arXiv:2206.07731v1 [quant-ph] 15 Jun 2022

Controlling gain with loss: Bounds on localizable entanglement in multi-qubit systems

Jithin G. Krishnan, Harikrishnan K. J. and Amit Kumar PalDepartment of Physics, Indian Institute of Technology Palakkad, Palakkad 678 623, India

(Dated: June 17, 2022)

We investigate the relation between the amount of entanglement localized on a chosen subsystem of a multi-qubit system via local measurements on the rest of the system, and the bipartite entanglement that is lost duringthis measurement process. We study a number of paradigmatic pure states, including the generalized GHZ, thegeneralized W, Dicke, and the generalized Dicke states. For the generalized GHZ and W states, we analyticallyderive bounds on localizable entanglement in terms of the entanglement present in the system prior to themeasurement. Also, for the Dicke and the generalized Dicke states, we demonstrate that with increasing systemsize, localizable entanglement tends to be equal to the bipartite entanglement present in the system over a specificpartition before measurement. We extend the investigation numerically in the case of arbitrary multi-qubit purestates. We also analytically determine the modification of these results, including the proposed bounds, insituations where these pure states are subjected to single-qubit phase-flip noise on all qubits. Additionally, westudy one-dimensional paradigmatic quantum spin models, namely the transverse-field XY model and the XXZmodel in an external field, and numerically demonstrate a quadratic dependence of the localized entanglementon the lost entanglement. We show that this relation is robust even in the presence of disorder in the strength ofthe external field.

I. INTRODUCTION

In the last three decades, entanglement [1, 2] have beenestablished as the key resource in quantum information pro-cessing tasks, including quantum teleportation [1, 3, 4], super-dense coding [1, 5–7], and quantum cryptography [8, 9]. Con-cepts related to entanglement theory have also been used inareas that are seemingly different from quantum informationtheory, such as in probing gauge-gravity duality [10–13], inunderstanding time as an emergent phenomena from entan-glement [14–16], and even in studying systems like photo-synthetic complexes [17] that are important from biologi-cal point of view. These have motivated enormous experi-mental advancements in creating and manipulating entangledstates in the laboratory using various substrates, namely, pho-tons [18–20], trapped ions [21–23], cold atoms [24–26], su-perconducting qubits [27, 28], and nuclear magnetic resonancemolecules [29]. Moreover, quantum many-body systems [30]have emerged as the natural choice for implementing quantuminformation processing tasks, and the necessity of studying theentanglement properties of these systems has also been real-ized [31, 32].

Entanglement over a subsystem A of a composite quan-tum system in the state ρ can be quantified by computing anappropriate entanglement measure E over the reduced stateρA = TrB [ρ] of the subsystem A, where B is the rest ofthe system [1, 2]. While this approach has been successfulin a wide variety of multiparty quantum states [1, 2, 33], thereexist states like N -qubit Greenberger-Horne-Zeilinger (GHZ)states [34], graph states [35], and stabilizer states in quantumerror correcting codes [36, 37] for which the partial trace-based avenue may lead to vanishing entanglement measure onthe state ρA. In such situations, one may take a measurement-based approach, where non-zero bipartite or multipartite en-tanglement can be localized on the subsystem A in the post-measured state of the system by performing measurements onB [38]. In the case of a multi-qubit system, this leads to thedefinition of the localizable entanglement [39–41], defined asthe maximum average entanglement localized over A via local

single-qubit projection measurements on all qubits in B, givenby

〈EA〉 = max∑k

pkE(ρkA). (1)

Here, k labels the measurement outcomes corresponding tothe post-measured states ρk occurring with probability pk(∑k pk = 1), where ρkA = TrB [ρk]. Depending on the pos-

sible partitions in A, the entanglement measure E computedover A post-measurement can be either a bipartite [39–41], ora multipartite [42] measure. Apart from successfully charac-terizing entanglement in GHZ and GHZ-like states such as thegraph [35] and the stabilizer states [43, 44], localizable en-tanglement and related ideas have been immensely useful indefining the correlation length in one-dimensional (1D) quan-tum spin models [39–41, 45], in characterizing quantum phasetransitions in the cluster-Ising [46, 47] and cluster-XY mod-els [48] in terms of entanglement, and in entanglement perco-lation through quantum networks [49].

The measurement on the subsystem B completely decou-ples B from A, leading to

ρ→ ρ = ρA ⊗ ρB , (2)

implying a complete loss in entanglement over all qubits be-longing to the different subsystems A and B. A natural ques-tion that arises is whether and how the entanglement 〈EA〉 lo-calized on A via measurements on B depends on the entangle-ment that is lost during the same measurement process. Morespecifically, considering a bipartition A1 : A2 of A (i.e., anoverall tripartition A1 : A2 : B of the multi-qubit system, seeFig. 1), and a bipartite entanglement measure E, we ponderon the following question: Does any relation exist between thebipartite entanglement 〈EA〉 ≡ 〈EA1A2

〉 localized over thesubsystem A via single-qubit projection measurements on allqubits in B, and the entanglement over different bipartitionsprior to the measurement, namely, EA1A2:B , EA1:A2B , andEA2:A1B , that are lost during the measurement process lead-ing to 〈EA1A2

〉? On one hand, answer to this question maygive rise to constraints on the entanglement localizable over

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A2

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B

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A1

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B

FIG. 1. Consider a multi-qubit system divided into three subsystems, A1, A2, and B, with bipartite entanglement EA1A2:B , EA1:A2B , andEA2:A1B over different bipartitions. Measurement on all qubits in B, denoted byMB , decouples the subsystem B from the qubits in A ≡A1A2.

the subsystem of a multi-qubit system in terms of the entangle-ment present in the system prior to measurement – a situationthat is of fundamental interest from the perspective of completecharacterization of the system via entanglement. On the otherhand, such a study may also aid in estimating the localizableentanglement prior to the measurement via the information onthe bipartite entanglement present in the system. The latter isadvantageous from a practical point of view, specially in situa-tions where performing the measurements and optimizing overall possible measurements may turn out to be difficult.

A few results exist in this direction. Note that the mono-tonicity of E [1, 33, 50–52] implies that

〈EA1A2〉 ≤ min [EA1:A2B , EA2:A1B ] . (3)

It has been shown that the inequality (3) can be tightenedto an equality in the case of asymptotic pure state distilla-tion [53, 54]. At the single copy level, it has also been shownthat for all three-qubit pure states with each of A1, A2, and Bbeing a qubit, the inequality (3) becomes an equality via somemeasurement on B for a specific choice of entanglement mea-sure [55]. In the case of multi-qubit pure states, investigationon the relation between localizable multipartite entanglement〈EA〉 and the multipartite entanglement, as quantified by themultiparty entanglement measure E, over the state prior to themeasurement has also been made [42]. However, we are stillfar from a systematic and complete understanding of the prob-lem in the case of arbitrary multi-qubit quantum states.

In this paper, we study the interplay between 〈EA1A2〉,EA1:A2B , EA2:A1B , and EA1A2:B in multi-qubit systems,where each of the partitions A1, A2, and B may consist ofmultiple qubits, choosing negativity [56–60] as the entangle-ment measure. We start with the investigation of a numberof paradigmatic N -qubit pure states, including the general-ized GHZ states [34, 61], the generalized W states [61, 62],Dicke states [63–66], and generalized Dicke states [42], andanalytically derive bounds of 〈EA1A2

〉 in terms of EA1:A2B ,EA2:A1B , and EA1A2:B . In the cases of arbitrary pure states ofN -qubits, we numerically investigate whether it is always pos-sible to exceed the loss in bipartite entanglement via localiza-tion performed through measurement. We extend our investi-gation to the pure states subjected to the single-qubit phase flipnoise [67, 68] of Markovian [69] and non-Markovian [70–72]

type, and discuss the modifications of the bounds obtained forpure states due to the presence of noise. We also look into theground states obtained from one-dimensional (1D) interactingquantum spin models, both in the presence and absence of dis-order [73]. More specifically, we consider the 1D transverse-field XY model [74–79] and the 1D XXZ model in an externalfield [80–85], where disorder can be present in the strength ofthe field. We numerically demonstrate a quadratic dependenceof the localizable entanglement on the entanglement lost dur-ing measurement in the ground states of the ordered and thedisordered models, and demonstrate that the relation is robustagainst the presence of disorder in the field-strength.

The rest of the paper is organized as follows. In Sec. II,we formally define the localizable entanglement, and computeit, along with the bipartite entanglements present in the un-measured states, in the case of paradigmatic multi-qubit purestates. We also present the numerical data corresponding to ar-bitrary N -qubit pure states, and discuss the implications of thedata. The modifications of the results for the pure state dueto subjecting the states to single-qubit Markovian and non-Markovian phase flip channels are discussed in Sec. III. Thestudy of the localizable and the lost entanglement in the groundstates of ordered and disordered 1D quantum spin models canbe found in Sec. IV. Sec. V presents the outlook and conclud-ing remarks.

II. MULTI-QUBIT PURE STATES

In this section, we consider a number of paradigmatic multi-qubit pure states, and investigate the possible correlation be-tween the localizable entanglement and the bipartite entangle-ment that is lost due to measurement. Note that the maximumvalue of 〈EA1A2

〉 as well as the optimal measurement basis onB providing this value depends on the choice of E. For theresults reported in this paper, we choose negativity [56–60] asthe entanglement measure. See Appendix A for a brief defini-tion.

