Max-Planck-Institut f ur Mathematik in den Naturwissenschaften Leipzig - Characterizing multipartite entanglement by violation of CHSH inequalities
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Max-Planck-Institut
für Mathematik
in den Naturwissenschaften
Leipzig
Characterizing multipartite
entanglement by violation of CHSH
inequalities
by
Ming Li, Hui-Hui Qin, Chengjie Zhang,
Shu-Qian Shen, Shao-Ming Fei, and Heng Fan
Preprint no.: 44 2020Characterizing multipartite entanglement by violation of CHSH inequalities
Ming Li1,5 , Huihui Qin2 , Chengjie Zhang3 , Shuqian Shen1 , Shao-Ming Fei4,5 , and Heng Fan6
1
School of Science, China University of Petroleum, Qingdao 266580, China
2
Beijing Computational Science Research Center, Beijing 100193, China
3
School of Physical Science and Technology, Soochow University, Suzhou, 215006, China
4
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
5
Max-Planck-Institute for Mathematics in the Sciences, Leipzig 04103, Germany
6
Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Entanglement of high-dimensional and multipartite quantum systems offer promising perspectives
in quantum information processing. However, the characterization and measure of such kind of en-
tanglement is of great challenge. Here we consider the overlaps between the maximal quantum mean
values and the classical bound of the CHSH inequalities for pairwise-qubit states in two-dimensional
subspaces. We show that the concurrence of a pure state in any high-dimensional multipartite
system can be equivalently represented by these overlaps. Here we consider the projections of an
arbitrary high-dimensional multipartite state to two-qubit states. We investigate the non-localities
of these projected two-qubit sub-states by their violations of CHSH inequalities. From these vi-
olations, the overlaps between the maximal quantum mean values and the classical bound of the
CHSH inequality, we show that the concurrence of a high-dimensional multipartite pure state can
be exactly expressed by these overlaps. We further derive a lower bound of the concurrence for any
quantum states, which is tight for pure states. The lower bound not only imposes restriction on the
non-locality distributions among the pairwise qubit states, but also supplies a sufficient condition
for distillation of bipartite entanglement. Effective criteria for detecting genuine tripartite entan-
glement and the lower bound of concurrence for genuine tripartite entanglement are also presented
based on such non-localities.
PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w
Introduction.— Quantum entanglement has been one
of the most remarkable resource in quantum theory.
Multipartite and high-dimensional quantum entangle-
ment has become increasingly important for quantum
communication[1, 2]. Recently, a growing interest has
been devoted to investigation of such kind of quantum
resource [3–7]. In [8] the authors have derived a general
theory to characterize those high-dimensional quantum
states for which the correlations cannot simply be simu-
lated by low-dimensional systems.
The Bell inequalities[9] are of great importance for un-
derstanding the conceptual foundations of quantum the-
ory as well as for investigating quantum entanglement, as
Bell inequalities can be violated by quantum entangled
states. One of the most important Bell inequalities is FIG. 1: The concurrence of any two-qutrit pure state is equal
the Clauser-Horne-Shimony-Holt (CHSH) inequality[10] to the overlaps between the maximal quantum mean values
for two-qubit systems. In [11] Horodeckis have present- and the classical bound of the CHSH inequalities for nine pair
ed the necessary and sufficient condition of violating the of qubit states. Thus entanglement can simply be simulated
by the violation of CHSH inequalities of qubit pairs. The
CHSH inequality by an arbitrary mixed two-qubit state.
result holds for any pure states.
In [12, 13] we have discussed the trade-off relation of
CHSH violations for multipartite-qubit states based on
the norms of Bloch vectors.
