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

Abstract

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We introduce a family of anisotropic two-qutrit states (AITTSs). These AITTSs are expressed as $\rho_{aiso}=p\left|\psi_{\left(\theta,\phi\right)}\right\rangle\left\langle\psi_{\left(\theta,\phi\right)}\right|+(1-p)\frac{1_{9}}{9}$ with $\left|\psi_{\left(\theta,\phi\right)}\right\rangle=\sin\theta\cos\phi\left|00\right\rangle+\sin\theta\sin\phi\left|11\right\rangle+\cos\theta\left|22\right\rangle$ and $1_{9}=\sum_{j,k=0}^{2}\left|jk\right\rangle\left\langle jk\right|$. For a given $p\in[0,1]$, 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 ($\rho_{noise}=\frac{1_{9}}{9}$). 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 $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ has the Schmidt number 3, the AITTS has the possibility of exhibiting Bell nonlocality in proper parameter range.

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Keywords:

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I Introduction

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In various quantum fields, it is necessary to utilize quantum resources to leverage their advantages. Then, what are quantum resources?1 ; 2 All those quantum properties, we think, having the ability of going beyond classical ones in performing technological tasks, can be regarded as quantum resources. Many properties, such as nonclassicality3 ; 4 ; 5 ; 6 , non-Gaussianity7 ; 8 ; 9 ; 10 , entanglement11 ; 12 ; 13 , steering14 ; 15 ; 16 , Bell nonlocality17 ; 18 , Wigner negativity9 ; 19 ; 20 , contextuality21 ; 22 ; 23 ; 24 ; 25 , and so on, are the quantum resources. Many researchers have studied theoretical advantages and experimental applications of theses properties. Of course, all these properties have their respective certification/quantification ways and are somewhat correlated with each other26 ; 27 ; 28 ; 29 ; 30 .

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Mathematically, quantum states describing quantum systems can be represented in various ways such as state vectors, density operators, wave functions, etc 31 ; 32 . As a fundamental tool in quantum mechanics and quantum optics, Wigner function33 ; 34 ; 35 provides a quantum phase-space representation for quantum state in terms of position and momentum, analogous to classical phase space. However, Wigner function is often called as a quasi-probability distribution because it can take negative values. This feature can be quantified by the volumn of the negative part, i.e., Wigner negativity36 ; 37 ; 38 . Physically, Wigner negativity is a rigorous non-classical maker of quantum states39 , which reveals intrinsically non-classical behaviors (e.g., superposition, interference and entanglement)40 ; 41 ; 42 . Some studies have reported that Wigner negativity is the necessary resource for quantum computing43 ; 44 .

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As Hudson’s theorem45 established, for a continuous-variable system, the Wigner function of a pure state is non-negative if and only if it is a Gaussian state. While the Wigner function of a mixed state is non-negative if and only if it is a convex mixture of Gaussian states46 . Hudson’s theorem was extended into finite-dimensional systems by Gross47 . They showed that, the Wigner function of a pure state is non-negative if and only if it is a stabilizer state. While the Wigner function of a mixed state is non-negative if and only if it is a convex mixture of stabilizer states48 . In these ways, we can understand that what kind of quantum states can exhibit Wigner negativities.

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One can distinguish continuous-variable from discrete-variable quantum state by observing the Wigner function. Compared to the discrete case, people are more familiar with continuous Wigner function. With the development of quantum information, people have become increasingly enthusiastic about studying discrete Wigner functions (DWFs) in the past two decades. The DWFs have become useful tools of studying finite-dimensional quantum states49 ; 50 . In 2004, Gibbon, Hoffman, and Wootter developed the Wigner functions and investigated a class of DWFs51 . Subsequently, Galvao conjectured that the discrete Wigner negativity was necessary for quantum computation speedup52 . In 2017, Kocia and Love studied the DWFs for qubits53 . In 2024, Wootters studied the DWFs for two-qubit states, in order to interpret symplectic linear transformation in phase space54 . Recently, Antonopoulos and his co-workers presented a grand unification for all DWFs55 .

