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Cross-Cut Rank Is Not Monotone Under Stabilizer Extension

Claim/Theorem

The intrinsic cross-cut stabilizer rank

\[ \chi_L(\mathcal S) \]

is not monotone under enlarging the stabilizer space. That is, there exist stabilizer spaces

\[ \mathcal S \subsetneq \mathcal T \]

on the same qubit set and a cut \(L\) such that

\[ \chi_L(\mathcal T) < \chi_L(\mathcal S). \]

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.2109.14599
  • 10.48550/arXiv.0805.2199