Bipartite Entanglement, Partial Transposition and the Uncertainty Principle for Finite-Dimensional Hilbert Spaces

TL;DR

This paper links partial transposition to a sign change in the Wigner function for finite-dimensional systems and formulates an uncertainty relation that detects entanglement, extending continuous-variable results to discrete systems.

Contribution

It introduces an uncertainty relation for discrete two-particle systems that detects entanglement via violation after partial transposition, generalizing continuous-variable methods.

Findings

Partial transposition corresponds to a sign change in the Wigner function.

Violation of the uncertainty relation indicates entanglement in discrete systems.

The method detects entanglement in Werner states depending on the parameter r.

Abstract

We first show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter $r$ of a Bell state $|\Phi^{+} \rangle$ and the completely incoherent state, $\hat{\rho}_r = r |\Phi^{+} \rangle \langle \Phi^{+}| + (1-r)\frac{\hat{\mathbb{I}}}{N^2}$: we find that for $r_0 < r < 1$, where $r_0$ is obtained as a function of the dimensionality $N$, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.