Three-Tangle of a General Three-Qubit State in the Representation of Majorana Stars

Abstract

Majorana stars, the 2j spin coherent states that are orthogonal to a spin-j state, offer a geometric representation of the quantum state and many interesting quantum characteristics. In particular, the genuine tripartite entanglement - the three-tangle of a symmetric three-qubit state, which can be mapped to a spin-3/2 state, is measured by the normalized product of the distance between the Majorana stars. However, the Majorana representation cannot applied to general non-symmetric n-qubit states. We show that after a series of SL(2,C)(2,\mathbb{C}) transformations, non-symmetric three-qubit states can be transformed to symmetric three-qubit states, while at the same time the three-tangle is unchanged. Thus the genuine tripartite entanglement of general three-qubit states has the geometric representation of the associated Majorana stars. We further discuss the extension to the Majorana-star representation of the entanglement of more qubits.

Full text

PHYSICAL REVIEW A 101, 032318 (2020)
Three-tangle of a general three-qubit state in the representation of Majorana stars
Chon-Fai Kam and Ren-Bao Liu
Department of Physics, Centre for Quantum Coherence, and Institute of Theoretical Physics,
The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
(Received 4 February 2020; accepted 19 February 2020; published 13 March 2020)
Majorana stars, the 2 jspin coherent states that are orthogonal to a spin- jstate, offer a visualization of general
quantum states and may disclose deep structures in quantum states and their evolutions. In particular, the genuine
tripartite entanglement—the three-tangle of a symmetric three-qubit state, which can be mapped to a spin-3/2
state—is measured by the normalized product of the distance between the Majorana stars. However, the Majorana
representation cannot be applied to general nonsymmetric n-qubit states. We show that, after a series of SL(2,C)
transformations, nonsymmetric three-qubit states can be transformed to symmetric three-qubit states, while at
the same time the three-tangle is unchanged. Thus the genuine tripartite entanglement of general three-qubit
states has the geometric representation of the associated Majorana stars. The symmetrization and hence the
Majorana star representation of certain genuine high-order entanglement for more qubits are possible for some
special states. In general cases, however, the constraints on the symmetrization may prevent the Majorana star
representation of the genuine entanglement.
DOI: 10.1103/PhysRevA.101.032318
I. INTRODUCTION
In his 1932 seminal paper, Majorana studied the dynamics
of a general spin- jstate in a time-varying magnetic field
and derived a compact formula for the transition probability
[1]. He generalized Bloch’s representation of a spin-1/2 state
as a single point on a unit sphere to a constellation of 2 j
unordered points on unit sphere, known as the Majorana
representation. For a general spin- jstate expressed in the |jm
basis, |ψ=j
m=−jcm|jm, one may introduce a coherent-
state representation of |ψas [2]
j,n|ψ=
j

m=−j2j
j+mcos θ
2j−msin θ
2eiφj+m
cm,
(1)
where n≡(θ,φ) and |j,nis the spin- jcoherent state [2]
directed in the direction of n. Hence, the overlap between the
general spin- jstate and the spin coherent state which directed
in the antipodal direction of nis
j,−n|ψ=sin θ
22j
P(z),(2a)
P(z)≡
j

