An Information-Theoretic Characterization of Optimal Value-Readout in Response-Register Quantum Oracles

Response-register quantum oracles admit two complementary operational interpretations: value readout and phase kickback.

Although their computational equivalence is well understood, the quantitative relation between these two operational interpretations has lacked an exact characterization.

We prove that, for finite Abelian response groups, the optimal single-query value-readout probability is exactly the normalized Rényi-1/2 effective Fourier support of the response state.

This identity provides an exact information-theoretic characterization of optimal value-readout capability, gives the Rényi-1/2 effective Fourier support a direct operational interpretation in the oracle setting, and yields a tight phase--value complementarity theorem together with an explicit family of saturating response states.