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It is well known that, the qubit, as a two-state (or two-level) system, is the basic storage unit of quantum information56 . However, more and more practical quantum protocols require high-dimensional storage units57 ; 58 . This trend triggers considerable researches related with the qudits. As the name suggests, the qudit is the $d$-state (or $d$-level) physical system, corresponding to $d$-dimensional mathematical model. For some technological tasks, qudits perhaps may be more efficient than qubits. In the current era, the internet has become indispensable in our daily lives. Subsequently, the quantum internet59 ; 60 ; 61 came into being, which has aroused extensive research interests of scientists. The main characteristic of quantum internet is to distribute and share information among many sites at a certain distance. Therefore, quantum internet must be realized in multipartite scenarios. Above mentioned reasons are driving the advances in multipartite and high-dimensional systems62 ; 63 ; 64 .

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Every knows that entanglement is the crucial resource to achieve quantum advantageous. In recent years, many groups are dedicated to studying the entanglement for high-dimensional systems65 ; 66 ; 67 ; 68 ; 69 ; 70 . In addition, Bell nonlocality becomes another current hot research topic. Since Bell proposed the original idea of using inequalities to witness nonlocality71 , many researchers have conducted extensive researches on nonlocality. Most of works focus on two aspects: one is to construct different inequalities by changing measurement scenarios72 ; 73 ; 74 , and the other is to explore the Bell nonlocality for various multipartite and high-dimensional quantum systems75 ; 76 . Recently, Fonseca and his co-workers made a survey the Bell nonlocality of entangled qudits77 . In this regard, we particularly emphasize that, Collins, Gisin, Linden, Massar, and Popescu developed an approach to construct Bell inequalities for any bipartite high-dimensional quantum systems78 . These approach-related Bell inequalities were called the CGLMP-inequalities by later researchers. In the context of the CGLMP-inequalities, many researchers have conducted a large number of studies on Bell non-locality79 ; 80 ; 81 ; 82 .

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As the simplest model of the multipartite and high-dimensional systems, two-qutrit states are often chosen as examples to conduct researches on quantum properties83 ; 84 . In fact, two-qutrit states are just bipartite three-dimensional states, which can be used in various physical platforms85 ; 86 ; 87 . In 2012, Gruca, Laskowski, and Zukowski reported the nonclassicality for pure two-qutrit entangled states88 . On the other hand, noises inevitably affects the properties of quantum states. For instance, Roy and his co-workers found that the white noise will affect the robustness of higher-dimensional nonlocality89 . Lifshitz compared and analyzed noise-robustness in various self-testing protocols90 .

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Combinating pure two-qutrit states with white noises, we introduce a family of anisotropic two-qutrit states (AITTSs), which are the extension of the isotropic two-qutrit state. To the best of our knowledge, these AITTSs and their detailed properties are not studied completely in previous works. We will explore entanglement, Wigner negativity and Bell-nonlcality for the AITTSs. The paper is organized as follows: In Sec.II, we introduce the AITTSs. In Sec.III, we analyze their entanglement in terms of an appropriate witness. In Sec.IV, we analyze their DWFs, and then study their Wigner negativities. In Sec.V, we study their Bell nonlocality, by checking the violation of the CGLMP inequality. We conclude in the last section.

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II Anisotropic two-qutrit states

