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A secondary index for non-Fredholm operators associated with quantum walks

Toshikazu Natsume, Ryszard Nest

TL;DR

This work extends the Suzuki–Tanaka index theory to quantum walks on a binary tree by formulating a secondary K-theoretic index for the non-Fredholm chirality operator $Q_+$ and computing it via a trace pairing on the Cantor boundary. The authors develop a six-term K-theory framework showing $A/\mathcal{K}\cong C^*(V)\otimes C(K)$ and introduce a secondary symbol in $K_0([A,A])$, whose numerical value $\text{s-ind}\,Q_+$ is given by the Cantor measure difference $\tau(\chi_+)-\tau(\chi_-)$, capturing topological information about non-Fredholm operators in this quantum-walk model. The invariant is computable as a di-adic integer and generalizes to parameterized families with $(p,q)$, highlighting a deep connection between noncommutative geometry, quantum walks on trees, and Cantor-boundary topology. This provides a principled, calculable topological invariant for non-Fredholm chirality operators in discrete quantum dynamics with Cantor-boundary compactifications.

Abstract

We study an analogue of chirality operators associated with quantum walks on the binary tree. For those operators we introduce a K-theoretic invariant, an analogue of the index of Fredholm operators, and compute its values in the ring of di-adic integers

A secondary index for non-Fredholm operators associated with quantum walks

TL;DR

This work extends the Suzuki–Tanaka index theory to quantum walks on a binary tree by formulating a secondary K-theoretic index for the non-Fredholm chirality operator and computing it via a trace pairing on the Cantor boundary. The authors develop a six-term K-theory framework showing and introduce a secondary symbol in , whose numerical value is given by the Cantor measure difference , capturing topological information about non-Fredholm operators in this quantum-walk model. The invariant is computable as a di-adic integer and generalizes to parameterized families with , highlighting a deep connection between noncommutative geometry, quantum walks on trees, and Cantor-boundary topology. This provides a principled, calculable topological invariant for non-Fredholm chirality operators in discrete quantum dynamics with Cantor-boundary compactifications.

Abstract

We study an analogue of chirality operators associated with quantum walks on the binary tree. For those operators we introduce a K-theoretic invariant, an analogue of the index of Fredholm operators, and compute its values in the ring of di-adic integers
Paper Structure (7 sections, 16 theorems, 63 equations)

This paper contains 7 sections, 16 theorems, 63 equations.

Table of Contents

  1. introduction
  2. One-dimensional case
  3. Quantum walk on the binary tree
  4. The binary tree
  5. Noncommutative geometric approach
  6. The index theorem
  7. a generalization

Key Result

Theorem 1.1

The chirality operator $Q_+$ is Fredholm if and only if $|a(\pm\infty)| \ne \frac{1}{\sqrt{2}}$, and the Fredholm index is given by

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: W
  • Theorem 2.2: W
  • Proposition 2.3
  • proof
  • Theorem 2.4: Wold-von Neumann decomposition
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • ...and 17 more