# Entangling Power of Symmetric Two-Qubit Quantum Gates

D. Morachis Galindo  Departamento de Física, Centro de Nanociencias y
Nanotecnología, Universidad Nacional Autónoma de México, Apartado Postal 14,
22800 Ensenada, B.C., México  Jesús A. Maytorena  Departamento de Física,
Centro de Nanociencias y Nanotecnología, Universidad Nacional Autónoma de
México, Apartado Postal 14, 22800 Ensenada, B.C., México

###### Abstract

The capacity of a quantum gate to produce entangled states on a bipartite
system is quantified in terms of the entangling power. This quantity is
defined as the average of the linear entropy of entanglement of the states
produced after applying a quantum gate over the whole set of separable states.
Here we focus on symmetric two-qubit quantum gates, acting on the symmetric
two-qubit space, and calculate the entangling power in terms of the
appropriate local-invariant. A geometric description of the local equivalence
classes of gates is given in terms of the $\mathfrak{su}(3)$ Lie algebra root
vectors. These vectors define a primitive cell with hexagonal symmetry on a
plane, and through the Weyl group the minimum area on the plane containing the
whole set of locally equivalent quantum gates is identified. We give
conditions to determine when a given quantum gate produces maximally entangled
states from separable ones (perfect entanglers). We found that these gates
correspond to one fourth of the whole set of locally-distinct quantum gates.
The theory developed here is applicable to three-level systems in general,
where the non-locality of a quantum gate is related to its capacity to perform
non-rigid transformations on the Majorana constellation of a state. The
results are illustrated by an anisotropic Heisenberg model, the Lipkin-
Meshkov-Glick model, and two coupled quantized oscillators with cross-Kerr
interaction.

Geometric phase, Uhlmann, Topological, Mixed States, Spin-j Particle.

UhlmannJ dmorachisgalindo February 2021

##  I Introduction

Entanglement is a purely quantum mechanical phenomenon that is essential to
achieve universal quantum computation based on interacting qubits
systemsAudrescht (2007). Quantum logic gates are the building blocks to
perform quantum algorithms, where the generation of entangled states from a
separable set of states is mandatory to achieve the desired resultsHorodecki
_et al._ (2009).

Most of the proposed quantum computer architectures are based on multi-qubit
processors. Nevertheless, there are also proposals that use higher dimensional
systems called qudits, which have the advantage of reducing the number of
physical entities required to perform calculationsMuthukrishnan and Stroud
(2000). Among these are three-level systems, called qutrits, which are the
smallest systems that may exhibit purely quantum correlations such as
contextualityJerger  _et al._ (2016), and they have been used to construct
three-level quantum gatesDogra  _et al._ (2018); Randall  _et al._ (2018).
Qutrits may be emulated by a two-qubit system symmetric under particle
exchange. This allows us to think of many three-level transformations in terms
of operations on the symmetric two-qubit symmetric space.

Within the Majorana representation Majorana (1932); Devi  _et al._ (2012),
symmetric two-qubit states appear as two points (“the stars”) on the unit
sphere. It can be shown that the distance between the two stars maps
bijectively to the concurrenceLiu and Fu (2016). States with maximally
separated stars correspond to Bell states, while states with stars at the same
position are separable. This allows to think of any transformation between
this kind of states as rigid or non-rigid motions of the associated Majorana
constellation, where the latter (former) does (does not) change the
entanglement of states. Whenever there is no place for confusion, we will
refer to a two-qubit symmetric state (space) as symmetric state (space) only.

Since many transformations on symmetric (three-level) spaces involve changing
the entanglement (distance between the Majorana stars), it is important to
quantify the capacity of quantum gates to generate it. Such a quantity is
called the entangling powerZanardi  _et al._ (2000). It is defined as the
average linear entropy of the states produced by a quantum gate $\hat{V}$
acting on the manifold of all separable states. For general two-qubit gates
(TQQGs) the entangling power can be compactly written in terms of a two-qubit
gate local invariantBalakrishnan and Sankaranarayanan (2010) and sets values
to classify TQQGs as perfect entanglersRezakhani (2004); Balakrishnan and
Sankaranarayanan (2010), that is to say, a quantum gate that at least produces
a Bell state out of a separable state. Nevertheless, these expressions do not
quantify the entangling power of gates acting irreducibly on the symmetric
subspace, called symmetric quantum gates (SQGs), because they involve a
contribution from separable non-symmetric states. Here we will derive the
appropriate expression of the entangling power for SQGs and find an onset
value above which they can be classified as perfect entanglers.

As noted by Zhang et al.Zhang  _et al._ (2003), distinct TQQGs can be put
together into sets whose elements differ by local transformations, called
local equivalence classes (LECs). By group theoretical methods, the authors
were able to represent all LECs of TQQGs classes on a tetrahedron, and showed
that half of its volume is occupied by perfect entanglers. Motivated by
Reference Zhang  _et al._ , 2003, we develop a geometric description of the
LECs of symmetric gates. We found that these are characterized by a periodic
set of points on a plane, displaying hexagonal symmetry with lattice vectors
determined by the $\mathfrak{su}(3)$ algebra root vectors. This allows us to
identify a minimum extension where all inequivalent LECs of SQGs are located,
known as the Weyl chamber, and to show that one fourth of it is occupied by
perfect entanglers. This geometric approach as well as the entangling power
concept are relevant to study operations in general three-level systems.

This paper is organized as follows. In sections II and III we present brief
descriptions of the Majorana representation and of the Cartan decomposition of
TQQGs and how it is related to SQGs, respectively. In section IV the
appropriate local-invariant for SQGs is defined. Section V is devoted to
analyse the entangling power and use it to classify SQGs as perfect
entanglers. In section VI the developed formalism is applied to some example
models involving two interacting spinsAbliz  _et al._ (2006), the three-level
Lipkin-Meshkov-Glick model from nuclear physicsLipkin  _et al._ (1965), and
two coupled quantized oscillators through the cross-Kerr effect of quantum
opticsBhattacharya and Shi (2013). Section VII is devoted to conclusions.
Appendix A includes the derivation of the entangling power formula while
appendix B contains a theorem which allows to classify SQGs as perfect
entanglers.

##  II Majorana stellar representation

The Majorana representation is a geometric depiction of quantum states
contained in a finite Hilbert space, which can give insight into their
entangling propertiesLiu and Fu (2016). The idea behind its construction is to
obtain a complex polynomial out of the probability amplitudes that define a
state for some fixed basis. The roots of such a polynomial can be represented
by points in the Argand diagram, and mapped into a sphere by stereographic
projectionMajorana (1932); Liu and Fu (2016). For a quantum state
$\ket{\psi}=\sum a_{k}\ket{k}$ in a $n$ dimensional Hilbert space, the
Majorana polynomial is given by

$\displaystyle\sum\limits_{k=1}^{n}$
$\displaystyle\frac{(-1)^{k}a_{k}}{\sqrt{(n-k)!k!}}z^{n-k}=0.$ (1)

The solutions $\\{z_{k}\\}$ lie on the complex plane, and their projection
onto the Riemann sphere is made by the following association
$z_{k}=\tan(\theta_{k}/2)e^{i\phi_{k}}$. Each root $z_{k}$ is called a
Majorana star, while the whole set of roots is denoted as the Majorana
constellation of the quantum state $\ket{\psi}$. In fact, any two quantum
states with the same constellation are in fact equivalent, up to a global
phase, which makes this representation a good description of their projective
space.

The $n+1$ dimensional Hilbert space has a bijection onto the space of $n$
qubits with particle permutation symmetry. This implies that symmetric-qubit
states have associated Majorana constellations. The $z_{k}$ roots define the
components of a ket state in the symmetric space, by the relation

$\displaystyle\ket{\psi}$
$\displaystyle=\frac{1}{\sqrt{n!}N_{n}}\sum\limits_{P}^{n}\ket{u_{P(1)},u_{P(2)},...,u_{P(n)}},$
(2)

where $\ket{u_{i}}$ represents the 1-qubit state $(\cos\frac{\theta_{i}}{2}\
e^{i\phi_{i}}\sin\frac{\theta_{i}}{2})^{T}$; the $P$ symbol denotes all the
possible permutations of the $\ket{u_{i}}$ states and $N_{n}$ is a
normalization coefficientLiu and Fu (2014). Thus, the Majorana constellation
serves as the mapping between an $n+1$ dimensional Hilbert space and the space
of $n$ qubit symmetric wavefunctions.

For symmetric two-qubit states, the Majorana constellation consists of two
stars. As shown in Reference Liu and Fu, 2016, their concurrence is
proportional to the square of the chordal distance between the stars. This
implies that states with zero concurrence have their Majorana stars on the
same position, or equivalently said, have degenerate stars. On the contrary,
maximally entangled states have stars occupying antipodal positions on the
sphere. For mono-partite three-level systems we may also speak of entangled
states as those whose Majorana stars are not coincident.

On the same line of thought, three-level transformations will be referred to
as entangling as long as they can produce a state with non-degenerate Majorana
stars from a state with a degenerate constellation. Fig. 1 illustrates a
separable state (degenerate constellation) which, after being acted on by a
SQG ends up as a fully entangled state (antipodal Majorana stars). Examples of
these gates in three-level systems are phase gates and the $SWAP_{12}$ and
$SWAP_{23}$, which have found applications in qutrit-based quantum
computingDogra  _et al._ (2018).

Figure 1: Action of a quantum gate on a separable initial state (with
concurrence $C=0$) which ends up as a Bell state ($C=1$) on the Majorana
sphere.

##  III Cartan decomposition of unitary two-qubit transformations

The whole set of transformations on the Hilbert space of two-qubits can be
classified as local and non-local. Local operations are those physical
processes that act separately only on one component of the bipartite system
and, as a consequence, do not change the entanglement properties of the state.
Local two-qubit gates can always be written as tensor product of one-qubit
operations:

$\displaystyle\hat{V}^{(12)}$
$\displaystyle=\hat{V}^{(1)}\otimes\hat{V}^{(2)},$ (3)

which belong to the $SU(2)\otimes SU(2)$ Lie group. We will restrict $\hat{V}$
to denote transformations on the two-qubit space, not necessarily symmetric.