m=−j
(−1)j+m2j
j+mcmzj+m,(2b)
where z≡cot θ
2eiφis the stereographic image of nfrom
the north pole onto the equatorial plane. Using the funda-
mental theorem of algebra, the Majorana polynomial P(z)
may be factorized as P(z)=(−1)2jcj2j
k=1(z−zk). Majo-
rana stars are the inverse stereographic image nkof zk.
Hence, there are in general 2 jdirections −nkon the unit
sphere where −nk,j|ψvanishes. In particular, for spin
coherent states, the Majorana polynomial has the form
P(z)=(cos θ
2−sin θ
2e−iφz)2j, which is associated with 2 j
degenerated stars in the direction of n.
Majorana’s representation of general spin states and his
transition-probability formula were rediscovered several times
by Bloch [3], Salwen [4], Meckler [5], and Schwinger [6].
Majorana’s representation was known to mathematicians as
“canonical decomposition” of a totally symmetric spinor,
and the associated Majorana stars are called “principal null
directions” in spinor theory [7]. In 1960, Penrose developed
a spinor approach to general relativity and gave an elegant
proof of Petrov’s classification of gravitational fields based on
degeneracy configuration of the “principal null directions” of
a gravitational spinor [8]. Several decades later, after being
aware of Majorana’s work, Penrose brought it to wider at-
tention via his popular book [9]. Since then, research based
on Majorana’s representation gradually emerged. Zimba and
Penrose used Majorana’s representation of spin-3/2 states
to provide a simplified proof of Bell’s nonlocality theorem
[10]. Inspired by Penrose’s works, Hannay studied statistics
of Majorana stars for random spin states, and discovered a
simple formula for the pair correlation function in the large-j
limit [11]. Two years later, Hannay derived a general formula
of Berry’s phase for spin states by using Majorana’s represen-
tation [12] and applied the spin-1 formula to the polarization
of light [13]. Later, Dennis discovered a simple geomet-
ric interpretation of polarization singularities in nonparaxial
waves in terms of Majorana representation [14] and gave an
alternative proof of Maxwell’s multipole representation of
spherical functions using Majorana stars [15].
The visualization of the quantum states as a constellation
on a unit sphere may be highly valuable in the classifica-
tion of quantum states and their evolution. It has been used
to reveal a beautiful connection between the most sensitive
states under small rotations around arbitrary axes in quan-
tum metrology and the platonic solids [16–19] and has been
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CHON-FAI KAM AND REN-BAO LIU PHYSICAL REVIEW A 101, 032318 (2020)
applied to classifying novel phases in spinor Bose-Einstein
condensates [20–22].
Remarkably, Majorana’s representation of spin states finds
application in quantum information science. Bastin et al. gave
a simple classification of entanglement between symmetric
N-qubit states under stochastic local operations and classical
communication (SLOCC) via the degeneracy configuration of
the associated Majorana stars [23,24]. In subsequent works,
Markham et al. [25,26] showed that three types of entan-
glement measures—the geometric measure of entanglement,
the logarithmic robustness of entanglement, and the relative
entropy of entanglement are equivalent when the distribution
of Majorana stars obey certain symmetries. Subsequently,
Majorana’s representation was also used to provide insight
into quantum geometric phases and the dynamics of quantum
spins [27,28], and to study the anticoherence of symmetric
qubit states [29,30]. Based on these developments, Majorana’s
representation of general spin states becomes a valuable tool
for visual display of multipartite entanglement between sym-
metric qubit states.
Entanglement is a resource that is unique to quantum
information [31], which cannot be increased by local oper-
ations when the systems are distributed over spatially sep-
arated locations [32]. For two-qubit pure states, the en-
tanglement may be measured by Wootters’ concurrence
C[33], which varies monotonically from 0 to 1 when the
state changes from separable to maximally entangled. In
particular, for symmetric two-qubit states, which may be
written as |ψ=|n1⊗|n2+|n2⊗|n1, Wootters’ concur-
rence becomes C=sin2θ12
2/(1 +cos2θ12
2)[34], where |nk≡
cos θk
2|0+sin θk
2eiφk|1in the computational basis, and θ12 ≡
cos−1(n1·n2)∈[0,π] is the spherical distance between the
Bloch vectors n1and n2. The two unordered points n1and
n2(Majorana stars) completely determines a symmetric two-
qubit state and thus the entanglement. For three-qubit pure
states, the entanglement between the parties are measured by
five independent local unitary invariants: C12 ,C13,C23,κ, and
τ3[35,36], where Cij are the pairwise concurrence between
the parties iand j,κis the Kempe invariant [35], and τ3is
the three-tangle, which measures the genuine tripartite entan-
glement [36]. In particular, for symmetric three-qubit states,
which are written as |ψ=σ∈S3|nσ(1)⊗|nσ(2)⊗|nσ(3),
we have τ3=4
3(i<jsin θij
2/i<jcos2θij
2)2[34], where S3
