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Exact Separation of Words via Trace Geometry

Zeyu Chen, Junde Wu

Abstract

A basic question in the theory of two-state measure-once quantum finite automata (MO-QFAs) is whether two distinct input words can be separated with certainty. In the setting considered here, this exact separation problem reduces to a trace-vanishing question in \(SU(2)\): given distinct positive words \(u\) and \(v\), find matrices \(A,B\in SU(2)\) such that the evaluated trace of \(u^{-1}v\) is zero. The central difficulty lies in the genuinely nonabelian regime where \(u\) and \(v\) have the same abelianization, so the obvious commutative information disappears and the fine structure of the word must be connected to the geometry of representations. This paper develops a slice-driven framework for that task and proves exact separation for every hard positive-word difference covered by four explicit certified conditions, thereby reducing the problem to a sharply delimited residual super-degenerate class. The method extracts algebraic data from the positive-word difference and uses them to select explicit low-dimensional families in \(SU(2)^2\) on which the trace becomes computable. On the algebraic side, the metabelian polynomial is decomposed into explicit interval blocks determined by prefix statistics, and a suitable slope specialization preserves nontrivial information. On the analytic side, the paper derives a computable quadratic trace identity on a visible one-parameter family and complements it with a Laurent-matrix sum-of-squares identity in a parallel algebraic model. These certified criteria are already strong in numerical experiments. This paper also shows that no method based only on finitely many finite-image tests can be universal.

Exact Separation of Words via Trace Geometry

Abstract

A basic question in the theory of two-state measure-once quantum finite automata (MO-QFAs) is whether two distinct input words can be separated with certainty. In the setting considered here, this exact separation problem reduces to a trace-vanishing question in \(SU(2)\): given distinct positive words and , find matrices \(A,B\in SU(2)\) such that the evaluated trace of is zero. The central difficulty lies in the genuinely nonabelian regime where and have the same abelianization, so the obvious commutative information disappears and the fine structure of the word must be connected to the geometry of representations. This paper develops a slice-driven framework for that task and proves exact separation for every hard positive-word difference covered by four explicit certified conditions, thereby reducing the problem to a sharply delimited residual super-degenerate class. The method extracts algebraic data from the positive-word difference and uses them to select explicit low-dimensional families in \(SU(2)^2\) on which the trace becomes computable. On the algebraic side, the metabelian polynomial is decomposed into explicit interval blocks determined by prefix statistics, and a suitable slope specialization preserves nontrivial information. On the analytic side, the paper derives a computable quadratic trace identity on a visible one-parameter family and complements it with a Laurent-matrix sum-of-squares identity in a parallel algebraic model. These certified criteria are already strong in numerical experiments. This paper also shows that no method based only on finitely many finite-image tests can be universal.

Paper Structure

This paper contains 18 sections, 26 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notations
  4. Fox calculus and the metabelian polynomial
  5. Trace-zero witnesses
  6. Metabelian structure and slope-visible SU(2) slices
  7. Metabelian persistence and slope specializations
  8. An explicit one-parameter family and the quadratic Fox term
  9. A Laurent-matrix identity for the trace deficit
  10. Explicit slice families and trace-zero witnesses
  11. Explicit specializations of the one-parameter family
  12. The case |J|=1: classification and separation
  13. Character coordinates and further explicit tests
  14. Finite-menu viewpoint and limits of finite-image methods
  15. Why fixed finite-image completions cannot work
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $\overline{(\cdot)}:\mathbb{Z}[F_2]\to \mathbb{Z}[T^{\pm1},S^{\pm1}]$ be the abelianization homomorphism determined by $a\mapsto T$ and $b\mapsto S$. For $w\in F_2$, define If $w\in F_2'$, then $B_w(1,S)=0$. Equivalently, there exists a unique Laurent polynomial $M_w(T,S)\in \mathbb{Z}[T^{\pm1},S^{\pm1}]$ such that The Laurent polynomial $M_w(T,S)$ is called the metabelian polynomial of $w$.

Theorems & Definitions (53)

  • Theorem 2.1: cf. Fox53
  • Theorem 2.2: cf. LyndonSchupp77
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 3.4
  • ...and 43 more