Revealing Genuine Optical-Path Entanglement

transcript

F. Monteiro‡,1 V. Caprara Vivoli‡,1 T. Guerreiro,1 A. Martin,1

J.-D. Bancal,2 H. Zbinden,1 R. T. Thew,1, ∗ and N. Sangouard3, †

1Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543

3Department of Physics, University of Basel, CH-4056 Basel, Switzerland(Dated: February 13, 2015)

How can one detect entanglement between multiple optical paths sharing a single photon? Weaddress this question by proposing a scalable protocol, which only uses local measurements wheresingle photon detection is combined with small displacement operations. The resulting entanglementwitness does not require post-selection, nor assumptions about the photon number in each path.Furthermore, it guarantees that entanglement lies in a subspace with at most one photon peroptical path and reveals genuinely multipartite entanglement. We demonstrate its scalability andresistance to loss by performing various experiments with two and three optical paths. We anticipateapplications of our results for quantum network certification.

PACS numbers: 03.65.Ud, 03.67.Hk

Optical path entanglement – entanglement betweenseveral optical paths sharing a single photon – is oneof the simplest forms of entanglement to produce. Itis also a promising resource for long-distance quantumcommunication where the direct transmission of photonsthrough an optical fiber is limited by loss. In this con-text, loss can be overcome by using quantum repeaters,which require the creation and storage of entanglement insmall-distance links and subsequent entanglement swap-ping operations between the links. Among the differentquantum repeater schemes, those using path-entangledstates |1〉A|0〉B + |0〉A|1〉B , where a single photon is de-localized into two nodes A and B are appealing - theyrequire fewer resources and are less sensitive to mem-ory and detector efficiencies compared to repeater ar-chitectures based e.g. on polarization entanglement [1].Many ingredients composing these networks have beenexperimentally demonstrated, including path entangle-ment based teleportation [2], entanglement swapping [3],purification [4], quantum storage [5, 6] and an elementarynetwork link [7].

A natural question is how this body of work could serveto extend known point-to-point quantum repeaters toricher geometries for quantum networks? FIG. 1 presentsa possible solution: A single photon incident on a multi-port coupler generates entanglement overN output paths(see FIG. 1a), due to its non-classical nature [8]. Thesmall network created in this way can be entangled withother, potentially distant, networks via entanglementswapping operations using 50/50 beam-splitters and sin-gle photon detectors – a single detection is then enoughto entangle the remaining 2N-2 nodes (see FIG. 1b).Such 2D networks could open up new perspectives for

‡ These authors contributed equally to this work.∗Electronic address: robert.thew@unige.ch†Electronic address: nicolas.sangouard@unibas.ch

FIG. 1: Proposal to build up 2D networks over long distances.a) Networks made with neighboring nodes are made with N -path entangled states. b) These local networks can be con-nected remotely by means of entanglement swapping opera-tions resulting in a large scale network.

multi-user quantum information processing including se-cret sharing [9] or secure multi-party quantum computa-tion [10] as well as for experiments simulating quantummany-body system dynamics [11].

A central challenge, however, is to find an efficient, yettrustworthy, way to certify the functioning of these net-works, i.e. how to characterize path entanglement in adistributed scenario using only local measurements. Onemight do this by using several copies of path-entangledstates, as is the case for standard quantum repeaterschemes [12], however, doing so is resource demandingand addresses a restrictive class of applications – thoseaccepting post-selection. State tomography has alsobeen realized [13] to characterize two-path entangledstates but the exponential increase in measurementswith the number of subsystems makes the tomographicapproach impractical for detecting the entanglementin large multipartite systems like quantum networks.Recently, an entanglement witness for bipartite pathentangled states has been proposed and demonstrated,that is based on a Bell inequality combined with localhomodyne detections [14, 15]. However, it is not clearhow this approach can be extended to more than twopaths as even for three parties we know of no Bell

inequality that can be violated for W-like states withmeasurements lying on the equator of the Bloch sphere.

In this letter we propose an entanglement witnessspecifically developed to reveal path entanglement in dis-tributed systems. It relies on an accurate description ofmeasurement operators and assumes that each path isdescribed by a single mode. However, it does not requirepost-selection, nor assumptions about the photon num-ber of the measured state, hence, it reveals entanglementin a trustworthy manner. Moreover, it only makes useof local measurements and easily scales to multipartitesystems.

