Multipartite entanglement measures via Bell basis measurements Jacob L. Beckey1 2Gerard Pelegrí3Steph Foulds4and Natalie J. Pearson3 1JILA NIST and University of Colorado Boulder Colorado 80309 USA

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Multipartite entanglement measures via Bell basis measurements

Jacob L. Beckey,1, 2 Gerard Pelegrí,3Steph Foulds,4and Natalie J. Pearson3

1JILA, NIST and University of Colorado, Boulder, Colorado 80309, USA

2Department of Physics, University of Colorado, Boulder, Colorado 80309, USA

3Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, UK

4Physics Department, Durham University, South Road, Durham, DH1 3LE, UK

We show how to estimate a broad class of multipartite entanglement measures from Bell basis

measurement data. In addition to lowering the experimental requirements relative to previously

known methods of estimating these measures, our proposed scheme also enables a simpler analysis

of the number of measurement repetitions required to achieve an -close approximation of the mea-

sures, which we provide for each. We focus our analysis on the recently introduced Concentratable

Entanglements [Beckey et al. Phys. Rev. Lett. 127, 140501 (2021)] because many other well-known

multipartite entanglement measures are recovered as special cases of this family of measures. We

extend the deﬁnition of the Concentratable Entanglements to mixed states and show how to con-

struct lower bounds on the mixed state Concentratable Entanglements that can also be estimated

using only Bell basis measurement data. Finally, we demonstrate the feasibility of our methods by

realistically simulating their implementation on a Rydberg atom quantum computer.

Introduction. The precise control over quantum sys-

tems demonstrated in the past two decades has enabled

rapid progress in the experimental study of quantum en-

tanglement [1,2]. Entanglement plays an important role

in enabling emerging quantum technologies to outper-

form their classical counterparts, with the degree and

type of entanglement within the state determining its

usefulness for a given task. Consequently the empirical

characterization of entanglement is a problem of ubiq-

uitous interest in quantum information science. While

bipartite entanglement is well understood theoretically

[1,3] and is routinely estimated in experimental set-

tings, multipartite entanglement remains challenging to

understand theoretically and probe experimentally [2].

When these considerations are coupled with the exponen-

tial scaling of the Hilbert space of multipartite systems,

which makes quantum state tomography intractable at

scale [4,5], it is clear that there is a need for more exper-

imentally eﬃcient methods of multipartite entanglement

quantiﬁcation.

Recently, the authors of Ref. [6] conjectured that the

output probabilities of the so-called parallelized c-SWAP

test, shown in Fig. 1, could be used to construct a well-

deﬁned multipartite entanglement measure. The authors

of Ref. [7] then generalized this conjecture and proved

that a whole family of multipartite entanglement mea-

sures could be constructed using the output probabilities

of this circuit, depending on which ancilla qubits are mea-

sured. The resultant family of measures was dubbed the

Concentratable Entanglements (CEs), and it was shown

that many well-known multipartite entanglement mea-

sures could be recovered as special cases of this general

family. Since their introduction, several interesting prop-

erties and applications of the CEs have also been studied

[8–10]. We also note that the n-tangle [11], another well-

studied entanglement monotone, can be estimated via the

parallelized c-SWAP test [7], and that the parallelized c-

SWAP test was recently generalized to qudit and optical

states [12].

From Fig. 1(a), it is clear that the n-qubit c-SWAP test

requires nToﬀoli gates as well 3nqubits (2 copies of the

the quantum state of interest and nancilla qubits). The

most promising platform for implementing the c-SWAP

test is Rydberg atom systems [13,14] due to their native

ability to implement Toﬀoli gates [15–26]. However, to

make the CEs and related measures as accessible as pos-

sible, a method of estimating them that is experimentally

feasible on all hardware platforms is needed. This work

addresses this problem by introducing a method of esti-

mating many multipartite entanglement measures from

Bell basis measurement data – an ancilla-free scheme

that only requires one- and two-qubit gates acting on

two copies of the quantum state of interest.

