Two-particle bosonic-fermionic quantum walk via 3D integrated photonics

Linda Sansoni,1 Fabio Sciarrino,1, 2 Giuseppe Vallone,1, 3 Paolo Mataloni,1, 2

Andrea Crespi,4, 5 Roberta Ramponi,4, 5 and Roberto Osellame4, 5

1Dipartimento di Fisica, Sapienza Universita di Roma, Piazzale Aldo Moro, 5, I-00185 Roma, Italy2Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche (INO-CNR), Largo Enrico Fermi, 6, I-50125 Firenze, Italy

3Department of Information Engineering, University of Padova, I-35131 Padova, Italy4Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche

(IFN-CNR), Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy5Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy

Quantum walk represents one of the mostpromising resources for the simulation of phys-ical quantum systems [1], and has also emergedas an alternative to the standard circuit model forquantum computing [2]. Up to now the experi-mental implementations have been restricted tosingle particle quantum walk [3, 4], while very re-cently the quantum walks of two identical photonshave been reported [5]. Here, for the first time,we investigate how the particle statistics, eitherbosonic or fermionic, influences a two-particle dis-crete quantum walk. Such experiment has beenrealized by adopting two-photon entangled statesand integrated photonic circuits [6]. The polariza-tion entanglement was exploited to simulate thebunching-antibunching feature of non interactingbosons and fermions. To this scope a novel three-dimensional geometry for the waveguide circuit isintroduced, which allows accurate polarization in-dependent behaviour, maintaining a remarkablecontrol on both phase and balancement.

In the framework of quantum information processing,quantum walk has attracted much attention in the lastfew years [2]. Quantum walk is an extension of the clas-sical random walk: a walker on a lattice “jumping” be-tween different sites with a given probability. The fea-tures of the quantum walker are interference and super-position which lead to a non-classical dynamic evolu-tion. Two different cases may be considered: discrete-and continuous-time quantum walks [1]. In discrete-timequantum walk one or more quantum particles evolve ona graph, with their evolution governed by their internalquantum coin states. On the other hand, in a continuous-time quantum walk, there are no coin operations and theevolution is defined entirely in the position space. Theproperties of these two walks have shown several simi-larities and, in specific conditions, it is possible to mapthe discrete-time quantum walk to the continuous-timequantum walk [7]. However, the discrete quantum walkexhibits a higher flexibility due to the possibility of tailor-ing the quantum coin properties to investigate differentdynamic scenarios [3, 4]. By endowing the walker withquantum properties, many new interesting effects appear:quantum walks allow the speed-up of search algorithms[8] and the realization of universal quantum computation[2]. Moreover it has been recently shown that quantum

j=-4

T=1 T=2 T=3 T=4

j=0

j=+1

j=+2

j=+3

j=+4

j=-1

j=-2

j=-3

Fermions

Bosons

or

T=0J=8

J=7J=6

J=5J=4

J=3J=2

J=1BS

(a) (b) (c)

FIG. 1: (a) Unidimensional quantum walk: depending on theresult of the coin toss the walker moves upward (U) or down-ward (D). (b) Scheme of an array of beam splitters (BSs) fora four-steps quantum walk. Vertical dashed lines indicate thesteps T of the quantum walk and the horizontal strips repre-sent the position |j〉 of the walker. In an array with an even(odd) number of steps the output ports J are grouped intothe even (odd) final positions |j〉 of the walker. (C) differentbehaviours of bosons and fermions on a BS.

walks with a large number of sites exhibit a highly non-trivial dynamics, including localization and recurrence[9]. Within this scenario, a possible application is the in-vestigation on biophysical systems, like the energy trans-fer process within photosynthesis [10].

Single-particle quantum walks yield an exponentialcomputational gain with respect to random walks of clas-sical particles; it can be noted that they have an exactmapping to classical wave phenomena and therefore theycan be implemented using purely classical resources. Onthe other hand, quantum walks of more than one indis-tinguishable particle can provide an additional computa-tional power that scales exponentially with the resourcesemployed. This could be used to improve simulation per-formances in complex tasks, e.g. the graph isomorphismproblem [11]. However, they need “quantum” resourcesto be implemented, since classical theory no longer pro-vides a sufficient description.

