Entanglement Negativity of Spin-Orbit Correlations in a general Qubit-Qudit Setup

TL;DR

This paper derives the eigenvalue spectrum of the partially transposed density matrix for pure bipartite states in a 2 x n system and explores its application to spin-orbit entanglement in protons, linking it to gluon helicity PDFs.

Contribution

It provides a complete analytical spectrum for the partial transpose of pure bipartite states in a general qubit-qudit system and applies this to study spin-orbit entanglement in protons.

Findings

Eigenvalues of the partial transpose are explicitly derived.

Negativity relates to gluon helicity PDFs and proton polarization.

Only one negative eigenvalue exists for the spectrum.

Abstract

We present the complete eigenvalue spectrum of the partially transposed density matrix for a pure bipartite quantum state acting on a generic $2 \otimes n$ Hilbert space. The spectrum contains four non-zero eigenvalues, as, \begin{eqnarray} \lambda_{1,2}=\pm \sqrt{A}, ~~~ \lambda_{3,4}= \frac{1}{2}(1\pm\sqrt{1-4 A}), \nonumber \end{eqnarray} where $A$ is the determinant of the reduced density matrix (traced over the larger subspace). As $0 \leqslant A \leqslant1/4$, only one is negative among the four non-trivial eigenvalues. Within this qubit-qudit framework, we further studied the negativity as a measure of entanglement for the case of spin-orbit correlation of partons inside a proton. The entanglement negativity for spin-orbit correlations is found to be related to the gluon helicity PDF and the Hermitian angle of the associated Hilbert space for linearly polarized protons.