For three qubits, the eight fine projectors are \(P_a={\left|a\right\rangle}{\left\langle a\right|}\). Grouping addresses according to the parity of their Hamming weight gives
This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children.
Reading the drawing
The upper row contains the four even addresses and the lower row the four odd addresses. The note distinguishes the ranks of the coarse projectors from the ranks of their fine children.
Drawing labels and data 11 labels
Read the original labels as selectable text. Tables preserve grid relationships; other short labels are grouped compactly.
- even
- \({\left|000\right\rangle}\)
- \({\left|011\right\rangle}\)
- \({\left|101\right\rangle}\)
- \({\left|110\right\rangle}\)
- odd
- \({\left|001\right\rangle}\)
- \({\left|010\right\rangle}\)
- \({\left|100\right\rangle}\)
- \({\left|111\right\rangle}\)
Each coarse projector has rank four. All eight fine projectors have rank one.
Refinement is not measurement
The fine packet description and the coarse packet description each reconstruct the full vector when every component is retained. Replacing a state by the mixture \(\sum_aP_a\rho P_a\) is instead a dephasing channel. That operation removes coherence between the fine packets and can differ from dephasing only between the even and odd sectors.
For instance, a superposition of \({\left|000\right\rangle}\) and \({\left|011\right\rangle}\) lies entirely in the even packet. Its internal relative phase survives a parity measurement but is lost by a complete measurement in the computational basis if outcomes are forgotten. The two measurements use projectors from the same refinement hierarchy but expose different information.
The exact check verifies all eight addresses, their parity grouping, the rank of both coarse projectors, and equality with the tensor expression above. The general residue version replaces parity by total label modulo \(N\) and groups the tensor descendants by their summed residue.
Read this as
A coarse packet is a whole subspace. Internal coherence survives a change of address description. It is lost only when an operation discards it.