A counterexample to Bu--Gu--Jaffe's expected mixed-state inverse sumset estimate

We refute the mixed-state inverse sumset estimate expected in Remark 45, equation (58), of Bu, Gu and Jaffe, Quantum Ruzsa Divergence to Quantify Magic (arXiv:2401.14385v3). A computationally diagonal equal mixture of two basis states in dimension seven has quantum-doubling constant sqrt(2) and relative entropy log(7/2) from its mean state, while the proposed bound is at most (1/2)log(7). The contradiction uses exact inequalities and persists for every tensor power. For the same family, we verify that the weaker estimate (59) has a positive denominator and holds strictly, and give the equality case of the companion entropy-growth statement for computationally diagonal states.