# Genuine Entanglement detection via Projection map in multipartite system

Bivas Mallick<EMAIL_ADDRESS>S. N. Bose National Centre For Basic
Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, India  Sumit Nandi
<EMAIL_ADDRESS>Purandarpur High School, Purandarpur, West Bengal
731129, India

###### Abstract

We present a formalism to detect genuine multipartite entanglement by
considering _projection_ map which is a positive but not completely positive
map. _Projection_ map has been motivated from “no-pancake theorem” which
repudiates the existence of a quantum operation that maps the Bloch sphere
onto a disk along its equator. The not-complete positivity feature of
_projection_ map is explored to investigate its credibility for certifying
biseparability in multipartite quantum systems. We have lifted _projection_
map to derive a separability criterion in order to ascertain biseparability in
tripartite scenario. We have shown that _projection_ map can detect both
inequivalent SLOCC classes of genuine entanglement in tripartite scenario
_i.e._ W state and GHZ state. Also, we have shown that _projection_ map is
robust against white noise. We also construct a general framework to certify
genuine multipartite entanglement for arbitrary N-qubit states by lifting
_projection_ map. The efficacy of our framework is further explored to detect
quadripartite GHZ state.

##  I Introduction

The enigmatic phenomenon entanglement, originated from the famous EPR paradox,
paves a wonderful way for performing various information processing protocols,
and thus, calls for a considerable attention for enriching the underlying
theoretical foundation. The most fundamental and challenging topic in this
direction is to compose a well defined framework for detection of entanglement
in multipartite systems _i.e._ , when the given system involves macroscopic
number of particles. Indeed, multipartite entanglement has been shown to be
very useful for information processing protocols Cleve  _et al._ (1999);
Raussendorf and Briegel (2001); DUN  _et al._ (2003); Durt  _et al._ (2004).

To determine whether a given state is entangled has remained a central topic
for research. There exists necessary separability criterion for certifying
bipartite entanglement. Positive but not completely positive map, such as
transposition map, is a convenient tool for detecting entanglement in such a
paradigmatic situation. The state which remains a valid physical state after
the application of partial transposition is known to be separable whereas the
one which fails to be a valid density matrix is entangled. Also any entangled
state yields negative eigenvalues after it undergoes partial transposition
map. In this way positive partial transposition (PPT) criteria is necessary
and sufficient for detection of bipartite entanglement in $2\otimes 2$ and
$2\otimes 3$ system Horodecki  _et al._ (1996). However, multipartite
entanglement is much different notionally as well as operationally from
bipartite entanglement Gühne and Tóth (2009). In the former case there are, in
principle, infinite number of ways to partition the constituents into non-
overlapping groups. Naturally, multipartite entanglement is less canonical
than its bipartite counterpart. The ubiquitous feature of multipartite
entanglement allows infinitely many different SLOCC classesVerstraete  _et
al._ (2002); de Vicente  _et al._ (2013) that elucidate the vast structures of
multipartite entangled state. Extensive research in this direction for
characterizing multipartite entanglement Horodecki  _et al._ (2009); Michael
Walter (2016) has been carried out. Naturally, multipartite entanglement
detection also received a lot of attention. To generalize positive map based
entanglement detection criterion, the authors Horodecki  _et al._ (2001)
constructed a particular N-qubit map which is positive on all product states
of $N-1$ particles. Along the similar line a significant generalisation of
partial transposition criterion was presented in Chen and Wu (2002) to probe
multipartite entanglement. Bloch representation of bipartite state has been
found to be an useful scheme to detect entanglement. This particular
representation has been explored to formulate multipartite entanglement
detection in Hassan and Joag (2007). Another approach to detect multipartite
entanglement is to construct witness operators Gühne and Hyllus (2003);
Bourennane  _et al._ (2004); Tóth and Gühne (2005) which yield positive
expectation value for all separable states, and its negative expectation value
indicates the presence of entanglement. All these techniques require
estimation of large number of parameters growing rapidly with the number of
qubits making multipartite entanglement detection difficult even for a meagre
system. Moreover, in contrast to bipartite case many conceptual hinges appear,
and the concept of genuine multipartite entanglement (GME) becomes more
counterintuitive. _For example_ , it has been shown in Bennett  _et al._
(1999) that a state is separable in all of its bipartite cut, yet, not fully
separable. In contrast, it is shown that a multipartite state is entangled
across all of its bipartitions, yet, the state is not genuinely multipartite
entangled Liang  _et al._ (2014); Horodecki  _et al._ (1995). Thus, it is a
challenging task to determine whether a given multipartite state is genuinely
entangled or not, in fact it is shown to be NP hard Gurvits (2003). As an
outgrowth of earlier works Clivaz  _et al._ (2017); Huber and Sengupta (2014),
here we propose a framework which provides a necessary criterion for
biseparability in multipartite scenario. The straightforward extension of the
application of not completely positive map is not feasible to detect genuine
multipartite entanglement (GME), since, it is only capable of detecting
entanglement across its different bipartitions. In Clivaz  _et al._ (2017);
Huber and Sengupta (2014), the authors circumvent this difficulty by
developing a tool considering convex combinations of not completely positive
maps for detection of GME. Such construction reveals positivity on all
biseparable states, but not positive on the set of genuine entangled states.
The authors presented multipartite not-positive maps obtained from
transposition map via a lifting procedure.