In general, TQQGs $\in SU(4)$ that cannot be written as in (3) are called non-
local. There is a very concise way of writing every element of $SU(4)$ given
by the Cartan decomposition of the group. Namely, for every $\hat{V}\in
SU(4)$, we have the following identityKnapp (1996)

$\displaystyle\hat{V}$ $\displaystyle=\hat{K}_{1}\hat{A}\hat{K}_{2},$ (4)
$\displaystyle\hat{A}$
$\displaystyle=\exp\left[\frac{i}{2}\sum\limits_{k}c_{k}\hat{\sigma}^{(1)}_{x_{k}}\otimes\hat{\sigma}^{(2)}_{x_{k}}\right],$
(5)

The $\hat{K}$ factors belong to the $SU(2)\otimes SU(2)$ Lie group, hence they
are local. As usual, the $\hat{\sigma}_{x_{k}}$ operators denote the Pauli
matrices, with $k=1,2,3$ and $x_{k}=x,y,z$. The $\hat{A}$ factor contains the
non-local part of the quantum gates, and is given by the exponential of linear
combinations of the operators
$\hat{\sigma}^{(1)}_{x_{i}}\otimes\hat{\sigma}^{(2)}_{x_{i}}$. TQQGs that
differ only by a $\hat{K}$ factor are said to be in the same local equivalence
class. These set of operators span the Cartan subalgebra of the $SU(4)$ Lie
group, which is a maximally commuting subalgebra of $\mathfrak{su}(4)$Zhang
_et al._ (2003). It is seen that the $(c_{1},c_{2},c_{3})$ coordinates have a
period of $\pi$ each, and thus the topological structure of the local
equivalence classes is a $3$-torusZhang  _et al._ (2003). The
$\boldsymbol{c}=(c_{1},c_{2},c_{3})$ point will be called geometrical point
hereafter. For a more detailed discussion of the Cartan decomposition of
$SU(4)$ and its algebra, namely the $\mathfrak{su}(4)$ Lie algebra, see
Reference Zhang  _et al._ , 2003.

There is a special case of TQQGs that act irreducibly on the symmetric and
anti-symmetric two-qubit ket spaces. Therefore, if $\hat{V}^{(r)}$ is an
element of such a special set, in which $(r)$ denotes a reducible
representation, it has a matrix form

$\displaystyle\hat{V}^{(r)}$ $\displaystyle=\left(\begin{matrix}\hat{U}&0\\\
0&1\\\ \end{matrix}\right),$ (6)

where $\hat{U}\in SU(3)$ acts on any symmetric linear combination of the
computational basis; $\hat{U}$ is thus the SQG we are interested in. The last
factor acts on the anti-symmetric Bell state
$\ket{\phi^{-}}=\frac{1}{\sqrt{2}}(\ket{01}-\ket{10})$.

Since reducible gates are a subgroup of $SU(4)$, the Cartan decomposition
holds for all elements of the form (6). Also, for reducible TQQGs, the Cartan
decomposition is composed of reducible factors. To see this, first let us note
that $\hat{A}$ is reducible, as will be seen in the next section. With this,
it is readily shown that a sufficient condition for $\hat{V}$ to be reducible
is that $\hat{K}$ be reducible. To show that reducibility of $\hat{V}$ implies
the reducibility of $\hat{K}$ consider the product

$\displaystyle\left(\begin{matrix}\hat{U}&0\\\ 0&1\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}K^{(s)}_{1}&K^{\prime}_{1}\\\
K^{\prime\prime}_{1}&K^{(a)}_{1}\end{matrix}\right)\left(\begin{matrix}A^{(s)}&0\\\
0&A^{(a)}\end{matrix}\right)\left(\begin{matrix}K^{(s)}_{2}&K^{\prime}_{2}\\\
K^{\prime\prime}_{2}&K^{(a)}_{2}\end{matrix}\right).$

The upper and lower off-diagonal elements (primed and doubled-primed) are
three-dimensional column and row vectors, respectively. The $(s)$ and $(a)$
upper indices denote the $3\times 3$ matrix acting on the symmetric subspace
and the scalar acting on the anti-symmetric subspace, severally. By explicit
evaluation of the right hand side of the equation above, it is seen that
$K^{\prime}_{1}=K^{\prime}_{2}=0$ and
$K^{\prime\prime}_{1}=K^{\prime\prime}_{2}=0$, which implies that the
$\hat{K}$ factors are also reducible. The $\hat{K}$ factors for SQGs are
forcefully of the form
$e^{-i\frac{\theta}{2}\boldsymbol{\hat{n}}\cdot\boldsymbol{\hat{\sigma}}}\otimes
e^{-i\frac{\theta}{2}\boldsymbol{\hat{n}}\cdot\boldsymbol{\hat{\sigma}}}$,
with $\boldsymbol{\hat{n}}$ and $\theta$ a unit vector and a rotation angle.
It can be shown that the $\hat{K}^{(s)}$ factors belong to the $SU(2)$ group
in the spin-$1$ representationN. Jeevanjee (2015), which are seen as $SO(3)$
rotations on the Majorana sphereDogra  _et al._ (2018); Liu and Fu (2016).

##  IV Local invariants and equivalence classes of symmetric quantum gates

TQQGs that are equivalent up to a local gate factor (see eq.(4)) have the same
local invariants Makhlin (2002); Zhang  _et al._ (2003). Local invariants for
two-qubit quantum gates are determined by the set of eigenvalues of

$\displaystyle\hat{M}$ $\displaystyle=\hat{V}^{T}_{B}\hat{V}_{B},$ (7)

where the label $B$ indicates that the gate $\hat{V}$ is expressed in the Bell
basis $B=\\{\ket{\psi^{+}},\ket{\psi^{-}},\ket{\phi^{+}},\ket{\phi^{-}}\\}$.
The transformation matrix between the Bell states and the computational basis,
ordered as $\\{\ket{00},\ket{01},\ket{10},\ket{11}\\}$, is

$\displaystyle\hat{Q}^{\dagger}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\begin{matrix}1&0&0&1\\\ 0&i&i&0\\\
i&0&0&-i\\\ 0&1&-1&0\\\ \end{matrix}\right),$ (8)

where the definition of the Bell states is evident from the matrix above; note
that the last row corresponds to the anti-symmetric one. The Lie algebra of
the local components of two-qubit quantum gate is isomorphic to the Lie
algebra of the $SO(4)$ groupZhang  _et al._ (2003), through the map defined by
eq.(8). Thus, any two-qubit quantum gate in the Bell basis, whose
decomposition is given by (4), becomes

$\displaystyle\hat{V}_{B}$ $\displaystyle=\hat{O}_{1}\hat{F}\hat{O}_{2},$ (9)

where $\hat{O}_{1,2}=\hat{Q}^{\dagger}\hat{K}_{1,2}\hat{Q}\in SO(4)$, and
$\hat{F}=\hat{Q}^{\dagger}\hat{A}\hat{Q}$ which is diagonal in this basis. The
Bell states $\ket{\psi^{+}},\ket{\psi^{-}},\ket{\phi^{+}},\ket{\phi^{-}}$ are
thus eigenstates of the $\hat{A}$ matrix, with respective eigenvalues

$\displaystyle\lambda_{1}$ $\displaystyle=e^{i(c_{1}-c_{2}+c_{3})/2},$ (10a)
$\displaystyle\lambda_{2}$ $\displaystyle=e^{i(c_{1}+c_{2}-c_{3})/2},$ (10b)
$\displaystyle\lambda_{3}$ $\displaystyle=e^{i(-c_{1}+c_{2}+c_{3})/2},$ (10c)
$\displaystyle\lambda_{4}$ $\displaystyle=e^{-i(c_{1}+c_{2}+c_{3})/2}.$ (10d)

The eigenvalues of the matrix $\hat{M}$ are determined by the quantities
$(\operatorname{Tr}\,{\hat{M}})^{2}$ and
$(\operatorname{Tr}\,{\hat{M}})^{2}-\operatorname{Tr}\,\hat{M}^{2}$, which in
turn serve to define the local invariants of two qubit quantum gates, namely

$\displaystyle G_{1}$
$\displaystyle=\frac{1}{16}(\operatorname{Tr}\,\hat{M})^{2},$ (11a)
$\displaystyle G_{2}$
$\displaystyle=\frac{1}{4}[(\operatorname{Tr}\,\hat{M})^{2}-\operatorname{Tr}\,\hat{M}^{2}]\
.$ (11b)

Thus, distinct TQQGs having the same local invariants are said to be locally
equivalent.

Reducible TQQGs can be expressed as in eq.(6) and their action on the
symmetric subspace only depends on $\hat{U}$. The local invariant of the
symmetric part of the gate is determined by the eigenvalues of

$\displaystyle\hat{m}$ $\displaystyle=\hat{U}^{T}_{B}\hat{U}_{B},$ (12)

and is then independent of the $\lambda_{4}$ eigenvalue. Since we have
considered special unitary gates, this implies

$\displaystyle c_{1}+c_{2}+c_{3}=0.$ (13)

Had we regarded general unitary gates, removing the extra phase factor would
lead to the same condition above, thus all LECs can be located in the plane
defined in (13).

Figure 2: Phase plot of $\operatorname{Tr}\hat{m}$. The
$\boldsymbol{\alpha}_{1}$ and $\boldsymbol{\alpha}_{2}$ are the root vectors
of the $\mathfrak{su}(3)$ Lie algebra. Every pair of antiparallel short arrows
lies in a plane belonging to the set that generate the Weyl group. The inset
shows a subcell divided into six slices, each one corresponding to a Weyl
chamber.

The secular equation of the matrix $\hat{m}$ is given by

$\displaystyle\lambda^{3}-\operatorname{Tr}(\hat{m})\lambda^{2}-\operatorname{Tr}^{*}(\hat{m})\lambda-1$
$\displaystyle=0,$ (14)

where we have used (9) and (10) to simplify the related factor
$\operatorname{Tr}^{2}(\hat{m})-\operatorname{Tr}(\hat{m}^{2})$. Thus, for
SQGs the eigenvalues of $\hat{m}$ are determined by its trace. The argument of
$\operatorname{Tr}\hat{m}$ is plotted in Fig.2.

