Quantum Entanglement & Purity Testing: A Graph Zeta Function Perspective

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.

arXiv comment13 pages, 10 figures, to appear in Physics Letters A (2023)

References in corpus (11)

• Separability Criterion for Density Matrices
• Separability of Mixed States: Necessary and Sufficient Conditions
• A complete family of separability criteria
• Distinguishing separable and entangled states
• The laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states
• Detecting multipartite entanglement
• Applications of coherent classical communication and the Schur transform to quantum information theory
• Some families of density matrices for which separability is easily tested
• Testing product states, quantum Merlin-Arthur games and tensor optimisation
• Combinatorial approach to multipartite quantum systems:basic formulation
• Cycle Index Polynomials and Generalized Quantum Separability Tests