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A single-qutrit state can be described in the Hilbert space spanned by three bases $\{\left|0\right\rangle,\left|1\right\rangle,\left|2\right\rangle\}$, with $\left|0\right\rangle=(\begin{array}[]{ccc}1&0&0\end{array})^{T}$, $\left|1\right\rangle=(\begin{array}[]{ccc}0&1&0\end{array})^{T}$, and $\left|2\right\rangle=(\begin{array}[]{ccc}0&0&1\end{array})^{T}$. Consequently, a two-qutrit state can be described in the nine-dimensional space spanned by nine bases, i.e., {$\left|00\right\rangle$, $\left|01\right\rangle$, $\left|02\right\rangle$, $\left|10\right\rangle$, $\left|11\right\rangle$, $\left|12\right\rangle$, $\left|20\right\rangle$, $\left|21\right\rangle$, $\left|22\right\rangle$}. We assume that the two-qutrit state is shared by qutrit A and qutrit B, with $\left|jk\right\rangle=\left|j\right\rangle_{A}\otimes\left|k\right\rangle_{B}$ ($j,k\in\mathbb{Z}_{3}=\{0,1,2\}$). In general, pure two-qutrit states can be expressed as $\left|\psi_{pure}\right\rangle=\sum_{j,k=0}^{2}c_{jk}\left|jk\right\rangle$ with $c_{jk}\in\mathbb{C}$ and $\sum_{j,k=0}^{2}\left|c_{jk}\right|^{2}=1$. For instance, Liang et al. discussed the properties for some pure two-qutrit states, such as $(\left|00\right\rangle+i\left|22\right\rangle)/\sqrt{2}$, $(\left|11\right\rangle+i\left|22\right\rangle)/\sqrt{2}$, and $(i\left|02\right\rangle+i\left|12\right\rangle+\left|10\right\rangle+\left|12\right\rangle)/2$91 .

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Many researchers have been conducted on the properties of various isotropic two-qudit states64 ; 65 ; 92 ; 93 . In the case of $d=3$, we can express the isotropic two-qutrit state as

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$$\rho_{iso}=p\left|\Phi_{3}^{+}\right\rangle\left\langle\Phi_{3}^{+}\right|+(1-p)\frac{1_{9}}{9}.$$ (1)

Here, $\left|\Phi_{3}^{+}\right\rangle=\left(\left|00\right\rangle+\left|11\right\rangle+\left|22\right\rangle\right)/\sqrt{3}$ is the maximally entangled two-qutrit state (i.e., qutrit Bell state). And, $\frac{1_{9}}{9}=\rho_{noise}$ denotes the two-qutrit white noise, with the $9\times 9$ identity matrix $1_{9}=\sum_{j,k=0}^{2}\left|jk\right\rangle\left\langle jk\right|$. From the form, $\rho_{iso}$ is a mixed state composed of $\left|\Phi_{3}^{+}\right\rangle\left\langle\Phi_{3}^{+}\right|$ with ratio $p$ and $\rho_{noise}$ with ratio $1-p$. In a sense, the parameter $p$ denotes is the probability that $\left|\Phi_{3}^{+}\right\rangle$ is unaffected by noise.

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If $\left|\Phi_{3}^{+}\right\rangle$ of $\rho_{iso}$ in Eq.(1) is replaced by $\left|\psi_{\left(\theta,\phi\right)}\right\rangle=\sin\theta\cos\phi\left|00\right\rangle+\sin\theta\sin\phi\left|11\right\rangle+\cos\theta\left|22\right\rangle$, we introduce anisotropic two-qutrit states with the form

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$$\rho_{aiso}=p\left|\psi_{\left(\theta,\phi\right)}\right\rangle\left\langle\psi_{\left(\theta,\phi\right)}\right|+(1-p)\frac{1_{9}}{9}.$$ (2)

For the convenience of writing, these states are abbreviated as AITTSs. And, we assume that they are adjustable within $\theta\in[0,\pi]$ , $\phi\in[0,2\pi]$ and $p\in[0,1]$. Two extreme scenarios will happen, that is, $\rho_{aiso}\rightarrow\rho_{noise}$ if $p=0$ and $\rho_{aiso}\rightarrow\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ if $p=1$. If $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ in Eq.(2) is further replaced by arbitrary $\left|\psi_{pure}\right\rangle$, the anisotropic character will be stronger.

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In the Hilbert space of two-qutrit systems, $\rho_{aiso}$ can be expanded as