We define the SQG local invariant as

$\displaystyle G$
$\displaystyle=\frac{1}{9}\left[\operatorname{Tr}(\hat{m})\right]^{2}.$ (15)

The norm of $G$ can be compactly written in terms of the $(c_{1},c_{2},c_{3})$
vector as

$\displaystyle|G|$
$\displaystyle=1-\frac{4}{9}\left[\sin^{2}(c_{12})+\sin^{2}(c_{13})+\sin^{2}(c_{32})\right],$
(16)

where $c_{ij}$ is a short-hand notation for $c_{i}-c_{j}$. Thus, distinct SQGs
having the same value of $G$ are said to be locally equivalent.

The whole extension of the $O$-plane (13) has more information than is
actually needed, given the periodicity the matrix $\hat{A}$ (eq.(5)) up to
local gate factors $\hat{K}$. Consider the vectors
$\boldsymbol{\alpha}_{1}=(-\pi,0,\pi)$ and
$\boldsymbol{\alpha}_{2}=(0,\pi,-\pi)$ lying on the $O$-plane. The $\hat{A}$
matrix is obviously periodic along these directions. Also, the angle between
them is $2\pi/3$. Thus, SQGs whose geometrical point differ by a
$n\boldsymbol{\alpha}_{1}+m\boldsymbol{\alpha}_{2}$ translation
($n,m\in\mathbb{Z}$) are locally equivalent, and the whole set of local
equivalent classes can be found within the hexagonal primitive cell spanned by
$\boldsymbol{\alpha}_{1}$ and $\boldsymbol{\alpha}_{2}$, which is displayed in
Fig. 2 by the area between these vectors and the dotted lines. The vectors
$\boldsymbol{\alpha}_{1}$ and $\boldsymbol{\alpha}_{2}$ are the root vectors
of the $\mathfrak{su}(3)$ Lie algebra, and the set
$\\{\pm\boldsymbol{\alpha}_{1},\pm\boldsymbol{\alpha}_{2},\pm(\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2})\\}$
(solid-arrows in Fig. 2) form the root space of the algebraHall (2015).

Consider the reflection matrix
$(\hat{\sigma}_{\boldsymbol{\hat{\beta}}})_{ij}=\delta_{ij}-2\beta_{i}\beta_{j}/\beta^{2}\
(i,j=x,y,z)$ on the plane normal to the unit vector
$\boldsymbol{\hat{\beta}}$, which is obtained from any vector lying between
two successive root vectors (dotted-line arrows). It can be checked that the
effect of $\hat{\sigma}_{\boldsymbol{\hat{\beta}}}$ is to permute and complex-
conjugate the eigenvalues of the $\hat{m}$ matrix. From Fig. 2 it is seen that
reflection on the plane normal to the vertical $\boldsymbol{\hat{\beta}}$
vectors interchanges the triangles composing the unit cell depicted there.
This means that knowledge of $\operatorname{Tr}\hat{m}$ on a subcell
determines its value on the other subcell by complex conjugation, and thus all
LECs can be located in just one subcell.

Reflection on the planes normal to the root vectors generates the Weyl group
of $\mathfrak{su}(3)$; the corresponding reflection matrices are given by
$\hat{\sigma}_{\boldsymbol{\hat{\alpha}}}$, where $\boldsymbol{\hat{\alpha}}$
is a normalized root vector. The action of this group on the $\hat{A}$ matrix
is permuting its eigenvalues, and thus leaves the character of $\hat{m}$
invariant. As in the general two-qubit caseZhang  _et al._ (2003), the Weyl
group allows to define a minimum extension containing the whole set of local
equivalence classes, called the Weyl chamber Zhang  _et al._ (2003); Hall
(2015). By bisecting one triangle of the primitive cell by the planes normal
to the root space we get the inset of Fig. 2. Every slice of the triangle
contains all the local-equivalence classes of SQGs up to complex conjugation
and represents a Weyl chamber. Thus, by means of the
$\hat{\sigma}_{\boldsymbol{\hat{\beta}}}$ and
$\hat{\sigma}_{\boldsymbol{\hat{\alpha}}}$ reflections, we have reduced to a
minimum the extension needed to locate all distinct LECs. We will take
advantage of this in the next section to obtain the ratio of perfect
entanglers to all the possible SQGs.

##  V Entangling power of symmetric two-qubit quantum gates

###  V.1 Expression and properties

Figure 3: Entangling power $ep$ of SQGs on the $O$-plane. The parallelogram
spanned by the root vectors $\boldsymbol{\alpha_{1}},\boldsymbol{\alpha_{2}}$
corresponds to the primitive cell. The smaller section defined by
$\boldsymbol{\beta_{1}},\boldsymbol{\beta_{2}}$ contains all the possible
values $ep$.

The entangling power of a quantum gate acting on a bipartite system is a
measure of the ability for these gates to create entangled states from the set
of all bipartite separable states. In the general two-qubit caseZanardi  _et
al._ (2000), the entangling power of a gate $\hat{V}$ (4) is defined as the
average of the linear entropy of entanglement
$E(\ket{\psi})=1-\operatorname{Tr}(\hat{\rho}^{2}_{1})$, over the set of all
separable symmetric two-qubit states with a uniform probability,

$\displaystyle ep(\hat{A})$
$\displaystyle=\overline{E(\hat{V}\ket{\psi_{1}}\otimes\ket{\psi_{2}})},$ (17)

with the bar indicating such an average. Note that we have written $ep$ as a
function of $\hat{A}$, since local transformations do not change the
entanglement of a quantum state. The entanglement power is very informative,
given that it can be compactly expressed in terms of the two-qubit $|G_{1}|$
local-invariant and can be used as an indicator to whether a quantum gate is a
perfect entangler or notBalakrishnan and Sankaranarayanan (2010).
Nevertheless, for SQGs acting on symmetric states only, this expression of the
entangling power is not adequate since it takes into account all two-qubit
separable states, not necessarily symmetric. Hence, we need to restrict the
entangling power definition to symmetric states in order to obtain the correct
expression. Accordingly, we define the entangling power of SQGs as

$\displaystyle ep(\hat{U})$
$\displaystyle=\overline{E(\hat{U}\ket{u}\otimes\ket{u})}.$ (18)

where $\ket{u}$ is a 1-qubit state, as in the section II.

We are now going to derive an explicit formula for this expression. First of
all, let us consider a uniform distribution of initial $\ket{u,u}$ states, for
which the Majorana constellation consists of two stars in the same position.
Referring to appendix A for the details in the derivation, the entangling of
SQGs is

$\displaystyle ep(\hat{U})$ $\displaystyle=\frac{3}{10}\left(1-|G|\right),$
(19)

with $|G|$ given by expression (16). It is remarkable that the entangling
power obtained can be so compactly expressed in terms of the local invariant
$|G|$, which only depends on the trace of matrix $\hat{m}$ (12). This result
resembles that of the entangling power for general two-qubit gatesBalakrishnan
and Sankaranarayanan (2010), namely $ep=2(1-|G_{1}|)/9$.

The function $ep(c_{1},c_{2},c_{3})$ (19) presents minimum and maximum
magnitudes at the geometrical points $\boldsymbol{c}=(0,0,0)$ and
$(-\pi/3,0,\pi/3)$, which are zero and $3/10$, respectively. As Fig. 3
suggests, these extreme values are also reached in additional points
$\boldsymbol{c}$, obtained through symmetry operations which are translations
along $\boldsymbol{\beta}$ vectors and $C_{6}$ rotations about the $(1,1,1)$
axis. Note also that $ep$ is invariant under translations along this same
vector, which means that the same pattern as that shown in Fig. 3 is displayed
in planes parallel to the $O$-plane (13).

###  V.2 Perfect entanglers

A TQQG $\hat{V}$ is a perfect entangler if it is capable of producing a fully
entangled state from a separable one. The condition for this is that the
convex hull of eigenvalues of the $\hat{M}$ matrix contains the origin in the
$\boldsymbol{c}$-spaceZhang  _et al._ (2003). For SQGs the same condition
holds applied to the corresponding matrix $\hat{m}$. The proof of this goes
along the same lines as in the general case Zhang  _et al._ (2003); in order
to make the paper more self-contained we sketch it in appendix B.

Figure 4: Convex hull of the matrix $\hat{m}$ associated to SQGs for several
entanglement capabilities. The vertices on the unitary circle are defined by
the phase of the eigenvalues of $\hat{m}$. (a) Perfect entangler with a
maximum value of $ep$. (b) Perfect entangler with a minimum value of $ep$. (c)
Not a perfect entangler. The entangling character of the gate is geometrically
determined by the location of the origin, inside (perfect) or outside (non-
perfect) of the convex hull.

Fig. 4 shows the convex hulls of eigenvalues for three distinct cases. The
eigenvalues of unitary matrices all have unit length, and thus the circle in
the figure is unitary. The vertices of the triangles are defined by the phase
of $\lambda_{i}^{2}$, which are the eigenvalues of $\hat{m}$. In (a) the
eigenvalues are separated by $2\pi/3$. This case corresponds to the maximum
value $ep=3/10$ of the entangling power, since
$\lambda^{2}_{1}+\lambda^{2}_{2}+\lambda^{2}_{3}=0$ and $|G|=0$. The case in
(b) also represents a convex hull for a perfect entangling SQG, with the
requirement that $\lambda^{2}_{i}=-\lambda^{2}_{j}$. This makes $|G|=1/9$ and,
as a consequence,the entangling power is then $4/15$. In fact, the $SWAP_{12}$
($SWAP_{23}$) gateDogra  _et al._ (2018) have this value of $ep$, hence all
SQGs with the same entangling power are locally equivalent to the $SWAP_{12}$.
Since any deformation of such a convex hull such that it no longer contains
the origin makes $ep$ less than $4/15$, this values is the minimum such that
the corresponding SQGs are perfect entanglers. A SQG with this convex hull can
be built with coefficients $(0,\pi/2,\pi/2)$. Other coefficients
$(c_{1},c_{2},c_{3})$ satisfying this condition can be obtained through
symmetry transformations on the geometric point just given. The case when the
convex hull does not contain the origin is depicted in (c), the SQGs not being
perfect entanglers ($ep<4/15$).

