Quantum Entanglement & Purity Testing: A Graph Zeta Function Perspective

TL;DR

This paper introduces a novel approach to quantum entanglement and purity testing by linking density matrices to graph zeta functions, revealing new mathematical insights and equivalences in quantum state analysis.

Contribution

It establishes a connection between quantum state properties and graph zeta functions, providing a new mathematical framework for entanglement testing.

Findings

The bipartite pure state separability algorithm is equivalent to the unity condition of zeta function coefficients.

Nonzero eigenvalues of a density matrix correspond to singularities of its zeta function.

Examples demonstrate the practical application of the theoretical results.

Abstract

We assign an arbitrary density matrix to a weighted graph and associate to it a graph zeta function that is both a generalization of the Ihara zeta function and a special case of the edge zeta function. We show that a recently developed bipartite pure state separability algorithm based on the symmetric group is equivalent to the condition that the coefficients in the exponential expansion of this zeta function are unity. Moreover, there is a one-to-one correspondence between the nonzero eigenvalues of a density matrix and the singularities of its zeta function. Several examples are given to illustrate these findings.