambda_{n}\,\rho(\lambda_{1},\lambda_{2},...\lambda_{n})N_{1},\,\textmd{where}$
$\displaystyle
N_{1}=p(o_{1}|y_{1},\lambda_{1})\Pi_{j=2}^{n}p(\vec{\mathfrak{o}}_{j}|\lambda_{j-1},\lambda_{j})p(o_{n+1}|y_{n+1},\lambda_{n})$

Notations appearing in Eq.(II.2) are detailed below:

  * •

$\forall j,$ $\Lambda_{j}$ labels the set of all possible values of local
hidden variable $\lambda_{j}.$

  * •

$y_{1},y_{n+1}$$\in\\{0,1\\}$ denote measurements of $\mathcal{A}_{1}$ and
$\mathcal{A}_{n+1}$ respectively.

  * •

$o_{1},o_{n+1}$$\in$$\\{\pm 1\\}$ denote outputs of $\mathcal{A}_{1}$ and
$\mathcal{A}_{n+1}$ respectively.

  * •

$\forall j,$ $\vec{\mathfrak{o}}_{j}$$=$$(o_{j1},o_{j2})$ labels four outputs
of input $y_{j}$ for $o_{ji}$$\in$$\\{0,1\\}$

Correlations are $n$-local if those satisfy both Eqs.(3,II.2). So any set of
$n+1$ partite correlations that do not satisfy both of these constraints are
termed as non $n$-local.

$n$-local inequality[21] corresponding to this network scenario is:

$\displaystyle\sqrt{|I|}+\sqrt{|J|}\leq 1,\,\textmd{where}$ $\displaystyle
I=\frac{1}{4}\sum_{y_{1},y_{n+1}}\langle
O_{1,y_{1}}O_{2}^{0}.....O_{n}^{0}O_{n+1,y_{n+1}}\rangle$ $\displaystyle
J=\frac{1}{4}\sum_{y_{1},y_{n+1}}(-1)^{y_{1}+y_{n+1}}\langle\small{O_{1,y_{1}}O_{2}^{1}...O_{n}^{1}O_{n+1,y_{n+1}}}\rangle\,\,\textmd{with}$
$\displaystyle\langle
O_{1,y_{1}}O_{2}^{i}.....O_{n}^{i}O_{n+1,y_{n+1}}\rangle=\sum_{\mathcal{D}}(-1)^{\mathfrak{o}_{1}+\mathfrak{o}_{n+1}+\mathfrak{o}_{2i}+...\mathfrak{o}_{ni}}N_{2},$
$\displaystyle\textmd{\small{where}}\,N_{2}=\small{p(\mathfrak{o}_{1},\vec{\mathfrak{o}}_{2},...,\vec{\mathfrak{o}}_{n},\mathfrak{o}_{n+1}|y_{1},y_{n+1})},\,i=0,1$
$\displaystyle\textmd{\small{and}}\,\mathcal{D}=\\{\mathfrak{o}_{1},\mathfrak{o}_{21},\mathfrak{o}_{22},...,\mathfrak{o}_{n1},\mathfrak{o}_{n2},\mathfrak{o}_{n+1}\\}$
(5)

Violation of Eq.(II.2) ensures non $n$-local nature of corresponding
correlations.

Figure 1: _Schematic diagram of $n$-local linear network[21]_

###  II.3 Quantum Linear $n$-local Network Scenario

In the $n$-local network, let $\mathcal{S}_{i}(i$$=$$1,2,...,n)$ generate an
arbitrary two qubit state $\varrho_{i}.$ Each of the central parties
$\mathcal{A}_{i}(i$$=$$2,3,...,n)$ thus receives two qubits: one of
$\varrho_{i-1}$ and another of $\varrho_{i}.$ $\mathcal{A}_{1}$ and
$\mathcal{A}_{n+1}$ receive single qubit of $\varrho_{1}$ and $\varrho_{n}$
respectively. Let each of the central parties perform projection in Bell basis
$\\{|\psi^{\pm}\rangle,|\phi^{\pm}\rangle\\}$, often referred to as Bell state
measurement(BSM[13]). Let $M_{i}$ denote BSM of central party
$\mathcal{A}_{i}.$ Let each of the extreme parties perform projective
measurements in any one of two arbitrary directions:
$\\{\vec{m}_{0}.\vec{\sigma},\vec{m}_{1}.\vec{\sigma}\\}$ for
$\mathcal{A}_{1}$ and
$\\{\vec{n}_{0}.\vec{\sigma},\vec{n}_{1}.\vec{\sigma}\\}$ for
$\mathcal{A}_{n+1}$ with
$\vec{m}_{0},\vec{m}_{1},\vec{n}_{0},\vec{n}_{1}$$\in$$\mathcal{R}^{3}.$ Under
this measurement settings, non $n$-local correlations are detected by
violation of Eq.(II.2) if[30]:

$\sqrt{\Pi_{i=1}^{n}t_{i11}+\Pi_{i=1}^{n}t_{i22}}>1$ (6)

with $t_{i11},t_{i22}$ denoting largest two singular values of correlation
tensor ($T_{i}$) of $\varrho_{i}\,(i$$=$$1,2,...,n).$ In case Eq.(6) is
violated nothing can be concluded about $n$-local behavior of the
correlations.

##  III Noisy $n$-local linear network

Consider $n$-local linear network(Fig.1). Entire procedure in the network can
be divided into two phases: Preparation Phase and Measurement Phase. Former
phase further comprises of two parts: generation and distribution of
entanglement. For analysis of non $n$-locality in the noisy network, errors
are considered in all these stages. In case the network is used for
distribution of entanglement, ideally pure entanglement is to be distributed
from each source $S_{i}(i$$=$$1,2,...,n)$. However, errors in the preparation
phase lead to distribution of mixed two qubit state $\varrho_{i}$ among the
parties $A_{i}$ and $A_{i+1}\forall i.$
We first analyze the correlations considering error in measurement stage.
Precisely speaking, under measurement imperfections, closed form of upper
bound of $n$-local inequality(Eq.(II.2)) is derived for arbitrary two qubit
states. This form is further utilized in exploiting non $n$-locality under
effect of errors in preparation phase.

###  III.1 Imperfection In Measurements

As discussed in subsec.II.3, each of
$\mathcal{A}_{2},\mathcal{A}_{3},...,\mathcal{A}_{n}$ performs BSM. Now let
the devices fail to detect particles with some probability. Let
$\beta_{i}$$\in$$[0,1]$ characterize imperfection in measurement operator
$M_{i}$ in the sense that it fails to detect with probability $1-\beta_{i}.$
Measurement operator $M_{i}^{noisy}$(say) of $\mathcal{A}_{i}$ thus turns out
to be a POVM with the elements $\\{M_{ij_{1}j_{2}}^{noisy}\\}$ given by:

$\displaystyle M_{i,00}^{noisy}$ $\displaystyle=$
$\displaystyle\beta_{i}|\phi^{+}\rangle\langle\phi^{+}|+\frac{1-\beta_{i}}{4}\mathbb{I}_{2\times
2}$ $\displaystyle M_{i,01}^{noisy}$ $\displaystyle=$
$\displaystyle\beta_{i}|\phi^{-}\rangle\langle\phi^{-}|+\frac{1-\beta_{i}}{4}\mathbb{I}_{2\times
2}$ $\displaystyle M_{i,10}^{noisy}$ $\displaystyle=$
$\displaystyle\beta_{i}|\psi^{+}\rangle\langle\psi^{+}|+\frac{1-\beta_{i}}{4}\mathbb{I}_{2\times
2}$ $\displaystyle M_{i,11}^{noisy}$ $\displaystyle=$
$\displaystyle\beta_{i}|\psi^{-}\rangle\langle\psi^{-}|+\frac{1-\beta_{i}}{4}\mathbb{I}_{2\times
2},\,\forall i=2,3,...,n$ (7)

Now, it may be noted that in case $\mathcal{A}_{i}$ performs perfect BSM then
$\\{|\phi^{\pm}\rangle\langle\phi^{\pm}|,|\psi{\pm}\rangle\langle\psi^{\pm}|\\}$
is the set of possible projectors. $\forall i$$=$$2,3,...,n,$ denoting
$M_{i,00}^{ideal},M_{i,01}^{ideal},M_{i,10}^{ideal},M_{i,11}^{ideal}$ as the
measurement operators corresponding to the BSM projectors, POVM elements of
imperfect BSM(Eq.(III.1)) can be represented as:

$M_{i,j_{1}j_{2}}^{noisy}=\beta_{i}M_{i,j_{1}j_{2}}^{ideal}+\frac{1-\beta_{i}}{4}\mathbb{I}_{2\times
2}$ (8)

Similar to imperfection in measurement settings of central parties, let each
of the two extreme parties also use imperfect detecting devices. For party
$\mathcal{A}_{1},$ let $\mu$$\in$$[0,1]$ parametrizes faulty measurement
device. For single qubit projection, such a device fails to detect any output
with probability $1-\mu.$ POVM resulting due to imperfection in
$\vec{m}_{k}.\vec{\sigma}$ thus has two elements
$\\{P_{kj}^{noisy}\\}_{j=0,1}$ given by:

$\displaystyle P_{k0}^{noisy}$ $\displaystyle=$
$\displaystyle\mu\mathcal{O}^{+}+\frac{1-\mu}{2}\mathbb{I}_{2}$ $\displaystyle
P_{k1}^{noisy}$ $\displaystyle=$
$\displaystyle\mu\mathcal{O}^{-}+\frac{1-\mu}{2}\mathbb{I}_{2},\,k=0,1$ (9)

where $\mathcal{O}^{+}(\mathcal{O}^{-})$ denote projection operator
corresponding to $+1(-1)$ eigen value. $\mathcal{O}^{\pm}$ denote projectors
corresponding to perfect projective measurement. Labeling
$P_{k0}^{ideal},P_{k1}^{ideal}$ as the projectors corresponding perfect
measurement $\vec{m}_{k}.\vec{\sigma},$ alternate representation of POVM
elements(Eq.(III.1)) is given by:

$P_{ki}^{noisy}=\mu P_{ki}^{ideal}+\frac{1-\mu}{2}\mathbb{I}_{2},\,i,k=0,1$
(10)

Similarly for $\mathcal{A}_{n+1},$ let $1-\nu$ denote failure probability in
$\vec{n}_{k}.\vec{\sigma}.$ Elements of corresponding POVM are given by:

$\displaystyle Q_{k0}^{noisy}$ $\displaystyle=$
$\displaystyle\nu\mathcal{Q}^{+}+\frac{1-\nu}{2}\mathbb{I}_{2}$ $\displaystyle
Q_{k1}^{noisy}$ $\displaystyle=$
$\displaystyle\nu\mathcal{Q}^{-}+\frac{1-\nu}{2}\mathbb{I}_{2},\,k=0,1$ (11)

with $\mathcal{Q}^{+}(\mathcal{Q}^{-})$ denote projection operator
corresponding to $+1(-1)$ eigen value.
We now put forward the criterion that suffices to detect non $n$-locality when
all the parties are performing imperfect measurements. For
$\mathcal{A}_{n+1},$ analogue of representation given by Eq.(10) is:

$Q_{ki}^{noisy}=\nu Q_{ki}^{ideal}+\frac{1-\nu}{2}\mathbb{I}_{2},\,i,k=0,1$
(12)

Theorem.1: With each source $\mathcal{S}_{i}$ generating an arbitrary two
qubit state and all the parties performing imperfect measurements, sufficient
criterion for detecting non $n$-locality in a linear $n$-local network is
given by:

$\sqrt{\Pi_{i=1}^{n}t_{i11}+\Pi_{i=1}^{n}t_{i22}}>\frac{1}{(\mu\nu\Pi_{j=2}^{n}\beta_{j})^{\frac{1}{2}}}.$
(13)

Proof: See Appendix.
Eq.(13) being a sufficient detection criterion, violation of the same gives no
definite conclusion regarding simulation of non $n$-local correlations in
corresponding noisy network. Above criterion points out the effect of the
imperfection parameters over the usual non $n$-locality criterion(Eq.(6)).
Comparing right hand side of both Eqs.(6,13) it is observed that if at least
one of the detectors turns out to be imperfect with some non zero probability
then that reduces chances for generation of non $n$-locality in the noisy
network compared to the ideal situation. Moreover, if any of the detectors
used in the network always fails to detect, i.e., corresponding success
probability turns out be $0$ then above criterion(Eq.(13)) can never be
satisfied.