The principle of the witness is the following: N dis-tant observers share a state ρ describing N optical paths.Assuming that each path is completely described by asingle mode of the electromagnetic field, the aim is notonly to say whether the overall state is entangled, butalso to check that entanglement lies in a subspace withat most one photon in each mode and to check thatρ⋂ini≤1 is genuinely entangled. The subscript i labels

the observer 1, 2, ....N and ni is a non negative integerdescribing the photon number in the optical path i. Todo this, each observer uses a measurement combining asmall displacement operation and a single photon de-tector, a measurement initially proposed in Refs. [16–18] and demonstrated in [19]. In the qubit subspace{|0〉, |1〉}, the POVM elements corresponding to clickand no-click events of such a measurement can be seenas non-extremal projective measurements on the Blochsphere whose direction depends on the amplitude andphase of the displacement [20]. In other words, if oneconsiders non-photon-number-resolving (NPNR) detec-tors with a quantum efficiency η and a small displace-

ment D(α) = eαa†i−α

?ai operating on the mode i, thecorresponding observable is given by

�ηα = D†(α)(

2(1− η)a†iai − 1

)D(α) (1)

if one assigns the outcome +1 when the detector doesnot click and −1 when it clicks. If the measured statebelongs to the subspace with at most one photon andwith η = 1, �0 (the superscript is omitted when η = 1)corresponds to the Pauli matrix σz, i.e. a qubit mea-surement along the z direction. Similarly, for α = 1and α = i, �1 and �i are a good approximation to qubitmeasurements along x and y, respectively. We use thisanalogy to build up a fidelity-based entanglement wit-ness of the form ZN = N(2N |WN 〉〈WN | − 1), where

WN = 1√N

∑Ni=1 |01, ..., 1i, ...0N 〉 refers to the state in-

volving N modes sharing a single photon. We approxi-

mate this expression by the operator

N∑m=1

(N − 2m)�⊗m0 ⊗ 1⊗N−m

N−2∑m=0

�⊗m0 ⊗ 1⊗N−2−m ⊗ (�α ⊗ �α + �iα ⊗ �iα)

+sym. (2)

which only involves measurements of the form (1). �⊗m0 ⊗1⊗N−m stand for a measurement in which the first mpaths are measured with the observable �0 and the N−mremaining ones are traced out. ”sym.” indicates that weadd terms corresponding to permutations over all paths.

To make our witness suitable for experiments, we focuson the case where the displacements are phase averagedso that the relative phase of displacements is randombut the phase of each displacement which respect to thestate on which it operates is well controlled. Under thisassumption, the statistics on outcomes obtained by mea-suring m paths with �αeiφ is the same for any φ. Hence,our witness reduces to

ΠNi=1e

ia†iaiφ) ( N∑

(N − 2m)�⊗m0 ⊗ 1⊗N−m

N−2∑m=0

�⊗m0 ⊗ 1⊗N−2−m ⊗ �α ⊗ �α

+sym)(

ΠNi=1e

−ia†iaiφ)

where φ is averaged out. In order to detect entanglementwith ZN , it is suffisant to compare its value to the

maximum value zmaxppt,1 = 1

∫ 2π

0dφTr[ZNρ] that it can

take if the projection of the state ρ in the {0, 1} subspaceρ⋂ini≤1 has a positive partial transposition (PPT) with

respect to a single party. Indeed, the observation ofa value of ZN larger than zmax

ppt,1 implies by the Perescriterion [21, 22] that the measured state is entangledand that the entanglement lies in the qubit subspace.Since finding zmax

ppt,1 constitutes a linear optimizationproblem with semidefinite positive constraints, it can becomputed efficiently (see Supplemental Material). Simi-larly, comparing the value of ZN to zmax

ppt , the maximumvalue of zmax

ppt,s further optimized over all possible PPTs,reveals genuine multipartite entanglement.

As an example, consider the value that the witnesswould take, zW , in a scenario without loss and involving astate WN in which N optical paths share a single photon.We can compare this to the value zmax

ppt that would beachieved without genuine entanglement in the {|0〉 , |1〉}subspace. We show in the Supplemental Material that

zW − zmaxppt = 2N+3N − 1

N|α|2e−2|α|2 (4)

which is positive for all N. The proposed witness thushas the capability to reveal genuine entanglement of WN

λLOλD1

Entanglement

FIG. 2: Three different set-ups used to test the proposed en-tanglement witness for two and three parties: a) The heraldedstate can be tuned from maximally entangled to separableby the half-wave-plate (HWP) λE before the first polarizingbeam-splitter (PBS). The local oscillator is introduced at theother port of the PBS such that in each arm, the coherentstate and the single photon have orthogonal polarization. Thedisplacement operation is performed by rotating the HWPsat λD1 and λD2. b) The single photon and coherent state areinput earlier in the set-up with orthogonal polarizations. Theinput loss can be varied to study the robustness of the witness.This set-up can be easily modified, by adding a 30/70 beam-splitter and another (dashed) arm, allowing us to herald andprobe a tripartite W-state.

states for any path number. In practice, the value of αis optimized to make the difference zW − zmax

ppt as largeas possible.