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$$\rho_{aiso}=\left(\begin{array}[]{ccccccccc}\kappa_{1}&0&0&0&\tau_{1}&0&0&0&\tau_{2}\\ 0&\epsilon&0&0&0&0&0&0&0\\ 0&0&\epsilon&0&0&0&0&0&0\\ 0&0&0&\epsilon&0&0&0&0&0\\ \tau_{1}&0&0&0&\kappa_{2}&0&0&0&\tau_{3}\\ 0&0&0&0&0&\epsilon&0&0&0\\ 0&0&0&0&0&0&\epsilon&0&0\\ 0&0&0&0&0&0&0&\epsilon&0\\ \tau_{2}&0&0&0&\tau_{3}&0&0&0&\kappa_{3}\end{array}\right)$$ (3)

with $\epsilon=(1-p)/9$, $\kappa_{1}=p\sin^{2}\theta\cos^{2}\phi+\epsilon$, $\kappa_{2}=$ $p\sin^{2}\theta\sin^{2}\phi+\epsilon$, $\kappa_{3}=$ $p\cos^{2}\theta+\epsilon$, $\tau_{1}=(p\sin^{2}\theta\sin 2\phi)/2$, $\tau_{2}=(p\sin 2\theta\cos\phi)/2$, $\tau_{3}=(p\sin 2\theta\sin\phi)/2$.

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For $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$, we would like to give more detailed explanations. Formally, $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ is the special case of $\left|\psi_{pure}\right\rangle$ with $c_{00}=\sin\theta\cos\phi$, $c_{11}=\sin\theta\sin\phi$, $c_{22}=\cos\theta$, and $c_{01}=c_{02}=c_{10}=c_{12}=c_{20}=c_{21}=0$. Here, we define three coefficients ($c_{00}$, $c_{11}$, and $c_{22}$) of $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ by referring the conversion between spherical coordinates (Radius $r=1$, polar angle $\theta$, and azimuthal angle $\phi$) and Cartesian coordinates ($x=c_{00}$, $y=c_{11}$, and $z=c_{22}$). In terms of Schmidt number(Sn)94 ; 95 ; 96 ; 97 determined by the coefficients, $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ may be classified into the following possible Schmidt decompositions.

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(Sn-1) If there is only one non-zero coefficient, $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ will be the Sn=1 states, including $\left|\psi_{(\pi/2,0)}\right\rangle=\left|00\right\rangle\equiv\left|S_{1}^{(1)}\right\rangle$, $\left|\psi_{(\pi/2,\pi/2)}\right\rangle=\left|11\right\rangle\equiv\left|S_{1}^{(2)}\right\rangle$, $\left|\psi_{\left(0,\phi\right)}\right\rangle=\left|22\right\rangle\equiv\left|S_{1}^{(3)}\right\rangle$.

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(Sn-2) If there are two non-zero coefficients, $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ will be the Sn=2 states, including $\left|\psi_{\left(\pi/2,\phi\right)}\right\rangle=\cos\phi\left|00\right\rangle+\sin\phi\left|11\right\rangle$, $\left|\psi_{\left(\theta,0\right)}\right\rangle=\sin\theta\left|00\right\rangle+\cos\theta\left|22\right\rangle$, and $\left|\psi_{\left(\theta,\pi/2\right)}\right\rangle=\sin\theta\left|11\right\rangle+\cos\theta\left|22\right\rangle$. Note that we must ensure the condition of two non-zero coefficients. Among these Sn=2 states, $\left|\psi_{(\pi/2,\pi/4)}\right\rangle=(\left|00\right\rangle+\left|11\right\rangle)/\sqrt{2}\equiv\left|S_{2}^{(1)}\right\rangle$, $\left|\psi_{(\pi/4,0)}\right\rangle=(\left|00\right\rangle+\left|22\right\rangle)/\sqrt{2}\equiv\left|S_{2}^{(2)}\right\rangle$, and $\left|\psi_{(\pi/4,\pi/2)}\right\rangle=(\left|11\right\rangle+\left|22\right\rangle)/\sqrt{2}\equiv\left|S_{2}^{(3)}\right\rangle$ are the maximally entangled Sn=2 states. Others are the non-maximally entangled Sn=2 states, such as $\left|\psi_{(\pi/2,\pi/6)}\right\rangle=\frac{\sqrt{3}}{2}\left|00\right\rangle+\frac{1}{2}\left|11\right\rangle)\equiv\left|S_{2}^{(4)}\right\rangle$, $\left|\psi_{(\pi/6,0)}\right\rangle=\frac{1}{2}\left|00\right\rangle+\frac{\sqrt{3}}{2}\left|22\right\rangle)\equiv\left|S_{2}^{(5)}\right\rangle$, and $\left|\psi_{(\pi/6,\pi/2)}\right\rangle=\frac{1}{2}\left|11\right\rangle+\frac{\sqrt{3}}{2}\left|22\right\rangle\equiv\left|S_{2}^{(6)}\right\rangle$.