In this paper, we have considered a novel map called _projection_ map Sudha
_et al._ (2021) to detect genuine multipartite entanglement. This map stems
from the “no-pancake theorem” Blume-Kohout  _et al._ (2010) which states that
there is no qubit quantum channel that kills one component of polarization,
thereby, mapping entire Bloch sphere onto a disk touching the sphere. As
mentioned earlier, such not complete positive map is necessary in the context
of entanglement detection. We have shown _projection_ map plays phenomenal
role in detecting multipartite entanglement which has not been discussed
earlier. Firstly, we explore its potential usage for detecting genuine
entanglement in tripartite scenario. We have shown that _projection_ map can
detect both inequivalent SLOCC classes of tripartite genuine entanglement.
Also our proposed map is more robust than the previous studiesClivaz  _et al._
(2017); Chimalgi  _et al._ (2023) to detect both the noisy W and noisy GHZ
state in tripartite scenario. We have the lifted _projection_ map to construct
a general framework to detect genuine entanglement in multipartite scenario.
The efficacy of our framework is further investigated to detect quadripartite
GHZ state.

The plan of the paper is organised as follows. In (Sec.II), we have shown that
_projection_ map can be used to detect bipartite entanglement. We set up our
formalism for detecting multipartite entangled states via _projection_ map in
(Sec.III). Then we conclude in (Sec.IV) along with some interesting future
perspectives.

##  II Detection of Bipartite Entanglement via _projection_ map

Let us consider a bipartite quantum system in the finite dimensional Hilbert
space $\mathcal{H}$ of dimension $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. We
denote the set of $d\times d$ complex matrices by $M_{d}$ and let $M_{d}^{+}$
be a convex set of positive semi-definite elements in $M_{d}$. Thus
$M_{d}^{+}$ defines a space of states of $d$-level quantum system. We also
denote a vector space of finite dimensional bounded operators acting on
${\mathcal{H}}_{n}$ by $B({\mathcal{H}}_{n})$. A linear map
$\Lambda:B({\mathcal{H}}_{n})\rightarrow B({\mathcal{H}}_{n})$ is positive if
$\Lambda[X]\geq 0$ for all $X\geq 0$ Størmer (1963). Additionally, if
$\Lambda$ is trace-preserving, then it maps quantum states represented by
density operators in $M_{d}^{+}$ into quantum states. Denoted
$I_{n}:B({\mathcal{H}}_{n})\rightarrow B({\mathcal{H}}_{n})$ as an identity
map acting on $n$ dimensional Hilber space, we say that $\Lambda$ is
completely positive if for all $n$, the induced map $I_{n}\otimes\Lambda$ is
positive. The positive maps are regarded as an important ingredient for
envisaging separability problem. We can intuitively understand that the
product states are always mapped into positive operators by positive maps
Horodecki  _et al._ (1996). It also follows for the convex combination of
product states. In what follows, any density operator
$\sigma\in\mathcal{H}_{d_{1}}\otimes\mathcal{H}_{d_{2}}$ is separable if and
only if $(I_{d_{1}}\otimes\Lambda)\sigma$ is positive for any positive map
$\Lambda$. On the other hand, if a given state $\rho$ is inseparable, then
there exists a positive map such that ($I_{d_{1}}\otimes\Lambda)\rho$ is not
positive. However, not all positive maps possess this particular property.
Transposition maps are not completely positive map, and often used for the
detection of inseparable states in $2\otimes 2$ and $2\otimes 3$ system. It is
to be explored further the structure of such positive maps which might enable
detection of entanglement in arbitrary dimension Woronowicz (1976). In the
present paper, we shall explore the novelty of _projection_ map denoted by
$\varmathbb{P}$ towards detection of multipartite entanglement.

We start by showing that $\varmathbb{P}$ is a positive map on $M_{2}$. The
state space $\Omega_{2}$ of a qubit is often visualised by the Bloch sphere
representation in $\mathbb{C}^{2}$ as follow

$\rho=\frac{1}{2}\Big{(}I_{2}+\sum_{i=1}^{3}p_{i}\sigma_{i}\Big{)},\hskip
18.06749pt\rho\in{\Omega_{2}}\\\ $ (1)

where $I_{2}$ is the $2\times 2$ identity matrix and $\sigma_{1}$,
$\sigma_{2}$, $\sigma_{3}$ are pauli matrices written explicitly as

$\sigma_{1}=\begin{bmatrix}0&1\\\ 1&0\\\
\end{bmatrix},\sigma_{2}=\begin{bmatrix}0&-i\\\ i&0\\\
\end{bmatrix},\sigma_{3}=\begin{bmatrix}1&0\\\ 0&-1\\\ \end{bmatrix}$

The component of Bloch vectors $p_{i}$ have the following properties:
$p_{i}\in\mathbb{R}^{3}$, $p_{i}=Tr(\rho\sigma_{i})$ and
$\sum_{i=1}^{3}|p_{i}|^{2}\leq 1$. As mentioned earlier, the _projection_ map
arises from generalization of “no-pancake theorem” which states that the image
of the Bloch sphere cannot be a disk that touches the sphere Blume-Kohout  _et
al._ (2010). _projection_ map annihilates $\sigma_{3}$ component while
preserving the other Bloch components. Let us take the action of map
$\varmathbb{P}$ as Sudha  _et al._ (2021):

$(p_{1},p_{2},p_{3})\rightarrow(p_{1},p_{2},0)$

We write the qubit density matrix $\rho$ explicitly in the standard basis
$\ket{0}=(1,0)^{T}$ and $\ket{1}=(0,1)^{T}$ as follow

$\rho=\begin{bmatrix}\frac{1+p_{3}}{2}&\frac{p_{1}-ip_{2}}{2}\\\\[5.69046pt]
\frac{p_{1}+ip_{2}}{2}&\frac{1-p_{3}}{2}\\\ \end{bmatrix}$ (2)