When the measured state is not entirely containedin the {|0〉 , |1〉} subspace, contributions from higherphoton numbers can increase the value zmax

ppt . To geta valid bound in this regime, we used autocorrelationmeasurements in each mode. They give a bound onthe probability of having more than one photon in each

path (p(i)c denotes this bound for mode i) and avoid

making assumptions about the photon number. Thecomputation of zmax

ppt is then slightly modified to take the

value of p(i)c into account (see Supplemental Material).

Importantly, the autocorrelation measurements areperformed locally with a beam-splitter and two photondetectors. Overall, the number of measurements requiredto reveal genuine entanglement between N paths scales

quadratically (N2

2 + N2 + 1), which shows a much more

favorable scaling compared to the exponential scaling oftomographic approaches.

We now report on a series of experiments demonstrat-ing the feasibility of our witness. We prepare entanglednetworks made with 2 or 3 paths by sending single

photons onto beam-splitters, see FIG. 2. The photonsare prepared using a heralded single photon source(HSPS) based on a bulk PPLN nonlinear crystal [23].The crystal is pumped by a pulsed laser at 532 nm in theps regime with a repetition rate of 430 MHz producingnon-degenerate photons at 1550 nm and 810 nm viaspontaneous parametric down conversion. The telecomphoton is filtered down to 200 pm and subsequentlydetected by an InGaAs single photon avalanche diodes(SPAD), producing pure heralded single photons at810 nm – the purity is verified by measuring the secondorder autocorrelation function g2(0) [24]. To ensure ahigh fidelity entangled state, the pair creation prob-ability per pulse is limited to 10−3, to minimize theeffect of double pairs, the photons are coupled with 90 %efficiency [25] and the overall system transmission isoptimized. We herald single-photon states at a rate of∼ 8 kHz.

The measurements are performed by combiningdisplacement operations and single photon detection.The local oscillator for the displacements is generated ina similar PPLN nonlinear crystal pumped by the same532 nm pulsed laser as well as a 1550 nm telecom CWlaser – this ensures a high degree of indistinguishabilitybetween the HSPS and the local oscillator, which isconfirmed by measuring a Hong-Ou-Mandel interferencedip between the two sources, where the visibility isonly limited by the statistics of the two sources [24].Custom gated Silicon SPADs are then used to detectthe photons at 810 nm [26]. The detectors operateat 50 % efficiency and have a dark-count probabilityof 10−2 per gate, for a gate width of approximately 2.3 ns.

To determine the value of the witness, which reducesto

Z2 = Π2i=1e

ia†iaiφ(

2�α ⊗ �α − �0 ⊗ �0

)e−ia

†iaiφ (5)

in the bipartite case, we measure click/no-click (c/0)events in the two paths and calculate the correspond-ing probabilities, P00, P0c, Pc0, Pcc, as well as the boundson the probabilities for having more than one photon

in each path, p(1)c and p

(2)c . The correlators of the form

{�α′⊗�α′} in (5) then correspond to P00+Pcc−P0c−Pc0,for α′ = α or 0. We first block the single photon from go-ing to the set-up and apply the displacement operators inboth arms, validating that |α| corresponds to the desiredvalue. Experimentally, |α| is such that Pc ∼ P0 locally,see Supplemental Material. Secondly we allow the sin-gle photon to go to the set-up and record the correlatorswith, and without, the displacement operations. An au-tomated series of measurements is performed, integratingover 1 s for each setting, and is repeated as many times

as needed to have good statistics. The values for p(1)c and

p(2)c are dominated by detector noise due to operating the

detectors at such high efficiencies so as to maximize theglobal efficiency of the measurements. These values are

50/50 40/60 30/70 20/80 10/90 0/100Beam-splitter Ratio

FIG. 3: Observed value for the bipartite entanglement witness(relatively to the PPT bound) as a function of the beamsplit-ter ratio. Concretely, the half waveplate λE in Fig 2a) isrotated which changes the state from a maximally entangledto a separable state (50/50 - 0/100 splitting ratio, respec-tively). The blue band is obtained from a theoretical model,taking into account the set-up’s global transmission, the char-acteristics of sources and gated detectors and the value for|α| ∼ 0.83 in the displacement operations.

used to determine the observed value of ZN labelled zexpρ

and the maximum value zmaxppt that would be obtained if

the projection of the measured state in the {|0〉 , |1〉} sub-space has a positive partial transpose (see SupplementalMaterial).

