Quantifying entanglement of a two-qubit system via measurable and invariant moments of its partially transposed density matrix

TL;DR

This paper presents a practical method to directly measure the negativity of two-qubit states using accessible moments of the partially transposed density matrix, linking it to invariants and analyzing noise effects.

Contribution

It introduces a novel experimental approach to quantify two-qubit entanglement through measurable moments and invariants, enhancing practical entanglement detection.

Findings

Negativity can be calculated from three moments of the partially transposed density matrix.

Negativity expressed as a function of six invariants derived from the density matrix.

Analysis of noise impact on entanglement estimation using the proposed method.

Abstract

We describe a direct method to determine the negativity of an arbitrary two-qubit state in experiments. The method is derived by analyzing the relation between the purity, negativity, and a universal entanglement witness for two-qubit entanglement. We show how the negativity of a two-qubit state can be calculated from just three experimentally accessible moments of the partially transposed density matrix of a two-photon state. Moreover, we show that the negativity can be given as a function of only six invariants, which are linear combinations of nine invariants from the complete set of 21 fundamental and independent two-qubit invariants. We analyze the relation between these moments and the concurrence for some classes of two-qubit states (including the X states, as well as pure states affected by the amplitude-damping and phase-damping channels). We also discuss the possibility of using the universal entanglement witness as an entanglement measure for various classes of two-qubit states. Moreover, we analyze how noise affects the estimation of entanglement via this witness.