Experimental realization of local-to-global noise transition in a two-qubit optical simulator
Claudia Benedetti, Valeria Vento, Stefano Olivares, Matteo G. A. Paris, Simone Cialdi
EExperimental realization of local-to-global noise transition in a two-qubit opticalsimulator
Claudia Benedetti, Valeria Vento, Stefano Olivares, Matteo G. A. Paris, Simone Cialdi Dipartimento di Fisica "Aldo Pontremoli", Università degli Studi di Milano, I-20133 Milano, Italia
We demonstrate the transition from local to global noise in a two-qubit all-optical quantumsimulator subject to classical random fluctuations. Qubits are encoded in the polarization degreeof freedom of two entangled photons generated by parametric down-conversion (PDC) while theenvironment is implemented using their spatial degrees of freedom. The ability to manipulate withhigh accuracy the number of correlated pixels of a spatial-light-modulator and the spectral PDCwidth, allows us to control the transition from a scenario where the qubits are embedded in localenvironments to the situation where they are subject to the same global noise. We witness thetransition by monitoring the decoherence of the two-qubit state.
Quantum simulators are controllable quantum sys-tems, usually made of qubits, able to mimic the dynamicsof other, less controllable, quantum systems [1, 2]. Quan-tum simulators make it possible to design and control thedynamics of complex systems with a large number of de-grees of freedom, or with stochastic components [3–7].In turn, open quantum systems represent a fundamentaltestbed to assess the reliability and the power of a quan-tum simulator. The external environment may be de-scribed either as a quantum bath, or a classical randomfield which, in generale, lead to different system evolu-tions. However, in the case of pure dephasing, the effectsof a quantum bath are equivalent to those provoked byrandom fluctuations [8]. For this reason, together withthe fact that it is an ubiquitous source of decoherencethat jeopardizes quantum features, dephasing noise playsa prominent role in the study of open quantum systems.Pioneering works on the controlled simulation of single-qubit dephasing channels appeared few years ago [9, 10],whereas the realisation of multi-qubit simulators is stillmissing. In fact, the simulation of multi-qubit systems isnot a mere extension of the single-qubit case since com-posite systems present features that are absent in thesingle-component case, e.g. entanglement [11–15]. More-over, multipartite systems allow us to analyze the effectsof a global source of noise against those due to local en-vironments. Understanding the properties of the local-to-global (LtG) noise transition is in turn a key task inquantum information, both for quantum and classical en-vironments, since it sheds light on the mechanisms gov-erning the interaction between the quantum system andits environment, providing tools to control decoherence[16–21].We present here an all-optical implementation of thewhole class of two-qubit dephasing channels arising fromthe interaction with a classically fluctuating environment.The qubits are encoded in the polarization degree of free-dom of a photon-pair generated by parameteric-down-conversion (PDC), while the spatial degrees of freedomare used to implement the environment. Different realiza-tions of the noise are randomly generated and imprinted on the qubits through a spatial-light modulator (SLM).The ensemble average is then performed by collectingthe photons with a multimode fiber. With our simula-tor there is no need to work at cryogenic temperaturesand we are able to simulate any conceivable form of theenvironmental noise, independently on its spectrum.In particular, here we exploit our simulator to demon-strate the transition from a local-environment scenario,where each qubit is subject to an independent source ofnoise, to a global environment where both qubits feel thesame synchronous