Let us consider an N -qubit system, S, where the qubits arelabelled as 1, 2, · · · , N . As discussed in Sec. I, we considera tripartition A1 : A2 : B of the system, and perform single-qubit rank-1 projection measurements on all qubits inB. With-

3

out any loss in generality, we assume thatB holds n(< N−1)qubits, and A = A1 ∪ A2 consists of the rest N − n qubits.We label the qubits in B as 1, 2, · · · , n, and the qubits in A asn+ 1, n+ 2, · · · , N − 1, N . In this situation, 〈EA1A2〉 (Eq. 1)takes the form

〈EA1A2〉 = max

2n−1∑k=0

pkE(ρkA1A2), (4)

where ρkA1A2= TrB

[ρk], ρk =

[MkρMk†] /pk, and pk =

Tr[MkρMk†], with Mk being the measurement operation

corresponding to the measurement outcome k, and the max-imization is performed over the set of all possible single-qubitrank-1 projection measurements on all qubits in B. Note thatthe maximum value of 〈EA1A2

〉 can not exceed the maximumvalue of the chosen entanglement measure, Emax, implying〈EA1A2

〉 ≤ Emax.The measurement operators {Mk} can be written as Mk =

IA1A2⊗P kB , with the projectors on the qubits inB correspond-

ing to the measurement outcome k given by P kB = |bk〉〈bk|,where ki = 0, 1 and k ≡ k1k2 · · · kn is a multi-index. Here,|bk〉 = ⊗∀i∈B |bki〉, and

|b0〉i = cosθi2|0〉+ eiφi sin

θi2|1〉 ,

|b1〉i = sinθi2|0〉 − eiφi cos

θi2|1〉 , (5)

on all qubits i ∈ B, where θi, φi ∈ R, 0 ≤ θi ≤ π,0 ≤ φ ≤ 2π. Unless otherwise stated, we maintain thesenotations in all subsequent calculations. Also, we use A andA1A2 interchangeably, since A ≡ A1 ∪A2 = A1A2.

A. Generalized GHZ states

We start with the N -qubit generalized GHZ (gGHZ)state [34], given by

|gGHZ〉 = a0 |0〉⊗N + a1 |1〉⊗N , (6)

where a0 and a1 are complex numbers, i.e., a0, a1 ∈ C, andthe state is normalized, implying |a0|2 + |a1|2 = 1. For thegGHZ state, we present the following proposition.� Proposition I. For any tripartition A1 : A2 : B of an N -qubit gGHZ state,

〈EA1A2〉 = EA1A2:B = EA1:A2B = EA2:A1B . (7)

Proof. Partial transposition of ρ = |gGHZ〉〈gGHZ| with re-spect to the subsystems B leads to

ρTB = |a0|2(|0〉〈0|)⊗N + |a1|2(|1〉〈1|)⊗N+a0a

∗1(|0〉〈1|)⊗N−n(|1〉〈0|)⊗n

+a∗0a1(|1〉〈0|)⊗N−n(|0〉〈1|)⊗n, (8)

with non-zero eigenvalues |a0|2, |a1|2,±|a0||a1|. Thereforethe entanglement between partition A1A2 and partition B, asquantified by negativity, is given by

EA1A2:B = 2|a0|√

1− |a0|2. (9)

We now compute localizable entanglement, 〈EA1A2〉, forthe gGHZ state. Application of the measurement operatorMk corresponding to the measurement outcome k on |gGHZ〉leads to

Mk|gGHZ〉 = |gGHZk〉A1A2⊗ |bk〉B (10)

where

|gGHZk〉 =1√pk

(a0f

k0 |0〉⊗n + a1f

k1 |1〉⊗n

),

pk = |a0|2|fk0 |2 + |a1|2|fk1 |2. (11)

Here,

fk0 =∏i∈B

fki0 =∏i∈B〈bki |0〉i,

fk1 =∏i∈B

fki1 =∏i∈B〈bki |1〉i (12)

are functions of 2n real parameters {θi, φi}, i = 1, · · · , n.The negativity, EkA1:A2

, can now be computed for each post-measured states |gGHZk〉 on A1A2 as

EkA1:A2=

2|a0|√

1− |a0|2|fk0 ||fk1 |pk

. (13)

The localizable entanglement across the bipartition A1 : A2 ofA is, therefore,

〈EA1:A2〉 = 2|a0|

√1− |a0|2

[max

2n−1∑k=0

|fk0 ||fk1 |].

(14)

In order to perform the maximization, we note that |fk0,1|,and subsequently pk andEkA1:A2

, are independent of {φi}, i =1, · · · , n, thereby reducing the maximization problem to oneinvolving n real parameters, {θi}, i = 1, · · · , n. Moreover,since |fk0,1| ≥ 0, and

|fk0 ||fk1 | =1

2n

∏∀i∈B

sin θi, (15)

one obtains

maxθi

2n−1∑k=0

|fk0 ||fk1 | =1

2n

2n−1∑k=0

maxθi

∏∀i∈B

sin θi

= 1, (16)

where the maximization takes place for θi = π2 ∀i ∈ B, im-

plying that a σx measurement on all qubits in B is optimal forobtaining the maximum 〈EA1A2〉 as

〈EA1A2〉 = 2|a0|

√1− |a0|2. (17)

It is easy to see from the symmetry of the gGHZ state thatEA1A2:B = EA1:A2B = EA2:A1B , leading to

〈EA1A2〉 = EA1A2:B = EA1:A2B = EA2:A1B . (18)

Hence the proof.

4

Therefore, in the case of the N -qubit gGHZ states, it is notpossible to exceed the bipartite entanglement present in thestate prior to the measurement via localizable entanglement.Note also that for the N -qubit gGHZ state, (3) is an equality.

B. Generalized W states

Next, we focus on the N -qubit generalized W states [61,62], given by

|gW〉 =

N∑i=1

ai |0〉⊗(i−1) |1〉i |0〉⊗(N−i)

, (19)

where ai ∈ C ∀i ∈ {1, 2, · · · , N}, satisfying the normaliza-tion condition

∑Ni=1 |ai|2 = 1. Let us first consider a subset

of the N -qubit gW states given in Eq. (19) where the coeffi-cients ai are real, i.e., ai ∈ R. We first focus on the correlationbetween 〈EA1A2

〉 and EA1A2:B ≡ EAB . Negativity over anybipartition AB of the system is given by1

EA1A2:B = 2

∣∣∣∣∣∣[(

n+m∑i=n+1

a2i +

N∑i=n+m+1

a2i

)n∑i=1

a2i

] 12

∣∣∣∣∣∣ ,(20)

where we have assumed that qubits 1, 2, · · · , n constitute thesubsystemB, and qubits n+1, n+2, · · · , N form the subsys-tem B. On the other hand, computation of 〈EA1A2

〉 involvesapplication of Mk, k = 0, 1, · · · , 2n − 1, on the n qubits in Bof |gW〉 (see Sec. II A), leading to

Mk|gW〉 = |bk〉B ⊗ |ψk〉A1A2(21)

with

|ψk〉 = ck0 |0〉⊗(N−n)

+

N−n∑i=1

cki |0〉⊗(i−1) |1〉i |0〉⊗(N−n−i)

, (22)

where ck0 = fk0 /√pk, and cki = fki /

√pk, i = 1, 2, · · · , n,

with pk being the probability of obtaining the measurementoutcome k, such that

∑N−ni=0 |cki |2 = 1, ensuring normal-

ization. Note further that among the coefficients cki , i =0, 1, · · · , N − n, only ck0 is complex (see Appendix B for ex-plicit examples with the cases of single- and two-qubit mea-surements). Using this, and writing negativity in the state |ψk〉over the bipartition A1 : A2 can be written for all k as2

EkA1A2= 2

∣∣∣∣∣∣[(

n+m∑i=n+1

a2i

)N∑

n+m+1

a2i

] 12

∣∣∣∣∣∣ . (23)

1 For gW states with real coefficients, this can be obtained by observing thepatterns of the negative eigenvalues of the partially transposed density ma-trix of the smaller systems. Eq. (20) is also verified numerically for largegW states with real coefficients as well as complex coefficients, where inthe case of the latter, a2i is replaced with |ai|2.

2 For gW states with real coefficients, this can be obtained following the sameapproach as for Eq. (20). For gW states with complex coefficients, a2i s arereplaced by |ai|2s.

Here, we have assumed, without any loss in generality, thatthe subsystem A1 (A2) is constituted of the qubits n + 1, n +2, · · · , n + m (qubits n + m + 1, n + m + 2, · · · , N ). SinceEA1A2 = E(ρkA1A2

) is independent of θ, φ, localizable entan-glement over the subsystem A ≡ A1A2 is given by

〈EA1A2〉 = EkA1A2(24)

We now propose the following for an N -qubit gW state withreal coefficients.� Proposition II. In the space (EA1A2:B , 〈EA1A2

〉), the lo-calizable entanglement 〈EA1A2

〉 of anN -qubit normalized gWstate with real coefficients is upper bounded by the line

〈EA1A2〉 =1

2

(1 +

√1− E2

A1A2:B

), (25)

where EA1A2:B is the bipartite entanglement over the bipar-tition A1A2 : B in the state prior to measurement on all thequbits in B.

Proof. For ease of calculation, we focus on 〈EA1A2〉2 givenby (using Eq. (23) and normalization of the gW state)

〈EA1A2〉2 = 4

(1−

n∑i=1

a2i −n+m∑i=n+1

a2i

)n+m∑i=n+1

a2i , (26)

which, for fixed a =∑ni=1 a

2i , is a single-parameter function

F (x) = 4x(1 − a − x) of x, where x ≡ ∑n+mi=n+1 a

2i . The

function F (x) has the maximum value (1 − a)2, occurring atx = (1 − a)/2. This implies

∑n+mi=n+1 a

2i = (1 − a)/2 for

the maximum of 〈EA1A2〉 = 1 − a. Note also from Eq. (20)

thatEA1A2:B = 2√a(1− a). Eliminating a and subsequently

solving for 〈EA1A2〉, one obtains Eq. (25).

The following Corollaries can be obtained straightforwardlyfrom Proposition I.�Corollary II.1. The family of gW states with real coefficientsthat satisfy Eq. (25) are given by

n+m∑i=n+1

a2i =

N∑i=n+m+1

a2i =1

2

(1−

n∑i=1

a2i

). (27)

Proof. For a fixed value of∑ni=1 a

2i , the maximization condi-

tion for 〈EA1A2〉 is given by

∑n+mi=n+1 a

2i = 1

2

(1−∑n

i=1 a2i

).