###  III.2 Noisy Entanglement Generation

Let us first discuss an ideal entanglement generation procedure[15]. Without
loss of any generality, we consider the ideal generation of
$|\phi^{-}\rangle\langle\phi^{-}|.$ Let $\varrho$$=$$|01\rangle\langle 01|$ be
the state at each source $\mathcal{S}_{i}.$ To generate entanglement, Hadamard
gate ($\mathcal{H}$) is applied on first qubit. Considering first qubit as
control qubit, C$\small{-}$NOT ($\mathcal{C}\small{-}\mathcal{NOT}$) gate is
then applied resulting in generation of the Bell state
$|\phi^{-}\rangle\langle\phi^{-}|$[15]. Ideally, each of
$\mathcal{S}_{1},\mathcal{S}_{2},..,\mathcal{S}_{n}$ is supposed to generate
$|\phi^{-}\rangle\langle\phi^{-}|.$

However in practical situations imperfections in preparation devices lead to
generation of mixed entangled states. Such imperfection results from erroneous
applications of Hadamard and/or C$\small{-}$NOT
($\mathcal{C}\small{-}\mathcal{NOT}$). At each source $\mathcal{S}_{i}$, let
$\alpha_{i}$ and $\delta_{i}$ denote the imperfection parameters
characterizing $\mathcal{H}$ and $\mathcal{C}\small{-}\mathcal{NOT}$
respectively. $\forall i$$=$$1,2,...,n,$ starting from
$\varrho_{i}$$=$$|01\rangle\langle 01|,$ noisy Hadamard gate generates[41]:

$\displaystyle\varrho^{{}^{\prime}}_{i}$ $\displaystyle=$
$\displaystyle\alpha_{i}(\mathcal{H}\otimes\mathbb{I}_{2}\varrho\mathcal{H}^{\dagger}\otimes\mathbb{I}_{2})+\frac{1-\alpha_{i}}{2}\mathbb{I}_{2}\otimes\varrho_{2i},\,\textmd{with}\,\alpha_{i}\in[0,1]$
$\displaystyle\textmd{and}\,\,\varrho_{2i}=\textmd{Tr}_{1}(\varrho_{i}).$
$\displaystyle=$ $\displaystyle\frac{1}{2}(|00\rangle\langle
00|+|10\rangle\langle 10|)-\frac{\alpha_{i}}{2}(|00\rangle\langle
10|+|10\rangle\langle 00|)$

Subjection of $\varrho^{{}^{\prime}}_{i}$ to noisy
$\mathcal{C}\small{-}\mathcal{NOT}$ gives[41]:

$\displaystyle\varrho^{{}^{\prime\prime}}_{i}$ $\displaystyle=$
$\displaystyle\delta_{i}(\mathcal{C}\small{-}\mathcal{NOT}\varrho^{{}^{\prime}}_{i}(\mathcal{C}\small{-}\mathcal{NOT})^{\dagger})+\frac{1-\delta_{i}}{4}\mathbb{I}_{2}\otimes\mathbb{I}_{2}$
(15) $\displaystyle=$
$\displaystyle\frac{1}{4}(\sum_{i,j=0}^{1}(1+(-1)^{i+j}\delta_{i})|ij\rangle\langle
ij|-2\alpha_{i}\delta_{i}(|11\rangle\langle 00|+$
$\displaystyle|00\rangle\langle 11|))$

Correlation tensor of $\varrho^{{}^{\prime\prime}}_{i}$ is
$\textmd{diag}(-\alpha_{i}\delta_{i},\alpha_{i}\delta_{i},\delta_{i}).$ In
case $\mathcal{S}_{i}$ distributes $\varrho_{i}^{{}^{\prime\prime}}$ and the
parties perform imperfect measurements, non $n$-locality is observed if:

$\sqrt{\Pi_{i=2}^{n}\delta_{i}\beta_{i}\mu\nu\delta_{1}(1+\Pi_{j=1}^{n}\alpha_{j})}>1\,$
(16)

###  III.3 Noisy Quantum Communication

Let us now consider that communication of $\varrho_{i}^{{}^{\prime\prime}}$
from $\mathcal{S}_{i}$ to respective parties is occurring through noisy
channels. Such a communication affects generation of non $n$-local
correlations for obvious reasons. To analyze effect of such noise parameters
over $n$-locality detection, we are considering a few standard noisy
channels[42].

####  III.3.1 Amplitude Damping Channel

$\forall i$$=$$1,2,...,n,$ let $\gamma^{amp}_{i},\xi_{i}^{amp}$ characterize
channels connecting $\mathcal{S}_{i}$ with $\mathcal{A}_{i}$ and
$\mathcal{A}_{i+1}$ respectively. Two qubits of
$\varrho^{{}^{\prime\prime}}_{i}$(Eq.(15)) are thus passed through two
different amplitude damping channels. Let
$\varrho_{i}^{{}^{\prime\prime\prime}}$ denote corresponding noisy state. Any
amplitude damping channel(parametrized by $\gamma^{amp},$ say) is represented
by Krauss operators $|0\rangle\langle
0|$$+$$\sqrt{1-\gamma^{amp}}|1\rangle\langle 1|$ and
$\sqrt{\gamma^{amp}}|0\rangle\langle 1|.$ Correlation tensor of
$\varrho_{i}^{{}^{\prime\prime\prime}}$ is given by
$\textmd{diag}(-\alpha_{i}\delta_{i}\sqrt{D_{i}^{amp}},\alpha_{i}\delta_{i}\sqrt{D^{amp}_{i}},\delta_{i}D^{amp}_{i}+\gamma^{amp}_{i}\xi^{amp}_{i})$
where $D^{amp}_{i}$$=$$(1-\gamma^{amp}_{i})(1-\xi^{amp}_{i}).$ Using the
closed form of $n$-local bound(Eq.(13)) under imperfect measurement context
non $n$-locality is detected if:

$\displaystyle\sqrt{\Pi_{i=2}^{n}\beta_{i}\mu\nu\textmd{Max}(2F_{1},F_{2})}>1\,\,$
$\displaystyle,\,\textmd{where}$ $\displaystyle
F_{1}=\Pi_{j=1}^{n}\alpha_{j}\delta_{j}\sqrt{(1-\gamma^{amp}_{j})(1-\xi^{amp}_{j})}$
and $\displaystyle
F_{2}=\Pi_{j=1}^{n}\alpha_{j}\delta_{j}\sqrt{(1-\gamma^{amp}_{j})(1-\xi^{amp}_{j})}+$
$\displaystyle\Pi_{j=1}^{n}(\delta_{j}(1-\gamma^{amp}_{j})(1-\xi^{amp}_{j})+\gamma^{amp}_{j}\xi^{amp}_{j})$
(17)

####  III.3.2 Phase Damping Channel

$\forall i$$=$$1,2,...,n,$ Let $\gamma^{ph}_{i},\xi_{i}^{ph}$ characterize
channels connecting $\mathcal{S}_{i}$ with $\mathcal{A}_{i}$ and
$\mathcal{A}_{i+1}$ respectively. Let $\varrho_{i}^{{}^{\prime\prime\prime}}$
denote corresponding noisy state. Krauss operators corresponding to phase
damping channel(having noise parameter $\gamma^{ph},$ say) are given by
$|0\rangle\langle 0|$$+$$\sqrt{1-\gamma^{ph}}|1\rangle\langle 1|$ and
$\sqrt{\gamma^{ph}}|0\rangle\langle 1|.$ Correlation tensor of
$\varrho_{i}^{{}^{\prime\prime\prime}}$ is given by
$\textmd{diag}(-\alpha_{i}\delta_{i}\sqrt{D_{i}^{ph}},\alpha_{i}\delta_{i}\sqrt{D_{i}^{ph}},\delta_{i})$
where $D^{ph}_{i}$$=$$(1-\gamma^{ph}_{i})(1-\xi^{ph}_{i}).$ In this case non
$n$-locality is detected if:

$\displaystyle\sqrt{\Pi_{i=2}^{n}\beta_{i}\mu\nu\textmd{Max}(G_{1},G_{2})}>1\,\textmd{where}\qquad\qquad$
, $\displaystyle
G_{1}=2\Pi_{j=1}^{n}\alpha_{j}\delta_{j}\sqrt{(1-\gamma^{ph}_{j})(1-\xi^{ph}_{j})}\,\textmd{and}$
$\displaystyle
G_{2}=\Pi_{j=1}^{n}\alpha_{j}\delta_{j}\sqrt{(1-\gamma^{ph}_{j})(1-\xi^{ph}_{j})}+\Pi_{j=1}^{n}\delta_{j}$
(18)

After analyzing non $n$-locality detection in presence of noise we next
introduce the notion of persistency in this context

##  IV Persistency of non $n$-locality

Discussion in sec.III clearly points out dependence of upper bound of
$n$-local inequality over noise parameters. Closer observation of different
relations derived therein gives rise to the intuition that increasing length
of network hinders simulation of non $n$-local correlations. Formal
characterization of such an interpretation will be provided in this section.
In this context, let us now consider that for each of the three categories of
errors discussed in sec.III, noise parameters remain identical. To be precise:

  * •

each of $n$ noisy sources is
identical:$(\alpha_{i},\delta_{i})$$=$$(\alpha,\delta)\textmd{(}say)$
$\forall i$$=$$1,2,...,n$

  * •

parties are interconnected via identical noisy quantum channels

  * •

single parameter characterizing imperfection in measurements of central
parties
$\mathcal{A}_{2},...,\mathcal{A}_{n}:\,\beta_{2}$$=$$...$$=$$\beta_{n}$$=$$\beta\textmd{(}say).$

Under such assumptions Eq.(16) becomes:

$\sqrt{\mu\nu\delta^{n}\beta^{n-1}(1+\alpha^{n})}>1.$ (19)

We now define persistency of non $n$-locality for each of three types of
errors individually.

###  IV.1 $1^{st}$ Type of Persistency of non $n$-locality

Definition.1 $1^{st}$ type of persistency of non $n$-locality
$\mathcal{P}_{I}$(say) may be defined as the maximum number($n$) of
independent identical sources that can be connected so as to form a linear
$n$-local network where Eq.(II.2) detects non $n$-locality under the
assumption that each source distributes two qubit mixed entangled state
through noiseless quantum channels and all parties perform perfect
measurements.

Above definition can be interpreted as a measure of maximum length of an
entanglement distribution network in which non $n$-local correlations can be
detected when the sources fail to generate pure entanglement. The measure is
given in terms of independent sources as length of any network can be
specified by it. If $\mathcal{P}_{I}$$=$$m$ for a noisy network, addition of
even a single source(generating mixed entanglement) to the network will result
in generation of $(m$$+$$1)$ partite correlations whose non $(m$$+$$1)$-local
feature cannot be detected by Eq.(II.2).

Let us now consider a noisy network where error in entanglement generation is
the only source of noise. For $\beta$$=$$\mu$$=$$\nu$$=$$1,$ Eq.(19) becomes:

$\sqrt{\delta^{n}(1+\alpha^{n})}>1$ (20)

For any specified values of $\alpha,\delta,$ $\mathcal{P}_{I}$ is given by:

$\mathcal{P}_{I}=\lfloor n_{I}\rfloor$ (21)

where $n_{I}$ denotes the upper bound of $n$ given by Eq.(20). Variation of
$\mathcal{P}_{I}$ with that of $(\alpha,\delta)$ is plotted in Fig.2. For an
example, let $\alpha$$=$$\delta$$=$$0.9.$ Eq.(20) gives:

$n<n_{I}=4.567$ (22)

So $\mathcal{P}_{I}$$=$$4$ in this case.

Figure 2: _Variation of $1^{st}$ Type of Persistency of non $n$-locality with
that of variables parametrizing error in entanglement generation is shown
here._

###  IV.2 $2^{nd}$ Type of Persistency of non $n$-locality

Definition.2 $2^{nd}$ type of persistency of non $n$-locality
$\mathcal{P}_{II}$(say) may be defined as the maximum number($n$) of
independent identical sources that can be connected so as to form a linear
$n$-local network where non $n$-locality is detected by Eq.(II.2) when
communication over identical noisy quantum channels is the only source of
error in the network.
Let $\mathcal{N}$ denote a linear $n$-local network configuration for some
fixed value of $n$$=$$m\textmd{(say)}$ such that each of $m$ identical sources
distributes pure entanglement through identical noisy channels and all parties
perform perfect measurements. Let corresponding correlations turn out to be
non $m$-local. Now let $\mathcal{N}$ be extended to an $(m$$+$$1)$-local
network $\mathcal{N}^{{}^{\prime}}$ under prevalent conditions.
$\mathcal{P}_{II}$ turns out to be $m$ if Eq.(II.2) fails to detect non
$(m$$+$$1)$-locality(if any) of corresponding correlations.

