Cross-Cut Rank Is Not Monotone Under Stabilizer Extension¶
Claim/Theorem¶
The intrinsic cross-cut stabilizer rank
is not monotone under enlarging the stabilizer space. That is, there exist stabilizer spaces
on the same qubit set and a cut \(L\) such that
Concrete example:
- take two qubits with cut \(L=\{1\}\) and \(R=\{2\}\),
-
let
\[ \mathcal S=\operatorname{span}\{11\}, \]so \(\mathcal S\) is generated by one cross-cut stabilizer,
-
let
\[ \mathcal T=\operatorname{span}\{11,\ 10\}. \]Then \(\mathcal S\subsetneq \mathcal T\), but
\[ \chi_L(\mathcal S)=1, \qquad \chi_L(\mathcal T)=0. \]Indeed, using [[cross-cut-stabilizer-rank-rank-formula.md]]:
-
for \(\mathcal S\), one has \(\operatorname{rank}(H_L)=1\), \(\operatorname{rank}(H_R)=1\), and \(\operatorname{rank}(H)=1\), hence
\[ \chi_L(\mathcal S)=1+1-1=1; \] -
for \(\mathcal T\), with generator matrix
\[ H= \begin{bmatrix} 1 & 1\\ 1 & 0 \end{bmatrix}, \]one has \(\operatorname{rank}(H_L)=1\), \(\operatorname{rank}(H_R)=1\), and \(\operatorname{rank}(H)=2\), hence
\[ \chi_L(\mathcal T)=1+1-2=0. \]So adding an \(L\)-local stabilizer can destroy the earlier cross-cut rank contribution.
-
For Conjecture 3 this is a real proof-theoretic warning: one cannot lower-bound the global intrinsic cut rank merely by exhibiting a favorable subfamily of disjoint local blocks inside the full stabilizer space. Any globalization of [[small-side-local-cut-gives-full-local-cross-rank.md]] must control the entire row space, not just a convenient subspace.
Dependencies¶
- [[cross-cut-stabilizer-rank-rank-formula.md]]
Conflicts/Gaps¶
- This is a negative structural fact, not a lower bound.
- It rules out a tempting monotonicity shortcut, but does not say what replacement argument should work.
- The explicit-family frontier is therefore tighter: globalizing local cut-rank contributions requires a theorem about the full stabilizer space or parity-check matroid, not just a selected subset of rows.
Sources¶
10.48550/arXiv.2109.1459910.48550/arXiv.0805.2199