Measuring Entanglement in a Photonic Embedding Quantum Simulator
Juan C. Loredo, Marcelo P. Almeida, Roberto Di Candia, Julen S. Pedernales, Jorge Casanova, Enrique Solano, Andrew G. White
MMeasuring Entanglement in a Photonic Embedding Quantum Simulator
J. C. Loredo, ∗ M. P. Almeida, R. Di Candia, J. S. Pedernales, J. Casanova, E. Solano,
2, 4 and A. G. White Centre for Engineered Quantum Systems, Centre for Quantum Computer and Communication Technology,School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain Institut f¨ur Theoretische Physik, Albert-Einstein-Allee 11, Universit¨at Ulm, D-89069 Ulm, Germany IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain
Measuring entanglement is a demanding task that usually requires full tomography of a quantumsystem, involving a number of observables that grows exponentially with the number of parties.Recently, it was suggested that adding a single ancillary qubit would allow for the efficient measure-ment of concurrence, and indeed any entanglement monotone associated to antilinear operations.Here, we report on the experimental implementation of such a device—an embedding quantumsimulator—in photonics, encoding the entangling dynamics of a bipartite system into a tripartiteone. We show that bipartite concurrence can be efficiently extracted from the measurement ofmerely two observables, instead of fifteen, without full tomographic information.
Entanglement is arguably the most striking featureof quantum mechanics [1], defining a threshold betweenthe classical and quantum behavior of nature. Yet itsexperimental quantification in a given system remainschallenging. Several measures of entanglement involveunphysical operations, such as antilinear operations, onthe quantum state [2, 3], and thus its direct measure-ment cannot be implemented in the laboratory. As aconsequence, in general, experimental measurements ofentanglement have been carried out mostly via the fullreconstruction of the quantum state [4]. While thistechnique—called quantum state tomography (QST)—has been widely used when dealing with relatively low-dimensional systems [5, 6], it is known to become rapidlyintractable as the system size grows, being outside of ex-perimental reach in systems with ∼
10 qubits [7]. Thisdifficulty lies in having to measure an exponentially-growing number of observables, 2 N −
1, to reconstruct N -qubits. Such constraint can be relaxed somewhat byusing, for example, multiple copies of the same quan-tum state [8], prior state knowledge in noisy dynam-ics [9], compressed sensing methods [10], or measuringphases monotonically dependent on entanglement [11].However, measuring entanglement in scalable systems re-mains a challenging task.An efficient alternative is to embed the system dynam-ics into an enlarged Hilbert-space simulator, called em-bedding quantum simulator (EQS) [12, 13], where un-physical operations are mapped onto physical transfor-mations on the simulator. The price to pay, compara-tively small in larger systems, is the addition of only oneancillary qubit and, usually, dealing with more involveddynamics. However, measuring the entanglement of thesimulated system becomes efficient, involving the mea-surement of a low number of observables in the EQS, incontrast to the 2 N − Protocol.
We consider the simulation of two-qubitentangling dynamics governed by the Hamiltonian H = − gσ z ⊗ σ z , where σ z = | (cid:105)(cid:104) |−| (cid:105)(cid:104) | is the z-Pauli ma-trix written in the computational basis, {| (cid:105) , | (cid:105)} , and g is a constant with units of frequency. For simplic-ity, we let (cid:126) =1. Under this Hamiltonian, the concur-rence [2] of an evolving pure state | ψ ( t ) (cid:105) is calculatedas C = |(cid:104) ψ ( t ) | σ y ⊗ σ y K | ψ ( t ) (cid:105)| , where K is the complexconjugate operator defined as K | ψ ( t ) (cid:105) = | ψ ( t ) ∗ (cid:105) . Noticehere the explicit dependance of C upon the unphysicaltransformation K . We now consider the dynamics of theinitial state | ψ (0) (cid:105) =( | (cid:105) + | (cid:105) ) ⊗ ( | (cid:105) + | (cid:105) ) /
