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An information-theoretic three-dimensional (3D) resolution measure for the optical microscope is introduced. Based on the Cramer-Rao inequality, this resolution measure specifies a lower bound on the accuracy with which a given distance separating two objects in 3D space can be estimated from the acquired image. Useful in many applications, accurate determination of the distance of separation can, for example, help to characterize the interaction that occurs between two closely spaced biomolecules in a biological cell. In addition to presenting the underlying theory, we show that the resolution measure predicts that, by detecting a sufficient number of photons from an object pair, arbitrarily small distances of separation can be estimated with prespecified accuracy. Furthermore, we illustrate its dependence on properties such as the object pair's 3D spatial orientation. With estimations on simulated images, we show that the maximum likelihood estimator is capable of attaining the accuracy predicted by the resolution measure.