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the underlying principles of many quantum information protocols can be traced back to the concept of quantum random walk ( qrw ) , which since its first description have become a fundamental paradigm in quantum science @xcite .
the idea of qrws was originally conceived by aharonov _
@xcite as an extension of the well - known classical random walk ( crw ) @xcite .
the main distinguishing feature of a qrw , compare to a crw , is the possibility of interference between the multiple paths that can be simultaneously traversed by a quantum walker , enabling thus a faster spreading of the uncertainty of location of the walker than in the classical case @xcite .
the temporal evolution of a quantum system , such as a qrw , depends on the presence , and specific characteristics , of the environmental effects ( decoherence ) which can modify it @xcite . in most cases ,
the influence of decoherence during the evolution of a quantum walker transforms an originally pure state into a mixed state , lowering the uncertainty about the location of the walker as it propagates . in the limiting case , when all cross - interference terms between different lattice sites are completely erased ,
the state of pure diffusive classical propagation is reached @xcite .
qrws have been theoretically explored for the case of one - dimensional lattices @xcite , and experimentally implemented by means of different physical platforms , such as photon - based systems @xcite , optical lattices @xcite and waveguide arrays @xcite .
also , qrws have been implemented using trapped ions @xcite and nuclear magnetic resonance systems @xcite .
although the implementation of one - dimensional qrws has showed to be useful when describing several quantum information systems , there is a great interest in expanding the concept to multidimensional lattices . along these lines ,
two - dimensional qrws provide a powerful tool for modeling complex quantum information and energy transport systems @xcite .
notwithstanding , their realization represents a challenge because of the need of a four - level coin operation @xcite .
one way to overcome this drawback is to make use of different degrees of freedom of photons , such as polarization and orbital angular momentum , as it has been shown in @xcite .
another approach is to mimic the two - dimensional qrws evolution by performing two subsequent one - dimensional qrws @xcite .    here , we make use of the latter approach to put forward an experimental setup for the realization of two - dimensional qrws .
we include the environmental effects ( decoherence ) as pure dephasing by means of the introduction of random phase patterns , generated by a spatial light modulator ( slm ) , which can be different from site to site ( _ spatial disorder _ ) . by controlling the degree of decoherence
, we study the transition from the quantum ballistic spreading to the diffusive classical walk .
also , by adding static disorder , we show the possibility of observing anderson localization @xcite .
importantly , our proposal provides a versatile , highly - scalable experimental setup , which may be used as a tool for understanding quantum processes whose underlying physics can be somehow traced to the concept of random walks , such as energy transport in photosynthetic light - harvesting complexes @xcite and material band gap structures @xcite .
the structure of the article is as follows . in sec .
ii , we introduce the qrw model system , including dephasing , considered here
. the proposed experimental setup is described in sec .
numerical results are discussed in sec .
finally , we summarize our results in the conclusion .
a typical discrete quantum random walk comprises two operations : a coin - tossing operation and a shift operation . here
, the coin - tossing operation is performed in the hilbert space @xmath0 spanned by vectors @xmath1 , corresponding to the photon polarization .
the random walk is performed in the hilbert space @xmath2 , corresponding to the position of the photon in the transverse plane , spanned by vectors @xmath3 ( @xmath4 integers ) , which indicate sites ( @xmath4 ) in the transverse plane ( @xmath5 . the global quantum system thus evolve in the hilbert space @xmath6    the state of the system is described by the density matrix @xmath7 , which is transformed to a new density matrix each step @xmath8 via the map @xmath9 @xmath10 denotes the hadamard operator @xmath11 which acts on the polarization degree of freedom .
the operators @xmath12 and @xmath13 , which describe the walker s shift in the transverse dimensions @xmath14 and @xmath15 , independently , read as @xmath16 and @xmath17    the coupling of the quantum walker with the environment is described by pure dephasing @xcite .
the form of the unitary dephasing operator considered here can be written as @xmath18 where @xmath19 is a random phase matrix and @xmath20 is the pauli operator .
inspection of eq .