Let us first consider that the sources and the parties are interconnected via
identical amplitude damping channels:
$\gamma^{amp}_{i}$$=$$\xi_{i}^{amp}$$=\gamma_{amp}\textmd{(say)},\forall
i$$=$$1,2,,...,n.$ Non $n$-locality is detected in the network if:

$\displaystyle\sqrt{\textmd{Max}(2F_{1},F_{2})}$ $\displaystyle>$
$\displaystyle 1,\,\textmd{where}$ $\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle(1-\gamma_{amp})^{n}\textmd{and}$ $\displaystyle F_{2}$
$\displaystyle=$ $\displaystyle
F_{1}+((1-\gamma_{amp})^{2}+(\gamma_{amp})^{2})^{n}$

Eq.(IV.2) is obtained from Eq.(III.3.1) under assumption of noisy
communication as the only source of error in the $n$-local network. Eq.(IV.2)
provides upper bound on the number of sources $n.$ If $n_{II}^{amp}$ denote
corresponding upper bound of n, then $2^{nd}$ type of persistency of non
$n$-locality is given by:$\mathcal{P}_{II}=\lfloor n_{II}^{amp}\rfloor.$
Dependence of source count($n$) and hence that of $\mathcal{P}_{II}^{amp}$ on
noise parameter(see Fig.3) is given by Eq.(IV.2).

Let us now consider the noisy network where communication through identical
phase damping channels is the only source of noise. Setting
$\gamma^{ph}_{i}$$=$$\xi_{i}^{ph}$$=\gamma_{ph}\textmd{(say)},\forall
i$$=$$1,2,,...,n$ and rest of the parameters to be $1$ in Eq.(III.3.2), non
$n$-locality detection criterion is given by:

$\sqrt{1+(1-\gamma_{ph})^{n}}>1.$ (24)

Eq.(24) in turn gives $\mathcal{P}_{II}^{ph}$$=$$\lfloor n_{II}^{ph}\rfloor$
with $n_{II}^{ph}$ denoting upper bound of $n$ given by the detection
criterion(Eq.(24). Comparison of Eqs.(IV.2,24) indicates
$\mathcal{P}_{II}^{ph}$$>$$\mathcal{P}_{II}^{amp}$ for any fixed value of
noise parameter $\gamma_{amp}$$=$$\gamma_{ph}$$=$$\gamma$(see Fig.3).

---

(a)  (b)

Figure 3: _Decrease in $\mathcal{P}_{II}$, i.e., $\mathcal{P}_{II}^{amp}$ with
increasing noise level($\gamma_{amp}$) in amplitude damping channels is
plotted (sub fig.a). Similarly, for phase damping channel,
$\mathcal{P}_{II}^{ph}$ vs $\gamma_{ph}$ is plotted in sub fig.b._

###  IV.3 $3^{rd}$ Type of Persistency of non $n$-locality

Definition.3 $3^{rd}$ type of persistency of non $n$-locality may be defined
as the maximum number($n$) of independent identical sources that can be
connected so that non $n$-locality is detected by Eq.(II.2) in corresponding
network under the assumption that each source distributes pure entanglement
over noiseless channels and all the parties perform imperfect measurements.
Let $\mathcal{P}_{III}$ denote $3^{rd}$ type of persistency of non
$n$-locality.

Consider $n$-local network $\mathcal{N}$ for some fixed value of
$n$$=$$m\textmd{(say)}$ such that each of $m$ identical sources distributes
pure entanglement through noiseless channels. Let each of the extreme
parties($\mathcal{A}_{1},\mathcal{A}_{m+1}$) perform imperfect projective
measurements whereas each of central
parties($\mathcal{A}_{2},..,\mathcal{A}_{m}$) performs imperfect Bell basis
measurement. Under such measurement contexts, let corresponding correlations
turn out to be non $m$-local. Now let $\mathcal{N}$ be extended to an
$(m$$+$$1)$-local network $\mathcal{N}^{{}^{\prime}}$ by adding another
identical source $\mathcal{S}_{m+1}.$ In $\mathcal{N}^{{}^{\prime}},$ there
are $m$ central parties. With all the parties performing imperfect
measurements, if Eq.(II.2) fails to detect non $(m$$+$$1)$-locality(if any) in
$\mathcal{N}^{{}^{\prime}}$ then $\mathcal{P}_{III}$$=$$m.$
With imperfection in measurements considered as only source of error, non
$n$-locality detection criterion Eq.(16) becomes:

$\sqrt{2\mu\nu\beta^{n-1}}>1.$ (25)

If $n_{III}$ denote upper bound of $n$ in Eq.(25) then
$\mathcal{P}_{III}$$=$$\lfloor n_{III}\rfloor.$ With increase in imperfection,
$\mathcal{P}_{III}$ decreases(see Fig.4). For example, in case
$\mu$$=$$\nu$$=$$\beta$$=$$0.9,$ non $n$-locality is detected up to $n$$=$$5.$
Hence, $P_{III}$$=$$5.$

Figure 4: _Variation of $3^{rd}$ Type of Persistency of non $n$-locality with
that of imperfection in measurements is shown. Here imperfection in single
qubit projective measurements of the extreme parties are parametrized by a
single variable $\mu$$=$$\nu$$=$$\Delta$(say). _

Till now persistency of non $n$-locality has been analyzed under the presence
of only one type of noise at a time. However for practical purposes, it is
important to study the same when at least two of the three possible factors of
noise are present in the network. So generalization of the notion follows
below.

###  IV.4 Persistency of non $n$-locality

Definition.4 Persistency of non $n$-locality($\mathcal{P},$say) may be defined
as the maximum number($n$) of independent identical sources that can be
connected to form a network such that Eq.(II.2) detects non $n$-local
correlations when at least two of the three noise factors are present in the
network.

Above definition corresponds to the most general notion of persistency.
Consider $n$-local network $\mathcal{N}$ for some fixed value of
$n$$=$$m\textmd{(say)}$ under the assumption that at least two of the three
categories of noise are present in the network. For better understanding,
without loss of any generality, let sources distribute mixed entanglement
whereas parties perform imperfect measurements. Let non $m$-locality be
observed in the network. On extension of $\mathcal{N}$ to an $(m$$+$$1)$-local
network in presence of existing noise factors only, if Eq.(II.2) fails to
detect non $n$-locality for $n$$=$$m+1,$ then $\mathcal{P}$$=$$m$ for
$\mathcal{N}.$ At this point it must be noted that in order to measure
$\mathcal{P},$ extension of any network $\mathcal{N}$ to another one
$\mathcal{N}^{{}^{\prime}}$(say) by adding identical sources must be
considered under the assumption that noise factors of $\mathcal{N}$ and
$\mathcal{N}^{{}^{\prime}}$ remain invariant.

Clearly $\mathcal{P}$$<$$\mathcal{P}_{1},\mathcal{P}_{2},\mathcal{P}_{3}.$ Let
us now provide an example for further illustration.
Let us consider a noisy network where noise due to quantum communication may
occur due to use of phase damp channels. Non $n$-locality detection
criterion(Eq.(III.3.2)) is given by:

$\displaystyle\sqrt{\beta^{n-1}\mu\nu\textmd{Max}(2G_{1},G_{2})}$
$\displaystyle>$ $\displaystyle 1,\,\textmd{where}$ $\displaystyle G_{1}$
$\displaystyle=$ $\displaystyle\alpha\delta(1-\gamma_{ph})^{n}\textmd{and}$
$\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle(\alpha\delta(1-\gamma_{ph}))^{n}$ (26)
$\displaystyle+\delta^{n}$

Persistency of non $n$-locality for some specific values of noise parameters
are given in Table.1.

Table 1: Persistency of non $n$-locality in networks under variation of noise factors. Error in communication of qubits is considered to be due to use of phase damping channels. $1^{st}$ row give persistency of non $n$-locality($\mathcal{P}$) when there is no error in entanglement generation. $2^{nd}$ row gives $\mathcal{P}$ when parties perform perfect measurements and $3^{rd}$ row gives $\mathcal{P}$ when qubits pass through noiseless channel respectively. Last row gives $\mathcal{P}$ when all three forms of errors exist in the network. Error in | Noisy | Imperfect | $\mathcal{P}$
---|---|---|---
Entanglement | Commu- | Measurements |
Generation | nication |  |
$(\alpha,\delta)$$=$$(1,1)$ |  $\gamma$$=$$0.1$ |  $(\mu,\nu,\beta)$$=$$(0.94,0.93,0.92)$ | $4$
$(\alpha,\delta)$$=$$(0.94,0.93)$ |  $\gamma$$=$$0.1$ |  $(\mu,\nu,\beta)$$=$$(1,1,1)$ | $7$
$(\alpha,\delta)$$=$$(0.92,0.95)$ |  $\gamma$$=$$0$ |  $(\mu,\nu,\beta)$$=$$(0.92,0.94,0.95)$ | $9$
$(\alpha,\delta)$$=$$(0.92,0.95)$ |  $\gamma$$=$$0.12$ |  $(\mu,\nu,\beta)$$=$$(0.94,0.93,0.95)$ | $4$

##  V Discussions

Characterization of non $n$-local correlations in presence of three different
noise factors is provided in present work. For our purpose existing upper
bound(Eq.(6)) of $n$-local inequality(Eq.(II.2)) has been used as the
detection criterion of non $n$-locality. Persistency of non $n$-locality has
been introduced for analyzing decay of non $n$-local correlations with
increasing length of a noisy network. Considering persistency of $n$-locality
for each of three types of noise individually, the notion has been generalized
to the more practical situation when at least two of the error factors exist
in the network.

To this end, it must be pointed out that from experimental perspectives,
current study turns out to be a simple form of error analysis for exploiting
non $n$-locality. Though we have considered three broad categories of errors
that usually occur in any entanglement swapping based network scenario, yet
the discussion is oversimplified as any discussion on the technical
difficulties, associated with experimental realization of quantum
networks[43], lies beyond the scope of this manuscript.

In [44], the authors have pointed out multiple problems associated with
physical implementation of any network configuration based on entanglement
swapping. For instance, one of the most significant problem is the exponential
decrease in coherence of quantum states. Such decoherence of a quantum system
occurs due to long range distribution of entanglement over noisy channels and
also on being subjected to operations over considerably long span of time[43].
Methods such as entanglement purification[45, 46, 47, 48], concentration[45,
48, 52, 53, 54, 55], distillation[49, 50, 51] have been developed to
distribute entanglement along long chain of network With technological
advancement in the field of quantum information science, practical
implementations of these procedures with tolerable error rates have become
possible[47, 48, 52, 53, 54, 55, 49, 50, 51]. Apart from loss in coherence,
long range fiber based quantum communications becomes challenging due to
photon loss, noise in photon detection and many other factors[43]. Exponential
decrease of signal-to-noise ratio with increasing length of the fiber in
quantum key distribution protocols is one of the consequences of limitations
over long range quantum communication. Though remarkable technological
progress has been attained over years[56, 57, 58, 59, 60, 61], yet, till date,
there exist several limitations constraining quantum communication over large
distance.

Apart from the issues mentioned above, experimentalists face several other
challenges while implementing a network configuration[43]. Hence, from
experimental perspectives, it becomes important to consider at least some of
these crucial factors while making any form of error analysis in a network
scenario. However, our analysis has not included any such practical problem.
At this junction, it is needed to be mentioned that the sole aim of present
study is to introduce the notion of persistency of non $n$-locality in context
of exploiting non-classicality in linear network configuration. So the study,
in its present form, can be considered to be in a nascent stage which needs
gradual upgradation keeping pace with present day technology. Consequently, it
will be interesting to extend the study taking into consideration the
practical problems.