A similar question to [8] is that can we simulate high- The second goal of this paper is to characterize genuine
dimensional quantum entanglement by the violations multipartite entanglement (GME)[14] in high dimension-
of CHSH inequalities for pairwise-qubit states in two- al quantum systems. As one of the important type of en-
dimensional subspaces? We present here a positive solu- tanglement, GME offers significant advantage in quan-
tion to this problem (see Fig. 1). For simplicity, we call tum tasks comparing with bipartite entanglement [15].
a “two-qubit” state, obtained by projecting high dimen- In particular, it is the basic ingredient in measurement-
sional d1 ⊗ d2 bipartite space to 2 ⊗ 2 subspaces, a qubit based quantum computation [16], and is beneficial in
pair in the following. various quantum communication protocols, including se-2
cret sharing [17, 18], extreme spin squeezing [19], high X t stands for the transposition of X.
sensitivity in some general metrology tasks [20], quan- Distribution of high-dimensional entanglement in qubit
tum computing with cluster states [21], and multiparty pairs.— Let us first consider general d×d bipartite quan-
quantum network [22]. Despite its significance, detecting tum systems in vector space HAB = HA ⊗ HB with di-
and measuring such kind of entanglement turn out to be mensions dim HA = dim HB = d, respectively. Denote
quite difficult. To certify GME, an abundance of linear by LA B
α and Lβ the generators of special orthogonal groups
and nonlinear entanglement witnesses [23–31], general- SO(d). Let a⃗i , b⃗j and σi s denote unit vectors and Pauli
ized concurrence for genuine multipartite entanglement matrices, respectively. Set ⃗σ = (σ1 , σ2 , σ3 ). We define
[32–35], and Bell-like inequalities [36], entanglement wit- β
the operators Aα i (resp. Bj ) from Lα (resp. Lβ ) by re-
nesses were derived (see e.g. reviews [14, 37]) and a char-
placing the four entries on the positions of the nonzero
acterisation in terms of semi-definite programs was devel-
2 rows and 2 columns of Lα (resp. Lβ ) with the corre-
oped [38, 39]. Nevertheless, the problem remains far from
being satisfactorily solved. sponding four entries of the matrix a⃗i · ⃗σ (resp. b⃗j · ⃗σ ),
β
In this paper we investigate entanglement by consider- and keeping the other entries of Aα i (resp. Bj ) zero. We
ing the overlap between the maximal quantum mean val- then define the following CHSH type Bell operator:
ue and the classical bound of the CHSH inequality. The β β β β
overlap is used to derive a lower bound of concurrence for Bαβ = Aα
1 ⊗ B1 + A1 ⊗ B2 + A2 ⊗ B1 − A2 ⊗ B2 . (2)
α α α
any multipartite and high dimensional quantum states, † A B † B
Set yαβ = Tr{(LA α ) Lα ⊗ (Lα ) Lβ ρ}. If yαβ ̸= 0,
which is tight for pure states. Thus we show that the B †
we define ραβ = yαβ Lα ⊗ Lα ρ(LA
1 A B
α ⊗ Lα ) , γαβ (ρ) =
concurrence in any quantum systems can be equivalently
1
represented by the violations of the CHSH inequalities for yαβ max Tr{Bαβ ρ}, where the maximum is taken over all
qubit pairs. The lower bound not only imposes restric- the Bell operators Bαβ of the form(2). Otherwise we set
tion on the non-locality distributions among qubit pairs, ραβ = 0 and γαβ (ρ) = 0. We further define that
but also supplies a sufficient condition for bipartite dis-
tillation of entanglement. Criteria for detection genuine Qαβ (ρ) = max{γαβ
2
(ρ) − 4, 0}, (3)
tripartite entanglement (GTE) and lower bound of GTE
concurrence are further presented by the overlaps. We which will be called the CHSH overlaps of ρ. If we can
then show by examples that these criteria and the low- find a certain pair of αβ such that Qαβ (ρ) > 0, then
er bound can detect more genuine tripartite entangled the two qudit state ρ ∈ HAB must be nonlocal as a Bell
states than the existing criteria do. inequality is violated.