is the permutation group of order three. In other words, for
symmetric two- and three-qubit pure states, we may measure
the genuine entanglement in terms of the distances between
the Majorana stars on a unit sphere.
Now, one question naturally arises: can we have a star rep-
resentation for general two- and three-qubit pure states with-
out permutation symmetries? The benefits of such a represen-
tation are evident: it offers an intuitive approach to visualizing
the entanglement in terms of a three-dimensional geometry; it
also provides a simple way to obtain the entanglement—one
just calculates the distances between all Majorana stars on unit
sphere. As general nonsymmetric states do not possess a Ma-
jorana representation, the Majorana star representation cannot
be directly employed to formulate entanglement measures.
Nevertheless, if we can transform a general nonsymmetric
pure state to a symmetric state, while at the same time keeping
Wootters’ concurrence Cor the three-tangle τ3unchanged,
then the Majorana star representation can be applied. For two-
qubit states, Schmidt decomposition [37] exists and allows
one to express the state as a symmetric state without changing
its entanglement properties. Hence, the Majorana star repre-
sentation for general two-qubit pure states can be immediately
obtained from its Schmidt decomposition. Because Schmidt
decomposition does not exist for three-partite pure states, the
Majorana star representation for three-qubit states is not so
evident. However, we show in the following sections that,
after using Acín’s canonical form [38]—a type of generalized
Schmidt decomposition—one may have a star representation
of entanglement for general three-qubit states.
The organization of the paper is as follows: In Sec. II,
we discuss the Majorana star representation of spin states,
and the representation of entanglement in terms of Majorana
stars. In Sec. III, we explicitly construct a set of invert-
ible local transformations L=L1⊗L2⊗L3, which bring a
general nonsymmetric three-qubit pure state to a symmetric
one, where Li∈SL(2,C) are special linear transformations
of degree two. As the three-tangle τ3is an invariant under
special linear transformation [37], we may express the three-
tangle of a general three-qubit state in the constellation of
three Majorana stars. In Sec. IV, we discuss generalization
of such transformation to multipartite entangled pure states
and show that similar procedures can be applied to some but
not all n-qubit states with n4. In Sec. V, we discuss mixed
entanglement in the Majorana star representation, and use
the density matrix of a symmetric N-qubit GHZ state as an
example. Finally, in Sec. VI, we discuss the implications and
limitations of the current work.
II. MAJORANA REPRESENTATION OF SPIN STATES
The essence of the Majorana representation is that a spin- j
state can be written as a symmetric tensor product of N=2j
spin-1/2 states:
|ψ= 1
√N!AN
σ∈SN|nσ(1)⊗···⊗|nσ(N),(3)
where AN≡σ∈SNknk|nσ(k)is a normalization factor, SN
is the permutation group of order N, and |nkis a spin-1/2
state polarized along the direction nk.TheN=2jantipodal
directions −nkof the Majorana stars nkcorrespond to the
spin- jcoherent states |j,−nk≡|−nk⊗Nthat are orthog-
onal to the spin- jstate |ψ. The Majorana representation
of spin states can be rephrased as a theorem [7]: a 2 j-
dimensional complex projective space CP2j, which is the
state space of a spin- jstate, is homeomorphic to a 2 j-fold
symmetric tensor product of sphere SP2j(S2)[39,40], i.e., an
ordered tuple (a0,a1,...,aN) in a complex projective space
CPNis equivalent to an unordered tuple [n1,n2,...,nN]≡
{(n1,n2,...,nN)/∼|ni∈S2}, where ∼is an equivalence re-
lation defined by (n1,n2,...,nN)∼(nσ(1) ,nσ(2),...,nσ(N)).
For N=2, the Schmidt decomposition of a general two-
qubit pure state reads |ψ=μ1|00+μ2|11, where μ1≡
cos χand μ2≡sin χare the Schmidt coefficients, and χ∈
[0,π/4] is the Schmidt angle [38]. The entanglement be-
tween the two qubits, measured by Wootters’ concurrence
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THREE-TANGLE OF A GENERAL THREE-QUBIT STATE … PHYSICAL REVIEW A 101, 032318 (2020)
C, may be written in terms of the Schmidt coefficients: C≡
2μ1μ2=sin 2χ[37]. The entanglement is larger when the
Schmidt angle has a larger value. As the Schmidt decom-
position of a general two-qubit state is already symmetric
under permutation of qubits, it can be mapped to a spin-1
state which possesses two Majorana stars with latitudes θ1=
θ2=π−2arctan√tan χand longitudes φ1=π/2 and φ2=
3π/2. The Schmidt coefficients and the Wootters concurrence
can be expressed via the spherical distance θ12 =2θ1be-
tween the Majorana stars: μ1=1
2(√1+C+√1−C), μ2=
1
2(√1+C−√1−C) and C=sin2θ12
2/(1 +cos2θ12
2). The
entanglement between the qubits is larger when the spherical
distance between the stars has larger value. Separable states
correspond to two identical stars and maximally entangled
Bell states correspond to two antipodal stars on equator [34].
III. REPRESENTATION OF THREE-TANGLE USING
MAJORANA STARS
