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arXiv:2307.03321 · Physics Letters A

Quantum entanglement & purity testing: A graph zeta function perspective

Zachary P. Bradshaw, Margarite L. LaBorde

Published Jul 5, 2023• 0 citations
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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.

DOI
10.1016/j.physleta.2023.128993

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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.
Abstract Braunstein et al have started the study of entanglement properties of the quantum states through graph theoretical approach. Their idea was to start from a simple unweighted graph G and then they have defined the quantum state from the Laplacian of the graph G . A lot of research had already been done using the similar idea. We ask here the opposite one i.e can we generate a graph from the density matrix? To investigate this question, we have constructed a unital map ϕ such that ϕ ( ρ ) = L ρ + ρ , where the quantum state is described by the density operator ρ . The entries of L ρ depends on the entries of the quantum state ρ and the entries are taken in such a way that L ρ satisfies all the properties of the Laplacian. This make possible to design a simple connected weighted graph from the Laplacian L ρ . We show that the constructed unital map ϕ characterize the quantum state with respect to its purity by showing that if the determinant of the matrix ϕ ( ρ ) − I is positive then the quantum state ρ represent a mixed state. Moreover, we study the positive partial transpose (PPT) criterion in terms of the spectrum of the density matrix under investigation and the spectrum of the Laplacian associated with the given density matrix. Furthermore, we derive the inequality between the minimum eigenvalue of the density matrix and the weight of the edges of the connected subgraph of a simple weighted graph to detect the entanglement of d 1 ⨂ d 2 dimensional bipartite quantum states. Lastly, we have illustrated our results with few examples.
In this paper we give a method to associate a graph with an arbitrary density matrix referred to a standard orthonormal basis in the Hilbert space of a finite dimensional quantum system. We study related issues such as classification of pure and mixed states, Von Neumann entropy, separability of multipartite quantum states and quantum operations in terms of the graphs associated with quantum states. In order to address the separability and entanglement questions using graphs, we introduce a modified tensor product of weighted graphs, and establish its algebraic properties. In particular, we show that Werner's definition (Werner 1989 Phys. Rev. A 40 4277) of a separable state can be written in terms of graphs, for the states in a real or complex Hilbert space. We generalize the separability criterion (degree criterion) due to Braunstein et al (2006 Phys. Rev. A 73 012320) to a class of weighted graphs with real weights. We have given some criteria for the Laplacian associated with a weighted graph to be positive semidefinite.
Abstract We consider the zeta function associated with the time evolution matrix of a bipartite walk on a bipartite graph and derive a formula for its absolute zeta function. Furthermore, we give formulas for the absolute zeta functions of the zeta functions of the time evolution matrices of bipartite walks on the complete bipartite graph and an even cycle graph. As an application, we obtain formulas for the absolute zeta functions of the zeta functions of the Grover matrices of the complete bipartite graph and an even cycle graph.
It is well know that the problem of the description of separable states of finite dimensional quantum system is central for the quantum information theory. The aim of the present paper is to point out that the conjecture of An Min Wang [1] concerning the separability of pure states is valid for pure states of bipartite quantum systems. A general condition for separability of a quantum state could be in principle obtained from the measure of entanglement. For a bipartite pure state the natural measure of entanglement is given by the von Neumann entropy of the reduced density matrices. Then, a bipartite state is separable if and only if this measure of entanglement is equal to zero, i.e., if and only if the reduced states are pure states. The purity of any state, described by a density matrix, can be verified directly using the Bloch vector, associated with that state. One of the restrictions is on the value of the squared Euclidean norm of the Bloch vector. Exactly on this restriction is based the An Min Wang criterion for the separability of a bipartite quantum state. The other restrictions concern the symmetric product of the Bloch vector with itself. We shall prove that the equations of the second set of restrictions follows from the An Min Wang restrictions and from the equations fulfilled by the Fano parameters of any pure bipartite state. In order to prove the An Min Wang conjecture we shall use three different parametrizations for the pure states of bipartite quantum systems: the generalized Bloch vector parametrization [2-10]; the Fano parametrization [3-10]; the Schmidt parametrization [11,12], and the relations between them.