Different experimental implementations of single par-ticle quantum walks were performed with trapped atoms[12], ions [13, 14], energy levels in NMR schemes [15],photons in waveguide structures [16], and in a fiber loopconfiguration [3, 4]. Very recently Peruzzo et al. [5]demonstrated quantum walks of two identical photonsin an array of 21 evanescently-coupled waveguides in

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a SiOxNy chip. However, up to now no experimen-tal demonstration on how the particle statistics, eitherbosonic or fermionic, influences a two-particle quantumwalk has been reported.

In this work, we report on the first implementationof a quantum walk for entangled particles. By changingthe symmetry of entanglement we can simulate the quan-tum dynamics of the walks of two particles with bosonicor fermionic statistic. These results are made possibleby the adoption of novel geometries in integrated opti-cal circuits fabricated by femtosecond laser pulses, whichpreserved the indistinguishability of the two polarizationsas well as provided high phase accuracy and stability.

Optical implementation of quantum walks.Let us first briefly review some basic concepts on discretequantum walk. A particle doing a classical, symmetricrandom walk on the line may be described as follows.The particle starts, say, at the origin, and at every timestep, tosses a fair coin. Each of the two possible outcomesof the toss is associated with a distinct direction, “up”or “down”. The particle moves one step in the directionresulting from the toss (Fig. 1a). In the quantum gen-eralization the classical walker is replaced by a quantumparticle -such as an electron, atom or photon- with anadditional degree of freedom spanning a two-dimensionalspace and named the “quantum coin” (QC). At any giventime the particle may be in a superposition of the twobasis states, referred as up (|U〉) or down (|D〉) and rep-resenting the two “coin faces”. The QC state directs themotion of the particle and the stochastic evolution by anunitary process. A key difference with the classical case isthat the many possible paths of the quantum walker mayexhibit interference, leading to a very different probabil-ity distribution of finding the walker at a given location.The dynamics of the discrete quantum walk is given bythe following rules: at each step the state of the QC ischanged by a “coin tossing” represented by a 2×2 unitaryoperation C. The particle shifts up or down dependingon the internal QC state |U〉 or |D〉, respectively. Theevolution of the walk is then described by the followingstep operator:

E =∑j

|j − 1〉〈j| ⊗ |U〉〈U |+ |j + 1〉〈j| ⊗ |D〉〈D|, (1)

where |j − 1〉〈j| and |j + 1〉〈j| stand for the operatorswhich move the particle in the higher and lower position,respectively. The coherent action of the step operatorE and coin tossing C leads to entanglement between theposition and the internal degree of freedom. After severalsteps, the counterintuitive profile of the quantum walkprobability distribution emerges as a result of quantuminterference among multiple paths.

More complex probability distribution arises when twoparticles are injected into the same quantum walk. Animportant perspective is to combine in the same plat-form quantum walk and entangled states [17, 18]. In thiscase both the initial and the final state of the walkersare entangled states and, depending on the entanglement

symmetry, different final distributions can be observed.The possibility of changing the symmetry of the entan-gled state allows to simulate the quantum walk of twoparticles with integer or semi-integer spin (see Methods).The fermionic and bosonic behaviors, with the two parti-cles respectively avoiding each other or tending to bunchtogether, as imposed by the symmetrization postulate ofquantum mechanics, drastically influence the dynamicsof their quantum walk on the line. Such kind of cor-related walks involving quantum interference of severalwalkers may provide a speed-up of quantum algorithmsdelivering an exponential acceleration with the numberof correlated walkers.

The simulation of single particle quantum walks on aline can be implemented using single photon states, beamsplitters, phase shifters, and photodetectors [19, 20]. Thequantum dynamics is achieved by an array of balancedbeamsplitters (BS s) as shown in Fig. 1b, each verticalline of beam splitters representing a step of the quan-tum walk. Horizontal strips represent the position |j〉of the walker. If a photon, at time T and in the stripj, is incident downward |D〉 (upward |U〉) on the BS wecan represent its state as |j,D〉T (|j, U〉T ). The tran-sition from time T to time T + 1 is given by the BSoperator |j,D〉T → 1√

2(|j − 1, D〉T+1 − |j + 1, U〉T+1),

|j, U〉T → 1√2(|j + 1, U〉T+1 + |j − 1, D〉T+1). This oper-

ation simultaneously implements the coin (precisely theHadamard coin C = 1√

2

(1 11 −1

)) and step operator E.

Note that, if the particle starts at position |j = 0〉, ateven (odd) times it will occupy only even (odd) positions.