Now, the _projection_ map is defined as $\varmathbb{P}:M_{2}\rightarrow
M_{2}$,

$\varmathbb{P}(\rho)=\begin{bmatrix}\frac{1}{2}&\frac{p_{1}-ip_{2}}{2}\\\\[5.69046pt]
\frac{p_{1}+ip_{2}}{2}&\frac{1}{2}\\\ \end{bmatrix}$ (3)

To verify positivity of $\varmathbb{P}$, it suffices to show that all the
principal minors of $\varmathbb{P}(\rho)$ are positive. The diagonal elements
of $\varmathbb{P}(\rho)$ are the first order principal minor which are clearly
positive numbers, and the value of second order principal minor is
$\frac{1-(|p_{1}|^{2}+|p_{2}|^{2})}{4}$. Using the fact
$\sum_{i=1}^{3}|p_{i}|^{2}\leq 1$, it can be checked that
$\frac{1-(|p_{1}|^{2}+|p_{2}|^{2})}{4}\geq 0$, which proves the positivity of
our proposed map $\varmathbb{P}$.
∎

We proceed to show that $\varmathbb{P}$ is not a completely positive map. To
achieve this goal, we will make use of Choi Jamiolkowski isomorphism developed
in Jamiołkowski (1974); Choi (1975). In the computational basis $\ket{i}$, the
maximally entangled state is
$\ket{\phi}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$. By Choi- Jamiolkowski
isomorphism, it is sufficient to show that, the Choi matrix
$I\otimes\varmathbb{P}(\ket{\phi}\bra{\phi})$ is not positive semidefinite. By
computing the corresponding Choi matrix, we get

$C^{\varmathbb{P}}=\begin{bmatrix}\frac{1}{4}&0&\frac{1}{4}&\frac{1}{2}\\\\[2.84544pt]
0&\frac{1}{4}&0&\frac{1}{4}\\\\[2.84544pt]
\frac{1}{4}&0&\frac{1}{4}&0\\\\[2.84544pt]
\frac{1}{2}&\frac{1}{4}&0&\frac{1}{4}\\\\[2.84544pt] \end{bmatrix}\\\ $ (4)

We obtain a negative eigenvalue $-\frac{1}{2\sqrt{2}}$ of the above Choi
matrix which implies that $\varmathbb{P}$ is not completely positive map.
Therefore, $\varmathbb{P}$ can be used to detect entangled states.
∎

We shall now probe its efficacy for detecting entanglement in the presence of
noise. Consider, the Werner state given by:

$\rho_{w}=p|\phi\rangle\langle\phi|+\frac{1-p}{4}I_{2\otimes 2},$ (5)

where the parameter $p$ indicates the presence of white noise. $\rho_{w}$
reduces to maximally mixed separable state for $p=0$ whereas for $p=1$, it
denotes maximally entangled state. It is known that $\rho_{w}$ features
entanglement if $p>\frac{1}{3}$ and it violates Bell-CHSH inequality for
$p>\frac{1}{\sqrt{2}}$ Horodecki  _et al._ (1995). Now, one side action of
_projection_ map on $\rho_{w}$ produces the following Choi matrix:

$\big{(}I\otimes\varmathbb{P}\big{)}\rho_{w}=\begin{bmatrix}\frac{1}{4}&0&\frac{p}{4}&\frac{p}{2}\\\\[2.84544pt]
0&\frac{1}{4}&0&\frac{p}{4}\\\\[2.84544pt]
\frac{p}{4}&0&\frac{1}{4}&0\\\\[2.84544pt]
\frac{p}{2}&\frac{p}{4}&0&\frac{1}{4}\\\\[2.84544pt] \end{bmatrix}\\\ $ (6)

The four eigenvalues of $\big{(}I\otimes\varmathbb{P}\big{)}\rho_{w}$ are
$\frac{1}{4}(1-p-\sqrt{2}p)$, $\frac{1}{4}(1+p-\sqrt{2}p)$,
$\frac{1}{4}(1-p+\sqrt{2}p)$ and $\frac{1}{4}(1+p+\sqrt{2}p)$, respectively.
It can be checked easily for $p>0.415$, the eigenvalue
$\frac{1}{4}(1-p-\sqrt{2})$ is negative. Thus, the action of _projection_ map
yields a bound on the visibility parameter $p$ beyond which the Werner state
is entangled. In the context of thermodynamic work extraction protocol, the
authors Maruyama  _et al._ (2005) had presented a novel separability
criterion, and shown that $\rho_{w}$ is entangled if $p>0.60$. Remarkably, our
presented formalism is capable of detecting entanglement in Werner state for a
wider range of the visibility parameter.

##  III Genuine multipartite entanglement detection via _projection_ map

The notion of separability is less unambiguous in multipartite scenario
compared to its bipartite counterpart. For instance, separability may exist in
one particular bipartition amongst its numerous inequivalent partitions. Here
we consider two operationally distinct framework of separability. An N-partite
state is called fully separable iff it can be decomposed into its constituents

$\rho_{sep}=\sum_{i=1}^{N}p_{i}{\rho_{1}}^{i}\otimes....\otimes{\rho_{N}}^{i}$
(7)

where $\\{p_{i}\\}$ is a probability distribution satisfying
$\sum_{i=1}^{N}p_{i}=1$. It is quite analogous to what separability means in
bipartite scenario. If a state cannot be written in the above way, then we can
say that the system must contain some entanglement, since, some of its
constituents may still be separable. However, here we consider a particular
case, as follow: a state $\rho_{2-sep}$ is biseparable iff it can be
decomposed as

$\rho_{2-sep}=\sum_{A}\sum_{i}{p_{A}}^{i}{\rho_{A}}^{i}\otimes{\rho_{\bar{A}}}^{i}$
(8)

where $\rho_{A}$ denotes a quantum state for the subsystem defined by the
subset A and $\sum_{A}$ stands for the sum over all bipartitions $A|\bar{A}$.
An N-partite state which can not be decomposed as Eq.(8) is called genuine
multipartite entangled (GME) state Clivaz  _et al._ (2017). It suggests that
entanglement persists across its all possible $2^{N-1}-1$ inequivalent
bipartitions. We intend to detect these particular states by lifting the
positive map $\varmathbb{P}$. In this section, we shall introduce a
methodology to detect genuine multipartite entanglement via _projection_ map.

