Two-walker discrete-time quantum walks on the line with percolation
TTwo-walker discrete-time quantum walks on the line with percolation
L. Rigovacca and C. Di Franco
Quantum Optics and Laser Science Group, Imperial College London, Blackett Laboratory, SW7 2AZ London, UK
One goal in the quantum-walk research is the exploitation of the intrinsic quantumnature of multiple walkers, in order to achieve the full computational power of themodel. Here we study the behaviour of two non-interacting particles performing aquantum walk on the line when the possibility of lattice imperfections, in the formof missing links, is considered. We investigate two regimes, statical and dynamicalpercolation, that correspond to different time scales for the imperfections evolutionwith respect to the quantum-walk one. By studying the qualitative behaviour of threetwo-particle quantities for different probabilities of having missing bonds, we argue thatthe chosen symmetry under particle-exchange of the input state strongly affects theoutput of the walk, even in noisy and highly non-ideal regimes. We provide evidenceagainst the possibility of gathering information about the walkers indistinguishabilityfrom the observation of bunching phenomena in the output distribution, in all thosesituations that require a comparison between averaged quantities. Although the spreadof the walk is not substantially changed by the addition of a second particle, we showthat the presence of multiple walkers can be beneficial for a procedure to estimate theprobability of having a broken link. I n the field of quantum walks many results have been recently obtained, from both a theoretical [1–5] and anexperimental [6–14] points of view. Being the quantum analogue of the classical random walk, the quantum walk[15] is a simple yet interesting model that can be exploited in several ways. It has been proved that quantum walkscan be used to implement quantum search algorithms [16], and they can be considered as a universal computationalprimitive [17, 18]. They have applications in the simulation of biological processes [19], and an equivalence withquantum lattice gas models [20] has been proposed. It is worth to keep in mind that the scenario with only asingle walker is however classically simulable, for example using coherent light instead of a single-photon state in anoptical implementation [21, 22]. In order to have a true quantum behaviour, that could possibly lead to computationalprotocols achieving an exponential speedup over their classical counterparts, the complexity introduced by the presenceof more than one walker has to be considered (see for instance the short discussion in [23]). From this perspective,the study of the emerging features of multi-particle quantum walks, together with an analysis of their resilience tonon-perfect conditions (e.g. noisy or disordered environments), is of paramount importance. For example, it is wellknown that a quantum particle in a static disordered media cannot evolve arbitrarily far from its starting point, aphenomenon known as “Anderson localisation” [24]. Such behaviour has been observed also for particles performing aquantum walk, and their localisation length does not considerably change even in the presence of interactions [25–27].In this manuscript we focus on discrete-time quantum walks of non-interacting particles. This model is based on therepeated application of a single-step evolution operator, applied independently on both walkers. Such multi-particlescenario has been studied when a fixed unitary evolution is considered [21, 28–33]. Since this could not always be thecase, it is interesting to study what happens with less stringent requirements. In the following we will consider thepossibility of having missing links in the underlying structure (percolation graphs), modelling for example physicalsituations where one does not have perfect control over the lattice in which the walk takes place. Up to now, thisscenario has been investigated by several authors only in the single-particle case, approaching the problem fromdifferent perspectives: e.g. studying decoherence [34] or disordered systems [35], modelling transport phenomena [36],or describing asymptotic behaviours [37].The purpose of this paper is to understand how the possible absence of links (due, for instance, to a non-perfect ex-perimental setup) could influence the evolution of multiple walkers, a fundamental step toward a complete exploitationof the quantum-walk model. In order to do so, we study the walk of two non-interacting particles, using as underlyinglattice a simple line with bonds randomly missing with time. More precisely, on the one hand we want to see to whatextent signatures of the presence of multiple walkers survive when percolation is considered, while on the other handwe are searching for possible advantages arising when dealing with two particles instead of one. To achieve the firstgoal, we qualitatively describe how different quantities depend upon the probability of having missing bonds and onthe frequency with which the lattice changes. In particular, we will consider the final average distance between thewalkers and the probability of finding both particles at the same location or exactly in their initial position, showinghow their values strongly depend on the considered initial state (potentially entangled) even in noisy regimes withmany broken links. Concerning the advantages of having a second walker we show that, although the spread of thewalk is not considerably altered by the addition of a particle, there is a procedure to estimate the probability ofhaving a missing link whose performance is often improved by its presence. Furthermore, from our study it emerges a r X i v : . [ qu a n t - ph ] F e b that a signature of enhanced bunching between the walkers in the output probability distribution does not necessarilyimply the presence of entangled states symmetric under particle-exchange. Since the symmetry condition is requiredby quantum indistinguishability, this observation is a warning against the possibility of gathering information aboutindistinguishability from the mere presence of bunching effects, at least in all those situations that deal with a randomcomponent in the evolution by considering averaged quantities. This is an important feature that has to be kept inmind also when studying more general many-walker scenarios in future research.Let us point out that similar studies in more complex geometries, as higher-dimensional lattices [38] or graphs[39], could be of interest. However, being more computationally demanding, this is left as a possibility for futureinvestigations. In the remainder of this introduction we will review some basic facts about quantum walks andimperfect lattices. Single-particle discrete-time quantum walk without percolation.
