Experimental Quantum Networking Protocols via Four-Qubit Hyperentangled Dicke States

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Experimental Quantum Networking Protocols via Four-Qubit Hyperentangled Dicke States

A. Chiuri,1 C. Greganti,1 M. Paternostro,2 G. Vallone,3 and P. Mataloni1, 4

1Dipartimento di Fisica, Sapienza Universita di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy2Centre for Theoretical Atomic, Molecular, and Optical Physics,

School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom3Department of Information Engineering, University of Padova, I-35131 Padova, Italy

4Istituto Nazionale di Ottica (INO-CNR), L.go E. Fermi 6, I-50125 Firenze, Italy(Dated: September 25, 2012)

We report the experimental demonstration of two quantum networking protocols, namely quantum 1→3 tele-cloning and open-destination teleportation, implementedusing a four-qubit register whose state is encoded in ahigh-quality two-photon hyperentangled Dicke state. The state resource is characterized using criteria based onmultipartite entanglement witnesses. We explore the characteristic entanglement-sharing structure of a Dickestate by implementing high-fidelity projections of the four-qubit resource onto lower-dimensional states. Ourwork demonstrates for the first time the usefulness of Dicke states for quantum information processing.

PACS numbers: 42.50.Dv,03.67.Bg,42.50.Ex

Networking offers the benefits of connectivity and sharing,often allowing for tasks that individuals are unable to accom-plish on their own. This is known for computing, where gridsof processors outperform the computational power of singlemachines or allow the storage of much larger databases. Itshould thus be expected that similar advantages are transferredto the realm of quantum information. Quantum networking,where a given task is pursued by a lattice of local nodes shar-ing (possibly entangled) quantum channels, is emerging asa realistic scenario for the implementation of quantum pro-tocols requiring medium/large registers. Key examples ofsuch approach are given by quantum repeaters [1], non-localgates [2], scheme for light-mediated interactions of distantmatter qubits [3] and one-way quantum computation [4].

In this scenario, photonics is playing an important role:the high reconfigurability of photonic setups and outstand-ing technical improvements have facilitated the birth of a newgeneration of experiments (performed both in bulk optics and,recently, in integrated photonic circuits [5]) that have demon-strated multi-photon quantum control towards high-fidelitycomputing with registers of a size inaccessible until only re-cently [6–11]. The design of complex interferometers and theexploitation of multiple degrees of freedom of a single pho-tonic information carrier have enabled the production of inter-esting states, such as cluster/graph states, GHZ-like states and(phased) Dicke states [12–14], among others [15, 16]. Dickestates have been successfully used to characterize multipartiteentanglement close to fully symmetric states and its robust-ness to decoherence [14]. They are potentially useful resourcefor the implementation of protocols for distributed quantumcommunication such as quantum secret sharing [17], quan-tum telecloning (QTC) [18], and open destination teleporta-tion (ODT) [19, 20]. So far, such opportunities have onlybeen examined theoretically and confirmed indirectly [12, 13],leaving a full implementation of such protocols unaddressed.

In this Letter, we report the experimental demonstration of1→ 3 QTC and ODT of logical states using a four-qubit sym-metric Dicke state with two excitations realized using a high-

quality hyperentangled (HE) photonic resource [14, 21]. Theentanglement-sharing structure of the state has been charac-terized quantitatively using a structural entanglement witnessfor symmetric Dicke states [22, 23] and fidelity-based entan-glement witnesses for the three- and two-qubit states achievedupon subjecting the Dicke register to proper single-qubit pro-jections [13]. All such criteria have confirmed the theoreticalexpectations with a high degree of significance. As for theprotocols themselves, the qubit state to teleclone/teleport isencoded in an extra degree of freedom of one of the phys-ical information carriers entering such multipartite resource.This has been made possible by the use of a displaced Sagnacloop [24] [cf. Fig.1], which introduced unprecedented flexi-bility in the setting, allowing for the realization of high-qualityentangling two-qubit gates on heterogeneous degrees of free-dom of a photonwithin the Sagnac loop itself. The high fi-delities achieved between the experiments and theory (as largeas 96%, on average, for ODT) demonstrate the usefulness ofDicke states as resources for distributed quantum communi-cation beyond the limitations of a “proof of principle”. Ourscheme is well suited for implementing 1→N>3 QTC of log-ical states or ODT with more than three receivers via the real-ization of larger HE resources, which is a realistic possibility.