( [ eq:6 ] ) shows that @xmath19 represents a newly introduced phase difference between the horizontal and vertical polarizations at each site .
concerning this , we will consider three physically relevant scenarios , that can be easily implemented in the set - up proposed here ( see sec .
iii ) . in the general case ,
the phase differences @xmath19 are independent random variables , but with the same probability distribution .
moreover , the ensemble of phase differences @xmath19 can change from step @xmath8 to step @xmath21 . in the following
, we will refer to this case as a qrw influenced by _
dynamical spatial disorder_.        the easiest probability distribution that we can consider is an uniform probability distribution . if phases can be chosen arbitrarily between the extreme values @xmath22 and @xmath23 ,
there is a constant probability @xmath24 to obtain any phase in this interval .
@xmath25 is the maximal phase which we can have between the two orthogonal polarizations .
@xmath26 corresponds to the absence of any spatial disorder . if phases do not change during propagation ,
even though they might differ from site to site , i.e. @xmath27 , then we have _ static spatial disorder_. finally , if all phase differences are the same for all sites , but they can still change from one step to the following , we have _ dynamical dephasing without spatial disorder_.    the probability of detecting a photon in the site ( @xmath4 ) is @xmath28| i , j\rangle , \label{eq:8}\ ] ] where the density matrix that describes the whole system is traced out over the polarization degree of freedom @xmath29 .
the spreading of the uncertainty of photon location is characterized by dependence of the variance on the step index n    @xmath30    where @xmath31 represents the lattice site with indexes ( @xmath4 ) and @xmath32 is the mean position , i.e. , @xmath33 .
the main building block of the qrw setup is the multi - path mach - zehnder - like configuration shown in fig .
1 . it allows to make several runs of the qrw without the necessity of using a large amount of optical components , as it is the case , for instance , of the experiment described in @xcite .
a similar scheme , based on a single - path mach - zehnder - like configuration has been used @xcite . however in these cases , the walker moves in time , whereas in our proposal the walker moves in the two - dimensional transverse plane , offering a way to simplify the experimental implementation of the two - dimensional qrw .
as source of photons one can use a highly attenuated short coherent pulse , prepared by the combination of a photon source and attenuator ( at ) , generating the initial state @xmath34 where @xmath35 is the central site .
the duration of the pulse has to be sufficiently smaller than the time - of - fly through the setup in one cycle .
the transverse size of the gaussian beam profile of the pulse has to be carefully chosen , so that two adjacent sites are not overlapping in the space due to diffraction .
for instance , by making use of a gaussian beam of 2 mm beam waist , corresponding to a rayleigh range of 23.6 m ( for a wavelength @xmath36 = 532 nm ) , along with typical - sized optical components , we could in principle perform a qrw of approximately more than 20 steps .
the number of steps can be further improved by applying smaller beams together with a re - focusing system placed along the walker s path .
alternatively , a spontaneous parametric down - conversion source can be used , provided each down - converted photon is generated in a pure state .
then , the photon is transmitted via the beam splitter ( bs ) to the system .    to get a clearer picture of the working of the quantum random
walk , let us consider in detail the quantum state of the photon in its first passage through the system . first , the polarization state of the photon is changed by the half - wave plate ( hwp1 ) to @xmath37 $ ] , i.e. , the hadamard ( @xmath10 ) operation is applied . after this transformation , the photon is displaced by the horizontal beam displacer ( hbd ) along the @xmath14 axis according to its polarization , as described by the shift operator @xmath12 .
the photon is now in the state @xmath38.$ ] a second half - wave plate ( hwp2 ) implements a new hadamard transformation , which transforms the quantum state to @xmath39 .
the vertical beam displacer ( vbd ) shifts the position of the photon along the @xmath15 axis . after this
, the quantum state of the photon reads    @xmath40 . \label{eq:11}\end{aligned}\ ] ]    the spatial light modulator ( slm ) is used to introduce random phases , given by the phase - matrix @xmath19 . after the photon passes through the slm , the whole cycle is repeated with high probability .
if not , the photon escapes through the other output port of the beam - splitter , which directs it towards a highly sensitive ( single - photon sensitivity ) intensified ccd camera with integrated photo - multipliers , which allows to spatially resolve a weak signal with a high efficiency ( @xmath4190% ) @xcite .