Provided that all the optical devices used in the walkare polarization insensitive, the polarization degree offreedom may be exploited to entangle the photons in-jected into the BS arrays. Moreover by changing the en-tangled state from a symmetric one, such as the triplet,into an antisymmetric one, the singlet, it is possibleto mimic the quantum dynamics of two non-interactingbosonic and fermionic particles (see Fig. 1c). It must benoticed that the experimental realization of such a net-work of BS s is exceedingly difficult with bulk optics, evenfor a small number of steps. The implementation of Tsteps requires 1

2T (T+1) BS s. Aligning this quadraticallygrowing number of elements is a challenging task. Fur-thermore, for a correct operation of the quantum walk,the phase introduced by the optical paths, passing fromeach beam splitter to the following, must be controlledand stable.3D integrated quantum circuits.Our approach exploits an integrated waveguide architec-ture, which allows to concentrate a large number of opti-cal elements on a small chip and to achieve intrinsic phasestability due to the monolithic structure. In a waveguideimplementation BS s are replaced by directional couplers(DC s) i.e. structures in which two waveguides, broughtclose together for a certain interaction length, couple byevanescent field.

To realize the integrated optical circuits we adopted

3

-175 μm -170 μm -165 μm

H

V

7.2 μm

8.4

μm

7.7 μm

8.3

μm

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80

C V/ C

H

Angle θ [ ]

CV/C

H

Angle θ [°]

a)

b) c)

d)

yx

xzy

y

xθWG1

WG2

e)e)

FIG. 2: Integrated optical circuits. a) Measured intensity profile for the guided modes with polarization V and H, at806nm wavelength. The 1/e2 dimensions are reported. b) Schematic cross-section of the interaction region of a directionalcoupler, where the two waveguides (WG1 and WG2) lies on a plane tilted by an angle θ. c) Ratio of the estimated couplingcoefficient for polarization V (CV ) and polarization H (CH), for directional couplers fabricated with different angles θ but fixedinteraction length 3mm and distance 11µm between the waveguides. The fitting line is a guide to the eye. d) Schematic of thenetwork of directional couplers fabricated for implementing a 4 steps quantum walk. The color coding indicates the writingdepth of the waveguides, which is varying from point to point. e) 3D representation of the basic cell of the network, which actsas a Mach-Zehnder interferometer

the femtosecond laser writing technology [21, 22]. Briefly,nonlinear absorption of focused femtosecond pulses is ex-ploited to induce permanent and localized refractive in-dex increase in transparent materials. Waveguides aredirectly fabricated in the bulk of the substrate by trans-lation of the sample at constant velocity with respectto the laser beam, along the desired path. Since it isa single-step and maskless process, this technique allowsrapid and cost-effective prototyping of new devices. Fur-thermore, it has intrinsic three-dimensional possibilitieswhich have indeed been exploited in this work.

In a previous work we have demonstrated that fem-tosecond laser technology can produce high-quality

waveguides able to support polarization entangled pho-ton states [6]. In particular, an integrated DC wasshown, which allowed high fidelity filtration of the singletBell state. In these femtosecond laser written waveg-uides, birefringence is low and does not affect signifi-cantly the coherence of the photons. Anyway the guidedmodes for the two polarizations are still slightly different(as shown in Fig. 2a) and this results in a residual po-larization dependence in the properties of the fabricatedDCs. In fact, the coupling coefficient depends on theoverlap integral between the two guided mode profilesand is indeed quite sensitive to even small differences inthe mode dimensions [23]. Let us consider a DC built

4

LC

WPBBO

SMF MMF

coinc.

BS

LC controller

LaserPC

FIG. 3: Experimental setup. Sketch of the experimental setup, it can be divided into three parts. (i) The source:polarization entangled photon pairs at wavelength λ = 806nm were generated via spontaneous parametric down conversion in a1.5mm β-barium borate crystal (BBO) cut for type-II non-collinear phase matching, pumped by a CW laser diode with powerP = 50mW [24]. Separable polarization states can be prepared by inserting polarizing beam splitters (PBSs) and waveplates(WPs). A quarter waveplate (QWP) and a half waveplate (HWP) inserted on the two photon paths allow generation of anykind of entangled states. A delay line (DL) is inserted to control the temporal superposition of the photons, which are thencoupled to single mode fibers (SMFs) and injected into the integrated device. Interference filters (IF) determine the photonbandwidth ∆λ = 6nm. A liquid crystal device (LC) is inserted in one of the two paths in order to set the phase φ. (ii) Integratedquantum walk circuit realized by ultrafast laser writing technique. (iii) The measuring apparatus: the chip output is divided bya beam splitter (BS) and magnified through a set of two lenses: the first one, with focal f1 = 50 mm, collimates the emergingbeam, the second one, f2 = 150 mm, focuses the light onto a multimode fiber (MMF). The photons coupled to multimodefibers are then delivered to single photon counting modules (SPCMs). The MMFs are mounted on motorized translation stagesin order to select an arbitrary combination of two output ports and measure two-photon coincidences. Polarization controllers(PC) are used before the chip to compensate polarization rotations induced by the fibers.

