The Groverian Measure of Entanglement for Mixed States

Abstract

The Groverian entanglement measure introduced earlier for pure quantum states [O. Biham, M.A. Nielsen and T. Osborne, Phys. Rev. A 65, 062312 (2002)] is generalized to the case of mixed states, in a way that maintains its operational interpretation. The Groverian measure of a mixed state of n qubits is obtained by a purification procedure into a pure state of 2n qubits, followed by an optimization process based on Uhlmann's theorem, before the resulting state is fed into Grover's search algorithm. The Groverian measure, expressed in terms of the maximal success probability of the algorithm, provides an operational measure of entanglement of both pure and mixed quantum states of multiple qubits. These results may provide further insight into the role of entanglement in making quantum algorithms powerful.