Quantum direct communication protocol using recurrence in $k$-cycle quantum walks

The ability of quantum walks to evolve in a superposition of distinct quantum states has been used as a resource in quantum communication protocols. Under certain settings, the $k$-cycle discrete-time quantum walks (DTQW) are known to recur to its initial state after every $t_r$ steps. We first present a scheme to optically realize any $k$-cycle DTQW using $J$-plate, orbital angular momentum (OAM) sorters, optical switch, and optical delay line. This entangles the polarization and OAM degrees of freedom (DoF) of a single photon. Making use of this recurrence phenomena of $k$-cycle DTQW and the entanglement generated during the evolution, we present an alternate quantum direct communication protocol. The recurrence and entanglement in the $k$-cycle walk are effectively used to retrieve and secure the information, respectively, in the proposed protocol. We investigate the security of the protocol against intercept and the resend attack. We also quantify the effect of amplitude damping and depolarizing noises on recurrence and mutual information between polarization and the OAM DoF of a single photon. Finally, we indicated an optimal attack strategy by which an Eavesdropper can tamper part of the message without revealing her presence. However, when the quantum communication channel is less noisy, any attempt by the Eavesdropper to tamper the message would end up in exposing her to the receiver.

Figure 1

Probability of walker returning to the initial position  ($x=1$), $P_1\left(t\right)$, is plotted as a function of steps ($t$) for a five-cycle DTQW. The walker returns to the initial position once after every 60 steps.

Figure 2

(a) Shown here is the optical realization of $k$-cycle DTQW. An incoming single photon (ISP) with OAM value ($\ell$) between ${\ell }_{\min }$ and ${\ell }_{\max }$ is first sent to the coin operator $\widehat{C}\left(\rho \right)$ [see Eq. (1)], which is a half-wave-plate (HWP) rotated through an angle $\beta /2$, with $\beta ={tan}^{-1}\left(\sqrt{(1-\varrho )/\varrho }\right)$. The shift operator ${\widehat{S}}_k$ is realized using a $J$-plate, represented by $\mathbf{J}(-\varphi ,\varphi ,0)$ [see Eq. (6)], two OAM sorters (OAM-S), thin lenses $L_1$ and $L_2$, two spiral phase plates (SPP) with $\ell =\pm k$, and two fiber couplers (FC). Having realized both coin and shift operations, the photon is now sent through an optical switch (OS) and optical delay line (ODL) before being passed through the HWP once again for the next step. Upon performing the desired number of steps, the photon is finally sent to the quantum channel (QC) through the use of OS. (b) Shown here is the Mach-Zehnder (MZ) $+$ SPP setup which sorts the incoming OAM modes. When an ISP, represented by $|\psi \rangle$ [see Eq. (10)] passes through a HWP, a $J$-plate, and the MZ $+$ SPP setup, one-step in the five-cycle DTQW is performed by the ISP. First MZ setup sorts odd and even OAM modes. The SPP with $\ell =-1$ converts all odd OAM modes to even OAM modes. Subsequently, the remaining 3 MZ setups and 4 SPPs with appropriate $\ell$ values ensure the modulo five-addition of the five-cycle DTQW. Now the states $|{\psi }_2\rangle , |{\psi }_5\rangle , |{\psi }_6\rangle , |{\psi }_7\rangle$, and $|{\psi }_8\rangle$ are fiber-coupled together and sent to OS.

Figure 3

(a) Shown here is the schematic outline of the communication protocol. Here, bold arrows denote quantum channel and broken arrows denote classical channels. (b) Shown here is the optical setup which adds (mod $k$ addition) an OAM value between ${\ell }_{\min }$ and ${\ell }_{\max }$ to the incoming single photon (ISP) through the use of a spatial light modulator (SLM).

Figure 4

Shown here is the joint probability distribution, $P(\ell ,t)$, of finding the walker at position $\ell$ after $t$ steps corresponding to a five-cycle DTQW with coin parameter $\varrho =(5-\sqrt{5})/10$. The probability of finding the walker at a given $t$ over all $\ell$ values is 1. Because we want the sum over all possible $\ell$ and $t$ for $P(\ell ,t)$ to be 1 [see Eq. (11)], we divided the probability distribution by $(t_r-1)$.

Figure 5

(a) Shown here is the probability of the five-cycle walker returning to the position $x=1, P_1\left(t\right)$, after every step $t$ for various levels of amplitude damping noise parameter ${\gamma }_a$ [see Eq. (15)]. (b) Shows how the mutual information, $I_{\mathrm{mu}}\left(t\right)$ [see Eq. (14)], varies as a function of $t$ for the same amount of amplitude damping noise parameters. In (c), (d) we plotted $P_1\left(t\right)$ and $I_{\mathrm{mu}}\left(t\right)$ for various levels of depolarizing noise parameter ${\gamma }_d$ [see Eq. (16)], respectively.

Figure 6

This graph plots $P\left(y\right|x)$ versus $y$ [see Eq. (18)] for the choices $N=100$ and $x=60$. $P\left(y\right|x)$ looks like a Gaussian distribution centered at $y=20$.