2. Under theseconditions one can calculate the resulting concurrence atany time t as C = | sin(2 gt ) | . (1)The target evolution, e − iHt | ψ (0) (cid:105) , can be embedded ina 3-qubit simulator. Given the state of interest | ψ (cid:105) , thetransformation | Ψ (cid:105) = | (cid:105) ⊗ Re | ψ (cid:105) + | (cid:105) ⊗ Im | ψ (cid:105) , (2)gives rise to a real-valued 3-qubit state | Ψ (cid:105) in the cor-responding embedding quantum simulator. The decod-ing map is, accordingly, | ψ (cid:105) = (cid:104) | Ψ (cid:105) + i (cid:104) | Ψ (cid:105) . The phys-ical unitary gate σ z ⊗ I transforms the simulator stateinto σ z ⊗ I | Ψ (cid:105) = | (cid:105)⊗ Re | ψ (cid:105)−| (cid:105)⊗ Im | ψ (cid:105) , which after thedecoding becomes (cid:104) | Ψ (cid:105)− i (cid:104) | Ψ (cid:105) =Re | ψ (cid:105)− i Im | ψ (cid:105) = | ψ ∗ (cid:105) .Therefore, the action of the complex conjugate op-erator K corresponds to the single qubit rotation σ z ⊗ I [12, 14]. Now, following the same encoding rules: (cid:104) ψ | OK | ψ (cid:105) = (cid:104) Ψ | ( σ z − iσ x ) ⊗ O | Ψ (cid:105) , with O an observable inthe simulation. In the case of O = σ y ⊗ σ y , we obtain C = |(cid:104) σ z ⊗ σ y ⊗ σ y (cid:105) − i (cid:104) σ x ⊗ σ y ⊗ σ y (cid:105)| , (3) a r X i v : . [ qu a n t - ph ] M a r H ( E ) = (cid:31) σ x ⊗ σ y ⊗ σ y (cid:29) (cid:31) σ z ⊗ σ y ⊗ σ y (cid:29) gσ (0) y ⊗ σ (1) z ⊗ σ (2) z Π (cid:31) Π (cid:31) Π (cid:31) Π C b) t − − H = (cid:31) I ⊗ σ z (cid:29) t − gσ (1) z ⊗ σ (2) z .... − − − (cid:31) I ⊗ σ x (cid:29) QST a) EQS (cid:31) σ y ⊗ σ y (cid:29)
15 observables2 observables gt FIG. 1. (a) Qubits 1 and 2 evolve via an entangling Hamil-tonian H during a time interval t , at which point quantumstate tomography (QST) is performed via the measurement of15 observables to extract the amount of evolving concurrence.(b) An efficient alternative corresponds to adding one extraancilla, qubit 0, and having the enlarged system—the embed-ding quantum simulator (EQS)—evolve via H ( E ) . Only twoobservables are now required to reproduce measurements ofconcurrence of the system under simulation. which relates the simulated concurrence to the ex-pectation values of two nonlocal operators in theembedding quantum simulator. Regarding the dy-namics, it can be shown that the Hamiltonian H ( E ) that governs the evolution in the simulator is H ( E ) = − σ y ⊗ (Re H )+ i I ⊗ (Im H ) [12]. Accordingly, inour case, it will be given by H ( E ) = gσ y ⊗ σ z ⊗ σ z .Our initial state under simulation is | ψ (0) (cid:105) =( | (cid:105) + | (cid:105) ) ⊗ ( | (cid:105) + | (cid:105) ) /
2, which requires, seeEq. (2), the initialization of the simulator in | Ψ(0) (cid:105) = | (cid:105)⊗ ( | (cid:105) + | (cid:105) ) ⊗ ( | (cid:105) + | (cid:105) ) /
2. Under these con-ditions, the relevant simulator observables, see Eq. (3),read (cid:104) σ x ⊗ σ y ⊗ σ y (cid:105) = sin (2 gt ) and (cid:104) σ z ⊗ σ y ⊗ σ y (cid:105) =0, fromwhich the concurrence of Eq. (1) will be extracted.Therefore, our recipe, depicted in Fig. 1, allows theencoding and efficient measurement of two-qubitconcurrence dynamics.To construct the described three-qubit simulator dy-namics, it can be shown (see Supplemental Material)that a quantum circuit consisting of 4 controlled-signgates and one local rotation R y ( φ )=exp ( − iσ y φ ), as de-picted in Fig. 2(a), implements the evolution operator U ( t )=exp [ − ig ( σ y ⊗ σ z ⊗ σ z ) t ], reproducing the desireddynamics, with φ = gt . This quantum circuit can be fur-ther reduced if we consider only inputs with the ancillaryqubit in state | (cid:105) , in which case, only two controlled-signgates reproduce the same evolution, see Fig. 2 (b). Thisreduced subspace of initial states corresponds to simu-lated input states of only real components. Experimental implementation.
