Demonstration of a photonic router via quantum walks

As illustrated in figure 1(a), a standard model of a one-dimensional QW consists of a walker carrying a coin which is flipped before each step [71–74]. The time- and position-dependent coin rotation for the tth step $\sum_x\vert x\rangle\langle x\vert\otimes C_{x,t}$ (here $C_{x,t}\in SU(2)$ with the coin-state basis $\{\vert 0\rangle,\vert 1\rangle\}$) is applied, followed by a conditional position shift $S = \sum_x\vert x-1\rangle\langle x\vert\otimes\vert 0\rangle\langle 0\vert+\vert x+1\rangle\langle x\vert\otimes\vert 1\rangle\langle 1\vert$ due to the outcome of the coin rotation. The unitary operation for the tth step is $U_t = S\sum_x\vert x\rangle\langle x\vert\otimes C_{x,t}$.

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The quantum data are encoded in the initial coin state $\alpha\vert 0\rangle+\beta\vert 1\rangle$ ($|\alpha|^2+|\beta|^2 = 1$) and sent to an arbitrary position on a one-dimensional lattice on demand. Suppose the walker starts from the original position $\vert x = 0\rangle$. The formalized description of the semi-quantum routing task can be written as

$$\begin{align} \vert \psi(t)\rangle=\prod_t U_t\vert \psi(0)\rangle=\sum_x A_x\vert x\rangle\otimes\left(\alpha\vert 0\rangle+\beta\vert 1\rangle\right). \end{align} \tag{ 1 }$$

That is, after t steps, the signal information encoded in the coin state can be routed from the input mode x = 0 to an arbitrary coherent superposition of the output modes $x = \pm (t-2),\pm(t-4),\pm(t-6),\ldots,\pm1$ (or 0) for an odd number of steps t (or even t) in a deterministic way.

We can divide the routing task into three stages as illustrated in figure 1(a). First, we apply the coin rotation $C_{0,1} = \sigma_x$ to the initial coin state and it is flipped into $\beta\vert 0\rangle+\alpha\vert 1\rangle$. Second, for the ith step ($2\leqslant i\lt t$), the position-dependent coin rotation

$$\begin{align} C_{x=\pm(i-1),i}= \begin{pmatrix} \cos\theta_{\pm(i-1),i} & \sin\theta_{\pm(i-1),i}\\ \sin\theta_{\pm(i-1),i} & -\cos\theta_{\pm(i-1),i} \end{pmatrix} \end{align} \tag{ 2 }$$

is applied to the coin for $x = \pm(i-1)$ and $\unicode{x1D7D9}$ is applied for other cases, and θ is the parameter of coin rotation. Third, for the last step t, the coin rotation $C_{\pm(t-1),t} = \sigma_x$ is applied to the coin for $x = \pm(t-1)$ and $\unicode{x1D7D9}$ is applied for other cases. The parameters A_x satisfy the following relations

$$\begin{align} &A_{-(i-2)}=\sin\theta_{1,2}=(-1)^i\prod_{j=1}^{i-2}\cos\theta_{-j,j+1},\nonumber\\ &A_{-(i-4)}=\cos\theta_{1,2}\sin\theta_{2,3}\nonumber\\ &=(-1)^{i-1}\prod_{j=1}^{i-3}\cos\theta_{-j,j+1}\sin\theta_{-(i-2),i-1},\nonumber\\ &\cdots\nonumber\\ &A_{i-4}=\prod_{j=1}^{i-3}\cos\theta_{j,j+1}\sin\theta_{i-2,i-1}=-\cos\theta_{-1,2}\sin\theta_{-2,3},\nonumber\\ &A_{i-2}=\prod_{j=1}^{i-2}\cos\theta_{j,j+1}=\sin\theta_{-1,2}. \end{align} \tag{ 3 }$$

Next, we extend the algorithm to that for an entanglement router based on a QW in figure 1(b), which consists of two walkers and two coins [75, 76]. The routing task in each direction is the same as the one-dimensional routing task in equation (1). The basis state is of the form $\vert x,y,c_x,c_y\rangle$ describing the position (x, y) of the walkers and the corresponding coin states with $c_{x,y}\in\{0,1\}$. The time evolution for each step is then

$$\begin{align} U_t\otimes U_t=S_{xy}\sum_{x,y}\vert x,y\rangle\langle x,y\vert\otimes C_{x,t}\otimes C_{y,t}, \end{align} \tag{ 4 }$$

