Witnessing entanglement without entanglementwitness operatorsLuca Pezzèa,b,c, Yan Lid, Weidong Lid,1, and Augusto Smerzia,b,c

aQuantum Science and Technology in Arcetri, 50125 Florence, Italy; bNational Institute of Optics, National Research Council, 50125 Florence, Italy; cEuropeanLaboratory for Non-Linear Spectroscopy, 50125 Florence, Italy; and dInstitute of Theoretical Physics and Department of Physics, Collaborative InnovationCenter of Extreme Optics, Shanxi University, Taiyuan 030006, China

Edited by Anton Zeilinger, University of Vienna, Vienna, Austria, and approved July 26, 2016 (received for review March 7, 2016)

Quantum mechanics predicts the existence of correlations betweencomposite systems that, although puzzling to our physical intuition,enable technologies not accessible in a classical world. Notwithstand-ing, there is still no efficient general method to theoretically quantifyand experimentally detect entanglement of many qubits. Here wepropose to detect entanglement by measuring the statistical responseof a quantum system to an arbitrary nonlocal parametric evolution.Wewitness entanglement without relying on the tomographic reconstruc-tion of the quantum state, or the realization of witness operators. Theprotocol requires two collective settings for any number of parties andis robust against noise and decoherence occurring after the imple-mentation of the parametric transformation. To illustrate its userfriendliness we demonstrate multipartite entanglement in differentexperiments with ions and photons by analyzing published data onfidelity visibilities and variances of collective observables.

quantum entanglement | entanglement detection | quantum technology |Fisher information | trapped ions

Acentral problem in quantum technologies is to detect andcharacterize entanglement among correlated parties (1–3).

A most popular approach is based on the implementation ofentanglement witness operators (EWs). EW is a Hermitian op-erator W such that Tr½ρsepW�≥ 0 for all separable states ρsepand Tr½ρW�< 0 for, at least, one entangled state ρ (4–8). Thepower of this method relies on the algebraic fact that for eachmultipartite entangled state there exists (at least) one EW thatrecognizes it (4). EWs are device-dependent: Experimentalmischaracterization may lead to false positives, namely, tothe unwitting realization of operators Wexp ≠W such thatTr½ρsepWexp�< 0 also for some separable states, therefore signalingentanglement in states that are only classically correlated (9, 10).The same problem arises, even more dramatically, when trying todetect entanglement via the tomographic reconstruction of thequantum state (10–15).Device-independent entanglement witness operators (DIEWs)

can be constructed by exploiting Bell-like inequalities testing thecorrelations between measurement data. These are obtained fordifferent optimized configurations and demand that the partiesbe addressed locally and do not interact during operations andmeasurements. DIEWs recognize an important class of entangledstates without relying on any hypothesis about the measurementactually performed (16–19). The specific configurations requiredto witness entanglement are only known for particular cases; theextension to an arbitrary state can vary from computationally hardto prohibitive. A recent experiment (20) has exploited Bell-basedDIEWs to demonstrate genuine multipartite entanglement upto six trapped ions. Cross-talk among the particles affected theDIEW of larger systems.An alternative approach to detect entanglement that does not

require the construction of a witness operator has been proposed(21–23). This exploits the Fisher information (or “statisticalspeed”) quantifying how quickly two slightly different quantumstates become statistically distinguishable under a parametrictransformation that, as in the case of DIEW, is local. However,

local probing preserves the amount of entanglement in thesystem and belongs to a restricted subset of all possible unitarytransformations. As a consequence, this method recognizesan important but relatively small class of entangled states.Furthermore, the strict condition of local probing might notbe satisfied experimentally, thus preventing the use ofthis method.In this paper we demonstrate that a quite broader class of

