Eigenvalue measurement of topologically protected edge states in split-step quantum walks

We study topological phenomena of quantum walks by implementing a novel protocol that extends the range of accessible properties to the eigenvalues of the walk operator. To this end, we experimentally realise for the first time a split-step quantum walk with decoupling, which allows for investigating the effect of a bulk-boundary while realising only a single bulk configuration. The experimental platform is implemented with the well-established time-multiplexing architecture based on fibre-loops and coherent input states. The symmetry protected edge states are approximated with high similarities and we read-out the phase relative to a reference for all modes. In this way we observe eigenvalues which are distinguished by the presence or absence of sign flips between steps. Furthermore, the results show that investigating a bulk-boundary with a single bulk is experimentally feasible when decoupling the walk beforehand.

The concept of quantum random walks extends the model of classical random walks by taking the effects of interference into consideration. It has been established for two different scenarios, namely the continuous-time and the discrete-time version. The continuous-time quantum walk is defined by a Hamiltonian including nearest neighbour hopping, while discrete-time quantum walks (DTQW) describe the discrete time evolution of a walker with an internal degree of freedom on a lattice. An extensive account on the relation of discrete-and continuous-time quantum walks from view-point of mathematical physics can be found in [23,24]. The interference effects allow quantum walks to spread quadratically faster than their classical counterparts, which recently raised much interest in quantum walks as a computational resource [25][26][27]. Moreover, quantum walks exhibit a rich variety of single-particle quantum effects such as Landau-Zener tunnelling [28], the Klein paradox [29] or the formation of molecules [30]. Recently, a uniform framework for coupling external gauge fields to quantum walks has been established [31] which contains the one-dimensional electric walks studied in [32] as a special case. Possible experimental implementations include nuclear magnetic resonance [33,34], trapped ions [35,36] and atoms [37,38]. Considering photonic systems, translating the walker in the spatial degree of freedom might seem most straight-forward [39][40][41][42][43][44][45][46], however, implementations utilising time as the external degree of freedom outperform spatiallly multiplexed systems in terms of resource efficiency and stability [47][48][49][50][51][52][53]. Other possible degrees of freedom include spectral distributions or orbital angular momentum [54,55].
So far, the experimental investigation of topological phenomena has been focused on the demonstration of edge states, i.e. eigenstates of the systems, while eigenvalues of the walk operator have not been measured. Employing a phase-reference method, in which we selectively interfere components of walker's wavefunction assigned to a certain step, position and polarisation with a reference of well-controlled phase, we are for the first time able to measure not only the intensities related to the eigenfunctions, but also the signs related to the eigenvalues of the walk operator. This work also constitutes the first experimental implementation of a split-step quantum walk with decoupling as proposed in [56,57], which allows for investigating the effect of a bulkboundary while realising only a single bulk configuration. Such a scheme thus not only requires a smaller set of different coin angles that have to be implemented, but also reduces the size of the space occupied by the walk, freeing positions which can now be used for routing of the phase-reference. We implement this split-step quantum walk system exhibiting a bulk-boundary with decoupling by making use of a photonic platform [47] that allows for the read-out of the spatial and the coin degree of freedom as well as for dynamic coin operations [58].
In our work, we build on the comprehensive topological classification framework for infinite 1D lattices laid out in [57]. It predicts the emergence of symmetry protected edge states at the boundary of bulks with different symmetry indices. These symmetry protected eigenfunctions of the walks operator belong to the eigenvalues ±1, which lie in the band gap of the system. Whereas the minimal number of these edge states is an invariant quantity, their associated eigenvalue (+1 or −1) may change under local perturbations and hence the eigenvalues of edge states for half-chain quantum walks are not robust against local perturbation. We will investigate this behaviour for an experimental realisation of the split-step quantum walk.
The article is structured as follows: we start with the definition of a (DTQW) in section 2, followed by an introduction to the topological classification of 1D quantum walks in section 2.1, then present split-step quantum walks in section 2.3, the relation between chiral symmetry and topological phases in section 2.4, the concept of decoupling in section 2.5, the concrete settings in section 2.6 as well as the expected eigenfunctions and eigenvalues in section 2.7. We then turn to the experimental concepts, namely the physical implementation in section 3.1, the eigenstate distillation in section 3.2 and the phase-reference method in section 3.3. The results concerning the evolution of the wavefunction and the validation of the eigenvalues are presented in sections 4.1 resp. 4.2. Finally, the conclusion is drawn in section 5.