From the normalization of the gW states, Eq. (27) follows.

�Corollary II.2. For the family of gW states given by Eq. (27),EA1:A2B = EA2:A1B .

Proof. Similar to Eq. (20), entanglement over the bipartitionsA2 : A1B and A1 : A2B can be written as

EA1:A2B = 2

∣∣∣∣∣∣[(

n∑i=1

a2i +

N∑i=n+m+1

a2i

)n+m∑i=n+1

a2i

] 12

∣∣∣∣∣∣ ,(28)

5

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min{EA1:A2B , EA2:A1B}

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1A

2i

B.


1A

2i

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EA1A2:B

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L1

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L2

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L1


L2

FIG. 2. Generalized W states. Scatter plot of a sample of 107 Haar-uniformly generated 3-qubit gW states on the A. (EA1A2:B , 〈EA1A2〉)plane and the B. (min{EA1:A2B , EA2:A1B}, 〈EA1A2〉) plane. The lines L1 and L2 on the (EA1A2:B , 〈EA1A2〉) plane represent respectivelythe family of states following Eq. (25), andEA1A2:B = 1. On the other hand, the linesL1 andL2 on the (min{EA1:A2B , EA2:A1B}, 〈EA1A2〉)plane represent the family of states obeying Eqs. (30) and (31) respectively. All the axes in all figures are dimensionless.

and

EA2:A1B = 2

∣∣∣∣∣∣[(

n∑i=1

a2i +

n+m∑i=n+1

a2i

)N∑

i=n+m+1

a2i

] 12

∣∣∣∣∣∣ ,(29)

respectively. Clearly, for∑n+mi=n+1 a

2i =

∑Ni=n+m+1 a

2i ,

EA1:A2B = EA2:A1B .

� Proposition III. In (min{EA1:A2B , EA2:A1B}, 〈EA1A2〉)

space, the localizable entanglement 〈EA1A2〉 of an N -qubit

normalized gW state with real coefficients is upper-boundedby the line

〈EA1A2〉 = min{EA1:A2B , EA2:A1B}, (30)

and lower-bounded by the line

〈EA1A2〉2 − 2〈EA1A2

〉+ (min{EA1:A2B , EA2:A1B})2 = 0,

(31)

where EA1:A2B (EA2:A1B) is the bipartite entanglement overthe bipartition A1 : A2B (A2 : A1B) in the state prior tomeasurement on all the qubits in B.

Proof. The upper bound follows from the monotonicity of E(see Eq. (3)). On the other hand, note that from Eqs. (28) and(29),

E2A1:A2B = 4

(n+m∑i=n+1

a2i

)n∑i=1

a2i + 〈EA1A2〉2, (32)

and

E2A2:A1B = 4

(N∑

i=n+m+1

a2i

)n∑i=1

a2i + 〈EA1A2〉2, (33)

respectively, where we have used Eqs. (23) and (24). Letus proceed by assuming EA2:A1B ≥ EA1:A2B , implying

min{EA1:A2B , EA2:A1B} = EA1:A2B , and∑Ni=n+m+1 a

2i ≥∑n+m

i=n+1 a2i . For a fixed a =

∑n+mi=n+1 a

2i , 〈EA1A2〉

is minimum if∑Ni=n+m+1 a

2i is minimum, leading to∑N

i=n+m+1 a2i = a, and subsequently 〈EA1A2

〉 ≥ 2a andEA1:A2B = 2

√a(1− a). Eliminating a, we obtain the equa-

tion of the lower bound as

〈EA1A2〉2 − 2〈EA1A2〉+ E2A1:A2B = 0. (34)

Similarly, assuming EA1:A2B ≥ EA2:A1B , one can also provethat

〈EA1A2〉2 − 2〈EA1A2

〉+ E2A2:A1B = 0. (35)

Eqs. (34) and (35) lead to Eq. (31).

Note 1. Note here that Propositions II and III, and the relatedCorollaries are proved for the subclass of gW states with realcoefficients, and it is therefore logical to ask whether the sameresults apply to the three-qubit gW states with complex coeffi-cients. While analytical calculation is difficult for a genericN -qubit gW state due to increase in the number of state parame-ters, we numerically check the applicability of these results forN -qubit gW states with complex coefficients, and find them tobe valid. Therefore, for generic gW states, Propositions II, III,and Corollaries II.1 and II.2 can be straightforwardly updatedby replacing a2i with |ai|2 for all i = 1, · · · , N . For demon-strations of these results, see Figs. 2A-B. Specific examplesof the family of states described in Eq. (27) can be found inAppendix C.Note 2. Note in Fig. 2A that apart from the line given byEq. (25), the Haar uniformly [86, 87] chosen gW states arealso bounded by the lines (a) EA1A2:B = 0, (b) 〈EA1A2

〉 = 0,and (c) EA1A2:B = 1, which are not shown explicitly inthe figure, and which are obtained from the fact that 0 ≤EA1A2:B , 〈EA1A2:B〉 ≤ 1. It is worthwhile to note that the

6


1A

2i


EA1A2:B


1A

2i


EA1A2:B

A. B.

FIG. 3. Dicke states. A. Scatter plots of |D(N,N1)〉 for N ≤ 35 on the (EA1A2:B , 〈EA1A2〉) plane, with all possible values of N1 =1, · · · , N − 1. B. Scatter plot of Haar-uniformly generated gD states of N = 3, N = 4, and N = 5 qubits on the (EA1A2:B , 〈EA1A2〉) plane,where each of the subsystems B and A1 is constituted of one qubit only. For each value of N , a sample of 105 states are used. All the axes inall figures are dimensionless.

family of gW states, for which EA1A2:B = 1, are given by

4

n∑i=1

a2i = 1/

N∑i=n+1

a2i . (36)

The significance of these states will be clear in Sec. III B. Forexamples of such states, see Appendix C.

C. Dicke states

We now consider the class of symmetric states that remaininvariant under permutation of parties. More specifically, wefocus on the Dicke states [63–66] of N -qubits, where N0

qubits are in the ground state |0〉, and the rest N1 = N − N0

qubits are in the excited state |1〉. A Dicke state with N1 ex-cited qubits can be written as

|D(N,N1)〉 =1√(NN1

) ∑i

Pi(|0〉⊗N−N1 |1〉⊗N1

), (37)

where for a fixed N , N1 = 0, 1, 2, · · · , N . Here, {Pi} is theset of all possible permutations of N0 (N1) qubits at ground(excited) state, such that N0 + N1 = N . Note that |D(N, 0)〉and |D(N,N)〉 are product states, while |D1〉 and |DN−1〉 areidentical to N -qubit W states, or its local unitary equivalents,the results for which can be determined as special cases of thegW states discussed in Sec. II B. Note also that in the case ofn = m < N/2 (n = m < (N − 1)/2) where N is even(odd), with m being the size of the subsystem A1, symme-try of the Dicke states under qubit permutations suggests thatEA1:A2B = EA1A2:B . We focus on the scenario where n = 1,and without any loss in generality, measure on qubit 1. Nega-tivity for Dicke states are computable over 1:rest bipartition as[88] (see also [89, 90])

EA1A2:B = maxi,j,i 6=j

1(NN1

)√(N − 1

N1 − i

)(N − 1

N1 − j

), (38)

where B is constituted of one (measured) qubit, and i, j =0, 1.

In the case of |D(N,N1)〉, the normalized post-measuredstates are given by

|D〉 =1√p

[√N −N1

Ncos

θ

2D(N − 1, N1)

+

√N1

Ne−iφ sin

θ

2D(N − 1, N1 − 1)

], (39)

|D⊥〉 =1√p⊥

[√N −N1

Nsin

θ

2D(N − 1, N1)

−√N1

Ne−iφ cos

θ

2D(N − 1, N1 − 1)

], (40)

with

p = cos2θ

2− N1

Ncos θ, (41)

p⊥ = sin2 θ

2+N1

Ncos θ. (42)

Determination of the general form of negativity over a biparti-tion of states of the form |D〉 , |D⊥〉, and the subsequent ana-lytical optimization is a difficult task. However, our numericalanalysis suggests that the optimization of localizable negativ-ity, in the current situation, always takes place in the σz basis.Using this information, |D〉 , |D⊥〉 and p, p⊥ become

|D〉 = |D(N − 1, N1)〉 ,|D⊥〉 = |D(N − 1, N1 − 1)〉 , (43)

and

p = (N −N1)/N ; p⊥ = N1/N, (44)

7

respectively, resulting in

〈EA1A2〉 = maxi,j,i 6=j

N −N1

N(N−1N1

)√√√√(N − 2

N1 − i

)(N − 2

N1 − j

)

+ maxi,j,i 6=j

N1

N(N−1N1−1

)√√√√( N − 2

N1 − 1− i

)(N − 2

N1 − 1− j

).

(45)

It is easy to numerically check that EA1A2:B ≥ 〈EA1A2〉,

which approaches equality as N increases. Note also thatthe variation of 〈EA1A2

〉 with EA1A2:B is non-monotonic forlower values of N1, resulting in a larger difference between〈EA1A2

〉 and EA1A2:B at higher N . However, these featuresdisappear as N1 increases. See Fig. 3A.

a. Generalized Dicke states. Using the Dicke states, onecan define anN -qubit permutation-symmetric state in the formof a generalized Dicke (gD) state [42], as

|D(N)〉 =

N∑N1=0

aN1 |D(N,N1)〉 , (46)

where aN1 ∈ C, and∑NN1=0 |aN1 |2 = 1. Due to a large num-

ber of state parameters, analytical calculation is difficult forgD states. However, as in the case of the Dicke states, the per-mutation symmetry can be used here also to have EA1:A2B =EA1A2:B in the case of n = m < N/2 (n = m < (N − 1)/2)for even (odd) N . This implies that it is sufficient to look intothe relation between 〈EA1A2

〉 and min{EA1:A2B , EA2:A1B},which is given by (3). Moreover, our numerical results sug-gest that min{EA1:A2B , EA2:A1B} = EA1:A2B for all Haaruniformly generated gD states, which leads to the upper boundof 〈EA1A2

〉 as 〈EA1A2〉 ≤ EA1A2:B (see Fig. 3B). It is clear

from the scatter diagrams that as N increases, the differencebetween EA1A2:B and 〈EA1A2

〉 decreases, and considerablylarger fraction of states are found to obey 〈EA1A2

〉 = EA1A2:B

in the situation where each of the subsystems B and A1 holdsonly one qubit.

D. Arbitrary multi-qubit pure states

In this section, we investigate the question posed in Sec. Ifor arbitrary states of systems with arbitrary number of qubits.

a. Three-qubit systems. While analytical calculation ofthe relevant quantities is difficult for large number of qubits,in relation to Proposition II, some analytical results can be de-rived for the three-qubit systems (N = 3) with n = m = 1.Note that the three-qubit gW states are a subset of the three-qubit W class states, which, together with the three-qubitGHZ class states, form the complete set of three-qubit purestates [61]. We explore whether the bound in Proposition IIalso applies to the three-qubit W class states given by [61]

|ψW〉 = a0 |000〉+ a1 |100〉+ a2 |010〉+ a3 |001〉 , (47)

where∑3i=0 |ai|2 = 1 and ai ∈ C, i = 0, 1, 2, 3. For ease

of discussion, we consider the qubits 1, 2, and 3 to be subsys-tems B, A1, and A2, respectively. Also, we first consider the

subclass of the W-class states with real coefficients only, i.e.,ai ∈ R, i = 0, 1, 2, 3, and numerically verify that the negativ-ity over different bipartitions of the three-qubit W-class statesare given by

E1:23 = 2

∣∣∣∣a1√a22 + a23

∣∣∣∣ , (48)

E2:13 = 2

∣∣∣∣a2√a21 + a23

∣∣∣∣ , (49)

E3:12 = 2

∣∣∣∣a3√a21 + a22

∣∣∣∣ . (50)

Further, performing local projection measurements in the ba-sis {|b0〉 , |b1〉} on the qubit 1, one obtains the post-measuredstates on qubits 2 and 3 to be in the same form as in Eq. (22)with fk0 and fk1 (k = 0, 1) given in Appendix B. Similar to thethree-qubit gW states, negativity of the post-measured statesare independent of the measurement-basis, and are given by2|a2||a3|. This leads to

〈E23〉 = 2|a2||a3|. (51)

We are now in a position to present the following propositionfor the W class states of three qubits.� Proposition IV. In the space (E1:23, 〈E23〉), the localizableentanglement 〈E23〉 of a 3-qubit normalized W-class state withreal coefficients is upper bounded by the line

〈E23〉 =1

2

(1 +

√1− E2

1:23

), (52)

where E1:23 is the bipartite entanglement over the bipartition1 : 23 in the state prior to measurement on the qubit 1.

Proof. Similar to the case of the gW states, we can write〈E23〉2 as

〈E23〉2 = 4a22(1− a20 − a21 − a22). (53)

For a fixed value of a20+a21 = a2, the maximum of 〈E23〉2, andtherefore of 〈E23〉 occurs at a22 = (1 − a2)/2, the maximumvalue of 〈E23〉 being (1 − a2). Also, E1:23 = 2

∣∣a1√1− a2∣∣.

Note that the maximum value of 〈E23〉 as well as E1:23 havetwo free parameters, a0 and a1, constrained by a20 + a21 beinga constant, a2. Eliminating a1 from 〈E23〉 and E1:23, followedby solving for 〈E23〉 leads to

〈E23〉 =1

2

[(1− a20) +

√(1− a20)2 − E2

1:23

], (54)

where 0 ≤ a20 ≤ a2. Further maximization w.r.t. a0 impliesa0 = 0, leading to Eq. (52).

Similar to the three-qubit gW states, the following Corollar-ies originate from Proposition IV.� Corollary IV.1. The family of W class states with real coef-ficients that satisfy Eq. (52) are given by

a22 = a23 = (1− a21)/2. (55)

8

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L1


L2

<latexit sha1_base64="fLLfPTPFFZEis0JmO5romU12eFU=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1gEVyWpRV0WRXBZwT6gCWUyvW2HTiZhZiKUUPBX3LhQxK3f4c6/cZpmoa0HLpw5517m3hPEnCntON9WYWV1bX2juFna2t7Z3bP3D1oqSiSFJo14JDsBUcCZgKZmmkMnlkDCgEM7GN/M/PYjSMUi8aAnMfghGQo2YJRoI/XsI48TMeSAb3tp9XzqyezVs8tOxcmAl4mbkzLK0ejZX14/okkIQlNOlOq6Tqz9lEjNKIdpyUsUxISOyRC6hgoSgvLTbP0pPjVKHw8iaUponKm/J1ISKjUJA9MZEj1Si95M/M/rJnpw5adMxIkGQecfDRKOdYRnWeA+k0A1nxhCqGRmV0xHRBKqTWIlE4K7ePIyaVUr7kWldl8r16/zOIroGJ2gM+SiS1RHd6iBmoiiFD2jV/RmPVkv1rv1MW8tWPnMIfoD6/MHowOVSw==</latexit> hE23i

<latexit sha1_base64="i2Juc5B2V7n8H0dv7zen71t8VP0=">AAAB73icbVBNSwMxEJ3Ur1q/qh69BIvgqezWouKpKILHCvYD2qVk02wbms2uSVYoS/+EFw+KePXvePPfmLZ70NYHA4/3ZpiZ58eCa+M43yi3srq2vpHfLGxt7+zuFfcPmjpKFGUNGolItX2imeCSNQw3grVjxUjoC9byRzdTv/XElOaRfDDjmHkhGUgecEqMldq3vdS9qpxNesWSU3ZmwMvEzUgJMtR7xa9uP6JJyKShgmjdcZ3YeClRhlPBJoVuollM6IgMWMdSSUKmvXR27wSfWKWPg0jZkgbP1N8TKQm1Hoe+7QyJGepFbyr+53USE1x6KZdxYpik80VBIrCJ8PR53OeKUSPGlhCquL0V0yFRhBobUcGG4C6+vEyalbJ7Xq7eV0u16yyOPBzBMZyCCxdQgzuoQwMoCHiGV3hDj+gFvaOPeWsOZTOH8Afo8wfqJo8/</latexit>

E1:23<latexit sha1_base64="yH0TsMpHDCZhwxWPqVKtFJICG/E=">AAACAnicbZDLSsNAFIYnXmu9RV2Jm2ARXEhJ2qLSVVEElxXsBZoQJtNJO3RmEmYmQgnBja/ixoUibn0Kd76N0zYLbf1h4OM/53Dm/EFMiVS2/W0sLa+srq0XNoqbW9s7u+befltGiUC4hSIaiW4AJaaE45YiiuJuLDBkAcWdYHQ9qXcesJAk4vdqHGOPwQEnIUFQacs3D11GuJve+Gml7lSzMw3VulPJ3Mw3S3bZnspaBCeHEsjV9M0vtx+hhGGuEIVS9hw7Vl4KhSKI4qzoJhLHEI3gAPc0csiw9NLpCZl1op2+FUZCP66sqft7IoVMyjELdCeDaijnaxPzv1ovUeGllxIeJwpzNFsUJtRSkTXJw+oTgZGiYw0QCaL/aqEhFBApnVpRh+DMn7wI7UrZOS/X7mqlxlUeRwEcgWNwChxwARrgFjRBCyDwCJ7BK3gznowX4934mLUuGfnMAfgj4/MHK4KWAg==</latexit>

min{E2:13, E3:12}


B.<latexit sha1_base64="Yn4h9kr5Bofdggv6gXJFQgHIhaA=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E1DJoY2ER0XxAcoS9zVyyZG/v2N0TwpGfYGOhiK2/yM5/4ya5QhMfDDzem2FmXpAIro3rfjuFldW19Y3iZmlre2d3r7x/0NRxqhg2WCxi1Q6oRsElNgw3AtuJQhoFAlvB6Gbqt55QaR7LRzNO0I/oQPKQM2qs9HDX83rlilt1ZyDLxMtJBXLUe+Wvbj9maYTSMEG17nhuYvyMKsOZwEmpm2pMKBvRAXYslTRC7WezUyfkxCp9EsbKljRkpv6eyGik9TgKbGdEzVAvelPxP6+TmvDKz7hMUoOSzReFqSAmJtO/SZ8rZEaMLaFMcXsrYUOqKDM2nZINwVt8eZk0z6reRfX8/rxSu87jKMIRHMMpeHAJNbiFOjSAwQCe4RXeHOG8OO/Ox7y14OQzh/AHzucPy4eNfQ==</latexit>

L1


L2


<latexit sha1_base64="yH0TsMpHDCZhwxWPqVKtFJICG/E=">AAACAnicbZDLSsNAFIYnXmu9RV2Jm2ARXEhJ2qLSVVEElxXsBZoQJtNJO3RmEmYmQgnBja/ixoUibn0Kd76N0zYLbf1h4OM/53Dm/EFMiVS2/W0sLa+srq0XNoqbW9s7u+befltGiUC4hSIaiW4AJaaE45YiiuJuLDBkAcWdYHQ9qXcesJAk4vdqHGOPwQEnIUFQacs3D11GuJve+Gml7lSzMw3VulPJ3Mw3S3bZnspaBCeHEsjV9M0vtx+hhGGuEIVS9hw7Vl4KhSKI4qzoJhLHEI3gAPc0csiw9NLpCZl1op2+FUZCP66sqft7IoVMyjELdCeDaijnaxPzv1ovUeGllxIeJwpzNFsUJtRSkTXJw+oTgZGiYw0QCaL/aqEhFBApnVpRh+DMn7wI7UrZOS/X7mqlxlUeRwEcgWNwChxwARrgFjRBCyDwCJ7BK3gznowX4934mLUuGfnMAfgj4/MHK4KWAg==</latexit>

min{E2:13, E3:12}

C.