with two waveguides brought close along the horizon-tal (x ) direction, which is the most common geometryadopted for these devices. The vertically (V) polarizedmodes have a smaller width, along the x axis, with re-spect to the horizontally (H) polarized modes. This im-plies a lower evanescent field coupling of the two waveg-uides, and thus a different splitting ratio, for the V polar-ization with respect to the H polarization. When severaldevices are cascaded, as in the case of a DC array forimplementing a quantum walk, small differences in thesplitting ratios would accumulate and in the end affectthe indistinguishability of the two polarizations, whichis necessary to implement reliable quantum simulationswith polarization entangled photons.

Femtosecond laser waveguide writing allows one to pro-duce couplers with different three-dimensional geome-tries, thus tailoring the polarization behaviour of the de-vice. For example, if the two waveguides are broughtclose along the vertical (y) direction, the polarizationdiversity would be opposite to the previous case. Infact, along the y axis the V polarized mode is slightlygreater than the H polarized one. More in general, ifthe two waveguides lies along an arbitrary direction inthe xy plane (see Fig. 2b) an intermediate situation canbe observed. We have thus fabricated several DC s withdifferent angles θ for the interaction region, but fixedinteraction length and spacing between the waveguides.We then estimated the coupling coefficients for H or V

polarized inputs. As shown in Fig. 2c, there exists anangle for which the ratio between the two coefficients isone, i.e. the coupler becomes polarization insensitive,thus enabling the realization of a polarization indepen-dent quantum walk. In order to realize an experimentalimplementation of a discrete quantum walk with photons,we fabricated a network of DC s, with the configurationshown in Fig. 2d. All the DC s are realized with thetilted geometry described above, in which in the inter-action region the two waveguides are brought at 11µmdistance, at an angle of 62◦, guaranteeing the polariza-tion independence. The length of the interaction regionis chosen as L = 2.1mm in order to obtain a balancedsplitting ratio.

In the interaction region the two waveguides are atdifferent depths in the glass. To connect one coupler tothe following, we designed a structure where the waveg-uides continuously vary the depth, as shown with thecolor codes in Fig. 2d. In fact, the basic cell of thenetwork, depicted in Fig. 2e, acts as a Mach-Zehnderinterferometer. For the correct operation of the quan-tum walk all the interferometers present in the networkmust be phase balanced. This is intrinsically achievedwith the highly symmetric three-dimensional geometryimplemented in the network (Fig. 2d and 2e). To couplethe device with a single-mode fiber array, the two centralwaveguides of the structure start with an initial separa-tion of 250µm, while at the output the waveguides are

5

-4 -3 -2 -1 0 1 2 3 40,00

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-4 -3 -2 -1 0 1 2 3 40,00

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HV+-Theo

(b)

Out

put p

roba

bilit

yO

utpu

t pro

babi

lity

(a)

QW output j

FIG. 4: Measured output probability distributions and the-oretical expectations for a 4 steps quantum walk of a singleparticle injected into mode kA (a) and mode kB (b). Theexperimental results demonstrate that the DCs network leadsto the same quantum dynamics independently from the initialpolarization state and in agreement with the theory.