Theoretical background
Discrete time quantum walks describe the discrete time evolution of a walker with an internal degree of freedom on the lattice. We here focus on the spatially one-dimensional case where at each site of the lattice the internal degree of freedom is described by the finite-dimensional coin space d  . Consequently, the overall Hilbert space is . A quantum walk W is a unitary operator on  which obeys a locality condition and determines the discrete time evolution of a state t  y ñ Î | ( ) by The locality of W means that in a single time step an initially localised state can jump only by a finite number of lattice sites. We remark that the classification in [57] allows for more general notions of locality, but bounding the jump length is sufficient for this paper. A constructive way to define quantum walks, which is handy for experimental realisations and automatically fulfils the locality condition, is the concept of coined quantum walks. For these the walk operator W is build from a finite sequence of conditional shifts and coin operations: Denoting by basis vectors of the position and coin space, respectively, conditional shifts and coin operations act on basis vectors of  by The coin operation C acts locally in each cell, whereas the conditional shift S j (S j † ) shifts the j-component of the coin space to the right (left) while leaving the other internal degrees of freedom unchanged. Both operators are unitaries, such that W is also guaranteed to be unitary. In this definition, the maximal jump-length of the walk is bounded from above, by the number of shift operations in the sequence constituting W.
In our experimental realisation the coin-space is two-dimensional and the basis vectors are implemented by horizontal and vertical polarisation, which we denote by Hñ | and V ñ | , respectively. The quantum walks which are modelled in the experiment are of the form ( ) † † with possibly different coins C 1 and C 2 , which will be specified later in sections 2.3 and 2.6.
Note, that in general one might define a one-dimensional quantum walk abstractly as a unitary on a doubly infinite direct sum of possibly different finite dimensional Hilbert spaces : The locality condition is then stated as W x y L 0 for all , with , 4 x y x x y y for some L < ¥. The general theory for the topological classification of symmetric quantum walks, which will be sketched in the next section, demands an even less strict definition of locality. There the matrix elements W x y y y á ñ | are only assumed to decay sufficiently fast with x y -| |.

Topological classification of one-dimensional quantum walks
With this setting in mind, let us sketch the topological classification of symmetric quantum walks given in [57].
Below we apply the results provided here to discuss the topological classification of the experimentally implemented model. A crucial assumption in [57] is that the quantum walks under consideration are 'symmetric', i.e. there is a group of discrete, involutive symmetries which either commute or commute up to taking the adjoint with the walks.
Concretely, we distinguish the following unitary and anti-unitary symmetries: The groups generated by these symmetries either contain only one or all three, since any two of them multiply to the third, and they constitute the symmetry types of the tenfold way [57,59,60]. By the above relations, the ±1-eigenspaces of symmetric quantum walks are invariant under these symmetries. Consequently, we assume the bulk systems under consideration to be gapped at these symmetry-invariant points, similar to the Hamiltonian setting 4 .
The topological classification of symmetric quantum walks is concerned with two questions: (1) which symmetric quantum walks may be continuously connected without breaking either the symmetries, the gap condition or locality, and (2) which walks may be transformed into each other by perturbing one of them locally. In contrast to the continuous time setting, these questions are fundamentally different: In the unitary discretetime setting there are local perturbations that cannot be contracted, which is due to the extra symmetry invariant point in the spectrum at −1 [57]. The classification in [57] answers these questions by assigning the three symmetry indices si , si -+ , and sı  to each symmetric quantum walk. Here, si  characterise the ±1-eigenspaces and their values cannot be changed by continuous symmetric deformations. On the other hand, the right symmetry index sı  is a bulk invariant that characterises the evolution asymptotically far on the right half-chain. It is also invariant under continuous deformations, but in contrast to si  also cannot be changed by locally perturbing the system.