Resource production and state characterization.-The build-ing block of our experiment is the source of two-photon four-qubit polarization-path HE states developed in [21, 25] andused recently to test multi-partite entanglement, decoherenceand general quantum correlations [14, 26, 27]. Such apparatushas been modified as described in the Supplementary Informa-tion [SI] [29] to produce the HE state|ξ〉abcd=[|HH〉ab(|rℓ〉 −|ℓr〉)cd+ 2|VV〉ab|rℓ〉]cd/

√6. Here, we have used the encoding

{|H〉, |V〉}≡{|0〉, |1〉}, with H/V the horizontal/vertical polariza-tion states of a single photon, and{|r〉, |ℓ〉}≡{|0〉, |1〉}, whererandℓ are the path followed by the photons emerging from theHE stage [29]. Qubitsa, c (b, d) are encoded in the polariza-tion and momentum of photon A (B). State|ξ〉 is turned into afour-qubit two-excitation Dicke state|D(2)

4 〉=(1/√

6)∑6

j=1 |Π j〉(with |Π j〉 the elements of the vector of states constructed by

2

FIG. 1: a) Scheme for the|ξ〉 → |D(2)4 〉 conversion. The spatial qubits

experience the Hadamard gatesHc,d implemented through a polariza-tion insensitive beam splitter (BS1). A controlled-NOT (controlled-PHASE) gateCX=|0〉i〈0|⊗11j+ |1〉i〈1|⊗σx

j (CZ=|1〉i〈1|⊗11j+ |0〉i〈0|⊗σz

j) is realized by a half-wave plate (HWP) with axis at 45◦ (0◦) withrespect to the vertical direction (i= c,d, j=a, b). The control (target)qubit of such gate is the path (polarization) degree of freedom (DOF).b) & c) Displaced Sagnac loop for the realization of the QTC/ODTprotocol. Panelb) [c)] shows the path followed by the upper [lower]photon A [B]. The glass platesφA,B,X allow us to vary the relativephase between the different paths within the interferometer.d) Cir-cuit for 1→3 QTC and ODT. Qubits{a,b, c,d} are prepared in|D(2)

4 〉while X should be cloned/teleported. For QTC, theCXXb gate is com-plemented by the projection ofX (b) on the eigenstates ofσx (σz),so as to perform a BM. For QTC (ODT), operation O is a local PauligateP determined by the outcome of the BM according to the giventable. For ODT (with, say, receiver qubitc), the operations in thedashed boxes should be removed.

taking all the permutation of 0’s and 1’s in|0011〉) by meansof unitaries arranged as specified in Ref. [14] [cf. Fig.1 a)]. Inthe basis of the physical information carriers, the state reads|D(2)

4 〉=[|HHℓℓ〉 + |VVrr〉 + (|VH〉 + |HV〉)(|rℓ〉 + |ℓr〉)]/√

6.The fidelity of the protocols depends on the quality of thisstate, as will be clarified soon. We have thus tested the close-ness of the experimental state to|D(2)

4 〉 and characterized itsentanglement-sharing structure.

First, we have ascertained the genuine multipartite entan-gled nature of the state at hand by using tools designed toassess the properties of symmetric Dicke states [22, 23, 28].We have considered the multipartite entanglement witness

Wm = [2411+ J2xSx + J2

y Sy + J2z(3111− 7J2

z)]/12, (1)

which is specific of |D(2)4 〉 [23] and requires only three

measurement settings. Here,Sx,y,z=(J2x,y,z − 11)/2 with

Jx,y,z=∑

i∈Q σx,y,zi /2 collective spin operators, ˆσ j ( j=x, y, z) the

j-Pauli matrix andQ = {a, b, c, d}. The expectation valueof Wm is positive on any bi-separable four-qubit state, thusnegativity implies multipartite entanglement. Its experimentalimplementation allows to provide a lower bound to the statefidelity with the ideal Dicke state asFD(2)

4≥ (2 − 〈Wm〉)/3.

