Exploring entanglement, Wigner negativity and Bell nonlocality for anisotropic two-qutrit states

We introduce a family of anisotropic two-qutrit states (AITTSs). These AITTSs are expressed as $\rho _{aiso}=p\left\vert \psi _{\left( \theta,\phi \right) }\right\rangle \left\langle \psi _{\left( \theta ,\phi \right)}\right\vert +(1-p)\frac{1_{9}}{9}$ with $\left\vert \psi _{\left( \theta,\phi \right) }\right\rangle =\sin \theta \cos \phi \left\vert00\right\rangle +\sin \theta \sin \phi \left\vert 11\right\rangle +\cos\theta \left\vert 22\right\rangle $ and $1_{9}=\sum_{j,k=0}^{2}\left\vert jk\right\rangle \left\langle jk\right\vert $. For a given $p\in \lbrack 0,1]$, these states are adjustable in different ($\theta ,\phi $) directions. In the case of ($\theta ,\phi $) = ($\arccos (1/\sqrt{3}),\pi /4$), the AITTS will reduce to the isotropic two-qutrit state $\rho _{iso}$. In addition, the AITTSs are severely affected by the white noise ($\rho _{noise}=1_{9}/9$). Three properties of the AITTSs, including entanglement, Wigner negativity and Bell nonlocality, are explored detailedly in the analytical and numerical ways. Each property is witnessed by an appropriate existing criterion. Some of our results are summarized as follows: (i) Large entanglement does not necessarily mean high Wigner negativity and strong Bell nonlocality. (ii) A pure state with a large Schmidt number does not necessarily have a greater Wigner negativity. (iii) Only when $\left\vert\psi _{\left( \theta ,\phi \right) }\right\rangle $ has the Schmidt number 3, the AITTS has the possibility of exhibiting Bell nonlocality in proper parameter range.