Emergence of symmetry protected edge states
An important physical consequence of this topological classification of symmetric quantum walks is the bulkboundary correspondence: whenever two bulks are joined spatially, the absolute value of the difference of their right symmetry indices sı  is a lower bound for the number of symmetry protected edge states which appear near the interface region. These edge states have eigenvalues ±1 and are therefore stable against continuous deformations that respect the symmetries. Since these symmetry protected edge states are the only eigenfunctions observed in the concrete examples below, we occasionally refer to them simply as eigenfunctions.
In order to provide experimental evidence for the emergence of symmetry protected edge states one typically measures the position distribution of a time evolved initial state. This is, however, not sufficient to reveal their topological nature since the emergence of localised states near a local perturbation is a typical phenomenon [30,62]. Instead, in this publication we demonstrate a direct eigenvalue measurement to give evidence that the observed edge states are indeed symmetry protected.
The appearance of symmetry protected edge states does not depend on how the crossover between two symmetric systems is designed. A particularly simple scenario in which such edge states emerge is given by decoupling a walk operator W locally to a walk W W W L R ¢ = Å which has zero transition amplitudes between the right and left half chain of the cut. Such decouplings exist for every walk and every symmetry type and can even be chosen to be a continuous perturbation [57]. Depending only on the right half-chain, the right symmetry

Split-step quantum walks
Emblematic for coined quantum walks is the so-called split-step quantum walk [20]. It is defined on to the right and V ñ | to the left, respectively. The coin operation rotates the internal degree of freedom by an angle θ around the x s -axis. Note that the split-step walk is typically defined with rotations around the σ y -axis. Since these two implementations are unitarily equivalent and in the experiment described below we implement the σ x -rotation, we take (5) with coin (6) as the definition of the splitstep walk. The walk (5) is of symmetry type BDI, i.e. it is chiral, time-reversal and particle-hole symmetric with each symmetry squaring to the identity. Moreover, it exhibits a rich structure of topological phases, and therefore has become the working example in many publications concerning topological effects in quantum walks [9,20,21,61,63]. Figure 1 shows the different topological phases of the split-step quantum walk. The interactive web tool in [64] allows the user to explore how the eigenfunctions and the symmetry indices of the split-step walk change with modifications of the parameters θ 1 and θ 2 as well as for different decouplings.
The quantum walk we implement in the experiment described below is a model related to the split-step walk. It is defined by † denotes the bidirectional conditional shift. Since this walk protocol contains two such shift operations, a walker which is initially localised on even (odd) positions, never leaves the even (odd) sub-lattice. Thus, on each of these sublattices the walk (7) implements effectively a split-step walk with doubled jump length. Since our initial states will always be localised at x=0, we restrict considerations to the even sub-lattice from now on. Rescaling this even sub-lattice by x the effective walk we implement in the experiment is the split-step walk defined in (5).

Chiral symmetry and topological phases
Being of symmetry type BDI the effective split-step walk has chiral symmetry, i.e. there exists a unitary g with This symmetry operator is given by where the 2 q -dependence of g is due to the choice of the local bases: conjugating with C(θ 2 /2) gives C C 2 2 For translation invariant chiral symmetric walks with 2  g = + , the symmetry index sı  is given by the winding number of a certain matrix block of the walk operator in momentum space. Since the derivation of the formula would exceed the discussion here, we only sketch the relevant relations. The complete derivation can be found in [61, proposition 3.9]. Being translation invariant, the split-step quantum walk gets mapped to a continuous matrix valued function W k  ( ) of the quasi momentum k by the discrete Fourier transformation : , . If we now switch to a chiral eigenbasis (e.g. by conjugating with as a complex valued function of kä[−π, π). The corresponding values of the index are shown in the parameter plane in figure 1. Compare also [64] for a visualisation of the winding function (9). Note, that there the order of θ 1 and θ 2 is interchanged and the coin rotations are defined as R exp i y q q s = ( ) ( ), resulting in a clockwise π/2 rotation of the parameter plane in figure 1.