Let us briefly review the concept of discrete-time quantum walk on the line. This can be thought as the quantum counterpart of a classical random walk, whereat each step the walker can move left or right with a certain fixed probability. In this scenario, each step can beconsidered as composed of two parts: a (possibly biased) coin toss and a shift in the direction associated with therandom output. Initially introduced in [15], for a more detailed discussion on the quantum-walk model we refer to[1, 2].Quantum mechanically the position of the walker is described by orthonormal quantum states {| i (cid:105)} i , that span theHilbert space H P , while the coin can be taken into account by adding a two-dimensional Hilbert space H C , so thatthe global system can be described in H P ⊗ H C . The coin toss can be realised by means of a fixed unitary operationapplied to the C component of the system. Throughout this manuscript, we will use for this purpose the Hadamardmatrix H C = 1 √ (cid:18) − (cid:19) , (1)expressed in the standard basis of the coin {|↑(cid:105) , |↓(cid:105)} , whose states correspond to the classical idea of “head” and“tail”. This matrix represents the simplest choice for an unbiased walk, with no direction privileged by the unitaryevolution. Although it has been shown that the Hadamard coin leads to all possible quantum walks if different startingstates are considered [40], such equivalence is not guaranteed when percolation is introduced. Our choice has hence tobe considered as motivated by the sake of simplicity. The second part of a single step corresponds to the shift of thewalker in the lattice, toward a direction depending on the coin output. Formally, this can be written as the followingoperator acting non-trivially on the whole space H P ⊗ H C : S = (cid:88) i (cid:16) | i + 1 (cid:105) (cid:104) i | ⊗ |↑(cid:105) (cid:104)↑| + | i − (cid:105) (cid:104) i | ⊗ |↓(cid:105) (cid:104)↓| (cid:17) , (2)where we wrote the position component in front of the coin component. We will keep the same order throughout themanuscript.A single step of the evolution can therefore be obtained by applying the operator U ≡ S ( P ⊗ H C ) (3)to the vector | ψ in (cid:105) ∈ H P ⊗ H C , representing the initial state of the system. In order to obtain a different result withrespect to a classical random walk, a measurement is performed in the position degree of freedom only after a certainnumber of steps N . The probability distribution characterising the result will be P [ | ψ in (cid:105) ] ( i ) = Tr (cid:104) ( | i (cid:105) (cid:104) i | ⊗ C ) U ( N ) | ψ in (cid:105) (cid:104) ψ in | U ( N ) † (cid:105) , (4)where the subscript “1” reminds us that it represents the probability associated with a single walker. Withoutpercolation, the N -step evolution operator U ( N ) is simply obtained by exponentiating N times the single-step operator U defined in Eq. (3). After the measurement, that instance of quantum walk will be considered as concluded withoutput given by (4). It is well known that the average distance travelled by a classical random walker scales with thenumber N of steps as √ N , while for a quantum walker it scales linearly with it [2]. A typical plot for the probabilitydistribution of a quantum walk can be observed in Fig. 1a.Notice that all coefficients appearing in the evolution operators (1) and (2) are real. This implies that the realand imaginary parts of the initial state (expanded in the basis {| i (cid:105) |↑(cid:105) , | i (cid:105) |↓(cid:105)} ) evolve independently. Therefore, aconjugation of the state amplitudes does not change the final probability distribution, a fact that will be used lateron. (a) (b) (c) FIG. 1: Average probability distribution for a quantum walk on the line after N = 300 steps, without missing bonds ( a ) andwith dynamical ( b ) or statical ( c ) percolation (different ranges for the position values were chosen in order to increase thereadibility of the plots). In order to have a symmetric distribution we started from the origin with a coin state given by Eq.(11). In the last two plots, we considered lattices characterised by a percolation parameter p = 0 .
75 and we averaged over 10 different outputs. Percolation in 1D quantum walks.
The situation previously described changes if we consider a lattice whosebonds can be randomly missing with time. Typically, in a percolation graph each bond is actually present only witha fixed probability 0 ≤ p ≤
1, called “percolation parameter”, and the structure is periodically changed respectingsuch probabilistic constraint. For more information on percolation lattices and their applications we refer to [41, 42].Depending on the frequency of the changes, there can be two extreme regimes of percolation: statical (different latticesfor different walks) or dynamical (different lattice at each step). The first scenario can be associated with a time scalefor the lattice-imperfections evolution that is long with respect to the typical time required to finish a single walk, butsmall with respect to the average time needed between different runs of the apparatus. On the other hand, the regimeof dynamical percolation appears when the imperfections in the lattice evolve so quickly that the missing bonds canchange position even between consecutive steps of the same walk. In the following both these extreme cases will beconsidered.In the presence of percolation, the evolution outlined in Eq. (3) must be changed to avoid crossing of missing bonds.To achieve this the step operator applied on the i -th site is chosen as one of the following: • S = | i + 1 (cid:105) (cid:104) i | ⊗ |↑(cid:105) (cid:104)↑| + | i − (cid:105) (cid:104) i | ⊗ |↓(cid:105) (cid:104)↓| , • S + = | i (cid:105) (cid:104) i | ⊗ |↓(cid:105) (cid:104)↑| + | i − (cid:105) (cid:104) i | ⊗ |↓(cid:105) (cid:104)↓| , • S − = | i + 1 (cid:105) (cid:104) i | ⊗ |↑(cid:105) (cid:104)↑| + | i (cid:105) (cid:104) i | ⊗ |↑(cid:105) (cid:104)↓| , • S ± = | i (cid:105) (cid:104) i | ⊗ |↓(cid:105) (cid:104)↑| + | i (cid:105) (cid:104) i | ⊗ |↑(cid:105) (cid:104)↓| ,respectively when both neighbouring bonds are present, when the following or the previous one is missing, or whenthey are both missing.Every time the lattice changes, the missing-bonds positions are chosen randomly, so that eventually the outputprobability distribution has to be averaged over many different lattice sequences. As a result, the ballistic spreadingtypical of a quantum walk is lost, and the final probability distribution approaches that of a classical random walk[34, 35], similarly to what happens with other decoherence models [3, 43, 44]. An example can be seen in Fig. 1, wherethe outcomes of a 300-step walk are shown in different percolation regimes. While in the standard quantum walkonly odd (or even) positions can be occupied, there is no such constraint when the aforementioned strategy of timeevolution with missing bonds is adopted. Notice also how in the dynamical case the spread of the Gaussian-like shapeddistribution of the walker is much larger than the one of the statical output probability, which is narrowly peaked. Thisis a typical feature emerging in the two percolation regimes after a certain number of steps given approximately by L ( p ) /
2, where L ( p ) = p/ (1 − p ) is the average length of a connected segment of the line, representing the length scaleover which percolation becomes relevant. After this point the walker evolves with a diffusive behaviour in a dynamicalpercolation regime [34], while in the statical one it remains always localised within a segment: the subsequent averageleads to the narrowly peaked distribution of Fig. 1c, independently of the number of steps. Non-interacting particles.