As discussed in Sec. I, the Majorana representation cannot
be directly applied to general nonsymmetric states. However,
we may still find a set of local transformations which send
nonsymmetric states to symmetric ones without changing
the three-tangle, i.e., the global entanglement of three-qubit
states. For a general three-qubit state |ψ0=ijk|ijk, Acín’s
canonical form, the generalized Schmidt decomposition of
three-qubit states, reads [38]
|ψ=λ0|000+λ1eiϕ|100+λ2|101+λ3|110+λ4|111,
(4)
where 0 ϕπ, and λ0,λ1,λ2,λ3, and λ4are non-negative
real numbers satisfying 4
i=0λ2
i=1. The relation between
Acín’s canonical form and the coefficients ijk can be spec-
ified as follows: let T0and T1be two matrices with elements
(Ti)jk ≡ijk, and let U1,U2, and U3be three unitary matrices
satisfying [38]
T
i≡
j
(U†
1)ijTj,s.t. det T
0=0,(5a)
U†
2T
0U3=λ00
00
,U†
2T
1U3=λ1eiϕλ2
λ3λ4,(5b)
then |ψ=U∗
1⊗U∗
2⊗U3|ψ0, where M∗denotes a matrix
with complex-conjugated entries. U∗
1,U∗
2, and U3are unitary
matrices, and thus the net transformation U∗
1⊗U∗
2⊗U3is
a local unitary. As local unitary transformations do not alter
the degree of entanglement, Acín’s canonical form preserves
entanglement.
The three-tangle of a general three-qubit state |ψ0is
proportional to the hyperdeterminant of the third-order tensor
(3) ≡[ijk], which may be specified as [37]
τ3(|ψ0)≡4|Det((3) )|,(6)
where Det((3) ) is Cayley’s hyperdeterminant defined by [41]
Det((3) )≡
000 011
100 111+
010 001
110 1012
−4
000 001
100 101·
010 011
110 111
.(7)
Cayley’s hyperdeterminant is a homogeneous polynomial
of degree four. Under invertible local operations |˜
ψ=
L1⊗L2⊗L3|ψ, it transforms with a determinantal factor,
Det( ˜
(3) )=[det(L1)]2[det(L2)]2[det(L3)]2Det((3) ), where
L1,L2, and L3are invertible matrices [37]. When L1,L2,
and L3are special linear transformations of degree two, i.e.,
two-by-two matrices of determinant 1, the hyperdeterminant
and the three-tangle become invariants: Det( ˜
(3) )=Det((3))
and τ3(|˜
ψ)=τ3(|ψ). Dür et al. showed that two states have
the same kind of entanglement if both of them can be obtained
from the other by means of stochastic local operations and
classical communications (SLOCC) [42]. They proved that
two states are equivalent under SLOCC if they are related
by invertible local transformations. In other words, the
three-tangle τ3of general three-qubit states is an SLOCC
invariant [42].
Using Acín’s canonical form, Eq. (4), the three-tangle
τ3(|ψ) reads τ3(|ψ)=4λ2
0λ2
4. As we are interested in states
for which τ3(|ψ)= 0, we assume λ0= 0 and λ4= 0. To
transform general nonsymmetric three-qubit states to sym-
metric three-qubit states, we consider the following SL(2,R)
transformation on the third qubit:
M≡γ0
gγ−1,g≡λ2γ−1−λ3γ
λ4
,(8)
so that, after the transformation, the three-qubit state is invari-
ant under permutation of the second and third qubits,
|ψ≡I2⊗I2⊗M|ψ(9a)
=γλ
0|000+γ
λ4|100+λ2
2+λ2
4
λ4γ|1nn,(9b)
where I2denotes the two-by-two identity matrix, |n≡
(λ2
2+λ2
4)−1/2(λ2|0+λ4|1), γis a constant which will be
determined later, and ≡λ1λ4eiϕ−λ2λ3vanishes when
|ψcan be split into two orthogonal product states, |ψ=
γλ
0|000+(γλ
4)−1(λ2
2+λ2
4)|1nn. To proceed further, we
consider the following SL(2,C) transformation on the first
qubit:
M≡ab
cd
,a≡1
λ2
4−λ2
λ0
,
b≡−λ4
λ0
,c≡λ2λ4,d≡λ2
4,(10)
so that after the transformation, the three-qubit state is invari-
ant under permutation of all three qubits
|ψ≡M⊗I2⊗I2|ψ=A(|000+y|nnn),(11)
where A=γλ
0λ−2
4and y=γ−2λ2
4λ−1
0(λ2
2+λ2
4)3/2. Equation
(11) is in Mandilara’s canonical form for pure symmetric
states [43], which may be further simplified by performing
the following SL(2,R) transformation on all the three qubits
M ≡10
g1,g≡−λ2
λ4
,(12)
so that, after the transformation, the three-qubit state has the
form
|ψ≡M ⊗M ⊗M|ψ(13a)
=γλ
0λ−2
4|000+γ−1λ3
4|111.(13b)
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CHON-FAI KAM AND REN-BAO LIU PHYSICAL REVIEW A 101, 032318 (2020)
We now fix the parameter γby requiring ψ|ψ=1.
A direct calculation yields |ψ=ν1|000+ν2|111, where
γ≡λ−1
0λ2
4ν1and
ν1≡1
2+1
21−4λ2
0λ2
41/2,
ν2≡1
2−1
21−4λ2
0λ2
41/2.
As ν1ν2, we may write |ψ=cos ϑ|000+sin ϑ|111,
which is similar to the Schmidt decomposition of general two-
qubit states, where ϑ≡cos−1ν1∈[0,π/4]. Hence, the three-
tangle of the three-qubit state |ψmay be expressed in terms
of ϑas τ3(|ψ)=sin2(2ϑ)∈[0,1], which is an increasing
function of ϑon [0,π/4].
After the transformations M,M, and M, Acín’s
canonical form of three-qubit states becomes a symmetric
state, which may be written in terms of the symmetric
basic states: |ψ=3
i=0ai|S(3)
i, where |S(3)
0≡|000,
|S(3)
1≡3−1
2(|001+|010+|100), |S(3)
2≡3−1
2(|011+
|101+|110), and |S(3)
3≡|111. Here, the coefficients
aiare given by a0=cos ϑ,a1=a2=0, and a3=sin ϑ.
Majorana’s star representation may be introduced via
the one-to-one correspondence between the symmetric
basic states and the conventional |jmbasis states, i.e.,
|S(2 j)
j−m↔|jm, so that we have |ψ=j
m=−jcm|jmwith
cm=aj−m(j=3/2). Then the Majorana polynomial of |ψ
is obtained via the general formula introduced in Sec. I:
P(z)=
2j

