Entanglement Detection with Variational Quantum Interference: Theory and Experiment

Abstract

Entanglement detection is a fundamental task in quantum information science, serving as a cornerstone for quantum benchmarking and foundational studies. With an increasing qubit number that can be effectively controlled, there is a pressing need for a scalable and robust detection protocol which requires minimal resources while maintaining high detection capability. By integrating the Positive Partial Transposition criterion with variational quantum interference, we propose an entanglement detection protocol that requires moderate classical and quantum computation resources. We numerically show that this protocol achieves a high detection capability with shallow quantum circuits, surpassing some widely-used entanglement detection methods. The protocol also exhibits strong resilience to circuit noise, ensuring its applicability across different physical platforms. We further demonstrate the protocol experimentally on an eight-photon linear-optical platform, where it successfully detects the entanglement of a three-qubit mixed state that is inaccessible to conventional entanglement witnesses. By combining quantum interference with classical optimization, our protocol provides a scalable and resource-efficient route toward practical entanglement detection.

Figures (6)

• Figure 1: Overview of iPPT protocol. For a target state $\rho$, the initial step involves preparing a reference state $\sigma(\theta)$ using the variational circuit (light green box). Subsequently, Bell state measurements (BSMs) are carried out on each corresponding qubit pair from the target state $\rho$ and the reference state $\sigma(\theta)$. Lines in blue and black represent qubits of subsystems $A$ and $B$ of target and reference states. BSM modules in blue and black collect data associated with the Bell states $\ket{\Phi^+}=(\ket{00}+\ket{11})/\sqrt{2}$ and $\ket{\Psi^-}=(\ket{01}-\ket{10})/\sqrt{2}$, respectively. The collected data are post-processed to determine the entanglement property of the target state $\rho$, with a negative result of $\Tr(\rho\sigma^{\mathrm{T}_A})$ indicating the existence of entanglement.
• Figure 2: Entanglement detection capability of the iPPT protocol with various depth ($D$) of the reference state preparation circuits, in comparison with the purity and fidelity protocols. For each point, we randomly sample 1000 six-qubit mixed states and verify their entanglement using different protocols. The $y$-coordinate stands for the success probability for verifying the entanglement of the sampled six-qubit mixed state. The larger values of $x$-coordinate term, $k$, correspond to the lower expected purity of the sampled mixed states.
• Figure 3: Experimental design and results. (a) The three-qubit target state $\rho$ and reference state $\sigma(\theta)$ are prepared with two pairs of entangled photons, respectively. Three BSMs are performed between three pairs of photons from the reference and target state. (b) Ideal and experimental value of $\Tr(\rho\sigma^{\mathrm{T}_A})$ with respect to the variable phase parameter $\Delta \theta$.
• Figure 4: Schematic of experimental setup for iPPT protocol. The pulsed ultraviolet pump laser successively passes through four separated C-BBO crystals from left to right and generate four EPR entangled photon pairs, $\ket{\mathrm{EPR}}=(\ket{HH}+\ket{VV})/\sqrt{2}$. The first two EPR sources, EPR-1 and EPR-2, are used to prepare the reference state. The last two EPR sources, EPR-3 and EPR-4, are used to prepare the to-be-verified target state. The mixing of each component state of the target state is achieved by a one-dimensional linear motor ("motor" in schematic) that randomly introducing the polarization flipping. The three pairs of photons from reference state and target state are introduced to three BSM device, which are composed of PBSs, HWPs, and single-photon detectors. SC-YVO4 and TC-YVO4 are YVO$_4$ crystals used for spatial compensation (SC) and temporal compensation (TC) of each EPR source.
• Figure 5: Illustration of the variational circuit for preparing the reference state.