Support and weight of an operator on a finite register

Fix an ordered physical register ${\mathcal{H}}_P={⨂}_{j<n}{\mathcal{H}}_j$ with finite-dimensional factors. For $S\subseteq [n]:=\{0,\dots ,n-1\}$, form ${\mathcal{H}}_S$ and ${\mathcal{H}}_{S^c}$ in the inherited order, and let ${\mathrm{\Pi }}_S:{\mathcal{H}}_P\to {\mathcal{H}}_S\otimes {\mathcal{H}}_{S^c}$ be the declared bijective wire permutation. An operator $E\in \mathrm{End}({\mathcal{H}}_P)$ is supported in $S$ if there is $E_S\in \mathrm{End}({\mathcal{H}}_S)$ with

${\mathrm{\Pi }}_{S}E{\mathrm{\Pi }}_S^{-1}=E_S\otimes I_{S^c}\mathrm{.}$

Its weight is $\mathrm{wt}(E):=\min \{\mathrm{\mid }S\mathrm{\mid }:E\text{ is supported in }S\}$. The minimum exists because the full set $[n]$ supports every operator and $[n]$ has finitely many subsets. Let ${\mathcal{H}}_{\mathrm{\varnothing }}=\mathbb{C}$; scalar identity operators have weight zero. The weight is invariant under reordering the tensor factors by their declared wire permutation (Operator support weight is invariant under wire permutation ↗).

Here $E_S\otimes I_{S^c}$ is the tensor map of Tensor products of finitely many linear maps. The product factors may have arbitrary finite dimensions; qubits are the special case $\dim {\mathcal{H}}_j=2$ for every $j$.

The register tensor pairing and exact permutation convention are supplied by Ordered tensor registers, their inner product, and wire permutations and Finite wire permutations are unitary basis permutations. Their current bodies were inspected; Operator support weight is invariant under wire permutation ↗ proves the resulting weight invariance.