To test the bipartite witness as a function of theamount of entanglement, the single photon and localoscillator are combined at different ports of a polarizingbeam-splitter (PBS) ensuring that they leave in thesame spatial mode with orthogonal polarizations, seeFIG. 2a. A half-wave-plate (HWP) λE placed in thesingle photon input arm is used to adjust the splittingratio in the two output modes and the subsequent am-plitudes for the entangled state. �α are performed via arotation of the wave-plates λD1, λD2 (< 1 degree) beforethe final PBSs. The amplitude of the displacement|α| ∼ 0.83 is set to maximize zexp

ρ − zmaxppt . FIG. 3

shows the result as a function of the beam-splitter ratio,from maximally entangled (50/50) to a separable state(0/100). The shaded line is obtained from a theoreticalmodel with independently measured system parameters,with the associated uncertainty (see SupplementalMaterial). The theory and experimental results are inexcellent agreement and prove that the proposed wit-ness can reveal even very small amounts of entanglement.

To prove the robustness of this witness against loss,and demonstrate the scalability, we introduce a differentexperimental configuration, FIG. 2b, with only a 50/50beam-splitter, to generate maximally entangled states,and the single photon and local oscillator are combinedearlier in the set-up. We can then introduce loss to theinput state, thus increasing the mixedness of the state.FIG. 4 shows the value of our witness of entanglement as

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35Total Transmission

FIG. 4: Observed value for the bipartite entanglement witness(relatively to the PPT bound) as a function of loss. The to-tal transmission consists of the photon coupling, transmissionthrough the system, and detector efficiencies.

we increase loss. The starting point has a slightly largervalue than in FIG. 3, due to a slightly better transmis-sion, n.b. the maximum transmission of &30 % includesphoton coupling, transmission and detection efficiency.Here we see that the witness is capable of revealing en-tanglement even in the case of high loss, or similarly, forlow detection efficiency.

Finally, by adding a 30/70 beam-splitter and anotherarm, dashed line in FIG. 2b, we herald tripartitestates. If we assume perfect transmission and detectorswith unit efficiency, we expect a maximum value forzexpρ − zmax

ppt ∼ 7.63, where a value greater than zeroindicates the presence of genuine entanglement. Byapplying our model, similarly to the bipartite case, againwith |α| ∼ 0.83, but with a total transmissions in eacharm of 0.19± 0.002, we expect to find a theoretical valueof 0.99 (see Supplemental Material). We found a valueof 0.99±0.10 that agrees with our model and shows aclear violation, thus revealing genuine tripartite pathentanglement.

In conclusion, we have shown an entanglement witnesssuited for path entangled states that is robust andscalable, providing the means for the characterizationof genuine multipartite entanglement distributed overcomplex quantum networks. The co-propagation of thelocal oscillator with the path entangled state overcomesthe potential problem of distributing a phase reference,which also has the added advantage that it couldbe exploited for stabilisation and synchronisation ofdistributed networks. Interestingly, our witness providesa trustworthy means to reveal entanglement, withoutthe need to make assumption about the number ofphotons in each path. A possible extension would be tomake it fully device independent through the violationof a Bell inequality, which would require higher overallefficiencies [20, 27].

Acknowledgments

The authors would like to thank M. Ho, P. Sekatski andF. Frowis for stimulating discussions. This work was sup-ported by the Swiss NCCR QSIT, the Swiss National Sci-ence Foundation SNSF (grant PP00P2-150579), the EUproject SIQS and Chist-Era: DIQIP and Qscale, as wellas the Singapore Ministry of Education (partly throughthe Academic Research Fund Tier 3 MOE2012-T3-1-009)and the Singapore National Research Foundation.

Appendix

A. PPT bound for an arbitrary number of qubits

In this first paragraph, we give details on the resultpresented in Eq. (4) of the main text. Our witness hasthe following form

ZN = ΠNi=1e

ia†iaiφ( N∑m=1

(N − 2m)�⊗m0 ⊗ 1⊗N−m

N−2∑m=0

�⊗m0 ⊗ 1⊗N−2−m ⊗ �α ⊗ �α

+sym)e−ia

†iaiφ. (6)

The question we address is what is the maximal value

Tr[ 12π

∫ 2π

0dφZNρ] that ZN can take if the projection of

the state ρ in the {0, 1} subspace ρ⋂ini≤1 stays positive

under partial transposition with respect to some bipar-tition. In this section, we consider the case where eachmode is described by qubits ρ = ρ⋂

ini≤1, i.e. they con-

tain at most one photon each. This condition is relaxedin the next section. The threshold value that we look forcan be obtained from the following optimization

zmaxppt,1 = max

ρ⋂ini≤1

∫ 2π

dφZNρ⋂ini≤1] s.t.

