We study the capacity of the primitive diamond relay channel and derive an improved upper bound compared with existing upper bounds by employing the Csiszár--Körner--Marton sum identity and Gallager-type identification. For Gaussian primitive diamond relay channels, we show that our proposed bound coincides with the existing strengthened Gaussian upper bound of Wu, Özgür, Peleg, and Shamai. For discrete memoryless channels, our bound can be strictly tighter than both the cut-set bound and the existing upper bound, and we assess the improvement using different channel instances, together with a decode--forward/compress--forward time-sharing inner bound.

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