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Part c gives the integral representationThe integrand vanishes on the diagonal $y=x$. Thus the assumed off-diagonal Peskun ordering impliesfor every $f\in L^2(\pi )$.

On the mean-zero subspace, the variational characterization of the spectral gap of a positive reversible kernel isIt follows immediately thatEquivalently, the energy inequality says $K⪯Q$ in the Löwner order on $L_0^2(\pi )$. Positivity permits the operator monotonicity of the square root and hence $K^{1/2}⪯Q^{1/2}$; the spectral representations in the question identify the top spectral values and give the same gap inequality.

Put $\rho =p{\rho }_1+(1-p){\rho }_2$. The operator inequality $\rho \ge p{\rho }_1$ and the operator monotonicity of logarithm give, on the support of ${\rho }_1$,Consequently,The corresponding inequality from $\rho \ge (1-p){\rho }_2$ yieldsAdding these inequalities and using $S(\rho )=-\mathrm{Tr}(\rho \log \rho )$ proves the entropy bound for a binary mixture:Singular states follow by adding a positive multiple of the identity and taking a limit; the endpoint cases use $0\log 0=0$.

The two forms of ${\omega }_{AB}$ found in part (i) giveTherefore, in the Löwner order,