1. INTRODUCTION

Entanglement, an intriguing characteristic of quantum systems, plays the critical role in 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 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 [5–10]. For example, the genuine multipartite entanglement of the ground state in the quantum spin model [6,11] can be enhanced by disorder. Another example is the dynamic disorder that can enhance entanglement generation in quantum walks (QW) [8–10].

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 [17–19]. 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 [21–24]. 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 in a deterministic way, the coin-position entanglement [21–23] 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 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] have shown 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 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 arrays of wave guides [31,32]. Unlike transport behaviors, which have been sufficiently studied just by measuring the final probability distribution, the entanglement properties still need 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 [33–35] have been made to improve the scalability. For example, all-fiber-based QWs 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 protocols [38] (in Ref. [39], the authors measured the relative phase, 0 or $\pi$, 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 a large scale with the ability of full wave function reconstruction.

2. THEORETICAL IDEA

The Hilbert space of a QW is $H=H_C\otimes H_P$, where $H_C$ is a two-dimensional Hilbert space spanned by $\{|\uparrow ⟩,|\downarrow ⟩\}$ and $H_P$ is an infinite dimensional Hilbert space spanned by a set of orthogonal vectors $|j⟩(j\in \mathbb{Z})$. 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 $\mathcal{U}(T)\equiv \prod _{t=0}^{T}U(t)$, where $U(t)=S·(C(t)\otimes I_P)$ with $I_P$ is the identity operator in $H_P$ and $C(t)$ is the coin tossing in $H_C$. The shift operator $S=\sum _j|\uparrow ⟩⟨\uparrow |\otimes |j+1⟩⟨j|+|\downarrow ⟩⟨\downarrow |\otimes |j-1⟩⟨j|$ describes the conditional displacement in the 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 $\mathbb{C}$ with a certain probability distribution (in particularly, homogeneous distribution). According to the literature [8,9], the type of dynamic disorder (including type of $\mathbb{C}$) is not important. Without loss of generality, in our experiment, we assumed that $\mathbb{C}$ consists of a Hadamard operator ($H$) and a Fourier operator ($F$), where

The global time evolution operator for a single sample is also unitary in a QW, and the final state $|\mathrm{\Psi }(t)⟩$ after a $t$-step walk remains pure if the initial state is pure. The general form of $|\mathrm{\Psi }(t)⟩$ can be written as $\sum _j[a(j,t)|\uparrow ⟩+b(j,t)|\downarrow ⟩]\otimes |j⟩$, where $a(j,t)$ and $b(j,t)$ are complex numbers with the normalization condition $\sum _j[{|a(j,t)|}^2+{|b(j,t)|}^2]=1$. With the unitarity factor, the coin-position entanglement in a QW can be defined by the von Neumann entropy

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 cannot reach its maximum 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 its maximum regardless of the initial states, as shown in Fig. 1(d).

 

Fig. 1. Illustrations for (a) an ordered QW, where only one kind of coin is employed (red rectangle), and for (b) a dynamically disordered QW, where two kinds of coin are employed (red and green rectangles). The coin-position entanglement as a function of the initial state parameters $\{\theta ,\varphi \}$ for (c) an ordered Hadamard QW and (d) the dynamically disordered QW with the coin tossing $S_C^0$ (defined in the text). The dependency of the coin-position entanglement on the initial state is significantly higher for ordered QW compared to the dynamically disordered one.

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3. EXPERIMENTAL SETUP AND RESULTS

The experimental setup is shown in Fig. 2 and a more detailed description is given in Supplement 1. Single photons generated from spontaneous parametric downconversion (SPDC) are adopted as the herald walker. These kinds of coin states are initialized by sending them through the polarizer PBS1-HWP1-QWP1 (see Fig. 2). The state $|\uparrow ⟩(|\downarrow ⟩)$ 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 half-wave plate (HWP, with its optical axis oriented at $\pi /8$) and quarter-wave plate (QWP, with its optical axis oriented at $-\pi /4$), respectively. The reduced density matrix ${\rho }_C$ in Eq. (2) is equal to $\sum _j{p}_j{\rho }_j$. Equation $p_j={|a(j)|}^2+{|b(j)|}^2$ is the probability in site $j$, and ${\rho }_j$ is the local density matrix in site $j$. In the experiment, ${\rho }_j$ 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 upconversion single-photon detectors.