FIG. 4. W-class states. Scatter plot of a sample of 107 Haar-uniformly generated three-qubit W-class states on the A. (E1:23, 〈E23〉) and the B.(min{E2:13, E3:12}, 〈E23〉) plane. The lines L1 and L2 bounding the states correspond to Eq. (52) and E1:23 = 1 respectively. C. A sampleof 107 Haar-uniformly generated GHZ-class states are also found to be bound by the same upper and lower bounds as proposed in PropositionIII. Both the axes of all the figure are dimensionless.

Proof. The proof of this Corollary follows from the maximiza-tion condition of 〈E23〉, and the normalization of the W-classstate.

� Corollary IV.2. For the family of W-class states given byEq. (55), E2:13 = E3:12.

Proof. The proof of this corollary follows from identifying thestates satisfying Eq. (55) as the three-qubit gW states, and fromthe proof of Corollary II.2.

� Proposition V. In (min{E2:13, E3:12}, 〈E23〉) space, the lo-calizable entanglement 〈E23〉 of a three-qubit normalized W-class state with real coefficients is upper-bounded by the line

〈E23〉 = min{E2:13, E3:12}, (56)


〈E23〉2 − 2〈E23〉+ (min{E2:13, E3:12})2 = 0, (57)

where E2:13 (E3:12) is the bipartite entanglement over the bi-partition 2 : 13 (3 : 12) in the state prior to measurement onthe qubit 1.

Proof. Similar to the proof of Proposition III, the upper boundfollows from the monotonicity of E (see Eq. (3)). To provethe lower bound, we start by assuming E2:13 ≥ E3:12 andmin{E2:13, E3:12} = E3:12, which, by virtue of Eqs. (49) and(50), implies a22 ≥ a23. For a fixed a3 and a0, 〈E23〉 is mini-mum if a2 is minimum, leading to a2 = a3, and subsequently〈E23〉 ≥ 2a23. On the other hand, exploiting normalization ofthe W-class state, E3:12 = 2

∣∣∣a3√1− a20 − a23∣∣∣. Eliminating

a3 from E3:12 and the minimum of 〈E23〉, we obtain

〈E23〉2 + E23:12 − 2〈E23〉(1− a20) = 0. (58)

Similar to Proposition V, it can be shown that 〈E23〉 attains aminimum for a0 = 0, leading to Eq. (57).

Note 3. Similar to the case of N -qubit gW states, in this casealso, we numerically verify that the Propositions IV and V re-main valid in the case of generic three-qubit states from Wclass with complex coefficients. The bounds in the case ofthree-qubit W class states are demonstrated in Fig. 4A-B.Note 4. We also investigate the three-qubit GHZ class states,given by

|ψGHZ〉 =

7∑i=0

ci |φi〉 , (59)

with ci ∈ C and {|φi〉 ; i = 0, 1, · · · , 7} being the standardproduct basis in the Hilbert space of three qubits. While an-alytical investigation is difficult due to a large number of pa-rameters involved in these states, our numerical analysis doesnot provide any evidence of the existence of an upper bound of〈E23〉 on the (E1:23, 〈E23〉) plane. However, our investigationinvolving a sample of 107 Haar-uniformly generated GHZ-class states did not find any example that violates the lowerbound proposed in Proposition V (see Fig. 4C for a demon-stration). While this does not analytically prove the validityof the lower bound, this implies that Proposition IV has thepotential to distinguish between the three-qubit W class statesfrom the GHZ class states.

b. Larger systems. In order to study the correla-tion between localizable entanglement 〈EA1A2〉 andthe bipartite entanglement lost due to the measure-ment, namely, EA1A2:B , EA1:A2B , and EA2:A1B in thecase of pure states on N -qubits, we look into how thestates are distributed on the (EA1A2:B , 〈EA1A2〉) and the(min [EA1:A2B , EA2:A1B ] , 〈EA1A2〉) spaces. To investigatethis, we define the followings.

δ1 = 〈EA1A2〉 − EAB , (60)

δ2 = 〈EA1A2〉 −min{EA1B:A2

, EA2B:A1}. (61)

Note that δ1 ≥ 0 (δ2 ≥ 0) implies 〈EA1A2〉 ≥ EA1A2:B

(EA1A2〉 ≥ min{EA1B:A2

, EA2B:A1}), which is representa-

tive of a situation where one can, on average, localize at least

9

EA1A2:B (min{EA2:A1B , EA1:A2B}) amount of entanglementvia local projection measurements on the qubits in B. Thepercentages ofN -qubit states for which δ1 > 0 are included inTable I for different combinations ofN andm, keeping n = 1,where in each case, a sample of 105 Haar-uniformly generatedpure states are considered. Clearly, the percentage of states forwhich δ1 > 0 increases overall with the increase in the num-ber of qubits, implying that the number of states for which〈EA1A2

〉 > EA1A2:B are more for larger systems. However,the percentage of states for which δ1 = 0, up to our numeri-cal accuracy, is negligibly small for all cases of (N,n,m). Onthe other hand, as expected, δ2 > 0 does not occur for anymulti-qubit pure states, as it would imply a violation of (3).Moreover, for a very small fraction of states, δ2 = 0. Thisfraction overall increases with an increase in the number ofqubits in the system.

III. QUANTUM STATES UNDER PHASE-FLIP NOISE

It is now logical to ask whether and how the results reportedin Sec. II are modified if the multi-qubit system is subjected tonoise [67]. In this paper, we consider a situation where qubitsof the multi-qubit system are sent through independent phase-flip channels [67, 68]. These phase-flip channels can be ei-ther Markovian [69], or non-Markovian [70–72]. Using theKraus operator representation, the evolution of the quantumstate ρ0 = |ψ〉〈ψ| under these phase-flip channels are givenby

ρ =∑α

Kαρ0K†α, (62)

such that∑αK

†αKα = I , I being the identity operator in the

Hilbert space of the multi-qubit system, and Kraus operators{Kα}, takes the form Kα =

√pαK

′α with

K ′α =

N⊗i=1

K ′αi; pα =

N∏i=1

pαi . (63)

Here, the index α ≡ α1α2 · · ·αN is interpreted as a multi-index,

∑αipαi

= 1, and K ′αiare the single-qubit Kraus op-

erators, the form of which depends on the type of noise underconsideration. In the case of the phase-flip noise, αi ∈ {0, 1},K ′αi=0 = Ii, and K ′αi=1 = σzi , with

pαi=0 = 1− q

2, pαi=1 =

q

2(64)

in the Markovian case [69], and

pαi=0 =(

1− q

2

)(1− αq

2

),

pαi=0 =[1 + α

(1− q

2

)] q2

(65)

in the non-Markovian case [70–72]. Here, q is the noisestrength (0 ≤ q ≤ 1), and α is the non-Markovianity param-eter (0 ≤ α ≤ 1). For ease of discussion, from now onward,we denote entanglement in the noiseless scenario with a su-perscript “0”. For example, 〈EA1A2

〉0 and E0AB denote the

localizable entanglement over the subsystem A and the bipar-tite entanglement over the bipartition A : B of the system inthe case of ρ0.

A. Generalized GHZ states

We start our discussions with the N -qubit gGHZ state sub-jected to phase-flip noise on all qubits. In this situation, weprove the following proposition.� Proposition VI. For any tripartition A1 : A2 : B of an N -qubit gGHZ state under uncorrelated phase-flip channel on allqubits,

〈EA1A2〉 = EA1A2:B = EA1:A2B = EA2:A1B , (66)

irrespective of whether the noise is Markovian, or non-Markovian.

Proof. The N -qubit gGHZ state, under the Markovian phase-flip noise on all qubits, takes the form

ρ =(|a0|2(|0〉〈0|)⊗N + |a1|2(|1〉〈1|)⊗N

)+(1− q)N

(a0a∗1(|0〉〈1|)⊗N + a∗0a1(|1〉〈0|)⊗N

).(67)

Partial transposition of ρ with respect to the subsystems Bleads to

ρTB =(|a0|2(|0〉〈0|)⊗N + |a1|2(|1〉〈1|)⊗N

)+(1− q)Na0a∗1(|0〉〈1|)⊗N−n(|1〉〈0|)⊗n+(1− q)Na∗0a1(|1〉〈0|)⊗N−n(|0〉〈1|)⊗n, (68)

with non-zero eigenvalues |a0|2, |a1|2,±(1 − q)N |a0||a1|.Therefore the entanglement between partition A and partitionB, as quantified by negativity [56–60], is given by

EA1A2:B = 2(1− q)N |a0|√

1− |a0|2= (1− q)NE0

A1A2:B . (69)

To calculate the localizable entanglement over the subsys-tem A with bipartition A1 : A2, we proceed as in the case ofProposition I, and write the post-measured states on A as

ρkA = TrB[(MkρMk†) /pk] , (70)

which, written explicitly, takes the form

ρkA =1

pk

[{|a0|2|fk0 |2(|0〉〈0|)⊗N−n

+|a1|2|fk1 |2(|1〉〈1|)⊗N−n}

+(1− q)N{a0a∗1fk0 f

k∗1 (|0〉〈1|)⊗N−n

+a∗0a1fk1 f

k∗0 (|1〉〈0|)⊗N−n)

}], (71)

with

pk =(|a0|2|fk0 |2 + |a1|2|fk1 |2

). (72)

Partial transposition of ρkA over any bipartition A1 : A2, andsubsequent calculation of negativity followed by the optimiza-tion of average negativity over A1 : A2 yields

〈EA1:A2〉 = 2(1− q)N |a0|√

1− |a0|2[

max

2n−1∑k=0

|fk0 ||fk1 |],

(73)

10

A.


1A

2i


EA1A2:B

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q = 0

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q = 0.1

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q = 0.2


1A

2i

B.