entangled states can be detected by the statistical distinguish-ability of two quantum states connected by a transformation thatis nonlocal, namely, that couples different qubits (Fig. 1). Ourresults show that the statistical speed of an initially classicallycorrelated system is strictly upper-bounded even if the trans-formation generates entanglement. A higher speed witnessesentanglement on the initial state. This can open the way to studymultipartite entanglement near quantum phase transition criticalpoints by quenching the parameters of Ising-like Hamiltonians.Furthermore, because statistical distinguishability is directly re-lated to the sensitivity of parameter estimation, our study allowsus to characterize entanglement-enhanced precision measure-ments in many-body systems. The method also allows us to ex-plicitly take into account residual cross-talking between differentqubits. However, as will be explained below, our approach is notdevice-independent because it requires the experimental controlof the nonlocal transformation. However, any noise, decoher-ence, or finite efficiency readout occurring after the parametrictransformation (Fig. 1) does not lead to false positives. A char-acteristic trait of witnessing entanglement via statistical distin-guishability is its simplicity. We indeed demonstrate, by justelaborating on published experimental data (24–28), entangle-ment up to 14 ions and 10 photons, and, in agreement with Bell-based DIEW results reported in ref. 20, genuine multipartiteentanglement up to six ions.

Significance

The experimental detection and theoretical characterizationof entanglement in many-body systems is a mostly unsolvedproblem. Here we relate entanglement to the statistical speedmeasuring how quickly nearby states become distinguishableunder the action of an arbitrary many-body Hamiltonian. Themethod is remarkably simple: We witness multipartite entan-glement, just elaborating on published experimental data withions and photons. The unveiled connection between the sta-tistical speed and entanglement provides a unifying frame-work that can shed new light on quantum information scienceand quantum critical phenomena.

Author contributions: L.P., Y.L., W.L., and A.S. performed research and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. Email: wdli@sxu.edu.cn.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1603346113/-/DCSupplemental.

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ResultsWe consider N qubits in a state ρ and apply the collective unitarytransformation e−iHθ parameterized by a real number θ, where

H =XNi=1

αi2σðiÞm +

XNi, j=1

Vij

4σðiÞn σðjÞn . [1]

The coefficients αi (without loss of generality, 0≤ αi ≤ 1) accountfor (possibly) inhomogeneous linear couplings (29) or local/sub-groups operations on the parties, and Vij =Vji provides a nonlocalinteraction between different spins. Here, σðiÞn ≡ σðiÞ ·n is thePauli matrix for the ith particle and n is a versor in the Blochsphere. The unitary transformation is eventually followed by arbi-trary noise and decoherence that, in full generality, we model as acompletely positive trace-preserving map Λ that does not dependon θ. Finally, the output state is characterized by the statisticalprobability distribution PðμjθÞ of possible measurement results μof a readout observable M. The statistical distinguishability be-tween the two probability distributions Pðμjθ0Þ and PðμjθÞ is quan-tified by the Hellinger distance (30)

ℓðθ0, θÞ≡ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXμ

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPðμjθ0Þ

p−

ffiffiffiffiffiffiffiffiffiffiffiffiffiPðμjθÞ

p �2s, [2]

with the sum extending over all possible μ. ℓ is a statistical distance(31, 32): It ranges from zero, if and only if Pðμjθ0Þ=PðμjθÞ ∀μ, toits maximum value ℓ= 2

ffiffiffi2

p, if and only if Pðμjθ0Þ×PðμjθÞ= 0 ∀μ,

and it satisfies the triangular inequality. The Hellinger distance isproportional to the Euclidean distance

�� ffiffiffiffiffip0

p − ffiffiffiffiffipθ

p �� between theunit vectors

ffiffiffiffiffip0

p= f ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pðμjθ0Þp gμ and

ffiffiffiffiffipθ

p= f ffiffiffiffiffiffiffiffiffiffiffiffiffi

PðμjθÞp gμ. It is use-ful to introduce the notion of statistical speed:

υ≡dℓdθ

����θ0

, [3]

that is, the rate at which ℓðθ0, θÞ changes with θ around thereference point θ0. A Taylor expansion of Eq. 2 gives

υ2 =P

μ1

Pðμjθ0Þ�dPðμjθÞ

dθ

���θ0

�2, which coincides with the Fisher infor-

mation (33). The specific measurement observable and the para-metric transformation entering in Eqs. 2 and 3 via the conditionalprobabilities are arbitrary but, in practice, chosen so to efficiencydistinguish PðμjθÞ from Pðμjθ0Þ. Extracting the statistical speedrequires (at least) two settings, independent from the number ofparticles, the quantum state and the measurement observable. Astatistical speed higher than the bound