In this last introductory section we present the formalism adopted throughout thepaper to describe two non-interacting walkers. Using the same model of [29], the global Hilbert space of the systemis H = H ⊗ H , where each H k can be expanded as H k = H P k ⊗ H C k . The single-step operator characterisingthe evolution is chosen to be the single-particle operator U defined in Eq. (3) (or modified close to the missing bondsaccording to the previous rule) applied independently to both systems: U = U ⊗ U , (5)where the label represents the Hilbert space of application. This choice guarantees that both particles evolve inde-pendently, without any kind of interaction. If the evolution takes place on a perfect lattice, U ( N )12 is obtained from(5) via exponentiation, i.e., U ( N )12 = U N . This changes in the presence of dynamical percolation, because every step ischaracterised by a different step operator (associated with a different lattice).In particular, in this manuscript we want to focus on indistinguishable particles, satisfying boson-like or fermion-likestatistics or just being classically impossible to discriminate when detected at the end of the walk. A comparison ofthe corresponding results will allow us to distinguish the features due to the quantum nature of the particles fromthose coming from the averaging process. Two single-particle states | ψ (cid:105) and | ψ (cid:105) can be combined to describe twobosons or two fermions by taking as joint initial state their opportune symmetrisation: (cid:12)(cid:12) Sym ± ( ψ , ψ ) (cid:11) = | ψ (cid:105) ⊗ | ψ (cid:105) ± | ψ (cid:105) ⊗ | ψ (cid:105) (cid:114) (cid:16) ± (cid:12)(cid:12) (cid:104) ψ | ψ (cid:105) (cid:12)(cid:12) (cid:17) , (6)where the indeterminate form (cid:12)(cid:12) Sym − ( ψ, ψ ) (cid:11) is set to 0 by definition. The case of classically indistinguishableparticles, instead, can be modelled by considering an initial separable state | ψ (cid:105) ⊗ | ψ (cid:105) and by changing the mea-surement stage so as to make the final probability symmetric under particle-exchange. This is done by considering ameasurement involving the set of projectors Π ij = Π ji = 12 (cid:16) | ij (cid:105) (cid:104) ij | P ,P + | ji (cid:105) (cid:104) ji | P ,P (cid:17) , (7)that leads to a probability distribution P [ | ψ in (cid:105) ] ( i, j ) = Tr (cid:104) Π ij U ( N )12 | ψ in (cid:105) (cid:104) ψ in | U ( N ) † (cid:105) . (8)Notice that when dealing with bosons or fermions the addition of the symmetrised term in the projector does notaffect at all the output probability. With definition (8), the probability of finding one particle in position i and theother one in j (cid:54) = i is given by 2 P [ | ψ in (cid:105) ] ( i, j ), while the probability of finding them at the same location i is just P [ | ψ in (cid:105) ] ( i, i ). Such description is chosen in order to obtain an output distribution properly normalised to 1 on thewhole plane ( i, j ). Given two single-particle states | ψ k (cid:105) , the output probability distributions obtained from (8) inthe three cases (bosons, fermions or classically indistinguishable particles) can be labelled respectively by P ( ± )2 and P ( cl )2 . The dependence upon the states | ψ k (cid:105) , k = 1 , , will be dropped most of the time to simplify the notation,when there is no possibility of confusion. While the first two distributions cannot generally be expressed in terms ofsingle-particle outputs (4), for two classically indistinguishable particles the following equality holds: P ( cl )2 ( i, j ) = 12 P [ | ψ (cid:105) ] ( i ) P [ | ψ (cid:105) ] ( j ) + 12 P [ | ψ (cid:105) ] ( i ) P [ | ψ (cid:105) ] ( j ) , (9)as can be easily proved by using the factorised structure of both evolution (5) and projector (7).It is worth noticing that not every entangled state in H can be expressed in a symmetrised form as in (6). Anexample of bipartite state left out by that particular structure is (cid:12)(cid:12)(cid:12) ψ (NotSym) (cid:69) = | ij (cid:105) + | ji (cid:105)√ ⊗ |↑↓(cid:105) + |↓↑(cid:105)√ , (10)when i (cid:54) = j . In the following, however, we will restrict our investigation to states of a symmetrised form as in (6), orto separable states | ψ (cid:105) ⊗ | ψ (cid:105) , because we need a definition of | ψ k (cid:105) , k = 1 ,
2, to compare P ( ± )2 with P ( cl )2 . Moreover,in order to maximise the multi-particle effects in our interaction-free model, we will always consider states with alocalised common initial position: the origin. In addition to this, to avoid asymmetrical spreading, we will focusmainly on combinations of the two single-particle coin states | ϕ ± (cid:105) = |↑(cid:105) ± i |↓(cid:105)√ , (11) (a) (b) (c) FIG. 2: Output probabilities for a 15-step quantum walk of two bosons, fermions or classically indistinguishable particles[respectively in ( a ), ( b ) and ( c )], starting from the origin with the coin states given in Eq. (12). that lead to the symmetric evolution shown in Fig. 1. With these conditions, the corresponding coin states for bosons,fermions and classically indistinguishable particles read | φ + (cid:105) = |↑↑(cid:105) + |↓↓(cid:105)√ , | ψ − (cid:105) = |↑↓(cid:105) − |↓↑(cid:105)√ , | ψ S (cid:105) = | ϕ + (cid:105) ⊗ | ϕ − (cid:105) . (12)Notice that, with the initial state | ψ S (cid:105) , relation (9) further simplifies: being | ϕ + (cid:105) and | ϕ − (cid:105) in Eq. (11) the complexconjugate of each other, their single-particle output distributions are the same: P [ | ϕ + (cid:105) ] = P [ | ϕ − (cid:105) ] = P , (13)so that the joint probability P ( cl )2 ( i, j ) becomes P ( cl )2 ( i, j ) = P ( i ) P ( j ) . (14)Examples of the probability distributions obtained by using the initial states defined in Eq. (12) can be observed inFig. 2, where typical bunching and anti-bunching behaviours can be noticed (see [33] for an experimental realisation). Results
We can now finally address the problem of considering the quantum walk of two indistinguishable and non-interactingparticles in a one-dimensional percolation lattice, where every run of the walk (in the statical case) or even everystep (in a dynamical regime) takes place on a different underlying structure, randomly generated as described above.Considering a number A ∈ N of quantum walk realisations, the average output probability distribution reads P ( i, j ) = 1 A A (cid:88) a =1 P ( a )2 ( i, j ) , (15)where index a runs from 1 to A . The evaluation of such quantity for different input states will be the basic tooladopted to characterise and study the behaviour of the walk. Usually the plots in this manuscript are obtained byaveraging over a number of different outputs varying from A = 2000 to A = 5000, depending on the precision requiredand on the complexity of the simulation. The obtained results do not qualitatively change in a significant way in thatrange. Moreover, they do not strongly depend upon the number of steps considered, that will typically be N = 15(we will explicitly point out the features that depend upon the parity of N ). Therefore, we can use the obtainedinformation to draw conclusions about the general qualitative behaviour of the walk. Our discussion will include threesections, addressing three different issues.At first, we want to qualitatively describe the average output distribution obtained in different percolation regimesconsidering the symmetric input states of Eq. (12). The goal of this analysis is to gain a better intuitive understandingof the situation, pointing out the features that are due to the quantum interference between the particles (that arisesfor entangled initial coin states, associated with bosonic or fermionic statistics) and those effects whose presenceis rooted in the average appearing in Eq. (15). In particular, we show how a signature of enhanced bunching (a) (b) FIG. 3: P ( cl )2 ( i, j ) for a quantum walk of two walkers starting from the same position with coin state | ψ S (cid:105) [see Eq. (12)],measured after 15 steps. Panels ( a ) and ( b ) correspond respectively to a dynamical and statical percolation regime, with p = 0 . A = 1000. between the walkers in the output distribution does not necessarily imply the presence of entangled states symmetricunder particle-exchange. This is therefore a warning against the possibility of gathering information about quantumindistinguishability from the mere presence of bunching effects.In the second part of our discussion, we study how some quantities of interest depend upon the parameters of theproblem (percolation regimes, probability of missing bonds and input states). In particular we focus on the averagedistance between the positions in which the particles are detected ( D ) [29] and on the probability of finding themeither in the same location ( M ), also called “meeting probability” [30, 45], or exactly in the origin ( C ). On the onehand, this aims at capturing some particular aspects of the output by means of a few parameters that can be moreeasily interpreted and manipulated with respect to the whole probability distribution. Being the whole distributionimpossible to plot for higher dimensions of the lattice or for a larger number of particles, in future research this kind ofapproach will become more and more common. From this perspective, our analysis aims at highlighting the qualitativetrends that appear in the aforementioned quantities in different scenarios of interest, whose identification will ease theinterpretation of future, more complex, results. On the other hand, we are interested to see if the behaviour of thesequantities strongly depends upon the considered input state [chosen among (12)], implying the relevance that inter-particle quantum interference maintains even in very noisy regimes. This fact, observed in particular for the meetingprobability M , can be interpreted as a signature of resilience of multi-particle effects to structural imperfections.Finally, in the third and last section, we search for possible advantages arising when considering a multi-particlequantum walk with respect to its single-particle counterpart. At first we consider the single-particle spread, showingthat it cannot be improved by the addition of another particle. However, inspired by the qualitative behaviour of C that was previously found, we show how such quantity can be used in a procedure to estimate the percolationparameter (probability for each link to be present) whose performance can be improved by the presence of a secondwalker. A. Analysis of bunching events.
