Building separable approximations for quantum states via neural networks

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.

13 pages, 7 figures We welcome any comments on references for Tables 1 and 2. Comments on update #1: added more results to the figures, more precise numerical results in table 1 and 2, added references and comparison with previous work. Comments on update #2: added more analytical and numerical results. Comments on update #3: Removed duplicated references

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• Certifying Quantum Separability with Adaptive Polytopes
• Neural Network Approach to the Simulation of Entangled States with One Bit of Communication
• Probabilistic state synthesis based on optimal convex approximation
• Quantifying Quantum Steering with Limited Resources: A Semi-supervised Machine Learning Approach
• Unveiling the nonclassicality within quasi-distribution representations through deep learning
• Hybrid of Gradient Descent And Semidefinite Programming for Certifying Multipartite Entanglement Structure
• Separable ellipsoids around multipartite states
• Variational-toolbox-based separability detection of multiqubit states
• Trade-off between Bagging and Boosting for quantum separability-entanglement classification