υsep ≡ maxjψ sepi,Λ,M

υ, [4]

witnesses entanglement in the state ρ. υsep is obtained by maxi-mizing the statistical speed over all pure separable states��ψ sepi=

��ψ ð1Þi⊗ . . . ⊗��ψ ðNÞi, all possible maps Λ, and all (in-

cluding noisy and biased) measurements M. Because of the con-vexity of the Fisher information (21), the bound 4 holds not onlyfor product pure states but also for any statistical mixture ofproduct states (i.e., for arbitrary classically correlated statesρsep =

Pλqλ��ψ sep,λihψ sep,λ

��, with qλ ≥ 0). The lower bound in Eq.4 is the maximum quantum statistical speed of separable states,υsep =maxjψ sepiυQ. For a generic state ρ, the quantum statisticalspeed is defined as υQ ≡maxMυ and its square can be calculatedas υ2Q =Tr½ρL2� (32), where L is the symmetric logarithmic de-rivative defined via the relation dρ

dθ=Lρ+ ρL

2 (32, 33). Recently, ithas been shown that the quantum statistical speed is linked to thedynamic susceptibility (34) and it is thus available in condensed-matter experiments. Notice that the quantum statistical speed canonly remain constant or decrease under the action of a θ-indepen-dent map Λ following the unitary transformation (30). We remarkthat the Λ can be nonlocal, namely, it can generate entanglement.Because the bound 4 is obtained by maximizing over all general-ized readout, no false positives are possible in the presence ofarbitrary noise and decoherence affecting the state after theparametric transformation.As derived in Materials and Methods, the quantum statistical

speed of a pure product state��ψ sepi probed by the Hamiltonian

in Eq. 1 is

υ2Q = υ20 + υ21 + υ22, [5]

where

υ20 =XNi=1

α2i

�1−DσðiÞmE2�

, [6]

υ21 = 2Xi, j=1i≠ j

Vijαihn ·m−

DσðiÞnED

σðiÞmEiD

σðjÞnE, [7]

and

υ22 =XNi, j=1i≠ j

V 2ij

2

1−DσðiÞnE2D

σðjÞnE2

+XNi, j, l=1i≠ j≠ l

VijVil

1−DσðiÞnE2D

σðjÞnED

σðlÞnE. [8]

For local Hamiltonians (Vij = 0), the maximization in Eq. 4 is readilydone: The optimal states have hσðiÞm i= 0 ∀i, giving υ2sep =

PNi=1α

2i . This

generalizes the bound υ2sep =N discussed in refs. 21 and 35 for ho-mogeneous coupling αi = 1. For nonlocal Hamiltonians (Vij ≠ 0) the

A

B C

Fig. 1. Statistical distinguishability and statistical speed. (A) N parties pre-pared in a quantum state ρ undergo a unitary nonlocal transformation with θa tunable parameter. The completely positive trace-preserving map Λ includesarbitrary θ-independent decoherence effects. (B) The probability distributionPðμjθÞ is obtained by collecting the measurement results μ for different valuesof the parameter, here chosen to be θ0 (red line) and θ0 + δθ (green line). (C) Toquantify the statistical distinguishability between the two distributions weintroduce unit vectors

ffiffiffiffiffiffip0

p= fPðμjθ0Þ1=2gμ (red) and

ffiffiffiffiffiffiffipδθ

p= fPðμjθ0 + δθÞ1=2gμ

(green) and measure the Euclidean distance among them: ℓ= 2�� ffiffiffiffiffiffi

p0p − ffiffiffiffiffiffiffi

pδθp ��

(dashed line), Eq. 2. The statistical speed, Eq. 3, is an entanglement witness.

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bound depends on the explicit form of Vij and αi. It can be calculatedeither numerically or, as shown below, analytically in many cases ofinterest. In the following we consider, as an example, the Ising modelhaving nearest-neighbor interaction Vij = γ

δj,i+1 + δj,i−12 and n ·m= 1.