r=0
(−1)2j−rCr
2jarz2j−r(14a)
=sin ϑ−cos ϑz3.(14b)
The Majorana polynomial of |ψhas three distinct roots
zk≡(tan ϑ)1/3ei2kπ/3,(k=0,1,2). Let us denote the di-
rections of the three Majorana stars on unit sphere as nk.
Hence, |ψhas three Majorana stars distributed evenly
on the southern hemisphere with the same latitude θ=
2 arccot[(tan ϑ)1/3] and longitudes φ1=0, φ2=2π/3, and
φ3=4π/3. Then the three-tangle τ3(|ψ) can be evaluated
by the angles between the directions of the stars [34]:
τ3|ψ=4
3i<jsin θij
2
i<jcos2θij
22
=1
3i<jdij
12 −i<jd2
ij 2
,
(15)
where θij ≡cos−1(n1·n2) and dij ≡2sinθij
2are the angle
and chordal distances between niand nj, respectively. As
the three Majorana stars of |ψdistributed evenly on the
southern hemisphere with the same latitude, the chordal dis-
tance between any two Majorana stars are the same, i.e.,
d12 =d13 =d23 ≡d, which yields τ3(|ψ)=1
27 d6/(4 −
d2)2, which is an increasing function of don [0,2]. Using
elementary geometry, we obtain d=2√3(R−1+R)−1, where
R=(tan ϑ)1/3is the amplitude of the roots zk. A direct cal-
culation yields again τ3(|ψ)=4(R−3+R3)−2=sin2(2ϑ).
As the chordal distance between any two Majorana stars
of |ψis an entanglement monotone of genuine tripartite
entanglement, the genuine tripartite entanglement is higher
FIG. 1. The Majorana representation for generalized GHZ
states |gGHZ≡a|000+b|111, and generalized Wstates
|gW≡c|001+d|010+e|100. Here, a≡cos ϑ,b≡sin ϑ,and
ϑ=π/10.
when the Majorana stars are closer to the equator—the closer
to the equator, the higher the entanglement. As an exam-
ple, a generalized Greenberger-Horne-Zeilinger (GHZ) state
|gGHZ≡a|000+b|111with a three-tangle 4a2b2is al-
ready symmetrized and hence is represented by three distinct
Majorana stars distributed evenly on the southern hemisphere
with the same latitude [see Fig. 1(a)].
For a generalized Wstate |gW≡c|001+d|010+
e|100, the three-tangle vanishes, and hence the above trans-
formations M,M, and M may not be used directly. But
we may still apply a set of SL(2,R) transformations to
symmetrize it, while keeping the three-tangle unchanged.
We may apply the following SL(2,R) transformation on the
first qubit:
T≡α0
0α−1,α≡e
d,(16)
so that, after the transformation, the generalized W
state becomes |gW≡T⊗I2⊗I2|φ=c√e/d|001+
√de(|010+|100|), which is symmetric with respect to
the first two qubits. Similarly, we may apply the following
SL(2,R) transformation on the third qubit:
T≡β0
0β−1,β≡c
d,(17)
so that, after the transformation, one obtains |gW≡I2⊗
I2⊗T|gW=√ce(|001+|010+|100), which is the
symmetric Wstate. The associated Majorana polynomial has
three roots 0 and ∞, where ∞is a double root. Hence, a
generalized Wstate is represented by three Majorana stars
on unit sphere—one is located at the south pole and two
degenerate ones are located at the north pole. It shows that
the appearance of a pair of degenerate Majorana stars on the
unit sphere indicates the vanishing of the three-tangle for a
general three-qubit state [see Fig. 1(b)].
IV. FOUR-QUBIT STATES AND BEYOND
In the last section, we constructed a series of SL(2,C)⊗3
transformations which symmetrize a general three-qubit state
without changing its genuine tripartite entanglement. We now
show that similar procedures can be applied to some but not
all n-qubit states with n4.
032318-4

