Numerical evidence for a bipartite pure state entanglement witness from approximate analytical diagonalization

TL;DR

This paper introduces a numerical method to estimate bipartite pure state entanglement using an approximate analytical diagonalization that simplifies calculations and relates directly to the Log Negativity, providing exact results for certain classes of states.

Contribution

It presents a new approximate analytical diagonalization technique to evaluate entanglement witnesses directly from wavefunction coefficients, avoiding eigenvalue computations.

Findings

The entanglement witness exactly matches Log Negativity for pure states with positive Hermitian amplitude matrices.

The approximate Log Negativity formula provides a lower bound for general states and is exact for two-qubit pure states.

The method extends to certain mixed states, maintaining exactness for specific pure state decompositions.

Abstract

We show numerical evidence for a bipartite $d\times d$ pure state entanglement witness that is readily calculated from the wavefunction coefficients directly, without the need for the numerical computation of eigenvalues. This is accomplished by using an approximate analytic diagonalization of the bipartite state that captures dominant contributions to the negativity of the partially transposed state. We relate this entanglement witness to the Log Negativity, and show that it exactly agrees with it for the class of pure states whose quantum amplitudes form a positive Hermitian matrix. In this case, the Log Negativity is given by the negative logarithm of the purity of the amplitudes consider as a density matrix. In other cases, the witness forms a lower bound to the exact, numerically computed Log Negativity. The formula for the approximate Log Negativity achieves equality with the exact Log Negativity for the case of an arbitrary pure state of two qubits, which we show analytically. We compare these results to a witness of entanglement given by the linear entropy. Finally, we explore an attempt to extend these pure state results to mixed states. We show that the Log Negativity for this approximate formula is exact on the class of pure state decompositions for which the quantum amplitudes of each pure state form a positive Hermitian matrix.