When calculated over the resource that we have created inthe lab, we achieveWm= − 0.341± 0.015, which leads toFD(2)

4≥ (78± 0.5)%. The genuine multipartite entangled na-

ture of our state is corroborated by another significant test:we consider the witness testing bi-separability on multipartite

symmetric, permutation invariant states like our|D(2)4 〉 [13, 28]

Wcs(γ) = b4(γ)11− (J2x + J2

y + γJ2z) (γ∈R). (2)

Hereb4(γ) is the maximum expectation value of the collec-tive spin operatorJ2

x+J2y+γJ2

z over the class of bi-separablestates of four qubits and can be calculated for any value ofthe parameterγ. [28]. Finding 〈Wcs(γ)〉<0 for someγ im-plies genuine multipartite entanglement. The direct evaluationshows that already forγ = −0.12 the witness is negative bymore than one standard deviation and by more than fifteen forγ = −2.5 (cf. SI [29]).

These results, although indicative of high quality of the re-source produced, are not exhaustive and further evidence isneeded. In order to provide an informed and experimentallynot-demanding analysis on the state being generated, we havedecided to resort to indirect yet highly significant evidence onits properties. In particular, we have exploited the interestingentanglement structure that arises from|D(2)

4 〉 upon subjectingpart of the qubit register to specific single-qubit projections. Infact, by projecting one of the qubits onto the logical|0〉 and|1〉states, we maintain or lower the number of excitations in theresulting state without leaving the Dicke space, respectively.Indeed, we achieve|D(2)

3 〉 = (|011〉 + |101〉 + |110〉)/√

3 when

projecting onto|0〉, while |D13〉=(|100〉 + |010〉 + |001〉)/

√3

is obtained when the projected qubit is found in|1〉. Need-less to say, these are genuinely tripartite entangled states, as itcan be ascertained by using the entanglement witness formal-ism. For this task we have used the fidelity-based witness [30]WD(k)

3= (2/3)11−|D(k)

3 〉〈D(k)3 | (k = 1, 2), whose mean is pos-

itive for any separable and biseparable three-qubit state,is−1/3 when evaluated over|Dk

3〉 and whose optimal decom-position (cf. SI [29]) requires five local measurement set-tings [30, 31]. We have implemented the witness for statesobtained projecting qubitd (i.e. momentum of photon B),achieving〈Wexp

D(1)3

〉= − 0.21±0.01 and〈Wexp

D(2)3

〉= − 0.24±0.01

(the apex indicates their experimental nature) correspondingto lower bounds for the fidelity with the desired state of0.876±0.003 and 0.908±0.003, respectively.

Finally, by projecting two qubits onto elements of the com-putational basis, one can obtain elements of the Bell basis.In-deed, regardless of the projected pair of qubits,〈i j |D(2)

4 〉=|ψ+〉with {|ψ±〉=(|01〉±|10〉)/

√2, |φ±〉=(|00〉±|11〉)/

√2} the Bell

basis andi, j=0, 1. We have verified the quality of the re-duced experimental states achieved by projecting the Dickestate onto|10〉cd and|01〉cd using two-qubit quantum state to-mography (QST) [32] on the remaining two qubits. By findingfidelities>91% regardless of the projections operated, we canclaim to have a very good Dicke resource, which puts us in theposition to experimentally implement the quantum protocols.1→3 QTC and ODT.-Telecloning [18] is a communicationprimitive that merges teleportation and cloning to deliverap-proximate copies of a quantum state to remote nodes of anetwork. Differently, ODT [19] enables the teleportation ofa state to an arbitrary location of the network. Both re-quire shared multipartite entanglement. A deterministic ver-

3

sion of ODT makes use of GHZ entanglement [20], whilethe optimal resources for QTC are symmetric states havingthe form of superpositions of Dicke states withk excita-tions [15, 16, 18, 33]. Continuous-variable QTC was demon-strated in [35]. Although a symmetric Dicke state is known tobe useful for such protocols (ODT being reformulated proba-bilistically) [12], no experimental demonstration has yetbeenreported: in Ref. [12], only an estimate of the efficiency ofgeneration of a two-qubit Bell state between sender and re-ceiver was given, based on data for|D(2)

4 〉. Differently, oursetup allows to perform both QTC and probabilistic ODT.

