Theorem: Realization of every channel support family

Fix $N\ge 1$. There is a full-support compiled tower such that, for every family

a binary measurement channel ${\Lambda }_M$ satisfies

Thus every order in $\{1,\dots ,N,\infty \}$ is realizable at the question and arbitrary-channel levels.

Proof. Prepare $|+{⟩}^{\otimes N}$ and designate all $N$ qubits. The classes are singleton, every branch is supported, and $p_r=2^{-N}$.

For each nonempty $D\subseteq \mathcal{N}$, define

where $1_D$ flips precisely the bits in $D$. These are nonzero Hermitian operators with mutually disjoint matrix-unit supports, hence are Hilbert–Schmidt orthogonal. Localization gives

The Boolean-lattice transformation on nonempty subsets is triangular with nonzero diagonal, so the $Z_J$, $J\ne \varnothing$, are linearly independent over the reals.

Their real Gram matrix $G_{JK}=\mathrm{tr}(Z_J{Z}_K)$ is therefore positive definite. Set $a_J=1$ for $J\in \mathcal{F}$ and $a_J=0$ otherwise. Choose a Hermitian

so that $\mathrm{tr}(H{Z}_J)=a_J$. Take $t>0$ with $t\Vert H{\Vert }_{\infty }\le 1$, and put

This is an effect. Since $Z_J$ is traceless for nonempty $J$,

Corollary cor:question-order gives the prescribed support. For the empty family, $M=1/2$ suffices.