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Local Quotient Image Span Controls Rank Accumulation

Claim/Theorem

Let \(H\) be the binary qubit-support matrix of the chosen local-generator presentation of a Quantum Tanner code, with row space

\[ S=\operatorname{rowspan} H. \]

Write

\[ H=\sum_v H(v) \]

in the sense of [[quantum-tanner-local-generator-blowup.md]], where each local block \(H(v)\) is supported on the root neighborhood \(Q(v)\) and has row space

\[ S(v)=\operatorname{rowspan} H(v). \]

Fix a qubit cut \(L \sqcup R = Q\) and set

\[ B(L):=S_L+S_R, \]

where \(S_L\) and \(S_R\) are the subspaces of \(S\) supported entirely on \(L\) and \(R\).

Define the local global-quotient image

\[ W_v(L) := \frac{S(v)+B(L)}{B(L)} \;\le\; \frac{S}{B(L)}. \]

Then:

  1. The global intrinsic cut rank is exactly the span of these local quotient images:
\[ \lambda_{M(H)}(L) = \chi_L(S) = \dim \frac{S}{B(L)} = \dim\Big(\sum_v W_v(L)\Big). \]
  1. If one defines the surviving local image mass
\[ \sigma_H(L):=\sum_v \dim W_v(L), \]

then

\[ \lambda_{M(H)}(L)\le \sigma_H(L)\le \mu_H(L), \]

where \(\mu_H(L)\) is the local crossing mass from [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]].

  1. Consequently, the local-to-global accumulation problem splits into two distinct losses:

  2. survival loss:

\[ \mu_H(L)\rightsquigarrow \sigma_H(L), \]

where local cross-cut classes die after passing to the full quotient by \(B(L)\); - overlap loss:

\[ \sigma_H(L)\rightsquigarrow \lambda_{M(H)}(L), \]

where the surviving images \(W_v(L)\) overlap inside the global quotient.

So any theorem of the form

\[ \lambda_{M(H)}(L)\ge c\,\mu_H(L) \]

must control both losses, not merely the support geometry of the neighborhoods \(Q(v)\).

A concrete packed surrogate that always lower-bounds the global cut rank is therefore

\[ \nu_H(L) := \max\Big\{ \sum_{v\in F}\dim W_v(L) \;:\; \sum_{v\in F} W_v(L)\ \text{is a direct sum} \Big\}, \]

for which

\[ \nu_H(L)\le \lambda_{M(H)}(L). \]

This is the sharpest currently supported replacement for raw local crossing mass: the true accumulation bottleneck is not just how many local blocks are crossed, or how little they overlap on qubits, but how many of their quotient images survive and remain independent in \(S/B(L)\).

Proof sketch:

  1. By the local-generator construction, \(S=\sum_v S(v)\).
  2. Quotienting by \(B(L)\) gives
\[ \frac{S}{B(L)} = \sum_v \frac{S(v)+B(L)}{B(L)} = \sum_v W_v(L), \]

which proves the span formula. 3. Since

\[ S(v)_L+S(v)_R \subseteq S(v)\cap B(L), \]

one has

\[ \dim W_v(L) = \dim S(v)-\dim(S(v)\cap B(L)) \le \dim S(v)-\dim(S(v)_L+S(v)_R) = \chi_L(S(v)). \]

Summing over \(v\) gives \(\sigma_H(L)\le \mu_H(L)\). 4. Finally,

\[ \dim\Big(\sum_v W_v(L)\Big)\le \sum_v \dim W_v(L), \]

which yields \(\lambda_{M(H)}(L)\le \sigma_H(L)\).

Construction-level evidence from explicit Quantum Tanner instances with \(\Delta=3\) and the Appendix C local matrices of 10.48550/arXiv.2508.05095:

  • D_4 / [36,8,3]:
\[ \lambda=5,\qquad \sigma=16,\qquad \mu=19. \]

All 16 local blocks are crossed on the sampled balanced cut, their pairwise neighborhood overlaps lie in {0,2,3}, and the maximum disjoint crossed subfamily has size 4. - D_6 / [54,11,4]:

\[ \lambda=12,\qquad \sigma=32,\qquad \mu=32. \]

The sampled balanced cut crosses 24 local blocks, with pairwise overlaps in {0,1,2,3}, and the maximum disjoint crossed subfamily has size 6. - D_8 / [72,14,4]:

\[ \lambda=13,\qquad \sigma=35,\qquad \mu=45. \]

The sampled balanced cut crosses 32 local blocks, again with pairwise overlaps in {0,1,2,3}, and the maximum disjoint crossed subfamily has size 8.

These explicit instances show that bounded pairwise overlap and abundant crossed local blocks do not by themselves force strong rank accumulation: the decisive issue is the dimension of the span of the quotient images \(W_v(L)\).

Dependencies

  • [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]]
  • [[stabilizer-cut-rank-functional.md]]
  • [[cut-rank-is-interface-state-dimension.md]]
  • [[quantum-tanner-local-generator-blowup.md]]
  • [[cross-cut-rank-not-monotone-under-stabilizer-extension.md]]

Conflicts/Gaps

  • The surrogate \(\nu_H(L)\) is exact but not yet combinatorial. It isolates the right object, but no current source on disk proves \(\nu_H(L)=\Omega(|Q|)\) for balanced cuts in the Quantum Tanner family.
  • The D4/D6/D8 evidence is computational and sample-based, not yet a theorem over the whole family.
  • The current node does not show how to infer direct-sum or large-span behavior of the quotient images \(W_v(L)\) from the square-neighborhood overlap pattern alone.
  • So the exact remaining invariant is now clear: one needs a family theorem forcing linearly many independent local quotient images in \(S/B(L)\), not merely linearly many crossed neighborhoods.

Sources

  • 10.48550/arXiv.2109.14599
  • 10.48550/arXiv.0805.2199
  • 10.48550/arXiv.2202.13641
  • 10.48550/arXiv.2508.05095