random fluctuations. There are twodifferent mechanisms that may lead to this transition.The first one appears when two local environments be-come correlated due to the action of some external agent,and one moves from local to global noise as the two envi-ronments become fully correlated. In the second scenario,the two qubits are placed in the same environment, butat a distance that is much larger than the correlationlength of the noise. As the distance between the qubitsis reduced, they start to feel similar environments, un-til they are within the correlation length of the environ-ment and thus subject to the same common source ofnoise. The two situations are illustrated in Fig. 1. In thefirst case, two initially different environments (cottages)become gradually identical as far as correlations are es-tablished (by putting bricks), whereas in the second casethe two qubits (persons) are initially far apart, but theyend up feeling the same environment (cottage) as long astheir distance is reduced.The dephasing map of two non-interacting qubits aris-ing from a classical environment is generated by the di-mensionless Hamiltonian: H ( t ) = X ( t ) σ (1) z ⊗ I (2) + X ( t ) I (1) ⊗ σ (2) z , (1)where σ z is the Pauli matrix, I the identity matrix, X ( t ) is a stochastic process and the labels 1 and 2 denote thetwo qubits. Since our aim is to give a proof-of-principle ofthe LtG transition and we are not interested at this stagein the specific form of the noise, we fix the stochasticprocess to be a random telegraph noise (RTN). It followsthat X k ( t ) = ± is a dichotomous variable which jumps a r X i v : . [ qu a n t - ph ] F e b (b) (a) (a) FIG. 1: Cartoon description of the two scenarios leading tothe LtG transition. On the left: LtG transition after estab-lishing correlations between the two environments. On theright: LtG transition as the qubits approach each other. between two values with a certain switching rate γ thatdetermines the correlation length of the noise throughthe autocorrelation function C ( t ) = (cid:104) X ( t ) X (0) (cid:105) = e − γt .The symbol (cid:104) . . . (cid:105) denotes the ensemble average over allpossible realizations of the RTN. In Eq. (1) it is possibleto identify two complemtary regimes: If X ( t ) and X ( t ) are two identical but independent processes, then we arein the presence of local environments and each qubit issubject to its own noise. On the other hand, if X ( t ) = X ( t ) , at all times, then the fluctuations are synchronized(perfectly correlated) and the qubits interact with thesame global environment. The generated dynamics inthese two scenarios are very different and this can bewitnessed, for example, by looking at the behavior ofentanglement. What happens in between these regimesis unexplored territory. Theoretical model − In order to address the local-to-global noise transition, we first need to computethe two qubits dynamics in the presence of classi-cal noise. Starting from an initial Bell state | ψ (cid:105) = √ ( | HH (cid:105) + | VV (cid:105) ) , the system density matrix is obtainedas ρ ( t ) = (cid:104) U ( t ) ρ U † ( t ) (cid:105) , where ρ = | ψ (cid:105)(cid:104) ψ | , U ( t ) =exp (cid:2) − i (cid:82) H ( s ) ds (cid:3) = exp[ − iϕ ( t )] , ϕ ( t ) = (cid:82) H ( s ) ds is thenoise phase. The elements of ρ ( t ) in the polarizationbasis {| HH (cid:105) , | HV (cid:105) , | VH (cid:105) , | VV (cid:105)} are: ρ ( t ) = ρ ( t ) = and ρ ( t ) = ρ ∗ ( t ) = Γ( t ) , and all other elements arezero. The coherence factor Γ( t ) depends on the natureand the correlations of the noises. In particular, it wasshown that for local (LE) and global environments (GE)the coherence factor takes the forms respectively: Γ LE ( t ) = (cid:104) e iϕ ( t ) (cid:105) Γ GE ( t ) = (cid:104) e iϕ ( t ) (cid:105) (2)where the averages of the exponential moments are givenby (cid:104) e m iϕ ( t ) (cid:105) = e − γt (cosh δ m t + γ/δ m sinh δ m t ) with δ m = (cid:112) γ − m . The entanglement E between the qubits is FIG. 2: Schematic diagram of the experimental setup. A -nm cw laser diode generates a pump beam which passesthrough a half-wave plate ( λ/ ), a polarizing beam-splittercube (PBS), and another half-wave plate. Then it is colli-mated by a telescopic system composed by two lenses L t and L t . The beam passes through a series of compensationcrystals and then it interacts with two 1-mm long BBO crys-tals generating photons centered at nm via PDC. Eachbranch passes through a lenses L with focal f , the spatial lightmodulator (SLM) and a polarizer (P). Photons are finally fo-cused into two multimode fibers through the couplers C and C : the first is directly linked to an homemade single-photoncounting module, the second is sent to a spectral selector andthen to the counting module given by E ( t ) = | Γ( t ) | in both LE and GE cases.The realization of the qubit state ρ ( t ) requires the si-multaneous generation of a large number of stochastictrajectories of the noise. Our experimental apparatusallows us to obtain the average over the realizations inparallel, exploiting the spatial and the spectral degreesof freedom of the photons. In particular, in our experi-mental setup the following state is generated | ψ SE ( t ) (cid:105) = 1 √ (cid:90) dx dx f ( x , x ) (cid:104) | H x H x (cid:105) + e i [ ϕ ( x ,t )+ ϕ ( x ,t )] | V x V x (cid:105) (cid:105) (3)where f ( x , x ) is the spatial correlation function be-tween the two photons, the ϕ k ( x k , t ) ’s are the noisephases, and | P x P x (cid:105) denotes a state where the pho-ton 1(2) has polarization P (P ) and is in position x ( x ). In this scenario, the stochastic trajectories are en-coded in the spatial degree of freedom and the state ρ ( t ) is obtained by tracing out x and x . Finally, as we willshow below, we employ the spectral degree of freedom todefine the degree of spatial correlation between the twophotons. Experimental realization − Our experimental setup isschematically depicted in Fig. 2. The pump is generatedfrom a 405-nm cw InGaN laser diode. The laser beampasses through an amplitude modulator, composed bya half-wave plate and a polarizing beam-splitter (PBS),and then through another half-wave plate to set the po-larization. Subsequently, the polarized 405-nm beamgoes through a telescopic system composed by two lenses L t and L t with respective focal lengths f t = 100 mmand f t = 75 mm, with the double purpose of collimatingthe beam and optimizing its size in order to maximize thedetection efficiency. After the telescopic system there arethree 1-mm long crystals that compensate the delay timeintroduced by the PDC crystals. In order to generatethe entangled state in the polarization we use two mmlong crystals of beta-barium borate (BBO) [22]. On eachbranch of the PDC, a lens L with focal length f = 200 mm at 810 nm is placed at a distance f from the BBO.On the Fourier’s plane (at distance f from the lens L )there is the SLM which is a 1D liquid crystal mask with640 pixels of width 100 µ m/pixel. Each pixel imprinta computer-generated phase on the horizontal-polarizedcomponent. Photons then pass through polarizers andare finally focused into a multimode fiber through thecouplers C and C and into a photon-counting module.The signal beam, before the detection, passes through aspectral selector, which consists of two gratings and twolenses building an optical 4f system. In the Fourier planeof this apparatus, we use a mechanical slit in order toselect the desired spectral width [1].Since each pixel of the SLM has a finite width, wemay substitute the integral with a sum over the pix-els positions in Eq. (3). Then, by taking the par-tial trace over the spatial degrees of freedom, we ob-tain ρ S ( t ) = ( | HH (cid:105)(cid:104) HH | + | VV (cid:105)(cid:104) VV | + p Γ( t ) | HH (cid:105)(cid:104) VV | + p Γ ∗ ( t ) | VV (cid:105)(cid:104) HH | ) where p is a parameter