In Supporting Information we also report the results for theLipkin–Meshkov–Glick model where Vij = γ. The states thatmaximize Eq. 5 are

jψi= ⊗N

i=1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1+DσðiÞnE

2

vuut  j↑ii + e−iφi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−DσðiÞnE

2

vuut  j↓ii, [9]

where j↑i and j↓i are eigenstates of σn, φi are arbitrary phases,and hσðiÞn i as a function of γ is reported in Fig. 2A. Fig. 2Bshows υ2sep as a function of γ. The numerical analysis in thehomogeneous case αi = 1 reveals that, for γ smaller than acritical value γc, the speed υ2Q is maximized when hσðiÞn i are allequal. In particular, for γ � 1, υ2Q is the highest when hσðiÞn i= γ∀i, giving υ2sep=N = 1+ 5

4γ2 +Oðγ4Þ (solid blue line in Fig. 2B).

The value γc = 0.7302 is found analytically, as discussed inMaterials and Methods. For γ > γc, υ2QðγÞ is maximized by alter-nating hσðiÞn i= 1 and hσði+1Þn i= 0. In this limit, we find υ2sep=N =12+ γ + 1

2γ2 (solid blue line in Fig. 2B). An upper bound to υ2sep,

valid in the inhomogeneous case (αi ≠ 0) and for every γ,can be obtained by maximizing each term in Eq. 5 separately.This gives

υ2sep ≤XNi=1

α2i + γ  max

Xodd  i

αi,Xeven  i

αi

!+ γ2

N2, [10]

which is shown as the dashed red line in Fig. 2B.It is important to emphasize that different Hamiltonians H

detect different subsets of entangled states. Local Hamiltonians,

H0 = 12

PNi=1σ

ðiÞm , are well-suited to detect entangled states that

are symmetric under particle exchange. For instance, the squarequantum statistical speed of the Dicke state j↑i⊗N=2−νj↓i⊗N=2+ν

probed by a local Hamiltonian H0 is υ2Q =N2=2− 2ν2 +N (herewe consider n perpendicular to m). That is υ2Q > υ2sep =N forν≠ ±N=2: All Dicke states are detected as entangled except the(separable) spin polarized. Entangled nonsymmetric states arebetter detected by nonlocal Hamiltonians. Let us consider, for in-stance, the state of N spins jχi= ðj↑↓i⊗N=2 + j↑i⊗N=2j↓i⊗N=2Þ= ffiffiffi

2p

.It is possible to demonstrate that, when applying e−iH0θ, the quan-tum speed of jχi is smaller than the bound υ2sep =N for N > 6, evenwhen optimizing the directionm in the Hamiltonian H0. Therefore,jχi cannot be detected as entangled when probed by only localHamiltonians. Conversely, when probed by a nonlocal nearest-neighbor Hamiltonian H1 = 1

4

PNi=1σ

ðiÞn σði+1Þn , the state jχi has a

square quantum statistical speed equal to N2=4−N + 1 that sur-passes the bound υ2sep =N=2 if N ≥ 6. jχi can thus be detected asentangled when probed by the Ising Hamiltonian. The opposite isalso true. For instance the Greenberger–Horne–Zeilinger (GHZ)state jφi= ðj↑i⊗N + j↓i⊗NÞ= ffiffiffi

2p

has a null statistical speed whenprobed with H1. Nevertheless jφi can reach a statistical speedN2 >N and it can thus be detected as entangled when probed bythe local Hamiltonian H0.Beside the possibility to witness a larger class of entangled

states, the measurement of the statistical speed generated bynonlocal Hamiltonians allows to take into account the residualcoupling among neighboring spins. This, in contrast, can limit theexperimental implementation of Bell-based DIEWs, in particularwhen dealing with a large number of ions (20).

ApplicationsOur method to witness entanglement requires one to experi-mentally extract the statistical speed. We show below that thiscan be obtained from the visibility of fringe oscillating as afunction of θ, from moments of the probability distribution or,more generally, by exploiting a basic relation between the sta-tistical speed and the Kullback–Leibler entropy. In the absenceof experimental data obtained with a nonlocal probing Hamil-tonian, we apply our protocol to extract the statistical speed frompublished data in ions and photons experiments where a localtransformation was used. In this case, the above method can beextended as a witness of multiparticle entanglement (22, 23):The inequality

υ2 > sk2 + r2, [11]

signals ðk+ 1Þ-partite entanglement (i.e., among N parties, atleast k are entangled), where s is the largest integer smaller thanor equal to N=k and r=N − sk. In particular υ2 > ðN − 1Þ2 + 1,obtained from Eq. 11 with k=N − 1, is a witness of genuineN-partite entanglement.