EA1A2:B

B.<latexit sha1_base64="n3HzFNrj/wBQLriJZBZKr1VUj7Y=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexKUC9C0IvHiOYByRJmJ51kyOzsOjMrhCWf4MWDIl79Im/+jZNkD5pY0FBUddPdFcSCa+O6305uZXVtfSO/Wdja3tndK+4fNHSUKIZ1FolItQKqUXCJdcONwFaskIaBwGYwupn6zSdUmkfywYxj9EM6kLzPGTVWun+8crvFklt2ZyDLxMtICTLUusWvTi9iSYjSMEG1bntubPyUKsOZwEmhk2iMKRvRAbYtlTRE7aezUyfkxCo90o+ULWnITP09kdJQ63EY2M6QmqFe9Kbif147Mf1LP+UyTgxKNl/UTwQxEZn+TXpcITNibAllittbCRtSRZmx6RRsCN7iy8ukcVb2zsuVu0qpep3FkYcjOIZT8OACqnALNagDgwE8wyu8OcJ5cd6dj3lrzslmDuEPnM8fzreNfw==</latexit>

q = 0

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q = 0.1

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q = 0.2

FIG. 5. Generalized W states under Markovian phase-flip channels. Modification of boundaries of gW states proposed in Propositions II andIII on the A. (EA1A2:B , 〈EA1A2〉) and the B. (min{EA1:A2B , EA2:A1B}, 〈EA1A2〉) plane due to Markovian phase-flip noise of noise strengthq = 0, 0.1, 0.2 (see also Fig. 2). The modified boundaries are given in Propositions VI and VII. Note that the trivial boundaries correspondingto zero entanglement lines are not shown. All the axes on all figures are dimensionless.

as in Eq. (14). The maximization is similar to that shown inthe proof of Proposition I, leading to

〈EA1:A2〉 = (1− q)N 〈EA1A2

〉0 = EA1A2:B . (74)

Also, from the symmetry of ρ, EA1A2:B = EA1:A2B =EA2:A1B , leading to Eq. (66).

Same line of calculations would follow for the non-Markovian phase-flip channel, leading to

EA1A2:B = |1− f(q, α)|N E0A1A2:B ,

〈EA1A2〉 = |1− f(q, α)|N 〈EA1A2〉0, (75)

with

f(q, α) = q{

1 + α(

1− q

2

)}, (76)

which implies 〈EA1A2〉 = EA1A2:B . Similar proof followsfor EA1:A2B , and EA2:A1B also, resulting in Eq. (66) for thenon-Markovian phase-flip channel. Hence the proof.

Note 5. A comparative discussion on the variation of entan-glement with q in the cases of the Markovian and the non-Markovian phase-flip channels is in order here. Note that inthe former case, entanglement decays monotonically with q,as indicated from the (1 − q)N dependence, while the decayfastens exponentially with increasing number of qubits. It alsoindicates that entanglement vanishes asymptotically with in-creasing q, attaining zero value only at q = 1. Similar featuresare also present in the case of the non-Markovian channel, ex-cept one where in contrast to entanglement vanishing only atq = 1 in the former case, entanglement vanishes at a finitecritical q in the latter, given by

qc =1

α

(1 + α−

√1 + α2

). (77)

For q > qc, entanglement revives again. Note that qc is amonotonically decreasing function of α, which, in the limitα→ 0 (the Markovian limit), goes to 1.

B. Generalized W states

We now focus on theN -qubit gW states under the phase-flipnoise on all qubits. Analytical investigation of 〈EA1A2

〉 as wellas bipartite entanglement over the unmeasured state in suchcases is difficult due to the increasing number of state as wellas optimization parameters. However, our numerical investiga-tion suggests that irrespective of the tripartitionA1 : A2 : B ofthe N -qubit system, the optimization of 〈EA1A2

〉 always takesplace via σz measurement on all qubits i ∈ B. This resultcan be utilized to determine the dependence of 〈EA1A2

〉 andEA1A2:B on the noise strength q, and extend the PropositionsII and III, as follows.

a. Markovian phase-flip channels. In the case of theMarkovian phase-flip channels, we obtain3

〈EA1A2〉 = (1− q)2〈EA1A2

〉0, (78)EA1A2:B = (1− q)2E0

A1A2:B , (79)

Given these results, we present the following Proposition.� Proposition VII. In the space (EA1A2:B , 〈EA1A2

〉), the lo-calizable entanglement 〈EA1A2

〉 of anN -qubit normalized gWstate subjected to Markovian phase-flip channels of the samestrength, q, on all qubits is bounded by the lines

〈EA1A2〉 =

1

2

[(1− q)2 +

√(1− q)4 − E2

A1A2:B

], (80)

and

EA1A2:B = (1− q)2, (81)

where EA1A2:B is the bipartite entanglement over the bipar-tition A1A2 : B in the state prior to measurement on all thequbits in B.

3 These expressions are determined for systems of small sizes, and areverified numerically for larger systems with different combinations of(N,n,m).

11


1A

2i


1A

2i


1A

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1A

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EA1A2:B

FIG. 6. Generalized W states under non-Markovian phase-flip noise. The variation of 〈EA1A2〉 as a function of q (Eq. (84)), with〈EA1A2〉0 = 1 and α = 0.9, exhibiting monotonic decay followed by a vanishing at qc (Eq. (77)), and a subsequent revival for q > qc.In the entire range 0 ≤ q ≤ 1, five values of noise strengths, namely a. q = 0.1, b. q = 0.2, c. q = 0.4, d. q = 0.9, and e. q = 1.0 are chosen,and the modification of the boundaries on the gW states, as given in Proposition II, with varying q is demonstrated. The region accessible by thegW states on the (EA1A2:B , 〈EA1A2〉) decreases at first, vanishes at q = qc, and then revives again. The axes on all figures are dimensionless.

Note 6. It is worthwhile to note that the line given in Eq. (81)corresponds to the family of states, described by Eq. (36), sub-jected to the single-qubit phase-flip channels on all qubits.Note 7. Proposition VI implies that the area on the(EA1A2:B , 〈EA1A2〉) plane confining the noisy gW statesshrinks with increasing noise strength q, and vanishes at q = 1.This is demonstrated in fig. 4A.

Noting that EA1:A2B and EA2:A1B also have similar depen-dence on q as EA1A2:B , Proposition III can be extended to thecase of gW states under phase-flip noise, as follows.� Proposition VIII. In (min{EA1:A2B , EA2:A1B}, 〈EA1A2〉)space, the localizable entanglement 〈EA1A2〉 of an N -qubitnormalized gW state subjected to the phase-flip channel ofstrength q on all qubits is upper-bounded by the line

〈EA1A2〉 = min{EA1:A2B , EA2:A1B}, (82)


〈EA1A2〉2 − 2(1− q)2〈EA1A2

〉+ (min{EA1:A2B , EA2:A1B})2 = 0, (83)

whereEA1:A2B (EA2:A1B) is the entanglement over the bipar-tition A1 : A2B (A2 : A1B) in the state prior to measurementon all the qubits in B.

b. Non-Markovian phase-flip channels. A similar ap-proach can also be taken in the case of the non-Markovianchannels, where entanglement has the following dependenceon q and α:

〈EA1A2〉 = |1− f(q, α)|2 〈EA1A2

〉0, (84)

EA1A2:B = |1− f(q, α)|2E0A1A2:B , (85)

EA1:A2B = |1− f(q, α)|2E0A1:A2B , (86)

EA2:A1B = |1− f(q, α)|2E0A2:A1B , (87)

with f(q, α) given in Eq. (76). Using these, one can straight-forwardly obtain the results on the bounds on the gW stateswhen subjected to non-Markovian phase-flip channels, by re-placing the (1 − q)2 factors with |1− f(q, α)|2. To keep thetext uncluttered, we refrain from writing these Propositions ex-plicitly.

Similar to the Markovian case, for a fixed value of α, thearea on the EA1A2:B − 〈EA1A2

〉 plane confining the noisygW states shrinks with increasing q in the case of the non-Markovian phase flip channel also. However, in contrast to theMarkovian case, the area vanishes at a critical noise strengthqc, given in Eq. (77), and then revives again for q > qc (seeFig. 4B for a demonstration). It is worthwhile to note thatthe decay of entanglement in the case of the gW states underMarkovian and non-Markovian phase-flip channel is indepen-dent of the number of qubits in the system, as opposed to thecase of the gGHZ states, where the dependence is exponentialin N .

C. Numerical results

The complexity of the states obtained via applying single-qubit phase-flip noise to all qubits of a multi-qubit states pre-vents analytical investigation into the relation between the lo-calizable end the destroyed entanglement in most of the cases.In this subsection, we discuss the numerical results obtainedfor the mixed states generated via subjecting three-qubit W-class states, N -qubit Dicke and gD states, and arbitrary N -qubit pure states to Markovian and non-Markovian phase-flipchannels.