Bell basis measurements have played a crucial role in

quantum information theory since the advent of proto-

cols like quantum teleportation and superdense coding

[27–29]. More recently, Bell basis measurements have

been implemented experimentally to estimate bipartite

concurrences [30,31], non-stabilizerness (i.e. magic) [32],

entanglement dynamics in many-body quantum systems

[33–36], and even to demonstrate quantum advantage in

learning from experiments [37]. These recent experiments

corroborate the claim that our methods are feasible on

today’s hardware.

A limitation recently highlighted in Ref. [8] is that CEs

were only well-deﬁned on pure states. We address this

limitation by ﬁrst deﬁning the CEs for mixed state inputs

and then introducing lower bounds on these quantities

which also depend only on Bell basis measurement data,

thus making them readily accessible experimentally.

This work is organized as follows. We ﬁrst construct

unbiased estimators, which depend only on Bell basis

measurement data, for all entanglement measures com-

putable using the parallelized c-SWAP test, thus recover-

ing all results in Refs. [6,7] while using fewer resources.

We then derive expressions showing how many measure-

ment repetitions are needed to obtain an -close approxi-

mation of these measures with high probability. Next, we

arXiv:2210.02575v2 [quant-ph] 12 Oct 2022

extend the CEs to mixed states and introduce a family of

lower bounds for the mixed state CEs which allow one to

probe the multipartite entanglement of mixed quantum

states, thus generalizing Refs. [7,38–40]. Finally, we

demonstrate the feasibility of our methods by carrying

out realistic, noisy experiments on a simulated Rydberg

system. Background material, proofs, and simulation de-

tails can be found in the Supplementary Material [41].

CEs via Parallelized c-SWAP circuit. To appreciate

the utility of the Bell basis measurement scheme, one

must ﬁrst understand the CEs and how they can be esti-

mated via the parallelized c-SWAP test. Thus, we begin

by deﬁning the CEs.

Let |ψi ∈ (C2)⊗ndenote a pure state of n-qubits.

Further, denote the set of labels of the qubits as S=

{1,2, . . . , n}. Throughout, we will let s⊆ S be any sub-

set of the nqubits with P(s)the associated power set

(i.e. the set of all subsets of s, which has cardinality

2|s|). With our notations in place, we can deﬁne the CE.

Deﬁnition 1 (Ref. [7]).For any non-empty set of qubit

labels s∈ P(S)\ {∅}, the Concentratable Entanglement

is deﬁned as

C|ψi(s)=1−1

2|s|X

α∈P(s)

tr ρ2

α,(1)

where the ρα’s are reduced states of |ψihψ|obtained by

tracing out subsystems with labels not in α. For the trivial

subset, we take tr ρ2

∅:= 1.

When s=S, the sum in Def. 1is simply a uniform

average of subsystem purities. This matches the intu-

ition that highly entangled pure states should have highly

mixed (low purity) reduced states. Although many inter-

esting properties of the CE are summarized in Ref. [7],

we need only one more detail to motivate this current

work. Namely, the fact that the CE can be estimated

from the output probabilities of the parallelized c-SWAP

test via

C|ψi(s)=1−X

z∈Z0(s)

p(z),(2)

where z∈ {0,1}ndenotes a length nbitstring, p(z)the

probability of obtaining said bitstring, and Z0(s)the set

of all bitstrings with zeroes in the indices of s. As one

can see from Fig. 1, the parallelized c-SWAP test requires

3nqubits and nToﬀoli gates, which, on most platforms,

must be further broken down into one- and two-qubit

gates [42]. Although some hardware platforms, like Ry-

dberg atoms, can implement Toﬀoli gates natively with

high ﬁdelity [26], it would be preferable to eliminate the

3-qubit gates altogether. This is exactly what the Bell

basis method achieves while simultaneously reducing the

qubit requirements from 3nto 2n. Before seeing how

this is done, we introduce some background on Bell basis

measurements and introduce the required notation.

Bell basis measurements. Suppose we carry out M

rounds of Bell basis measurements. For each round

⊗

⋮

|0⟩

|ψ⟩

⋮

H H

SWAP

a)b)H H

H H

{

FIG. 1. c-SWAP circuits. a) Equivalent representations

of the single qubit controlled-SWAP circuit. b) The n-qubit

parallelized c-SWAP circuit can be used to probe a pure state

|ψi’s entanglement [6,7].

m∈ {1, . . . , M}, this consists of performing a Bell ba-

sis measurement on the k-th test and copy qubit for each

k∈ {1, . . . , n}, as shown pictorially in Fig. 2. Measuring

the k-th test and copy qubit in the Bell basis results in

one of the four Bell states as the post measurement state

B(m)

k∈|Φ+ihΦ+|,|Φ−ihΦ−|,|Ψ+ihΨ+|,|Ψ−ihΨ−|.