THREE-TANGLE OF A GENERAL THREE-QUBIT STATE … PHYSICAL REVIEW A 101, 032318 (2020)
Let us denote a general four-qubit state as |ψ≡
ijkl|ijkl. For SLOCC transformation SL(2,C)⊗4, there ex-
ists a set of four independent polynomial invariants. The first
polynomial invariant of degree two is Cayley’s hyperdetermi-
nant defined by [44]
H≡00001111 −0001 1110 −00101101 +00111100
−01001011 +0101 1010 +01101001 −01111000 ,
(18)
and the other two independent polynomial invariants of degree
four are two determinants given by [44]
L≡
0000 0100 1000 1100
0001 0101 1001 1101
0010 0110 1010 1110
0011 0111 1011 1111

,
M≡
0000 1000 0010 1010
0001 1001 0011 1011
0100 1100 0110 1110
0101 1101 0111 1111

.(19)
The invariants Land Mare closely related to the two-
qubit reduced density matrix of the original four-qubit state:
|L|2=det ρ12 and |M|2=det ρ13, where ρ12 ≡Tr34 (|ψψ|)
and ρ13 =Tr24 (|ψψ|)[45]. If one denotes (x,y,z,t)≡
ijkl ijklxiyjzktland gxy(x,y)≡det(∂2/∂zi∂tj), one may
define a 3 ×3matrixGxy by [44]
gxy(x,y)≡x2
0x0x1x2
1Gxy⎛
⎜
⎝
y2
0
y0y1
y2
1
⎞
⎟
⎠,(20)
For any pair of variables (u,v), one defines Dμν ≡det (Gμν ),
then the last polynomial invariant of degree six is a de-
terminant given by D≡Dxt. In particular, for a four-qubit
symmetric state, one obtains 0001 =0010 =0100 =1000,
0011 =0101 =0110 =1001 =1010 =1100, and 0111 =
1011 =1101 =1110. Hence, one obtains H=β−4,L=
M=0, and
D=
αδ
δβη
ηγ

,(21)
where α≡00000011 −2
0001,β≡0000 1111 −2
0011,
γ≡00111111 −2
0111,δ≡0000 0111 −00010011,
≡00010111 −2
0011, and η≡0001 1111 −00110111 .
The vanishing of the invariants Land Mfor symmetric
states can be expected from their unique properties [46]: the
invariant Lis odd under permutation of the first two qubits
(or the last two qubits), and the invariant Mis odd under
permutation of the first and the third qubits (or the second and
the fourth qubits). Hence, both Land Mvanish identically for
permutation-symmetric states.
Verstraete [47] showed that a general four-qubit state can
always be transformed into one of nine distinct SLOCC
classes, where each of which is a representative of states
interconvertible under SLOCC operations. We may consider
the generic class out of the nine SLOCC classes, whose
representative is
|Gabcd ≡a+d
2(|0000+|1111)+a−d
2(|0011
+|1100)+b+c
2(|0101+|1010)
+b−c
2(|0110+|1001),(22)
which may also be written as |Gabcd =a|+++
b|+++c|−−+d|−−, where |±and |±are
the maximally entangled two-qubit Bell states. The represen-
tative of the generic SLOCC class contains the four-qubit
GHZ state |GHZ4≡ 1
√2(|0000+|1111) and the Einstein-
Podolsky-Rosen (EPR) pair state |G1000=|++as spe-
cial cases. It also contains the four-qubit cluster state [48]
|φ4=1
2(|0000+|0011+|1100−|1111) as a special
case, which can be explicitly obtained from the SL(2,C)⊗4
transformation
|φ4=e−iπ/80
0eiπ/8⊗4
|Gabcd ,(23)
where a=(i+e−iπ/4)/2, b=c=0, and d=(i−e−iπ/4)/2.
The polynomial invariants for the generic SLOCC
class are [44]H=1
2(a2+b2+c2+d2), L=abcd,
M=[(c−d
2)2−(a−b
2)2][( a+b
2)2−(c+d
2)2], and D=
−1
4(ad −bc)(ac −bd )(ab −cd ). Hence, any states in
the generic SLOCC class may be transformed into symmetric
states only when the conditions (i) abcd =0 and (ii)
c−d=±(a−b)ora+b=±(c+d) are fulfilled.
It rules out the possibility that the EPR pair state and
four-qubit cluster state can be symmetrized by SL(2,C)⊗4
transformations. For example, for c=0 and a−d=b
|Gb+d,b,0,d
=b+2d
2(|0000+|1111)+b
2(|0011+|1100
+|0101+|1010+|0110+1001),(24)
where the associated Majorana polynomial has the form
P(z)=b+2d
2z4+3bz2+b+2d
2, which has four roots given by
zk≡±
−3b±9b2−(b+2d)2
b+2d.(25)
As another example, for a=1/√3, d=ω/√3, b=ω2/√3,
and c=0, we obtain an entangled four-qubit state which
maximizes the average Tsillas-qentropy E(q)
2for q>2
[49], |L≡ 1
√3(|+++ω|−−+ω2|++), where
ω≡e2πi/3,E(q)
2≡1
3(E(q)
(AB)(CD)+E(q)
(AC )(BD)+E(q)
(AD)(BC )),
E(q)≡1
1−q(Tr ρq
r−1) is the Tsallis-qentropy, and
032318-5