The typical behaviour of the average output distribution P ( cl )2 for classicallyindistinguishable particles, defined by evaluating Eq. (15) with the separable input state | ψ S (cid:105) of Eq. (12), is plottedin Fig. 3 for dynamical and statical percolation. As can be appreciated, the two results are qualitatively dissimilar.In particular, we can notice how the distribution is more spread out over the allowed region in the dynamical case,while in the statical regime it is concentrated in a few particular points, mainly along the diagonal. This result canbe interpreted by keeping in mind that in the first case the walkers spread in a diffusive way, while in every run ofa statical regime they are localised on a small segment around the origin. The final distribution is then obtained byaveraging the output of each run, whose detection probability is similar to Fig. 2c. This leads to the diagonal peaksvisible in Fig. 3b, where the corresponding off-diagonal peaks of Fig. 2c disappeared because it is very unlikely tohave a connected segment that extends far away from the origin in both directions. Similar considerations could bemade for entangled coin states, representing boson-like and fermionic-like statistics, if instead of Fig. 2c we consideras underlying coherent evolutions for example those given in Fig.s 2a and 2b. From this analysis it emerges thatthe so called “bunching events”, associated with the detection of two particles at the same location, depend on theinterplay between statistical average and effects of quantum interference between multiple walkers. In the followingwe want to formalise these considerations.In order to properly define a peak (or a valley) in an output distribution, we need a reference probability to useas comparison. The peaks (or valleys) due to inter-particle quantum interference can be evidenced by comparing theaverage probabilities P ( ± )2 , associated with the symmetrisation of two single-particle input states | ψ (cid:105) and | ψ (cid:105) [seeEq. (6)], with the average probability P ( cl )2 , corresponding to classically indistinguishable particles starting from aseparable input state obtained by taking the tensor product of the same | ψ (cid:105) and | ψ (cid:105) . Indeed, for a single run of thequantum walk, whose evolution is labelled by a , we can expand the probability of finding both particles in the samelocation as P ( ± )( a )2 ( j, j ) = P ( cl )( a )2 ( j, j )1 ± | (cid:104) ψ | ψ (cid:105) | ± (cid:12)(cid:12)(cid:12) (cid:104) ψ | U ( N ) † ( a ) | j (cid:105) P (cid:104) j | U ( N )( a ) | ψ (cid:105) (cid:12)(cid:12)(cid:12) ± | (cid:104) ψ | ψ (cid:105) | . (16)From this, it follows that at least when (cid:104) ψ | ψ (cid:105) = 0 [which is the case in the main situation of interest here, where | ψ , (cid:105) = | (cid:105) | ϕ ± (cid:105) ] the elements on the diagonal are larger (for bosons) or smaller (for fermions) than the diagonalelements in the corresponding probability for classically indistinguishable particles, as can be expected. Since thisproperty holds for every index a , it is still true for the average probability distribution. This effect in a regime ofdynamical percolation can be observed in Fig. 4.On the other hand, the effect of the average can be understood by comparing P ( cl )2 with the symmetrised product ofthe two single-particle average distributions P [ | ψ i (cid:105) ]. Surprisingly, in some situations the averaging procedure mightinduce an enhancement of the probability of bunching events, leaving a signature in the output distribution similarto the interference effect previously analysed. To see this, we can expand the probability P ( a )1 [ | ψ k (cid:105) ] around its meanvalue as P ( a )1 [ | ψ k (cid:105) ] = P [ | ψ k (cid:105) ] + δ ( a ) [ | ψ k (cid:105) ] , (17)so that, for every lattice site j , the differences δ ( a ) [ | ψ k (cid:105) ] average to zero:1 A A (cid:88) a =1 δ ( a ) [ | ψ k (cid:105) ] ( j ) = 0 , (18)giving, for the average detection probability of two classically indistinguishable particles, P ( cl )2 ( i, j ) = 12 (cid:0) P [( | ψ (cid:105) )] ( i ) P [( | ψ (cid:105) )] ( j ) + P [( | ψ (cid:105) )] ( j ) P [( | ψ (cid:105) )] ( i ) (cid:1) + 12 A A (cid:88) a =1 (cid:16) δ ( a ) [( | ψ (cid:105) )] ( i ) δ ( a ) [( | ψ (cid:105) )] ( j ) + δ ( a ) [( | ψ (cid:105) )] ( j ) δ ( a ) [( | ψ (cid:105) )] ( i ) (cid:17) . (19)Usually when | ψ (cid:105) and | ψ (cid:105) lead to different probability distributions, or when off-diagonal terms i (cid:54) = j are considered,the second line in (19) has not a fixed sign, so for A (cid:29) i = j for states with the same final distribution (e.g. | (cid:105) | ϕ + (cid:105) and | (cid:105) | ϕ − (cid:105) ). In thiscase the second line in (19) is always positive, being a sum of squared terms. These contributions hence add up tocreate the peaks over the first term, which in the example provided is just equal to the product P ( i ) P ( j ) (see Eq.(14) and Fig. 5 for the resulting plots in a regime of dynamical percolation).We expect that a similar behaviour can arise also with more complicated lattice structures affected by percolation.Therefore, whenever one deals statistically with such imperfections, it is useful to keep in mind that the mere av-eraging process could lead to an enhancement of the diagonal entries of the average probability distribution. Beingaware of this fact can avoid the misinterpretation of such phenomenon as a bunching signature due to the quantumindistinguishability of the multiple walkers considered. B. Qualitative analysis of three significant quantities in the presence of percolation.