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(Sn-3) If there are three non-zero coefficients, $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ will be the Sn=3 states, such as $\left|\psi_{(\arccos(1/\sqrt{3}),\pi/4)}\right\rangle=\left|\Phi_{3}^{+}\right\rangle\equiv\left|S_{3}^{(1)}\right\rangle$ and $\left|\psi_{(\arccos(1/\sqrt{3}),\pi/6)}\right\rangle=\frac{1}{\sqrt{2}}\left|00\right\rangle+\frac{1}{\sqrt{6}}\left|11\right\rangle+\frac{1}{\sqrt{3}}\left|22\right\rangle\equiv\left|S_{3}^{(2)}\right\rangle$. Since $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ will reduce to $\left|\Phi_{3}^{+}\right\rangle$ if $\left(\theta,\phi\right)=(\arccos(1/\sqrt{3}),\phi=\pi/4)$, $\rho_{aiso}$ in this case will reduce to $\rho_{iso}$, together with $\kappa_{1}=\kappa_{2}=\kappa_{3}=(2p+1)/9$ and $\tau_{1}=\tau_{2}=\tau_{3}=p/3$. It should be noted that $\left|S_{3}^{(1)}\right\rangle$ is just $\left|\Phi_{3}^{+}\right\rangle$, together with $\arccos(1/\sqrt{3})\simeq 0.955317$ and $\pi/4\simeq 0.785398$.

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In our following work, we often use above mentioned eleven states (abbreviated the Sn=n state as $\left|S_{n}^{(i)}\right\rangle$) as examples of $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ to study our considered properties.

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III Entanglement of AITTSs

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In this section, we shall quantify entanglement for AITTSs by virtue of negativity under partial transposition98 ; 99 . Performing partial transposition in part A (or part B) for $\rho_{aiso}$, we obtain

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$$\rho_{aiso}^{T_{A}}=\rho_{aiso}^{T_{B}}=\left(\begin{array}[]{ccccccccc}\kappa_{1}&0&0&0&0&0&0&0&0\\ 0&\epsilon&0&\tau_{1}&0&0&0&0&0\\ 0&0&\epsilon&0&0&0&\tau_{2}&0&0\\ 0&\tau_{1}&0&\epsilon&0&0&0&0&0\\ 0&0&0&0&\kappa_{2}&0&0&0&0\\ 0&0&0&0&0&\epsilon&0&\tau_{3}&0\\ 0&0&\tau_{2}&0&0&0&\epsilon&0&0\\ 0&0&0&0&0&\tau_{3}&0&\epsilon&0\\ 0&0&0&0&0&0&0&0&\kappa_{3}\end{array}\right).$$ (4)

The matrix of Eq.(14) has nine eigenvalues $\lambda_{1}=\kappa_{1}$, $\lambda_{2}=\kappa_{2}$, $\lambda_{3}=\kappa_{3}$, $\lambda_{4}=\epsilon-\tau_{1}$, $\lambda_{5}=\epsilon+\tau_{1}$, $\lambda_{6}=\epsilon-\tau_{2}$, $\lambda_{7}=\epsilon+\tau_{2}$, $\lambda_{8}=\epsilon-\tau_{3}$, and $\lambda_{9}=\epsilon+\tau_{3}$. Adding up all these eigenvalues ($\sum_{j=1}^{9}\lambda_{j}$) gives $\mathrm{Tr}(\rho_{aiso}^{T_{A}})=\mathrm{Tr}(\rho_{aiso}^{T_{B}})=1$ as expected.