###  III.1 Genuine tripartite entanglement detection

In order to achieve this goal, we start with the simplest tripartite scenario
by considering the map

$\Phi_{3}(\rho)=[\varmathbb{P}_{1}\otimes I_{2}\otimes
I_{3}+I_{1}\otimes\varmathbb{P}_{2}\otimes I_{3}+I_{1}\otimes
I_{2}\otimes\varmathbb{P}_{3}+\kappa_{3}\mathbb{I}.Tr](\rho)$ (9)

where $\rho$ is an arbitrary state in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ and $\kappa_{3}$ is
a constant. It is to be noted that $\kappa_{3}$ plays a crucial role in
determining the state $\rho$. For this reason, $\kappa_{3}$ has to be chosen
such that

$\Phi_{3}(\rho)\geq 0$ (10)

_iff_ $\rho$ is biseparable and can be expressed as Eq.(8). In the light of
the above discussion, we present the following theorem

######  Theorem 1.

For $\kappa_{3}=\frac{1}{\sqrt{2}}$, $\Phi_{3}$ is positive on all biseparable
state

$\Phi_{3}(\rho_{2-sep})\geq 0$ (11)

###### Proof.

Following Eq.(8), we write a biseparable state in the most general form as

$\rho_{2-sep}=p_{1}\rho_{1}\otimes\rho_{23}+p_{2}\rho_{2}\otimes\rho_{13}+p_{3}\rho_{3}\otimes\rho_{12}$
(12)

Now we apply the map $\Phi_{3}$ on $\rho_{2-sep}$ to obtain the corresponding
Choi matrix

$\displaystyle\Phi_{3}(\rho_{2-sep})=(\varmathbb{P}_{1}\otimes
I_{23})(p_{1}\rho_{1}\otimes\rho_{23}+p_{2}\rho_{2}\otimes\rho_{13}+p_{3}\rho_{3}\otimes\rho_{12})+$
$\displaystyle(\varmathbb{P}_{2}\otimes
I_{13})(p_{1}\rho_{1}\otimes\rho_{23}+p_{2}\rho_{2}\otimes\rho_{13}+p_{3}\rho_{3}\otimes\rho_{12})+$
$\displaystyle(\varmathbb{P}_{3}\otimes
I_{12})(p_{1}\rho_{1}\otimes\rho_{23}+p_{2}\rho_{2}\otimes\rho_{13}+p_{3}\rho_{3}\otimes\rho_{12})+\frac{1}{\sqrt{2}}I$

In order to prove our theorem, we have to show that $\Phi_{3}(\rho_{2-sep})$
is always positive semidefinite. For this purpose, we evaluate the minimum
eigenvalue of a given two qubit density matrix $\rho$ subjected to partial
_projection_ map denoted as $\lambda_{min}(\rho^{\mathbb{P}})$ Rana (2013).
Let, the spectral decomposition of $\rho$ is: $\rho=\sum_{k}\gamma_{k}P_{k}$,
where $P_{k}=|\psi_{k}\rangle\langle\psi_{k}|$ and
$\ket{\psi_{k}}=\sum_{i=0,1}\sqrt{\lambda_{i}}\ket{ii}$. The eigenvalues of
$(I\otimes\mathbb{P})|\psi_{k}\rangle\langle\psi_{k}|$ are given by
$\frac{1}{4}\big{(}1\pm\sqrt{(1-\lambda_{2})^{2}+10(1-\lambda_{2})\lambda_{2}-8\sqrt{2}(1-\lambda_{2})\lambda_{2}+\lambda_{2}^{2}}\big{)}$
and
$\frac{1}{4}\big{(}1\pm\sqrt{(1-\lambda_{2})^{2}+10(1-\lambda_{2})\lambda_{2}+8\sqrt{2}(1-\lambda_{2})\lambda_{2}+\lambda_{2}^{2}}\big{)}$
respectively. It can be checked that the minimum eigenvalue turns out to be
$-\frac{1}{2\sqrt{2}}\approx-0.35$. Now,

$\displaystyle\lambda_{min}(\rho^{\mathbb{P}})$ $\displaystyle\geq$
$\displaystyle\sum_{k}\gamma_{k}\lambda_{min}(P_{k}^{\mathbb{P}})$ (14)
$\displaystyle=$
$\displaystyle\sum_{k}\gamma_{k}\lambda_{min}(I\otimes\mathbb{P}|\psi_{k}\rangle\langle\psi_{k}|)$
(15) $\displaystyle=$ $\displaystyle\sum_{k}\gamma_{k}(-\frac{1}{2\sqrt{2}})$
(16) $\displaystyle=$ $\displaystyle-\frac{1}{2\sqrt{2}}$ (17)

Next we evaluate minimum eigenvalue of $\Phi_{3}(\rho_{2-sep})$. Denoted again
$\lambda_{min}$ the minimum eigenvalue of $\Phi_{3}$, we write