(i) ρ⋂ini≤1 ≥ 0,

(ii) Tr ρ⋂ini≤1 = 1,

(iii) ρT1⋂ini≤1 ≥ 0.

The two first conditions insure that ρ⋂ini≤1 is a physical

state. The last one demands that the state remainspositive under partial transposition with respect to thefirst path. This is a linear optimization with semidefinitepositive constraints which can be efficiently calculatednumerically. If a physical state made with qubits leadsto a larger value than zmax

ppt,1, we conclude that thecondition (iii) does not hold, i.e. its partial transpositionover the first path has at least one negative eigenvalue

2 3 4 5 6 7 8 9 10

Number of Paths

FIG. 5: Value that the witness takes with a WN state zW rela-tively to the maximal value with states that are not genuinelyentangled zmax

ppt as a function of the number of optical paths.The absolute value of the difference zW−zmax

ppt does not matteras ZN could be multiplied by any values. However, a positivedifference zW − zmax

ppt > 0 reveals genuine entanglement.

and is thus entangled. If we further optimize over allpossible bipartitions, we obtain a bound zmax

ppt , fromwhich one can witness genuine entanglement betweenthe N optical paths.

Consider for concreteness the case where the testedstate is the WN state. We can easily show that zW =

Tr[ 12π

∫ 2π

0dφZN |WN 〉〈WN |] is given by

zW = (2N − 1)N + 2N+1(N − 1)e−|α|2

×(|α|2(4e−|α|

2 − 1)− 1).

Furthermore, a basic measurement of the photon numberdistribution in each path would show that the probabil-ity of having zero photon in all paths or more than onephoton per path is null (see below for the realization ofsuch measurement). Together with the condition (iii),this leads to conditions on the coherence terms of thetested state, e.g.

| 〈10...0| ρ |01...0〉 |2 ≤ p00...0p11...0 = 0. (8)

The conditions (i), (ii) (iii) together with the previouslymentioned constraints allows us to get an analytical ex-pression for the PPT bound with respect to the first path

zW − zmaxppt,1 = 2N+3 (N − 1)

N|α|2e−2|α|2 . (9)

More generally, if we consider the entanglement betweenthe first m modes and the other (N−m) modes by chang-ing the condition over the partial transposition, we obtain

zW − zmaxppt,m = 2N+3m(N −m)

N|α|2e−2|α|2 (10)

which gives the threshold for a m vs. (N −m) bisepara-ble state. This bound is identical for all such bipartitions

because our witness and the conditions (i)-(ii) are invari-ant under exchange of parties. The threshold for genuinemultipartite entanglement can be deduced by minimizingzW − zmax,m

ppt over m. This leads to

zW − zmaxppt = 2N+3N − 1

N|α|2e−2|α|2 (11)

which is always positive for non-zero |α|. FIG. 1 is theresult of the optimization of the previous expression overα as a function of the number of parties. This showsthat our witness can detect genuine entanglement of WN

states for any number of parties.

B. PPT bound for qudits

We consider the general case where no assumption ismade about the number of photons in each path (noqubits). If the measured state is described by a singlemode, it can be written as

ρ⋂ini≤1 ρcoh

ρ†coh ρ⋃ini≥2

, (12)

where ρ⋂ini≤1 denotes the block with at most one photon

per partie as before, ρ⋃ini≥2 is the block in which at least

one party has more than one photon and ρcoh denotes thecoherence terms between these two blocks. Importantly,the only term in ρcoh leading to a non-zero contributionto our witness are those between ρ⋂

ini≤1 and ρ⋃

ini=2.

Given the maximal algebraic value zalg that ZN can take,we have

zN ≤Tr

∫ 2π

ρ⋂ini≤1 ρcoh 0

ρ†coh ρ⋃ini=2 0

+ zalgp

ni ≥ 2

where p

ni ≥ 2

)is the probability to have more than

one photon in at least one path. This probability canbe upper bounded using a 50/50 beam-splitter followedby two detectors in each path. Indeed, let us considerthe reduced state describing, say, the first path ρ1. Itsdiagonal elements in the Fock basis can be written as∑n≥0 p

