Proposition: Anisotropic storage comparison

Suppose a declared CPTP storage map acts on the relevant Bloch components by

without adding an affine displacement or mixing these axes.

For the fixed square encoding and $X,Z$ decoder,

Optimizing preparations while retaining these decoder axes gives

Thus this restricted optimized comparison exceeds $3/4$ exactly when

Proof. For the fixed square, the two query advantages are ${\alpha }_x/(2\sqrt{2})$ and ${\alpha }_z/(2\sqrt{2})$. Average them.

For optimized preparations, the variable term in the score is

Each summand is maximized by a unit vector parallel to $(s{\alpha }_x,t{\alpha }_z)$, giving the displayed value. At ${\alpha }_x={\alpha }_z=0$, the value is $1/2$.