Robust hyperentanglement self testing

TL;DR

The paper tackles certifying multi-DOF hyperentanglement in a device-independent manner. It introduces a general hyperentanglement self-testing framework implemented via independent two-dimensional CHSH tests in each DOF and two-step swap isometries to extract the target hyperentangled Bell state, specifically for polarization-spatial-mode encodings. The key contributions are (i) a protocol that self-tests all 16 polarization-spatial-mode hyperentangled Bell states, (ii) the derivation of anti-commuting relations that enable the isometries, and (iii) a robust extension that bounds state fidelity under non-ideal CHSH violations, with explicit fidelity expressions. The work is significant for certifying complex hyperentangled sources in future high-capacity quantum networks and is designed to be compatible with current experimental capabilities.

Abstract

Hyperentanglement, which refers to entanglement encoded in two or more independent degrees of freedom (DOFs), is a valuable resource for the future high-capacity quantum network. Certifying hyperentanglement sources work as intended is critical for the hyperentanglement-based quantum information tasks. Self testing is the strongest certification method for quantum state and measurement under minimal assumptions, even without any knowledge of the devices' inner workings. However, the existing self testing protocols all focus on one-DOF entanglement, which cannot self test the multi-DOF entanglement. In the paper, we propose a hyperentanglement self testing framework. We take the self testing for the polarization-spatial-mode hyperentangled Bell states as an example. The self testing is based on the violation of two-dimension CHSH test in each DOF independently. The two-step swap isometry circuits are proposed for self testing the entanglement in spatial-mode and polarization DOFs, respectively. All the sixteen polarization-spatial-mode hyperentangled Bell states can be self tested. Our hyperentanglement self testing framework has three advantages. First, it is a general hyperentanglement self testing framework, and can be extended to self test multi-DOF hyperentanglement and multipartite hyperentanglement. Second, it can provide the robust hyperentanglement self testing and establish the relation between the lower bound of fidelity and the imperfect violation of Bell-like inequality in each DOF. Third, it is feasible with current experimental technology. Our hyperentanglement self testing framework provides a promising way to certify complex hyperentanglement sources, and has potential application in future high-capacity quantum network.

Figures (5)

• Figure 1: The linear optical apparatuses for the CHSH measurements of the polarization-spatial-mode hyperentanglement chen2003allyang2005all. (a) (b) (c) (d) correspond to $\sigma_{x}^S\sigma_{z}^P$, $\sigma_{x}^S\sigma_{x}^P$, $\sigma_{z}^S\sigma_{x}^P$, and $\sigma_{z}^S\sigma_{z}^P$ measurement bases, respectively. The polarization beam splitter (PBS) can totally transmit the photon in $|h\rangle$ and reflect the photon in $|v\rangle$. The 50:50 beam splitter (BS) and quarter wave plate (QWP) are used to perform the Hadamard (H) operations in the spatial-mode DOF and polarization DOF, respectively.
• Figure 2: The theoretical swap isometry circuit in the one-DOF scenario. $H$ represents the Hadamard operation. $Z$ and $X$ represent the $\sigma_z$ and $\sigma_x$ operations in the controlled-Z (CZ) and controlled-not (CNOT) modules, respectively MaYaorobust0robust00.
• Figure 3: The swap isometry circuit in the spatial-mode DOF. Two auxiliary photons are required in the spatial modes a' and b'. $H_S$ represents the Hadamard operation in the spatial-mode DOF. After all the operations, the parties perform the BSM in spatial-mode DOF on the auxiliary photons in $a_1"b_1"a_2"b_2"$ modes, respectively.
• Figure 4: The swap isometry circuit in the polarization DOF. The physical hyperentangled state and two auxiliary photons pass through this swap isometry circuit. $H_P$ represents the Hadamard operation in the polarization DOF. After all the operations, the parties perform the BSMs in the polarization DOF on the auxiliary photons in the spatial modes $A_1'B_1'$ and $A_2'B_2'$, respectively.
• Figure 5: The lower bounds of the fidelity $F_P$ and $F_t$ as a function of the parameter $\epsilon_P$ with $\epsilon_P=\epsilon_S$.

Theorems & Definitions (1)

• Definition 1