quantifying theentanglement in the initial state ρ S (0) = p ρ +(1 − p ) ρ mix ,where ρ mix = ( | HH (cid:105)(cid:104) HH | + | VV (cid:105)(cid:104) VV | ) . In our case p isclose to 1 [24] and the procedure we use to purify thestate is described in [3, 4]. The decoherence function Γ( t ) depends on the spatial correlations between the twophotons and on the stochastic realizations: Γ( t ) = (cid:88) jk | f jk | e i [ ϕ ( x j ,t )+ ϕ ( x k ,t )] . (4)where the distribution | f jk | = N exp (cid:26) − j − j ) − ( k − k )] n w n cp (cid:27) × exp (cid:26) − j − j ) w p − k − k ) w p (cid:27) (5)takes into account the size of the coupled PDC w p andspatial correlation between the photons (i.e. the numberof correlated pixels) w cp (See the supplementary materialfor details and the detivation). The first factor is a super-Gaussian of order n , while j and k are the central pixes on the SLM for each PDC branch. Finally N is a nor-malization factor in order to assure that (cid:80) jk | f jk | = 1 .It is now clear that we may simulate the LtG transitionusing two different strategies, either by controlling therealizations of the noise on the two paths of the PDC, orby tuning the number of correlated pixels. Results − Let us start by explaining the role of theSLM in encoding the stochastic process into the pixels.We figuratively divide the SLM in two parts, both madeof 320 pixels. The first set is dedicated to the first qubitand the pixels are indexed by an integer j that goes from to . The second part is dedicated to the second qubitand the pixels are labeled by k that goes from to .Called d the width of the pixel, the two positions x and x are: x j = d j , x k = d (640 − k ) . This allows us todirectly consider the simmetry of the spatial correlationsbetween the photons in the notation.The first step in order to send the same noise on thecorrelated pixels in the two parts of the mask is to ex-perimentally find out the central pixels j and k . Thecentral pixels are the reference for the definition of thephases ϕ and ϕ . In particular we set: ϕ ( x ,j +∆ , t ) = ϕ ( x ,k +∆ , t ) = ϕ (∆ , t ) (6)where ∆ is an integer shift with respect the referencesand ϕ is a phase function defined over points. Thefirst method we use to simulate te LtG transition consistsin introducing an integer shift δ on the array of phases ϕ imprinted on the second side of the SLM. Setting δ = 0 the correlated pixels see the same noise and we mimic thecase where the environments are fully correlated. When δ is increased, the two environments become progressivelyless correlated. It follows, that as the value of δ is de-creased from a large value to zero, we obtain the LtGtransition. The function Γ is now a function of δ and wehave: Γ( δ, t ) = (cid:88) jk | f jk | e i [ ϕ ( x j ,t )+ ϕ ( x k + δ ,t )] . (7)Since this technique is effective for spatial correlationlengths w cp smaller or equal with respect to the typi-cal spatial variation of the function ϕ , we use a spectralwidth of ∆ λ = 15 nm, resulting in a w cp of about pix-els [24] and use a function ϕ that changes value every pixels. By reducing the spectrum width it is possibleto obtain smaller values of w cp at the price of reducingcounts and, in turn, increasing fluctuations.At first, let us consider the experimental realization ofthe local-to-global transition in the case of the RTN with γ = 0 using the δ shift-technique. Upon imposing δ = 0 we obtain two fully correlated environments, and usingEq. (2) we have | Γ GE ( t ) | = |(cid:104) e iϕ ( t ) (cid:105)| = | cos(4 t ) | . Indeed,due to the RTN noise, the only two possible values for ϕ are t and − t . The blue and the red curves in Figure3b. are respectively | Γ | and | Re { Γ }| obtained theoreti-cally using the experimental values for w cp and w p and Experimental data Simulated data Theoretical curve