Statistical Speed from Dichotomic Measurements.We consider herethe simplest (but experimentally relevant) case where the mea-surement results can only take two values, μ=±1. In this case, Eqs.2 and 3 simplify to ℓ2 = 8½1− ffiffiffiffiffiffiffiffiffiffiffiffi

P0Pδθp

−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1−P0Þð1−PδθÞ

p � and

υ2 =1

P0ð1−P0Þ�∂Pθ

∂θ

����θ0

�2

, [12]

respectively, where P0 ≡Pð+1jθ0Þ and Pδθ ≡Pð+1jθ0 + δθÞ. Forinstance, if

A

B

Fig. 2. Witness of entanglement with the Ising Hamiltonian. (A) Mean spinvalues hσðiÞn i for the separable state maximizing υ2Q, as a function of γ.(B) Maximum statistical speed of separable states, υ2sep (dots), probed by theIsing Hamiltonian. Entanglement is witnessed by a statistical speed υ2 > υ2sep,in the gray region. The solid blue lines are analytical limits discussed in themain text. The dashed line is the upper bound Eq. 10.

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Pð±1jθÞ= 1±V cosNθ

2, [13]

where 0≤V ≤ 1 is the visibility of the oscillating probabilities, wecan straightforwardly calculate Eq. 12, obtaining

υ2 =V 2N2sin2ðNθÞ1−V 2cos2ðNθÞ. [14]

It is thus possible to detected entanglement when V > 1ffiffiffiN

p . Noticethat, with an increasing number of qubits N the required mini-mum visibility to detect entanglement decreases. Genuine

N-partite entanglement is detected when V >

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1− 1

N

�2+ 1

N2

r,

which requires a visibility increasing with N.

Statistical Speed from Average Moments. The probability of differentmeasurement results are not always available, but only some aver-aged moments hμiθ =

PμμPðμjθÞ. We can extend the notion of

Hellinger distance and statistical speed to the probability distribu-tion PðμjθÞ, where μ= 1

m

Pmi=1μi and μ1, . . . μm are measurement

results. We find ℓ2mom = 4P

μðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPðμjθ0Þ

p−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPðμjθ0 + δθÞp Þ2 and

υ2mom =Xμ

1Pðμjθ0Þ

�dPðμjθÞ

dθ

����θ0

�2

, [15]

where the sum extends over all possible values of μ. Using aCauchy–Schwarz inequality it is possible to demonstrate (Sup-porting Information) that υmom ≤ υ

ffiffiffiffim

p. For m � 1, the central

limit theorem provides

PðμjθÞ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m

2πðΔμÞ2θ

se−mðμ−hμiθÞ2

2ðΔμÞ2θ , [16]

where ðΔμÞ2θ =P

μðμ− hμiθÞ2PðμjθÞ. To the leading order in m,replacing Eq. 16 into Eq. 15, we obtain

υ2mom =m

ðΔμÞ2θ

�dhμiθdθ

����θ0

�2

. [17]

The entanglement criteria thus becomes υ2mom=m> υ2sep. Whenprobing with local Hamiltonians, the inequality υ2mom=m> sk2 + r2witness ðk+ 1Þ-partite entanglement from the experimental mea-surements of average moments. These bounds generalize to arbi-trary observables the bounds to detect entanglement (36) andmultipartite entanglement (37) from the estimation of the meancollective spin (1, 38).

Witnessing Multipartite Entanglement in Trapped-Ion Experiments.Several recent efforts have been devoted to create GHZ statesðj↑i⊗N + j↓i⊗NÞ= ffiffiffi