Now that we can classify SQGs as perfect entanglers or not according the
geometrical point $\boldsymbol{c}$, we are at a position to calculate the
ratio of perfect entanglers to non-perfect entanglers. To do this we will
calculate the area on the Weyl chamber whose geometrical points correspond to
perfect entangling SQGs. Let us restrict the $(c_{1},c_{2},c_{3})$ to the Weyl
chamber, as shown on Fig. 5. Comparison with Fig. 3 indicates the following
association: $\boldsymbol{v}=\boldsymbol{\beta}_{1}+\boldsymbol{\beta}_{2}$,
$\boldsymbol{v}_{x}=(\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2})/2$ and
$\boldsymbol{v}_{y}=(\boldsymbol{\alpha}_{1}-\boldsymbol{\alpha}_{2})/6$. As
before, the $\boldsymbol{\alpha}_{1}$ and $\boldsymbol{\alpha}_{2}$ root
vectors are taken as $\pi(-1,0,1)$ and $\pi(0,1,-1)$. Hence, the vectors on
Fig. 5 are $\boldsymbol{v}=\frac{\pi}{3}(-2,1,1)$,
$\boldsymbol{v}_{x}=\frac{\pi}{2}(-1,1,0)$, and
$\boldsymbol{v}_{y}=\frac{\pi}{6}(-1,-1,2)$.

Figure 5: Weyl chamber of local-equivalence classes. The blue area contains
all the perfect entanglers, and it is one fourth of the triangle.

Any geometrical point on the Weyl chamber shown in Fig. 5 can be expressed as
$\boldsymbol{c}=s_{1}\boldsymbol{v}_{x}+s_{2}\boldsymbol{v}_{y}$, where
$s_{1},s_{2}\in[0,1]$ and $s_{2}\leq s_{1}$. Upon multiplying the
$\lambda^{2}$ eigenvalues by a total phase, its convex hull gets rotated, and
does not affect whether or not it contains the origin. Thus, regarding
$c_{1}+c_{2}+c_{3}=0$ and setting $\lambda^{2}_{2}=0$ (see eqs.(10)), we have

$\displaystyle\phi_{1}$ $\displaystyle=\pi(s_{1}+s_{2}),$ (20)
$\displaystyle\phi_{3}$ $\displaystyle=\pi(s_{2}-s_{1}),$ (21)

where $\phi_{1}$ and $\phi_{3}$ are the phase angles of $\lambda^{2}_{1}$ and
$\lambda^{2}_{3}$, respectively. With this, a SQG is a perfect entangler if
and only if the following condition holds:

$\displaystyle 0\leq\phi_{1}\leq\pi\hskip 14.22636ptAND\hskip
14.22636pt-\pi\leq\phi_{3}\leq-\pi+\phi_{1},$ (22)

where all phases are equivalent modulo $2\pi$. The case $\phi_{1}>\pi$ always
yields non-perfect entanglers by the imposed conditions on $s_{1}$ and $s_{2}$
(see the discussion below). Let us analyse all sections of the Weyl chamber to
determine whether they are composed of perfect entanglers or not.

Area 1. This area is constrained to the $(s_{1},s_{2})$ coordinates: $0\leq
s_{1}<1/2$ and $0\leq s_{2}\leq s_{1}$. These inequalities imply that
$s_{1}+s_{2}<1$, hence $0\leq\phi_{1}<\pi$, leaving us in the domain of (22).
By substituting $\phi_{3}$ into the right side of (22) we get $s_{1}\geq 1/2$,
which is a contradiction given the imposed conditions on $s_{1}$ and $s_{2}$.
Thus, all geometrical points in this section of the Weyl chamber do not
correspond to perfect entanglers.

Areas 2 and 3. For these regions we have the following restriction on the
$(s_{1},s_{2})$ coordinates: $1/2<s_{1}\leq 1$ and $1-s_{1}<s_{2}\leq s_{1}$.
These imply $\phi_{1}>\pi$ and $-\pi<\phi_{3}<0$. Thus, both $\lambda^{2}_{1}$
and $\lambda^{2}_{3}$ are on the lower half of the unit circle (see Fig.4.c),
none of them at $\pi$. The convex hull does not contain the origin and the
geometrical points do not correspond to perfect entanglers.

Blue area. In this case, the $(s_{1},s_{2})$ coordinates are constrained by
the inequalities: $1/2\leq s_{1}\leq 1$ and $0\leq s_{2}\leq 1-s_{1}$. This
implies $\pi\geq\phi_{1}\geq\pi/2$ and consequently we must focus on
expression (22). Inserting eqs.(20) and (21) into the right inequality of (22)
we obtain $-1\leq s_{2}-s_{1}\leq-1+s_{1}+s_{2}$. $-1\leq s_{2}-s_{1}$ holds
trivially, while the right hand side of the last inequality implies $s_{1}\geq
1-s_{1}$, which holds since we are considering $1\geq s_{1}\geq 1/2$.
Expression (22) holds in this case and all the geometrical points contained in
the blue area correspond to perfect entanglers. This section occupies one
fourth of the Weyl chamber. For this reason, the perfect entanglers are one
fourth of the total SQGs.

It is worth noting at this point that the geometric picture of LECs of SQGs
does not trivially arises from that of TQQGs. For example, the Weyl Chamber of
TQQGs is a tetrahedron that, without loss of generality, has one vertex on the
origin $O$. One of its edges, called the $OA_{3}$ edgeZhang  _et al._ (2003);
Balakrishnan and Sankaranarayanan (2009), along geometrical points of the form
$(c,c,c)$, contains a point $P$ which represents a perfect entangler.
Nevertheless, for SQGs, the convex hull on any geometrical point of the form
$(c,c,c)$ does not contain the origin, as can be seen from inspection of
eqs.(10) and, as a result, the entangling power is zero along the $OA_{3}$
edge. This fundamental difference between the geometry of LECs of SQGs and
TQQGs proves that the former is not just a trivial special case of the latter.

##  VI Examples

In this section we will apply the theory developed so far to three distinct
physical models with three-dimensional Hilbert spaces. We calculate the
entangling power as a function of an independent parameter for the three
models and find conditions on them to obtain perfect entanglers. The linear
entropy on the Majorana sphere is computed, where each point on it corresponds
to a state with degenerate Majorana constellation and the color indicates the
value of the entanglement measure of the final state. We comment on some
features of the spacial distribution of entanglement on the Majorana sphere.

###  VI.1 Anisotropic Heisenberg model with no cross-terms

The anisotropic Heisenberg model of two interacting spins is represented by
the HamiltonianAbliz  _et al._ (2006)

$\displaystyle\hat{H}_{H}$
$\displaystyle=-\frac{1}{2}\left(I_{x}\hat{\sigma}^{(1)}_{x}\hat{\sigma}^{(2)}_{x}+I_{y}\hat{\sigma}^{(1)}_{y}\hat{\sigma}^{(2)}_{y}+I_{z}\hat{\sigma}^{(1)}_{z}\hat{\sigma}^{(2)}_{z}\right),$
(23)

where $I_{i}$ are the spin coupling constants. This Hamiltonian is composed of
the Cartan subalgebra elements of $\mathfrak{su}(4)$, and hence has a
reducible representation into symmetric and anti-symmetric subspaces. The
symmetric part of the time evolution operator is, in the Bell basis,

$\displaystyle\hat{U}_{H}$
$\displaystyle=\left(\begin{matrix}e^{i(I_{x}-I_{y}+I_{z})t/2}&0&0\\\
0&e^{i(I_{x}+I_{y}-I_{z})t/2}&0\\\
0&0&e^{i(-I_{x}+I_{y}+I_{z})t/2}\end{matrix}\right).$ (24)

From eq.(23), the $\boldsymbol{c}$ coordinate vector is $(I_{x},I_{y},I_{z})$.
The entangling power becomes

$\displaystyle ep$
$\displaystyle=\frac{2}{15}\left[\sin^{2}(I_{xy}t)+\sin^{2}(I_{yz}t)+\sin^{2}(I_{xz}t)\right],$
(25)

where $I_{ij}=I_{i}-I_{j}$. Note that for equal spin coupling constants the
entangling power is zero, which means that the isotropic Heisenberg model does
not have any entangling power on the symmetric two-qubit subspace.

Fig. 6(a) shows the entangling power as a function of $\omega t$ for the
choice of parameters $I_{y}=0,I_{x}=-I_{z}=\omega$ (left panel). The maximum
values are located at $\omega t=\pi/3,2\pi/3$, as one would expect, while the
minimum $ep$ for which the quantum gate is a perfect entangler is located at
$\omega t=\pi/2$.

The sphere on the right shows the linear entropy on the Majorana sphere; we
have chosen $\omega t=\pi/3$ in order to obtain maximum entanglement. There
are zones on which the quantum gate does not produce entanglement (red spots),
and zones where the initial states become symmetric Bell states (blue spots).
Even though it is not fully depicted here, there are exactly four low-
entanglement zones and four high-entanglement zones, which form a tetrahedron
on the sphere. As far as we have numerically checked, this tetrahedron
distribution of entanglement on the Majorana sphere is a general feature of
SQGs with maximum entangling power.

Figure 6: Left: entangling power for (a) the anisotropic Heisenberg model, (b)
the Lipkin-Meshkov-Glick model and (c) the cross-Kerr interaction. The
horizontal-dashed line indicates the lower bound for perfect entanglers
($ep=4/15$). Right: Linear entropy of the final states obtained by applying
$\hat{U}$ to the initial states with degenerate Majorana stars.

###  VI.2 Lipkin-Meshkov-Glick model

The Lipkin-Meshkov-Glick model was firstly proposed to study the many body
problem in nuclear physicsLipkin  _et al._ (1965), and has also been useful to
model the physics of molecular solidsGarg (1993); Campos and Hirsch (2011) and
critical phenomena in Bose-Einstein condensatesRibeiro  _et al._ (2007). The
Hamiltonian describing the interaction is given by

$\displaystyle\hat{H}_{L}$
$\displaystyle=B\hat{J}_{z}+g_{1}\hat{J}^{2}_{z}-g_{2}\hat{J}^{2}_{x}.$ (26)

This Hamiltonian commutes with the total angular momentum operator
$\boldsymbol{J}^{2}$, which allows us to fix the $j$ value to unity, for which
the model belongs to the $\mathfrak{su}(3)$ Lie algebra. This model can be
written in matrix form as

$\displaystyle\hat{H}_{L}$
$\displaystyle=\left(\begin{matrix}B+g_{1}-g_{2}/2&0&-g_{2}/2\\\ 0&-g_{2}&0\\\
-g_{2}/2&0&-B+g_{1}-g_{2}/2\end{matrix}\right).$ (27)

The eigenvalues oh $\hat{H}_{L}$ are: $\lambda_{0}=-g_{2}$ and
$\lambda_{\pm}=g_{1}-g_{2}/2\pm\sqrt{B^{2}+g^{2}_{2}/4}$. The corresponding
quantum gates of this system are given by $\exp{(-i\hat{H}_{L}t)}$.

