Dynamic-Disorder-Induced Enhancement of Entanglement in Photonic Quantum Walks

Entanglement generation in discrete time quantum walks is deemed to be another key property beyond the transport behaviors. The latter has been widely used in investigating the localization or topology in quantum walks. However, there are few experiments involving the former for the challenges in full reconstruction of the final wave function. Here, we report an experiment demonstrating the enhancement of the entanglement in quantum walks using dynamic disorder. Through reconstructing the local spinor state for each site, von Neumann entropy can be obtained and used to quantify the coin-position entanglement. We find that the enhanced entanglement in the dynamically disordered quantum walks is independent of the initial state, which is different from the entanglement generation in the Hadamard quantum walks. Our results are inspirational for achieving quantum computing based on quantum walks.


INTRODUCTION
Entanglement, an intriguing character of quantum systems, plays the critical role in the quantum information processing [1], such as quantum key distribution [2] and quantum computing [3]. However, quantum entanglement is so fragile that it can easily be destroyed by the noises and the environment. Systems become disordered [4] due to the inhomogeneous environmental conditions and other parameters in the system, which are impossible to control experimentally. Intuitively, one may expect that such disorder would reduce the entanglement of a given system, which is indeed true for a large variety of systems. In fact for some systems, disorder can enhance the entanglement [5][6][7][8][9][10]. For example, the genuine multipartite entanglement of ground state in the quantum spin model [6,11] can be enhanced by disorder. Another example is the dynamic disorder that can enhance the entanglement generation in quantum walks (QW) [8][9][10].
QW on lattices and graphs [12] is a quantum generalization of the classical random walks [13], which may play as universal quantum computers [14,15], quantum simulators [16] and platforms to investigate topological phases [17][18][19]. The behavior of QW, especially the ballistic behavior [20] of the transport properties, is dramatically different from its classical counterpart due to the superposition principle and has been extensively investigated.
Besides the probability distribution, the entanglement properties of QW have also been theoretically studied [21][22][23][24]. It is the genuine quantum feature in QW since there is no classical counterpart. The entanglement (coin-position entanglement) here is different from its original definition for multiple parties. It is actually entanglement between two modes sharing a single particle [25], which has been widely used in some crucial quantum information protocols [26]. In the ordered QW where the quantum coin is fixed during the whole evolution process or changes with a deterministic way, the coin-position entanglement [21][22][23] is highly dependent on the initial state and usually never achieves its maximal value.
Entanglement in QW will be affected by dynamic disorder, in which the quantum coin is independent of the site j and at the same time is randomly chosen for each step. Or it can be affected by the static disorder, in which the quantum coin is fixed for all the time and at the same time is randomly chosen for each site j. Entanglement in QW can also be affected by the combination of both, dynamic and static disorder. Theoretical investigations [8,9] showed that the entanglement is reduced by the static disorder while the dynamic disorder induces enhancement of the entanglement independent of the initial states and even with the appearance of the static disorder.
Linear optics is a good platform for implementing QW and thus many technologies have been developed: spatial displacers [18], orbital angular momentum (OAM) [27], time multiplexing [28,29], integrated optical circuits [30] and array of wave guides [31,32]. Unlike transport behaviors, which have been sufficiently studied just by measuring the final probability distribution, the entanglement properties are still needed to be studied in both ordered and disordered QW. The experimental challenges are twofold: how to reach large-scale QW (the disorder-induced entanglement enhancement can only be demonstrated in the asymptotic limit) and how to reconstruct the wave function in both the coin and position space [23].
Different efforts [33][34][35] have been made to improve the scalability. For example, all fiberbased QW have reached 62 steps with high fidelity and low loss [36]. Only recently, the final wave function in a one-dimensional QW of a single cesium (Cs) atom has been obtained by the local quantum state tomography [37], and complete reconstruction of wave function was achieved in OAM [27] and time multiplexing protocol [38] (in Ref. [39] the authors measured the relative phase, 0 or π, between the neighboring sites). In this paper,we report an experiment for demonstrating the enhancement of entanglement generation in QW by the dynamic disorder. This experiment is based on our recently developed novel compact platform for genuine single photon QW in large scale with the ability of full wave function reconstruction.

