Mermin polynomials for non-locality and entanglement detection in Grover's algorithm and Quantum Fourier Transform

The non-locality and thus the presence of entanglement of a quantum system can be detected using Mermin polynomials. This gives us a means to study non-locality evolution during the execution of quantum algorithms. We first consider Grover's quantum search algorithm, noticing that states during the execution of the algorithm reach a maximum for an entanglement measure when close to a predetermined state, which allows us to search for a single optimal Mermin operator and use it to evaluate non-locality through the whole execution of Grover's algorithm. Then the Quantum Fourier Transform is also studied with Mermin polynomials. A different optimal Mermin operator is searched for at each execution step, since in this case nothing hints us at finding a predetermined state maximally violating the Mermin inequality. The results for the Quantum Fourier Transform are compared to results from a previous study of entanglement with Cayley hyperdeterminant. All our computations can be repeated thanks to a structured and documented open-source code that we provide.