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This paper presents an information-theoretic approach to address the phasor measurement unit (PMU) placement problem in electric power systems. Different from the conventional `topological observability' based approaches, this paper advocates a much more refined, information-theoretic criterion, namely the mutual information (MI) between PMU measurements and power system states. The proposed MI criterion not only includes observability as a special case, but also rigorously models the uncertainty reduction on power system states from PMU measurements. Thus, it can generate highly informative PMU configurations. The MI criterion can also facilitate robust PMU placement by explicitly modeling probabilistic PMU outages. We propose a greedy PMU placement algorithm, and show that it achieves an approximation ratio of (1-1/e) for any PMU placement budget. We further show that the performance is the best that one can achieve, in the sense that it is NP-hard to achieve any approximation ratio beyond (1-1/e) . Such performance guarantee makes the greedy algorithm very attractive in the practical scenario of multi-stage installations for utilities with limited budgets. Finally, simulation results demonstrate near-optimal performance of the proposed PMU placement algorithm.