(1)n |n〉〈n|. The probability p

(1)c that this state leads

to a two-fold coincidence after the beam-splitter is givenby

p(1)c =

∑n≥2

2n(2n−1 − 1)p(1)

n ≥1

∑n≥2

p(1)n =

(1)n≥2 (14)

where p(1)n≥2 is the probability to have strictly more

than one photon in the path 1. Since p

ni ≥ 2

)≤∑N

i=1 p(i)n≥2, we have

ni ≥ 2

)≤ 2(p(1)

c + p(2)c + ...+ p(N)

c ). (15)

The previous formula shows how to bound the probabilityfor being outside the qubit subspace {|0〉 , |1〉}⊗N withlocal measurements involving a 50/50 beam-splitter andtwo photon detectors. From Eqs. (13) and (15), we candeduce the PPT value in the general case of qudits fromthe following optimization

zmaxppt,1 ≤ max

ρ⋂ini≤2

∫ 2π

dφZNρ⋂ini≤2

)+2zalg(p(1)

c + p(2)c + ...+ p(N)

c ) s.t. (16)

(i) ρ⋂ini≤2 ≥ 0,

(ii) Tr ρ⋂ini≤2 ≤ 1,

(iii) ρT1⋂ini≤1 ≥ 0.

Note first that the optimization is performed over the setof physical states (condition (i)) having two photons atmaximum in each path because the only coherence termshaving a non-zero contribution lie in this subspace. Tokeep the optimization general, the trace of the state isrequired to be smaller than, or equal to, one (condition(ii)). The last condition ensures that the projection ofthe state with at most one photon in each path has apositive partial transpose when the transposition is takenover the first path. By changing the PPT condition (iii)

to ρTm⋂ini≤1 ≥ 0 where Tm denotes partial transposition

with respect to the bipartition m, we get a PPT boundfor this bipartition zmax

ppt,m. The value of the PPT boundwe are interested in is zmax

ppt = maxm zmaxppt,m. Importantly,

the difference between zW and zmaxppt is made larger

by using the results of the joint probability to haveclicks/no-clicks without displacement and adding thecorresponding constraints in the optimization procedure,cf below. Overall, the number of required measurements(1 +CN2 ) +N = N2/2 +N/2 + 1 has a quadratic scalingin the number of paths and is thus suited to provegenuine entanglement in multiple-path states.

C. Witnessing path entanglement with non-unitefficiency detectors

Note that so far we have assumed that the photon de-tectors have unit efficiencies. How can we use this witnessin practice, when non-unit efficiency detectors are used?

First, note that a detector with efficiency η can be seenas a unit-efficiency detector preceded by a beam-splitterwith a transmission efficiency η. Let Ubs be the unitarycorresponding to this beam-splitter. Let also D(α) be theunitary associated to the displacement operation with ar-gument α. We have

D(α√η)Ubs = UbsD(α) (17)

meaning that the inefficiency of the detector can bemodeled as a beam-splitter operating before the dis-placement operation provided that the amplitude ofthe displacement is reduced by

√η. Let us consider

the configuration where the loss operates before thedisplacement. The optimizations performed so far allowsone to conclude that the state after the loss is entangled.This implies that the state was already entangled beforethe loss as a separable state remains separable underloss. Our witness thus proves entanglement whennon-unit efficiency detectors are used provided thatthe displacement accounts for the reduced detectionefficiency.

D. Bipartite case

We now give an explicit expression of the PPT boundfor the bipartite case. For N = 2, we take the followingwitness

Z2 = Π2i=1e

ia†iaiφ(

2�α ⊗ �α − �0 ⊗ �0

)e−ia

†iaiφ. (18)

Note that we have removed a factor 2 from the definition(6). To get the PPT bound zmax

ppt , first note thatthe maximal (algebraic) value for Z2 is 3 and since

p (n1 ≥ 2 ∪ n1 ≥ 2) ≤ 2(p(1)c + p

(2)c ), we have

zmaxppt ≤ max

ρn1≤2∩n2≤2

∫ 2π

dφZ2ρn1≤2∩n2≤2

)+6(p(1)

c + p(2)c )

1. ρn1≤2⋂n2≤2 ≥ 0,

2. Tr(ρn1≤2⋂n2≤2) ≤ 1,

3. ρT2

n1≤1⋂n2≤1 ≥ 0,

4. 1− p (n1 ≥ 2⋃n2 ≥ 2) = Tr(ρn1≤1

⋂n2≤1).