Decoupling
To construct a decoupling for the split-step walk we make use of its special form. Replacing in (5) the local coin C(θ 1 ) at x=0 by a reflective coin, i.e. a coin whose diagonal elements are zero, decouples the resulting walk between x=0 and x=−1. To reproduce this decoupled split-step setting as an effective sublattice walk of the experimentally accessible walk, we need to replace the coin C(θ 1 ) at x=−1 in (7).
The decoupling coins within the parameter regions are the rotations by angles θ=±π/2, for which (6) indeed has only zeros on the diagonal. Whenever a walk in a non-trivial phase, i.e. with non-vanishing sı  is decoupled, exponentially localised eigenstates are predicted at the interface by bulk-boundary correspondence. As remarked above, however, it depends on the specific decoupling whether these eigenfunctions correspond to the eigenvalue +1 or −1. These two possibilities can be distinguished by an additional (walk specific) invariant, In both cases θ 1 is chosen to decouple the walk on the even sub-lattice. We do not need to specify the left half chain here, since we are only interested in edge states of the decoupled walk located to the right. Hence, only the coin configuration for x 0 > determines the phase of the effective walk, and we infer from the phase diagram in figure 1 that in both settings the corresponding symmetry index is W sı 1 , which predicts the emergence of edge states.

Eigenfunctions and eigenvalues
To compute the eigenfunctions of W in both settings, we first note that W and g commute on the ±1eigenspaces of W. Therefore, we can jointly diagonalize W and g on these eigenspaces. The Therefore, by the condition 1 m < | | in our setting the eigenfunction located to the right of the bulk must have chirality χ=−1.
In order to determine a we have to take into account the boundary condition. Choosing the solutions of the eigenvalue equation with χ=−1 leads to the following equations for a: Note that the first equation is actually independent of a whenever a 0 ¹ . Hence, the first equation rules out one of the two possibilities in (18). We get λ=−1 for θ 1 =π/2 (setting A) and λ=1 for θ 1 =−π/2 (setting B). In both settings, however, we must have a=1 for the second equation to be satisfied.

Time-multiplexing setup
The realisation of the two described settings and the direct measurement of the eigenvalues require a stable, flexible experimental platform, as a phase-stable evolution incorporating a dynamic coin operation has to be ensured over a sufficiently large number of steps. Our system of choice relies on a well-established timemultiplexed architecture utilising fibre loops for DTQW [47,48,50]. The dynamic coin operation implemented with a fast-switching electro-optic modulator (EOM) makes it suitable for a wide range of experiments, including the investigation of topological phenomena [22,58]. Previous photonic implementations allowed for accessing topological invariants associated with probability distributions or amplitudes within a certain step of the walk [9,22,[65][66][67][68][69]. However, topological properties can also manifest themselves in the emergence of eigenstates with associated eigenvalues that are revealed by the phase relation between the walker's wavefunctions for two consecutive steps. So far, this phase relation has not been investigated experimentally. By interfering the walker with a reference of fixed phase, we are now able to probe this feature as well.
Our implementation relies on a photonic walker implemented by an attenuated coherent laser pulse with its polarisation representing the internal (coin) degree of freedom. In this way, our system makes use of the equivalence of coherent light and a single quantum particle when propagating in a linear optical network [70]. Figure 3 shows the physical implementation of the quantum walk setup: Our time-multiplexing architecture relies on translating the external (position) degree of freedom of the walker into the time domain by splitting the pulses up spatially, routing them through fibres of different length and subsequently merging the two paths again. This translation in time corresponds to the shift operation according to (2). The ratio of this probabilistic splitting taking place at polarising beam splitters (PBS) is determined by the polarisation state of the walker. This internal degree of freedom is acted upon by coin operations (see (6) ) implemented with static (Soleil-Babinet compensator (SBC)) and dynamic (EOM based on Pockels cell, EOM) polarisation optics. The remarkable characteristic of the EOM is that its switching speed allows for addressing individual positions within the walk. This dynamic coin enables us to implement an alternating coin needed for the split-step scheme (see (10) and (11)) as well as the reflecting and transmitting coin operations indispensable in directing the reference pulse along a certain path (see section 3.3). Note that implementing a protocol according to (7) means that an actual step in the quantum walk requires two roundtrips through the setup. The polarisation degree of freedom can also be accessed in the read-out process, since our detection unit comprising another PBS and 2 avalanche photo-diodes is polarisation-resolving.