FIG. 3: Local-to-global noise transition by translation of therealizations ϕ ( t ) for the RTN with γ = 0 and ∆ λ = 15 nm.The green curves represent the experimental data, while theblue curves are the corresponding simulations for | Re[Γ( t )] | .The red curve is | Γ( t ) | . In the left panel we show results for δ = 3 , i.e. the two environments are not correlated. The rightpanel is δ = 0 , i.e. fully correlated environments. pixels respectively. The comparison between the redand the blue curves shows that the Γ function is real, asit would be in the ideal case with an infinite number ofrealizations. In order to obtain this result it is necessaryto select the ϕ function with a balanced number of posi-tive and negative realization at time t = 0 . When δ = 3 (Figure 3a) the environments are not correlated. In thiscase the two qubits see two different phases (indeed theshift δ makes the two quantities ϕ and ϕ completelydifferent at the correlated positions). Using Eq. (2) wehave | Γ LE ( t ) | = (cid:104) e iϕ ( t ) (cid:105) = cos(2 t ) . Notice that the sec-ond peak in Figure 3b does not reach the value 1 due tothe undersampling of the noise realizations [6].The second strategy to obtain the LtG transition con-sists in increasing the number of correlated pixels w cp while fixing δ = 0 and the number of repeated pixels inthe function ϕ . In order to increase w cp we increase thewidth of the PDC spectrum by acting on the spectralselector [24]. When the spectral width is ∆ λ = 15 nm,the number of correlated pixels is equal to the numberof repeated pixel in the ϕ function and the two quibitsee the same environment. By progressively increasingthe value of ∆ λ , w cp becomes larger than the numberof repeated pixels in ϕ and the two qubits see differentenvironments. Indeed, on the two photons are imprinteddifferent phases.In Fig. 4, we show the experimental realization of thetransition, using both techniques. In the left column the δ -shift method is used, on the right column the transi-tion is obtained changing the PDC spectral width. Wenote that moving from local noises to a global noise therising of the new peaks are evident. Here γ = 0 . inorder to show a case with a non stationary RTN noise.Fluctuations in the experimental data are mostly due tothe strong dependence on the central pixel. Indeed, thecenter of the PDC beam may be shifted from the cen-ter of the pixel itself for a fraction of the pixel’s length:this has been taken into account in the simulations but itchanges from one measure to another and it not possible Strategy 1 Strategy 2
FIG. 4: Experimental implementation of the transition fromglobal to local noise in the case of RTN with γ = 0 . . In theleft column the δ shift-strategy is used. On the right columnthe transition is obtained by changing the spectral width ∆ λ . to estimate it with a sufficient precision. Conclusions − We have experimentally demonstratedthe transition from local-to-global decoherence in an all-optical two-qubit quantum simulator subject to classicalnoise. We exploited the spatial degrees of freedom of thePDC photons to implement the noise realizations whilethe photons polarizations encoded the two qubits. Inparticular, thanks to the high control of the PDC widthand of the spatial correlations among pixels of the SLM,we have been able to implement two different strategiesfor the noise transition, either involving the building ofcorrelations between environments or the tuning of thePDC spectral width. Besides RTN, that we used as atestbed for our simulator, any kind of classical noise maybe implemented, making our scheme suitable to simu-late a wide range of dynamics involving super- and semi-conducting qubits that are of the utmost importance forquantum technologies.Our results also paves the way to the realization ofmany-qubit simulators, and open up to the chance of ex-plore the dynamics of multi-partite entanglement as wellas to study the robustness of quantum features againstdecoherence. In multi-qubit systems the LtG transitiontakes a broader meaning with sub-groups of qubits thatmay feel the same noise while others are subject localfluctuations. In this case spatial correlations becomes thekey element that governs the dynamics [28]. More gener-ally, the ability to monitor and control the LtG transitionis a fundamental step in understanding decoherence, es-pecially in the context of reservoir engineering, where thenoise is tailored or to improve performances of specificprotocols [29–31]. [1] I. Buluta, F. Nori, Science , 108 (2009)[2] I. M. Georgescu, S. Ashhab, F. Nori, Rev. Mod. Phys. ,153 (2014).