Almost zero fine-structure splitting or degeneracy of the exciton spin state is a prerequisite for the production of a polarization-entangled photon pairs from the biexciton decay of a quantum dot, which is necessary for some schemes of quantum cryptography and quantum information processing. We have studied fine-structure splittings in single self-assembled quantum dots by using a photocurrent spectroscopy, whose resolution is higher than 1 μeV. We show the photocurrent spectroscopy technique as a good tool to characterize the fine-structure splittings with high accuracy for the entanglement generation. Using the technique, we have found a very narrow splitting of less than 3 μeV, which is limited by a time-dependent fluctuation of the photocurrent peak.