Building separable approximations for quantum states via neural networks

TL;DR

This paper introduces a neural network-based method to approximate the closest separable quantum states to a given target state, providing bounds on entanglement and efficient results for bipartite and multipartite systems.

Contribution

The authors develop a neural network approach to find separable approximations of quantum states, offering a practical and scalable solution to a difficult problem in quantum information.

Findings

Accurately approximates closest separable states for bipartite systems up to dimension 10.

Provides bounds and analytic insights for isotropic and Werner states.

Efficiently extends to multipartite states, recovering known bounds and discovering new ones.

Abstract

Finding the closest separable state to a given target state is a notoriously difficult task, even more difficult than deciding whether a state is entangled or separable. To tackle this task, we parametrize separable states with a neural network and train it to minimize the distance to a given target state, with respect to a differentiable distance, such as the trace distance or Hilbert--Schmidt distance. By examining the output of the algorithm, we obtain an upper bound on the entanglement of the target state, and construct an approximation for its closest separable state. We benchmark the method on a variety of well-known classes of bipartite states and find excellent agreement, even up to local dimension of $d=10$, while providing conjectures and analytic insight for isotropic and Werner states. Moreover, we show our method to be efficient in the multipartite case, considering different notions of separability. Examining three and four-party GHZ and W states we recover known bounds and obtain additional ones, for instance for triseparability.