CHON-FAI KAM AND REN-BAO LIU PHYSICAL REVIEW A 101, 032318 (2020)
ρr≡TrB|ψAB ψAB |is the reduced density matrix for a
given bipartite state |ψAB. As the conditions abcd =0 and
a+b+c+d=0 are both fulfilled, it is possible to find
an SL(2,C)⊗4transformation to symmetrize the maximally
entangled state |L. Let us consider the following SL(2,C)⊗4
transformation:
|L≡I2⊗I2⊗01
−10
⊗2
|L
=1−ω
√3(|0000+|1111)−ω2
√3(|0011+|1100
+|0101+|1010+|0110+|1001),(26)
so that after the transformation, the maximally entangled
state is invariant under permutation of qubits. The symmetric
state |Lis associated with a Majorana polynomial P(z)=
1−ω
√3z4−6ω2
√3z2+1−ω
√3, which has four distinct roots given by
zk≡±
3ω2±2√3ω
1−ω.
As a final example, for the four-qubit GHZ state |GHZ4≡
|G1
√200 1
√2, one has P(z)=(z4+1)/√2 and zk=ei(2k+1)π/4
(k=0,1,2,3), which corresponds to four Majorana stars
distributed evenly on the equator, with azimuthal angles φk=
(2k+1)π/4, respectively.
Miyake [50] showed that general multiqubit states un-
der stochastic local operations and classical communication
(SLOCC) can be classified by multidimensional determinants
[41] similar to Cayley’s hyperdeterminant. As multidimen-
sional determinants are invariant under SL(2,C)ntransfor-
mation, it seems that we may symmetrize a general n-qubit
state without changing its global entanglement properties.
However, for n-qubit states with n>4, one may construct
polynomial invariants for SLOCC transformation which are
odd under permutations of two qubits. For example, for
a five-qubit state |ψ≡ijklm|ijklm, the unique invariant
Fof degree six can be explicitly written as [51]
F=i1j1k1l1m1i2j2k2l2m2i3j3k3l3m3i4j4k4l4m4i5j5k5l5m5i6j6k6l6m6
×k1k2m1m2j1j3l1l3i1i4l2l4m3m4i2i5l5l6m5m6
×i3i6k3k6j4j6k4k5,(27)
where ij is the antisymmetric tensor with 01 =−10 =1 and
00 =11 =0. As the SLOCC invariant Fis an odd function
under qubit permutations, it vanishes identically for symmet-
ric five-qubit states. Hence, it prevents the symmetrization of
general five-qubit states with a nonvanishing value of F.
V. ENTANGLED MULTIPARTITE MIXED STATES
In this section, we briefly discuss the extension of Ma-
jorana star representation to multipartite mixed states. The
Majorana-like geometric representation of symmetric mixed
states, or equivalently mixed spin states, was first introduced
by Ramachandran and Ravishankar in 1986 [52], in which
they constructed a set of Majorana stars for the 2 jFano statis-
tical tensor parameters which characterize a spin- jassembly.
To begin with, one needs the spherical tensor representation
of a general spin- jdensity matrix [53],
ρ=
2j

k=0
k

q=−k
ρkqTkq ≡
2j

k=0
ρk·Tk,(28)
where ρkq ≡Tr(ρT†
kq), and Tkq are the irreducible tensor op-
erators of rank kin the 2 j+1 dimensional spin space with
projection qalong the axis of quantization, which can be ex-
plicitly expressed in terms of the Clebsch-Gordan coefficients
as Tkq ≡j
m,m=−j(−1)j−mCkq
jm,j−m|jmjm|and satisfy the
relations T†
kq =(−1)qTk−qand Tr(T†
kqTkq)=δkkδqq. Each
vector ρkis associated with 2kMajorana stars defined as the
stereographic projection of the roots of the polynomial [54,55]
P(k)(z)≡
k