In what follows,we want to see how a few quantities depend upon the adopted percolation parameter p and the particle-exchangesymmetry encoded in the input states, for both dynamical and statical percolation regimes. In particular, we willfocus on two walkers starting from the same position with initial coin states | φ + (cid:105) , | ψ − (cid:105) or | ψ S (cid:105) , defined in Eq. (12).For every considered quantity Q and for a set of equally spaced percolation parameters p , we will average the valueobtained for a single lattice realisation Q ( a ) over A = 5000 different configurations: Q ( p ) = 1 A A (cid:88) a =1 Q ( a ) ( p ) . (20) (a) P (+)2 ( i, j ) (b) P ( − )2 ( i, j ) (c) P ( cl )2 ( i, j )(d) Difference ( a ) − ( c ) (e) Difference ( b ) − ( c ) FIG. 4: Average output probability distributions for a two-particle 15-step quantum walk affected by dynamical percolation( p = 0 . A = 2000). We considered boson-like ( a ), fermion-like ( b ), and classically indistinguishable ( c ) combinations of twosingle-particle states | ϕ ± (cid:105) that spread symmetrically from the origin. Panels ( d ) and ( e ) evidence the contribution to bunchingevents due to inter-particle quantum interference. (a) P ( cl )2 ( i, j ) (b) P ( i ) P ( j ) (c) Difference ( a ) − ( b ) FIG. 5: P ( cl )2 ( i, j ), P ( i ) P ( j ), and their difference are plotted respectively in ( a ), ( b ), and ( c ), for a quantum walk of N = 15steps starting from the origin with an initial coin state | ψ S (cid:105) , defined in Eq. (12). A dynamical percolation regime with p = 0 . A = 2000 different evolutions is considered. This figure shows how the statistical averaging procedure mightenhance bunching events in the average probability distribution of a multi-particle quantum walk. The uncertainty associated with such average will be then given by the standard error σ Q ( p ) = 1 √ A (cid:118)(cid:117)(cid:117)(cid:116) A − A (cid:88) a =1 (cid:2) Q ( a ) ( p ) − Q ( p ) (cid:3) . (21) B1. Final average distance between the particles.
A global property that can be analysed is the average (a) (b)
FIG. 6: Average output distance between the two particles as a function of the percolation parameter p , for a 15-step quantumwalk starting from the origin with three different coin states [see Eq. (12)]. ( a ) and ( b ) correspond respectively to dynamicaland statical percolation regimes. The average is calculated over A = 5000 different lattice realisations and the errors areevaluated as the standard errors on the mean values obtained. distance between the positions where the two walkers are detected. Formally such quantity can be expressed as D = (cid:88) ij | j − i | P ( i, j ) . (22)Its numerical evaluation for different percolation parameters p is plotted in Fig. 6, for a 15-step quantum walk. Ascan be expected, for any value of the percolation parameter, this quantifier for fermions is larger than the one forbosons, with an intermediate behaviour for the separable coin | ψ S (cid:105) . This result agrees with the discussion in [29],that considered this same quantity without percolation.If we consider the dependence upon p in the dynamical percolation regime, we can see that the average distancebetween bosons increases approximately in a linear way, with a small decrease just before p = 1. This can beinterpreted as a trade-off between two trends: on the one hand with less missing bonds the two walkers can moveon a larger segment of the line, increasing the likelihood of finding them further away; on the other hand, if the twoparticles are less disturbed, their distribution approaches the one obtained without percolation (shown in Fig. 2a),characterised by a high degree of bunching. For fermions, instead, the same considerations add up increasing theaverage distance of separation, since they tend to show an anti-bunched distribution in the limit p → L available to the walkers in a regime of statical percolation [34]. B2. Meeting probability.
Another global quantity, easier to obtain with respect to the previous one as it involvesa smaller number of measurements, is the overall probability for the two walkers to be on the same position at theend of the walk. Such quantity, also called meeting probability [30, 45], corresponds to the sum of the diagonal termsin the output probability distribution P ( i, j ) in Eq. (8): M = (cid:88) j P ( j, j ) . (23)The results obtained for a 15-step quantum walk are plotted in Fig. 7 for both dynamical and statical percolation.We can notice how in the first scenario the meeting probability decreases for p (cid:46) .
92 and slightly increases afterthis point, while in the statical case it is monotonically decreasing for all values of p . The minimum position in thedynamical case seems to depend on the number of steps, moving slowly toward p = 1 when this is increased. However,even if this could look like a finite-size effect, we pushed simulations up to 80 steps, and have strong numericalevidence that the non-monotonic behaviour would still be present even for a much larger number of steps. We can0 (a) (b) FIG. 7: Average meeting probability as a function of the percolation parameter p , for a 15-step quantum walk starting from theorigin with three different coin states [see Eq. (12)]. ( a ) and ( b ) correspond respectively to dynamical and statical percolationregimes. The average is calculated over A = 5000 different lattice realisations and the errors are evaluated as the standarderrors on the mean values obtained. obtain an intuitive explanation for this fact by comparing the output of the walk in a perfect lattice (Fig. 2) with the“spreading” effect that dynamical percolation has on the output probability distribution (see Fig. 3a). Starting fromthe ideal case of p = 1, the addition of small amounts of imperfections makes the diagonal peaks of Fig. 2 spread,leading to a diminished meeting probability. This effect is much less important in a statical percolation regime, ascan be appreciated in Fig. 3b. This is because in every lattice realisation the walk is coherent on a small segment ofthe line around the origin, and the percolation only averages such small-scale coherent evolutions. The overall resultis that in this case the meeting probability depends monotonically upon p (see Fig. 7b).From Fig. 7 we can also see how the quantum statistics of the particles heavily affects the meeting probability,even for small values of p . This shows that a signature of bosonic or fermionic multi-particle effects survives also in anoisy regime where the quantum walk tends to its classical counterpart. B3. Probability of finding both walkers in the origin.
Finally, we consider a “local” property, i.e. that can beobtained through a measurement in a single position: the probability of finding both walkers in their initial startingpoint. In the following we will label it as C = P (0 , . (24)In addition to being more easily obtained with respect to the previous quantities, this one differs also because itdepends on the parity of the number of steps N considered. Indeed, when p = 1, C is rigorously zero if N is odd,but this could not be the case if N is even. Being mostly interested in this quantity for its monotonic and regularbehaviour, that can be exploited to estimate the percolation parameter p (see next section), we will focus only on oddnumbers of steps in order to have a quantity varying in the maximum range [0 , C will be close to 1 for small values of p , since both walkers are constrained by the missing bonds tostay close to the starting point, and it decreases with increasing p while the walk spreads further away, reaching zerofor p = 1 (for N odd). This behaviour is numerically confirmed by Fig. 8, that shows how C depends upon p for a15-step quantum walk affected by dynamical or statical percolation. We can see how in the first regime C ( p ) goesquickly to zero as soon as the imperfections are reduced. This can be explained with the diffusive behaviour of thewalkers, that are allowed to quickly populate regions with i (cid:54) = j in the P ( i, j ) plot (e.g. see Fig. 3a). The situationis different in the statical case, where the walkers move away from the origin in a more controlled way (e.g. see Fig.3b), based once again on the typical length of connected lattice around the origin. The resulting plot for C ( p ) ismonotonic with almost constant slope in the whole range [0 , C. Search for multi-particle advantages.