[3] D. Jaksch, P. Zoller, Ann. Phys. , 52(2005).[4] I. Kassal, J.D. Whitfield, A. Perdomo-Ortiz, M.-H. Yung,and A. Aspuru-Guzik, Annu. Rev. Phys. Chem. , 185(2011).[5] A. Aspuru-Guzik, P. Walther, Nat. Phys. , 285 (2012).[6] R. Blatt, C. F. Roos, Nat. Phys. bf 8, 277 (2012).[7] A. A. Houck, H. E. Türeci, and J. Koch, Nat. Phys. , 292(2012).[8] D. Crow and R. Joynt, Phys. Rev. A , 042123 (2014).[9] S. Cialdi, M. A. C. Rossi, C. Benedetti, B. Vacchini, D.Tamascelli, S. Olivares, and M. G. A. Paris, Appl. Phys.Lett. , 081107 (2017).[10] Z.-D. Liu, H. Lyyra, Y.-N. Sun, B.-H. Liu, C.-F. Li, G.-C. Guo, S. Maniscalco and J, Piilo, Nat. Comm. , 3453(2018).[11] T. Yu and J. H. Eberly, Phys. Rev. B , 165322 (2003).[12] C. Benedetti, F. Buscemi, P. Bordone, M. G. A. Paris,Int. J. Quantum Inf. 10, 1241005 (2012).[13] C. Benedetti, F. Buscemi, P. Bordone, M. G. A. Paris,Phys. Rev. A. , 052328 (2013).[14] A. C. S. Costa, M. W. Beims, and W. T. Strunz, Phys.Rev. A , 052316 (2016).[15] S. Daniotti, C. Benedetti, and M. G. A. Paris, Eur. Phys.J. D , 208 (2018).[16] D. P. S. McCutcheon, A. Nazir, S. Bose, and A. J. Fisher,Phys. Rev. A , 022337 (2009).[17] A. Streltsov, H. Kampermann, and D. Bruß, Phys. Rev.Lett. , 170502 (2011).[18] R. Lo Franco, B. Bellomo, S. Maniscalco and G. Com-pagno, Int. J. Mod. Phys. B , 1245053 (2013).[19] C. Addis, P. Haikka, S. McEndoo, C. Macchiavello, andS. Maniscalco, Phys. Rev. A , 052109 (2013).[20] M. A. C. Rossi, C. Benedetti and M. G. A. Paris, Int. J.Quantum Inf. , 1560003 (2014).[21] F. Galve, A. Mandarino, M. G. A. Paris, C. Benedettiand R. Zambrini Sci. Rep. , R773 (1999).[23] A.Smirne, S. Cialdi, G. Anelli, M. G. A. Paris, B. Vac-chini , Phys. Rev. A , 012108 (2013).[24] See supplemental material.[25] S. Cialdi, D. Brivio, M. G. A. Paris, Appl. Phys. Lett. , 041108 (2010).[26] S. Cialdi, D. Brivio, M. G. A. Paris, Phys. Rev. A ,042322 (2010).[27] M. A. C. Rossi, C. Benedetti, S. Cialdi, D. Tamascelli,S. Olivares, B. Vacchini, M. G. A. Paris, Int. J. QuantumInf. , 1740009 (2017).[28] M. A. C. Rossi, C. Benedetti, M. Borrelli, S. Maniscalco,M. G. A. Paris, Phys. Rev. A 96, 040301 (2017).[29] J. F. Poyatos, J. I. Cirac, and P. Zoller, Phys. Rev. Lett.77, 4728 (1996). [30] D. de Falco, D. Tamascelli, J. Phys. A: Math. Theo. (22) 225301 (2013).[31] S. McEndoo, P. Haikka, G. de Chiara, M. Palma, S. Man-iscalco, EPL , 60005 (2013).[32] S.Cialdi, D. Brivio, A. Tabacchini, A. M. Kadhim, M. G.A. Paris, Opt. Lett. , 3951 (2012).[33] A. K. Jha, R. W. Boyd, Phys. Rev. A. , 013828 (2010). Supplementary material for "Experimental realization of local-to-global noise transition in a two-qubitoptical simulator" by C. Benedetti et al
Two-photon spatial correlations − Figure S.1 shows the geometrical configuration of the two photons generatedby PDC. The two angles θ and θ are the angular shifts with respect the PDC central angle θ defined by the phase-matching condition. The coordinates x and x are respectively the positions and the references on the SLMplane. The arrows represent the orientations we use for the axes. The two-photon state can be written [1]: | ψ (cid:105) = 1 √ (cid:90) dω dθ dθ ˜ A (∆ k ⊥ ) Sinc (∆ k (cid:107) L/ (cid:104) | H, θ , ω (cid:105) | H, θ , − ω (cid:105) + e i Φ( θ ,θ ) | V, θ , ω (cid:105) | V, θ , − ω (cid:105) (cid:105) , (S.1)where, up to first order in frequency and angle, ∆ k (cid:107) = − ω p θ ( θ + θ ) / (2 c ) and ∆ k ⊥ = ω p ( θ − θ ) / (2 c ) + 2 θ ω/c . ω p is the pump frequency ( nm) and ω is the frequency shift with respect the PDC central frequency ω p / ( nm), c is the speed of light and the phase term Φ( θ , θ ) is due to the different optical paths followed by the pairs ofphotons generated in the first and in the second crystal. FIG. S.1: The geometry of the parametric downconversion scheme