where $S_{xy} = S_{x}\otimes S_{y} = \sum_{x,y}(\vert x-1,y-1\rangle\langle x,y\vert\otimes\vert 00\rangle\langle 00\vert+\vert x-1,y+1\rangle\langle x,y\vert \otimes\vert 01\rangle\langle 01\vert+\vert x+1,y-1\rangle$ $\langle x,y\vert\otimes\vert 10\rangle\langle 10\vert+\vert x+1,y+1\rangle\langle x,y\vert\otimes\vert 11\rangle\langle 11\vert)$, and the choices of the coin operators $C_{x,t}$ and $C_{y,t}$ are same as those for the one-dimensional QW in equation (2). The quantum data are encoded in the initial coin state $\vert \psi\rangle_\textrm c = \alpha\vert 01\rangle+\beta\vert 10\rangle$ and sent to the position (x, y) on demand, deterministically and coherently. The entanglement routing task can be written as

$$\begin{align} \prod_t U_t\otimes U_t\vert 0,0\rangle\otimes\vert \psi\rangle_\textrm c =\sum_{x,y} A_{x}A_{y}\vert x,y\rangle\otimes\vert \psi\rangle_\textrm c, \end{align} \tag{ 5 }$$

where $x,y\in\{-(t-2),-(t-4),\ldots,t-4,t-2\}$ after the tth step in the evolution. The parameters A_x and A_y satisfy the relations in equation (3). The user informs two walkers of the demand of A_x and A_y according to equation (5), and then two walkers can precisely adjust the coin parameters θ according to equation (3). That is, after t steps of a two-dimensional QW, the signal information encoded in the entangled state $\vert \psi\rangle_\textrm c$ is routed from the position $(0,0)$ to (x, y) or a superposition of all output modes, where $x,y = \pm (t-2),$ $\pm(t-4),\pm(t-6),\ldots,\pm1$ (or 0) for an odd number of steps t (or even t).

In this algorithm, all the five standards of building a successful quantum router are met. An arbitrary single-qubit (two-qubit) state can be routed from an input mode to an arbitrary output mode via QWs with single walker (two walkers). The signal state is not changed under routing and the signal information encoded in the coin state is kept undisturbed. During the routing process, the signal information is routed to a coherent superposition of the output modes on demand [37, 77], which is completely different from the task of quantum state transfer, transferring a quantum state from one input mode to one output mode. No postselection is required, nor is ancilla. Thus, our semi-quantum router works in a deterministic way.

The experimental setup for realization of a deterministic photonic router is illustrated in figure 2, consists of an entangled photon source, two cascaded interferometric networks and measurement devices. For generating polarization-entangled photons, a type-II periodically poled potassium titanyl phosphate (PPKTP) crystal in a Sagnac interferometer configuration is used, which is pumped by a continuous wave (CW) diode laser operating at 405 nm. Two interference filters with a bandwidth of 3 nm are used to filter the photons. The state parameter is controlled simply by the polarization of the pumping laser which is tuned by an ultraviolet half-wave plate (HWP). The measured pump power at the input face of the PPKTP crystal is 2.71 mW and the coincidence count rate of 14 000 pair per second is achieved. The coin states are represented by the horizontal $\vert H\rangle$ and vertical $\vert V\rangle$ polarization states of the photon pairs, and the walker states are encoded in their spatial modes. The initial coin state is prepared in one of the Bell states $\vert \Psi^+\rangle = (\vert HV\rangle+\vert VH\rangle)/\sqrt{2}$ with fidelity $0.941\pm0.007$.

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Each photon of an entangled photon pair is steered into the optical modes of a cascaded interferometric network formed by a series of birefringent calcite beam displacers (BDs) and HWPs. The conditional position shift S_x (S_y ) is implemented by a BD, whose axis is cut so that vertically polarized photons are directly transmitted, and horizontally polarized ones move up a lateral displacement into a neighboring mode and interfere with the vertically polarized photons in the same mode. The site-dependent coin rotations $C_{x,t}$ or $C_{y,t}$ for the tth step can be realized via HWPs with certain setting angles placed in mode x or y. The angles of the first HWP (H $^{x(y)}_0$) is set to 45^∘ to realize σ_x and the angles of the other HWPs (H $^{x(y)}_{\pm (t-1)}$) are set to $\frac{1}{2}\theta^{x(y)}_{\pm(t-1),t}$ to realize $C_{x = \pm(t-1),t}$ ($C_{y = \pm(t-1),t}$) in equation (2) (see supplemental materials for details). Output photons are detected using avalanche photo-diodes (APDs) whose coincidence signals, monitored using commercially available counting logic, are used to collect two single-photon events. The walker position probabilities are obtained by normalizing the coincidence counts on each mode with respect to the total count for each respective step. A tomographic reconstruction of the coin state is performed to demonstrate a perfect router.