We encode a three- a)b) R y ( φ ) exp (cid:31) − i (cid:30) σ (0) y ⊗ σ (1) z ⊗ σ (2) z (cid:29) φ (cid:28) R y ( φ ) exp (cid:31) − i (cid:30) σ (0) y ⊗ σ (1) z ⊗ σ (2) z (cid:29) φ (cid:28) .. .... .... .. FIG. 2. Quantum circuit for the embedding quan-tum simulator. (a) 4 controlled-sign gates and onelocal rotation R y ( φ ) implement the evolution operator U ( t )=exp (cid:16) − igσ (0) y ⊗ σ (1) z ⊗ σ (2) z t (cid:17) , with φ = gt . (b) A reducedcircuit employing only two controlled-sign gates reproducesthe desired three-qubit dynamics for inputs with the ancil-lary qubit in | (cid:105) . qubit state in the polarization of 3 single-photons. Thelogical basis is encoded according to | h (cid:105)≡| (cid:105) , | v (cid:105)≡| (cid:105) ,where | h (cid:105) and | v (cid:105) denote horizontal and vertical polariza-tion, respectively. The simulator is initialized in the state | Ψ(0) (cid:105) = | h (cid:105) (0) ⊗ (cid:0) | h (cid:105) (1) + | v (cid:105) (1) (cid:1) ⊗ (cid:0) | h (cid:105) (2) + | v (cid:105) (2) (cid:1) / φ = gt iscontrolled by the angle φ/ (cid:104) σ x ⊗ σ y ⊗ σ y (cid:105) isobtained from measuring the 8 projection combina-tions of the {| d (cid:105) , | a (cid:105)}⊗{| r (cid:105) , | l (cid:105)}⊗{| r (cid:105) , | l (cid:105)} states, where | d (cid:105) =( | h (cid:105) + | v (cid:105) ) / √ | r (cid:105) =( | h (cid:105) + i | v (cid:105) ) / √
2, and | a (cid:105) and | l (cid:105) are their orthogonal states, respectively. To implementthese polarization projections, we employed Glan-Taylorprisms due to their high extinction ratio. However, onlytheir transmission mode is available, which required eachof the 8 different projection settings separately, extendingour data-measuring time. The latter can be avoided bysimultaneously registering both outputs of a projectivemeasurement, such as at the two output ports of a polar- (0)( π/ t h =1t v =0 : :QWP APD : t v =1 / h =1 : ( φ/ ( π/
8) ( π/ PPBS1 PPBS2 ( π/ t h =1 / v =1 : : HWP ( θ ) (GT) FIG. 3. Experimental setup. Three single-photons with wavelength centered at 820 nm are injected via single-mode fibers intospatial modes 0, 1 and 2. Glan-Taylor (GT’s) prisms, with transmittance t h =1 ( t v =0) for horizontal (vertical) polarization,and half-wave plates (HWP’s) are employed to initialize the state. Controlled two-qubit operations are performed based ontwo-photon quantum interference at partially polarizing beam-splitters (PPBS’s). Projective measurements are carried outwith a combination of half-wave plates, quarter-wave plates (QWP’s) and Glan-Taylor prisms. The photons are collected viasingle-mode fibers and detected by avalanche photodiodes (APD’s). izing beam splitter, allowing the simultaneous recordingof all 8 possible projection settings. Thus, an immediatereconstruction of each observable is possible.Our source of single-photons consists of four-photonevents collected from the forward and backward pairemission in spontaneous parametric down-conversion in a beta -barium borate (BBO) crystal pumped by a 76 MHzfrequency-doubled mode-locked femtosecond Ti:Sapphirelaser. One of the four photons is sent directly to anavalanche photodiode detector (APD) to act as a trig-ger, while the other 3 photons are used in the proto-col. This kind of sources are known to suffer from un-desired higher-order photon events that are ultimatelyresponsible of a non-trivial gate performance degrada-tion [18–20], although they can be reduced by decreasingthe laser pump power. However, given the probabilisticnature and low efficiency of down-conversion processes,multi-photon experiments are importantly limited by lowcount-rates, see Supplemental Material. Therefore, in-creasing the simulation performance quality by loweringthe pump requires much longer integration times to ac-cumulate meaningful statistics, which