The function ˜ A comes from the integration along the transverse coordinate [2] and it is the Fourier transform ofthe pump spatial amplitude, the Sinc function comes from the integration along the longitudinal coordinate insidethe crystal ( L is the crystal length). ∆ k (cid:107) and ∆ k ⊥ are the shifts with respect the phase-matching condition of thelongitudinal and the transverse momentum of the two photons. Finally, we obtain Φ = 0 by using the purificationmethod explained in [3, 4]. Due to the pump spot dimension ( w pump ≈ . mm) and the crystal length ( mm)employed in our experimental configuration, the angular correlations depend mainly on the function ˜ A and in turnby ∆ k ⊥ . Considering that ∆ k ⊥ is a function of ω , it is simple to get that by enlarging the PDC spectrum width weprogressively lose the angular correlations. Moreover, when the spectrum width goes to , the angular correlationsdepend directly on the width of the pump spot via its Fourier transform ˜ A . In order to obtain both the PDC width w p and the spatial correlations, we define: F ( θ , θ ) = (cid:82) dω | ˜ A (∆ k ⊥ ) Sinc (∆ k (cid:107) L/ | . Before proceeding with the cal-culation, we have to switch from angular to spatial coordinates. Due to the fact that the lens f is placed at distance f both from the crystals and the SLM, we have: ∆ x , = x , − x , = f θ , and we can write F as a function of ∆ x , .To estimate the PDC width w p we define the function: F p (∆ x ) = (cid:82) d ∆ x F (∆ x , ∆ x ) . This function is thesame for the two paths of the PDC and it is well approximated by a Gaussian profile. It gives the probability todetect a photon vs the spatial coordinates, so its width is the PDC width. From a numerical approach we obtain w p ≈ pixels. This number is confirmed by a direct measure of the PDC profile. We note also that this width isdirectly connected with the Sinc function and is only weakly dependent on the collection spatial efficiency. This is aconsequence of the fact that in our experimental scheme the PDC cone is forced to remain in a little area due to thepresence of the lens f .Now we can face the derivation of the correlation length w cp . This quantity is of fundamental importance inour work and it gives the probability to detect a photon within a definite interval when the other photon is foundin a definite position (a pixel in our case). In particular, it is clear that in order to define a proper ϕ function,we need an experimental apparatus able to generate a correlation length of only few pixels and for this reasonwe use the configuration with the lenses f between the crystals and the SLM. The correlation length w cp has twocontributions, one connected directly to the function F and the other one connected with the pump dimension. Thefirst contribution is the width ˜ w pc of the function F cp = F (∆ x , . About this function it is important to say thatthis width doesn’t change if we integrate the position 2 along the dimension of one pixel. ˜ w cp increases with thespectrum width and the profile of F cp is well reproduced by a Gaussian when the spectrum width is smaller than nm and it is well reproduced by a super-Gaussian with n = 4 for bigger spectrum width. This result depends bythe fact that with our spectrum selector we obtain a quasi rectangular profile of the PDC spectrum. The secondcontribution is related with the pump spot dimension. About this we have to consider that the PDC is generatednot only in one point in the transversal direction but along the pump profile [5]. The point is