In our experiment, we realize six-step QWs with two walkers. The first task is to route the quantum information encoded in the entangled coin state coherently to total 25 output modes. The final state of the QW evolves to a product state of an equal superposition state of the walkers and the unchanged coin state, i.e. $\frac{1}{5}\sum_{x,y = \pm 4, \pm2, 0}\vert x,y\rangle\otimes\vert \Psi^+\rangle$. For each output mode, we perform a tomographic reconstruction of the coin state to demonstrate a perfect router. The resultant fidelities [78]

$$\begin{align} F=\left(\text{Tr}\sqrt{\sqrt{\rho_f}\rho_{\textrm {in}}\sqrt{\rho_f}}\right)^2 \end{align} \tag{ 6 }$$

between the reduced coin state ρ_f and the initial coin state ρ_in are shown in figure 3(a), ranging from $0.788\pm0.009$ to $0.878\pm0.006$, which indicate an efficient demonstration of a photonic router. The position distribution of the QW after six steps is shown in figure 3(b). We obtain the experimental results of the probabilities ranging from $0.035\pm0.001$ to $0.047\pm0.001$, which agree well with the theoretical prediction 0.04. In figures 3(c) and (d), real and imaginary parts of the experimentally reconstructed density matrix of the reduced coin state at the output mode $(0,-4)$, which merely indicate that the final coin state has been routed from the input mode $(0,0)$ to the output mode $(0,-4)$ with the coin state well preserved during the process.

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The second task is to route the signal information from the input mode $(0,0)$ coherently to two output modes A $(0,0)$ and B $(0,2)$ with equal probabilities. Thus, the final state of the QW is $\frac{1}{\sqrt{2}}(\vert 0,0\rangle+\vert 0,2\rangle)\otimes\vert \Psi^+\rangle$. For each mode, we reconstruct the reduced coin state via 2-qubit quantum state tomography. We also perform a 3-qubit tomographic measurement to reconstruct the final state of walker-coin $\rho = \frac{1}{2}(\vert A\rangle+\vert B\rangle)(\langle A\vert+\langle B\vert)\otimes \vert \Psi^+\rangle\langle \Psi^+\vert$, in which we define $\vert A\rangle = \vert 0,0\rangle$ and $\vert B\rangle = \vert 0,2\rangle$ as a qubit. The experimentally reconstructed density matrices of the reduced coin state at either of two output modes are shown in figure 4, so is that of the final state of walker-coin. By comparing the experimentally reconstructed state of the walker-coin and its theoretical prediction, the fidelity is $0.907\pm0.002$. By tracing out the coin state, the fidelity of the position-qubit state is as high as $0.989\pm 0.001$, which confirms coherence between the two output modes. The fidelities of the experimentally reconstructed coin states compared to the initial coin state ρ_in are $0.929\pm0.003$ and $0.926\pm0.003$, respectively.

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A photonic router is described by a completely positive map which routes an arbitrary initial state to the corresponding final state. In quantum process tomography, a fixed set of basis operators is chosen so that the map is identified with a process matrix χ. Here we take the progress of the third task in which the signal information is routed from the input mode $(0,0)$ to the output mode $(0,2)$ as an example and experimentally reconstruct χ through the maximum likelihood technique. We choose the basis operators $O_1\otimes O_2$ ($O_{1,2} = \unicode{x1D7D9}, \sigma_x,\sigma_y,\sigma_z$), which require measurements on the evolved states of the sixteen different initial states $\{\vert HH\rangle,\vert HV\rangle,\vert HD\rangle,\vert HL\rangle,\vert VH\rangle,\vert VV\rangle,\vert VD\rangle,\vert VL\rangle, \vert DH\rangle,\vert DV\rangle,\vert DD\rangle,\vert DL\rangle,\vert LH\rangle,\vert LV\rangle,\vert LD\rangle,\vert LL\rangle\}$ (here $\vert D\rangle = \frac{1}{\sqrt{2}}(\vert H\rangle+\vert V\rangle)$ and $\vert L\rangle = \frac{1}{\sqrt{2}}(\vert H\rangle+\textrm i\vert V\rangle)$). Real and imaginary parts of the reconstructed process matrix elements are shown in figures 5(a) and (b). The photonic router preserves the coin state. Thus, in the ideal case the process equals an identity operator. The experimentally reconstructed process matrices χ_exp are characterized by the process fidelity [79]

$$\begin{align} F_\textrm p=\frac{\text{Tr}(\chi^\dagger_{\textrm {th}}\chi_{\textrm {exp}})}{\sqrt{\text{Tr}(\chi^\dagger_{\textrm {exp}}\chi_{\textrm {exp}})\text{Tr}(\chi^\dagger_{\textrm {th}}\chi_{\textrm {th}})}}. \end{align} \tag{ 7 }$$

In our experiment, the process fidelity is $0.935\pm0.001$, which indicates a successful photonic router.

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Finally, we choose total 16 product states as the initial coin states, which are routed from $(0,0)$ to $(0,2)$ after the evolution of the 6-step QW. Our results in figure 5(c) exhibit all the fidelities are larger than $0.929\pm0.001$, which indicate that our routing algorithm also works for product states.