We start discussing the 1→3 QTC scheme based on|D(2)4 〉,

which is a variation of the protocol given in Ref. [18]. We con-sider the qubit state to clone|α〉X = α|0〉X+β|1〉X (|α|2+|β|2=1),held by aclient X. The agents of aserver composed ofqubits{a, b, c, d} and sharing the Dicke resource agree on theidentification of aport qubit p.The state of pair (X, p) un-dergoes a Bell measurement (BM) performed by implement-ing a controlled-NOT gateCXXp followed by a projectionof X (b) on the eigenstates of ˆσx (σz). They publicly an-nounce the results of their measurement, which leaves us with⊗

j∈StcP j(α|D(1)

3 〉+β|D(2)3 〉)Stc⊗|ψ+〉Xp, whereStc={a, b, c, d}/

p is the set of server’s qubits minusp, |D(k)3 〉 is a three-qubit

Dicke state withk=1, 2 excitations and the gatesP j (identi-cal for all the qubits inStc) are determined by the outcomeof the BM, as illustrated in Fig.1 d). The protocol is nowcompleted and the client’s qubit is cloned into the state of theelements ofStc. To see this, we trace out two of the elementsof such set and evaluate the state fidelity between the densitymatrixρr of the remaining qubitr and the client’s state, whichreadsF (θ)=[9− cos(2θ)]/12, whereα= cos(θ/2). Clearly, thefidelity depends on the state to clone, achieving a maximum(minimum) of 5/6 (2/3) at θ = π/2 (θ = 0, π). This exceedsthe value 7/9 achieved by a universal symmetric 1→ 3 clonerdue to the state-dependent nature of our protocol.

We now introduce the ODT protocol. As for QTC, this isformulated as a game with a client and a server. The clientholds qubitX, into which the state|α〉X to teleport is en-coded. The elements of the server share the|D(2)

4 〉 resource.The client decides which partyr of the server should receivethe qubit to teleport (r and p can be any of{a, b, c, d}, andris chosen at the last step of the scheme). Unlike QTC, theclient performs aCXXp. At this stage the information on thequbit to teleport is spread across the server, and the clientde-clares who will receive it. Depending on his choice, the mem-bers inSodt={a, b, c, d}/{r, p} project their qubits onto|01〉Sodt,getting [α(|001〉 + |010〉)Xpr + β(|111〉 + |100〉)Xpr] ⊗ |01〉Sodt.The scheme is completed by a projection onto| + 1〉Xp with|+〉=(|0〉 + |1〉)/

√2 [34].

Experimental implementations of1→3 QTC.- The setup inFig. 1 b) andc), which represents a significant improvementover the scheme used in [14], allows for the implementation ofboth the protocols. The shown displaced Sagnac loop and theuse of the lower photon B allow us to add the client’s qubitto the computational register. This is encoded in the sense

FIG. 2: (Color online)a) Experimental QTC: for an input state|1〉X,after the BM on (X,b) with outcome|φ+〉Xb, the ideal output state is(2|0〉 j〈0| + |1〉 j〈1|)/3, ∀ j=a, c, d [left column of the panel]. The stateof qubit j after the experimental QTC, has very large overlap withthe theoretical state. The right column of the panel shows the exper-imental single-qubit density matrices.b) Theoretical QTC fidelityand experimental density matrices of the clone (qubita) for variousinput states. We show the fidelities between the experimental inputstates and clones (associated uncertainties determined byconsider-ing Poissonian fluctuations of the coincidence counts). Thedashedline shows the theoretical fidelity for pure input states of the client’squbit. The dashed area encloses the values of the fidelity achieved fora mixed input state ofX and the use of an imperfect Dicke resourcecompatible with the states generated in our experiment (cf.SI [29])

of circulation of the loop by such field: modes|r〉 and |ℓ〉 ofphoton B impinge on different points of beam splitter BS2,so that the photon entering the Sagnac loop can follow theclockwise path, thus being in the|�〉≡|0〉 state, or the coun-terclockwise one, being in|〉≡|1〉 (photon A does not passthrough BS2). The probability|α|2 of being in the former (lat-ter) state relates to the transmittivity (reflectivity) of BS2. Thisprobability is varied using intensity attenuators interceptingthe output modes of BS2. At this stage, the state of the reg-ister is |D(2)

