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

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.

Figure 1

The relation between the theoretical precise values $N_{\mathrm{theory}}$ and “experimental” noisy values $N_{\mathrm{experiment}}$ of the negativity for ${10}^4$ density matrices randomly generated in a Monte Carlo simulation. For each density matrix, all these moments ${\Pi }_n$ were calculated and then a random noise $\delta {\Pi }_n\in [0,\Delta {\Pi }_n]$ was added. The maximal noise $\Delta {\Pi }_n$ was set to (a) ${10}^{-3}{\Pi }_n$ and (b) ${10}^{-2}{\Pi }_n$. This figure demonstrates that the value of the experimentally estimated negativity is very sensitive to noise, especially if this value is approaching 0.

Figure 2

Same as in Fig. 1 but for the relation between the theoretical $\left(w_{\mathrm{theory}}\right)$ and experimental $\left(w_{\mathrm{experiment}}\right)$ values of the UWE $w$, given in Eq. (12). This figure demonstrates how the value of the experimentally estimated witness is sensitive to noise.

Figure 3

Relation between the negativity $N$ and the universal entanglement witness $w$ for various states as defined in Table 1. (a) The relations $N_n\left(w\right)$ for states, given in the $n\text{th}$ case in Table 1. (b), (c), and (d) The shaded areas depict the relation $N_n\left(w\right)$ for two-parameter states given in the cases for $n=4,5,6$, respectively. The covered area lies between the dashed curve corresponding to the lower bound $f\left(w\right)$, defined in Eq. (16), and the solid curve corresponding to the upper bound $\sqrt[4]{w}$. In all these panels, the shaded areas do not cover the whole space between the boundaries. The whole area is covered only in case 7 which, for brevity, is not presented here. Strictly speaking, the whole area is not covered in panel (c).