Manuscript En01-A@1 · En01-A.tex · proposition · 558b972b

Proposition: State interference complex

Operator-valued Möbius hierarchy and three orders · lines 1658–1690 · proof included or linked

En01-A@1#prop:statecomplex

For nonempty $J$ ,

Z_{J} \neq = 0 ⟺ J \subseteq D (r, r^{'}) for some supported pair .

Moreover,

∥ Z_{J} ∥_{HS}^{2} = r \neq = r^{'} ordered J \subseteq D (r, r^{'}) \sum p_{r} p_{r^{'}} .

Thus the state support, after adjoining the empty face, is the simplicial complex generated by supported branch difference sets.

Proof. The branch matrix units are Hilbert–Schmidt orthonormal:

tr [(∣ Φ_{r} ⟩ ⟨ Φ_{r^{'}} ∣)^{†} ∣ Φ_{s} ⟩ ⟨ Φ_{s^{'}} ∣] = δ_{r s} δ_{r^{'} s^{'}} .

Hence no distinct terms in (eq:localization) cancel globally. The norm formula and support characterization follow. Subset closure is immediate.

Displayed formulas in this passage

unlabeled · line 1660
$Z_{J} \neq = 0 ⟺ J \subseteq D (r, r^{'}) for some supported pair .$
unlabeled · line 1667
$∥ Z_{J} ∥_{HS}^{2} = r \neq = r^{'} ordered J \subseteq D (r, r^{'}) \sum p_{r} p_{r^{'}} .$
unlabeled · line 1679
$tr [(∣ Φ_{r} ⟩ ⟨ Φ_{r^{'}} ∣)^{†} ∣ Φ_{s} ⟩ ⟨ Φ_{s^{'}} ∣] = δ_{r s} δ_{r^{'} s^{'}} .$