4 〉abcd ⊗ (α|�〉 + eiφx√

1− |α|2|〉)X, whereφx ischanged by tilting the glass-plate in the loop. TheCXXp gatehas been implemented with qubitX as the control, qubitb(i.e. the polarization of photon B) as the portp and taking aHWP rotated at 45◦ with respect to the optical axes, placedonly on the counterclockwise circulating modes of the Sagnacloop [37]. The second passage of the lower photon in BS2

allows to project qubitX on the eigenstates of ˆσxX. To com-

plete the Bell measurement on qubits (X, p) we have placed aHWP and a PBS before the detector in order to project qubitp on the eigenstates of ˆσz

p. The remaining qubits (a, c andd)embody three copies of the qubitX. Their quality has beentested by performing QST over the reduced states obtained bytracing over any two qubits. Pauli operators in the path DOFhave been measured using the second passage of both pho-tons through BS1. The glass platesφA,B allowed projectionsonto 1√

2(|r〉 + eiφA(B) |ℓ〉)c(d). To perform QST on the polariza-

tion DOF we used an analyzer composed of HWP, QWP andPBS before the photo-detector. To trace over polarization,weremoved the analyzer. To trace over the path, a delayer wasplaced on either|r〉 or |ℓ〉 coming back to BS1, thus making

4

them distinguishable and spoiling their interference.In Fig. 2 a) we show the experimental results obtained for

the input states|1〉X, whenp=b. QST on qubitj=a, c, d showsan almost ideal fidelity with the theoretical state, uniformlywith respect to labelj, thus proving the symmetry of QTC.Our setup allows us to teleclone arbitrary input states. To illus-trate the working principles and efficiency of the telecloningmachine, we have considered the logical states|0〉X and|+〉Xand|1〉X (i.e. we tookθ≃0, π/2 andπ) and measured the cor-responding copies in qubita (i.e. the polarization of photonA). States|0〉X and|1〉X were generated by selecting the modesin the displaced Sagnac. In the first (second) case we consid-ered only modes|�〉 (|〉), while |+〉X was generated usingboth modes and adjusting the relative phase with the glass-plate φX (by varying this phase, we can explore the wholephase-covariant case). Although the experimental resultsarevery close to the expectations forF (θ) [cf. Fig. 2 b)], somediscrepancies are found forθ = π/2. In particular, the the-ory seems to underestimate (overestimate) the experimentalfidelity of telecloning close toθ = π/2 (θ = 0, π). Theseeffects are due to the mixedness of theX state entering theSagnac loop as well as the suboptimal fidelity between the ex-perimental resource and|D(2)

4 〉. In fact, the experimental inputstate corresponding toθ ≃ π/2 has fidelity 0.91± 0.02 withthe desired|+〉X due to depleted off-diagonal elements in itsdensity matrix (cf. SI [29]). We have thus modelled the tele-cloning of dephased client states based on the use of a mixedDicke channel of sub-unit fidelity with|D(2)

4 〉. The details arepresented in Ref. [29]. Here we mention that, by includingthe uncertainty associated with the estimatedFD(2)

4, we have

determined aθ-dependent region of telecloning fidelities intowhich the fidelity between the experimental state of the clonesand the input client state falls. As shown in Fig.2 b), this pro-vides a better agreement between theory and data.Experimental implementations of ODT.-In ODT the clientholds qubitX, which is added to the computational regis-ter using the Sagnac loop. The client’s qubit has been tele-ported to the server’s elementsa andb (i.e. the polarizationof photons A and B). The necessaryCXXp gate has been im-plemented, as above, by takingX as the control andp=b asthe target qubit. The server’s elements{c, d} have been pro-jected onto|01〉cd and |10〉cd. Depending on the chosen re-ceiver (eithera or b), the scheme is implemented by project-ing onto|+1〉Xa(b) and performing QST of the teleported qubitb(a). While the projection onto|+〉X has been realized usingthe second passage of the lower photon through BS2, a pro-jection onto|1〉a(b) is achieved projecting the physical qubitonto |V〉a(b). In TableI we report the experimental results ob-tained for several measurement configurations and teleporta-tion channels. In SI [29] we provide the reconstructed densitymatrices of qubits{X, a, b} for each configuration used.Conclusions and outlook.-We have implemented QTC andODT of logical states using a four-qubit symmetric Dickestate. We have realized a novel setup based on the well-testedHE polarization-path states and complemented by a displacedSagnac loop. This allowed the encoding of non-trivial in-

TABLE I: Experimental fidelities between the teleported qubit (a orb) and the state of qubitX (determined byθ). Uncertainties resultfrom associating Poissonian fluctuations to the coincidence counts.