$\displaystyle\lambda_{min}(\Phi_{3})=\lambda_{min}\Big{(}(\varmathbb{P}_{1}\otimes
I_{23})(p_{1}\rho_{1}\otimes\rho_{23}+p_{2}\rho_{2}\otimes\rho_{13}+p_{3}\rho_{3}\otimes\rho_{12})+$
$\displaystyle(\varmathbb{P}_{2}\otimes
I_{13})(p_{1}\rho_{1}\otimes\rho_{23}+p_{2}\rho_{2}\otimes\rho_{13}+p_{3}\rho_{3}\otimes\rho_{12})+$
$\displaystyle(\varmathbb{P}_{3}\otimes
I_{12})(p_{1}\rho_{1}\otimes\rho_{23}+p_{2}\rho_{2}\otimes\rho_{13}+p_{3}\rho_{3}\otimes\rho_{12})+\frac{1}{\sqrt{2}}I\Big{)}$
$\displaystyle\geq\Big{(}(\varmathbb{P}_{1}\otimes
I_{23})(\underbrace{p_{1}\rho_{1}\otimes\rho_{23}}_{\geq
0}+\underbrace{p_{2}\rho_{2}\otimes\rho_{13}}_{\geq-\frac{1}{2\sqrt{2}}}+\underbrace{p_{3}\rho_{3}\otimes\rho_{12}}_{\geq-\frac{1}{2\sqrt{2}}})+$
$\displaystyle(\varmathbb{P}_{2}\otimes
I_{13})(\underbrace{p_{1}\rho_{1}\otimes\rho_{23}}_{\geq-\frac{1}{2\sqrt{2}}}+\underbrace{p_{2}\rho_{2}\otimes\rho_{13}}_{\geq
0}+\underbrace{p_{3}\rho_{3}\otimes\rho_{12}}_{\geq-\frac{1}{2\sqrt{2}}})+$
$\displaystyle(\varmathbb{P}_{3}\otimes
I_{12})(\underbrace{p_{1}\rho_{1}\otimes\rho_{23}}_{\geq-\frac{1}{2\sqrt{2}}}+\underbrace{p_{2}\rho_{2}\otimes\rho_{13}}_{\geq-\frac{1}{2\sqrt{2}}}+\underbrace{p_{3}\rho_{3}\otimes\rho_{12}}_{\geq
0})+\underbrace{\frac{1}{\sqrt{2}}I}_{\geq\frac{1}{\sqrt{2}}}\Big{)}$
$\displaystyle\geq
2(\sum_{i=1}^{3}p_{i})(-\frac{1}{2\sqrt{2}})+\frac{1}{\sqrt{2}}$
$\displaystyle\geq 0$

In the last line, we have used the fact $\sum_{i=1}^{3}p_{i}=1$. It concludes
the proof of our theorem. ∎

It is to be noted that, this value of $\kappa_{3}$ is not optimal. Optimality
can be achieved by considering a biseparable state, for example
$\ket{{\phi}^{+}}\otimes\ket{0}$ where, $\ket{{\phi}^{+}}$ is maximally
entangled state in ${\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2}$, and thereby
applying the given map $\Phi_{3}$ on it. The corresponding value of
$\kappa_{3}$ turns out to be $0.5$.

Now, it suffices to show that lifting of _projection_ map acts as a non-
trivial map to detect genuine entangled states in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Therefore, we must
find _at least_ one entangled state $(\rho)$ for which $\Phi_{3}(\rho)$ is not
positive semidefinite. As we know, there are two SLOCC inequivalent classes in
tripartite scenario namely the W and GHZ classes. First we consider GHZ state
which is a genuine entangled state

$\ket{GHZ}=\frac{1}{\sqrt{2}}(\ket{000}+\ket{111}).$ (19)

We evaluate corresponding $\Phi_{3}(\rho)$ for GHZ state using the map defined
in Eq.(9)

$\Phi_{3}(\rho)=\begin{bmatrix}\frac{5}{4}&0&0&\frac{1}{4}&0&\frac{1}{4}&\frac{1}{4}&\frac{3}{2}\\\\[1.13791pt]
0&\frac{3}{4}&0&0&0&0&0&\frac{1}{4}\\\\[1.13791pt]
0&0&\frac{3}{4}&0&0&0&0&\frac{1}{4}\\\\[1.13791pt]
\frac{1}{4}&0&0&\frac{3}{4}&0&0&0&0\\\ 0&0&0&0&\frac{3}{4}&0&0&\frac{1}{4}\\\
\frac{1}{4}&0&0&0&0&\frac{3}{4}&0&0\\\\[0.85355pt]
\frac{1}{4}&0&0&0&0&0&\frac{3}{4}&0\\\\[0.85355pt]
\frac{3}{2}&\frac{1}{4}&\frac{1}{4}&0&\frac{1}{4}&0&0&\frac{5}{4}\\\
\end{bmatrix}$ (20)

where $\rho=|GHZ\rangle\langle GHZ|$. Interestingly, the above matrix produces
a negative eigenvalue given by $-0.41$ by taking the optimal value of $k_{3}$.
It is to be mentioned that, previous studies in this direction by taking
transposition map into consideration Clivaz  _et al._ (2017); Chimalgi  _et
al._ (2023), was only able to detect GHZ state after performing $\sigma_{x}$
rotation onto the map. Therefore, our proposed map rids us of applying an
additional unitary operation to detect GHZ state. ∎

Further to investigate the efficacy of the map against noise, we consider
noisy GHZ state as follow

$\rho_{noisy}=x|GHZ\rangle\langle GHZ|+\frac{1-x}{8}{I}_{8}.$ (21)

The map $\Phi_{3}$ can detect $\rho_{noisy}$. It can be checked that the
threshold value of $x$ for certifying GME of the given state turns out to be
$x>0.69$ by taking optimal $k_{3}$. Therefore, optimal $\kappa_{3}$ provides a
more robust detection criterion than transposition map based GME criterion
found in Clivaz  _et al._ (2017).