This model is a three-level system, and as such can be mapped to the symmetric
two-qubit space. The isomorphism is given by mapping the angular momentum kets
$\ket{1,1},\ket{1,0}$ and $\ket{1,-1}$ to the symmetric states
$\ket{0,0},\frac{1}{\sqrt{2}}(\ket{0,1}+\ket{1,0}),\ket{11}$, respectively.
With this, the transformation matrix from the spin-$1$ angular momentum basis
to the symmetric Bell-basis is given by

$\displaystyle\hat{T}$ $\displaystyle=\left(\begin{matrix}1&0&1\\\
0&\frac{i}{\sqrt{2}}&0\\\ i&0&-i\end{matrix}\right).$ (28)

The definition of the Bell states should be evident from the matrix above.
With this, the absolute square of the local invariant G, is given by

$\displaystyle|G_{L}|$
$\displaystyle=\frac{1}{9}[1+4G_{1}\cos(2(g_{1}+g_{2}/2)t)+4G^{2}_{1}]\ .$
(29)

where

$\displaystyle G_{1}$ $\displaystyle=1-\frac{g^{2}_{2}\sin^{2}(Rt)}{2R},$
$\displaystyle R$ $\displaystyle=\sqrt{B^{2}+g^{2}_{2}/4}.$

The entangling power is readily obtained through eq.(19).

Fig. 6(b) shows the entangling power for the choice of parameters
$-2B/7=g_{1}/2=g_{2}/4=\omega$, where $\omega$ is a fixed frequency. The
entangling power displays an oscillating behaviour where the maximum $ep$ is
reached at values of $\omega t$ closed to $\frac{\pi}{4}(n+1/2)$ for some
integer $n\geq 0$. On the right panel the linear entropy on the Majorana
sphere for $\omega t=\pi/8$ is plotted. The colors on the sphere follow the
tetrahedron-like patterns, as we have numerically checked for SQGs with
maximum entangling power.

###  VI.3 Cross-Kerr Interaction

The non-linear cross-Kerr effect in quantum optics is model by the
HamiltonianBhattacharya and Shi (2013)

$\displaystyle\hat{H}_{CK}$
$\displaystyle=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger}\hat{b}+\omega_{CK}\hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b}.$
(30)

This Hamiltonian conserves the total number of excitations
$\hat{N}=(\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b})$. By transforming
the Hamiltonian via the Schwinger operators of angular momentumSakurai and
Napolitano (1994), it is easily seen that $\hat{N}/2$ equals the total angular
momentum $j$ of the resulting Hamiltonian. Let us work in the $j=1$
representation. The cross-Kerr Hamiltonian becomes

$\displaystyle\hat{H}_{CK}$
$\displaystyle=(\omega_{a}-\omega_{b})\hat{J}_{z}-g_{CK}\hat{J}^{2}_{z},$

The cross-Kerr interaction can be modeled as a Lipkin-Meshkov-Glick model
after the identification $B=\omega_{a}-\omega_{b},g_{1}=-g_{CK},g_{2}=0$. The
entangling power is then easily obtained from eqs. (29) and (19) and the
transformation matrix (28),

$\displaystyle ep_{CK}$ $\displaystyle=\frac{4}{15}\sin^{2}(g_{CK}t).$

Fig. 6(c) shows the entangling power for a choice of parameters
$B=g_{1}/2=\omega$. The entangling power has an oscillating behaviour that
never reaches the maximum possible value. The eigenvalues of the $\hat{m}$
matrix are $e^{-ig_{CK}t},e^{-ig_{CK}t},1$. There is a pair of degenerate
eigenvalues, and thus the convex hull contains the origin only when the
vertices are on antipodal positions on the unit circle, which gives the
minimum $ep$ for which $\hat{U}_{CK}$ is a perfect entangler. The linear
entropy on the Majorana sphere is depicted on the right, which is very
different from the spheres on panels (a) and (b). The antipodal high
entangling spots on the sphere with a low entangling zone between them is a
general feature of quantum gates with $ep=4/15$, as we have seen from distinct
numerical calculations, without regards on the details of the Hamiltonian.

##  VII Conclusions

We have given a geometric classification of the LECs of SQGs, which turns out
to be a plane with hexagonal symmetry. There we have identified the Weyl
chamber, which is the minimum area that contains all distinct LECs. This
geometric description contrasts with the general two-qubit case, for which the
geometry is three-dimensional. The entangling power for SQGs is obtained in
terms of the local-invariant. As was done in the general two-qubit case in
reference Zhang  _et al._ (2003), we give conditions for which a SQG is a
perfect entangler, and have found that the perfect entanglers are 1/4 of all
possible SQGs. Along this line, it is also found that perfect entanglers must
have $ep\geq 4/15$. It is stressed that LECs of SQGs do not arise trivially
from those of TQQGs, since there exist geometrical points corresponding to
perfect entanglers in the latter case (TQQGs) that yield gates with zero $ep$
in the former (SQGs). The theoretic framework just developed can be applied to
any three-level system, despite whether being bi-partite or not. The
entangling power then refers to the capability of quantum gates to perform
transformations that do not act as $SO(3)$ rotations on the Majorana
constellation as, for example, some phase gates, $SWAP_{12}$ and $SWAP_{23}$
gatesDogra  _et al._ (2018). Finally, the theory just developed was applied to
three physical models, namely, the anisotropic Heisenberg ModelAbliz  _et al._
(2006), the Lipkin-Meshkov-Glick modelLipkin  _et al._ (1965) and two coupled
quantized oscillators with cross-Kerr interactionBhattacharya and Shi (2013)
and found some conditions on the Hamiltonian parameters to generate perfect
entangling SQGs. Additional examples might include solid-state systems, or
optical analogues, like assemblies of three quantum dots with few electrons
used to study coherent control of quantum statesBrandes (2005), Landau-Zener-
Stückelberg interferometryGallego-Marcos  _et al._ (2016) or quantum transport
Maldonado  _et al._ (2018), among other properties.

##  VIII Acknowledgements

D.M.G. acknowledges financial support from CONACyT (México).

##  Appendix A Derivation of the entangling power for SQGs

To derive an explicit expression for the entangling power of SQGs we first
compute $\operatorname{Tr}[\hat{\rho}^{2}_{1}]$ and then integrate it over the
unit sphere. The pure state density matrix obtained after applying a SQG
$\hat{U}$ to a symmetric two-qubit separable state $\ket{u,u}$ is given by
$\hat{U}\ket{u,u}\bra{u,u}\hat{U}^{\dagger}$. Since we are interested in this
quantity in order to calculate the entropy of entanglement, the local
contributions to the quantum gate $\hat{U}$ can be omitted, hence

$\displaystyle\hat{\rho}$
$\displaystyle=\hat{A}\ket{u,u}\bra{u,u}\hat{A}^{\dagger}.$

Now, by expressing $\ket{u,u}$ in the Bell basis through the matrix
$\hat{Q}^{\dagger}$ (8), we find

$\displaystyle\hat{Q}^{\dagger}$
$\displaystyle\ket{u,u}=\frac{1}{\sqrt{2}}\left(\begin{matrix}a&ib&-ic&0\end{matrix}\right)^{T},$

where $a=\cos\phi-i\sin\phi\cos\theta$, $b=i\sin\theta$ and
$c=-i\sin\phi+\cos\phi\cos\theta$. By transforming the $\hat{A}$ matrix to the
Bell basis too, and after some manipulations we get the density matrix

$\displaystyle\hat{\rho}$
$\displaystyle=\frac{1}{4}\left(\begin{matrix}|A|^{2}&AB^{*}&AB^{*}&AC^{*}\\\
A^{*}B&|B|^{2}&|B|^{2}&BC^{*}\\\ A^{*}B&|B|^{2}&|B|^{2}&BC^{*}\\\
A^{*}C&BC^{*}&BC^{*}&|C|^{2}\\\ \end{matrix}\right),$

where the complex coefficients in the matrix elements are
$A=\lambda_{1}a-i\lambda_{3}c$, $B=\lambda_{2}b$ and
$C=\lambda_{1}a+i\lambda_{3}c$, with the $\lambda_{i}$ factors given in eqs.
(10). After reducing this matrix on one of the subsystem, we obtain

$\displaystyle\operatorname{Tr}\hat{\rho}^{2}_{1}$
$\displaystyle=\frac{1}{16}\left[(|A|^{2}+|B|^{2})^{2}+(|B|^{2}+|C|^{2})^{2}\right.$
$\displaystyle\hskip 99.58464pt\left.+2|AB^{*}+BC^{*}|^{2}\right].$

What remains is to average this expression over all the possible symmetric
separate states (degenerate Majorana constellations), where each state is
equally likely to be obtained,

$\displaystyle\overline{\operatorname{Tr}\hat{\rho}^{2}_{1}}=\frac{1}{4\pi}\int_{\mathcal{A}}\left[\operatorname{Tr}\hat{\rho}^{2}_{1}\right]\sin\theta\,d\theta
d\phi,$

with integration performed over the surface $\mathcal{A}$ of the unit sphere.
After a long but straightforward algebra, we obtain from (18)

$\displaystyle ep$
$\displaystyle=\frac{2}{15}[\sin^{2}(c_{12})+\sin^{2}(c_{23})+\sin^{2}(c_{31})],$

which can be recast in the form (19) by using the local invariant expression
(16).

##  Appendix B Condition for a perfect entangler

In order to determine the values of $ep$ for which a SQG is a perfect
entangler, we need first a theorem based on the eigenvalues of the matrix
$\hat{m}$. The proof runs very similar as that given in Reference Zhang  _et
al._ , 2003 for the general two-qubit case, with the appropriate changes to
focus on the symmetric case. First, we recall the definition of a convex hull.

