Colloquium

Grover's Algorithm and entanglement

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Grover's search algorithms, both the original one and the fixed-point one, can be used for making up maximally entangled states. A measure of entanglement with respect to a bipartite partition of $n$-qubit is defined and taken for sequences of states generated by respective Grover algorithms. If one wishes to make up a maximally entangled state, one should like to choose it among maximally entangled states which are nearest from the initial state. To this end, the set of nearest maximally entangled states are determined with respect to the natural Riemannian metric on the state space.

Contrarily, if the target state is separable as in the usual search algorithm, the Grover algorithms generate sequences of entangled states approaching the separable target state. A split Grover algorithm is proposed, which is so modified as to generate sequences of separable states only.