Eigenstate distillation
Since the exponentially localised eigenstate cannot be excited directly with an initial state just occupying one position, we need a method to prepare an approximate eigenstate of the system. In order to do so, we make use of the spreading behaviour of the split-step quantum walk. In systems for which the translational invariance is only broken at the edge, the eigenstates are exponentially localised [61]. If we choose an initial state near a boundary and let the system evolve for some time, the components of the state which have no overlap with the localised state will propagate away from the boundary, while the content which overlaps will stay. Consequently, we concentrate our study onto the three positions near the boundary, which is justified due to the exponential decay of the state (see equation (21)). By renormalizing the remaining state, we prepare an approximate eigenfunction. For quantifying how closely we have approached the theoretically expected distribution, we use the similarity which is obtained by summing up the square roots of the products of the theoretical and experimental probabilities for the relevant positions within the step that is examined. It can possibly assume values between 0 (no overlap of intensities) and 1 (perfect overlap of intensities) Figure 4 illustrates how the dynamic coin operation can be harnessed to implement a split-step quantum walk followed by a phase readout via the interference method: by applying a mixing coin at the initial position (marked Ĉ ) the walker is split into a vertical component travelling through the shorter fibre, i.e. being translated to the left in the schematic and a horizontal component running through the longer fibre, which corresponds to a translation to the right. The part going to the left constitutes the reference. In order to prevent it from mixing with the split-step walk taking place on the right as well as from losing intensity to positions that are not on the desired path, the EOM switches to identity (marked T for transmission in figure 4) on the positions where the reference is found. For the phase read-out the reference needs to interfere with the light having undergone the walk, so the travelling direction is inverted by switching a reflection (marked R in figure 4) in the middle of the propagation.