ultimately limitsthe number of measured experimental settings.As a result of these higher-order noise terms, a sim-ple model can be considered to account for non-perfectinput states. The experimental input n -qubit state ρ exp can be regarded as consisting of the ideal state ρ id withcertain probability ε , and a white-noise contribution withprobability 1 − ε , i.e. ρ exp = ερ id +(1 − ε ) I n / n . Since thesimulated concurrence is expressed in terms of tensorialproducts of Pauli matrices, the experimentally simulated concurrence becomes C exp = ε | sin(2 gt ) | .In Fig. 4, we show our main experimental results fromour photonic embedding quantum simulator for one cycleof concurrence evolution taken at different pump pow-ers: 60 mW, 180 mW, and 600 mW—referred as to 10%,30%, and 100% pump, respectively. Figure 4 (a) showstheoretical predictions (for ideal pure-state inputs) andmeasured fractions of the different projections involvedin reconstructing (cid:104) σ x ⊗ σ y ⊗ σ y (cid:105) and (cid:104) σ z ⊗ σ y ⊗ σ y (cid:105) for 10%pump at gt = π/
4. From measuring these two observables,see Eq. (3), we construct the simulated concurrence pro-duced by our EQS, shown in Fig. 4 (b). We observe agood behavior of the simulated concurrence, which pre-serves the theoretically predicted sinusoidal form. Theoverall attenuation of the curve is in agreement with theproposed model of imperfect initial states. Together withthe unwanted higher-order terms, we attribute the ob-served degradation to remaining spectral mismatch be-tween photons created by independent down-conversionevents and injected to inputs 0 and 2 of Fig. 3—at whichoutputs 2 nm band-pass filters with similar but not iden-tical spectra were used.We compare our measurement of concurrence via oursimulator with an explicit measurement from state to-mography. In the latter we inject one down-convertedpair into modes 0 and 1 of Fig. 3. For any value of t ,set by the wave-plate angle φ , this evolving state has thesame amount of concurrence as the one from our simula-tion, they are equivalent in the sense that one is relatedto the other at most by local unitaries, which could beseen as incorporated in either the state preparation or h rr h r l h l r h ll v rr v r l v l r v ll F r a c t i on h rr h r l h l r h ll v rr v r l v l r v ll F r a c t i on (cid:31) σ z ⊗ σ y ⊗ σ y (cid:29) ( gt = π/ d rr d r l d l r d ll a rr a r l a l r a ll F r a c t i on d rr d r l d l r d ll a rr a r l a l r a ll F r a c t i on (cid:31) σ x ⊗ σ y ⊗ σ y (cid:29) ( gt = π/ a) Π (cid:31) Π (cid:31) Π (cid:31) Π (cid:31) gt C three-qubit EQS b) FIG. 4. (a) Theoretical predictions (top) and experimentally measured (bottom) fractions involved in reconstructing (cid:104) σ x ⊗ σ y ⊗ σ y (cid:105) (left) and (cid:104) σ z ⊗ σ y ⊗ σ y (cid:105) (right), taken at gt = π/ C = C pp | sin(2 gt ) | , where C pp is the maximum concurrence extracted for a given pump power (pp): C =0 . ± . C =0 . ± .
03 and C =0 . ± . within the tomography settings.Figure 5 shows our experimental results for the de-scribed two-photon protocol. We extracted the con-currence of the evolving two-qubit state from overcom-plete measurements in quantum state tomography [4].A maximum concurrence value of C =1 is predicted in Π (cid:31) Π (cid:31) Π (cid:31) Π (cid:31) gt C two-qubit QST FIG. 5. Concurrence measured via two-qubit quantum statetomography (QST) on the explicit two-photon evolution,taken at 10% (blue), 30% (green), and 100% (red) pump pow-ers. The corresponding curves indicate C = C pp | sin(2 gt ) | , with C pp the maximum extracted concurrence for a given pumppower (pp): C =0 . ± . C =0 . ± . C =0 . ± . the ideal case of perfect pure-state inputs. Exper-imentally, we measured maximum values of concur-rence of C =0 . ± . C =0 . ± .
002 and C =0 . ± . C =0 . ± . Discussion.