When �0 ⊗ �0 is measured, one accesses the joint prob-abilities of having click/no-click without displacement.Let Pij be the joint probability to have outcomes i andj respectively, i, j = 0 for no-click and i, j = c forclick. The Pijs provide upper bounds of the diagonalterms of the measured state p00 = P00, p01 ≤ P0c,p10 ≤ Pc0 and p11 ≤ Pcc. Similarly, from the local

probabilities of having more than one photon, we have

p02 ≤ 2p(2)c , p20 ≤ 2p

(1)c , p12 ≤ 2p

(2)c , p21 ≤ 2p

and p22 ≤ 2p(1)c . Together with the condition 1. and

3. (implying e.g. |〈10|ρn1≤2⋂n2≤2|01〉|2 ≤ p01p10 and

|〈10|ρn1≤2⋂n2≤2|01〉|2 ≤ p00p11 respectively), these con-

straints provide the following upper bound on zmaxppt in

the regime α ≥ 0.45

zmaxppt ≤

(2(−1 + 2e−|α|

2(−1 + 2e−|α|

2)(−1 + 2e−|α|

2 |α|2)

(P0c + Pc0)

(2(−1 + e−|α|

2 |α|4)2

)p(1)c

2(−1 + 2e−|α|

2)(−1 + e−|α|

2 |α|4)

(p(1)c + p(2)

+ 16|α|2e−2|α|2(√

p(1)c p

(2)c |α|4 +

√P00Pcc

(2)c Pcc +

(1)c Pcc +

(1)c p

)|α|2

This analytical bound is used and compare to the ob-served value of our witness. If the latter is larger thanthe former, we can safely conclude that the tested stateis entangled.

E. Theoretical model

The aim of this section is to detail the model that hasbeen used to reproduce the experimental results (bluelines in FIG. 3 and 4 of the main text). The startingpoint is the pair production in two different modes bymeans of a spontaneous down conversion source and sub-sequent photon detection of one of two modes. The stateconditioned to a click on the heralding detector is givenby [20]

ρh =1−R2

T 2g (1−R2

1− T 2g

− 1− T 2g

1−R2hT

i.e. a difference between two thermal states ρth(n)with n mean photons. Tg = tanh(g), g being thesqueezing parameter, a† and a stand for bosonic op-erators and Rh =

√1− ηh where ηh is the heralding

efficiency. Importantly, a thermal state can be writ-ten as a mixture of coherent states |γ〉 . Concretely,

ρth (n) =∫d2γP n(γ) |γ〉〈γ| with P n(γ) = 1

πne− |γ|

n .The correlators that we want to calculate can thus beobtained from the behavior of a coherent state. A beam-splitter splits a coherent state into two coherent states,i.e. |γ〉 → |

√Rγ〉a |

√Tγ〉b, where T and R are, respec-

tively, the transmittivity and reflectivity. A displacement

D(α) on a coherent state |γ〉 gives another coherent statewith mean photon number |γ + α|2. Together with

xa†a

2 |γ〉 = e−(1−x)|γ|2

2 |√xγ〉 , (21)

we easily obtain the probability to get no click in bothsides from a thermal state ρth(n) knowing the amplitudeof the local displacement α

Pα00 =e−2|α|2+

nηtη1+nηtη

(√R+√T )2|α|2

1 + nηtη(22)

where ηtη is the overall efficiency (from the source to thedetector, including the detection efficiency). Followingthe same line of thought for Pαcc, P

α0c, P

αc0, we find

Tr(ρth(n)�α ⊗ �α) =1 + 4e−2|α|2+

nηtη1+nηtη

(√R+√T )2|α|2

1 + nηtη

−2e−

|α|21+ηtηnR

1 + ηtηnR− 2

e−|α|2

1+ηtηnT

1 + ηtηnT.

From this last expression, we deduce the correlator forthe state (20)

Tr (ρh�α ⊗ �α) =1−R2

T 2g (1−R2

h)× (23)

1− T 2g

)�α ⊗ �α

− 1− T 2g

1−R2hT

)�α ⊗ �α

Finally, the value taken by our entanglement witnessis obtained from 2 Tr(ρh�α ⊗ �α) − Tr(ρh�0 ⊗ �0).An independent characterization of the source to-gether with the measurements of efficiencies ηh andηtη shows a very good agreement between our modeland the results of the experiments (see FIG. 3 and 4of the main text). The error associated to both thetheoretical models in FIG. 3 and 4 of the main texttake into account the error on the value of α plus theerror in the transmission of the set-up for FIG. 3 and inthe beam-splitter ratio that prepares the state for FIG. 4.

