A fixed POVM need not attain ensemble Holevo information

Given: The equiprobable ensemble of the computational basis states $\mathrm{\mid }0⟩,\mathrm{\mid }1⟩$ on a qubit, and the fixed $X$-basis POVM with effects $\mathrm{\mid }+⟩⟨+\mathrm{\mid },\mathrm{\mid }-⟩⟨-\mathrm{\mid }$, where $\mathrm{\mid }\pm ⟩:=(\mathrm{\mid }0⟩\pm \mathrm{\mid }1⟩)\mathrm{/}\sqrt{2}$.

Finite classical-quantum states and Holevo information: the ensemble Holevo information is the entropy of the average state minus the average output entropy.

Finite mutual information: for finite classical variables, mutual information measures dependence and equals zero when the joint law is the product of its marginals.

Finite POVMs and their outcome law: the outcome probability on state $\rho$ for POVM effect $E_y$ is $\mathrm{tr}(\rho E_y)$.

The finite Holevo measurement bound: every finite POVM has classical mutual information at most the ensemble Holevo information.

Density operator and Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely: the pure qubit basis projectors and their average $I\mathrm{/}2$ are density operators with the displayed spectra.