###### Definition 1

(Convex hull) The convex hull of $n+1$ points
$\boldsymbol{p}_{0},\boldsymbol{p}_{1},...,\boldsymbol{p}_{n}\in\mathbb{R}^{n}$
is given by the set of all vectors of the form
$\sum\limits^{n}_{i=0}\theta_{i}\boldsymbol{p}_{i}$, where $\theta_{i}$ are
non-negative real numbers satisfying $\sum\limits^{n}_{i=0}\theta_{i}=1$.

Now, we state and prove the following theorem.

###### Theorem 1

(Perfect entanglers) A symmetric two-qubit gate $\hat{U}$ is a perfect
entangler if and only if the convex hull of eigenvalues of $\hat{m}$ contains
zero.

Proof: As explained in section III, a general symmetric two-qubit gate can be
decomposed as $\hat{U}=\hat{K}^{(s)}_{1}\hat{A}^{(s)}\hat{K}^{(s)}_{2}$, where
$\hat{K}^{(s)}\in SU(2)$ in the spin-$1$ representation. Given a symmetric
separable two-qubit state $\ket{\psi_{o}}=(a\ b\ b\ c)^{T}$, it is obtained
for the concurrenceAudrescht (2007) of the state $\hat{U}\ket{\psi_{o}}$ that
$C(\hat{U}\ket{\psi_{o}})=C(\hat{A}\ket{\psi_{o}})$, and thus, for $\hat{U}$
to be a perfect entangler, $\hat{A}$ has to be a perfect entangler; we have
also used $\hat{A}\ket{\psi_{o}}=\hat{A}^{(s)}\ket{\psi_{o}}$, since
$\ket{\psi_{o}}$ is symmetric. Explicitly,
$C(\hat{A}\ket{\psi_{o}})=\overline{\bra{\psi_{o}}}\hat{A}^{T}\hat{P}\hat{A}\ket{\psi_{o}}$,
where $\hat{P}=-\hat{\sigma}_{y}\otimes\hat{\sigma}_{y}$. This expression can
be rewritten in terms of Bell states as
$C(\hat{A}\ket{\psi_{o}})=\overline{\bra{\psi_{o}}\hat{Q}})\hat{F}^{2}(\hat{Q}^{\dagger}\ket{\psi_{o}}$,
where we have used the matrix $\hat{Q}$ (8) and the result
$\hat{Q}^{T}\hat{P}\hat{Q}=\mathds{1}$; the operator $\hat{F}$ is
$\hat{Q}^{\dagger}\hat{A}\hat{Q}$. Let
$\hat{Q}^{\dagger}\ket{\psi_{o}}=\ket{\phi}$. Since $\ket{\psi_{o}}$ is a non-
entangled state, we have
$C(\ket{\psi_{o}})=\overline{\bra{\psi_{o}}}\hat{P}\ket{\psi_{o}}=\overline{\bra{\phi}}\hat{Q}^{T}\hat{P}\hat{Q}\ket{\phi}=\overline{\bra{\phi}}\mathds{1}\ket{\phi}=0$,
which implies

$\displaystyle\phi^{2}_{1}+\phi^{2}_{2}+\phi^{2}_{3}=0,$ (31)

where only three expansion coefficients appear, since $\ket{\psi_{o}}$ is a
symmetric state, and thus has no projection on the antisymmetric axis. For
$\hat{A}$ to be a perfect entangler, the concurrence of
$\hat{A}\ket{\psi_{o}}$ must equal unity, which along with the normalization
condition yields
$|\phi_{1}^{2}\lambda^{2}_{1}+\phi_{2}^{2}\lambda^{2}_{2}+\phi_{3}^{2}\lambda^{2}_{3}|=|\phi^{2}_{1}\lambda^{2}_{1}|+|\phi^{2}_{2}\lambda^{2}_{2}|+|\phi^{2}_{3}\lambda^{2}_{3}|$,
where the eigenvalues $\lambda_{i}$ are given in eq.(10). This equation holds
if and only if there exists a number $\theta\in[0,2\pi)$, such that
$\phi_{j}^{2}\lambda^{2}_{j}=|\phi_{j}|^{2}e^{i2\theta},\,j=1,2,3,4$. With the
abovementioned and eq.(31), we get

$\displaystyle\phi^{2}_{1}+\phi^{2}_{2}+\phi^{2}_{3}$
$\displaystyle=e^{i2\theta}\left(\frac{|\phi_{1}|^{2}}{\lambda^{2}_{1}}+\frac{|\phi_{2}|^{2}}{\lambda^{2}_{2}}+\frac{|\phi_{3}|^{2}}{\lambda^{2}_{3}}\right),$
$\displaystyle=e^{i2\theta}\left(|\phi_{1}|^{2}\overline{\lambda^{2}_{1}}+|\phi_{2}|^{2}\overline{\lambda^{2}_{2}}+|\phi_{3}|^{2}\overline{\lambda^{2}_{3}}\right),$
$\displaystyle=0.$

By complex conjugation of the last equality, it follows that the convex hull
of the eigenvalues of $m(\hat{U})$ contains the origin. The converse statement
can be done following Reference Zhang  _et al._ , 2003.

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# Characterizing Exceptional Points Using Neural Networks

Md. Afsar Reja  Solid State and Structural Chemistry Unit, Indian Institute of
Science, Bangalore 560012, India  Awadhesh Narayan<EMAIL_ADDRESS>Solid
State and Structural Chemistry Unit, Indian Institute of Science, Bangalore
560012, India

###### Abstract

One of the key features of non-Hermitian systems is the occurrence of
exceptional points (EPs), spectral degeneracies where the eigenvalues and
eigenvectors merge. In this work, we propose applying neural networks to
characterize EPs by introducing a new feature – _summed phase rigidity_ (SPR).
We consider different models with varying degrees of complexity to illustrate
our approach, and show how to predict EPs for two-site and four-site gain and
loss models. Further, we demonstrate an accurate EP prediction in the
paradigmatic Hatano-Nelson model for a variable number of sites. Remarkably,
we show how SPR enables a prediction of EPs of orders completely unseen by the
training data. Our method can be useful to characterize EPs in an automated
manner using machine learning approaches.

Introduction– In recent years, the exploration of non-Hermitian systems has
been gaining wide interest Moiseyev (2011); Bender (2007); Ashida  _et al._
(2020); Bergholtz  _et al._ (2021); Kawabata  _et al._ (2019); Banerjee  _et
al._ (2022). This is due to a large number of inherent rich physical
phenomena, such as exceptional points (EPs), non-Hermitian skin effect (NHSE),
non-Bloch band theory, exotic topological phases, and extended symmetry
classes, which have no counterpart in the contemporary Hermitian realm. One of
the most intriguing characteristics of non-Hermitian systems is the existence
of EPs – spectral degeneracy points at which not only the eigenvalues merge
but also the eigenstates coalesce simultaneously Heiss (2012); Kato (2013). At
EPs, the Hamiltonian matrix becomes defective. EPs are an inherent
characteristic of non-Hermitian physics. They serve a pivotal role in defining
non-Hermitian topological phases and the associated phase transitions. EPs can
also manifest as exceptional rings Zhen  _et al._ (2015) and lines Budich  _et
al._ (2019) within the parameter space of non-Hermitian systems, and they are
closely tied to PT (parity-time) symmetry-breaking phase transitions Heiss
(2012). Non-Hermitian Bloch bands form knots and EPs play a crucial role in
the transition between different knots Hu and Zhao (2021). In addition to
their fundamental significance, EPs have already found practical applications,
including enhanced sensing capabilities at higher-order EPs Hodaei  _et al._
(2017), applications in optical microcavities Chen  _et al._ (2017), and the
development of directional lasing technologies Peng  _et al._ (2016). EPs have
been successfully observed and studied in various experimental setups,
spanning fields such as optics Miri and Alù (2019); Parto  _et al._ (2020),
photonics Özdemir  _et al._ (2019), electric circuits Choi  _et al._ (2018);
Stehmann  _et al._ (2004), and acoustics Shi  _et al._ (2016); Zhu  _et al._
(2018).

Due to its diverse applications and unique learning capabilities, machine
learning (ML) is rapidly being adopted by researchers as a novel tool. In
particular, ML has the potential to uncover new physics without prior human
assistance. The growing trend of the application of ML techniques is not only
restricted to different areas of condensed matter physics Bedolla  _et al._
(2020), but are also being explored in other areas such as particle physics,
cosmology, and quantum computing Carleo  _et al._ (2019). In past years, ML
techniques are being applied with outstanding accuracy to study topological
phases Deng  _et al._ (2017); Zhuang  _et al._ (2022); Scheurer and Slager
(2020); Tibaldi  _et al._ (2023) and topological invariants Zhang  _et al._
(2018); Rodriguez-Nieva and Scheurer (2019), classification of topological
phases Long and Zhang (2023); Scheurer and Slager (2020), and phase
transitions Carrasquilla and Melko (2017) in the Hermitian realm. Very
recently, the first applications of ML techniques to non-Hermitian systems
have been undertaken. For instance, Narayan et al. Narayan and Narayan (2021),
Cheng et al. Cheng and Yu (2021) and Zhang et al. Zhang  _et al._ (2021) have
undertaken the study of non-Hermitian topological phases using convolutional
neural networks (NNs). Yu et al. have used an unsupervised method, namely
diffusion maps, to explore such phases Yu and Deng (2021). Non-Hermitian
topological phases in phononics have been studied using manifold clustering
Long  _et al._ (2020). Furthermore, Araki and co-authors have analysed NHSE in
an ML framework Araki  _et al._ (2021). Moreover, ML approaches have been
recently explored to investigate various non-Hermitian experimental platforms.
Examples include, physics-graph-informed ML to study second-order NHSE in
topoelectrical circuits Shang  _et al._ (2022), principal component analysis
and NNs to explore non-Hermitian photonics Ahmed  _et al._ (2023), and
diffusion maps to analyse non-Hermitian knotted phases in solid state spin
systems Yu  _et al._ (2022).

In this work, we propose and demonstrate the characterization of EPs using
NNs. We introduce a new feature, which we term _summed phase rigidity_ (SPR),
which allows an unambiguous characterization of higher-order EPs. As an
illustration of our approach, starting with simple models, we have categorised
the EPs in various systems with increasing complexity with the help of NNs. In
particular, we show how EPs can be distinguished in two- and four-site models
by means of NNs. Furthermore, our NN construction and SPR enables an accurate
prediction of the order of the EPs. Finally, using the celebrated Hatano-
Nelson model for a variable number of sites, we demonstrate an accurate
prediction of EPs and show how SPR allows a prediction of EPs of orders
completely unseen by the training data. Our approach is useful for the
characterization of EPs in an automated manner using ML techniques.

