Exact Separation of Words via Trace Geometry
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913079085760512 |
|---|---|
| author | Chen, Zeyu Wu, Junde |
| author_facet | Chen, Zeyu Wu, Junde |
| contents | A basic question in the study of measure-once quantum finite automata is whether two distinct input words can be separated with certainty. The exact separation problem reduces to a trace-vanishing question in \(SU(2)\). The main difficulty lies in the genuinely nonabelian regime, where \(u\) and \(v\) have the same abelianization. This paper develops a slice-driven framework that converts algebraic invariants of the word -- prefix statistics, metabelian polynomials, and slope specializations -- into explicit low-dimensional families in \(SU(2)^2\) on which the trace-vanishing question can be analyzed effectively. A quadratic trace-deficit identity on a principal one-parameter family provides the main algebraic-to-geometric bridge. Building on this framework, the paper establishes three core certified slice criteria: a dihedral criterion, equivalently readable through a signed \(a\)-count; a quaternionic criterion; and a local one-row criterion. Together with a supplementary interior-point test and a binary-dihedral slice, these results sharply reduce the unresolved portion of the problem to a residual super-degenerate class, while also clarifying the limitations of certification strategies based only on finitely many finite-subgroup evaluations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29411 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact Separation of Words via Trace Geometry Chen, Zeyu Wu, Junde Formal Languages and Automata Theory A basic question in the study of measure-once quantum finite automata is whether two distinct input words can be separated with certainty. The exact separation problem reduces to a trace-vanishing question in \(SU(2)\). The main difficulty lies in the genuinely nonabelian regime, where \(u\) and \(v\) have the same abelianization. This paper develops a slice-driven framework that converts algebraic invariants of the word -- prefix statistics, metabelian polynomials, and slope specializations -- into explicit low-dimensional families in \(SU(2)^2\) on which the trace-vanishing question can be analyzed effectively. A quadratic trace-deficit identity on a principal one-parameter family provides the main algebraic-to-geometric bridge. Building on this framework, the paper establishes three core certified slice criteria: a dihedral criterion, equivalently readable through a signed \(a\)-count; a quaternionic criterion; and a local one-row criterion. Together with a supplementary interior-point test and a binary-dihedral slice, these results sharply reduce the unresolved portion of the problem to a residual super-degenerate class, while also clarifying the limitations of certification strategies based only on finitely many finite-subgroup evaluations. |
| title | Exact Separation of Words via Trace Geometry |
| topic | Formal Languages and Automata Theory |
| url | https://arxiv.org/abs/2603.29411 |