2p

with trapped ions (24–26). In ref. 24 thecreation of the state has been followed by a collective rotation⊗N

j=1eiπ2σ

ðjÞθ , with σðjÞθ = σðjÞx cos θ+ σðjÞy sin θ. The output state has

been characterized by dichotomic measurements of the parity,Π= ð−1ÞN↑, with N↑ being the number of qubits measured in oneof the two modes. The reported results are the oscillations of theaverage parity (and, therefore, of the probability to obtain the ±1result) as a function of θ (cf. Eq. 13), and we can directly analyzethe experimental data with our multipartite entanglement wit-ness. In Fig. 3, filled blue circles are obtained from data reportedin ref. 24 for N = f2 . . . 6,8,10,12,14g, open blue circles from thedata of ref. 25 for N = 3, and of ref. 26 for N = 4,5,6. We firstnotice that all data satisfy υ2 >N: We thus detect entanglementin all of the states created in refs. 24–26. The different coloredregions correspond to different k-partite entanglement detection

(delimited by solid thin lines given by Eq. 11). In particular,genuine N-partite entanglement is marked by the darker redregion that, from the data of ref. 24, is reached up to N = 6 ions.The maximum value of υ2 is obtained for N = 8 particles, corre-sponding to seven-partite entanglement. The number of entan-gled particles in the system slowly decreases for increasing N: Wehave four-partite entanglement for the states of N = 10 ions andthree-partite entanglement for the state of N = 12 and N = 14ions. It is interesting to notice that a recent experiment (20) hasinvestigated a Bell-based DIEW reporting genuine multipartiteentanglement up to N = 6 ions, which is in agreement withour finding.

Witnessing Multipartite Entanglement in Photon Experiments. Severalexperiments have demonstrated the creation of multipartite en-tanglement in photonic systems (27, 28, 39–41). In particular, ref. 28reports on the creation of a GHZ state up to 10 photons. After thecreation of the state by parametric down-conversion, a phase shift isapplied to each qubit, according to the scheme of Fig. 1. The state isfinally characterized by measuring the operator σ⊗N

x , whose meanvalue shows high-frequency oscillations, hσ⊗N

x i=V cosNθ (28).Noticing that ðΔσ⊗N

x Þ2 = 1− hσ⊗Nx i2, we can calculate the corre-

sponding statistical speed from Eq. 17 to obtain υ2momm = V 2N2sin2ðNθÞ

1−V 2cos2ðNθÞ.Also in this case, the witness of multipartite entanglement is solelybased on the visibility of the interference signal. Results are shownin Fig. 3 (filled squares). We witness four-partite entanglementfor the state of N = 8 photons, giving the highest value of the sta-tistical speed reached with photons. Because υ2mom is a lower boundof Eq. 3, the filled squares in Fig. 3 are a lower bound formultipartite entanglement.

Statistical Speed from the Kullback–Leibler Entropy. So far we haveextracted the statistical speed by fitting the experimental prob-abilities of the different detection events. This simple approachcan be implemented when the probabilities can be accuratelyfitted with a single parameter function, as in the ions and pho-tons experiments discussed above. In general, it might be nec-essary to extract the statistical speed directly from the bare data

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

10

20

30

40

50

2

NFig. 3. Witness of multipartite entanglement. Squared statistical speed as afunction of the number of qubits obtained analyzing published ions (circles)and photon (squares) experimental data: ref. 24, filled circles; ref 25 for N= 3and ref. 26 forN= 4,5,6, open circles; ref. 27, open square; ref. 28, filled squares.When not visible, error bars are smaller than the symbol size. The upper thickline is the upper bound υ2 =N2, and the lower thick line is the separabilitybound υ2 =N. The different lines are bound for k-partite entanglement, Eq. 11.In particular, the darker red region stands for genuine N-partite entanglement.

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without fitting the probability distribution. This can be done ex-perimentally by estimating the Kullback–Leibler (KL) entropy (42)

DKL =Xμ

Pðμjθ0Þln Pðμjθ0ÞPðμjθ0 + δθÞ. [18]

The KL entropy grows quadratically for small δθ with a coefficientproportional to the squared statistical speed (3), DKL = υ2δθ2=2. InFig. 4A we illustrate the method using experimental data of ref. 24.We focus on the case N = 8 and calculate DKL around θ0 ≈ π=ð2NÞaccording to Eq. 18. A quadratic fit provides υ2 = 44.6± 7.7, inagreement with the result (υ2 = 39.6± 0.8) obtained using Eq. 13(Fig. 3). The large error bars are due to the finite sample statisticsof the published data and can be reduced by increasing the samplesize and concentrating the measurements around a few phasevalues (rather than for the whole 2π interval).To supply to the lack of available experimental data and for