Now we consider tripartite W state which is a genuine entangled state,

$\ket{W}=\frac{1}{\sqrt{3}}(\ket{001}+\ket{010}+\ket{100}).$ (22)

To detect W state, we introduce the generalised map $\tilde{\Phi}_{3}$, where

$\tilde{\Phi}_{3}()=[\tilde{\sigma_{z}}\circ\varmathbb{P}_{1}\otimes
I_{2}\otimes I_{3}+I_{1}\otimes\tilde{\sigma_{z}}\circ\varmathbb{P}_{2}\otimes
I_{3}+I_{1}\otimes
I_{2}\otimes\tilde{\sigma_{z}}\circ\varmathbb{P}_{3}+\kappa_{3}\mathbb{I}.Tr]()$
(23)

The map $\tilde{\sigma}_{z}\circ\varmathbb{P}_{i}$ denotes projection followed
by a unitary operation $\sigma_{z}$. As local quantum operation such as
unitary application cannot increase entanglement, therefore $\tilde{\Phi}_{3}$
gives positive output on all biseparable states. However, we will immediately
discover that the modified version of projection map given by Eq.(23) would
also be a GME map as $\Phi_{3}(\ket{W}\bra{W})$ gives us a negative eigen
value -0.325 by taking optimal $k_{3}$. Therefore, our proposed _projection_
map can detect both SLOCC classes of genuine tripartite entangled states.

Furthermore, we consider noisy W state as follow

$\rho_{noisy}=x|W\rangle\langle W|+\frac{1-x}{8}{I}_{8}.$ (24)

To detect noisy W state with _projection_ map we apply the map as given in
Eq.(23). Using optimal $\kappa_{3}$, the threshold value of $x$ for certifying
GME of the given state turns out to be $x>0.73$. Therefore, our presented
criterion for detection of noisy W state is more robust than the previously
known $x>0.90$ Clivaz  _et al._ (2017).

###  III.2 N-partite generalization of _Projection_ map

We will now envisage the inseparability of an arbitrary N-partite state by
directly lifting $\varmathbb{P}$. We denote $A\subset\\{1,2,\dots,n\\}$ be the
proper subset of the parties. For a given N, there are $2^{N-1}-1$ number of
inequivalent ways to arrange all bipartitions. Hence lifting of
$\varmathbb{P}$ involves convex combination of all such terms. Let us define
the following map:

${\Phi_{N}}()=\left[\sum_{A}\varmathbb{P}_{A}\otimes{I}_{\bar{A}}+\kappa_{N}{I}.Tr\right]()$
(25)

Now we present the following theorem:

######  Theorem 2.

For $\kappa_{N}=\frac{1}{2\sqrt{2}}\times(2^{N-1}-2)$, $\Phi_{N}$ is positive
on all N-partite biseparable states $\rho_{2-sep}$ given by Eq.(8),

$\Phi_{N}(\rho_{2-sep})\geq 0$ (26)

proof: To prove the above theorem, we apply Eq.(25) on the biseparable state
given by Eq.(8) and evaluate minimum eigenvalue $(\Lambda_{min})$ of
$\Phi_{N}(\rho_{2-sep})$ .

$\displaystyle\Lambda_{min}(\Phi_{N}[\rho_{2-sep}])$ $\displaystyle=$
$\displaystyle\Lambda_{min}\Big{(}\sum_{A}\varmathbb{P}_{A}\otimes
I_{\bar{A}}+\tilde{\kappa}I.Tr\Big{)}[\rho_{2-sep}]$ (27) $\displaystyle=$
$\displaystyle\Lambda_{min}\Big{(}\sum_{A}\varmathbb{P}_{A}\otimes
I_{\bar{A}}+\tilde{\kappa}I.Tr\Big{)}\sum_{A^{\prime}}\sum_{i}{p_{A^{\prime}}}^{i}{\rho_{A^{\prime}}}^{i}\otimes{\rho_{\bar{A^{\prime}}}}^{i}$
$\displaystyle\geq$
$\displaystyle\sum_{A^{\prime}}\sum_{i}\Big{(}\sum_{A=A^{\prime}}p_{i}^{A^{\prime}}\underbrace{\Lambda_{min}(\varmathbb{P}_{A^{\prime}}\otimes
I_{\bar{A}^{\prime}}[\rho_{i}^{A^{\prime}}\otimes\rho_{i}^{\bar{A}^{\prime}}])}_{\geq
0}+\sum_{A\neq
A^{\prime}}p_{i}^{A^{\prime}}\underbrace{\Lambda_{min}(\varmathbb{P}_{A^{\prime}}\otimes
I_{\bar{A}^{\prime}}[\rho_{i}^{A^{\prime}}\otimes\rho_{i}^{\bar{A}^{\prime}}]}_{\geq-\frac{1}{2\sqrt{2}}})\Big{)}+(2^{N-1}-2)\times\frac{1}{2\sqrt{2}}\underbrace{\Lambda_{min}(I}_{=1})$
$\displaystyle\geq$ $\displaystyle 0$ (29)

Since, the sum $\sum_{A\neq A^{\prime}}$ involves $(2^{N-1}-2)$ terms, we
obtain the inequality in the last line. It is to be noted that we obtain the
corresponding tripartite map as given in Eq.(9) by substituting $N=3$ in our
generalised map for N-partite state given by Eq.(25).
∎