Projection θ Fidelity Projection θ Fidelity

cd〈10| 0 Fa=0.93±0.01 cd〈01| π Fa=0.98±0.01

cd〈10| 0 Fb=0.95±0.01 cd〈01| π Fb=0.97±0.01

cd〈01| 0 Fa=0.97±0.01 cd〈10| 1.46 Fa=0.92±0.02

cd〈01| 0 Fb=0.97±0.01 cd〈10| 1.46 Fb=0.98±0.01

cd〈10| π Fa=0.96±0.01 cd〈01| 1.37 Fa=0.97±0.02

cd〈10| π Fb=0.98±0.01 cd〈01| 1.37 Fb=0.96±0.02

put states in the computational register, and the performanceof high-quality quantum gates and protocols. Our results gobeyond state-of-the-art in the manipulation of experimentalDicke states and the realization of quantum networking.

Acknowledgments.–We thank Valentina Rosati for the con-tribution given to the early stages of this work. This workwas supported by EU-Project CHISTERA-QUASAR, PRIN2009 and FIRB-Futuro in ricerca HYTEQ, and the UK EP-SRC (EP/G004579/1).

5

FIG. 3: Sketch of the experimental setup used to produce the HEresource state|ξ〉abcd. The setup is discussed fully in the body of thetext.

SUPPLEMENTARY INFORMATION ON: EXPERIMENTALQUANTUM NETWORKING PROTOCOLS VIA

FOUR-QUBIT HYPERENTANGLED DICKE STATES

In this supplementary Information we provide further de-tails on both the theoretical and experimental results and anal-ysis reported in the main Letter.

RESOURCE PRODUCTION AND STATECHARACTERIZATION

Here we describe the source of hyperentanglement thathas been used as the building block of our experiment. Asremarked in the text of the Letter, we use the encodings{|H〉, |V〉}≡{|0〉, |1〉}, with H/V the horizontal/vertical polariza-tion states of a single photon, and{|r〉, |ℓ〉}≡{|0〉, |1〉}, whererandℓ are the path followed by the photons emerging from theHE stage introduced and exploited in [14, 21, 26, 27].

We modify such setup so to prepare the HE resource|ξ〉abcd=[|HH〉ab(|rℓ〉 − |ℓr〉)cd+ 2|VV〉ab|rℓ〉]cd/

√6 introduced

in the main Letter. A sketch of the apparatus is shown inFig. 3. A Type-I nonlinearβ-barium borate crystal, pumpedby a vertically polarized laser field (wavelengthλp), generatesa polarization-entangled state given by the superpositionofthe spontaneous parametric down conversion (SPDC) signalsat degenerate wavelength produced by a double-pass scheme.The mask selects four spatial modes{|r〉, |ℓ〉}A,B (two for eachphoton), parallelized by lens L. QWP1,2 are quarter-waveplates. The first pass produces 2|VV〉|rℓ〉. The spatial modesare intercepted by two beam stoppers. QWP1 changes the po-larization into |VV〉 after reflection by mirror M. The latteralso reflects the pump, which produces the second-pass SPDCcontribution|HH〉(|rℓ〉 − |ℓr〉). The weight of this term in thefinal state|ξ〉 is determined by QWP2 [14].