separated by 70µm. The whole chip is 32mm long.Experimental quantum walks with entangled

states.To carry out and characterize the different quantumwalks we adopted the experimental apparatus reportedin Fig. 3. Different single photon and two photon stateswere injected in the network of DCs via the single modefibers A and B. By adopting a voltage controlled liquidcrystal device inserted on the mode kB we set the phasebetween the horizontal and vertical polarizations. Theoutput of the integrated device is collected by a suitabletelescope, splitted through a bulk beam splitter and thencoupled to two multimode fibers (MMFs). By indepen-dently translating the MMFs on the arms C and D weselect the output ports to detect, respectively I and J ,and measure the single photon signals SC(I) and SD(J)and two-photon coincidences CCD(I, J).Single-photon measurements: as a first measurement wecharacterized the quantum walk circuit with single pho-tons injected in either mode kA or kB . By measuring theoutput signals SC(I) we obtained the distributions givenin Fig. 4. In order to demonstrate the polarization insen-sitivity of our device, we repeated this measurement byinjecting light in different polarization states -horizontal,vertical, diagonal and antidiagonal- always observing thesame distribution, in agreement with the theoretical ex-pectation.

Note that the plotted distributions refer to the positionj of the walker (see Fig. 1b) and not to the output sitesJ of the BS array. Indeed, referring to Fig. 1b, for awalk with T ∗ step, the relation between the probabilitiesof photons emerging from one of the N = 2T ∗ outputs of

the BS array (PBSJ ) and the final position of the walker(pwalkj ) is:

pwalk−T∗ = PBS1 ,

pwalk−T∗+2k = PBS2k + PBS2k+1, k = 1, ..., T ∗ − 1

pwalkT∗ = PBS2T∗ ,

(2)

Two-photon entangled states distributions: as a secondstep we injected the two-photon state 1√

2(|H〉A|V 〉B +

eiφ|V 〉A|H〉B) where the transition from the singlet to thetriplet state of the Bell basis was performed by changingφ through rotations of HWP and QWP (see Fig. 3). Thedistribution of the triplet and singlet states |Ψ+〉 (φ = 0)and |Ψ−〉 (φ = π) emerging from the quantum walkwas reconstructed by measuring the coincidence countsCCD(I, J) for each combination of the indexes I and J .The measured bosonic and fermionic distributions com-pared with expected ones are reported in Fig. 5a-b for|Ψ+〉 and |Ψ−〉. As done for the measurements on thesigle photon quantum walk, we plotted the probabilitydistributions for the walkers to be in the final positionsi, j of the quantum walk which is related to the proba-bility of photons to emerge from the output ports I andJ of the BS array by a relation that can be easily foundas an extension of equation (2).

The presence of diagonal elements in the obtained dis-tribution for bosons (Fig. 5a) highlights the bunch-ing behaviour of a symmetric two-particle system, onthe other hand the fermionic distribution presents non-zero off-diagonal elements (Fig. 5b), confirming that ananti-symmetric two-particle system undergoes the anti-bunching phenomenon. Furthermore, by considering ageneric phase φ (different from 0 and π), it is possi-ble to simulate the behaviour of the quantum walk oftwo anyons, particles with a non semi-integer spin thatrepresent a generalization of fermions and bosons [25].Precisely, the entangled state |Ψφ〉 = 1√

2(|H〉A|V 〉B +

eiφ|V 〉A|H〉B) simulates two anyons characterized by cre-

ation operators satisfying cicj = eiφcjci and cic†j =

eiφc†jci + δij . These systems stress both bunching and

anti-bunching behaviours (i.e. diagonal and off-diagonalelements in the final distribution). As a further measure-ment, we therefore prepared some anyonic states |Ψφ〉, inparticular with φ = π

4 ,π2 ,

34π, and measured the output

probabilities. In Fig. 5c the distribution for φ = π2 is

reported as an example of an anyonic behaviour.The experimental data can be compared with

the theoretical distributions by the similarity S =

(∑i,j

√DijD′i,j)

2/∑i,j Dij

∑i,j D

′ij , which is a gener-

alization of the classical fidelity between two distribu-tions D and D′. We obtained Sbos = 0.982 ± 0.002 andSfer = 0.973±0.002 for the bosonic and fermionic quan-

tum walk and Sπ/4any = 0.987±0.002, S

π/2any = 0.988±0.001

and S3π/4any = 0.980±0.002 for the anyonic quantum walks

with φ = π4 ,

π2 ,

34π, respectively. The obtained results are

in good agreement with the expected behaviours.

6

BOSONS FERMIONS ANYONS(a) (b) (c)Theory

Experim

ent

FIG. 5: Two-particle quantum walks: ideal (top) and measured (bottom) distributions of (a) bosonic, (b) fermionic and (c)anyonic (with φ = π/2) quantum walks.