illustration purposes, here we implement a numerical MonteCarlo analysis of Eq. 18 to evaluate the role of δθ and the samplesize, taking, as a testing ground the parity measurements withprobability (13). In Fig. 4B we plot Eq. 18 in the case N = 8 andV = 0.787, around θ0 ≈ π=ð2NÞ. The figure shows that the qua-dratic behavior is obtained at sufficiently large δθ (SupportingInformation). For comparison, we also show the Hellinger dis-tance as a function of δθ, which can also be exploited to extractexperimentally the Fisher information (43). Notice that, forthese values of the visibility, due to higher-order terms, thequadratic approximation of DKL holds at larger values of δθ thanthe ℓ2 expansion.A source of noise in the extraction of υ2 is the limited statistics of

the measurement data (other sources of noise, such as detectionnoise and decoherence, result in a reduced visibility). Let us indicateas m the sample size (we consider m measurements performed at

phase θ0 andm measurements performed at phase θ= θ0 + δθ). Theexperiment gives access to frequencies rather than probabilities, andEq. 18 extends as ~DKL = f0ln

f0fδθ+ ð1− f0Þln 1− f0

1− fδθ, where f0 = neven,0=m

is the frequency of even parity results obtained at phase θ0 (andanalogous definition for fδθ). Numerically, we can calculate f0 andfδθ by a Monte Carlo sampling of the probabilities P0 and Pδθ,respectively. In the large-m limit, we can calculate statisticalfluctuations of the Hellinger distance by taking f0 =P0 + δf0 andfδθ =Pδθ + δfδθ, and then expanding ~DKL in Taylor series for smallδfδθ. We calculate the bias of the squared Hellinger distance,b≡ h~DKLi−DKL, where brackets indicate statistical averaging. InSupporting Information we show that the bias is positive but de-creases as b∼ 1=2m with the sample size m. Fig. 4C shows acomparison between a numerical Monte Carlo analysis and theanalytical prediction. Similarly, we can evaluate statistical fluctu-

ations of the KL entropy, Δ2 ~DKL = h~D2KLi− h~DKLi2. We obtain

Δ2 ~DKL ∼ 1=m, showing, also in this case, a scaling inversely pro-portional to the sample size. Details of our analytical calculationsare reported in Supporting Information, and a comparison withanalytical calculations is shown in Fig. 4D. Overall, the simulationsshow that few hundred measurements are sufficient to extract υ2with a small statistical bias and large signal-to-noise.

DiscussionThe statistical speed reveals and quantifies entanglement amongN parties. This requires one to probe a quantum state with ageneric but known parametric transformation. After the state hasbeen transformed, any coupling with a decoherence source or anonoptimal choice (or noisy implementation) of the readoutmeasurement does not lead to a false detection of entanglement.A false positive can be obtained as the consequence of a mis-characterization of the Hamiltonian H, or of the value of theparameter θ. Notice that the upper bound on the statistical speedυsep, Eq. 4, can be further maximized over all possible directionof the Pauli matrices acting on each qubit. This bound can beused in the presence of systematic errors in the direction of Paulimatrices of the Hamiltonian implemented experimentally. Theprotocol also includes generic nonlocal interactions due toexperimental tuning or accidental cross-talk effects. The numberof operations required to witness entanglement does not increasewith the number of parties: The statistical speed is extracted fromthe knowledge of, at least, two probability distributions obtained atnearby values of θ. Several experiments have shown the feasibilityof controlled collective parametric transformations with cold (44,45) and ultracold atoms (43, 46, 47), ions (24–26, 48, 49), photons(28, 41), and superconducting circuits (50). It should beemphasized that not all entangled states probed by Eq. 1 have astatistical speed larger than all possible separable states, even ina noiseless scenario and with optimized output measurements.However, the entangled states violating Eq. 4 with a nonlocalHamiltonian are those, and only those, overcoming the maximuminterferometric phase sensitivity limit achievable with separablestates and a phase-encoding transformation generated by thegiven Hamiltonian.

