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A general rate-distortion converse bound for entropy-constrained scalar quantization

We derive a lower bound on the smallest output entropy that can be achieved via scalar quantization of a source with given expected quadratic distortion. As the allowed distortion tends to zero, the bound converges to the output entropy achieved by a uniform quantizer, thereby recovering the result by Gish and Pierce that uniform quantizers are asymptotically optimal. The proposed derivation applies for any memoryless source that has a probability density function (pdf), a finite differential entropy, and whose integer part has a finite entropy. In contrast to Gish and Pierce, we do not require any additional constraints on the continuity or decay of the source pdf.