moreover , losses in the setup can be compensated by an increase of the amplitude of pulse .
the size of whole array of beams can be reduced by an auxiliary imaging system in order to fit on the limited size of sensitive area of ccd .    [ cols="^,^ " , ]     [ fig:5 ]    the photon position is less dramatic than in the case with dynamical spatial disorder . for the dynamical spatial disorder , @xmath42 for @xmath25 after 20 steps . on the contrary , for dynamical dephasing without spatial disorder , we have @xmath43 under the same conditions .
indeed , the @xmath8-dependence of the typical deviation , characteristic of the classical regime , it is not yet reached after @xmath44 steps , as is readily observed in fig .
4(b ) .      in the context of our discussion
, the anderson localization is the reduction of spreading of the uncertainty of the photon position @xcite . we will demonstrate that this effect can also be observed in the set - up considered here . in @xcite , anderson localization was observed in the transverse plane of a light beam passing through a crystal with random static fluctuations of the index of refraction . since the randomness in the index of refraction is affecting only the phase of the propagating beam , it is possible to imitate these phase - fluctuations with a slm , under the consideration of static spatial disorder , since anderson localization does not appear with dynamical spatial disorder .    in fig .
5(a ) we present a typical profile of the phase - matrix @xmath19 , independent of n , which leads to beam localization .
5(b ) shows the corresponding probability distribution of the photon position for this specific phase profile .
notice that it contains a strong peak located in the middle of the lattice .
the presence of anderson localization is confirmed in fig .
5(c ) , where we show the averaged probability distribution function for @xmath25 , exhibiting an exponential suppression of probabilities for sites distant from the center . for the sake of clarity , we also plotted in fig .
5(d ) two cuts of the averaged probability distribution along the @xmath45 and @xmath46 axes , to highlight this feature .
we have put forward a new , highly scalable and easily implemented experimental configuration to observe spatial two - dimensional random walks under a great variety of circumstances , by means of the implementation of two consecutive one - dimensional random walks .
the proposal makes use of only a small amount of simple optics components , and allows us to simulate many different quantum systems and protocols based on the quantum random walk concept . additionally , by carefully controlling the amount and type of disorder present in the system
, we have shown the effects of different environmental effects : dynamical spatial disorder , dynamical dephasing without spatial disorder and static spatial disorder .
the last case drove us to the observation of anderson localization .
the control of environmental effects is paramount importance in nearly all quantum systems . in some cases , it is even crucial to understand the dynamics experimentally observe .
for instance , in light harvesting complexes @xcite , the interplay between coherent evolution and noise is a critical ingredient , which allows to obtain a high efficiency energy transport .
the authors thank to dr .
jan soubusta for useful discussions .
this work was supported by project nos .
fis2010 - 14831 and fet - open 255914 ( phorbitech ) .
thanks the project fi - dgr 2011 of the catalan government .
this work has also supported in part and by projects cost oc 09026 , cz.1.05/2.1.00/03.0058 of the ministry of education , youth and sports of the czech republic and by project prf-2012 - 003 of palack university .
a.  peruzzo , m.  lobino , j.  c.  f.  matthews , n.  matsuda , a.  politi , k.  poulios , x.  zhou , y.  lahini , n.  ismail , k.  wrhoff , y.  bromberg , y.  silberberg , m.  g.  thompson , and j.  l.  obrien , science * 329 * , 1500 ( 2010 ) .
we put forward a new , versatile and highly - scalable experimental setup for the realization of discrete two - dimensional quantum random walks with a single - qubit coin and tunable degree of decoherence . 
 the proposed scheme makes use of a small number of simple optical components arranged in a multi - path mach - zehnder - like configuration , where a weak coherent state is injected . 
 environmental effects ( decoherence ) are generated by a spatial light modulator , which introduces pure dephasing in the transverse spatial plane , perpendicular to the direction of propagation of the light beam . by controlling the characteristics of this dephasing 
 , one can explore a great variety of scenarios of quantum random walks : pure quantum evolution ( ballistic spread ) , fast fluctuating environment leading to a diffusive classical random walk , and static disorder resulting in the observation of anderson localization .