ON ENTANGLEMENT WITNESSES FOR GENUINEMULTIPARTITE ENTANGLEMENT

Collective-spin operators are useful tools for the investi-gation of genuine multipartite entanglement, particularly forsymmetric, permutation invariant states. One can constructthe witness operator [38]

Wcs = bn11− (J2x + J2

y ), (3)

wherebn is the maximum expectation value ofJ2x + J2

y over

the class of bi-separable states ofn qubits. Finding〈Wsn〉<0

for a given state implies genuine multipartite entanglement.It can be the case that Eq. (3) fails to reveal the multipartitenature of a state endowed with a lower degree of symmetry.More flexibility can nevertheless be introduced by means of asuitable generalization such as

Wcs(γ) = bn(γ)11− (J2x + J2

y + γJ2z) (γ ∈ R). (4)

Negativity of 〈Wcs(γ)〉 over a given state guarantees multi-partite entanglement. The witness requires only three mea-surement settings and is thus experimentally very convenient.The bi-separability boundbn(γ) is now a function of param-eterγ and can be calculated numerically using the proceduredescribed in Ref. [28]. In general,bn(γ) < bn(0) for γ < 0.Consequently, we restrict ourselves to the case of negativeγ.

In Table II we provide the experimental values of〈J2x,y,z〉

through which we have evaluated Eq. (4), which is plottedagainstγ in Fig.4. While 〈Wcs(γ)〉 soon becomes negative asγ< − 0.1 is taken, the uncertainty associated with such expec-tation value, calculated by propagating errors in quadrature as

δ〈Wexpcs (γ)〉 =

√

∑

j=x,y

(δ〈J2j 〉)2 + γ2(δ〈J2

z〉)2, (5)

grows only very slowly withγ, therefore signaling an increas-ingly significant violation of bi-separability.

OPTIMAL DECOMPOSITION OF THE ENTANGLEMENTWITNESS FOR |D1

3〉

As discussed in the main body of of the Letter, we haveused a fidelity-based entanglement witness to characterizethegenuine tripartite entanglement content of the state achieved

TABLE II: Experimentally measured expectation values of collec-tive spin operators for the symmetric four-qubit Dicke state preparedin our experiment. The uncertainties are determined by associatingPoissonian fluctuations to the coincidence counts.

Expectation value (with uncertainty) Value

〈J2x〉 ± δ〈J2

x〉 2.568±0.015〈J2

y〉 ± δ〈J2y〉 2.617±0.011

〈J2z〉 ± δ〈J2

z〉 0.039±0.028

6

-2.5 -1.5 -0.5

-1.0

-0.6

-0.2

uncertainty

FIG. 4: Functional form of〈Wcs(γ) =〉 againstγ, as determinedby the measured expectation values of collective spin operators (cf.Table tavola). A negative value of〈Wcs(γ)〉 signals genuine multi-partite entanglement of the state experimental state underscrutiny.The associated experimental uncertainty [see Eq. (5)] increases onlyvery slowly as|γ| grows.

upon projecting one of the qubits onto a state of the logi-cal computational basis. Without affecting the generality ofour discussion, here we concentrate on the case of a qubit-projection on qubitd giving outcome|1〉d, thus leaving us withstate|D(1)

3 〉abc. The fidelity-based witness that we have imple-mented is given in the main Letter and is decomposed in fivemeasurement settings as [31]

WD(1)3=

124

{

1711+7σzaσ

zbσ

zc+3Π[σz

a11bc]+5Π[σaσb11c]

−∑

l=x,y

∑

k=±(11a+σ

za+kσl

a)(11b+σzb+kσl

b)(11c+σzc+kσl

c)}

(6)whereΠ[·] performs the permutation of the indices of its ar-gument. The decomposition is optimal in the sense thatWD(1)

3

cannot be decomposed with lesser measurement settings. Ex-perimentally, we have used the following rearrangement of theprevious expression

WD(1)3=

124

{

1311abc+3σzaσ

zbσ

zc−Π[σz

a11bc]+Π[σzaσ

zb11c]

−2Π[σxaσ

xb11c]−2Π[σy

aσyb11c]−2Π[σx

aσxbσ

zc]−2Π[σy

aσybσ

zc]}

,

(7)which was easier to implement with our setup.

ON THE EXPERIMENTAL MEASUREMENT OF THECLIENT’S QUBIT FOR QUANTUM TELECLONING

A few remarks are in order on the way the client’s qubitX isexperimentally measured in the actual implementation of thequantum telecloning protocol.