We start with a short introduction of the generators For a bipartite pure state ρAB = |ψ⟩⟨ψ| ∈ HAB ,
of special orthogonal group SO(d) and the CHSH Bel- √ concurrence is defined by [40, 41] C(|ψ⟩) =
the
l inequalities. The generators of SO(d) can be intro- 2 (1 − Trρ2A ), where ρA = TrB ρAB∑is the reduced den-
duced according to the transition-projection operators sity matrix. For a mixed state ρ = i pi |ψi ⟩⟨ψi |, pi ≥ 0,
∑
Tst = |s⟩⟨t|, where |s⟩, s = 1, · · · , d, are the orthonor- i pi = 1, the
∑ concurrence is defined as the convex-roof:
mal eigenstates of a linear Hermitian operator on Hd . C(ρ) = min i pi C(|ψi ⟩), minimized over all possible pure
Set Pst = Tst − Tts , where 1 ≤ s < t ≤ d. We get state decompositions.
a set of d(d−1) operators that generate SO(d). Such We are ready to represent concurrence in high dimen-
2
kind of operators(which will be denoted by Lα , α = sional systems by the CHSH overlaps Qαβ (|ψ⟩).
1, 2, · · · , d(d−1)
2 ) have d − 2 rows and d − 2 columns with Theorem 1 For any two qudit pure quantum state |ψ⟩ ∈
zero entries. For two-qubit quantum systems, the CHSH HAB , we have
Bell operators[10] are defined by
1∑ 2
C 2 (|ψ⟩) = yαβ Qαβ (|ψ⟩). (4)
ICHSH = A1 ⊗ B1 + A1 ⊗ B2 + A2 ⊗ B1 − A2 ⊗ B2 , (1) 4
αβ
∑
3 ∑
3
where Ai = a⃗i · ⃗σA = aki σA
k
, Bj = b⃗j · ⃗σB = blj σB
l
, Proof. For any two-qubit pure state |ϕ⟩ =
∑2
i,j=1 aij |ij⟩, the concurrence C(|ϕ⟩) and Qαβ (|ϕ⟩) are
k=1 l=1
a⃗i = (a1i , a2i , a3i ) and b⃗j = (b1j , b2j , b3j ) are real unit vec- preserved under any local unitary operations. Thus to
1,2,3
tors satisfying |a⃗i | = |b⃗j | = 1, i, j = 1, 2, σA/B are prove the theorem, we just∑2 need to consider∑
the Schmidt
2
Pauli matrices. The CHSH inequality says that if there decomposition of |ϕ⟩ = i=1 λi |ii⟩, where i=1 λ2i = 1.
16λ21 λ22
exist local hidden variable models to describe the sys- One computes C 2 (|ϕ⟩) = 4λ21 λ22 , and Q11 (|ϕ⟩) = (λ21 +λ22 )2
.
tem, the inequality |⟨ICHSH ⟩| ≤ 2 must hold. For any ∑2
By i=1 λ2i = 1, we get
two-qubit state ρ, one defines the matrix X with en-
tries xkl = Tr{ρσk ⊗ σl }, k, l = 1, 2, 3. Horodeckis 1
have computed in [11] the√maximal quantum mean value C 2 (|ϕ⟩) = Q11 (|ϕ⟩). (5)
4
γ = max |⟨ICHSH ⟩ρ | = 2 τ1 + τ2 , where the maximum
is taken for all the CHSH Bell operators ICHSH in Eq.(1), Then we consider two-qudit pure state |ψ⟩ =
∑d
i,j=1 aij |ij⟩, C (|ψ⟩) can be equivalently represented
2
τ1 , τ2 are the two greater eigenvalues of the matrix X t X,3
by [41, 44] directly generalized to multipartite case. An N -partite
pure state in H1 ⊗ H2 ⊗ · · · ⊗ HN is generally of the form,
∑ ∑
d ∑
d
C 2 (|ψ⟩) = |Cαβ (|ψ⟩⟨ψ|)|2 = 4 |aik ajl −ail ajk |2 , ∑
d
αβ i4
function of ρ and the summation of convex functions is
still a convex function, we have
∑ ∑
X +Y +Z ≤ pα (Xα + Yα + Zα ) ≤ 8 pα = 8. (14)
α α
The GTE concurrence for tripartite quantum systems
defined below is proved to be a well defined measure[32,
33]. For a pure state |ψ⟩ ∈ H123 , the GTE concurrence
is defined by
√
CGT E (|ψ⟩) = min{1 − Tr(ρ21 ), 1 − Tr(ρ22 ), 1 − Tr(ρ23 )},
FIG. 2: The concurrence of any three-qubit pure state is given where ρi is the reduced matrix for the ith subsystem.