We have shown experimentally that entan-glement measurements in a quantum system can be effi-ciently done in a higher-dimensional embedding quantumsimulator. The manipulation of larger Hilbert spaces forsimplifying the processing of quantum information hasbeen previously considered [24]. However, in the presentscenario, this advantage in computing concurrence origi-nates from higher-order quantum correlations, as it is thecase of the appearance of tripartite entanglement [25, 26].The efficient behavior of embedding quantum simula-tors resides in reducing an exponentially-growing numberof observables to only a handful of them for the extrac-tion of entanglement monotones. We note that in thisnon-scalable photonic platform the addition of one an-cillary qubit and one entangling gate results in countrates orders of magnitude lower as compared to directstate tomography on the 2-qubit dynamics. This meansthat in practice absolute integration times favor the di-rect 2-qubit implementation. However, this introducedlimitation escapes from the purposes of the embeddingprotocol and instead belongs to the specific technologyemployed in its current state-of-the-art performance.This work represents the first proof-of-principle ex-periment showing the efficient behavior of embeddingquantum simulators for the processing of quantum in-formation and extraction of entanglement monotones.This validates an architecture-independent paradigmthat, when implemented in a scalable platform, e.g.ions [7, 13], would overcome a major obstacle in the char-acterization of large quantum systems. The relevanceof these techniques will thus become patent as quan-tum simulators grow in size and currently standard ap-proaches like full tomography become utterly unfeasible.We believe that these results pave the way to the efficientmeasurement of entanglement in any quantum platformvia embedding quantum simulators.We thank M. A. Broome for helpful discussions.This work was supported by the Centre for Engi-neered Quantum Systems (Grant No. CE110001013)and the Centre for Quantum Computation and Com-munication Technology (Grant No. CE110001027).M. P. A. acknowledges support from the AustralianResearch Council Discovery Early Career Awards(No. DE120101899). A. G. W. was supported by theUniversity of Queensland Vice-Chancellor’s Senior Re-search Fellowship. J. C. acknowledge support from theAlexander von Humboldt Foundation, while R. D. C.,J. S. P., and E. S. from Basque Government IT472-10;Spanish MINECO FIS2012-36673-C03-02; UPV/EHUUFI 11/55; UPV/EHU PhD fellowship; PROMISCEand SCALEQIT EU projects.
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Following the main text, the evolution oper-ator associated with the embedding Hamiltonian H ( E ) = gσ y ⊗ σ z ⊗ σ z can be implemented via 4 control- Z gates ( CZ ), and a single qubit rotation R y ( t ). Thesegates act as CZ ij = | (cid:105)(cid:104) | ( i ) ⊗ I ( j ) + | (cid:105)(cid:104) | ( i ) ⊗ σ ( j ) z , (4) R iy ( t ) = e − iσ ( i ) y gt ≡ (cos( gt ) I ( i ) − i sin( gt ) σ ( i ) y ) , (5)with σ z = | (cid:105)(cid:104) |−| (cid:105)(cid:104) | , and σ y = − i | (cid:105)(cid:104) | + i | (cid:105)(cid:104) | . Theindices i and j indicate on which particle the operatorsact. The circuit for the embedding quantum simulatorconsists of a sequence of gates applied in the followingorder: U ( t ) = CZ CZ R y ( t ) CZ CZ . (6)Simple algebra shows that this expression can be recastas U ( t ) = cos( gt ) I (0) ⊗ I (1) ⊗ I (2) − i sin( gt ) σ (0) y ⊗ σ (1) z ⊗ σ (2) z = exp (cid:16) − igσ (0) y ⊗ σ (1) z ⊗ σ (2) z t (cid:17) , (7)explicitly exhibiting the equivalence between the gate se-quence and the evolution under the Hamiltonian of in-terest. II. Linear optics implementation