 

Fig. 2. Diagram of the experimental setup (additional details are given in Supplement 1). The system contains four parts: (1) Second-harmonic generation in BBO1 to obtain the ultraviolet pulse; (2) generation of heralded single photons by adopting the beam-like type SPDC in BBO2; (3) time-multiplexing quantum walks realized by birefringent crystals sketched by the inset at the bottom right corner; (4) ultra-fast detection of the arrival time of single photons with an upconversion single-photon detector. BBO, $\beta {-\mathrm{BaB}}_2{\mathrm{O}}_4$; L, lens; DM, dichroic mirror; PBS, polarization-dependent beam splitter; HWP, half-wave plate; QWP, quarter-wave plate; R, reflector; FC, fiber collimator; PMT, photomultiplier tubes; SPAD, single-photon avalanche diode.

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The initial state in our experiments is located at the original site ($|x=0⟩$), and the general state of the coin is $|\mathrm{\Psi }(0)⟩=[\cos (\theta /2)|\uparrow ⟩+e^{i\varphi }\sin (\theta /2)|\downarrow ⟩]\otimes |0⟩$, where $\theta \in [0,\pi ]$ and $\varphi \in [0,2\pi )$. 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 states: $\{\theta ,\varphi \}=\{51°,0°\}$, $\{51°,90°\}$, and $\{51°,180°\}$. 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: $S_E=0.739$ for $\varphi =0°$, $S_E=0.867$ for $\varphi =90°$, and $S_E=0.977$ for $\varphi =180°$ [dashed guided lines in Fig. 3(a)]. In our experiment, the entanglement almost approached the theoretical asymptotic vale for $\varphi =90°$ and $\varphi =180°$. At $\varphi =0°$, the experimental result was $0.665\pm 0.027$ 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 $F({\rho }_C^{\exp },{\rho }_C^{\mathrm{th}})=\mathrm{Tr}{\left[\sqrt{\sqrt{{\rho }_C^{\exp }}{\rho }_C^{\mathrm{th}}\sqrt{{\rho }_C^{\exp }}}\right]}^2$ with ${\rho }^{\exp (\mathrm{th})}$ representing the experimentally measured (theoretically predicted) density matrix, are larger than $0.986\pm 0.001$ for each initial state and step.

 

Fig. 3. (a) Coin-position entanglement measured (dots) for different initial states in an ordered Hadamard QW. The lines (solid red, dashed blue, and dotted yellow) are the theoretical predictions of the von Neumann entropies. The entanglement generated in an ordered QW shows a significant dependency on the initial state. Note that the entanglement involves periodical oscillation around an asymptotic value (guided by the dashed black horizontal line) in the limit of infinite steps with an amplitude decay. We have not considered the case $\{51°,270°\}$, which is the same as $\{51°,90°\}$. (b) Coin-position entanglement measured (dots) for different initial states in the dynamically disordered scenario. The lines (solid red, solid blue, solid yellow, and solid green) are the theoretical predictions. The von Neumann entropies are about 0.98 at 20 steps for all initial state clusters, which significantly differ from the ordered case, where they converge to different asymptotic values for different initial states. The periodical oscillation of entanglement with time degenerates. Only statistical errors are considered with total counts of 24,000 in 400 s.