In this final section we want to discuss whether the addition of a secondwalker to the walk can lead to an immediate advantage in some task or protocol. As a first check we consider thepossibility of having an improvement in the spread of the walk and show that this is not the case. For example,for every quantum statistics of the two particles involved there is always a single-particle state leading to a largerspread. We then consider a more specific task, namely the estimation of the percolation parameter affecting the lattice,proposing a two-walker procedure that in certain regimes shows an advantage over its single-particle counterpart.1 (a) (b)
FIG. 8: Average probability of finding both walkers in the origin, as a function of the percolation parameter p , for a 15-stepquantum walk starting from the origin with three different coin states [see Eq. (12)]. ( a ) and ( b ) correspond respectively todynamical and statical percolation regimes. The average is calculated over A = 5000 different lattice realisations and the errorsare evaluated as the standard errors on the mean values obtained. C1. Spread of the walk.
The intuitive idea of spreading for a single walker, starting from a localised position i ,can be well described by the quantity V = (cid:88) i ( i − i ) P ( i ) , (25)where the initial coin state is not explicitly written. Notice that, if the walker evolves symmetrically, such quantitycorresponds to the variance of the output distribution. How can we generalise this expression to the two-walker case?Also here, we are mostly interested in letting both particles start from the same lattice position. Indeed, since inour model there are no interactions, this situation is the one that could maximise the multi-particle effects. For thisreason, we will consider here only the case where both walkers start from the origin (with arbitrary coin states). Withthis hypothesis, the straightforward generalisation of (25) for a generic fixed unitary evolution labelled by a (i.e. fora given sequence of lattices) is V ( a )2 = 12 (cid:88) ij (cid:0) i + j (cid:1) P ( a )2 ( i, j ) . (26)Once two single-particle coin states have been fixed, we can label with V ( ± )( a )2 , V ( cl )( a )2 the spread corresponding tothe different statistics of the particles, obtained by using in (26) respectively P ( ± )( a )2 or P ( cl )( a )2 . In the same way,with an overbar we will label the value of the spread averaged over many different lattice sequences in a percolationregime. The overall factor 1 / V of Eq. (25), because from Eq. (27) it follows V ( cl )( a )2 = V ( a )1 ( c ) + V ( a )1 ( c )2 ≤ max k ∈{ , } V ( a )1 ( c k ) , (27)where two generic single-particle coin states | c (cid:105) , | c (cid:105) are considered. This is a simple consequence of the fact thattwo particles initialised in a separable state spread independently under the evolution in Eq. (5). The fact that wecannot distinguish them in the measurement leads only to the average that appears in Eq. (27).One can now wonder whether the presence of a proper quantum statistics, introduced by the symmetrisation of thevector describing the state, can somehow alter this result. In order to show that the situation does not considerablychange, we consider an initial state (cid:12)(cid:12)(cid:12) ψ ( ± ) in (cid:69) = | (cid:105) ⊗ (cid:12)(cid:12) Sym ± ( c , c ) (cid:11) , (28)whose spread after N steps (for a particular lattice sequence) can be written as V ( ± )( a )2 = (cid:88) k =1 λ ( ± ) k V ( a )1 (cid:16) ˜ c ( ± ) k (cid:17) . (29)2In this expression 0 ≤ λ ( ± ) k ≤ (cid:12)(cid:12)(cid:12) ˜ c ( ± ) k (cid:69) ∈ H C are the two eigenvalues and eigenvectors of the coin state P (cid:104) | Tr (cid:104)(cid:12)(cid:12)(cid:12) ψ ( ± ) in (cid:69) (cid:68) ψ ( ± ) in (cid:12)(cid:12)(cid:12)(cid:105) | (cid:105) P , (30)obtained by tracing out the Hilbert space of the second particle. Moreover, the odd combination (cid:12)(cid:12) Sym − ( c , c ) (cid:11) oftwo generic coin states | c (cid:105) , | c (cid:105) ∈ H C , as well as the even one (cid:12)(cid:12) Sym + ( c , c ) (cid:11) or separable one | c (cid:105) ⊗ | c (cid:105) of twoorthogonal coin states (cid:104) c | c (cid:105) = 0, yields the same spread as the two single-particle symmetrical-spreading states | ϕ ± (cid:105) [defined in Eq. (11)]: V ( − )( a )2 = V (+)( a )2 (cid:12)(cid:12)(cid:12) (cid:104) c | c (cid:105) =0 = V ( cl )( a )2 (cid:12)(cid:12)(cid:12) (cid:104) c | c (cid:105) =0 = V ( a )1 ( ϕ ± ) . (31)The proof of these facts can be found in Methods. Being | ϕ ± (cid:105) orthogonal coin states leading to the same probabilitydistribution, Eq.s (27) and (31) imply V ( a )2 ( φ + ) = V ( a )2 ( ψ − ) = V ( a )2 ( ψ S ) = V ( a )1 ( ϕ ± ) . (32)This means that there is no advantage, from the spread perspective, in considering two symmetrical-spreading particlesinstead of just one, no matter what the statistics over particle-exchange enforced in the input state by Eq. (6) is.From Eq. (31) it follows that only the bosonic statistics could a priori influence the spread, when two non-orthogonalsingle-particle coin states | c (cid:105) and | c (cid:105) are symmetrised. However, Eq. (29) implies that there will always exist asingle-particle state with a spread at least as large as the one found with two bosons, because V (+)( a )2 = (cid:88) k =1 λ (+) k V ( a )1 (cid:16) ˜ c (+) k (cid:17) ≤ max k ∈{ , } V ( a )1 (cid:16) ˜ c (+) k (cid:17) . (33)We emphasise that such states (cid:12)(cid:12)(cid:12) ˜ c (+) k (cid:69) could be different from both coin states | c (cid:105) and | c (cid:105) symmetrised at thebeginning, and that the optimal k could depend on the explicit lattice realisation labelled by “ a ”.All these results can be trivially extended to the spread averaged over many lattice configurations. In particular,for every initial coin states | c (cid:105) , | c (cid:105) one has V ( cl )2 = V ( c ) + V ( c )2 ≤ max k ∈{ , } V ( c k ) , (34) V (+)2 = (cid:88) k =1 λ (+) k V (cid:16) ˜ c (+) k (cid:17) ≤ max k ∈{ , } V (cid:16) ˜ c (+) k (cid:17) , V ( − )2 = V ( ϕ ± ) , (35)in terms of (cid:12)(cid:12)(cid:12) ˜ c (+) k (cid:69) and λ (+) k defined in Eq. (30). C2. Estimation of the percolation parameter.