Outlook.In conclusion, we presented the behavior of a discretequantum walk based on an integrated array of symmet-ric, polarization insensitive, directional couplers in whichtwo-photon polarization entangled states are injected.Exploiting the different statistics of singlet and triplet en-tangled states, such scheme allowed us to simulate howsymmetric and antisymmetric particles travels throughthe quantum walk. The insensitivity to photon polar-ization, high-accuracy in the phase control and intrinsicscalability of the integrated multi-DC network presentedin this work, pave the way to further advanced investiga-tions on complexity physics phenomena. For instance, byintroducing suitable static and dynamic disorder in thewalk it would be possible to simulate the interruption ofdiffusion in a periodic lattice. In these conditions, theAnderson localization could be observed [26–29]. Quan-tum random walk has also been proposed to simulatethe coherent propagation of electronic excitons througha network of chromophores [10]. By using an integratedDC network and varying the parameters of the system,transport efficiency can be analyzed. Then the interplaybetween quantum coherence and presence of dephasingnoise can be experimentally investigated, with particularattention to entanglement and the noise-assisted trans-port effect [30]. This approach could allow in the future

to analyze these phenomena in complex situations thatcan accordingly simulate “as close as possible” the com-plexity of real biological systems.

During the final revision of this manuscript, the simula-tion of quantum statistics with entangled photons withina continuous quantum walk has been reported online [31].

Aknowledgements

This work was supported by EU-Project CHISTERA-QUASAR and FIRB-Futuro in Ricerca HYTEQ.

Correspondence

Correspondence and requests for materialsshould be addressed to Fabio Sciarrino (email:fabio.sciarrino@uniroma1.it) and Roberto Osellame(roberto.osellame@ifn.cnr.it).

Methods

Chip fabrication and characterization - Waveg-uides were micromachined in borosilicate glass substrate

7

(Corning EAGLE2000) using a cavity-dumped Yb:KYWmode-locked oscillator, which delivers 300fs pulses at1MHz repetition rate. For the waveguide fabricationpulses with 220nJ energy were focused 170µm underthe glass surface, using a 0.6N.A. microscope objectivewhile the sample was translated at a constant speed of40mm/s by high precision, three-axis air-bearing stages(Aerotech FiberGlide 3D). Measured propagation lossesare 0.5dB/cm and coupling losses to single mode fiberarrays at the input facet are about 1dB. Birefringenceof these waveguides is on the order of B = 7 × 10−5, ascharacterized in [6].

Equivalence between entangled and boson-fermion quantum walk . Let us consider a genericT step single-photon quantum walk implemented as inFig. 1b. If the particle is injected at mode |J〉, with in-

put state a†J |0〉, the walk performs a unitary transforma-

tion on the creation operator, namely a†J →∑K UJKa

†K .

Let’s now consider two photons injected into the walk in a

polarization entangled state |Ψ±IJ〉 ≡1√2(a†Ib

†J ± a

†Jb†I)|0〉

where a† and b† are the horizontal and vertical polar-ization creation operators, respectively. The evolution ofthe walk becomes

|Ψ±IJ〉 →1√2

∑K,L

(UIKUJL ± UJKUIL︸ ︷︷ ︸ψ±

IJ,KL

)a†Kb†L|0〉 (3)

The probability of detecting one photon at K and theother at L (without measuring the polarization) is givenby

p±(I, J ;K,L) =

|ψ±IJ,KL|

2 for L 6= K

1

2|ψ±IJ,KL|

2 for L = K(4)

It is easy to show that p+(I, J ;K,L) and p−(I, J ;K,L)respectively correspond to the probabilities of detectingat positions K and L two bosons or fermions injected intothe modes I ad J of the quantum walk. In fact, if at theinput state we have two identical bosons characterized by

commuting creation operator d†I and d†J , the output state

is d†Id†J |0〉 →

12

∑K,L ψ

+ij,KLd

†Kd†L|0〉. The probability of

detecting one boson at K and the other at L is preciselyp+(I, J ;K,L). If at the input state we have two iden-tical fermions characterized by anticommuting creation

operator c†I we have c†Ic†J |0〉 →

12

∑K,L ψ

−IJ,KLc

†Kc†L|0〉

and the probability of detecting one fermion at K andthe other at L is p−(I, J ;K,L). Note that the dynamicof two-particle quantum walk cannot be reconstructedby multiplying the output probabilities (|UJK |2) of twosingle-particle walks.

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