(3)

For our purposes, we consider B(m)

kas a random variable

that takes values in the set of Bell basis projectors. For

each of the Mrounds, we eﬃciently store the qubit label,

k, and the corresponding measurement outcome B(m)

kin

classical memory, which one can then post-process in a

number of ways to obtain many entanglement measures

of interest, as we will show.

To understand the power of Bell basis measurements,

ﬁrst note that the Bell states are eigenstates of the SWAP

operator with eigenvalues ±1, so tr hFkB(m)

ki=±1for

all k, m, where Fkis the SWAP operator acting on the

k-th test and copy qubit. This connection between the

SWAP operator and the Bell basis is why SWAP tests

can be simulated by Bell basis measurement methods.

For instance, one can construct an unbiased estimator of

the purity of a single qubit as

E"1

m=1

tr hFB(m)i#=tr ρ2,(4)

where the expectation is taken with respect to the em-

pirical distribution resulting from Bell basis measurement

outcomes. This method has been utilized by many ex-

perimental groups to estimate quantum purities [33–36].

In fact, from the data in Refs. [33–36], one could esti-

mate all possible subsystem purities of n-qubit states by

extending the idea in Eq. 4[41,43].

Multipartite entanglement from Bell basis measure-

ment data. Our main results are concerned with the

ancilla-free simulation of the parallelized c-SWAP test.

The following theorems show how to recover all results

of the c-SWAP test without the need for ancillary qubits

or Toﬀoli gates, thus making the resulting entanglement

measures far more experimentally accessible.

First, we show the existence of a family of unbiased

estimators for the CEs which depend solely on Bell basis

measurement outcomes.

Theorem 1. The quantities

C|ψi(s) = 1 −1

m=1 Y

k∈s



1 + tr hFkB(m)

2

,(5)

are unbiased estimators of the Concentratable Entangle-

ments. That is, for all s⊆ S,

Ehˆ

C|ψi(s)i= 1 −1

2|s|X

α∈P(s)

tr[ρ2

α],(6)

where the expectation value is with respect to the probabil-

ity distribution induced by the Bell basis measurements.

This theorem implies that the circuit in Fig. 1b can

be completely simulated by a projective Bell basis mea-

surement on two copies of a state of interest. Many

well-known entanglement measures can be estimated us-

ing this result. By letting s={j}, one obtains an

estimate of 1

2(1 −tr ρ2

j), which, when averaged over

all j∈ S yields the entanglement measure from Refs.

[44,45]. At the opposite extreme, when s=S, one ob-

tains a CE which is related to the generalized concurrence

cn(|ψi), as deﬁned in Ref. [46,47], via the simple formula

C|ψi(S) = cn(|ψi)2/4. This realization implies that the

entanglement measure being explored in Ref. [6] was ex-

actly the generalized concurrence as deﬁned in Ref. [46].

In between these two extremes, many other well-deﬁned

measures of multipartite entanglement can be estimated,

all from the same measurement data.

There is still more one can learn from Bell basis mea-

surement data, however. For instance, we can state

a very similar theorem for the n-tangle, another well-

studied multipartite entanglement measure [11].

Theorem 2. The quantity

ˆτ(n)=2n

m=1

k=1 



1−tr hFkB(m)

2

,(7)

is an unbiased estimator of the n-tangle. That is,

Eˆτ(n)=τ(n),(8)

where the expectation value is with respect to the probabil-

ity distribution induced by the Bell basis measurements.

With this theorem, we have recovered all of the mea-

sures shown in Ref. [7] to be computable with the par-

allelized c-SWAP test. In addition to requiring fewer ex-

perimental resources, it is simple to determine how many

⋮⋮⋮

10011011010101

00111011010101

01011011010101

prepare

ρ⊗ρ

measure

repeat times

Concentratable !