by the CHSH overlaps of six pairs of qubit states. For mixed state ρ ∈ H123 , the GTE concurrence is then
defined by the convex roof
∑
|ψα ⟩ ∈ H123 are normalized pure states. ⟩ If all |ψ ⟩ α⟩ CGT E (ρ) = min pα CGT E (|ψα ⟩). (15)
are biseparable,
⟩ namely,
⟩ either |ψ α ⟩ = φ 1
α ⊗ φ 23
α or {pα ,|ψα ⟩}
⟩ ⟩ ⟩
|ψβ ⟩ = φβ ⊗ φβ or |ψγ ⟩ = φγ ⊗ φγ , where φγ
2 13 3 12 i
⟩ The minimum is taken over all pure ensemble decompo-
and φij γ denote pure states in Hid and Hid ⊗ Hjd respec- sitions of ρ. Since one has to find the optimal ensemble
tively, then ρ is said to be bipartite separable. Otherwise, for the minimization, the GTE concurrence is hard to
ρ is called genuine tripartite entangled. compute. In the following we present a lower bound of
1|23
For any ρ ∈ H123 , we define X = maxαβ Qαβ , Y = GTE concurrence in terms of Qαβ s.
2|13 3|12
maxαβ Qαβ and Z = maxαβ Qαβ .
Theorem 5 Let ρ ∈ H123 be a tripartite qudits quantum
Theorem 3 For any pure tripartite state |ψ⟩, state. Then one has
min{X, Y, Z} > 0 holds if and only if |ψ⟩ is gen- √ √
uine tripartite entangled. 1 ∑ ∑ p 2 p 2 d−1
CGT E (ρ) ≥ √ (yαβ ) Qαβ (ρ) − ,
6 2 p 3 d
αβ
Proof. According to the definition,⟩any bi-separable
⟩ (16)
pure⟩ state |ψ⟩
⟩ must be either
⟩ |ψ⟩⟩ = φ1 ⊗ φ23 or |ψ⟩ =
where the partitions p ∈ {1|23, 2|13, 3|12}.
φ2 ⊗ φ13 or |ψ⟩ = φ3 ⊗ φ12 . On the contrary, if |ψ⟩
is GTE (not bi-separable), then it must be not in any bi- Proof. We start the proof with a pure state. Let
separable form, which can be represented by violating all ρ = |ψ⟩⟨ψ| ∈ H123 be a pure quantum state. From the
the CHSH inequalities for any qubit pairs of |ψ⟩. This can result in Theorem 1, we have
be further represented by min{X, Y, Z} > 0 according to
√
the definition of X, Y, and Z. 1 ∑ 1|23 1|23 1
The sufficient and necessary condition for detecting 1 − trρ21 = √ ( (yαβ )2 Qαβ (|ψ⟩)) 2
2 2 αβ
GTE in Theorem 3 can be generalized to any pure multi-
partite quantum states. In the following we derive a suf-
ficient condition to detect GTE for any tripartite mixed and
quantum states. √ √
d−1
1 − trρk ≤
2 , k = 2, 3.
Theorem 4 If ρ ∈ H123 is bipartite separable, then d
Therefore,
X +Y +Z ≤8 (13)
√ √∑ √
always holds. Thus if (13) is violated, then ρ is of GTE. 1 ∑ 2 d−1
1− trρ21 ≥ √ p 2 p
(yαβ ) Qαβ (ρ) − .
6 2 p 3 d
αβ
Proof. ⟩ For any⟩ bipartite separable pure state, say,
|ψ⟩ = φ1 ⊗ φ23 , one gets X = 0, Y ≤ 4 and Z ≤ 4, Similarly, we get
which proves (13).
