Purity as a measure of entanglement between two distant mechanical mirrors

Characterizing and quantifying quantum entanglement in states of open quantum systems are at the heart of quantum information theory. In this work, we study the mixedness and entanglement of a two-mode symmetrical Gaussian state evolving in a thermal Markovian environment. The considered state is that of two movable end-mirrors of two distant Fabry-Perot cavities coupled via a two-mode squeezed light and driven by coherent lasers. We use the logarithmic negativity to consistently quantify the degree of entanglement between the two mechanical modes. The purity is employed as a witness of mixedness and characterize of entanglement in the two-mode mechanical bi-partition. By comparing the behavior of the logarithmic negativity with that of the purity under the effect of various parameters and based on the entanglement classification established in [G. Adesso, A. Serafini and F. Illuminati, Phys. Rev. Lett. 92, 087901 (2004)], we found that the purity can be used as a characterize and measure of entanglement in the two-mode symmetrical state when the squeezing parameter is fixed. The effects of thermal noise, squeezing and optomechanical coupling are also discussed in detail.