Phase-reference method
The light translated to the right in the initial splitting constitutes the input state of the split-step quantum walk with decoupling. The decoupling is realised by a reflection operation implemented on the decoupling Figure 4. Schematic of our implementation of a split-step quantum walk followed by a phase readout via the interference method. Squares with T denote a transmission operation, while R refers to reflection and Ĉ to mixing coins, for which the exact form can be different depending on setting and position.
position. The split-step walk (see (10) and (11)) incorporates a mixing coin (indicated by Ĉ in figure 4) on even positions and the identity coin, corresponding to a transmission operation, on uneven positions.
The scheme shown in figure 4, (a) brings the reference to interfere with the vertical light ending up at position 0 of the split-step walk. At the position where the light from the walk and reference interfere a mixing coin has to be applied. The ratio of the intensities at the two detection positions (marked by detectors) then allows obtaining information on the phase relation. All polarisations, positions and steps can be accessed analogously, as long as the proper routing of the reference and the pulse under investigation can be ensured. The basic principle remains the same in all runs, but the actual positions of the switchings differ. For this purpose, a flexible dynamic coin operation is indispensable.
We access two regimes with topologically different eigenvalues by applying a coin of either C e for an eigenvalue of −1 (setting A, (10)) or C e EV 1 i x · for an eigenvalue of +1 (setting B, (11)) and T e i 0 for the transmission as well as R e i x 2 = s -p  · for the reflection. All of these coins can be implemented with a combination of static SBC and dynamic EOM operations. The Thus we can deduce the phase difference between the reference and a certain component of the walker from the measured intensities at the read-out positions (marked by a detector symbol in figure 4): Note that the M-parameter given by this formula is sin r w a a -( ) , which is not an injective function. However, it still provides a clear distinction between cases in which we expect an eigenvalue of 1 and cases with an eigenvalue of −1.
Distinguishing these eigenvalues requires monitoring how the M-parameter changes from step to step. For an eigenvalue of −1 (setting A), the walker's phase α w is expected to change by π, while for an eigenvalue of 1 (setting B) it is either 0 or integer multiples of 2π. Since the phase of the reference α r can be assumed to be constant for all numbers of steps, monitoring the step-wise evolution of the M-parameter for the two settings should clearly reveal the different eigenvalues. In the experiment, the eigenvalues will be manifested in the relative intensities of horizontal and vertical light at the read-out positions. Accessing the eigenvalues requires the read-out of both polarisations for the three inner positions. Furthermore, the state of the walker over 3 steps (6, 7 and 8) is monitored. Since each measurement just yields information for a certain position, polarisation and step, obtaining the full information makes 18 individual measurement runs necessary. Observing the differences between settings A and B doubles the number of measurements required, so that eventually 36 data sets have to be taken.

Results
As outlined previously, the experiment aims at accessing the evolution of eigenvalues over three consecutive steps. As we are limited by losses to numbers of physical roundtrips around 22, the actual state of the walker will constitute an approximation of the ideal eigenstate. We quantify the quality of the approximation by calculating the similarity between the experimental intensity and the ideal eigenstate according to (22). Furthermore, the phase-reference method indicates the eigenvalue of the system under investigation.

Evolution of the wavefunction
As a certain proportion of the intensity is outcoupled in each roundtrip, we are able to monitor its temporal evolution. Figure 5 shows the evolution for six steps of a split-step quantum walk and the subsequent read-out of the vertical (a) resp. the horizontal component (b) at position 0. Note that in this nomenclature 0 corresponds to the innermost position of the eigenstate and not to the position at which the light pulse starts (see figure 4).
The plots illustrate how the reference is steered with transmission and reflection operations on the left-hand side (marked by a grey arrow) and is then brought to interference with either the vertical or the horizontal component (green arrow) of the walker evolving on the right-hand side. From the distribution of intensity between the two detection bins the M-parameter can be inferred according to equation (25). During the splitstep quantum walk up to roundtrip 13, the intensity on the right-hand side either concentrates near position 0 or runs out towards the right. In order to quantify the overlap of this intensity near position 0 with the theoretically expected eigenstate, we calculate the similarity according to (22) for the three innermost positions (see figure 6).
While in step 6 the similarity between experiment and the ideal eigenfunction exhibits values of 0.891±0.019 (eigenvalue −1) resp. 0.929±0.029 (eigenvalue +1), in step 8 these values attain 0.979±0.024 (eigenvalue −1) resp. 0.984±0.029 (eigenvalue +1), getting almost as close to the ideal value of 1 as the numerically predicted state. The numerical simulation accounts for the limited number of steps but no other experimental imperfections, it thus quantifies the effects of the finite system size in correspondence to the deviation from the ideal similarity of 1. The difference between the numerically and the experimentally determined similarity is accordingly due to further experimental imperfections such as slightly inhomogeneous losses or imperfect EOM switchings. The high values of the similarity give evidence for the successful outcome of the distillation process. Note that the difference between numerical and experimental values is slightly larger for the scenario with eigenvalue −1, presumably due to the fact that here the EOM needs to switch a larger coin angle and thus the applied voltage needs to be higher. Consequently, experimental imperfections, e.g. due to resonances of the Pockels cell used in the EOM, which are hard to quantify in an error model and subject to ongoing investigation, are increased for the eigenvalue −1 case.