Here we want to explicitly consider the possibility of usingsome of the quantities described in section B to infer the value of the percolation parameter p that characterises thelattice imperfections. In order to do so, let us consider a certain event E , happening at the measurement stage of thequantum walk: e.g. both particles at the same location, both in the origin, etc. The probability of such event willdepend on the explicit lattice realisation, and we will write it as P ( ξ ) E , where ξ runs over all the possible sequences oflattice configurations. Notice that here we are using ξ instead of the previous a , to emphasise the fact that we areconsidering an effective lattice realisation and not of a simulation of it. P ( ξ ) E has to be interpreted as the probability ofthe event considered, conditioned upon the ξ -th realisation of the lattice: P ( ξ ) E = P ( E | ξ ). Therefore, the probabilityof E can be written as P E = (cid:88) ξ P ( E | ξ ) P ( ξ ) = (cid:88) ξ P ( ξ ) E P ( ξ ) . (36)This quantity, although difficult to obtain analytically, can be numerically estimated. Indeed, for the event “bothparticles in the same place” (or “both particles in the origin”) it would correspond to M ( p ) (20) [or C ( p ) (24)] in thelimit of many different sequences of lattice configurations considered ( A → ∞ ). In the following we will generically3label this numerical estimate as P ( sim ) E , whose precision will depend on the number A adopted [see the standarderror in Eq. (21)]. On the other hand, if one had access to an experimental setup of such quantum walk affected bypercolation, after n runs of the experiment P E could be estimated via the quantity P ( est ) E = n E n , (37)where n E is the number of times in which E happened. This estimator is a stochastic variable, characterised byexpectation value and variance given by EV (cid:104) P ( est ) E (cid:105) = P E , VAR (cid:104) P ( est ) E (cid:105) = P E (1 − P E ) n . (38)This can be shown by considering P ( est ) E as the average of n independent and identically distributed variables, takingthe value 1 with probability P E (when E happens in that single run) and the value 0 otherwise.Our goal is to estimate the value of p through a measurement of P ( est ) E , that estimates P E with an errorVAR (cid:104) P ( est ) E (cid:105) / . Exploiting the linear propagation of errors and the explicit expression of the variance given inEq. (38), we can write the uncertainty in the estimation of p as δp = (cid:114) VAR (cid:104) P ( est ) E (cid:105)(cid:12)(cid:12)(cid:12) ∂P E ∂p (cid:12)(cid:12)(cid:12) = (cid:112) P E (1 − P E ) √ n (cid:12)(cid:12)(cid:12) ∂P E ∂p (cid:12)(cid:12)(cid:12) , (39)so that an upper bound δp ≤ (cid:15) yields a lower bound on the required number of runs n ≥ n ( (cid:15) ) min ( p ): n ≥ n ( (cid:15) ) min ( p ) = P E (1 − P E ) (cid:15) (cid:12)(cid:12)(cid:12) ∂P E ∂p (cid:12)(cid:12)(cid:12) . (40)The right-hand side can be estimated by using P ( sim ) E , numerically obtained by considering a total number of A (cid:29) P E (1 − P E ) (cid:39) P ( sim ) E (1 − P ( sim ) E ) , (41) ∂P E ∂p (cid:39) ∂P ( sim ) E ∂p , (42)where the second term is obtained from a polynomial fit of the numerical samples { P ( sim ) E ( p i ) } i . We point out that,despite Eq. (38), the expectation value of the right-hand side of (41) coincides with the left-hand side only in thelimit A → ∞ , making P ( sim ) E (1 − P ( sim ) E ) a biased but consistent estimator for P E (1 − P E ).In order to explicitly obtain the bound n ( (cid:15) ) min ( p ) for different kinds of quantum particles, we considered the event“both walkers in the origin” for a 7-step quantum walk in a regime of statical percolation. Indeed, having an almostconstant slope over the whole region p ∈ [0 , C ( p ) is the best quantity among those studied in subsection B toestimate the value of p in that regime. Notice that here the number of steps N considered is reduced (with respect tothe usual 15) to allow us to take a much larger value of A : for this analysis we randomly generated ∼ percolationlattices. This guarantees that the total number of simulated configurations A is much larger than the total numberof possible lattices, given by 2 · (cid:39) . · for the situation considered. We stress once again that in order to obtainplots qualitatively similar to Fig. 8b, leading to a good estimation strategy, it is important to consider an odd numberof steps.The single-walker counterpart of the event here considered corresponds to detecting the particle in the origin at theend of the walk, and it can be used to infer p with the same procedure previously outlined. Therefore, we can comparethe performances achieved using either one or two walkers initialised with different input exchange symmetries. InFig. 9 we plotted the event probability P ( sim ) E and the bound n ( (cid:15) =0 . min for the input coin states defined in Eq. (12),corresponding to boson-like, fermion-like statistics or to two classically indistinguishable particles. We can see howfor the 7-step walk considered, for values of the percolation parameter p (cid:46) .
82, there is an advantage in using twoquantum particles instead of one. In particular bosons are optimal in the range 0 . (cid:46) p (cid:46) .
82, while fermionsperform better below p (cid:39) .
38. We studied the dependence of such thresholds of bosons optimality p ≤ p ≤ p upon4 (a) (b) FIG. 9: ( a ) Simulated event probability P ( sim ) E of finding the single particle used (SP) or both particles [see Eq. (12)] in theorigin, as a function of the percolation parameter p . Due to the large number of simulations, the standard error is negligible. ( b )Corresponding lower bound on the number of required experimental runs n ( (cid:15) =0 . min as a function of the percolation parameter p . In both cases a 7-step quantum walk has been considered, averaging over A ∼ lattice sequences. For the evaluation of n ( (cid:15) ) min , we used polynomials of degree 5 to fit the points in ( a ) (see the discussion below Eq. (42) in the main text). N , considering walks with an odd number of steps between N = 3 and N = 11. We found that p ( p ) decreases(increases) slightly with increasing N , with p and p varying respectively in the ranges [0 . , .
