Generation of hyperentangled states and two-dimensional quantum walks using $J$ or $q$ plates and polarization beam splitters

A single photon can be made to entangle simultaneously in its different internal degrees of freedom (DoF)—polarization, orbital angular momentum (OAM), and frequency—as well as in its external DoF—path. Such entanglement in multiple DoF is known as hyperentanglement and provides additional advantage for quantum information processing. We propose a passive optical setup using $q$ plates and polarization beam splitters to hyperentangle an incoming single photon in polarization, OAM, and path DoF. By mapping polarization DoF to a two-dimensional coin state, and path and OAM DoF to two spatial dimensions, $x$ and $y$, we present a scheme for realization of a two-dimensional discrete-time quantum walk using only polarization beam splitters and $q$ plates ensuing the generation of hyperentangled states. The amount of hyperentanglement generated is quantified by measuring the entanglement negativity between any two DoF. We further show that hyperentanglement generation can be controlled by using an additional coin operation or by replacing the $q$ plate with a $J$ plate.

Figure 1

Probability distribution $\left[P\right(x,m\left)\right]$ of the 2D DTQW without explicit coin operation in position and OAM DoF, beginning with an initial state $[\left(\right|H\rangle +|V\rangle )/\sqrt{2}]\otimes |x=0\rangle \otimes |y=0\rangle$, after 50 steps. (a) Modified Pauli walk using $q$ plates and PBS [see Eq. (23)]. Note that the probability distribution is identical for the Pauli walk [see Eqs. (9) and (24)] which is realized using $q$- plates, PBS, and HWP. (b) Modified Pauli walk using $J$ plates and PBS for the choice $|u_1{\rangle =[1,-1]}^T/\sqrt{2}$ and $|u_2{\rangle =[1,1]}^T/\sqrt{2}$ in Eq. (23). (c) Modified Pauli walk using $J$ plates and PBS for the choice $|u_1\rangle ={[1,\sqrt{3}i]}^T/2$ and $|u_2\rangle ={[\sqrt{3},-i]}^T/2$ in Eq. (23).

Figure 2

Optical implementation of Pauli and modified Pauli walks. Both Pauli and modified Pauli walks with any orthogonal set of vectors $\left\{\right|u_1\rangle ,|u_2\rangle \}$ can be realized using $J$ plates and PBS [see Eqs. (9), (23), and (24)]. When $\left\{\right|u_1\rangle ,|u_2\rangle \}=\left\{\right|R\rangle ,|L\rangle \}$, every $J$ plate can be replaced with a $q$ plate ($q$ plate and HWP) to realize the modified Pauli walk (Pauli walk). $D_i$ denotes a detector unit placed at the position $x=i$. Each detector unit consists of a spatial light modulator (SLM), a single mode fiber (SMF), and a single photon detector (SPD). A single photon is sent through a $q$ plate (or $J$ plate) to the PBS placed at $x=0$ (shown with an arrow). Here, $J$ plate implements either one of the shift operators ${\widehat{S}}_{\mathit{\sigma },\mathrm{OAM}}^{\prime }$ [Eq. (22)] or ${\widehat{S}}_{\mathit{\sigma },\mathrm{OAM}}$ [Eq. (24)], and PBS implements the shift operator ${\widehat{S}}_{x,\mathrm{pos}}$ [see Eq. (23)]. Evidently, no explicit 2D coin operation is necessary to implement these type of quantum walks.

Figure 3

Entanglement negativity $\mathcal{N}$ [see Eq. (26)] plotted against the increasing number of steps between two DoF for various $J$-plate parameters $(\xi ,\zeta ,\theta )$ [see (21)]. Here, $J(\varphi ,0,-\pi /2,\pi /4)$ represents a $q$ plate, and the initial state is $[\left(\right|H\rangle +|V\rangle )/\sqrt{2}]\otimes |x=0\rangle \otimes |y=0\rangle$. (a) $\mathcal{N}$ between path and OAM DoF. (b) $\mathcal{N}$ between polarization and OAM DoF. Note that $\mathcal{N}$ between polarization and path DoF is identical to that of (b).

Figure 4

Entanglement negativity $\mathcal{N}$ [see Eq. (26)] between any two of the three DoF plotted against any one of the three $J$-plate parameters $(\xi ,\zeta ,\theta )$ [see (21)] after 25 steps. The initial state was taken to be $[\left(\right|H\rangle +|V\rangle )/\sqrt{2}]\otimes |x=0\rangle \otimes |y=0\rangle$. In (a), $\mathcal{N}$ between path and OAM DoF is considered, whereas in (b) that between polarization and OAM DoF is considered. In both (a) and (b) the black curve was obtained for the $J(\varphi ,\pi /2,\pi /2,\underline{\theta })$ and PBS combination, where $\underline{\theta }$ denotes that $\theta$ is varied from 0 through $\pi /2$ in steps of $\pi /180$. And the red curve was obtained for the $J(\varphi ,\underline{\xi },-\pi /2,\pi /4)$ and PBS combination, where $\underline{\xi }$ denotes that $\xi$ is varied from 0 through $\pi /2$ in steps of $\pi /180$. Note that the choice $J(\varphi ,0,-\pi /2,\pi /4)$ represents a $q$ plate and is encircled in both (a) and (b).