Entanglements

-tangle

Concurrences

Rényi entropies

post-process

estimate

A1B1

A2B2

AnBn

FIG. 2. Bell Basis Estimation Method. Experimentally,

one ﬁrst prepares the test state and an, ideally identical, copy

state. We denote the composite state ρ⊗ρ. Then the k-th

subsystems in the test and copy states are entangled using na-

tive one- and two-qubit gates. This converts a computational

basis measurement to a Bell basis measurement. The data

from Mrounds of this procedure is stored in classical mem-

ory which one can then post-process in a number of ways to

obtain many entanglement measures of interest.

rounds of Bell basis measurements are needed to achieve

an -close approximation of the estimators we have in-

troduced. We formalize this statement in the following

proposition, the proof of which follows directly from Ho-

eﬀding’s inequality from classical statistics.

Proposition 1. Let , δ > 0and M= Θ log 1/δ

2. Fur-

ther, let θ∈ {C|ψi(s), τ(n)}and let ˆ

θdenote the corre-

sponding estimator for θ. Then we have

ˆ

θ−θ< , (9)

with probability at least 1−δ.

This result, while simple and analytical, does not take

into account the underlying probability distribution, and

is thus not as tight as it could be. As we show in Fig.

4(b), using information about the underlying distribu-

tion, one ﬁnds numerically that Prop. 1often leads to

overestimates on the number of measurements needed to

obtain -close estimates of the quantities of interest.

Thus far, we have only considered estimating these

measures given two identical copies of a pure quantum

state. In order for these methods to be truly useful on

today’s hardware, we must extend to the measures to

mixed states.

CE for mixed states. The standard method of extend-

ing pure state entanglement measures to mixed states is

a so-called convex-roof extension [48,49]

Cρ(s) = inf X

piC|ψii(s),(10)

where the inﬁmum is over the set of decompositions of

the form ρ=Pipi|ψiihψi|, with Pipi= 1. Because

this optimization is generally diﬃcult, we would like to

avoid it. An alternative method is to ﬁnd lower bounds

for the mixed state CEs that depend only on Bell basis

measurement outcomes. This allows one to bound the

mixed state entanglement within the above framework

developed for estimating pure state entanglement.

We will construct the lower bounds on Cρ(s)using the

relationship between CEs and the bipartite concurrences

cα(|ψi)[50], as well as a known lower bound for the mixed

state bipartite concurrence [39]. Speciﬁcally, any CE can

be expressed in terms of bipartite concurrences as

C|ψi(s) = 1

2|s|+1 X

α(|ψi),(11)

where cα(|ψi) := p2(1 −tr [ρ2

α]). Then, because we can

use the known lower bound for each bipartite concurrence

in the sum, we can construct a lower bound for any CE

of interest. This is a generalization of the method used

in Ref. [40] in which the authors derive a lower bound on

the mixed state multipartite concurrence. For example,

the lower bound on Cρ(S)takes the form

ρ(S) = 1

2n+ (1 −1

2n)tr ρ2−1

2nX

α∈P(S)

tr ρ2

α.(12)

Because each term in this expression can be directly

estimated from Bell basis measurement data, it allows

one to quantify mixed state entanglement in the same

framework developed above for pure state entanglement.

We further note that, for high-purity states that are

common in today’s state of the art experiments, this

bound is very close to the pure state theoretical value,

as shown in Fig. 4. This can be seen by noting that

C|ψi(S)− C`

ρ(S) = (1 −2−n)(1 −tr ρ2),which is very

close to zero for nearly pure states [41]. With these

bounds in place, we turn to demonstrating the viabil-

ity of our proposed scheme via realistic Rydberg system

simulations.

