Tripartite entanglement measure under local operations and classical communication Xiaozhen Ge1Lijun Liu2and Shuming Cheng3 4 5 1Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong China

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Tripartite entanglement measure under local operations and classical communication

Xiaozhen Ge,1Lijun Liu,2, ∗and Shuming Cheng3, 4, 5, †

1Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

2College of Mathematics and Computer Science, Shanxi Normal University, Linfen 041000, China

3The Department of Control Science and Engineering, Tongji University, Shanghai 201804, China

4Shanghai Institute of Intelligent Science and Technology, Tongji University, Shanghai 201804, China

5Institute for Advanced Study, Tongji University, Shanghai, 200092, China

(Dated: December 5, 2023)

Multipartite entanglement is an indispensable resource in quantum communication and compu-

tation, however, it is a challenging task to faithfully quantify this global property of multipartite

quantum systems. In this work, we study the concurrence ﬁll, which admits a geometric inter-

pretation to measure genuine tripartite entanglement for the three-qubit system in [S. Xie et al.,

Phys. Rev. Lett. 127. 040403 (2021)]. First, we use the well-known three-tangle and bipartite

concurrence to reformulate this quantiﬁer for all pure states. We then construct an explicit example

to conclusively show the concurrence ﬁll can be increased under local operations and classical com-

munication (LOCC) on average, implying it is not an entanglement monotone. Moreover, we give a

simple proof of the LOCC-monotonicity of three-tangle and ﬁnd that the bipartite concurrence and

the squared can have distinct performances under the same LOCC. Finally, we propose a reliable

monotone to quantify genuine tripartite entanglement, which can also be easily generalised to the

multipartite system. Our results shed light on studying genuine entanglement and also reveal the

complex structure of multipartite systems.

I. INTRODUCTION

Entanglement, having no classical counterpart, is of

fundamental importance in quantum theory [1], and also

an essential resource in various quantum information pro-

cessing tasks, including cryptography [2, 3], teleporta-

tion [4, 5], dense coding [6, 7], secret sharing [8, 9],

metrology [10, 11], and computation [12]. Thereof, an

important problem arises about how to quantitatively

measure the degree of entanglement for the quantum sys-

tem. Typically, one entanglement measure is deﬁned as

some non-negative function which maps any quantum

state to a real number in the interval [0,1], and satis-

ﬁes a set of reasonable assumptions [1, 13, 14], such as

being zero for all non-entangled states and invariant un-

der local unitary operations. Within the resource theory

of entanglement [15, 16], it is further required to be non-

increasing under local operations and classical communi-

cation (LOCC) on average, hence being an entanglement

monotone of which entanglement never increases under

the free LOCC operations. Correspondingly, numerous

entanglement measures and monotones have been devel-

oped [17–22], especially for the low-dimensional bipartite

system, such as concurrence [23, 24] and logarithmic neg-

ativity [25].

It is challenging to ﬁnd proper measures for multipar-

tite entanglement due to the complicated partial separa-

bility structure of multipartite systems. Indeed, in or-

der to detect genuine multipartite entanglement which

is the key resource in multi-party information tasks,

the reliable measure needs to meet extra conditions,

∗lljcelia@126.com

†drshuming.cheng@gmail.com

like being zero for partial separable states and strictly

positive for genuinely entangled states [26, 27]. Thus,

there has been some commonly-used quantiﬁers, includ-

ing α-entanglement entropy [27] and generalized concur-

rence [28, 29] which can not detect all genuinely entan-

gled states, and few genuine multipartite entanglement

monotones [30–35]. In particular, the concurrence ﬁll,

with a nice geometric interpretation as the square root

of the concurrence triangle area, was recently proposed

to measure the degree of genuine entanglement for the

three-qubit system [36], together with an experimental

test [37]. Although it conforms almost all of the necessary

conditions mentioned above, a fundamental problem still

remains open about whether it is an entanglement mono-

tone, or equivalently, it admits the LOCC-monotonicity.

Here we ﬁrst establish the close connections between

the concurrence ﬁll and the well-known three-tangle [38]

and reduced bipartite concurrences [23, 24, 39, 40], of

some interest in its own right. Then, we solve the above

open problem via an explicit example to show that it

can be increased under LOCC on average, implying it

is not an entanglement monotone. Furthermore, a sim-

ple proof of the LOCC-monotonicity of three-tangle is

presented, which completes the proof in [41]. It is also

interesting to ﬁnd that the bipartite concurrence and the

squared can have distinct performances under the same

LOCC. Finally, we conjecture that the area of the tri-

angle with edges corresponding to the bipartite concur-

rences is an entanglement monotone and also propose a

reliable monotone to quantify genuine tripartite entan-

glement, which can be easily generalized to the general

multipartite systems.