THEORETICAL IDEA
The Hilbert space of a QW is H = H C ⊗ H P , where H C is a two-dimensional Hilbert space spanned by {|↑ , |↓ } and H P is an infinite dimensional Hilbert space spanned by a set of orthogonal vectors |j (j ∈ Z). A QW is given by a sequence of coin tossing followed by a conditional shift according to the coin state. The time evolution operator for a QW from t = 0 to T can be represented by a multi-step unitary operator U(T ) ≡ T t=0 U (t), where U (t) = S · (C(t) ⊗ I P ) with I P is the identity operator in H P and C(t) is the coin tossing in H C . The shift operator S = j |↑ ↑| ⊗ |j + 1 j| + |↓ ↓| ⊗ |j − 1 j| describes the conditional displacement in lattice, which generates the coin-position entanglement.
Generally, the coin tossing C(t) in a QW is time and position dependent. In this paper, C(t) is assumed to be site independent since we only consider the effect of the dynamic disorder. In a QW with dynamic disorder, the coin tossing C(t) is time dependent: for each step, it is randomly chosen from a set C with certain probability distribution (in particularly, homogeneous distribution). According to the literature [8,9], the type of dynamic disorder (including type of C) is not important. Without loss of generality, in our experiment, we (1) We also considered an ordered QW, in which the coin tossing is time independent, and we chose Hadamard gate all the time for comparison.
The global time evolution operator for a single sample is also unitary in a QW and the final state |Ψ(t) after a t-step walk remains pure if the initial state is pure. The general form of |Ψ(t) can be written as j [a(j, t)|↑ +b(j, t)|↓ ]⊗|j , where a(j, t) and b(j, t) are complex numbers with the normalization condition j [|a(j, t)| 2 + |b(j, t)| 2 ] = 1. With the unitarity factor, the coin-position entanglement in a QW can be defined by the von Neumann entropy where ρ C (t) = Tr P [ρ(t)] is the reduced density matrix of the coin with ρ(t) = |Ψ(t) Ψ(t)| and Tr P is taking the trace over position.
For a fixed initial state, with the increase in time, regardless of the ordered or disordered QW, the coin-position entanglement will be asymptotic to a stable value. Generally, this asymptotic value in an ordered QW can not reach maximal and is strongly dependent on the initial state. The entanglement after 20 steps with different initial states is shown in Fig. 1.(c). For a dynamically disordered QW, this asymptotical value is found [8,9] to reach maximal regardless of the initial states as shown in Fig. 1.(d).

EXPERIMENTAL SETUP AND RESULTS
The experimental setup is shown in Fig. 2 and the more detail description is given in supplementary material. Single photons generated from spontaneous parametric down conversion(SPDC) are adopted as the herald walker. These kind of coin states are initialized by sending them through the polarizer PBS1-HWP1-QWP1(see Fig. 2). The state |↑ (|↓ ) corresponds to a single horizontally polarized photon |H (|V ), which stands for the horizontal (vertical) polarization of the photon (walker). The QW device is composed of wave plates (for realizing coin tossing) and calcite crystals (for implementing conditional shift), and each step contains each one of them. H and F coin tossing was implemented by single HWP (with its optical axis oriented at π/8) and QWP (with its optical axis oriented at −π/4) respectively. The reduced density matrix ρ C in Eq. 2 is equal to j p j ρ j . Equation is the probability in site j, ρ j is the local density matrix in site j. In experiment, ρ j is obtained through local quantum state tomography (realized by the polarization analyzer QWP2-HWP2-PBS2 in Fig. 2). Meanwhile p j is directly given by the projection probabilities in the first two bases, |H H| and |V V |. The lattice is composed of arriving time of signal photons and the time interval is around 5 ps, which is challenging to detect with available commercial detectors. Therefore we constructed the up-conversion single photon detectors.