Two-site model– We first consider the simple two-site non-Hermitian model with
on-site gain and loss to illustrate our approach [see Fig. 1(a)]. The system
is described by the following Hamiltonian Zhang  _et al._ (2020)

$H_{2}=\begin{pmatrix}\omega_{0}+i\gamma&J\\\ J&\omega_{0}-i\gamma\\\
\end{pmatrix},$ (1)

where $\omega_{0}$ is the onsite potential, $J$ is the coupling between the
sites as shown in Fig. 1(a), and the non-Hermiticity is introduced by the gain
and loss term $\pm i\gamma$. The eigenvalues are given by
$\lambda_{\pm}=\omega_{0}\pm\sqrt{J^{2}-\gamma^{2}}$. For simplicity, we
choose $\omega_{0}=0$, and the EP occurs at $\gamma=\pm J$. Note that this
model can host EPs of order two only.

We briefly describe the concept of phase rigidity, based on which we will
classify the EPs. The phase rigidity, $r$, at any point in the parameter space
of a given Hamiltonian is defined as Müller and Rotter (2008); Eleuch and
Rotter (2016)

$r=\frac{\langle\Psi_{L}|\Psi_{R}\rangle}{\langle\Psi_{R}|\Psi_{R}\rangle},$
(2)

where $\Psi_{L}$ and $\Psi_{R}$ are left and right eigenstates of the non-
Hermitian Hamiltonian. Due to bi-orthogonality, $r$ takes a value of zero at
EPs and a value of unity far from the EP. Instead of $r$, which ranges from
zero to one, we propose to take the negative log of $r$, i.e., $-\ln{|r|}$.
This change of scale essentially enhances the separation between EPs and non-
EPs in the parameter space – this leads to a much-improved characterization of
EPs as we will show. So at an EP, this quantity takes a large positive value,
and far from an EP, it drops to zero.

Figure 1: Training of NN for the two-site model. (a) Illustration of the two-
site gain and loss non-Hermitian model. Here $J$ is the coupling between the
two sites and $\pm i\gamma$ denotes the gain and loss terms. (b) A NN with 2
hidden layers was constructed. The hidden layers consist of 16 and 4 neurons,
respectively. The total number of trainable parameters is 121. The loss curve
with epoch is plotted during training. The blue represents the training
dataset and the yellow for validation. It is clear from the loss curve that
our model is not over-fitted. (c) $-\ln(|r|)$ is plotted for the test dataset
in $J-\gamma$ plane on a color scale. A color bar values tending to unity
represent an EP and low values denote non-EP. (d) The corresponding predicted
value from the NN is shown. We see a good agreement between the actual and
predicted values in (c) and (d).

We choose 40,000 randomly generated points in the $\gamma-J$ plane for our
two-site model and compute $-ln|r|$. Then, in a similar manner, we produced
10,000 points such that $\gamma=J$ and combined them with the points from
above. Among this generated data, we kept 10$\%$ as test data, and the rest
are used as training data. We constructed a network consisting of two hidden
layers with 16 and 4 neurons per layer, respectively. The total number of
trainable parameters is 121. We used the rectified linear unit (ReLU) as the
activation function for the hidden layers. The loss (mean squared error loss)
curve for 200 epochs with a batch size of 32 is shown in Fig. 1(b). We note
that convergence is quite fast (within nearly 50 epochs) and the loss for the
test set does not fluctuate too much or deviate from the training set loss
curve, indicating the sign of a good fit.

After training is performed by optimizing various hyperparameters (see Table
LABEL:table:1), we predict the order of EPs for the test set. On the other
hand, for this simple illustrative case, we already know the true order of the
EPs. We compare the true and the predicted results for the same data points in
the $\gamma-J$ plane, as shown in Fig. 1(c) and Fig. 1(d). We observe that our
NN predictions are almost identical to the true values with an accuracy of
99$\%$, based on which we can easily classify whether a given point is an EP
or not. We also calculated the performance of the network by means of the
$R^{2}$ score. This is presented in Table LABEL:table:1 and is close to 0.968.

Table 1: Details of NN structures. The details of the constructed networks are presented for the different models. We note that for a fixed number (two) of layers to get a good $R^{2}$ score ($\approx 0.95$), one needs to increase the number of total trainable parameters by increasing the number of neurons per layer. This is because of the increasing complexity of models. Along with the $R^{2}$ score we have reported the mean absolute error (MAE) and the root mean square error (RMSE) for each model. Model |  No. of hidden layers(neurons) |  Total parameters |  $R^{2}$ score | MAE | RMSE
---|---|---|---|---|---
Two-site | 2 (16, 4) | 121 | 0.968 | 0.018 | 0.068
Four-site | 2 (32, 16) | 673 | 0.974 | 0.088 | 0.181
$N$ sites (Hatano-Nelson) | 2 (12, 4) | 105 | 0.998 | 0.057 | 0.087

Four-site model– Next, we consider a slightly more involved scenario in a
four-site model, which may host both second and fourth-order EPs. The
Hamiltonian reads Jaiswal  _et al._ (2021)

$H_{4}=\begin{pmatrix}i\delta&p&0&0\\\ p&i\gamma&q&0\\\ 0&q&-i\gamma&p\\\
0&0&p&-i\delta\end{pmatrix}.$ (3)

Here $p$ denotes the coupling between sites one and two, and between sites
three and four, $q$ is the coupling between sites two and three, $\pm i\delta$
and $\pm i\gamma$ are gain and loss terms, as shown schematically in Fig. 2
(a). The above Hamiltonian can host both second- and fourth-order EPs,
depending on values of $p$, $q$, $\delta$, and $\gamma$ in the parameter
space. For the rest of the discussion, we set $\gamma=1$. The EP-2 lies on a
surface in the parameter space satisfying the following condition

$p^{4}+\delta^{2}+2\delta p^{2}-\delta^{2}q^{2}=0.$ (4)

On the other hand, we obtain an EP-4 when both Equation 4 and the following
Equation 5 are satisfied simultaneously,

$1+\delta^{2}-2p^{2}-q^{2}=0.$ (5)

As compared to the previous case, i.e., a classification between an EP and a
non-EP, here we need to distinguish three types of points, EP-2, EP-4, and
non-EPs. To do this, we have designed a new feature, which we term the _summed
phase rigidity_ , SPR. This is defined as

$\text{SPR}=\sum_{k}-\ln{|r_{k}|}/\text{max}(-\ln{|r_{k}|)},$ (6)

where $k$ runs over all eigenstates of the given Hamiltonian. For an $N$-th
order EP, $\text{SPR}\approx N$ at the EP. The steps for constructing SPR are
illustrated in Table 2.

Table 2: Constructing SPR for two site model. To construct the SPR, we first calculate $-\ln|r|$ at the chosen points and then rescale by dividing them by the maximum value. After the rescaling, the sum of all rescaled $-\ln|r|$ gives the SPR. Note that SPR value is nearly the order of the EP at an EP and nearly zero far from the EPs. Point type | $J$ | $\gamma$ | $-\ln|r_{1}|$ | $-\ln|r_{2}|$ |  rescaled $-\ln|r_{1}|$ |  rescaled $-\ln|r_{2}|$ | SPR
---|---|---|---|---|---|---|---
Non-EP | 0.3 | 0.7 | 0.10 | 0.10 | 0.0036 | 0.0036 | 0.007
EP-2 | 0.5 | 0.5 | 27.34 | 27.34 | 0.99 | 0.99 | 1.98
EP-2 | 0.3 | 0.3 | 27.52 | 27.52 | 1 | 1 | 2
Non-EP | 0.5 | 0.8 | 0.25 | 0.25 | 0.009 | .009 | 0.018

We trained the network to classify the points in the parameter space according
to the SPR value. As such, we set the following cutoffs for the SPR – an SPR
value of nearly zero ($0<$ SPR $<1$) corresponds to the ordinary point (non-
EP), SPR between 1 and 2 reflects a second order EP (EP-2), and SPR above
three corresponds to a fourth order EP (EP-4). Note that in practical
computations, SPR takes a value lower than $N$ for an $N$-th order EP. This is
because our generated data points are chosen randomly and phase rigidity
depends on the detuning parameters, $\delta z$, which take the system away
from an EP. In general, $r\varpropto\delta z^{1/N}$. Nevertheless, we show
that SPR can classify the corresponding points with an outstanding accuracy,
and can thus be a very useful training feature.

Figure 2: Training of NN for the four-site model. (a) Schematic of the four-
site non-Hermitian model. Here $p$ and $q$ are the coupling constants between
the sites, and $\pm i\gamma$ and $\pm i\delta$ denote the gain and loss terms.
(b) An NN with 2 hidden layers was constructed. The hidden layers consist of
32 and 16 neurons, respectively. The total number of trainable parameters is
673. The loss curve with epoch during training is shown as the blue points for
the training data set and as the yellow points for validation set. It is clear
from the loss curve that our NN is not overfitted. (c) Histogram of test data
points from the parameter space ($p,q,\delta,\gamma$) according to the SPR
value. Points near SPR=0 are non-EP, i.e., ordinary points. SPR between one
and two represents the second-order EPs and SPR between three and four
represents the fourth-order EPs. (d) Same as (c) but now instead the predicted
value of SPR from the NN is plotted. We see good agreement between actual and
predicted values from (c) and (d).

We generated a data set with 50,000 points, such that it is a mixture of 5000
EP-2 and 5000 EP-4 points, with the rest being non-EPs. As before, we keep
aside 10% of data points as test data, and the rest is used as the training
data. The data points are generated by randomly picking a point from the
$(p,q,\delta)$ parameter space (with $\gamma=1$). The conditions for EPs are
established using Eq. 4 and Eq. 5. Next, we calculate SPR at these points as
described above. So finally the features for each data point become $p$, $q$,
$\delta$ and SPR. Our trained NN consists of two hidden layers with 32 and 16
neurons per layer respectively. The total number of trainable parameters is
673. In our training with batch size 32 and 200 epochs, the loss curve in Fig.
2 (b) shows that the NN is not overfitted and converges rapidly. After
training, we predict the SPR value for the test data set and plot it alongside
the actual value in Fig. 2(c) and (d). We note the excellent match between the
actual and predicted data. In particular, there is a clear distinction between
EPs of different orders. Overall, in estimating the actual order of EPs in the
test data set, we achieved an accuracy of 97.5%.