Eigenvalues
Having quantified the overlap of the measured intensities with the ideal eigenstates, the focus now shifts on measuring the eigenvalues of the walk operator. We therefore monitor the evolution of the M-parameter (see (25)) from step 6 to step 8 for a rotation angle 1 4 , for which theory predicts an eigenvalue λ=−1, and 3 4 B q p = , associated in theory with an eigenvalue of +1. This analysis is done for both horizontal and vertical polarisation, which require separate read-out procedures as explained above.  Figure 7 shows the evolution of the M-parameter (see (25)) for horizontal (a) and vertical (b) polarisation at position 0. Setting A with Eigenvalue −1 (orange markers) and setting B with Eigenvalue +1 (blue markers) are clearly distinguishable for both polarisations. However, the results for vertical polarisation exhibit larger error bars as we measure a more unbalanced ratio of the intensity of the walker and the intensity of the reference. This is due to the fact that the theoretical eigenstate as well as its experimental approximation show significantly less intensity in vertical polarisation than in horizontal polarisation, while the reference remains the same in both cases.
The amplitudes of the eigenstates decrease by a factor of (1 2 -) when the position is increased by one (see (21)). Accordingly, the errors get bigger when reading out position 1 (see figure 8). For horizontal polarisation ( figure 8 (a)) the results still reflect the theoretically expected behaviour with larger error bars than for position 0, while for vertical polarisation a quantification of the M-parameter is no longer possible in the eighth step, as the measured intensity of the walker is not significantly above the noise floor.

Error discussion
The eigenstate validation as well as the read-out of the M-parameter require the measurement of intensity distributions. These distributions are subjected to inhomogeneities of the coupling efficiencies and inaccuracies The similarity according to (22) between the ideal eigenstate and the numerically determined state for finite step numbers (blue markers, the same for both eigenvalues) as well as the similarity between the ideal state and the experimentally obtained state for both the regimes with eigenvalue +1 (green markers) and eigenvalue −1 (red markers). Note that all markers correspond to integer step numbers, even though they are slightly shifted on the horizontal axis for better readability. in the angles of the statically and dynamically implemented coins. Assuming errors of the coupling efficiencies of 2% and of the coin angle of 2°, we conduct a Monte-Carlo simulation in which we randomly generate 1000 different settings for these quantities within the assumed error range. For each of these settings we calculate the deviation of the resulting numeric intensity distribution from a reference intensity distribution. This is obtained when running the numerical simulation with the fit parameters allowing for the closest approximation of the experimental results. The error for the individual positions and polarisations is then calculated as the standard deviation of the randomly generated samples from the reference distribution. Eventually, the errors of the similarity and the M-parameter are determined via error propagation from the errors of the intensities, resulting in the error bars visible in figures 6-8.
As discussed in section 3.2, the eigenstates cannot be directly excited, but only be approximated via distillation during the evolution of the walker. The resulting deviations from the ideal eigenstates are quantified via the experimentally obtained similarities which are discussed in section 4.1.

Conclusion
We implemented a split-step quantum walk with decoupling over step numbers ranging from 6 to 8, requiring up to 22 roundtrips in the setup. At the end of the evolution, the states in the experiment approximate the localised eigenstates with similarities of up to 0.979±0.024 resp. 0.984±0.029. In addition, the phasereference method allows for measuring eigenvalues, clearly revealing two regimes with different eigenvalues for position 0. At position 1, lower intensities make the read-out more challenging, but still allow to clearly see the sign flip. The application of the phase-reference method relies on the correct implementation of the decoupling. Our setup does not only allow for decoupling, but also serves to demonstrate that the actual decoupling coin affects the eigenvalue. In our setup the possibilities to read out the internal degree as well as to dynamically apply different coins to different positions thus allow for investigating new aspects of topological quantum walks. As the eigenvalues of approximate eigenstates could not be measured in previous experiments, our experiment significantly extends the range of accessible topological signatures.