√ √∑ √
Now consider a mixed bipartite ∑ separable state 1 ∑ 2 d−1
with
∑ ensemble decomposition ρ = pα |ψα ⟩ ⟨ψα | with 1− trρ2k ≥ √ p 2 p
(yαβ ) Qαβ (ρ) − ,
6 2 p 3 d
pα = 1. By noticing that all X, Y and Z are convex αβ5
∑
where k = 2, 3. Then according to the definition of GME where
√ we have used x qx = 1 and inequality
∑ ∑ 2 √∑ ∑
concurrence, we derive
j xij ≥
2
i j( i xij ) .
√ √
1 ∑ ∑ p 2 p 2 d−1 Let us now consider an example to illustrate further
CGT E (|ψ⟩) ≥ √ (yαβ ) Qαβ (ρ) − .
6 2 p 3 d the significance of our result for detection of GTE.
αβ
(17) Example 1: Consider the quantum state ρ ∈ H1d ⊗
Now we consider a mixed state ρ ∈ ∑ H123 with the H2d ⊗ H3d ,
optimal
∑ ensemble decomposition ρ = x qx |ψx ⟩⟨ψx |,
x q x = 1, such that the GTE concurrence attains its
minimum. By (17) one gets 1−x
σ(x) = x|ψ⟩⟨ψ| + I, (18)
∑ d2
CGT E (ρ) = qx CGM E (|ψx ⟩)
x
√∑ √ ∑
d
1 ∑ 2 d−1 where |ψ⟩ = √1 |iii⟩ and I stands for the identity
≥ √ qx p 2 p
(yαβ (|ψx ⟩)) Qαβ (|ψx ⟩) − d
i=1
6 2 p,x 3 d operator.
αβ
√ √ By the positivity of X + Y + Z − 8, we get the ranges
1 ∑ ∑ p 2 p 2 d−1
≥ √ (yαβ ) Qαβ (ρ) − , of x for different d such that σ(x) is GTE (see table I).
6 2 p 3 d
αβ
TABLE I: Detection of GTE of σ(x) by Theorem 4 (Range 1), Theorem 5 (Range 2), Theorem in [50] (Range 3), Theorem 1
in [24, 28] (Range 4).
Dimension d=2 d=3 d=4
Range 1 x > 0.839708 x > 0.699544 x > 0.567035
Range 2 x > 0.788793 x > 0.731621 x > 0.705508
Range 3 x > 0.8532 x > 0.83485 x > 0.82729
Range 4 x > 0.87 x > 0.89443 x > 0.91287
The data in Table I show that Theorem 4 and 5 in this entangled[49]. Thus if
letter, independently, detect more genuine tripartite en-
tangled states than that in [50](by the lower bound of max Qpαβ (ρ⊗n ) > 0 (19)
multipartite concurrence), [28] and in [24](by the corre- αβ
lation tensor norms).
for a certain partition p, then there exists one subma-
The CHSH overlaps and distillation of entanglemen- trix of matrix ρ⊗n , which is entangled in a 2 × 2 space.
t.— The CHSH overlaps defined in (3) can be also ap- Hence we get that ρ is bipartite distillable in terms of
plied to distillation of entanglement. In [46] Dür has bipartition p. The constraint (19) is equivalent to the
shown that there exist some multi-qubit bound entan- strict positivity of the lower bound in (12). Note that
gled (non-distillable) states that violate a Bell inequali- maxαβ Qpαβ (ρ⊗n ) is generally not an invariant under lo-
ty. Acı́n further proves in [47] that for all states violating cal unitary operations on the state ρ. It is helpful to se-
this inequality there is at least one splitting of the par- lect proper local unitary operations to enhance the value
ties into two groups such that some pure state entangle- of maxαβ Qpαβ (ρ⊗n ) from 0 to a positive number. Since
ment can be distilled under this partition. The relation the separability is kept invariant under local unitary op-
between violation of Bell inequalities and bipartite dis- erations, we have that if maxU1 ,U2 ,··· ,Un maxαβ Qαβ (U1 ⊗
tillability of multi-qubit states is further studied in [48]. U2 ⊗ · · · ⊗ Un ρn U1† ⊗ U2† ⊗ · · · ⊗ Un† ) > 0 hold for proper
The lower bound (12) has also a close relationship with unitary Ui s, i = 1, .., n, then ρ is entangled and bipartite
bipartite distillation of any multipartite and high dimen- distillable.