Hatano-Nelson model with variable sites– Next, we present the generalized
approach to characterize the EPs based on the concept of SPR. We consider the
paradigmatic Hatano-Nelson model Hatano and Nelson (1996), which has served as
the inception ground for many of the central ideas of non-Hermitian physics.
The Hamiltonian for $L$ sites under open boundary conditions is given by the
following matrix

$H_{L}=\begin{pmatrix}0&t-\gamma/2&0&...&0\\\ t+\gamma/2&0&t-\gamma/2&...&0\\\
0&t+\gamma/2&0&...&0\\\ :&:&:&\ddots&:\\\ .&.&.&&.\end{pmatrix}.$ (7)

Here $t\pm\gamma/2$ denote non-reciprocal hopping terms, i.e., the unequal
left and right hopping strengths. In this model, higher order EPs occur at
$t=\pm\gamma/2$, and the order depends on the number of sites, i.e., $L$.
Therefore, if a point is an EP in the $t-\gamma$ parameter space, then the SPR
value will be nearly $L$, for an $L$ site model. For the sake of simplicity,
we change the variables as $t_{1}=t-\gamma/2$ and $t_{2}=t+\gamma/2$. In this
$t_{1}-t_{2}$ parameter space, EPs occur when one of $t_{1}$ or $t_{2}$ is
nearly zero and other one is nearly one.

Figure 3: Training of NN for variable site Hatano-Nelson model. (a) Schematic
diagram of the Hatano-Nelson model with the left-right asymmetric hopping
($t\pm\gamma/2$). (b) A neural network with 2 hidden layers was constructed.
The hidden layers consist of 12 and 4 neurons, respectively. The total number
of trainable parameters is 105. The loss curve shown as a function of the
epoch indicates that NN is not over-fitted or under-fitted. (c) Distribution
of test data set with actual SPR values for $L=3$ and $L=7$ sites. (d)
Distribution of the same data sets as (c) but with predicted SPR values from
the network. We note the agreement between actual and predicted SPR values.
Figure 4: SPR prediction of higher-order EPs of untrained Hatano-Nelson
models. (a) Actual data distribution for $L=4$ site Hatano-Nelson model. Here,
$3<$SPR$<4$ represents fourth-order EPs and SPR below one denotes ordinary
points. (b) The corresponding predictions from the NN, which was trained for
$L=3$ and $L=7$ sites only. (c) Actual data set for a mixture of $L=4$ and
$L=6$ sites HN models, where $3<$ SPR $<4$ denotes an EP of order 4 and $5<$
SPR $<6$ represents an EP of order 6. (d) Corresponding predicted values from
the NN. We discover that the NN is able to detect the order of the EP even
when there is a mixture of data for EPs of different orders. Moreover, the NN
is capable of predicting the EPs and their orders for those cases which were
not included in either the training or the test data sets.

Here we trained the Hatano-Nelson model with different sites. We have trained
the network with $L=3$ and $L=7$. We generated the data by choosing random
points in the $t_{1}-t_{2}$ parameter space and calculating the SPR. For
$L=3$, we generated 30,000 points, which are a mixture of 5,000 points such
that $t_{1}\approx 1$ and $t_{2}\approx 0$, 5,000 points such that
$t_{2}\approx 1$ and $t_{1}\approx 0$ and the rest are such that $t_{1}\approx
t_{2}$. Similarly, for $L=7$, 30,000 data points were generated with a similar
distribution. Therefore, finally, we have 60,000 data points with different
SPR values, i.e., nearly 3, 7 or 0 based on the values of $t_{1}$, $t_{2}$ and
$L$.

Among the generated data, 10% was kept as test data and the rest is used to
train the NN. We construct an NN with two hidden layers with 12 and 4 neurons,
respectively. The loss curve in Fig. 3 (b) shows that NN is not over-fitted.
After training, we predict the distribution of the test data set with respect
to SPR value as shown in Fig. 3 (d), where $6<$ SPR $<7$ represents EPs of
order seven, $2<$ SPR $<3$ denotes EPs of order three, and SPR $<1$ are non-EP
points. By comparing to the actual SPR distribution of the test data set in
Fig. 3 (c), we find good agreement between actual and predicted SPR value
distribution with an accuracy close to 99.9%. We, therefore, conclude that EPs
of different orders can be successfully classified by using SPR.

Whether our NN has learned the generalised property of SPR is a crucial
question. In order to understand this, we ask the already trained NN to
predict the EPs and their orders in models that were not included during
training. Fig. 4 (a) shows the distribution of true SPR data from the
$(t_{1},t_{2},L=4)$ parameter space for $L=4$ site Hatano-Nelson model. Here,
$3<$ SPR $<4$ represents EPs of order 4, and $0<$ SPR $<1$ are non-EPs. Now we
feed test this data set of $L=4$ sites to the already trained NN and the
distribution of data with the predicted SPR is shown in Fig. 4 (b).
Remarkably, we find that the NN, which was trained for $L=3$ and $L=7$ sites
is able to identify the EPs as well as their order with an accuracy greater
than 99.9%.

To further examine the robustness of our NN, we make the scenario more
complicated by adding random points from $(t_{1},t_{2},L=6)$ parameter space
to the existing $L=4$ data set. The resulting distribution of the SPR values
is shown in Fig. 4 (c), where $0<$ SPR $<1$ are non-EPs, $3<$ SPR $<4$ are EPs
of order 4 and $5<$ SPR $<6$ represent the EPs of order 6. The corresponding
distribution for the predicted SPR value is shown in Fig. 4 d. We note that
our trained NN for $L=3$ and $L=7$ sites is not only capable of predicting the
EPs and their orders for an untrained model, but it also has the ability to do
so for a mixture of data from different non-Hermitian models with EPs of
varying orders with an accuracy close to 99.9%.

Training Details of NN models– Before summarizing, we provide an overview of
the technical aspects of our simulations, which were applied consistently
across all NN models. We have followed a top-down approach to train the
models. We started with complicated models, i.e., a large number of
hyperparameters (number of layers and number of neurons in each layer) and
systematically reduced the complexity until the performance declined
significantly. This process resulted in the NN models which offer an optimal
balance between computational cost and performance. We initiated the training
process by employing the default Glorot uniform weight initializer, which
randomly initializes weights from a normal distribution and sets all biases to
zero. We selected the Adam optimizer for adaptive learning rate adjustment,
minimizing the mean square error between actual and predicted SPR. The
rectified linear unit (ReLU) activation function was utilized. Each model
underwent training for 200 epochs using mini-batch training with a batch size
of 32. No regularization techniques were applied. To monitor potential
overfitting, separate test data were employed for all models. It is important
to highlight that our machine learning models fundamentally operate as
regression models, predicting SPR values. Subsequently, the specific SPR range
is used to classify different orders of EPs.

Comparison with random forest regression– To compare with more straightforward
approaches we have trained a random forest (RF) regression model on the four-
site model data set 111We used the sklearn RandomForestRegressor class with
the following hyperparameters: (i) Number of trees (n_estimators)= 300, (ii)
the minimum number of samples required to split an internal node
(min_samples_split = 20, (iii) random_state = 0 to control the randomness, and
(iv) The number of jobs to run in parallel (n_jobs) = 3, while keeping other
parameters as default.. The data was split into 90% for training and 10% for
testing. These hyperparameters yielded the best fit to our data, resulting in
an $R^{2}$ score of 0.975. The performance metrics ($R^{2}$ score and RMSE) of
the RF model is presented in Table LABEL:table:3. The small difference in
performance metrics between training and test data indicates no overfitting.
The RF model achieves an accuracy of 92.5% which is lower than the
corresponding NN model.

Table 3: Performance of the RF model. We trained a random forest model with
300 trees. The slight difference in performance scores between the test and
training data suggests that the model is not overfitting.
Data |  $R^{2}$ score | RMSE
---|---|---
Train data | 0.975 | 0.176
Test data | 0.949 | 0.247

Discussion and summary– In summary, we have conveyed a data-centric approach
to characterize the EPs. Notably, our approach goes beyond existing machine-
learning applications in non-Hermitian physics, particularly in predicting
non-Hermitian topological phases Narayan and Narayan (2021); Cheng and Yu
(2021); Zhang  _et al._ (2021); Ahmed  _et al._ (2023) and NHSE Shang  _et
al._ (2022); Araki  _et al._ (2021). We have successfully demonstrated how ML
techniques can be used to predict EPs and EP orders in various models with
outstanding accuracy. For the sake of transparency, we have chosen simple yet
non-trivial models with increasing complexity. For the two-site model, we
trained a NN to classify EPs and non-EPs. Next, we proposed a new feature –
termed (SPR) -– and used it to distinguish between EPs of different orders. We
then generalized this procedure to the celebrated Hatano-Nelson model with
variable sites. Remarkably, we found that our NN models were able to predict
the true order of EPs with an accuracy greater than 99% even in cases with EPs
of orders completely unseen by the training data.

Looking ahead, our work may open up interesting avenues for future
explorations. Our techniques can be useful for studying the parametric
dependence of EPs in higher dimensions, which can become quite intricate,
especially in cases such as anisotropic EPs Ding  _et al._ (2018). The
behavior of EPs can differ based on the symmetries present in the system
Mandal and Bergholtz (2021). We envisage that generalizations of our framework
may assist in characterizing nontrivial behavior in such scenarios.
Additionally, EPs have intriguing connections to topological phases of non-
Hermitian systems, making our method potentially useful for studying their
topological properties. We hope our work motivates these promising
developments.

Acknowledgments– We have used Python Van Rossum  _et al._ (1995) and
TensorFlow Abadi  _et al._ (2015) for our computations, and we are grateful to
the developers. M. A. R. would like to thank Sourav Mal for useful
discussions, and would also like to thank Smoky, his cat, for being a calming
presence when he was writing this paper. M. A. R. is supported by a graduate
fellowship of the Indian Institute of Science. A. N. acknowledges a start-up
grant from the Indian Institute of Science.

Data availability– The data and code that support the findings of this study
are available at https://github.com/rejaafsar/EP_ML

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