q=−k
(−1)k+q2k
k+qρkqzk+q.(29)
The vector ρ0≡ρ00 does not have an associated constella-
tion of Majorana stars, and its value is fixed to (2 j+1)−1
by Tr ρ=1. Moreover, since ρis Hermitian, the condition
T†
kq =(−1)qTk−qimplies that ρ∗
kq =(−1)qρk−q. Hence, the
constellation of Majorana stars possesses antipodal symmetry,
i.e., P(k)(z)=(−1)kz2k[P(k)(−1/z∗)]∗. Unlike that for pure
spin states, the constellation for mixed spin states cannot
fully specify the states, and one needs the relative weights
of the irreducible representations. Let us denote ρk=rk˜
ρk
with respect to a normalized vector ˜
ρk, then the spin- jdensity
matrix ρmay be written as
ρ=1
2j+1+
2j

k=1
rk˜
ρk·Tk.(30)
Hence, any spin- jdensity matrix ρis specified by 2 jspheres
with radii rk, and each of which possesses a constellation of
2kMajorana stars.
As a first example, let us consider a spin-1/2 density matrix
ρ=1
2(1+r·σ)=1
2+ρ11T11 +ρ10T10 +ρ1−1T1−1, where
T10 =1
√2σz,T1±1=∓1
2σ±and ρ1≡(ρ11,ρ
10,ρ
1−1)=
1
2(−rx+iry,√2rz,rx+iry), which yields r1=r/√2.
The Majorana polynomial of the vector ρ1has the form
P(1)(z)=(−rx+iry)z2−2rzz+(rx+iry), which is
associated with a pair of Majorana stars rand −ron a
sphere of radius r/√2.
As another example, we consider the density matrix of
a symmetric N-qubit GHZ state in the |jmbasis, ρ=
|NGHZNGHZ|, where N=2jand |NGHZ≡ 1
√2(|jj+
|j−j). As the only nonzero matrix elements of ρ
is ρjj =ρj−j=ρ−jj =ρ−j−j=1
2, we obtain ρk0=1
2[1 +
(−1)k]Ck0
jjj−jand ρ2j−2j=(−1)2jρ2j2j=1
2(−1)2jC2j2j
jjjj.For
k<2j, the Majorana polynomial for the vector ρkis
P(k)(z)=(−1)k[2k
k]1/2ρk0zk, which has a multiple root 0 of
multiplicity kfor keven, and has a multiple root ∞of multi-
plicity kfor kodd. Hence, for k<2j, there are kdegenerate
Majorana stars at the north pole for keven, while there are
kdegenerate Majorana stars at the south pole for kodd. In
contrast, for k=2j, the Majorana polynomial has the form
032318-6

THREE-TANGLE OF A GENERAL THREE-QUBIT STATE … PHYSICAL REVIEW A 101, 032318 (2020)
P(2 j)(z)=1
2z4j+1
2[1 +(−1)2j]z2j+1
2(−1)2j, which has 4 j
distinct roots e2πin/(4 j)with n=0,1,...,4j−1for2jodd,
and has 2 jdouble roots eπim/(2 j)with m=0,1,...,2j−1
for 2 jeven.
VI. CONCLUSION
We have found a way to visually represent the genuine
tripartite entanglement of general three-qubit pure states. We
used Acín’s canonical form of general three-qubits states and
transformed it into a symmetric form similar to the Schmidt
decomposition of general two-qubit states via a series of
SLOCC transformations which keep the three-tangle invari-
ant. Based on Majorana’s representation of spin states, we
projected the symmetrized state onto a coherent state, and
obtained a set of three Majorana stars on a unit sphere which
distributes evenly on the southern hemisphere with the same
latitude. The genuine tripartite entanglement is then visually
represented by the chordal distance between any two Majo-
rana stars. Such a representation may become a useful tool in
the field of quantum computation and information.
Although our work is limited to the representation of
genuine tripartite entanglement of three-qubit states, the cur-
rent approach can be applied to some important four-qubit
states, including the four-qubit GHZ state and the entangled
four-qubit state which maximizes the average Tsallis-qen-
tropy for q>2. However, due to the fact that there exist
multipartite entangled states which cannot be symmetrized
by SLOCC transformations, the Majorana representation can
only be applied to some but not all n-qubit entangled states
with n4.
ACKNOWLEDGMENT
This work was supported by Hong Kong RGC/GRF
Project 14304117.
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