Due to slight unbalance at BS2 of Fig. 1c) of the main Let-ter, the blue and yellow paths in the Sagnac loop used in orderto encode qubitX are unbalanced. We have thus correctedfor such an asymmetry by first measuring the state of qubitXgenerated entering the loop only with|r〉 modes [i.e. the bluepath in Fig. 2c) andd)]. We have then done the same with

the |ℓ〉 modes (yellow paths). Finally, we have traced out thepath degree of freedom embodied by{|r〉, |ℓ〉} by summing upthe corresponding counts measured for every single projectionthat is needed for the implementation of single-qubit quantumstate tomography, therefore reinstating symmetry.

FIDELITY OF QUANTUM TELECLONING FOR MIXEDSTATES OF THE CLIENT

Here we provide a model for the solid (red) line of Fig. 2(b)accounting for the fidelity of quantum telecloning of a client’smixed state. The evaluation of the theoretical fidelity of tele-cloning given in the main Letter does not take into account themixed nature of the client’s state, as well as the non-idealityof the experimental Dicke channel used for the scheme. Asargued in the main Letter, these are the main sources of dis-crepancy between the experimental results and the theoreticalpredictions. Here we provide a simple model that includesthese imperfections and allows for a more faithful compari-son between theoretical predictions and experimental data.

Our starting point is the observation that mixed input statesof the client can correspond to telecloning fidelities larger thanthe theoretical values predicted byF (θ) = [9 − cos(2θ)]/12.This can be straightforwardly seen by running the quantumtelecloning protocol with a decohered state resulting fromtheapplication of a dephasing channel to a pure client’s state ofthe formα|0〉X + β|1〉X with α = cos(θ/2) as in the main Let-ter. This is illustrated in Fig. 2 of the main Letter. Quite in-tuitively, as the input client’s state loses tis coherences, thefidelity of telecloning improves. The second observation wemake is that the entangled channel used in our experiment,although being of very good quality, has a non-unit overlapwith an ideal Dicke resource. Taking into account the majorsources of experimental imperfections, along the lines of theinvestigation in [14], a reasonable description of the four-qubitresource produced in our experiment is the Werner-like state

ρD = p|D(2)4 〉〈D

(2)4 | + (1− p)11/16 (8)

with 0 ≤ p ≤ 1. The entangled Dicke component in suchstate is evaluated considering that our experimental estimatefor the lower bound on the state fidelity isFD(2)

4= (0.78± 0.5).

Moreover, we have checked that slight experimental imper-fections in the determination of the populations of the inputclient’s states (within the range observed experimentally) donot affect the overall picture significantly. We have thus in-corporated the effects of a coherence-depleted input states ofqubit X into the protocol for 1→ 3 quantum telecloning per-formed using a mixed Dicke resource as in Eq. (8). The de-phasing parameter used in the model for mixed client’s statehas been adjusted so that, atθ = π/2, we get the real part ofthe experimentally reconstructed off-diagonal elements of thedensity matrix of qubitX (fixed relative phases between|0〉Xand |1〉X do not modify our conclusions). The resulting statefidelity, shown in Fig. 2b) of the main letter, shows a verygood agreement with the experimental data.

7

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FIG. 5: We report the reconstructed density matrices of the telecloned states measured on qubita, for three different input client’s states (qubitX).

SINGLE-QUBIT QUANTUM STATE TOMOGRAPHY OFRECEIVERS’ STATES IN EXPERIMENTAL QTC AND ODT

In Fig. 5 (Fig. 6) we give the single-qubit density matrixobtained through quantum state tomography of the receiver’sstate in the QTC (ODT) protocol. The telecloned states re-ported in Fig.5 have been shown in Fig. 2b) of the mainletter. We have considered three different input client’s states.For each of them, we have measured the telecloned state onthe qubita.

The values of state fidelity included in the Figure6 arethose reported in Table I of the main Letter. We have con-sidered four different input client’s states. For each of them,we have projected the server’s elements onto either|01〉Sodt

or |10〉Sodt and taken qubita or b as he receiver. The corre-

sponding quantum state fidelities are evidently quite uniformand consistently above 90% (mean fidelity 0.96± 0.01), thusdemonstrating high-quality and receiver-oblivious ODT.

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FIG. 6: We report the reconstructed density matrices of the various receiver states for four different input client’s states (qubitX) and projectionsof the server’s qubits onto both|01〉Sodt and|10〉Sodt.

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