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The Grover quantum search algorithm is generalized to deal with an arbitrarymixed initial state. The probability to measure a marked state as a function oftime is calculated, and found to depend strongly on the specific initial state.The form of the function, though, remains as it is in the case of initial purestate. We study the role of the von Neumann entropy of the initial state, andshow that the entropy cannot be a measure for the usefulness of the algorithm.We give few examples and show that for some extremely mixed initial statescarrying high entropy, the generalized Grover algorithm is considerably fasterthan any classical algorithm.

Grover’s quantum search algorithm for an arbitrary initial mixed state