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This letter is concerned with the singular value characterization of fixed point-to-point millimeter wave channels with symmetric uniform circular antenna arrays (UCAs) at both the transmitter and receiver. We model the channel as a line-of-sight (LoS) multiple-input multiple-output (MIMO) one and derive an explicit expression for all channel singular values. The asymptotic analysis when the end-to-end distance or the number of antenna elements goes to infinity is then developed. Numerical examples are provided to verify the validity of our discussions. The results in this letter show that the achievable multiplex gain is limited under such a system setting, regardless of the numbers of antennas at the transmitter/receiver. This observation provides benchmarks for the practical performance of UCA-based gigabits wireless communications over millimeter wave channels.