Rydberg atom simulations with noise. In Fig. 3(a) we

illustrate the architectures that we propose for quantify-

ing the CE using the c-SWAP test and the Bell basis mea-

surement method in neutral atom systems. The c-SWAP

circuit is implemented by arranging each group of atomic

qubits {Ak, Bk, Ck}in an equilateral triangle, in such a

way that CZ and CCZ gates can be realized using the

Rydberg pulse sequences described in [26]. These global

unitaries are then transformed to CNOT and Toﬀoli gates

through the application of Hadamard gates to the target

qubit before and after the Rydberg pulses. The Bell basis

measurements are performed by applying Hadamard and

CNOT gates to the relevant pairs of qubits {Ak, Bk}and

then measuring in the computational basis. We model

the presence of experimental imperfections by substitut-

ing the ideal CZ and CCZ gates by non-unitary trans-

formations [41]. The application of these imperfect gates

on pure states results in phase errors and loss of norm,

which mimics the leakage of population outside of the

2 3 4 5 6 7 8

0.005

0.01

≤C(S)

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|Ci2

|Ai1

|Bi2

|Ai2

|Bi1

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Bell

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(a)

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(b)

2 3 4 5 6 7 8

0.025

0.05 c-SWAP

Bell

FIG. 3. c-SWAP vs. Bell Basis Method. a) Pictorial rep-

resentation of the geometry Rydberg atoms would be placed

in to implement either the c-SWAP or Bell methods for a two-

qubit state example. b) Top panel indicates the loss of norm

due to imperfect Rydberg pulses. Bottom panel shows the

error that this causes in estimation of the CE of a GHZ state.

In both cases, we see the Bell basis method outperforms the

c-SWAP.

computational basis under the application of the Ryd-

berg pulses. Since occurences of leakage can be detected

and discarded in the post-processing of the experimental

data, we re-normalize the state resulting from the appli-

cation of the non-unitary gates before computing its CE.

We keep track of the loss of norm for the purpose of es-

timating the number of repetitions required to achieve a

desired accuracy. For simplicity of notation, we denote

the CE computed over renormalized pure states as C(S).

The Bell measurement method oﬀers a substantial

practical advantage with respect to the c-SWAP test for

estimating C(S)due to its reduced requirement on the

number of copies and its signiﬁcantly lower total gate

count. To illustrate this, in Fig. 3we compare the re-

sults obtained when measuring with both methods the

CE for an n-qubit Greenberger–Horne–Zeilinger (GHZ)

state |GHZni=1

√2(|0i⊗n+|1i⊗n). In the lower plot of

Fig. 3(b) we show the relative discrepancy C(S)between

the value of C(S)obtained with each method and the an-

alytical result [6,7] as a function of n. We observe that

for all numbers of qubits the Bell measurement method

yields more accurate results than the c-SWAP test due

to the reduction in accumulated phase errors. The upper

plot of Fig. 3(b) shows that the loss of norm is smaller for

the Bell measurement method than for the c-SWAP test,

meaning that the former method would require fewer rep-

etitions to achieve a given level of accuracy than the lat-

ter.

Having established the superiority of the Bell measure-

ment method in the presence of experimental imperfec-

tions, we turn to investigating its performance for es-

timating C(S)for diﬀerent classes of highly entangled

states. In the lower plot of Fig. 4(a) we show the the-

10°310°210°1

95% CI

102

103

104

105

HoeÆding

CP, n=4

CP, n= 12

3 15

0.02

3 5 7 9 11 13 15

0.25

0.5

0.75

C(S)

Line

GHZ

(a)

(b)

1|h | i|2

2 12

0.01

3 5 7 9 11 13 15

10°3

≤C(S)

FIG. 4. Bell basis method with realistic Rydberg gates

a) Bottom panel: CE of GHZ, W and Line states, with solid

lines indicating theoretical values and dots representing the

results obtained with noisy Rydberg gates. Top panel: rela-

tive discrepancy between the theoretical and simulated values

of the CE. b) Number of measurements required as a function

of the desired size of the 95% CI on the precision of the esti-

mation of the CE for a Line state with diﬀerent numbers of

qubits. The inset shows the loss of norm for this state when

the CE is estimated with the Bell basis method.

oretical (solid lines) and simulated experimental (dots)

values of C(S)as a function of the number of qubits n

for GHZ, W and Line states (all of which admit analyt-

ical formulas which are given in the Supplementary Ma-

terials). We observe that values of C(S)remain clearly

distinguishable between the three states up to n= 15.