To illustrate the efficacy of our map $\Phi_{N}$ we now consider a particular
case for $N=4$. A quadripartite system involving four subsystems gives rise to
_seven_ inequivalent bipartitions occurring across $1|234$, $2|134$, $3|124$,
$4|123$, $12|34$, $13|24$, and $14|23$. Considering all these different
bipartitions of a quadripartite state, we define a map $\Phi_{4}(\rho)$ as
follow

$\begin{split}&\Phi_{4}(\rho)=[\varmathbb{P}_{1}\otimes I_{2}\otimes
I_{3}\otimes I_{4}+I_{1}\otimes\varmathbb{P}_{2}\otimes I_{3}\otimes
I_{4}+I_{1}\otimes I_{2}\otimes\varmathbb{P}_{3}\otimes I_{4}\\\
&~{}~{}~{}~{}~{}~{}+I_{1}\otimes I_{2}\otimes
I_{3}\otimes\varmathbb{P}_{4}+\varmathbb{P}_{1}\otimes\varmathbb{P}_{2}\otimes
I_{3}\otimes I_{4}+\varmathbb{P}_{1}\otimes
I_{2}\otimes\varmathbb{P}_{3}\otimes I_{4}\\\
&~{}~{}~{}~{}~{}~{}+\varmathbb{P}_{1}\otimes I_{2}\otimes
I_{3}\otimes\varmathbb{P}_{4}+\kappa_{4}\mathbb{I}.Tr](\rho).\end{split}$

where $\rho$ is an arbitrary state in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$
and $\kappa_{4}$ is a constant. We choose $\kappa_{4}$ in such that

$\Phi_{4}(\rho)\geq 0$ (30)

iff $\rho$ is biseparable and can be expressed in the form as given in Eq.(8).

Lemma1: For $\kappa_{4}=\frac{3}{\sqrt{2}}$, $\Phi_{4}$ is positive for all
4-partite biseparable states which can be expressed in form (8) i.e.

$\Phi_{4}(\rho_{2-sep})\geq 0$ (31)

Proof of the above Lemma follows from the proof of Theorem 2.

Now, to verify that $\Phi_{4}$ is a valid GME, we consider quadripartite GHZ
state as:

$\ket{GHZ}=\frac{1}{\sqrt{2}}(\ket{0000}+\ket{1111}).$ (32)

By computing the corresponding $\Phi_{4}(\rho)$, where
$\rho=|GHZ\rangle\langle GHZ|$ we obtain negative eigenvalue $-0.60$. Thus,
lifting of projection map is capable of detecting genuine quadripartite
entanglement. ∎

##  IV Conclusion

In quantum information theory, multipartite entanglement has been shown to be
a resource for various information processing protocols Cleve  _et al._
(1999); Raussendorf and Briegel (2001); DUN  _et al._ (2003); Durt  _et al._
(2004). In this work, we have showcased detection of multipartite entanglement
via _projection_ map which is a positive but not completely positive map. Our
approach towards detection of entanglement has many implications. Firstly, we
have constructed a suitable map by considering a convex combination of
_projection_ map. Our tripartite extension of _projection_ map detects both
the SLOCC inequivalent classes i.e. GHZ and W state. The map is also shown to
be more robust than previous studies Clivaz  _et al._ (2017); Chimalgi  _et
al._ (2023) for detecting noisy GHZ state and noisy W state. Further, we
construct a general framework to validate the notion of genuine multipartite
entanglement for arbitrary N-qubit states. We have demonstrated the potential
of our proposed map by considering a particular case of quadripartite state.
It turns out that our proposed map is a valid GME detection map which can
detect quadripartite GHZ state. As an interesting future direction, it would
be worthwhile to find whether multipartite bound entangled states Dür (2001);
Smolin (2001); Tóth and Gühne (2009) can be detected with _projection_ map.
Also, it would be worthwhile to explore further usages of _projection_ map in
higher dimensional quantum systems, specifically in the characterisation of
qudit entanglement detection. We need _three_ Bloch components to specify a
qubit state, whereas, the numbers rise to _eight_ for $d=3$. But, _projection_
map has an advantage, since it reduces corresponding Bloch components thereby
minimizing the number of parameters in the state space. This suggests the
feasibility of _projection_ map for further exploration towards manipulating
and characterising higher dimensional entanglement.

##  V Acknowledgement

The authors acknowledge Prof. A. S. Majumdar for his valuable opinions which
helped to develop this work. BM acknowledges Bihalan Bhattacharya, Ram krishna
Patra, Saheli Mukherjee and Ananda G. Maity for fruitful discussions. BM
acknowledges DST INSPIRE fellowship program for financial support.