In [62], entanglement swapping network has been used as Bell nonlocality
activation protocol. Key role of any such protocols is to generate Bell-CHSH
nonlocal quantum states starting from two or more Bell-CHSH local states. Over
years such type of protocols have been generalized so as to activate different
other notions of nonlocality[63, 64, 65]. Now, as already discussed in sec.I,
entanglement swapping network is a $n$-local network where each independent
source distributes entangled state. Present study on persistency of non
$n$-local correlations can thus be considered as one way of analyzing errors
in exploiting a particular form of non-classicality(non $n$-locality) of
$n+1$-partite correlations generated across any such network. Now, in place of
considering correlations across the entire network configuration, it will also
be interesting to study persistency of Bell nonlocality or any other notion of
non-classicality of the conditional states generated at the end of any such
activation network[62, 63].

Entire analysis is limited to noisy linear $n$-local networks only. It will be
interesting to exploit the same for any non linear configuration. Also in the
network scenarios considered here, each source distributes two qubit entangled
states. Analyzing decay of non classicality with growing imperfections in
network when each of the sources generate multipartite and/or higher
dimensional entangled states is a potential direction of future research.

##  VI Appendix

Proof of Theorem.1: Let us first consider the $n$-local inequality(Eq.(II.2))
for the noisy network:

$\displaystyle\sqrt{|I_{noisy}|}+\sqrt{|J_{noisy}|}\leq 1,\,\textmd{where}$
$\displaystyle I_{noisy}=\frac{1}{4}\sum_{y_{1},y_{n+1}}\langle
O_{1,y_{1}}O_{2}^{0}.....O_{n}^{0}O_{n+1,y_{n+1}}\rangle_{noisy}$
$\displaystyle
J_{noisy}=\frac{1}{4}\sum_{y_{1},y_{n+1}}(-1)^{y_{1}+y_{n+1}}\langle\small{O_{1,y_{1}}O_{2}^{1}...O_{n}^{1}O_{n+1,y_{n+1}}}\rangle_{noisy}\,\,\textmd{with}$
$\displaystyle\langle
O_{1,y_{1}}O_{2}^{i}.....O_{n}^{i}O_{n+1,y_{n+1}}\rangle_{noisy}=\sum_{\mathcal{D}}(-1)^{\mathfrak{o}_{1}+\mathfrak{o}_{n+1}+\mathfrak{o}_{2i}+...\mathfrak{o}_{ni}}N_{noisy},$
$\displaystyle\textmd{\small{where}}\,N_{noisy}=\small{p^{{}^{\prime}}(\mathfrak{o}_{1},\vec{\mathfrak{o}}_{2},...,\vec{\mathfrak{o}}_{n},\mathfrak{o}_{n+1}|y_{1},y_{n+1})},\,i=0,1$
$\displaystyle\textmd{\small{and}}\,\mathcal{D}=\\{\mathfrak{o}_{1},\mathfrak{o}_{21},\mathfrak{o}_{22},...,\mathfrak{o}_{n1},\mathfrak{o}_{n2},\mathfrak{o}_{n+1}\\}$
(27)

Different symbol $p^{{}^{\prime}}()$ has been used for probability terms so as
to discriminate those arising in noisy scenario from that in ideal scenario.
Let us denote the overall state in the network as
$\varrho$$=$$\otimes_{l=1}^{n}\varrho_{l}.$ Next we consider the expectation
terms given by Eq.(II.2). Without loss of any generality let us fix $i$$=$$0$
and fix the labeling of $(y_{1},y_{n+1})$ as $(0,0)$ and consider
corresponding expectation term $\langle
O_{1,0}O_{2}^{0}.....O_{n}^{0}O_{n+1,0}\rangle:$

$\displaystyle\langle O_{1,0}O_{2}^{0}.....O_{n}^{0}O_{n+1,0}\rangle_{noisy}$
$\displaystyle=$
$\displaystyle\sum_{\mathfrak{o}_{1},\mathfrak{o}_{21},\mathfrak{o}_{22},...,\mathfrak{o}_{n1},\mathfrak{o}_{n2},\mathfrak{o}_{n+1}}(-1)^{\mathfrak{o}_{1}+\mathfrak{o}_{n+1}+\mathfrak{o}_{20}+...+\mathfrak{o}_{n0}}\small{p(\mathfrak{o}_{1},\vec{\mathfrak{o}}_{2},...,\vec{\mathfrak{o}}_{n},\mathfrak{o}_{n+1}|0,0)}$
$\displaystyle=$
$\displaystyle\sum_{i,j=0}^{1}\sum_{g_{2},h_{2}=0}^{1}\sum_{g_{3},h_{3}=0}^{1}...\sum_{g_{n},h_{n}=0}^{1}(-1)^{i+j+g_{2}+g_{3}+...+g_{n}}\textmd{Tr}[P_{0i}^{noisy}\otimes_{k=2}^{n}M_{k,g_{k}h_{k}}^{noisy}\,Q_{0j}^{noisy}\varrho]\,ByEqs.(\ref{me1},\ref{me2},\ref{me3})$
$\displaystyle=$
$\displaystyle\sum_{i,j=0}^{1}\sum_{g_{3},h_{3}=0}^{1}...\sum_{g_{n},h_{n}=0}^{1}R_{2}$
$\displaystyle whereR_{2}$ $\displaystyle=$
$\displaystyle(-1)^{i+j+g_{3}+...+g_{n}}(\textmd{Tr}[P_{0i}^{noisy}M_{2,00}^{noisy}\otimes_{k=3}^{n}M_{k,g_{k}h_{k}}^{noisy}\,Q_{0j}^{noisy}\varrho]+\textmd{Tr}[P_{0i}^{noisy}M_{2,01}^{noisy}\otimes_{k=3}^{n}M_{k,g_{k}h_{k}}^{noisy}\,Q_{0j}^{noisy}\varrho]-$
(28)
$\displaystyle\textmd{Tr}[P_{0i}^{noisy}M_{2,10}^{noisy}\otimes_{k=3}^{n}M_{k,g_{k}h_{k}}^{noisy}\,Q_{0j}^{noisy}\varrho]-\textmd{Tr}[P_{0i}^{noisy}M_{2,11}^{noisy}\otimes_{k=3}^{n}M_{k,g_{k}h_{k}}^{noisy}\,Q_{0j}^{noisy}\varrho])$

Further simplifying $R_{2},$ we get:

$\displaystyle R_{2}$ $\displaystyle=$
$\displaystyle(-1)^{i+j+g_{3}+...+g_{n}}\textmd{Tr}[P_{0i}^{noisy}(M_{2,00}^{noisy}+M_{2,01}^{noisy}-M_{2,10}^{noisy}-M_{2,11}^{noisy})\otimes_{k=3}^{n}M_{k,g_{k}h_{k}}^{noisy}\,Q_{0j}^{noisy}\varrho]$
(29) $\displaystyle=$
$\displaystyle\beta_{2}(-1)^{i+j+g_{2}+g_{3}+...+g_{n}}\textmd{Tr}[P_{0i}^{noisy}\otimes
M_{2,g_{2}h_{2}}^{ideal}\otimes_{k=3}^{n}M_{k,g_{k}h_{k}}^{noisy}\,Q_{0j}^{noisy}\varrho]\,(usingEq.(\ref{me11}))$

Using Eq.(29) in Eq.(VI), we get:

$\langle
O_{1,0}O_{2}^{0}.....O_{n}^{0}O_{n+1,0}\rangle_{noisy}=\beta_{2}\sum_{i,j=0}^{1}\sum_{g_{2},h_{2}=0}^{1}\sum_{g_{3},h_{3}=0}^{1}...\sum_{g_{n},h_{n}=0}^{1}(-1)^{i+j+g_{2}+g_{3}+...+g_{n}}\textmd{Tr}[P_{0i}^{noisy}\otimes
M_{2,g_{2}h_{2}}^{ideal}\otimes_{k=3}^{n}M_{k,g_{k}h_{k}}^{noisy}\,Q_{0j}^{noisy}\varrho]$
(31)

Following similar approach of breaking sums over rest of the indices appearing
in Eq.(31), we have

$\langle
O_{1,0}O_{2}^{0}.....O_{n}^{0}O_{n+1,0}\rangle_{noisy}=\Pi_{i=2}^{n}\beta_{i}\mu\nu\sum_{i,j=0}^{1}\sum_{g_{2},h_{2}=0}^{1}\sum_{g_{3},h_{3}=0}^{1}...\sum_{g_{n},h_{n}=0}^{1}(-1)^{i+j+g_{2}+g_{3}+...+g_{n}}\textmd{Tr}[P_{0i}^{ideal}\otimes_{k=2}^{n}M_{k,g_{k}h_{k}}^{ideal}\,Q_{0j}^{ideal}\varrho]$
(32)

Following same procedure for each expectation term appearing in Eq.(VI), we
get

$\displaystyle I_{noisy}=\mu\nu\Pi_{i=2}^{n}\beta_{i}I$ $\displaystyle
J_{noisy}=\mu\nu\Pi_{i=2}^{n}\beta_{i}J$ (33)

Eq.(VI) thus gives:

$\sqrt{\mu\nu\Pi_{i=2}^{n}\beta_{i}}(\sqrt{|I|}+\sqrt{|J|})=1.$ (35)

Eq.(35) is the $n$-local inequality for a linear $n$-local network where the
parties perform imperfect measurements and each of the source generates an
arbitrary two qubit state. As upper bound of $n$-local inequality(Eq.(II.2))
in ideal linear $n$-local network is given by Eq.(6), clearly upper bound of
Eq.(35) is given
by:$\sqrt{\mu\nu\Pi_{j=2}^{n}\beta_{j}}\sqrt{\Pi_{i=1}^{n}t_{i11}+\Pi_{i=1}^{n}t_{i22}}.$
Non $n$-locality detection criterion is thus given by Eq.(13).
Proved

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# octonionic planes and real forms of $\text{G}_{2}$, $\text{F}_{4}$ and
$\text{E}_{6}$

D. Corradetti1, A. Marrani2, D. Chester3 and R. Aschheim3

###### Abstract

In this work we present a useful way to introduce the octonionic projective
and hyperbolic plane $\mathbb{O}P^{2}$ through the use of Veronese vectors.
Then we focus on their relation with the exceptional Jordan algebra
$\mathfrak{J}_{3}^{\mathbb{O}}$ and show that the Veronese vectors are the
rank-1 elements of the algebra. We then study groups of motions over the
octonionic plane recovering all real forms of $\text{G}_{2}$, $\text{F}_{4}$
and $\text{E}_{6}$ groups and finally give a classification of all octonionic
and split-octonionic planes as symmetric spaces.
MSC: 17C36, 17C60, 17C90, 22E15, 32M15
Keywords: Exceptional Lie Groups, Jordan Algebra, Octonionic Projective Plane,
Real Forms, Veronese embedding.

##  1 Introduction

The study of the exceptional Jordan algebra and its complexification has been
of interest in recent papers of theorethical physics. Todorov, Dubois-
Violette[17] and Krasnov[11] characterized the Standard Model gauge group
$\text{G}_{SM}$ as a subgroup of automorphisms of the exceptional Jordan
algebra $\mathfrak{J}_{3}\left(\mathbb{O}\right)$ while Boyle [4, 5] pointed
to its complexification
$\mathfrak{J}_{3}^{\mathbb{C}}\left(\mathbb{O}\right)$. An equivalent well
known view of the exceptional Jordan algebra is the one of projective
geometry, in which the automorphism group of
$\mathfrak{J}_{3}\left(\mathbb{O}\right)$ is the group of motions of the
octonionic projective plane[10]. Making use of Veronese coordinates we will
explore these relations and show how all real forms of $\text{F}_{4}$ and
$\text{E}_{6}$ can be recovered as group of motions of projective or
hyperbolic planes defined over division Octonions $\mathbb{O}$ or split
Octonions $\mathbb{O}_{s}$.

In sec. 2 we introduce the octonionic projective plane through the use of
Veronese coordinates, we define the projective lines and relate the
construction with the octonionic affine plane. In sec. 3 we show the
correspondance with the exceptional Jordan algebra
$\mathfrak{J}_{3}\left(\mathbb{O}\right)$ while in sec. 4 we show how real
forms of the exceptional Lie Groups $\text{E}_{6}$, $\text{F}_{4}$ and
$\text{G}_{2}$, arise as specific groups of collineations of the octonionic
projective plane. In the last section we proceed defining in a systematic way
all possible octonionic planes as symmetric spaces.

##  2 The Octonionic Projective Plane

Figure 1:  Multiplication rule of Octonions $\mathbb{O}$ as real vector space
$\mathbb{R}^{8}$ in the basis $\left\\{i_{0}=1,i_{1},...,i_{7}\right\\}$.