44] and [0 . , . n ( (cid:15) ) min → P E → P E →
1. The error VAR (cid:104) P ( est ) E (cid:105) / (and therefore δp ) is associated with a certain probability of finding the real value in that range, that for a smallnumber of measurements n is actually unknown. Only for n (cid:29) (cid:104) P ( est ) E (cid:105) / small enough for the function P E ( p ) to be considered approximatelylinear in that interval. Even if Eq. (40) seems to suggest otherwise, the scaling ∼ / √ n in Eq. (38) therefore requiresto take n ( (cid:15) ) min large enough to satisfy the previous conditions. Discussion
We studied the behaviour of a one-dimensional quantum walk when the possibility of having missing bonds is consid-ered together with the presence of a second walker. We focused on two percolation regimes, statical and dynamical,associated respectively with slowly and fast varying imperfections. Our numerical simulations have been averaged overmany (typically ∼ p ≤ p , fact that suggests the resilience of inter-particle interference to very noisy conditions.Searching for multi-walkers advantages, we considered at first the spread of the walk, focusing on the dependenceupon the second particle more than on the decoherence introduced by the percolation. Our analysis shows how theaddition of a second walker does not increase the spread, since there always exists a single-particle state with a spreadat least as large as the one obtained with two walkers. Moreover, for many cases of interest, i.e. for any couple offermions and for bosons or classically indistinguishable particles obtained by combining two orthogonal single-particlecoin states, the spread is not affected by the quantum statistics of the walkers. In these cases it is equal to the spreadof the two symmetric single-particle coin states | ϕ ± (cid:105) [defined in Eq. (11)].We also proposed a procedure to estimate the value of the percolation parameter that characterises the amountof lattice imperfections. Due to the monotonicity and the regularity observed in Fig. 8b, it is worth to apply suchstrategy considering the detection of both particles in the origin in a regime of slowly varying imperfections, i.e., ofstatical percolation. For values of p not too large, we showed how the performance of the proposed estimation isenhanced by the presence of a second walker, that allows us to need a smaller number of experimental measurementsin order to achieve a certain estimation precision.This investigation considered for the first time the effects of possible missing links (that could happen, for instance,in an experimental setting) on the features of walks where the quantum nature of the involved particles is relevant.Due to the importance of quantum walks for the purposes of quantum computation and simulation, and in particularfor the role that many-walkers complexity will have in the future, these results pave the way for a more completeunderstanding and a full exploitation of these models. Methods
Comparison between single-particle and two-particle spread.
Here we will provide a proof for Eq.s (29) and(31). In the following the term single-particle spread will always correspond to the quantity V obtained from Eq.(25) by setting i = 0 and we will use the shorthand notation P i and P ij to represent respectively the single-particleand the two-particle output probabilities to detect the walkers in positions i and j . We will also drop the dependenceupon the lattice realisation “ a ”.Let us begin by showing that given two orthogonal coin states | ˜ c (cid:105) , | ˜ c (cid:105) ∈ H C , characterising the initial conditionsof two single walkers localised in the origin, their average spread depends only upon the unitary evolution U ( N ) andnot on the basis {| ˜ c (cid:105) , | ˜ c (cid:105)} of H C . Indeed, being | ˜ c (cid:105) C (cid:104) ˜ c | + | ˜ c (cid:105) C (cid:104) ˜ c | = C , (43)from Eq. (4) the sum P i [ | ˜ c (cid:105) ] + P i [ | ˜ c (cid:105) ] does not depend on the choice of basis: P i [ | ˜ c (cid:105) ] + P i [ | ˜ c (cid:105) ] = Tr (cid:104) U ( N ) † | i (cid:105) P (cid:104) i | ⊗ C U ( N ) | i (cid:105) P (cid:104) i | ⊗ C (cid:105) , (44)property that holds also for the average between the two single-particle spreads. Therefore, given two particles startingfrom the origin with two orthonormal coin states | ˜ c (cid:105) , | ˜ c (cid:105) ∈ H C : V (˜ c ) + V (˜ c )2 = V ( ϕ ± ) , (45)being (cid:104) ϕ + | ϕ − (cid:105) = 0 and V ( ϕ + ) = V ( ϕ − ).We are now able to prove Eq. (29). Being P ij symmetric under the exchange i ↔ j , the two-particle spread V defined in Eq. (26) only depends upon the marginal sum (cid:80) j P ij : V ( ± )2 = (cid:88) i i (cid:88) j P ( ± ) ij . (46)Using the symmetry under particle exchange of the evolved state (cid:12)(cid:12)(cid:12) ψ ( ± ) f (cid:69) = U ( N )12 (cid:12)(cid:12)(cid:12) ψ ( ± ) in (cid:69) , (47)from Eq. (8) it follows (cid:88) j P ( ± ) ij = 12 (cid:88) j Tr , (cid:104) ( | ij (cid:105) (cid:104) ij | + | ji (cid:105) (cid:104) ji | ) (cid:12)(cid:12)(cid:12) ψ ( ± ) f (cid:69) (cid:68) ψ ( ± ) f (cid:12)(cid:12)(cid:12)(cid:105) = Tr (cid:110) | i (cid:105) (cid:104) i | · U ( N )1 Tr (cid:104)(cid:12)(cid:12)(cid:12) ψ ( ± ) in (cid:69) (cid:68) ψ ( ± ) in (cid:12)(cid:12)(cid:12)(cid:105) U ( N ) † (cid:111) (48)= (cid:88) k =1 λ ( ± ) k Tr (cid:104) | i (cid:105) (cid:104) i | · U ( N )1 | (cid:105) (cid:104) | ⊗ (cid:12)(cid:12)(cid:12) ˜ c ( ± ) k (cid:69) (cid:68) ˜ c ( ± ) k (cid:12)(cid:12)(cid:12) U ( N ) † (cid:105) = (cid:88) k =1 λ ( ± ) k P i (cid:104)(cid:12)(cid:12)(cid:12) ˜ c ( ± ) k (cid:69)(cid:105) , (49)6where the last two equalities follow from the definition of λ ( ± ) k and (cid:12)(cid:12)(cid:12) ˜ c ( ± ) k (cid:69) as eigenvalues and eigenvectors of (30),using T r (cid:104)(cid:12)(cid:12)(cid:12) ψ ( ± ) in (cid:69) (cid:68) ψ ( ± ) in (cid:12)(cid:12)(cid:12)(cid:105) = | (cid:105) P (cid:104) | ⊗ (cid:88) k λ ( ± ) k (cid:12)(cid:12)(cid:12) ˜ c ( ± ) k (cid:69) C (cid:68) ˜ c ( ± ) k (cid:12)(cid:12)(cid:12) . (50)The required equality in Eq. (29) then follows trivially from the spread definition (46).Eventually, it can be easily proved that for every | c (cid:105) , | c (cid:105) ∈ H C one has λ ( − ) k ≡ , k = 1 ,
2, and that the sameequality holds also for bosons if (cid:104) c | c (cid:105) = 0. Therefore, Eq. (31) is a consequence of Eq.s (45) and (27). Acknowledgements
We thank M. S. Kim for useful comments and discussions. L. R. was supported by the People Programme (Marie CurieActions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n ◦ Author Contributions
Both the authors made significant contributions to the concept, execution, interpretation, and preparation of thework.
Additional Information
Competing Financial Interests : The Authors declare no competing financial interests.
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