The initial state in our experiments is located at the original site (|x = 0 ) and the general state of the coin is and φ ∈ [0, 2π). In our experiment, the QW step number is limited to 20. First, we experimentally determined the key characteristics of the coin-position entanglement generated in the standard Hadamard QW (ordered QW). We chose three different initial The entanglement dynamics were experimentally obtained for each initial state ( Fig. 3(a)). Theoretically, the entanglement for a given initial state will approach an asymptotic value after several oscillations. In addition, the asymptotic value is strongly dependent on the initial state: Fig. 3(a)).
In our experiment, the entanglement almost approached the theoretical asymptotic vale for φ = 90 • and φ = 180 • . At φ = 0 • , the experimental result was 0.665 ± 0.027 after a 20-step QW and the entanglement was still oscillating. Besides, the ballistic transport behavior of ordered QW is shown in Fig. 4(a). During the experiment, the fidelities, defined as representing the experimentally measured (theoretically predicted) density matrix, are larger than 0.986 ± 0.001 for each initial state and step.
We further demonstrated that the dynamic disorder can enhance the coin-position entanglement. To achieve this, we first chose the initial coin state as {θ, φ} = {51 • , 0 • }, where the coin-position entanglement after a 20-step Hadamard QW is minimal (see Fig. 1(c) and Fig. 3(a)). We showed that the coin-position entanglement can be dramatically enhanced to about S E = 0.98 by the dynamically disordered coin tossing sequence S 0 C = F F HF HF HHF F F F F HF HHHHH (operated on the coin from left to right). Actually, the sequence S 0 C is one of the optimal sequences to enhance the entanglement for the initial state {51 • , 0 • } after 20 steps. S 0 C can also enhance the entanglement for any of the initial states (the theoretical enhanced entanglement with the sequence S 0 C for any initial state can be found in Fig. 1(d)). We checked the enhancement with three other initial states: The experimental results are shown in Fig. 3 which clearly shows that all the entanglements are improved and approach the asymptotic value faster than in the ordered scenario. More importantly, the entanglement approached the same value (about 0.98), which is almost equal to the theoretical maximal value of 1 regardless of the initial state. The fidelities are larger than 0.968±0.003 for each initial state and step in this scenario. Actually, the wave-function transport in a lattice will decelerate and show a sub-ballistic behavior in the presence of disorder [40]. The dynamic disorder can lead to a sub-ballistic transport behavior in a QW, and the transport trend depends on the choice of the two coin operations and the sequences [8,41]. In Fig. 4(a), we show the sub-ballistic transport behavior in a dynamically disordered QW. It is obvious that its spreading velocity is faster than a typical diffuse transport behavior in a classical random walk but slower than a ballistic transport behavior in an ordered QW.
Theoretically, the enhancement of coin-position entanglement is not dependent on the specific form of the coin tossing sequence when randomness is introduced, and the number of the steps is infinite. However, in our experiment, the total amount of step was limited to 20. In this case, the enhancement of entanglement is dependent on the sequence S C . The dependence of the final entanglement after a 20-step disordered QW with the initial state  , defined as S exp E = 12 i=1 S i E · P i 2 with P i being the rate of entanglement interval, to which S i E belongs, is about 0.923 ± 0.006, which is a significant improvement for the ordered QW.

CONCLUSIONS
In conclusion, we reported the first experiment to study the coin-position entanglement generation in discrete time quantum walks beyond the usual transport behaviors. We observed the initial state dependency of entanglement generation in ordered quantum walks.
This entanglement involves periodic oscillations with the amplitude decay around an asymptotic value, which is dependent on the initial state. More importantly, we found the coinposition entanglement can be significantly improved by the dynamic disorder for any initial state. Besides, we showed the sub-ballistic transport behavior in dynamically disordered quantum walks. Based on our experimental results, it seems that the entanglement power of the coin tossing sequence S C positively correlates with the complexity of the sequence.