Octonions $\mathbb{O}$ are, along with Real numbers $\mathbb{R}$, Complex
numbers $\mathbb{C}$ and Quaternions $\mathbb{H}$, one of the four _Hurwitz
algebras_ , more specifically are the only unital non-associative normed
division algebra. A pratical way to work with them is to consider their
$\mathbb{R}^{8}$ decomposition, i.e.,

$x=\sum\limits_{k=0}^{7}x_{k}\textbf{i}_{k}$ (1)

where $\left\\{\textbf{i}_{0}=1,\textbf{i}_{1},...,\textbf{i}_{7}\right\\}$ is
a basis of $\mathbb{R}^{8}$ and the multiplication rules are mnemonically
encoded in the _Fano plane_ (Fig. 1) along with $\textbf{i}_{k}^{2}=-1$ for
$k=1,...,7$. We then define the octonionic conjugate of $x$ as

$\overline{x}\coloneqq
x_{0}\textbf{i}_{0}-\sum\limits_{k=1}^{7}x_{k}\textbf{i}_{k}$ (2)

with the usual norm

$\left\|x\right\|^{2}=\overline{x}x=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}$
(3)

and the inner product given by the polarisation of the norm, i.e.,

$\left\langle
x,y\right\rangle=\left\|x+y\right\|^{2}-\left\|x\right\|^{2}-\left\|y\right\|^{2}=\overline{x}y+\overline{y}x.$
(4)

In respect to this norm the Octonions are a composition algebra, i.e.
$\left\|xy\right\|=\left\|x\right\|\left\|y\right\|$, which will be of
paramount importance in the following sections. Finally, we denote with
$\mathbb{O}_{s}$ the split-octonionic algebra, whose definition can be found
in [13, 14].

### The Projective Plane

It is a common practice defining a projective plane over an associative
division algebra starting from a vector space over the given algebra, e.g.
$\mathbb{R}^{n+1}$, and then define the projective space as the quotient

$\mathbb{R}P^{n}=\mathbb{R}^{n+1}/\sim$ (5)

where $x\sim y$ if $x$ and $y$ are multiple through the scalar field, i.e.
$\lambda x=y$, $\lambda\in\mathbb{R}$, $\lambda\neq 0$,
$x,y\in\mathbb{R}^{n+1}$. But, since the algebra of Octonions is not
associative, we have that
$x\left(\lambda\mu\right)\neq\left(x\lambda\right)\mu$ when
$\lambda,\mu\in\mathbb{O}$. If we then try to define the equivalence relation
as above, we then might have $x\sim y=x\lambda$, and $x\sim
z=x\left(\lambda\mu\right)$, but $z$ not related to $y$ since

$z=x\left(\lambda\mu\right)\neq\left(x\lambda\right)\mu=y\mu.$ (6)

Therefore the previous is not an equivalence relation and the quotient cannot
be defined. A method for overcoming such an issue is based on determining an
equivalent algebraic definition of the rank-one idempotent of the exceptional
Jordan algebra in order define points in the projective plane, but here we
want to use a direct and less known way to proceed making use of the Veronese
vectors.

### Veronese coordinates

Let $V\cong\mathbb{O}^{3}\times\mathbb{R}^{3}$ be a real vector space, with
elements of the form

$\left(x_{\nu};\lambda_{\nu}\right)_{\nu}=\left(x_{1},x_{2},x_{3};\lambda_{1},\lambda_{2},\lambda_{3}\right)$

where $x_{\nu}\in\mathbb{O}$, $\lambda_{\nu}\in\mathbb{R}$ and $\nu=1,2,3$. A
vector $w\in V$ is called _Veronese_ if

$\displaystyle\lambda_{1}\overline{x}_{1}$
$\displaystyle=x_{2}x_{3},\,\,\lambda_{2}\overline{x}_{2}=x_{3}x_{1},\,\,\lambda_{3}\overline{x}_{3}=x_{1}x_{2}$
(7) $\displaystyle\left\|x_{1}\right\|^{2}$
$\displaystyle=\lambda_{2}\lambda_{3},\,\left\|x_{2}\right\|^{2}=\lambda_{3}\lambda_{1},\left\|x_{3}\right\|^{2}=\lambda_{1}\lambda_{2}.$
(8)

Let $H\subset V$ be the subset of Veronese vectors. If
$w=\left(x_{\nu};\lambda_{\nu}\right)_{\nu}$ is a Veronese vector then also
$\mu w=\mu\left(x_{\nu};\lambda_{\nu}\right)_{\nu}$ is a Veronese vector, that
is $\mathbb{R}w\subset H$. We define the _Octonionic plane_ $\mathbb{O}P^{2}$
as the geometry having this one-dimensional subspaces $\mathbb{R}w$ as points,
i.e.

$\mathbb{O}P^{2}=\left\\{\mathbb{R}w:w\in
H\smallsetminus\left\\{0\right\\}\right\\}.$ (9)

######  Remark 2.1.

The point in the projective plane is defined as the equivalence class
$\mathbb{R}w$ of the Veronese vector $w$, but, in order to determine an
explicit relation between points in the projective plane and rank-one
idempotent elements of the Jordan algebra $\mathfrak{J}_{3}^{\mathbb{O}}$, we
will choose when as representative of the class the vector
$v=\left(y_{\nu};\xi_{\nu}\right)_{\nu}\in\mathbb{R}w$ such that
$\xi_{1}+\xi_{2}+\xi_{3}=1$. Then $\left(y_{\nu};\xi_{\nu}\right)_{\nu}$ are
called Veronese coordinates of the projective point.

### Projective lines

We then define _projective lines_ of $\mathbb{O}P^{2}$ as the vectors
orthogonal to the points $\mathbb{R}w$. Let $\beta$ be the bilinear form over
$\mathbb{O}^{3}\times\mathbb{R}^{3}$ defined as

$\beta\left(w_{1},w_{2}\right)=\sum\limits_{\nu=1}^{3}\left(\left\langle
x_{\nu}^{1},x_{\nu}^{2}\right\rangle+\lambda_{\nu}^{1}\lambda_{\nu}^{2}\right)$
(10)

where
$w_{1}=\left(x_{\nu}^{1};\lambda_{\nu}^{1}\right)_{\nu}$,$w_{2}=\left(x_{\nu}^{2};\lambda_{\nu}^{2}\right)_{\nu}\in\mathbb{O}^{3}\times\mathbb{R}^{3}$.
Then, for every Veronese vector $w$, corresponding to the point $\mathbb{R}w$
in $\mathbb{O}P^{2}$, we define a line $\ell$ in $\mathbb{O}P^{2}$ as the
orthogonal space

$\ell\coloneqq
w^{\perp}=\left\\{z\in\mathbb{O}^{3}\times\mathbb{R}^{3}:\beta\left(z,w\right)=0\right\\}.$
(11)

The bilinear form $\beta$ also defines the _elliptic polarity_ , i.e. the map
$\pi^{+}$ that corresponds points to lines and lines to points, i.e.

$\pi^{+}\left(w\right)=w^{\perp},\pi^{+}\left(w^{\perp}\right)=w$ (12)

where the orthogonal space to a vector is defined by the bilinear form
$\beta\left(\cdot,\cdot\right)$, so that

$\displaystyle\pi^{+}:$ $\displaystyle
w\longrightarrow\left\\{\beta\left(\cdot,w\right)=0\right\\}$ (13)
$\displaystyle\ell\longrightarrow w$ (14)

when $\ell$ is given by $\left\\{\beta\left(\cdot,w\right)=0\right\\}$.
Explicitly, $\beta\left(w_{1},w_{2}\right)=0$ when

$2\overline{x}_{1}^{1}x_{1}^{2}+2\overline{x}_{2}^{1}x_{2}^{2}+2\overline{x}_{3}^{1}x_{3}^{2}+\lambda_{1}^{1}\lambda_{1}^{2}+\lambda_{2}^{1}\lambda_{2}^{2}+\lambda_{3}^{1}\lambda_{3}^{2}=0.$
(15)

In addition to the elliptic polarity defined above, we then define the
_hyperbolic polarity $\pi^{-}$_, which still has

$\pi^{-}\left(w\right)=w^{\perp},\pi^{-}\left(w^{\perp}\right)=w$ (16)

but through the use of the bilinear form $\beta_{-}$ that has a change of sign
in the last coordinate, i.e. $\beta_{-}\left(w_{1},w_{2}\right)=0$ is given by

$2\overline{x}_{1}^{1}x_{1}^{2}+2\overline{x}_{2}^{1}x_{2}^{2}-2\overline{x}_{3}^{1}x_{3}^{2}+\lambda_{1}^{1}\lambda_{1}^{2}+\lambda_{2}^{1}\lambda_{2}^{2}-\lambda_{3}^{1}\lambda_{3}^{2}=0.$
(17)

A projective plane equipped with the hyperbolic polarity will be called
hyperbolic plane and denoted as $\mathbb{O}H^{2}$.

Figure 2: Representation of the affine plane: $\left(0,0\right)$ represents
the origin, $\left(0\right)$ the point at the infinity on the $x$-axis,
$\left(s\right)$ is the point at infinity of the line $\left[s,t\right]$ of
slope $s$ while $\left(\infty\right)$ is the point at the infinity on the
$y$-axis and of vertical lines $\left[c\right]$.

### The Affine Plane

The octonionic projective plane is also the completion of the octonionic
affine plane. The embedding of the affine plane can be explicited through the
use of Veronese coordinates defining the map that sends a point
$\left(x,y\right)$ of the affine plane to the projective point
$\mathbb{R}\left(x,\overline{y},y\overline{x};\left\|y\right\|^{2},\left\|x\right\|^{2},1\right)$,
i.e.

$\left(x,y\right)\mapsto\mathbb{R}\left(x,\overline{y},y\overline{x};\left\|y\right\|^{2},\left\|x\right\|^{2},1\right)$
(18)

which is an homeomorphism. To complete the affine plane, we then have to
extend the map to another set of coordinates, i.e.

$\displaystyle\left(x\right)$
$\displaystyle\mapsto\mathbb{R}\left(0,0,x;\left\|x\right\|^{2},1,0\right)$
(19) $\displaystyle\left(\infty\right)$
$\displaystyle\mapsto\mathbb{R}\left(0,0,0;1,0,0\right).$ (20)

######  Remark 2.2.

To show that the above is a Veronese vector and therefore that the map is well
defined, we made essential use of alternativity of the Octonions and fact that
Octonions are a composition algebra. In case of non-composition algebra,
though the definition of the projective and hyperbolic planes would still be
valid using Veronese coordinates, the geometry of these planes will not
satisfy the basic axioms of projective and affine geometry and therefore they
would have to be considered as "generalised" projective or hyperbolic planes.

Moreover, let $\left[s,t\right]$ be a line in the affine plane
$\mathbb{O}A^{2}$ of the form

$\left[s,t\right]=\left\\{\left(x,sx+t\right):x\in\mathbb{O}\right\\}$ (21)

where $s$ is the slope of the line. Then $\left[s,t\right]$ is mapped into the
projective line orthogonal to the vector
$\left(\overline{s}t,-\overline{t},-s;1,\left\|s\right\|^{2},\left\|t\right\|^{2}\right)$,
i.e.

$\left[s,t\right]\mapsto\mathbb{R}\left(\overline{s}t,-\overline{t},-s;1,\left\|s\right\|^{2},\left\|t\right\|^{2}\right)^{\perp}.$
(22)

Vertical lines $\left[c\right]$ that are of the form
$\left\\{c\right\\}\times\left(\mathbb{O}\right),$ are mapped into lines of
$\mathbb{O}P^{2}$ given by

$\left[c\right]\mapsto\mathbb{R}\left(-c,0,0;0,1,\left\|c\right\|^{2}\right)^{\perp}.$
(23)

Finally the line at infinity $\left[\infty\right]$ is mapped to the orthogonal
space of the vector

$\left[\infty\right]\mapsto\mathbb{R}\left(0,0,0;0,0,1\right)^{\perp}.$ (24)

##  3 The Exceptional Jordan Algebra $\mathfrak{J}_{3}\left(\mathbb{O}\right)$

The exceptional Jordan algebra $\mathfrak{J}_{3}\left(\mathbb{O}\right)$ is
the algebra of Hermitian three by three octonionic matrices with the Jordan
product

$X\circ Y=\frac{1}{2}\left(XY+YX\right).$ (25)

It is easy to see that $\mathfrak{J}_{3}\left(\mathbb{O}\right)$ is
commutative, i.e. $X\circ Y=Y\circ X$ and satisfies the _Jordan identity_

$\left(X^{2}\circ Y\right)\circ X=X^{2}\circ\left(Y\circ X\right).$ (26)

We then define the _bilinear form_

$\left(X,Y\right)=\frac{1}{2}\text{tr$\left(X\circ Y\right)$}$ (27)

the _quadratic form_ whose the previous bilinear form is a polarisation

$Q\left(X\right)=\frac{1}{2}\text{tr$\left(X^{2}\right)$}$ (28)

and the _Freudenthal product_ , i.e.

