Generation and detection of discrete-variable multipartite entanglement with multirail encoding in linear-optics networks

A linear-optics network is a multimode interferometer system, where indistinguishable photon inputs can create nonclassical interference that cannot be simulated with classical computers. Such nonclassical interference implies the existence of entanglement among its subsystems if we divide its modes into different parties. Entanglement in such systems is naturally encoded in multirail (multimode) quantum registers. For bipartite entanglement, a generation and detection scheme with multirail encoding was theoretically proposed [Wu and Hofmann, New J. Phys. 19, 103032 (2017)] and experimentally realized [Kiyohara et al., Optica 7, 1517 (2020)]. In this paper, we will take a step further to establish a theory for the detection of multirail-encoded discrete-variable genuine multipartite entanglement (GME) in fixed local-photon-number subspaces of linear-optics networks. We also propose a scheme for GME generation with both discrete-variable (single photons) and continuous-variable (squeezed states) light sources. This scheme allows us to reveal the discrete-variable GME in continuous-variable systems. The effect of photon losses is also numerically analyzed for the generation scheme based on continuous-variable inputs.

Figure 1

(a) Measurement in the ${\widehat{\mathrm{\Lambda }}}_{j_{1}l}\otimes \cdots \otimes {\widehat{\mathrm{\Lambda }}}_{j_{P}l}$ eigenbasis. (b) Generation and evaluation of genuine $P$-partite entanglement employing $P$-mode splitters.

Figure 2

Measurement statistics of the state $|{\mathrm{\Phi }}_{2,1,1}(r,x)\rangle$ with a squeezing factor of $r=1.15\left(10\text{dB}\right)$ and a displacement of $x=0$. (a) Measurement in the computational basis. (b) The measurement in the ${\widehat{\mathrm{\Lambda }}}_0\otimes {\widehat{\mathrm{\Lambda }}}_0\otimes {\widehat{\mathrm{\Lambda }}}_0$ eigenbasis. (c) The measurement in the ${\widehat{\mathrm{\Lambda }}}_{j_{A}l}\otimes {\widehat{\mathrm{\Lambda }}}_{j_{B}l}\otimes {\widehat{\mathrm{\Lambda }}}_{j_{C}l}$ eigenbasis, where the HW indices are chosen as $(j_A,j_B,j_C)=(1,4,4)$.

Figure 3

Evaluation of genuine multipartite entanglement of the (5,5,5)-mode (2,1,1)-photon state $|{\mathrm{\Psi }}_{2,1,1}(r,x)\rangle$ given in Eq. (52). (a) The expectation value $\langle {\widehat{V}}_{0,\kappa }\rangle$ of $|{\mathrm{\Psi }}_{2,1,1}(r,x)\rangle$ with a squeezing factor of 0.5 dB. The expectation value $\langle {\widehat{V}}_{0,\kappa }\rangle$ is evaluated in the (${\widehat{\mathrm{\Lambda }}}_{j_{A}l}\otimes {\widehat{\mathrm{\Lambda }}}_{j_{B}l}\otimes {\widehat{\mathrm{\Lambda }}}_{j_{C}l}$)-eigenbasis measurement settings with $(j_A,j_B,j_C)=(1,4,4)$ and $(j_A,j_B,j_C)=(1,1,2)$. (b) The expectation value $\langle {\widehat{V}}_{0,\kappa }\rangle$ of $|{\mathrm{\Psi }}_{2,1,1}(r,x)\rangle$ is evaluated in the (${\widehat{\mathrm{\Lambda }}}_l\otimes {\widehat{\mathrm{\Lambda }}}_{4l}\otimes {\widehat{\mathrm{\Lambda }}}_{4l}$)-eigenbasis measurement settings for different squeezing.

Figure 4

The expectation value $\langle {\widehat{V}}_{0,\kappa }\rangle$ with input squeezing factor $r=0.576\left(5\text{dB}\right)$. (a) The expectation value $\langle {\widehat{V}}_{0,\kappa }\rangle$ of $|{\mathrm{\Phi }}_{{\mathit{\nu }}_i|(2,1,1)}(r,x)\rangle$ with $\kappa =0$ and ${\mathit{\nu }}_{\mathrm{tot}}\in \{00000,10000,20000,11000,10100\}$. (b) The expectation value $\langle {\widehat{V}}_{0,\kappa }\rangle$ of $|{\mathrm{\Phi }}_{{\mathit{\nu }}_i|(2,1,1)}(r,x)\rangle$ with $\kappa >0$ and ${\mathit{\nu }}_{\mathrm{tot}}\in \{00000,10000,20000,11000,10100\}$. (c) The expectation value $\langle {\widehat{V}}_{0,\kappa }\rangle$ of ${\widehat{\rho }}_{2,1,1}(r,x;\epsilon )$ for different loss rates.

Figure 5

(a) A lossy photon number resolving measurement in LONs modeled by beam splitters interfering with the environment. (b) A circuit equivalent to (a). (c) A circuit equivalent to (b) after tracing out the environment.