Furthermore, as illustrated in the upper plot the relative

discrepancy between the theoretical and simulated val-

ues remains C(S).10−3for the range of nand states

considered. In Fig. 4b we show the size of the 95% Con-

ﬁdence Interval (CI) in the Maximum Likelihood Esti-

mation of the C(S)for a Line state of n= 4,12 qubits

computed with the Clopper-Pearson (CP) method as a

function of the total number of measurements M, as well

as the bound provided by Hoeﬀding’s inequality. The

CP method predicts a lower requirement in the num-

ber of measurements to achieve a given size of the CI

because it is tailored to the binomial probability distri-

bution that governs the statistics of C(S)measurements,

but Hoeﬀding’s inequality provides a useful bound which

is easy to compute analytically. The inset of Fig. 4(b)

shows the loss of norm as a function of the number of

qubits. Even for n= 15 the norm of the state remains

|hΨ|Ψi|2∼0.98, meaning that the number of experiment

repetitions would only need to be increased by .2% to

make up for the leakage outside of the computational

basis.

Conclusion. We have shown how to estimate the CEs

and n-tangle from Bell basis measurement data. We ex-

tended the deﬁnition of the CEs to mixed states and

showed how to estimate lower bounds on the mixed state

CE from Bell basis measurement data. Our methods si-

multaneously make these measures more experimentally

accessible, while also simplifying their associated theo-

retical analysis.

An interesting direction for future work would be to

compare, in terms of both theoretical sample complexity

and performance on real hardware, this Bell basis method

to local randomized measurements [51–56] and classical

shadows [57,58], two modern techniques that have been

applied to the study of other entanglement measures.

Acknowledgments

JLB was initially supported by the National Science

Foundation Graduate Research Fellowship under Grant

No. 1650115 and was partially supported by NSF grant

1915407. This material is based upon work supported

by the U.S. Department of Energy, Oﬃce of Science, Na-

tional Quantum Information Science Research Centers,

Quantum Systems Accelerator. JLB also acknowledges

helpful discussions with Hsin-Yuan (Robert) Huang,

Michael Walter, Graeme Smith, Guangkuo Liu, and

Louis Schatzki. GP and NJP are supported by the EP-

SRC (Grant No. EP/T005386/1) and M Squared Lasers

Ltd. SF is supported by a UK EPSRC funded DTG stu-

dentship (ref 2210204) and thanks Tim Spiller and Viv

Kendon for many useful discussions. GP and NJP ac-

knowledge fruitful discussions with Jonathan Pritchard

and Andrew Daley.

[1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki,

and Karol Horodecki, “Quantum entanglement,” Rev.

Mod. Phys. 81, 865–942 (2009).

[2] Nicolai Friis, Giuseppe Vitagliano, Mehul Malik, and

Marcus Huber, “Entanglement certiﬁcation from theory

to experiment,” Nature Reviews Physics 1, 72–87 (2018).

[3] Vlatko Vedral and Elham Kasheﬁ, “Uniqueness of the

entanglement measure for bipartite pure states and ther-

modynamics,” Phys. Rev. Lett. 89, 037903 (2002).

[4] Jeongwan Haah, Aram W. Harrow, Zhengfeng Ji, Xi-

aodi Wu, and Nengkun Yu, “Sample-optimal tomogra-

phy of quantum states,” IEEE Transactions on Informa-

tion Theory 63, 5628–5641 (2017).

[5] Ryan O’Donnell and John Wright, “Eﬃcient quantum

tomography,” (2015).

[6] Steph Foulds, Viv Kendon, and Tim Spiller, “The

controlled swap test for determining quantum entan-

glement,” Quantum Science and Technology 6, 035002

(2021).

[7] Jacob L. Beckey, N. Gigena, Patrick J. Coles, and

M. Cerezo, “Computable and operationally meaningful

multipartite entanglement measures,” Phys. Rev. Lett.

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MultipartiteentanglementmeasuresviaBellbasismeasurementsJacobL.Beckey,1,2GerardPelegrí,3StephFoulds,4andNatalieJ.Pearson31JILA,NISTandUniversityofColorado,Boulder,Colorado80309,USA2DepartmentofPhysics,UniversityofColorado,Boulder,Colorado80309,USA3DepartmentofPhysicsandSUPA,UniversityofStrathclyde,G...

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Multipartite entanglement measures via Bell basis measurements Jacob L. Beckey1 2Gerard Pelegrí3Steph Foulds4and Natalie J. Pearson3 1JILA NIST and University of Colorado Boulder Colorado 80309 USA

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