$X*Y=X\circ
Y-\frac{1}{4}\left(X\text{tr$\left(Y\right)$}+Y\text{tr$\left(X\right)$}\right)+\frac{1}{4}\left(\text{tr$\left(X\right)$}\text{tr$\left(Y\right)$}-\text{tr$\left(X\circ
Y\right)$}\right)I_{3}$ (29)

where $I_{3}=diag(+,+,+)$, along with the _symmetric trilinear form_

$\left(X,Y,Z\right)=\frac{1}{3}\left(X,Y*Z\right)$ (30)

and the _determinant_

$\text{det}\left(X\right)=\frac{1}{3}\left(X,X,X\right).$ (31)

Now, let $\mathbb{R}w$ be a point in the projective plane $\mathbb{O}P^{2}$,
related to a vector in $\mathbb{O}^{3}\times\mathbb{R}^{3}$ with Veronese
coordinates $w=\left(x_{\nu};\lambda_{\nu}\right)_{\nu}$ and consider the map
from $V\cong\mathbb{O}^{3}\times\mathbb{R}^{3}$ into the space of three by
three Hermitian matrices with octonionic coefficients, defined as

$\left(x_{\nu};\lambda_{\nu}\right)_{\nu}\mapsto\left(\begin{array}[]{ccc}\lambda_{1}&x_{3}&\overline{x}_{2}\\\
\overline{x}_{3}&\lambda_{2}&x_{1}\\\
x_{2}&\overline{x}_{1}&\lambda_{3}\end{array}\right).$ (32)

We then have that

$\text{det}\left(X\right)=\lambda_{1}\lambda_{2}\lambda_{3}-\lambda_{1}\left\|x_{1}\right\|^{2}-\lambda_{2}\left\|x_{2}\right\|^{2}-\lambda_{3}\left\|x_{3}\right\|^{2}+2\text{Re}\left(\left(x_{1}x_{2}\right)x_{3}\right)$
(33)

that, imposing the Veronese conditions translates to
$\text{det}\left(X\right)=0$.

Moreover, let $X^{\sharp}$ be the image of a non-zero element $X$ under the
adjoint ($\sharp$-)map of $\mathfrak{J}_{3}^{\mathbb{O}}$, which is given by
(cf. Example 5 of [12])

$X^{\sharp}:=\left(\begin{array}[]{ccc}\lambda_{2}\lambda_{3}-\left\|x_{1}\right\|^{2}&\overline{x_{2}}\overline{x_{1}}-\lambda_{3}x_{3}&x_{3}x_{1}-\lambda_{2}\overline{x_{2}}\\\
x_{1}x_{2}-\lambda_{3}\overline{x_{3}}&\lambda_{1}\lambda_{3}-\left\|x_{2}\right\|^{2}&\overline{x_{3}}\overline{x_{2}}-\lambda_{1}x_{1}\\\
\overline{x_{1}}\overline{x_{3}}-\lambda_{2}x_{2}&x_{2}x_{3}-\lambda_{1}\overline{x_{1}}&\lambda_{1}\lambda_{2}-\left\|x_{3}\right\|^{2}\end{array}\right).$
(34)

From this explicit expression, it is immediate to realize that the Veronese
conditions are equivalent to the vanishing of $X^{\sharp}$. Then, by the
Str$\left(\mathfrak{J}_{3}\left(\mathbb{O}\right)\right)$-invariant definition
of the rank of an element of $\mathfrak{J}_{3}\left(\mathbb{O}\right)$ [9],
one obtains that the Veronese conditions are equivalent to the rank-1
condition for an element of $\mathfrak{J}_{3}\left(\mathbb{O}\right)$.

Thus, from the knowledge of the orbit stratification of
$\mathfrak{J}_{3}^{\mathbb{O}}$ under the non-transitive action of its reduced
structure group
Str${}_{0}\left(\mathfrak{J}_{3}\left(\mathbb{O}\right)\right)\simeq\text{E}_{6(-26)}$,
it follows that the Veronese conditions for a non-zero element of
$\mathfrak{J}_{3}\left(\mathbb{O}\right)$ are equivalent to imposing that such
an element belongs to the (unique) rank-1 orbit of $\text{E}_{6(-26)}$ in
$\mathfrak{J}_{3}\left(\mathbb{O}\right)$ (cf. [small], and Refs. therein).

Now we want to show that in the rank-1 (unique) orbit of
$\mathfrak{J}_{3}\left(\mathbb{O}\right)$, idempotency is equivalent to the
condition of unitary trace. In order to do that, let us consider the element
$X^{2}$ and let us impose the condition of $X$ being of rank$=1$. Since this
condition is equivalent to the Veronese conditions, one obtains

$X^{2}={\scriptstyle\left(\begin{array}[]{ccc}\lambda_{1}^{2}+\lambda_{1}\lambda_{2}+\lambda_{1}\lambda_{3}&\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right)x_{3}&\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right)\overline{x_{2}}\\\
\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right)\overline{x_{3}}&\lambda_{2}^{2}+\lambda_{1}\lambda_{2}+\lambda_{2}\lambda_{3}&\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right)x_{1}\\\
\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right)x_{2}&\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right)\overline{x_{1}}&\lambda_{3}^{2}+\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{3}\end{array}\right)}$
(35)

from which it follows that $X^{2}=X$ if and only if

$\lambda_{1}+\lambda_{2}+\lambda_{3}=1$ (36)

i.e. if and only if $\text{tr}\left(X\right)=1$. Thus, the idempotency
condition for rank-1 elements of $\mathfrak{J}_{3}^{\mathbb{O}}$ is equivalent
to the condition of unitary trace.

##  4 Lie Groups of Type $\text{G}_{2}$, $\text{F}_{4}$ and $\text{E}_{6}$ as
Groups of Collineations

We are now interested in the motions and symmetries of the octonionic
projective plane. More specifically we are interested in _collineations_ that
are transformations of the projective plane that send lines into lines. If the
collineation preserves the elliptic polarity or the hyperbolic polarity is
then called _elliptic_ or _hyperbolic motion_. Elliptic and hyperbolic motions
are an equivalent characterization of the _isometries_ of the projective or
hyperbolic plane respectively, thus the elliptic motion group of the
projective plane will be indicated as $\text{Iso}\left(\mathbb{O}P^{2}\right)$
and the hyperbolic motion group as $\text{Iso}\left(\mathbb{O}H^{2}\right)$.

### Collineations of the Octonionic Projective Plane

A _collineation_ is a bijection $\varphi$ of the set of points of the plane
onto itself, mapping lines onto lines. It is straightforward to see that the
identity map is a collineation, as the inverse $\varphi^{-1}$ and the
composition $\varphi\circ\varphi^{\prime}$ are if $\varphi,\varphi^{\prime}$
are both collineation. Therefore the set
$\text{Coll}\left(\mathbb{O}P^{2}\right)$ of collineations is a group under
composition of maps. It also has a proper subgroup of order three generated by
the _triality collineation_ that permutes three special points of the
affine/projective octonionic plane, i.e. the origin of coordinate
$\left(0,0\right)$, the point at the origin of the line at infinity which has
coordinate $\left(0\right)$ and the point at infinity of the line at the
infinity which has affine coordinate $\left(\infty\right)$. In Veronese
coordinates these three points are images of the following vectors

$\displaystyle\left(0,0\right)$
$\displaystyle\longrightarrow\mathbb{R}\left(0,0,0;0,0,1\right)$ (37)
$\displaystyle\left(0\right)$
$\displaystyle\longrightarrow\mathbb{R}\left(0,0,0;0,1,0\right)$ (38)
$\displaystyle\left(\infty\right)$
$\displaystyle\longrightarrow\mathbb{R}\left(0,0,0;1,0,0\right)$ (39)

and the triality collineation $\tau$ is given by

$\left(x_{1},x_{2},x_{3};\lambda_{1},\lambda_{2},\lambda_{3}\right)\mapsto\left(x_{2},x_{3},x_{1};\lambda_{2},\lambda_{3},\lambda_{1}\right)$
(40)

that is a cyclic permutation of order three that leaves invariant the Veronese
vectors. This means that it induces a bijection $\tau$ on $\mathbb{O}P^{2}$
that is unseen by the bilinear form $\beta$ and therefore maps lines into
lines, since lines are constructed as the ortogonal space of a vector through
the bilinear form $\beta$.

Let us now consider the transformations $T_{a,b}$ of
$\mathbb{O}^{3}\times\mathbb{R}^{3}$ into itself defined on the Veronese
coordinates as

$\displaystyle x_{1}\longrightarrow$ $\displaystyle x_{1}+\lambda_{3}a$
$\displaystyle x_{2}\longrightarrow$ $\displaystyle
x_{2}+\lambda_{3}\overline{b}$ $\displaystyle x_{3}\longrightarrow$
$\displaystyle
x_{3}+b\overline{x_{1}}+\overline{x_{2}}\overline{a}+\lambda_{3}b\overline{a}$
$\displaystyle\lambda_{1}\longrightarrow$
$\displaystyle\lambda_{1}+\left\langle\overline{x}_{2},a\right\rangle+\lambda_{3}\left\|b\right\|^{2}$
(41) $\displaystyle\lambda_{2}\longrightarrow$
$\displaystyle\lambda_{2}+\left\langle\overline{x}_{1},a\right\rangle+\lambda_{3}\left\|a\right\|^{2}$
$\displaystyle\lambda_{3}\longrightarrow$ $\displaystyle\lambda_{3}.$

Those are in fact _translations_ on the affine plane corresponding to the
transformation
$\left(x_{1},x_{2}\right)\longrightarrow\left(x_{1}+a,x_{2}+b\right)$ and they
all induce collineations $T_{a,b}$ on $\mathbb{O}P^{2}$.

It can be shown that all collineations are generated by the interplay between
a translation and the conjugation of a power of the triality collineation,
i.e. are of the form

$T_{a,b},\,\,\,\tau T_{a,b}\tau^{-1},\,\,\,\tau^{2}T_{a,b}\tau^{-2}.$ (42)

From another perspective, collineations transform lines of $\mathbb{O}P^{2}$
in lines of $\mathbb{O}P^{2}$. This is equivalent to find all the linear
transformations $A$ of $V$ in itself such that the image of Veronese vectors
is still a Veronese vector $A\left(H\right)\subset H$. If this condition is
fulfilled, the linear transformation $A$ in $\text{End}\left(V\right)$ will
induce a collineation $\varphi$ on $\mathbb{O}P^{2}$, i.e.

$\begin{array}[]{ccc}\mathbb{O}^{3}\times\mathbb{R}^{3}&\overset{A}{\longrightarrow}&\mathbb{O}^{3}\times\mathbb{R}^{3}\\\
\uparrow&&\uparrow\\\
\mathbb{O}P^{2}&\overset{\varphi}{\longrightarrow}&\mathbb{O}P^{2}\end{array}$
(43)

Since all linear multiple of the transformation $A$ will produce the same
collineation $\varphi$, to have a bijection between linear transformations and
collineations we have to impose also $\text{det}\left(A\right)=1$. That is
that the group of collineation $\text{Coll}\left(\mathbb{O}P^{2}\right)$ is

$\text{SL}\left(V,H\right)\coloneqq\left\\{A\in\text{End}\left(V\right);A\left(H\right)\subseteq
H;\text{det}\left(A\right)=1\right\\}.$ (44)

If we also impose the preservation of the elliptic polarity, i.e. of the
bilinear form $\beta$, we will then have the group of elliptic motion
$\text{Iso}\left(\mathbb{O}P^{2}\right)$ that is

$\text{SU}\left(V,H\right)=\left\\{A\in\text{End}\left(V\right);A\left(H\right)\subseteq
H;\text{det}\left(A\right)=1;\text{tr}\left(A\right)=1\right\\}.$ (45)

Those two groups are in fact two exceptional Lie Groups, i.e.

