Tight lower bound on geometric discord of bipartite states

We use singular value decomposition to derive a tight lower bound for geometric discord of arbitrary bipartite states. In a single shot this also leads to an upper bound of measurement-induced nonlocality which in turn yields that for Werner and isotropic states the two measures coincide. We also emphasize that our lower bound is saturated for all 2 circle times n states. Using this we show that both the generalized Greenberger-Horne-Zeilinger and W states of N qubits satisfy monogamy of geometric discord. Indeed, the same holds for all N-qubit pure states which are equivalent to W states under stochastic local operations and classical communication. We show by giving an example that not all pure states of four or higher qubits satisfy monogamy.