Materials and MethodsDerivation of Eqs. 5–8. For pure states and unitary transformation e−iθH, thequantum statistical speed is given by υ2Q = 4ðΔHÞ2. Taking H=H0 +H1, υ2Q is givenby Eq. 5 with υ20 = 4ðΔH0Þ2, υ21 = 4ðÆfH0,H1gæ− 2ÆH0æÆH1æÞ and υ22 = 4ðΔH1Þ2.We detail here the calculation of υ22 for product pure states, whereH1 =

PNi,j=1

Vij

4 σðiÞn σðjÞn . We have υ22 =

Pi,j,k,l

VijVkl

4 ½ÆσðiÞn σðjÞn σðkÞn σðlÞn æ− ÆσðiÞn σðjÞn æÆσðkÞn σðlÞn æ�.Notice that the terms i= j and/or k= l do not contribute to υ22. When i≠ j andk≠ l, we have ÆσðiÞn σðjÞn æÆσðkÞn σðlÞn æ= ÆσðiÞn æÆσðjÞn æÆσðkÞn æÆσðlÞn æ, whereas ÆσðiÞn σðjÞn σðkÞn σðlÞn æ=ÆσðiÞn æÆσðjÞn æÆσðkÞn æÆσðlÞn æ only if the indexes i, j, k, l are all different. Therefore, onlyterms where at least two indexes are equal contribute to υ22. The terms k= i, l= jand k= j, l= i both contribute with

Pi,jV

2ij ð1− ÆσðiÞn æ2ÆσðjÞn æ2Þ=4. After straight-

forward algebra, taking into account all contributing terms, one arrives at Eq. 8.

A B

DC

Fig. 4. KL entropy and statistical speed. (A) KL entropy as a function of δθobtained from an analysis of the experimental data of ref. 24 for N= 8 ions.Blue dots are calculated using Eq. 18. The blue line is a parabolic fit,DKL = δθ2υ2=2. The colored region corresponds to multipartite entanglementlevel, with color scale as in Fig. 3. (B) Squared statistical distance, ℓ2 (red line),KL entropy, 2DKL (green line), and their common low-order approximation,δθ2υ2 (blue line), as a function of δθ. (C and D) Numerical simulation of the KLentropy (dots) as a function of the sample size m, for 2Nδθ=π =0.4. Solid linesare analytical predictions, valid for m � 1 (see text), for the statistical bias(C) and the statistical fluctuation of DKL (D). In B–D we used Eq. 13 for theprobability, with N= 8 and V = 0.787, consistently with the experimentaldata of A.

Pezzè et al. PNAS Early Edition | 5 of 6

PHYS

ICS

Repeating the same procedure for υ20 and υ21, where H0 =PN

i=1αi2σ

ðiÞm , one derives

Eq. 5.

Statistical Speed for the Ising Model. We provide here details on the analysisof the Ising model discussed in the main text. We consider the homogeneouscase αi = 1 and n=m. For γ ≤ γc we find numerically that υ2Q is maximized bytaking equal ÆσðiÞn æ for all i= 1, . . . ,N (Fig. 2A). The optimization is thus doneby replacing ÆσðiÞn æ= a in Eqs. 5 and 8. This provides the equation

υ2QN

=�1− a2

�+ 2γ

�a− a3

�+γ2

4

�1+ 2a2 −3a4

�, [19]

which can be maximized over a at fixed value of γ. The exact analyticalexpression is long and not reported here. For γ � 1 we find a= γ +Oðγ3Þ,

giving υ2sep=N= 1+ 54γ

2 +Oðγ4Þ. For γ > γc we obtain numerically that υ2QðγÞ ismaximized when ÆσðiÞn æ= 1 and Æσði+1Þn æ= 0, giving

υ2sepN

=12+ γ +

γ2

2. [20]

Indeed, in the limit γ � 1, Eq. 20 goes as ∼ γ2=2 and thus overcomes Eq. 19,which goes as ∼ γ2=3 at best, as obtained by maximizing ð1+ 2a2 − 3a4Þ=4over a. The value of γ for which Eq. 20 is equal to the maximum over a of Eq.19 gives the critical γc.

ACKNOWLEDGMENTS. This work was supported by the National NaturalScience Foundation of China Grant 11374197, Program for ChangjiangScholars and Innovative Research Team Grant IRT13076, and The HundredTalent Program of Shanxi Province (2012).

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