F. Tripartite case

For N = 3, the expression of the witness Z3 is

Z3 = Π3i=1e

ia†iaiφ(�0 ⊗ (1⊗ 1− 1⊗ �0 − 3�0 ⊗ �0)

+4 (1 + �0)⊗ �α ⊗ �α + sym)eia†iaiφ. (24)

where sym stands for the terms obtained by all possiblepermutations of modes 1, 2, and 3. Following the sameline of thought as for the bipartite case, we can show thatfor any α ≥ 0.67

zmaxppt ≤

(−3 + 24

(−1 + 2e−|α|

2)2)P000 +

(5 + 16

(−1 + 2e−|α|

2)(−1 + 2e−|α|

2 |α|2))× (P00c + P0c0 + Pc00)

1 + 4(−1 + e−|α|

2 |α|4)(−3 + 4e−|α|

+ e−|α|2 |α|4

))×(p(1)c + p(2)

c + p(3)c

)+ 64|α|2e−2|α|2

(|α|2

(3)c P0cc +

(3)c Pc0c +

(2)c P0cc +

(2)c Pcc0 +

(1)c Pc0c +

(1)c Pcc0

)+ |α|2

(1 + |α|2

)(√p

(3)c p

(2)c +

(3)c p

(1)c +

(2)c p

)+ max

(√P0c0P00c +

√P000Pc0c +

√P000Pcc0,

√Pc00P00c +

√P000P0cc +

√P000Pcc0,√

P0c0Pc00 +√P000Pc0c +

√P000P0cc

))+ 33

(2(p(1)c + p(2)

c + p(3)c

)−max

(p(1)c + p(2)

)− 1, 2

(p(1)c + p(3)

)− 1, 2

(p(2)c + p(3)

)− 1))

Observation of a value larger than zmaxppt allows one to

conclude that the measured state is genuinely entangled.G. Experiment

FIG. 2 shows the experimental scheme for thegeneration of heralded single photons and the local

1550 nm!

532 nm!

FIG. 6: The experimental set-up for the heralded single pho-ton source and the local oscillator. See text for details.

oscillator for the displacement operations. Most of theexperimental parameters are explained in the main text.We have two independent sources based on PPLN non-linear crystals, with type-0 quasi-phasematching. Theheralded single photon source (HSPS) uses a tunablefilter (F) with a 200 pm bandwidth that has around50% transmission for the central (telecom) wavelength.This is sufficiently narrow so as to herald the 810 nmphotons in pure states [24]. By stimulating the DFGprocess in the second non-linear crystal with a coherentCW laser at the same wavelength of the HSPS’s filter,energy conservation dictates that both the heraldedsingle photon and the local oscillator (LO) state areindistinguishable. A delay arm (not shown) allows forthe sychronization of the two sources.

To determine the value of |α| which maximizeszexpρ − zppt

max, the probabilities P00, P0c, Pc0, Pcc, as well

as p(1)c and p

(2)c are measured. By guessing that they are

obtained from a W state, these probabilities allow oneto estimate the overall efficiencies and hence the valuethat the witness would take for any α. Maximizing thedifference with (19) results in |α| = 0.83. We emphasizethat in practice, with non-unit efficiency detectors, thiscorresponds to a displacement with amplitude 0.83/

√η.

This means that the detection efficiency does not needto be known. The values of |α| are locally set to produceclick/no-click with the same probability independentlyof the detection efficiency.

The stability of the mean photon number for the LOstates is realized by a feedback loop, where a large frac-tion of the generated LO is sent, via a polarizing beam-splitter (PBS) to a power meter (PM) that then acts ona Pockels cell. The typical integration times, per point inFIG. 3 & 4 in the main text, are around 1.5 hours. Theerror bars are not Poissonian as they are dominated byfluctuations in the system, in particular, the generationof the LO, which depend on the feedback loop.

If we consider the measurement for the maximally en-tangled bipartite case as an example, the probabilityto have more than one photon in each arm are mea-

P00 P0c Pc0 Pcc0.0

babili

α = 0

α = 0.83 ± 0.01

FIG. 7: Measured probabilities of having click/no-click in thetwo arms of the bipartite experiment (FIG. 4 of the maintext with the maximal transmission efficiency (≥ 0.3)). Redcolumns give the probabilities measured without the displace-ment operation and the blue columns correspond to the prob-abilities measured when the displacement operator was ap-plied (|α| = 0.83± 0.01).

sured to be p(1)c = 10−4 and p

(2)c = 10−4. After mea-

suring the different click/no-click events we can deter-mine their probabilities - see the results presented inFIG. 7. These lead to zexp

ρ = −0.002, zmaxppt = −0.315,

i.e. zexpρ − zmax

ppt = 0.313 > 0. As the observed value

was larger than zpptmax, we can conclude that the measured

state is genuinely entangled.

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Revealing Genuine Optical-Path Entanglement

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