sional states. Note that a density matrix ρ is distillable Example 2: Consider the quantum state ρ ∈ Hd1 ⊗Hd2 ,
if and only if there are some projectors A, B that map
high-dimensional spaces to two-dimensional ones and a 1−x
certain number n such that the state A ⊗ Bρ⊗n A ⊗ B is ρ(x) = x|ψ⟩⟨ψ| +
d2
I, (20)6
∑
d
1-distillable (see table I, Range 1). Range 2 is derived
where |ψ⟩ = √1 |ii⟩ and I stands for the identity op-
d
i=1 by the reduction criterion (RC), as violation of RC is a
erator. sufficient condition of entanglement distillation[52, 53].
By the positivity of maxαβ Qαβ (ρ), one computes the
ranges of x for different d such that ρ is non-local and
TABLE II: Distillation of non-locality and entanglement for ρ(x) in Example 2:
Dimension d=2 d=3 d=4 d=5 d=6 d=7
Range 1 x > 0.707107 x > 0.616781 x > 0.546918 x > 0.491272 x > 0.445903 x > 0.408205
Range 2 x > 0.33333 x > 0.25 x > 0.2 x > 0.16667 x > 0.142857 x > 0.125
Example 3: Consider the quantum state ρ ∈ Hd1 ⊗ operator.
Hd2⊗ Hd3 ,
1−x
σ(x) = x|ψ⟩⟨ψ| + I, (21) To check the bipartite 1-distillability of σ(x), we com-
d2 pare maxαβ Qpαβ (ρ) with 0 for p = 1|23, 2|13, and 3|12.
∑
d One computes the ranges of x for different d such that ρ
where |ψ⟩ = √1
d
|iii⟩ and I stands for the identity is 1-distillable (see table II, Range 1).
i=1
TABLE III: Bipartite 1-distillation of entanglement for σ(x) in Example 3:
Dimension d=2 d=3 d=4 d=5
Range x > 0.54692 x > 0.34917 x > 0.23182 x > 0.16188
Conclusions and remarks.— In summary we have con- measure the multipartite concurrence in these system-
sidered the CHSH overlaps for quantum states. It has s and to investigate the roles played by the multipar-
been shown that the concurrence of any multipartite and tite concurrence in these quantum information process-
high dimensional pure states can be equivalently repre- ing. Our approach of the CHSH overlaps of qubit pairs
sented by the CHSH overlaps of a series of “two-qubit” can also be employed to investigate the distributions of
states. Based on the overlaps sufficient condition for dis- other quantum correlations in high dimensional systems.
tillation of entanglement have been obtained. As another Another important question that needs further discus-
application of the CHSH overlaps, we have further pre- sion is to find a criterion that discriminates W state and
sented criteria for detecting GME and lower bound of GHZ state, as GTE is a common property of W state and
GME concurrence for tripartite quantum systems. For GHZ state, but there is no local unitary transformation
tripartite pure states, a sufficient and necessary condition to relate them.
is derived to detect GME, while for tripartite mixed s- Acknowledgments This work is supported by the NS-
tates, we have obtained effective sufficient conditions and FC No. 11775306, 11701568, 11701128 and 11675113;
lower bounds for GME concurrence. An important ques- the Fundamental Research Funds for the Central U-
tion that needs further discussion is to find a criterion niversities Grants No.17CX02033A, 18CX02023A and
that discriminates W state and GHZ state. 19CX02050A; the Shandong Provincial Natural Science
Recently high dimensional bipartite systems like in N- Foundation No. ZR2016AQ06, ZR2017BA019, and Key
MR and nitrogen-vacancy defect center have been suc- Project of Beijing Municipal Commission of Education
cessfully used in quantum computation and simulation under No. KZ201810028042.
experiments[51]. Our results present a plausible way to7
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