The rest of this work is structured as follows. Sec. II

introduces basic notations and useful relations about the

concurrence ﬁll for all pure three-qubit states. In Sec. III,

arXiv:2210.06700v2 [quant-ph] 3 Dec 2023

LOCC

FIG. 1. For any pure three-qubit state |ψ⟩ABC , the concurrence triangle ∆ABC is composed of three sides described by the

three squared one-to-other concurrences C2(|ψ⟩i|jk). It leaves open in [36] that the square root of the area of this triangle,

named as the concurrence ﬁll, is a genuine entanglement monotone, or equivalently, the quantity Fas per (1) is non-increasing

under LOCC on average and thus satisﬁes the inequality (8). In this work, we construct an explicit example to show that the

concurrence ﬁll can violate the inequality (8), thus implying it is not an entanglement monotone.

we address the fundamental issue about whether the con-

currence ﬁll is an entanglement monotone or not, and

also examine the LOCC-monotonicity of three tangle

and bipartite concurrence. Sec. IV proposes a genuine

multipartite entanglement monotone for the multipartite

quantum system, and the conclusion is made in Sec. V.

II. CONCURRENCE FILL

For an arbitrary pure three-qubit state |ψ⟩ABC shared

by three parties A, B, and C, denote the concur-

rence between the bipartition iand jk by C(|ψ⟩i|jk) =

p2[1 −Tr [ρ2

i]] , with ρi= Trjk[|ψ⟩ijk⟨ψ|] and i, j, k =

A, B, C. It follows from the relation C2(|ψ⟩i|jk)≤

C2(|ψ⟩j|ik)+C2(|ψ⟩k|ij ) [42] that these three squared one-

to-other concurrences can be geometrically interpreted as

the lengths of three sides of a triangle, which is called the

concurrence triangle ∆ABC as shown in Fig. 1. The con-

currence ﬁll is deﬁned as the square-root of the area of

this concurrence triangle [36]

F(|ψ⟩ABC ) = PABC

3(PABC −2C2(|ψ⟩A|BC ))

(PABC −2C2(|ψ⟩B|AC ))(PABC −2C2(|ψ⟩C|AB))

(1)

with the perimeter

PABC =C2(|ψ⟩A|BC ) + C2(|ψ⟩B|AC ) + C2(|ψ⟩C|AB).

(2)

It has been shown in [36] the concurrence ﬁll (1) is useful

to quantify genuine entanglement for three-qubit states

as it satisﬁes almost all of the necessary conditions to

be an entanglement measure. It is also noted that the

perimeter PABC (2), known as global entanglement [43,

44], is a feasible measure, but is not genuine in the sense

that it can be non-zero for certain biseparable states.

Further, denoting reduced states ρij = Trk[|ψ⟩ijk⟨ψ|]

and using the relations [38, 45]

C2(|ψ⟩i|jk) = 2 1−Tr ρ2

i=τ+C2(ρij ) + C2(ρik),

(3)

we are able to derive

PABC = 3τ+ 2 C2(ρAB) + C2(ρAC ) + C2(ρBC ),(4)

and hence

PABC −2C2(|ψ⟩i|jk) = τ+ 2C2(ρjk).(5)

Here τis the three-tangle which quantiﬁes the tripartite

entanglement [38], and the concurrence of the reduced

two-qubit state ρij is given by max{0, λ1−λ2−λ3−λ4}

with {λi}the singular values of X⊤(σy⊗σy)Xwith ρij =

XX†[46].

Substituting Eqs. (4)-(5) into (1) immediately leads to

F(|ψ⟩ABC )

=τ+ 2 C2(ρAB) + C2(ρAC ) + C2(ρBC )/3

(τ+ 2C2(ρAB))(τ+ 2C2(ρAC ))(τ+ 2C2(ρBC ))

4.(6)

It indicates that the concurrence ﬁll can be fully deter-

mined by the well-known three tangle and bipartite con-

currence. By introducing the concurrence of assistance

for a two-qubit state as Ca(ρ) = Tr |X⊤(σy⊗σy)X|=

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TripartiteentanglementmeasureunderlocaloperationsandclassicalcommunicationXiaozhenGe,1LijunLiu,2,∗andShumingCheng3,4,5,†1DepartmentofAppliedMathematics,TheHongKongPolytechnicUniversity,HongKong,China2CollegeofMathematicsandComputerScience,ShanxiNormalUniversity,Linfen041000,China3TheDepartmentofCont...

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Tripartite entanglement measure under local operations and classical communication Xiaozhen Ge1Lijun Liu2and Shuming Cheng3 4 5 1Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong China

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