On the Limit of Superposition States

TL;DR

The paper constructs and analyzes a broad class of local quantum states on graph-based tensor algebras using Schur kernels valued in duals of C$^*$-algebras. It proves the existence of an infinite-volume limiting state for nets over arbitrary locally finite graphs and provides an explicit finite-volume correlation structure that factors through Schur kernels; in homogeneous settings, the limit often reduces to product-type states. On the lattice $\mathbb{Z}^{\nu}$, the authors establish mixing and $\alpha$-mixing properties for the limiting state under suitable conditions, highlighting ergodic behavior and potential links to entanglement and quantum Markov phenomena. The framework connects operator-algebraic tools with tensor-structured states to yield tractable criteria for ergodicity and entanglement in infinite-volume quantum systems.

Abstract

In this paper, we study the structure of a family of superposition states on tensor algebras. The correlation functions of the considered states are described through a new kind of positive definite kernels valued in the dual of C$^\ast$-algebras, so-called Schur kernels. Mainly, we show the existence of the limiting state of a net of superposition states over an arbitrary locally finite graph. Furthermore, we show that this limiting state enjoys a mixing property and an $α$-mixing property in the case of the multi-dimensional integer lattice $\mathbb{Z}^ν$.

Theorems & Definitions (20)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Lemma 1
• Theorem 3
• Corollary 1
• Corollary 2
• Lemma 2
• Lemma 3