The evolution of the reduced circuit is given by a R y ( t )rotation of qubit 0, followed by two consecutive control-Zgates on qubits 1 and 2, both controlled on qubit 0, seeFig. 6 (a). These logic operations are experimentally im-plemented by devices that change the polarization of thephotons, where the qubits are encoded, with transforma-tions as depicted in Fig. 6 (b). For single qubit rotations,we make use of half-wave plates (HWP’s), which shift thelinear polarization of photons. For the two-qubit gates,we make use of two kinds of partially-polarizing beamsplitters (PPBS’s). PPBS’s of type 1 have transmit-tances t h =1 and t v =1 / t h =1 / t v =1. Their ef-fect can be expressed in terms of polarization dependantinput-output relations—with the transmitted mode cor-responding to the output mode—of the bosonic creationoperators as a † ( i ) p,out = (cid:112) t p a † ( i ) p,in + (cid:112) − t p a † ( j ) p,in (8) a † ( j ) p,out = (cid:112) − t p a † ( i ) p,in − (cid:112) t p a † ( j ) p,in , (9) where a † ( i ) p,in ( a † ( i ) p,out ) stands for the i -th input (output)port of a PPBS with transmittance t p for p -polarizedphotons. Z Z bcdcbd σ Z σ X σ X a)b) hvhvhv t =13 t =13 t =13 t =13 Ry(t)Ry(t)
FIG. 6. (a) Circuit implementing the evolution opera-tor U ( t )=exp (cid:16) − igσ (0) y ⊗ σ (1) z ⊗ σ (2) z t (cid:17) , if the initial stateis | Ψ(0) (cid:105) = | (cid:105) (0) ⊗ (cid:16) | (cid:105) (1) + | (cid:105) (1) (cid:17) ⊗ (cid:16) | (cid:105) (2) + | (cid:105) (2) (cid:17) / Our circuit is implemented as follows: the first R y ( t )rotation is implemented via a HWP oriented at an angle θ = gt/ b h c h d h → b h c h d h b h c h d v → b h c h d v b h c v d h → b h c v d h b h c v d v → b h c v d v b v c h d h → b v c h d h b v c h d v → − b v c h d v b v c v d h → − b v c v d h b v c v d v → b v c v d v , (10)where b ≡ a † (0) , c ≡ a † (1) , and d ≡ a † (2) denote the creationoperators acting on qubits 0, 1, and 2, respectively. Thesepolarization transformations can be implemented witha probability of 1 /
27 via a 3-fold coincidence detectionin the circuit depicted in Fig. 6 (b). In this dual-railrepresentation of the circuit, interactions of modes c and d with vacuum modes are left implicit.The σ x and σ z single qubit gates in Fig. 6 (b) are im-plemented by HWP’s with angles π/ σ x : b h → b v , b v → b h (11) σ z : d h → d h , d v → − d v . (12)According to all the input-output relations involved, itcan be calculated that the optical elements in Fig. 6 (b)implement the following transformations b h c h d h → b h c h d h / (3 √ b h c h d v → b h c h d v / (3 √ b h c v d h → b h c v d h / (3 √ b h c v d v → b h c v d v / (3 √ b v c h d h → b v c h d h / (3 √ b v c h d v → − b v c h d v / (3 √ b v c v d h → − b v c v d h / (3 √ b v c v d v → b v c v d v / (3 √ , (13)if events with 0 photons in some of the three outputlines of the circuit are discarded. Thus, this linear opticsimplementation corresponds to the evolution of interestwith success probability P = (1 / (3 √ = 1 / III. Photon count-rates
Given the probabilistic nature and low efficiency ofdown-conversion processes, multi-photon experimentsare importantly limited by low count-rates. In our case,typical two-photon rates from source are around 150 kHzat 100% pump (two-photon rates are approx. linear withpump power), which after setup transmission ( ∼ / / ∼
100 mHz ( ∼ IV. Pump-dependence
To estimate the effect of power-dependent higher-orderterms in the performance of our protocols, we inspect thepump power dependence of extracted concurrence from
FIG. 7. Measured concurrence vs pump power. The concur-rence is extracted from both two-qubit quantum state tomog-raphy (QST) and the three-qubit embedding quantum simu-lator (EQS). Straight lines are linear fits to the data. Slopesoverlapping within error, namely − . ± . − . ± . both methods. Fig. 7 shows that the performances ofboth protocols decrease at roughly the same rate withincreasing pump power, indicating that in both meth-ods the extracted concurrence at 10% pump is close toperformance saturation.The principal difference between the two methods isthat in the three-qubit protocol one of the photons origi-nates from an independent down-conversion event and assuch will present a slightly different spectral shape dueto a difficulty in optimizing the phase-matching condi-tion for both forward and backward directions simulta-neously. To reduce this spectral mismatch, we used two2 nm filters at the output of the two spatial modes whereinterference from independent events occurs, see Fig. 8.Note that not identical spectra are observed. This lim-itation would be avoided with a source that presentedsimultaneous high indistinguishability between all inter-fering photons.
818 819 820 821 822 823 λ ( nm )0.00.20.40.60.81.0Transmission Filter AFilter B