## References

  * Cleve  _et al._ (1999) Richard Cleve, Daniel Gottesman, and Hoi-Kwong Lo, “How to share a quantum secret,” Physical review letters 83, 648 (1999).
  * Raussendorf and Briegel (2001) Robert Raussendorf and Hans J Briegel, “A one-way quantum computer,” Physical review letters 86, 5188 (2001).
  * DUN  _et al._ (2003) Thomas DUN, Nicolas J CERF, Nicolas GISIN, and Marek ZUKOWSKI, “Security of quantum key distribution with entangled qutrits,” Physical review. A 67, 012311–1 (2003).
  * Durt  _et al._ (2004) Thomas Durt, Dagomir Kaszlikowski, Jing-Ling Chen, and Leong Chuan Kwek, “Security of quantum key distributions with entangled qudits,” Physical Review A 69, 032313 (2004).
  * Horodecki  _et al._ (1996) M Horodecki, P Horodecki, and R Horodecki, “Separability of mixed quantum states: necessary and sufficient conditions phys,”  (1996).
  * Gühne and Tóth (2009) Otfried Gühne and Géza Tóth, “Entanglement detection,” Physics Reports 474, 1–75 (2009).
  * Verstraete  _et al._ (2002) Frank Verstraete, Jeroen Dehaene, Bart De Moor, and Henri Verschelde, “Four qubits can be entangled in nine different ways,” Physical Review A 65, 052112 (2002).
  * de Vicente  _et al._ (2013) Julio I de Vicente, Cornelia Spee, and Barbara Kraus, “Maximally entangled set of multipartite quantum states,” Physical review letters 111, 110502 (2013).
  * Horodecki  _et al._ (2009) Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki, “Quantum entanglement,” Reviews of modern physics 81, 865 (2009).
  * Michael Walter (2016) Jens Eisert Michael Walter, David Gross,  _Multipartite Entanglement_ , Chapter14 (Wiley, 2016).
  * Horodecki  _et al._ (2001) Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki, “Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps,” Physics Letters A 283, 1–7 (2001).
  * Chen and Wu (2002) Kai Chen and Ling-An Wu, “The generalized partial transposition criterion for separability of multipartite quantum states,” Physics Letters A 306, 14–20 (2002).
  * Hassan and Joag (2007) Ali Saif M Hassan and Pramod S Joag, “Separability criterion for multipartite quantum states based on the bloch representation of density matrices,” arXiv preprint arXiv:0704.3942 (2007).
  * Gühne and Hyllus (2003) Otfried Gühne and Philipp Hyllus, “Investigating three qubit entanglement with local measurements,” International Journal of Theoretical Physics 42, 1001–1013 (2003).
  * Bourennane  _et al._ (2004) Mohamed Bourennane, Manfred Eibl, Christian Kurtsiefer, Sascha Gaertner, Harald Weinfurter, Otfried Gühne, Philipp Hyllus, Dagmar Bruß, Maciej Lewenstein, and Anna Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Physical review letters 92, 087902 (2004).
  * Tóth and Gühne (2005) Géza Tóth and Otfried Gühne, “Detecting genuine multipartite entanglement with two local measurements,” Physical review letters 94, 060501 (2005).
  * Bennett  _et al._ (1999) Charles H Bennett, David P DiVincenzo, Tal Mor, Peter W Shor, John A Smolin, and Barbara M Terhal, “Unextendible product bases and bound entanglement,” Physical Review Letters 82, 5385 (1999).
  * Liang  _et al._ (2014) Yeong-Cherng Liang, Florian John Curchod, Joseph Bowles, and Nicolas Gisin, “Anonymous quantum nonlocality,” Physical review letters 113, 130401 (2014).
  * Horodecki  _et al._ (1995) Ryszard Horodecki, Pawel Horodecki, and Michal Horodecki, “Violating bell inequality by mixed spin-12 states: necessary and sufficient condition,” Physics Letters A 200, 340–344 (1995).
  * Gurvits (2003) Leonid Gurvits, “Classical deterministic complexity of edmonds’ problem and quantum entanglement,” in  _Proceedings of the thirty-fifth annual ACM symposium on Theory of computing_ (2003) pp. 10–19.
  * Clivaz  _et al._ (2017) Fabien Clivaz, Marcus Huber, Ludovico Lami, and Gláucia Murta, “Genuine-multipartite entanglement criteria based on positive maps,” Journal of Mathematical Physics 58 (2017).
  * Huber and Sengupta (2014) Marcus Huber and Ritabrata Sengupta, “Witnessing genuine multipartite entanglement with positive maps,” Physical review letters 113, 100501 (2014).
  * Sudha  _et al._ (2021) Sudha Sudha, BN Karthik, AR Usha Devi, and AK Rajagopal, “Canonical structure of a and b maps,” Quanta 10, 34–41 (2021).
  * Blume-Kohout  _et al._ (2010) Robin Blume-Kohout, Hui Khoon Ng, David Poulin, and Lorenza Viola, “Information-preserving structures: A general framework for quantum zero-error information,” Physical Review A 82, 062306 (2010).
  * Chimalgi  _et al._ (2023) Vaibhav Chimalgi, Bihalan Bhattacharya, Suchetana Goswami, and Samyadeb Bhattacharya, “Detecting entanglement harnessing lindblad structure,” Physica Scripta 98, 115117 (2023).
  * Størmer (1963) Erling Størmer, “Positive linear maps of operator algebras,”  (1963).
  * Woronowicz (1976) SL Woronowicz, “Nonextendible positive maps,” Communications in Mathematical Physics 51, 243–282 (1976).
  * Jamiołkowski (1974) A Jamiołkowski, “An effective method of investigation of positive maps on the set of positive definite operators,” Reports on Mathematical Physics 5, 415–424 (1974).
  * Choi (1975) Man-Duen Choi, “Positive semidefinite biquadratic forms,” Linear Algebra and its applications 12, 95–100 (1975).
  * Maruyama  _et al._ (2005) Koji Maruyama, Fumiaki Morikoshi, and Vlatko Vedral, “Thermodynamical detection of entanglement by maxwell’s demons,” Physical Review A 71, 012108 (2005).
  * Rana (2013) Swapan Rana, “Negative eigenvalues of partial transposition of arbitrary bipartite states,” Physical Review A 87, 054301 (2013).
  * Dür (2001) W Dür, “Multipartite bound entangled states that violate bell’s inequality,” Physical Review Letters 87, 230402 (2001).
  * Smolin (2001) John A Smolin, “Four-party unlockable bound entangled state,” Physical Review A 63, 032306 (2001).
  * Tóth and Gühne (2009) Géza Tóth and Otfried Gühne, “Entanglement and permutational symmetry,” Physical review letters 102, 170503 (2009).