Skip to content

Six-Qubit Witness Satisfies Direct Fsep

Claim/Theorem

Let f_{\mathcal S} be the 6-ary Boolean cut-rank function from [[six-qubit-stabilizer-cut-rank-escapes-modular-plus-fan-cone.md]], coming from the stabilizer matrix

\[ H= \begin{pmatrix} 1&0&0&1&1&1\\ 0&1&0&0&0&1\\ 0&0&1&0&0&1 \end{pmatrix}. \]

Then f_{\mathcal S} satisfies the direct higher-arity weighted-polymorphism condition

\[ F_{\mathrm{sep}} \in \mathrm{Mul}(\{f_{\mathcal S}\}), \]

where F_{\mathrm{sep}} is the Boolean 5-ary multimorphism from Živný-Cohen-Jeavons, Theorem 16 (p. 3354).

Equivalently: even without passing to any 4-ary minor, the original 6-qubit witness itself obeys the universal F_{\mathrm{sep}} inequality. Therefore the present 6-qubit witness is not separated from ordinary hidden-vertex graph-cut representability by the currently available F_{\mathrm{sep}} weighted-polymorphism method.

This is a derived exact finite theorem.

  1. By [[cross-cut-stabilizer-rank-rank-formula.md]], f_{\mathcal S} is the binary matroid connectivity function of the row space of H.

  2. The columns of H split as

    • P=\{1,4,5\}, three parallel copies of e_1,
    • Q=\{2,3,6\}, with column types e_2,e_3,e_1+e_2+e_3.

  3. Writing p(x):=\sum_{i\in P}x_i and q(x):=\sum_{i\in Q}x_i, one checks exactly that

    \[ f_{\mathcal S}(x) = \mathbf 1[x\notin\{0^6,1^6\}] + \mathbf 1[p(x)\in\{1,2\}]\,\mathbf 1[q(x)\in\{1,2\}]. \]

  4. For a 5-tuple (x^{(1)},\dots,x^{(5)}), decompose each assignment as its P-part and Q-part. Since F_{\mathrm{sep}} acts coordinatewise, the input-output data on P and Q decouple.

  5. For either 3-bit block (P or Q), each of the five inputs and five outputs falls into one of three categories: weight 0, weight 3, or mixed weight 1/2. Hence a block contributes only six 5-bit masks:

    • input masks for categories 0,3,\mathrm{mix},
    • output masks for categories 0,3,\mathrm{mix}.

  6. The F_{\mathrm{sep}} defect

    \[ \Delta := \sum_{j=1}^5 f_{\mathcal S}(y^{(j)}) - \sum_{i=1}^5 f_{\mathcal S}(x^{(i)}) \]

    depends only on those block masks, and simplifies to

    \[ \Delta = |I^P_0\cap I^Q_0|-|O^P_0\cap O^Q_0| + |I^P_3\cap I^Q_3|-|O^P_3\cap O^Q_3| + |O^P_{\mathrm{mix}}\cap O^Q_{\mathrm{mix}}| -|I^P_{\mathrm{mix}}\cap I^Q_{\mathrm{mix}}|. \]

  7. Exhaustive enumeration over all 8^5=32768 block 5-tuples produces only 768 distinct block summaries. Exhaustive search over all pairs of such summaries yields

    \[ \max \Delta = 0. \]

    Therefore no F_{\mathrm{sep}} violation exists, and f_{\mathcal S} satisfies the direct higher-arity F_{\mathrm{sep}} condition.

Consequences for the current frontier:

  • [[six-qubit-stabilizer-cut-rank-escapes-modular-plus-fan-cone.md]] is only a failure of the current constructive fan-cone route, not a failure of the F_{\mathrm{sep}} necessary condition;
  • [[six-qubit-witness-survives-all-four-ary-exact-minors.md]] was not merely a low-arity artifact: even the original 6-ary witness passes the direct F_{\mathrm{sep}} test;
  • any negative theorem against ordinary hidden-vertex graph-cut representability must therefore use an obstruction strictly stronger than the present fan-cone and F_{\mathrm{sep}} methods.

Dependencies

  • [[six-qubit-stabilizer-cut-rank-escapes-modular-plus-fan-cone.md]]
  • [[six-qubit-witness-survives-all-four-ary-exact-minors.md]]
  • [[cross-cut-stabilizer-rank-rank-formula.md]]

Conflicts/Gaps

  • This node still does not prove ordinary hidden-vertex graph-cut representability for the 6-qubit witness. F_{\mathrm{sep}} is a necessary condition, not a sufficient one in arity 6.
  • The grouped formula and direct F_{\mathrm{sep}} verification are derived computations, not verbatim theorem statements from the cited papers.
  • The result remains inside auxiliary-variable expressibility and does not by itself induce routing-style CD(T_n,G) semantics.

Sources

  • 10.1016/j.dam.2009.07.001
  • 10.48550/arXiv.2109.14599