Small-Side Local Cut Gives Full Local Cross Rank¶

Claim/Theorem¶

Let \(G\) be a full-row-rank binary matrix whose row space is a code \(L\subseteq \mathbf F_2^m\). Partition the coordinates as

\[ [m]=U\sqcup W, \]

and write

\[ r:=\operatorname{rank}(G)=\dim L. \]

If

\[ |U|<d(L^\perp) \qquad\text{and}\qquad |U|<d(L), \]

then the intrinsic cross-cut rank of the row space of \(G\) across the cut \(U|W\) is exactly

\[ \chi_U(\operatorname{rowspan}G)=|U|. \]

More symmetrically, if

\[ \min\{|U|,\ |W|\}<\min\{d(L),d(L^\perp)\}, \]

then

\[ \chi_U(\operatorname{rowspan}G)=\min\{|U|,\ |W|\}. \]

Proof sketch:

By [[dual-distance-gives-generator-puncture-rank.md]], the first inequality implies

\[ \operatorname{rank}(G_U)=|U|. \]
The second inequality implies that deleting the coordinates in \(U\) does not reduce the row rank:

if \(\operatorname{rank}(G_W)<r\), then some nonzero linear combination of the rows of \(G\) vanishes on \(W\), producing a nonzero codeword of \(L\) supported entirely in \(U\), contradicting \(|U|<d(L)\). Hence

\[ \operatorname{rank}(G_W)=r. \]
Applying [[cross-cut-stabilizer-rank-rank-formula.md]] to the row space of \(G\) gives

\[ \chi_U(\operatorname{rowspan}G) = \operatorname{rank}(G_U)+\operatorname{rank}(G_W)-r = |U|+r-r = |U|. \]

Quantum-Tanner corollary for the chosen local-generator blocks:

For each parity-i local block in [[quantum-tanner-theorem17-parity-expander.md]], the local row space is

\[ L_0=C_A\otimes C_B \qquad\text{or}\qquad L_1=C_A^\perp\otimes C_B^\perp. \]

Under the Theorem 17 hypotheses,

\[ d(L_i^\perp)\ge \delta\Delta \qquad\text{and}\qquad d(L_i)\ge \delta^2\Delta^2. \]

Therefore any local partition whose smaller side has size strictly less than \(\delta\Delta\) already has

\[ \chi_{\mathrm{local}}=\text{(smaller-side size)}. \]

So a lightly crossed root neighborhood is not merely “crossed”; it contributes the maximum possible local intrinsic cut rank.

This theorem is local. The global Quantum Tanner stabilizer space is not a direct sum of root neighborhoods, so one still needs a separate argument to globalize these local contributions without losing too much to overlaps.
The threshold is controlled by the dual distance of the local row space, which is \Theta(\Delta) rather than \Theta(\Delta^2). So the theorem only applies to genuinely light local crossings.
The result is again tied to the chosen local-generator presentation, not yet to arbitrary stabilizer bases of the same code.