In the spirit of the Kolmogorov complexity, the complexity of a binary sequence can be explicitly defined via Lempel-Ziv complexity C LZ [42,43]. The complexity of the twelve sequences (from 1 to 12) in the Fig. 5 are 3, 3, 5, 8, 7, 7, 6, 6, 6, 7, 7 and 6, respectively (details given in the supplementary). The complexity measure is qualitative, and our results qualitatively showed that the power of entanglement generation increased with the complexity of the sequence. When the sequence length increased, the complexity of random sequence also increased and the entanglement power of the sequence increased as a result as well. However, the complexity of periodic sequence will be eventually saturated, and the entanglement power will not increase. When the disordered sequence approached infinity, the complexity will be infinite and the entanglement power of all the disordered sequences will be the same and will be a maximal entanglement generator. Our experiment applies a way to explore the entanglement in quantum information.    IG. 6. Plot of the similarity for each step of QW. The similarity is defined as S = x P exp (x)P th (x), where P exp stands for the experimental probability distribution and P th is the corresponding theoretical prediction. The results for ordered QW are shown in the top panel and the results for dynamically disordered QW are presented in the bottom panel. The inset in each panel shows the coincidence counts after a 20-step walk for a initial state with θ = 51 • and φ = 0 • . In the ordered scenarios, the similarity becomes degenerate gradually as the number of steps increases, mainly for the decoherence. In the disordered scenarios, there are more occupations near the origin (in the middle of time delays), where the interference is more complicated, results the faster degeneration of the similarity compared to the ordered cases. Only statistical errors are considered with total counts 24 thousand in 4 hundred seconds. and one piece of calcite crystal. In the experiment, we have adopted 20 such sets, with only 1/4 of them shown in the figure. The initial state is prepared by an apparatus composed of a polarized beam splitter (PBS1), a half wave plate (HWP1) and a quartz wave plate (QWP1). The residual pump in the frequency-double process is adopted as the pump in the following frequency up conversion single photon detection with the retroreflector R1 for temporal matching. After the quantum walks is finished, the signal photons are collected into a short single mode fibre (10 cm long) by a fibre collimator (FC1) and then guided to the polarization analyzer composed of QWP2, HWP2 and PBS2 successively. Finally, the arriving time of signal photons is measured by scanning the pump laser and detecting the up conversion signals with a photomultiplier tubes (PMT). For reducing the scattering noise, BBO3 is cut for noncollinear up conversion and a spectrum filter based on a 4F system is constructed, where a prism is adopted for introducing the dispersion, a knife edge is used to block the long waves and the signal is reflected to the PMT with a pickup mirror.

Lempel-Ziv complexity
Lempel-Ziv (LZ) complexity can be introduced to estimate the randomness of finite sequences, in the spirit of the Kolmogorov complexity. The LZ complexity measure counts the number of distinct substrings (patterns) in a sequence when scanned from left to right and then parsed. Note that we only consider binary sources throughout this paper. The algorithm is as follows [44]: 1. Let S C = s 1 s 2 · · · s n denote a finite 0-1 symbolic sequence; S C (i, j) denotes a substring of S C that starts at position i and ends at position j, that is, when i j, S C (i, j) = s i s i+1 · · · s j and when i j, S C (i, j) = {} (null set); V (S C ) denotes the vocabulary of a sequence S C . It is the set of all substrings S C (i, j) of S C , (i.e., S C (i, j) for i = 1, 2 · · · n; i j). For example, let S C = 010, we then have V (S C ) = {0, 1, 01, 10, 010}.
2. The parsing procedure needs to scan the sequence S C from left to right. If S C (i, j) belongs to V (S C (1, j−1)), then S C (i, j) and V (S C (1, j−1)) is renewed to be S C (i, j+1) and V (S C (1, j)), respectively. Repeat the previous process until the renewed S C (i, j) does not belong to the renewed V (S C (1, j − 1)), then place a dot after the renewed S C (i, j) to indicate the end of a new sequence. Update S C (i, j) and V (S C (1, j − 1)) to S C (j + 1, j + 1)(the single symbol in the j + 1 position) and V (S C (1, j)), respectively, and the step 2 continues.
3. This parsing operation begins with S C (1, 1) and continues until j = n, where n is the length of the symbolic sequence S C .