12


1A

2i


1A

2i



EA1A2:B



EA1A2:B

A. B.


1A

2i


1A

2i

C. D.

FIG. 7. Dicke and generalized Dicke states under phase-flip noise. A., B. Scatter plots of N -qubit Dicke states, for upto N = 8, underMarkovian and non-Markovian phase-flip noise are shown on the (EA1A2:B , 〈EA1A2〉) planes for A. N1 = 1, and B. N1 = 2. C., D. On theother hand, scatter plots of samples of 107 gD states with C. N = 3 and D. N = 4 are shown on the (EA1A2:B , 〈EA1A2〉) plane. All axes inall of the figures are dimensionless.

a. Three-qubit states. Our numerical investigation of thethree-qubit W-class states subjected to Markovian as well asnon-Markovian phase-flip channels indicate that the variationsof the localizable entanglement 〈E23〉 as well as the bipartiteentanglement lost during measurement, namely, E1:23, E2:13,and E3:12, are identical to that described in Eqs. (78)-(79)(Markovian) and Eqs. (84)-(87) (non-Markovian). This im-plies that the mixed states obtained from the three-qubit W-class states, for a specific noise strength q, are bounded in asimilar way as described for the gW states under phase-flipnoise. On the other hand, while a similar numerical analysis isdifficult for the GHZ class states due to the increased numberof parameters, we observe that similar to the pure GHZ-classstates discussed in Note 4 (see also Fig. 4C.), the bounds pro-posed in Eqs. (82)-(83) hold also for three-qubit GHZ-classstates subjected to the Markovian phase-flip noise channels.

b. Dicke states and generalized Dicke states under noise.We also numerically investigate the N -qubit Dicke and gDstates subjected to Markovian and non-Markovian phase-flipchannels. Fig. 7A-B depict the scatter plots of the Dicke statesup to N = 8, and N1 = 1, 2, where Markovian and non-Markovian phase-flip channel is applied to all qubits. It is clear

N n m Fraction of states (in %) with δ1 > 0

q = 0.0 q = 0.1 q = 0.2 q = 0.3 q = 0.4

3 1 1 23.343% 23.608% 23.097% 21.530% 18.890%

4 1 1 38.704% 20.461% 15.757% 15.514% 15.336%

5 1 1 44.920% 14.213% 12.942% 10.531% 07.487%

5 1 2 100.00% 100.00% 100.00% 100.00% 93.402%

TABLE I. Variations of the fraction of Haar uniformly generatedthree-, four-, and five-qubit states for which δ1 > 0, as a func-tion of the noise parameter q, which assumes five values, q =0.0, 0.1, 0.2, 0.3, and 0.4, from left to right along the row for a spe-cific combination ofN , n, andm, in the case of the Markovian phase-flip noise. In all cases, we have kept n = 1, where n and m being thesizes ofB andA1 respectively. For the three-qubit system, only GHZclass states are considered. For each case, the percentages are deter-mined from a sample size of 105 Haar-uniformly generated states. Allthe quantities presented in the table are dimensionless.

from the figures that similar to the case of the Pure Dicke states(see Sec. II D), (a) the variation of 〈EA1A2

〉 with EA1A2:B re-mains non-monotonic, and (b) with increasing N1, the states

13

tend to the 〈EA1A2〉 = EA1A2:B line. Also, increasing thenon-Markovianity factor α generally tends to lower values of〈EA1A2〉 and EA1A2:B .

In the case of the gD states, our numerical investigationsuggests that while the upper bound 〈EA1A2

〉 ≤ EA1A2:B

remains valid even in the presence of phase-flip noise irre-spective of whether it is Markovian, or non-Markovian. How-ever, with increasing number of qubits, as in the case ofthe pure gD states, more states are concentrated close to the〈EA1A2〉 = EA1A2:B line, although the number of states forwhich 〈EA1A2〉 = EA1A2:B diminishes drastically. This isdemonstrated in Fig. 7C-D for gD states with N = 3, 4 qubits,respectively, under Markovian phase-flip noise. The resultsremain qualitatively the same even in the presence of non-Markovian phase-flip noise also.

c. Arbitrary pure states under noise. It is important tonote that the numerical investigation for arbitraryN -qubit purestates subjected to phase-flip channels is resource-intensiveeven for a small number of qubits due to the optimization in-volved in the computation of localizable entanglement, as thenumber of optimization parameter increases with increasing n,the size of the measured subsystemB. In this paper, we restrictourselves in reporting data for which B is constituted of onequbit only. Similar to the pure states of N -qubits describedin Sec. II D, we focus on δ1 and δ2 (Eqs. (60)-(61)) also forthe mixed states obtained by subjecting N -qubit arbitrary purestates to Markovian phase-flip channels. As expected, for allinvestigated values of q, no states are found for which δ2 ≥ 0,implying a violation of inequality (3), which is similar to thecase of the pure states (see Sec. II D). On the other hand, thepercentage of states for which δ1 > 0 are tabulated in TableI for different noise strengths in the case of the Markovianphase-flip channel. It is clear from the table that (a) for a fixedN with n = 1, the number of states for which δ1 > 0 overalldecreases with increasing q, and (b) for a specific q value > 0,such states overall decreases in number with increasing N , aslong as n and m are fixed at 1.

IV. INTERACTING 1D QUANTUM SPIN MODELS

It is natural to ask whether the bounds discussed in Sec. IIfor the pure states also exist in the ground states of paradig-matic quantum spin Hamiltonians. In order to investigate this,we focus on the one-dimensional (1D) quantum spin chainswith N spin-1/2 particles, governed by a Hamiltonian givenby [84]

H =

N∑i=1

[Jxyi,i+1

{(1 + γ)σxi σ

xi+1 + (1− γ)σyi σ

yi+1

}+Jzzi,i+1σ

zi σ

zi+1 + hiσ

zi

]. (88)

In Eq. (88), γ is the xy anisotropy parameter, hi is local mag-netic field strength corresponding to spin i, and Jxyi,i+1

(Jzzi,i+1

)represents the nearest-neighbor xy (zz) interaction strengths.Also, we assume periodic boundary condition (PBC) in thesystem, implying σx,y,zN+1 ≡ σx,y,z1 . A number of paradig-matic 1D quantum spin models can be represented by dif-

ferent special cases of H . In this paper we are interested intwo of them, namely, (a) transverse-field XY model (TXY)(0 < γ ≤ 1, Jzzi,i+1 = 0

)[74–79] (note that the transverse-

field Ising model (TIM) [78, 79, 91] is a special case of theTXY model with γ = 1), and (b) XXZ model with magneticfield (XXZ) (γ = 0) [80–85].

A. Ordered quantum spin models

In the case of the ordered quantum spin models where orderexists in all spin-spin couplings as well as the field strengths,we assume Jxyi,i+1 = Jxy > 0, Jzzi,i+1 = Jzz > 0, andhi = h > 0 for all i = 1, 2, · · · , N . In such models, wenumerically investigate the correlation between 〈EA1A2

〉 and{EA1A2:B , EA1B:A2

, EA2B:A1} in the ground state, which is

obtained via numerical diagonalization of H . We first con-sider the ordered AFM TXY model, and define g = h/Jxy asthe dimensionless field-strength. The model exhibits a quan-tum phase transition from an antiferromagnetic (AFM) phase(g < 1) to a paramagnetic (PM) phase (g > 1) at gc = 1,for all values of γ > 0 [31, 75–79, 92, 93]. Fig. 8A. depictsthe scatter plot of 〈EA1A2〉 − EAB corresponding to the or-dered TXY model for different values of N , where for eachN , values of n,m = 1. This, along with the PBC ensures thatEA1A2:B = EA1:A2B , while our numerical findings suggestthat for the ground states of these quantum spin models withn = m = 1, min{EA1:A2B , EA2:A1B} for all values of N .This implies that it is sufficient to investigate the dependenceof 〈EA1A2

〉 on EA1A2:B , which is demonstrated in Fig. 8A. Inorder to generate the scatter plot on the EA1A2:B − 〈EA1A2〉plane, the values of g are chosen to be deep inside the AFM(0.2 ≤ g ≤ 0.8) and the PM (1.2 ≤ g ≤ 1.8) phases. It isevident from the figure that 〈EA1A2〉 is positively correlatedwith EAB . Irrespective of the value of N , the data suggests aquadratic dependence of 〈EA1A2〉 on EA1A2:B , given by

〈EA1A2〉 = λ2E2A1A2:B + λ1EA1A2:B + λ0, (89)

where the values of λ0,1,2 can be obtained by fitting the numer-ical data to Eq. (89). In the present case, λ0 = −0.05± 0.001,λ1 = 1.09± 0.008, and λ2 = −0.05± 0.007. Our numericalinvestigation suggests that the form of Eq. (89) remains invari-ant with a change in the values of the xy anisotropy parameterγ.

In the case of the XXZ model in a magnetic field alongthe z direction with PBC, we assume Jzz = ∆Jxy , where∆ signifies the z-anisotropy parameter, and denote the dimen-sionless field-strength by g = h/Jxy . For −1 ≤ ∆ ≤ 1,the model undergoes a quantum phase transition from the XYphase to the ferromagnetic (FM) phase at the critical fieldstrength gc = ±(1 + ∆) [84]. Similar to the TXY model,we fix ∆ and investigate the correlation between 〈EA1A2

〉 andEAB deep inside the XY and the FM phases. The data ispresented in Fig. 8B., which indicates qualitatively similar re-sult as in the case of the TXY model. The fitted curve alsohas the form given in Eq. (89), with λ0 = −0.36 ± 0.045,λ1 = 1.92± 0.106, and λ2 = −0.70± 0.061. It is worthwhileto note that in the case of the XXZ model also, the overall re-

14


EA1A2:B


1A

2i


EA1A2:B



EA1A2:B


1A

2i


1A

2i


1A

2i

A. B.

C. D.

FIG. 8. 1D quantum spin models. Scatter plot of the ground states in 1D TXY model (A,C), and 1D XXZ model in an external field (B,D) onthe (EA1A2:B , 〈EA1A2〉) plane for different values of N . The ordered cases are presented in A and B, while the disordered cases are depictedin C and D. The variations of 〈EA1A2〉 as a function of EA1A2:B are fitted to Eq. (89). In the case of the TXY model, γ is taken to be 0.5 forboth ordered and disordered cases, while for the XXZ model, ∆ = 0.5. In the disordered case, for both models, σg = 0.05. All axes in allfigures are dimensionless.

lation between 〈EA1A2〉 and EA1A2:B remains unaltered with

a change in the value of ∆.

B. Disordered quantum spin models

In a disordered quantum spin model [73], the values of arelevant system parameter, such as g, are chosen from a Gaus-sian distribution, P (g), of fixed mean, 〈g〉, and fixed standarddeviation, σg . Here, σg represents the strength of the disorder,and each random value of g represents a random parameterconfiguration of the quantum spin model, describing a randomrealization of the system. For each random realization of thesystem, the quantity of interest, Q(g), can be computed. Asubsequent quenched average of Q(g) over a statistically largenumber of random realizations is given by

〈Q〉d =

∫P (g)Q(g)dg, (90)

where the subscript d represents a quenched average, and 〈Q〉is effectively a function of 〈g〉 and σg . Note that the corre-

sponding ordered result can be obtained as a special case atσg = 0. Note also that disorder can, in principle, be presentin a number of system parameters. In this paper, however, weconfine ourselves in situations where only one chosen systemparameter is disordered.