$\displaystyle\text{Coll}\left(\mathbb{O}P^{2}\right)\cong\text{SL}\left(V,H\right)$
$\displaystyle\cong E_{6}$ (46)
$\displaystyle\text{Iso}\left(\mathbb{O}P^{2}\right)\cong\text{SU}\left(V,H\right)$
$\displaystyle\cong F_{4}.$ (47)

The identification of this two group is done through a direct determination of
the generators as in [13]; instead, we will here follow Rosenfeld in [4]
focusing on the Lie algebra of the group of collineations on
$\mathbb{O}P^{2}$, i.e. $\mathfrak{coll}\left(\mathbb{O}P^{2}\right)$, which
is given by

$\mathfrak{coll}\left(\mathbb{O}P^{2}\right)=\mathfrak{g}_{2}\oplus\mathfrak{a}_{3}\left(\mathbb{O}\right)$
(48)

where $\mathfrak{g}_{2}\cong\mathfrak{der}\left(\mathbb{O}\right)$ and
$\mathfrak{a}_{3}$ are the three by three matrices on $\mathbb{O}$ with null
trace, i.e. $\text{tr}\left(A\right)=0$. The dimension count on the possible
generators of this algebra, since the only condition you have is to have null
trace, i.e. $\text{tr}\left(A\right)=0$, gives as only condition on

$A=\left(\begin{array}[]{ccc}a_{1}^{1}&a_{2}^{1}&a_{3}^{1}\\\
a_{1}^{2}&a_{2}^{2}&a_{3}^{2}\\\
a_{1}^{3}&a_{2}^{3}&a_{3}^{3}\end{array}\right)$ (49)

the condition on the trace, i.e.
$a_{3}^{3}=-\left(a_{1}^{1}+a_{2}^{2}\right)$, and therefore we have $8$
entries of dimension $8$ and $\text{dim}{}_{\mathbb{R}}\mathfrak{a}_{3}=64$.
We therefore have

$\text{dim}_{\mathbb{R}}\mathfrak{coll}\left(\mathbb{O}P^{2}\right)\cong
78=64+14.$ (50)

Since $\mathfrak{coll}\left(\mathbb{O}P^{2}\right)$ is a Lie group, simple and
of dimension 78, then it must be of $E_{6}$ type.

### Isometries of the Plane

Again, following Rosenfeld [4] we look at the elliptic motions of
$\mathbb{O}P^{2}$, which are the collineations that preserve also the polarity
$\pi^{+}$ or equivalently the form $\beta$; they are given by

$\mathfrak{iso}\left(\mathbb{O}P^{2}\right)=\mathfrak{g}_{2}\oplus\mathfrak{sa}\left(3\right)$
(51)

where we notated $\mathfrak{sa}\left(3\right)$ the skew-Hermitian matrices
with null trace. Here the elements of $\mathfrak{sa}\left(3\right)$ are of the
form

$A=\left(\begin{array}[]{ccc}a_{1}^{1}&a_{2}^{1}&-\overline{a}_{3}^{1}\\\
-\overline{a}_{2}^{1}&a_{2}^{2}&a_{3}^{2}\\\
a_{3}^{1}&-\overline{a}_{3}^{2}&a_{3}^{3}\end{array}\right)$ (52)

with $a_{ij}=\overline{a}_{ji}$, $a_{3}^{3}=-\left(a_{1}^{1}+a_{2}^{2}\right)$
and $\text{Re}\left(a_{1}^{1}\right)=\text{Re}\left(a_{2}^{2}\right)=0$. We
therefore have 3 coefficient of dimension $8$, $2$ entries of dimension $7$
and therefore $\text{dim}_{\mathbb{R}}\mathfrak{sa}\left(3\right)=38$ so that

$\mathfrak{\text{dim}_{\mathbb{R}}\mathfrak{iso}}\left(\mathbb{O}P^{2}\right)\cong
52=38+14$ (53)

and the group of elliptic motion $\text{Iso}\left(\mathbb{O}P^{2}\right)$,
being simple and of dimension 52, is of the $F_{4}$ type.

Moreover we can proceed as in [4] to find the collineations that preserve the
_hyperbolic polarity_ $\pi^{-}$ or equivalently the form $\beta_{-}$ we
previously defined. Here the element of the Lie algebra are of the form

$A=\left(\begin{array}[]{ccc}a_{1}^{1}&a_{2}^{1}&\overline{a}_{3}^{1}\\\
\overline{a}_{2}^{1}&a_{2}^{2}&a_{3}^{2}\\\
a_{3}^{1}&-\overline{a}_{3}^{2}&a_{3}^{3}\end{array}\right)$ (54)

with $a_{1}^{1}=\left(a_{2}^{2}+a_{3}^{3}\right)$ and
$\text{Re}\left(a_{1}^{1}\right)=\text{Re}\left(a_{2}^{2}\right)=0$, therefore
leading to the same count of the dimension of

$\mathfrak{\text{dim}_{\mathbb{R}}\mathfrak{iso}}\left(\mathbb{O}H^{2}\right)\cong
52=38+14$ (55)

deducing that $\text{Iso}\left(\mathbb{O}H^{2}\right)$ is again an
$\text{F}_{4}$ type group.

### Collineations with a Fixed Triangle or Quadrangle

We are now interested in studying the collineations $\varphi$ on the affine
plane that fix every point of $\triangle$, i.e.
$\varphi\left(\left(0,0\right)\right)=\left(0,0\right)$,
$\varphi\left(\left(0\right)\right)=\left(0\right)$ and
$\varphi\left(\left(\infty\right)\right)=\left(\infty\right)$.

######  Proposition 1.

The group $\Gamma\left(\triangle,\mathbb{O}\right)$ of collineations that fix
every point of $\triangle$ are transformations of this form

$\displaystyle\left(x,y\right)$
$\displaystyle\mapsto\left(A\left(x\right),B\left(y\right)\right)$ (56)
$\displaystyle\left(s\right)$
$\displaystyle\mapsto\left(C\left(s\right)\right)$ (57)
$\displaystyle\left(\infty\right)$ $\displaystyle\mapsto\left(\infty\right)$
(58)

where $A,B$ and $C$ are automorphisms with respect to the sum over
$\mathbb{O}$ and that satisfy

$B\left(sx\right)=C\left(s\right)A\left(x\right).$ (59)

###### Proof.

A collineation $\varphi$ that fixes $\left(0,0\right)$, $\left(0\right)$ and
$\left(\infty\right)$, also fixes the $x$-axis and $y$-axis and all lines that
are parallel to them. This means that the first coordinate is the image of a
function that does not depend on $y$ and the second coordinate is image of a
fuction that does not depend of $x$, i.e.
$\left(x,y\right)\mapsto\left(A\left(x\right),B\left(y\right)\right)$ and
$\left(s\right)\mapsto\left(C\left(s\right)\right)$. Now consider the image of
a point on the line $\left[s,t\right]$. The point is of the form
$\left(x,sx+t\right)$ and its image goes to

$\left(x,sx+t\right)\mapsto\left(A\left(x\right),B\left(sx+t\right)\right).$
(60)

If we want this to be a collineation, the points of the line
$\left[s,t\right]$ must all belong to the same line which can be easily
identified setting $x=0$, i.e. the image of $\left[s,t\right]$ is the line
that joins the points $\left(0,B\left(t\right)\right)$ and
$\left(C\left(s\right)\right)$. We now have that the condition for
$\left(A\left(x\right),B\left(sx+t\right)\right)$ to be in the image of
$\left[s,t\right]$ is

$B\left(sx+t\right)=C\left(s\right)A\left(x\right)+B\left(t\right).$ (61)

Now, if $B$ is an automorphism with respect to the sum over $\mathbb{O}$, we
then have the condition $B\left(sx\right)=C\left(s\right)A\left(x\right)$.
Conversely if $B\left(sx\right)=C\left(s\right)A\left(x\right)$ is true that
$B\left(sx+t\right)=B\left(sx\right)+B\left(t\right)$, and $B$ is an
automorphism with respect to the sum. ∎

Let us consider the quadrangle $\square$ given by the points
$\left(0,0\right)$, $\left(1,1\right)$, $\left(0\right)$ and
$\left(\infty\right)$, that is
$\square=\triangle\cup\left\\{\left(1,1\right)\right\\}$, and consider the
collineations that fix the $\square$. Since in addition to the previous case
we also have to impose

$\left(1,1\right)\mapsto\left(A\left(1\right),B\left(1\right)\right)=\left(1,1\right)$
(62)

then $C\left(1\right)=1$ and, therefore $A=B=C$ and therefore $A$ is an
automorphism of $\mathbb{O}$. We then have the following

######  Proposition 2.

The collineations that fix every point of $\square$ are transformations of the
type

$\displaystyle\left(x,y\right)$
$\displaystyle\mapsto\left(A\left(x\right),A\left(y\right)\right)$ (63)
$\displaystyle\left(s\right)$
$\displaystyle\mapsto\left(A\left(s\right)\right)$ (64)
$\displaystyle\left(\infty\right)$ $\displaystyle\mapsto\left(\infty\right)$
(65)

where $A$ is an automorphism of $\mathbb{O}$.

Moreover, since
$\text{Aut}\left(\mathbb{O}\right)=\text{G}_{2\left(-14\right)}$ and
$\text{Aut}\left(\mathbb{O}_{s}\right)=\text{G}_{2\left(2\right)}$ [18], we
have the following

######  Corollary 1.

The group of collineations $\Gamma\left(\square,\mathbb{O}\right)$ that fix
$\left(0,0\right)$, $\left(1,1\right)$, $\left(0\right)$ and
$\left(\infty\right)$ is isomorphic to $\text{Aut}\left(\mathbb{O}\right)$.
Therefore $\Gamma\left(\square,\mathbb{O}\right)$ is isomorphic to
$\text{G}_{2\left(-14\right)}$, while in the case of split octonions
$\mathbb{O}_{s}$ is isomorphic to $\text{G}_{2\left(2\right)}$.

It can be shown that the group of collineations
$\Gamma\left(\triangle,\mathbb{O}\right)$ is in fact the double cover of
$SO_{8}\left(\mathbb{R}\right)$, i.e.
$\text{Spin}_{8}\left(\mathbb{R}\right)$, that we define here as

$\text{Spin}\left(\mathbb{O}\right)=\left\\{\left(A,B,C\right)\in
O^{+}\left(\mathbb{O}\right)^{3}:A\left(xy\right)=B\left(x\right)C\left(y\right)\,\,\,\forall
x,y\in\mathcal{\mathbb{O}}\right\\}$ (66)

where $O^{+}$ is the connected component of the orthogonal group with the
identity.

######  Proposition 3.

The Lie algebra $Lie\left(\Gamma\left(\triangle,\mathbb{O}\right)\right)$ of
the group of collineation that fixes $\left(0,0\right),\left(0\right)$ and
$\left(\infty\right)$

$\mathfrak{tri}\left(\mathbb{O}\right)=\left\\{\left(T_{1},T_{2},T_{3}\right)\in\mathfrak{so}\left(\mathbb{\mathbb{O}}\right)^{3}:T_{1}\left(xy\right)=T_{2}\left(x\right)y+xT_{3}\left(y\right)\right\\}$
(67)

while the Lie algebra $Lie\left(\Gamma\left(\square,\mathbb{O}\right)\right)$
of the group of collineation that fixes $\left(0,0\right)$,
$\left(1,1\right)$, $\left(0\right)$ and $\left(\infty\right)$ is

$\mathfrak{der}\left(\mathbb{O}\right)=\left\\{T\in\mathfrak{so}\left(\mathbb{O}\right):T\left(xy\right)=T\left(x\right)y+xT\left(y\right)\right\\}.$
(68)

###### Proof.

$\Gamma\left(\triangle,\mathbb{O}\right)$ is a Lie group since it is a closed
subgroup of the Lie group of collineations. We will find directly its Lie
algebra considering the elements
$A,B,C\in\Gamma\left(\triangle,\mathbb{O}\right)$ in a neighbourhood of the
identity and writing them as

$\left(A,B,C\right)\longrightarrow\left(\text{Id}+\epsilon
T_{1},\text{Id}+\epsilon T_{2},\text{Id}+\epsilon T_{3}\right)$

where $T_{1},T_{2},T_{3}\in\mathfrak{so}\left(\mathbb{\mathbb{O}}\right)$.
Imposing the condition $A\left(xy\right)=B\left(x\right)C\left(y\right)$ and
then we obtain