that the spatialcoherence properties of the pump are directly transferred into the PDC. In a naive picture we can say that the singlemode of the pump is transferred into the single mode (defined by the direction θ ) of the PDC. This means that thelens f focuses this single mode on the SLM plane with a dimension w cp = λ fπw pump ≈ pixel where λ = 810 nm.Without focusing, w cp would be equal to the pump dimension, indeed in our case we have a well collimated pump.An alternative scheme would be to use a focused pump but we note that in this case we have to put the lens beforethe crystals obtaining in turn a bigger dimension on the SLM plane. Finally, considering these two contributions, wehave w cp = (cid:113) ( ˜ w cp ) + (cid:0) w cp (cid:1) . So we can write F (∆ x , ∆ x ) = e − x − ∆ x nwn cp e − x w p e − x w p (without consideringa normalization factor). In order to obtain the Equation 5 of the main text, we have only to demonstrate that | f jk | = F (∆ x , ∆ x ) . And we can easily to see that this equality is assured by the fact that phase Φ is not afunction of ω . Measure of the coherence factor − If the system is in the state ρ S ( t ) , and the polarizers are both at ◦ , the detection probability (considering the quantum efficiency QE=1) is: p ++ = 14 (1 + p Re { Γ } ) , while, if one polarizer is at ◦ and the other at ◦ , it results p + − = (1 − p Re { Γ } ) . Then, the coincidence countsfor second in the two cases are: (cid:40) N ++ = N (1 + p (Re { Γ } )) N + − = N (1 − p (Re { Γ } )) , where N is obtained directly from the experimental counts and it takes into account the spatial-spectral quantumefficiencies of the detection system. So we can infer information about Γ from the visibility: V = (cid:12)(cid:12)(cid:12)(cid:12) N ++ − N + − N ++ + N + − (cid:12)(cid:12)(cid:12)(cid:12) = p | Re { Γ }| In the ideal case (without undersampling effect [6]) Γ is a real quantity. This case can be experimentally recoveredtaking the array of noise phases ϕ with zero mean. In the graphs in Figures 3 and 4 (in the main text) we show thecomparison between the theoretical curves of | Re { Γ }| and | Γ | to put in evidence the effectiveness of this method. Measure of the number of correlated pixels − Let’s introduce in the first half of the SLM a rectangu-lar function which switches ± π/ every n r = 5 pixels, and in the second half the same function shifted by h .Therefore Γ is a function of h . By the measurements of N ++ = N ++ ( h ) and N + − = N + − ( h ) for h = − , − , ..., ,where each point is an average of measures and each measure has an acquisition time of s, we calculate V = V ( h ) ,as shown in the left panel of Fig. S.2. The visibility of V ( h ) is a decreasing function of the number of correlatedpixels w cp and it can be simulated as shown in the central panel of Fig. S.2, so that from the experimental valueof Vis ( V ( h )) one can extrapolate w cp . Repeating the experiment for different apertures of the spectral selector, we - - h ( h ) Δ w cp ( V ( h )) FIG. S.2: Left panel: The green dots represent the measured values of V ( h ) for ∆ λ = 15 nm and p = 0 . , while the blueline is the fitting sine wave. In the present case, it results Vis ( V ( h )) = 0 . ± . . Center panel: The blue dots represent thevalue of Vis ( V ( h )) obtained by simulation for ∆ λ = 15 nm and q = 0 . , while the blue dashed line is the fitting polynomialcurve. The red dot is the measured value of visibility. In the present case, the extrapolated number of correlated pixel is w cp = 3 . ± . . Right Panel: Number of correlated pixels as a function of the spectral width of the PDC: the experimentaldata in green, the simulated values in blue. obtain the number of correlated pixels as a function of ∆ λ (see the right panel of Fig. S.2). The experimental dataundergo a saturation for small ∆ λ because of the effect of the transversal width of the pump. 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