A General Framework for Constructing Local Hidden-state Models to Determine the Steerability

TL;DR

This work proposes a machine learning-based framework that employs batch sampling of measurements and gradient-based optimization to construct an optimal LHS model, and investigates the steerability of this class of states under arbitrary POVMs, and suggests that POVMs can offer an advantage over PVMs in revealing the steerability of such states.

Abstract

Not all entangled states can exhibit quantum steering, and determining whether a given entangled state is steerable is a crucial problem in quantum information theory. The main challenge lies in verifying the existence of a local hidden-state (LHS) model capable of reproducing all post-measurement assemblages generated by arbitrary measurements. To address this, we propose a machine learning-based framework that employs batch sampling of measurements and gradient-based optimization to construct an optimal LHS model. We validate our method by analyzing the steerability of two-qubit Werner and two-qutrit isotropic states. For Werner states, our approach saturates the analytical visibility bounds under three Pauli measurements, arbitrary projective measurements (PVMs), and arbitrary positive operator-valued measurements (POVMs). For isotropic states, we achieve the known analytical bounds under arbitrary PVMs. We further investigate the steerability of this class of states under arbitrary POVMs, and our results suggest that POVMs can offer an advantage over PVMs in revealing the steerability of such states.

Figures (5)

• Figure 1: Bipartite Scenarios. (a) The Steering Scenario: Alice and Bob share an unknown entangled state $\rho_{AB}$. Alice's device acts as an untrusted black box, where the measurements $M_{a|x}$ she performs are completely unknown; Bob's side is trusted. We can determine whether Alice can steer Bob's state by obtaining the assemblage $\{\sigma_{a|x}\}$ from Bob's side. (b) The LHS Model: This model describes a situation where a source sends a hidden variable $\lambda$ to Alice and a corresponding hidden state $\sigma(\lambda)$ to Bob, thereby generating the assemblage $\{\sigma_{a|x}\}$.
• Figure 2: Steerability of the two-qubit Werner states under three Pauli measurements. The parameters adopted for optimizing the LHS model are as follows: the order of orthonormal basis functions $D = 5$, the number of hidden variables $N_{\text{hidden}} = 8$, and the number of gradient descent iterations $N_{\text{steps}} = 10^5$. Theoretically, the higher the order D of the orthonormal basis functions and the more gradient descent iterations, the higher the result accuracy. However, practical verification shows that the currently selected parameters are sufficient to meet the requirements.
• Figure 3: Steerability of the two-qubit Werner states under PVMs. The parameters adopted for optimizing the LHS model are as follows: the order of orthonormal basis functions $D = 5$, the number of hidden variables $N_{\text{hidden}} = 200$, and the number of gradient descent steps $N_{\text{steps}} = 6 \times 10^5$. To verify that the optimal LHS model is not only applicable to the measurements used in training, we uniformly and randomly sampled a batch of measurements for testing, which confirms the model's universality.
• Figure 4: Steerability of the two-qubit Werner states under POVMs. The parameters adopted for optimizing the LHS model are as follows: the order of orthonormal basis functions $D = 2$, the number of hidden variables $N_{\text{hidden}} = 300$, and the number of gradient descent steps $N_{\text{steps}} = 10^5$. At each step of the gradient descent, $N_{\text{measures}}$ measurements are randomly selected to optimize the parameters, yielding the optimal $\lambda$ and $\sigma(\lambda)$. Finally, the loss value is computed using $N_{\text{measures\_test}}$ measurements.
• Figure 5: Steerability of the two-qutrit isotropic states under PVMs and POVMs. Red points show results under PVMs, with the following parameters: order of orthonormal basis functions $D = 2$, number of hidden variables $N_{\text{hidden}} = 100$, number of gradient descent steps $N_{\text{steps}} = 2 \times 10^5$, and $N_{\text{meas}} = 200$ randomly selected measurements per gradient step. Blue points correspond to results under general POVMs, with parameters: $D = 3$, $N_{\text{hidden}} = 200$, $N_{\text{steps}} = 2 \times 10^5$, and $N_{\text{meas}} = 300$.