$\displaystyle\left(\text{Id}+\epsilon T_{1}\right)\left(xy\right)$
$\displaystyle=\left(\text{Id}+\epsilon
T_{2}\right)\left(x\right)\left(\text{Id}+\epsilon T_{3}\right)\left(x\right)$
(69)

which, considering $\epsilon^{2}=0$, yields to

$T_{1}\left(xy\right)=T_{2}\left(x\right)y+xT_{3}\left(y\right).$ (70)

The second part of the proposition is obtained imposing $T_{1}=T_{2}=T_{3}=T$.
∎

We then have the following

$\displaystyle\Gamma\left(\triangle,\mathbb{O}\right)\cong$
$\displaystyle\text{Spin}\left(\mathbb{O}\right)\cong\text{Spin}_{8}\left(\mathbb{R}\right)$
(71) $\displaystyle\Gamma\left(\square,\mathbb{O}\right)\cong$
$\displaystyle\text{Aut}\left(\mathbb{O}\right)\cong\text{G}_{2\left(-14\right)}$
(72)

and, passing to Lie algebras, we obtain

$\displaystyle Lie\left(\Gamma\left(\triangle,\mathbb{O}\right)\right)\cong$
$\displaystyle\mathfrak{tri}\left(\mathbb{O}\right)\cong\mathfrak{so}\left(\mathbb{\mathbb{O}}\right)$
(73) $\displaystyle
Lie\left(\Gamma\left(\square,\mathbb{O}\right)\right)\cong$
$\displaystyle\mathfrak{der}\left(\mathbb{O}\right)\cong\mathfrak{g}_{2(-14)}.$
(74)

By considering the split Octonions $\mathbb{O}_{s}$, previous formulas yield
to

$\displaystyle\Gamma\left(\triangle,\mathbb{O}_{s}\right)\cong$
$\displaystyle\text{Spin}\left(\mathbb{O}_{s}\right)\cong\text{Spin}_{(4,4)}\left(\mathbb{R}\right)$
(75) $\displaystyle\Gamma\left(\square,\mathbb{O}\right)\cong$
$\displaystyle\text{Aut}\left(\mathbb{O}\right)\cong\text{G}_{2\left(2\right)}$
(76)

and, passing to Lie algebras, we obtain

$\displaystyle
Lie\left(\Gamma\left(\triangle,\mathbb{O}_{s}\right)\right)\cong$
$\displaystyle\mathfrak{tri}\left(\mathbb{O}_{s}\right)\cong\mathfrak{so}\left(\mathbb{\mathbb{O}_{s}}\right)\cong\mathfrak{so}_{4,4}$
(77) $\displaystyle
Lie\left(\Gamma\left(\square,\mathbb{O}\right)\right)\cong$
$\displaystyle\mathfrak{der}\left(\mathbb{O}\right)\cong\mathfrak{g}_{2(2)}.$
(78)

Resuming all the findings, following Yokota [18, p.105] in the definition of
real forms of $\text{E}_{6}$, we then have the following motion groups arising
from the octonionic and split-octonionic projective and hyperbolic plane, i.e.

Proj. Space | Collineation group | Isometry group | $\Gamma\left(\square\right)$
---|---|---|---
$\mathbb{O}P_{\mathbb{C}}^{2}$ | $\text{E}_{6}^{\mathbb{C}}$ | $\text{F}_{4}^{\mathbb{C}}$ | $G_{2}^{\mathbb{C}}$
$\mathbb{O}P^{2}$ | $\text{E}_{6\left(-26\right)}$ | $\text{F}_{4\left(-52\right)}$ | $G_{2(-14)}$
$\mathbb{O}_{s}P^{2}$ | $\text{E}_{6\left(6\right)}$ | $\text{F}_{4\left(4\right)}$ | $G_{2(2)}$
$\mathbb{O}_{s}H^{2}$ | $\text{E}_{6\left(2\right)}$ | $\text{F}_{4\left(4\right)}$ | $G_{2(2)}$
$\mathbb{O}H^{2}$ | $\text{E}_{6\left(-14\right)}$ | $\text{F}_{4\left(-20\right)}$ | $G_{2(-14)}$

##  5 Classification of the Octonionic Projective Planes

Thus, the space of rank-1 idempotent elements of
$\mathfrak{J}_{3}\left(\mathbb{O}\right)$ enjoys the following expression as
an homogeneous space

$\frac{\text{F}_{4(-52)}}{\text{Spin}_{9}},$ (79)

which is a compact Riemannian symmetric space, of (geodesic) rank $=1$ and of
real dimension

$\text{dim}_{\mathbb{R}}\left(\frac{\text{F}_{4(-52)}}{Spin_{9}}\right)=\text{dim}_{\mathbb{R}}\left(\text{F}_{4(-52)}\right)-\text{dim}_{\mathbb{R}}\left(\text{Spin}_{9}\right)=52-36=16,$
(80)

as expected from the number of degrees of freedom characterizing rank-1
idempotents of $\mathfrak{J}_{3}\left(\mathbb{O}\right)$ itself. Since the
unitary trace condition is imposed on top of Veronese conditions, the coset
(79) is a (proper) submanifold of
$\mathcal{O}_{\text{rank}\leavevmode\nobreak\
1}\left(\mathfrak{J}_{3}\left(\mathbb{O}\right)\right)$, i.e.

$\frac{\text{F}_{4(-52)}}{\text{Spin}_{9}}\subset\frac{\text{E}_{6(-26)}}{\text{Spin}_{9,1}\ltimes\mathbb{R}^{16}}\text{.}$
(81)

The space (79) of rank-1 idempotent (or, equivalently, trace-1) elements of
$\mathfrak{J}_{3}\left(\mathbb{O}\right)$ can be identified with the (compact
real form of the) octonionic projective plane $\mathbb{O}P^{2}$, which is the
largest octonionic (projective) geometry; this can also be hinted from the
fact that the tangent space to the coset $\text{F}_{4(-52)}/\text{Spin}_{9}$
transforms under the isotropy group $\text{Spin}_{9}$ as its spinor
irreducible representation $\boldsymbol{16}$, which can indeed be realized as
a pair of octonions [16] :

$\mathfrak{f}_{4(-52)}=\mathfrak{so}_{9}\oplus\mathbf{16}\Rightarrow
T\left(\frac{\text{F}_{4(-52)}}{\text{Spin}_{9}}\right)\simeq\mathbf{16\leavevmode\nobreak\
}\text{(of\leavevmode\nobreak\ }\text{Spin}_{9}\text{)}\simeq\mathbb{O\oplus
O}.$ (82)

Thus, one obtains that

${\textstyle\begin{array}[]{ccc}\mathcal{O}_{\text{rank}\leavevmode\nobreak\
1}\left(\mathfrak{J}_{3}\left(\mathbb{O}\right)\right)&\cong&\frac{\text{E}_{6(-26)}}{\text{Spin}_{9,1}\ltimes\mathbb{R}^{16}}\\\
\cup&&\cup\\\
\mathbb{O}P^{2}&\cong&\frac{\text{F}_{4(-52)}}{\text{Spin}_{9}}\end{array}}$
(83)

which gives an alternative definition of the octonionic projective plane.

Figure 3: Satake diagrams of the real forms of $\text{F}_{4}$, their character
$\chi$ and the corresponding octonionic plane whose they are the isometry
group.

From the table in the previous section, we can then classify all possible
octonionic planes. We start from the complexification of the Cayley plane

$\displaystyle\mathbb{O}P^{2}\left(\mathbb{C}\right)$ $\displaystyle\simeq$
$\displaystyle\frac{\text{F}_{4}\left(\mathbb{C}\right)}{Spin_{9}\left(\mathbb{C}\right)}$
(84)

and define three different real forms of the plane: a totally compact real
coset, that identifies with $\mathbb{O}P^{2}$; a totally non compact which is
$\mathbb{O}H^{2}$; a pseudo-Riemannian real coset that we will define as
$\mathbb{O}\widetilde{H}^{2}$. Those octonionic planes will be defined taking
as isometry group $\text{F}_{4\left(-52\right)}$ and
$\text{F}_{4\left(-20\right)}$, while the last real form
$\text{F}_{4\left(4\right)}$ will yield to projective planes on the split-
octonionic algebra $\mathbb{O}_{s}$, i.e.

$\displaystyle\mathbb{O}P^{2}$
$\displaystyle\simeq\frac{\text{F}_{4(-52)}}{Spin_{9}}$ (85)
$\displaystyle\mathbb{O}H^{2}$
$\displaystyle\simeq\frac{\text{F}_{4(-20)}}{Spin_{9}}$ (86)
$\displaystyle\mathbb{O}\widetilde{H}^{2}$
$\displaystyle\simeq\frac{\text{F}_{4(-20)}}{Spin_{8,1}}$ (87)
$\displaystyle\mathbb{O}_{s}\widetilde{H}^{2}$
$\displaystyle\simeq\mathbb{O}_{s}P^{2}\simeq\mathbb{O}_{s}H^{2}\simeq\frac{\text{F}_{4(4)}}{Spin_{5,4}}.$
(88)

Moreover, if we consider the type of the plane, i.e. the cardinality of non-
compact and compact generators $\left(\\#_{nc},\\#_{c}\right)$, and the
character $\chi$, i.e. the difference between the two,
$\chi=\\#_{nc}-\\#_{c}$, we then note that: the totally compact plane, i.e.
the classical _Cayley plane_ or the _octonionic projective plane_
$\mathbb{O}P^{2}$, is of type $(0,16)$ and character $\chi=16$; the totally
non-compact one, i.e. _hyperbolic octonionic plane_ $\mathbb{O}H^{2}$, is of
type $\left(16,0\right)$ and character $\chi=-16$; while the other two planes
named $\mathbb{O}\widetilde{H}^{2}$ and $\mathbb{O}_{s}\widetilde{H}^{2}$ are
of type $\left(8,8\right)$ and character $\chi=0$.

##  6 Conclusions

We have presented an explicit construction of the octonionic projective and
hyperbolic planes and showed how Lie groups of type $\text{G}_{2}$,
$\text{F}_{4}$ and $\text{E}_{6}$ arise naturally as groups of motion of such
planes. The fact that all different real forms of $\text{E}_{6}$,
$\text{F}_{4}$ and $\text{G}_{2}$ can be recovered from similar constructions
is of the uttermost physical importance since different physical theories
require different real forms of Lie groups. Compact and non-compact forms of
$\text{G}_{2}$ are notoriously known to be isomorphic to automorphisms of
Octonions and split Octonions. In this paper we show how they can be thought
as the subgroup of collineations that fix a quadrangle of the Projective plane
over Octonions and Split-Octonions. Different compact and non compact real
forms of $\text{E}_{6}$ and $\text{F}_{4}$ are related to different but
analogue geometric frameworks such as projective planes or hyperbolic planes
over the algebra of Octonions and split Octonions. Indeed while we recover
$\text{E}_{6\left(-26\right)}$ and $\text{F}_{4\left(-52\right)}$ as
collineation and isometry group of the octonionic projective plane
$\mathbb{O}P^{2}$, we have $\text{E}_{6\left(6\right)}$ and
$\text{F}_{4\left(4\right)}$ for the split case $\mathbb{O}_{s}P^{2}$. The
hyperbolic plane over Octonions and split-Octonions lead to
$\text{E}_{6\left(-14\right)}$ and $\text{F}_{4\left(-20\right)}$ in the
octonionic case $\mathbb{O}H^{2}$ or to $\text{E}_{6\left(2\right)}$ and
$\text{F}_{4\left(4\right)}$ in the split case $\mathbb{O}_{s}H^{2}$. The only
real form left out is the compact $\text{E}_{6\left(-78\right)}$ which is
obtained as isometry group of the complex Cayley plane or projective Rosenfeld
plane over Bioctonions $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$ [6].
Moreover, we have classified all octonionic and split-octonionic projective
planes as symmetric spaces.

##  7 Acknowledgments

The work of D.Corradetti is supported by a grant of the Quantum Gravity
Research Institute. The work of AM is supported by a “Maria Zambrano"
distinguished researcher fellowship, financed by the European Union within the
NextGenerationEU program.

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1Departamento de Matemática
Universidade do Algarve
Campus de Gambelas
8005-139 Faro, Portugal
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2Instituto de Física Teorica, Dep.to de Física,
Universidad de Murcia, Campus de Espinardo, E-30100, Spain
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3 Quantum Gravity Research,
Los Angeles, California, CA 90290, USA
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