Schrödinger as a Quantum Programmer: Estimating Entanglement via Steering

Quantifying entanglement is an important task by which the resourcefulness of a quantum state can be measured. Here we develop a quantum algorithm that tests for and quantifies the separability of a general bipartite state, by making use of the quantum steering effect, the latter originally discovered by Schr\"odinger. Our separability test consists of a distributed quantum computation involving two parties: a computationally limited client, who prepares a purification of the state of interest, and a computationally unbounded server, who tries to steer the reduced systems to a probabilistic ensemble of pure product states. To design a practical algorithm, we replace the role of the server by a combination of parameterized unitary circuits and classical optimization techniques to perform the necessary computation. The result is a variational quantum steering algorithm (VQSA), which is a modified separability test that is better suited for the capabilities of quantum computers available today. We then simulate our VQSA on noisy quantum simulators and find favorable convergence properties on the examples tested. We also develop semidefinite programs, executable on classical computers, that benchmark the results obtained from our VQSA. Our findings here thus provide a meaningful connection between steering, entanglement, quantum algorithms, and quantum computational complexity theory. They also demonstrate the value of a parameterized mid-circuit measurement in a VQSA and represent a first-of-its-kind application for a distributed VQA.