We start with the TXY model, choosing the field-strength h(and hence g, where Jxy is constant) to be the disordered sys-tem parameter. The quenched averaged LE, 〈〈EA1A2〉〉d, andthe bipartite entanglement, 〈EA1A2:B〉d, in the ground state ofthe system are numerically computed for different values ofthe disorder strength σg (0.01 ≤ σg ≤ 0.1), fixing 〈g〉 at sim-ilar values from the AFM and the PM phases of the model asthe values of g in the ordered case. In each of these computa-tions on an N -qubit system, both n and m are taken to be 1,similar to the ordered scenario. Our numerical data, exhibitedin Fig. 8C. and fitted to Eq. (89) with λ0 = −0.36 ± 0.024,λ1 = 1.87 ± 0.053, and λ2 = −0.52 ± 0.029, indicate thatthe relation between 〈EA1A2

〉 and EA1A2:B is qualitatively ro-bust against disorder in the field. Also, varying σg in the range0.01 ≤ σg ≤ 0.1 only changes the fitting parameters negligi-

15

bly. Similar analysis is performed for the XXZ model in anexternal field to arrive at a similar conclusion, where the data isfitted to Eq. (89) with λ0 = −0.26±0.093, λ1 = 1.69±0.222,and λ2 = −0.57± 0.130. See Fig. 8D.

V. CONCLUSIONS AND OUTLOOK

In this paper, we investigate dependence of the gain in theentanglement through localization over a group of qubits in amulti-qubit system via single-qubit projection measurementson the rest of the qubits on the amount of loss in bipartiteentanglement during these measurements. We probe a num-ber of paradigmatic N -qubit pure states, namely, the general-ized GHZ, generalized W, Dicke, and the generalized Dickestates. We derive analytical bounds for the generalized GHZand the generalized W states. We show that the gain is al-ways equal to the loss in the former, while in the latter, lowerand upper bounds of localizable entanglement can be derivedin terms of the bipartite entanglement present in the systemprior to the measurement process. In the case of the Dickeand the generalized Dicke states, a combination of analyticaland numerical investigations reveal that the localizable entan-glement tend to be equal to a component of the lost bipartiteentanglement when the number of qubits increases. Modifica-tions of these results, when the system is subjected to single-qubit Markovian and non-Markovian phase-flip channels, arealso discussed. We extend our study to the ground states ofthe 1D quantum spin models, namely, the transverse-field XYmodel and the XXZ model in an external field, and numer-ically demonstrate a quadratic dependence of the localizableentanglement over the bipartite entanglement in the groundstate prior to measurement, where measurement is restrictedto one qubit only, and the entanglement is always computedin the 1:rest bipartition. This dependence is shown to be qual-itatively robust even in the presence of disorder in the fieldstrength.

A number of possible avenues of future research emergefrom this study. Within the orbit of the results reported inthis paper, it is important to understand how the results ob-tained in the case of the pure states are modified when dif-ferent types of noise that are commonly occurring in ex-periments [94, 95], such as the bit-flip, depolarizing, andamplitude-damping [67, 68] noise are present in the system.However, investigating such noise channels may present newchallenges in deriving the appropriate bounds, if any, on lo-calizable entanglement. Also, in the case of the 1D quantumspin systems considered in this paper, it would be interestingto see the effect of the presence of disorder in the spin-spin in-teraction strengths along with a disordered field-strength. Be-sides, a plethora of quantum spin models are important fromthe perspective of quantum information theory [31, 32], andit is interesting to investigate whether a specific relation be-tween the localized and the lost entanglement, similar to theone in the case of the models described in this study, exist inthe ground states of these models in the presence and absenceof disordered interactions, as well as in situations where the

system is made open by allowing an interaction with the envi-ronment [96].

ACKNOWLEDGMENTS

We acknowledge the use of QIClib (https://github.com/titaschanda/QIClib) – a modern C++library for general purpose quantum information processingand quantum computing.

Appendix A: Negativity

The amount of entanglement between two partitions A andB of a bipartite quantum state ρAB can be quantified by a bi-partite entanglement measure [1, 2]. In this paper, we shall fo-cus on negativity [56–60, 97] as a bipartite entanglement mea-sure, which is defined as

EnegA:B = ||ρTB

AB || − 1, (A1)

which corresponds to the absolute value of the sum of negativeeigenvalues, λ, of ρTB

AB , given by

EnegA:B = 2

∣∣∣∣∣∑λi<0

λi

∣∣∣∣∣ . (A2)

Here, ||%|| = Tr√%†% is the trace norm of the density operator

%, computed as the sum of the singular values of %. The ma-trix ρTB

AB is obtained by performing partial transposition of thedensity matrix ρAB with respect to the subsystemB. Since weonly focus on negativity throughout this paper, we discard thesuperscript from EnegA:B , and denote the negativity between thepartitions A and B simply by EA:B .

Appendix B: Single- and two-qubit measurements

We first consider the single-qubit measurement in an N -qubit gW state, where (see Eq. (22))

f00 = a1e−iφ sinθ

2, f10 = −a1e−iφ cos

θ

2, (B1)

f0i = an+i cosθ

2, f1i = an+i sin

θ

2, (B2)

with i = 1, 2, · · · , N , and

p0 = a21 sin2 θ

2+ cos2

θ

2

N∑i=2

a2i ,

p1 = a21 cos2θ

2+ sin2 θ

2

N∑i=2

a2i , (B3)

such that 〈ψk|ψk〉 = 1 for k = 0, 1. On the other hand, inthe case of two-qubit projection measurements on an N -qubitgW state, the post-measured states are of the form given inEq. (22), with the coefficients fki as

https://github.com/titaschanda/QIClib

16

f00 = a1e−iφ1 sinθ12

cosθ22

+ a2e−iφ2 cosθ12

sinθ22, f10 = a1e−iφ1 sin

θ12

sinθ22− a2e−iφ2 cos

θ12

cosθ22,

f20 = −a1e−iφ1 cosθ12

cosθ22

+ a2e−iφ2 sinθ12

sinθ22, f30 = −a1e−iφ1 cos

θ12

sinθ22− a2e−iφ2 sin

θ12

cosθ22, (B4)

and

f0i = an+i cosθ12

cosθ22, f1i = an+i cos

θ12

sinθ22, f2i = an+i sin

θ12

cosθ22, f3i = an+i sin

θ12

sinθ22, (B5)

where i = 1, 2, · · · , N − n, and the probabilities of obtaining the measurement outcome k, k = 0, 1, 2, 3, are given by

p0 = a21 sin2 θ12

cos2θ22

+ a22 cos2θ12

sin2 θ22

+ cos2θ12

cos2θ22

N−n∑i=1

a2n+i,

p1 = a21 sin2 θ12

sin2 θ22

+ a22 cos2θ12

cos2θ22

+ cos2θ12

sin2 θ22

N−n∑i=1

a2n+i,

p2 = a21 cos2θ12

cos2θ22

+ a22 sin2 θ12

sin2 θ22

+ sin2 θ12

cos2θ22

N−n∑i=1

a2n+i,

p3 = a21 cos2θ12

sin2 θ22

+ a22 sin2 θ12

cos2θ22

+ sin2 θ12

sin2 θ22

N−n∑i=1

a2n+i. (B6)

For a three-qubit state belonging to the W class (seeEq. (47)), measurement on qubit 1 can be described in a waysimilar to that for the three-qubit gW state, with fk0 (k = 0, 1)given by

f00 = a0 cosθ

2+ a1e−iφ sin

θ

2,

f10 = a0 sinθ

2− a1e−iφ cos

θ

2, (B7)

and fk1,2 (k = 0, 1) given by

f01 = a2 cosθ

2, f02 = a3 cos

θ

2,

f11 = a2 sinθ

2, f12 = a3 sin

θ

2, (B8)

and

p0 =

∣∣∣∣a0 cosθ

2+ a1e−iφ sin

θ

2

∣∣∣∣2 + (a22 + a23) cos2θ

2,

p1 =

∣∣∣∣a0 sinθ

2− a1e−iφ cos

θ

2

∣∣∣∣2 + (a22 + a23) sin2 θ

2.

(B9)

Appendix C: Specific examples

In this section, we demonstrate a number of results dis-cussed in Secs. II and III using specific examples.

a. Upper bound for gW states. Consider the cases ofN = 3, 4, for which the family of states providing the upperbound of 〈EA1A2〉 in the case of gW states with real coeffi-

cients can be written as

|Ψ3〉 = a |100〉+

√1− a2

2(|010〉+ |001〉), (C1)

|Ψ4〉 = a |1000〉+

√1− a2

2|0100〉+ b |0010〉

+

√1− a2 − 2b2

2|0001〉 , (C2)

where a, b are real numbers. The behaviour of 〈EA1A2〉 as a

function of a, b is shown in Figs. 2A. and B. Also, 〈EA1A2〉

for the state |Ψ3〉 provides an upper bound for 〈EA1A2〉 cor-

responding to all three-qubit gW states with complex coef-ficients. This is demonstrated by the 〈EA1A2

〉 values cor-responding to 106 Haar-uniformly generated three-qubit gWstates lying below the upper bound.

b. Maximum bipartite entanglement for gW states. Theline EA1A2:B = 1 corresponds to the family of states given by

∑i∈B

a2i =

(4∑i∈A

a2i

)−1. (C3)

Specifically, for a three-qubit system, this family is given by

|Φ3〉 =

√1

2|100〉+ a |010〉+

√1

2− a2 |001〉 , (C4)

while for N = 4, n = m = 1,

|Φ4〉 =

√1

2|1000〉+ a |0100〉+ b |0010〉

+

√1

2− a2 − b2 |0001〉 . (C5)

17

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arXiv:2206.07731v1 [quant-ph] 15 Jun 2022

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