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Fig. 4. Measured trend (a) of second moment up to 20 steps (dots) for different initial states in ordered and dynamically disordered QWs. Ballistic transport behavior is observed in an ordered QW (solid red line, fitted as $0.29t^2$). In a dynamically disordered QW, a sub-ballistic transport behavior is observed (solid blue line, fitted as $0.6t^{1.54}$), and the dotted red line gives the theoretical predictions. The deep yellow line represents the typical diffuse transport behavior in classical random walks starting at the origin. The error bars are smaller than the point size with only consideration of statistical errors. (b) and (c) give the probability distributions after an 18-step QW for an initial state with $\theta =51°$ and $\varphi =0°$. It is seen clearly that the spreading velocity in (b) an ordered QW is significantly faster than (c) a dynamically disordered one.

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We further demonstrated that the dynamic disorder can enhance the coin-position entanglement. To achieve this, we first chose the initial coin state as $\{\theta ,\varphi \}=\{51°,0°\}$, where the coin-position entanglement after a 20-step Hadamard QW is minimal [see Figs. 1(c) and 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 $S_C^0=FFHFHFHHFFFFFHFHHHHH$ (operated on the coin from left to right). Actually, sequence $S_C^0$ is one of the optimal sequences to enhance the entanglement for the initial state $\{51°,0°\}$ after 20 steps. $S_C^0$ can also enhance the entanglement for any of the initial states [the theoretical enhanced entanglement with sequence $S_C^0$ for any initial state can be found in Fig. 1(d)]. We checked the enhancement with three other initial states: $\{51°,90°\}$, $\{51°,180°\}$, and $\{51°,270°\}$. The experimental results are shown in Fig. 3(b), 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 $0.968\pm 0.003$ 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 steps is infinite. However, in our experiment, the total number of steps 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 $\{51°,0°\}$ on the disordered sequence $S_C$ is shown in Fig. 5. Based on this dependence, the sequences can be divided into several different types (indicated in different colors in Fig. 5). We experimentally checked the enhancement of the entanglement after a 20-step QW with different types of sequences (two sequences in each type), and the experimental results are shown in Fig. 5. The disordered QW became more powerful than the ordered ones in terms of ability to generate entanglement. If we have no information about the sequence, the entanglement should be obtained by averaging all the sequences. Based on our experimental results, the average entanglement $⟨S_E^{\exp }⟩$, defined as $⟨S_E^{\exp }⟩=\sum _{i=1}^{12}{S}_E^i·\frac{P_i}{2}$, with $P_i$ being the rate of entanglement interval to which $S_E^i$ belongs, is about $0.923\pm 0.006$, which is a significant improvement for the ordered QW.

 

Fig. 5. Coin-position entanglement measured (opaque bars) for different sequences (forms are given in Supplement 1) in a dynamically disordered QW with the theoretical predictions shown by transparent bars. The inset shows the rate of sequences generating different $S_E\in \{0,1\}$ at 20 steps with the initial state $\{51°,0°\}$, whose entanglement is lowest (shown with the solid black line) in an ordered Hadamard QW. Note that almost 73% of sequences can generate $S_E>0.9$. In each entanglement interval (different color), we randomly choose two sequences. The theoretical average entanglement $⟨S_E⟩=0.924$ (guided by the black-dashed horizontal line) was calculated by averaging all sequences. Only statistical errors are considered.

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4. CONCLUSIONS

In conclusion, we reported the first, to our knowledge, experiment to study coin-position entanglement generation in discrete-time QWs beyond the usual transport behaviors. We observed the initial state dependency of entanglement generation in ordered QWs. This entanglement involves periodic oscillations with the amplitude decay around an asymptotic value, which is dependent on the initial state. More importantly, we found that the coin-position entanglement can be significantly improved by the dynamic disorder for any initial state. Besides, we showed the sub-ballistic transport behavior in dynamically disordered QWs. 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 Kolmogorov complexity, the complexity of a binary sequence can be explicitly defined via Lempel–Ziv complexity $C_{\mathrm{LZ}}$ [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 Supplement 1). 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 the random sequence also increased, and the entanglement power of the sequence increased as a result as well. However, the complexity of the periodic sequence will eventually be saturated, and the entanglement power will not increase. When the disordered sequence approaches 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.