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The entanglement of $\rho_{aiso}$ is calculated as

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$$\mathcal{E}\left(\rho_{aiso}\right)=\frac{1}{2}(\sum_{j=1}^{9}\left|\lambda_{j}\right|-1),$$ (5)

i.e., minus sum of all negative eigenvalues ($-\sum_{i}\lambda_{i}^{-}$, $\lambda_{i}^{-}<0$). As references, we list the following special values including $\mathcal{E}(\left|S_{1}^{(1)}\right\rangle)=\mathcal{E}(\left|S_{1}^{(2)}\right\rangle)=\mathcal{E}(\left|S_{1}^{(3)}\right\rangle)=0$, $\mathcal{E}(\left|S_{2}^{(1)}\right\rangle)=\mathcal{E}(\left|S_{2}^{(2)}\right\rangle)=\mathcal{E}(\left|S_{2}^{(3)}\right\rangle)=0.5$, $\mathcal{E}(\left|S_{2}^{(4)}\right\rangle)=\mathcal{E}(\left|S_{2}^{(5)}\right\rangle)=\mathcal{E}(\left|S_{2}^{(6)}\right\rangle)\simeq 0.481481$, $\mathcal{E}(\left|S_{3}^{(1)}\right\rangle)=1$, $\mathcal{E}(\left|S_{3}^{(2)}\right\rangle)\simeq 0.932626$, and $\mathcal{E}\left(\rho_{noise}\right)=0$.

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Figure 1 depicts the variation of entanglement $\mathcal{E}\left(\rho_{aiso}\right)$ versus $p$ for eleven $(\theta,\phi)$ cases. There are five curves in this figure. Each curve is illustrated as follows:

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(eL1) The first curve corresponds to the cases of $\left(\theta,\phi\right)=(\pi/2,0)$, $(\pi/2,\pi/2)$, $\left(0,\phi\right)$. It satisfy $\mathcal{E}\left(\rho_{aiso}\right)\equiv 0$ for any $p\in[0,1]$.

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(eL2) The second curve corresponds to the cases of $\left(\theta,\phi\right)=(\pi/2,\pi/6)$, $(\pi/6,0)$, $(\pi/6,\pi/2)$. It is a piecewise function line, satisfying $\mathcal{E}\left(\rho_{aiso}\right)=0$ in the interval of $0\leq p\lesssim 0.204202$ and $\mathcal{E}\left(\rho_{aiso}\right)\simeq 0.544124p-1/9$ in the interval of $0.204202\lesssim p\leq 1$.

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(eL3) The third curve corresponds to the cases of $\left(\theta,\phi\right)=(\pi/2,\pi/4)$, $(\pi/4,0)$, $(\pi/4,\pi/2)$. It is a piecewise function line, satisfying $\mathcal{E}\left(\rho_{aiso}\right)=0$ in the interval of $0\leq p\leq 2/11$ and $\mathcal{E}\left(\rho_{aiso}\right)=$ $11p/18-1/9$ in the interval of $2/11\lesssim p\leq 1$.

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(eL4) The fourth curve corresponds to the case of $\left(\theta,\phi\right)=(\arccos(1/\sqrt{3}),\pi/6)$. It is also a piecewise function line, satisfying $\mathcal{E}\left(\rho_{aiso}\right)=0$ in the interval of $0\leq p\lesssim 0.213939$, $\mathcal{E}\left(\rho_{aiso}\right)\simeq$ $0.519359p-1/9$ in the interval of $0.213939\lesssim p\lesssim 0.277926$, $\mathcal{E}\left(\rho_{aiso}\right)\simeq 0.919146p-2/9$ in the interval of $0.277926\lesssim p\lesssim 0.320377$, and $\mathcal{E}\left(\rho_{aiso}\right)\simeq$ $1.26596p-1/3$ in the interval of $0.320377\lesssim p\leq 1$.

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(eL5) The fifth curve corresponds to the case of $\left(\theta,\phi\right)=(\arccos(1/\sqrt{3}),\pi/4)$. It is also a piecewise function line, satisfying $\mathcal{E}\left(\rho_{aiso}\right)=0$ in the interval of $0\leq p\lesssim 0.25$ and $\mathcal{E}\left(\rho_{aiso}\right)\simeq 4p/3-1/3$ in the interval of $0.25\lesssim p\leq 1$.

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From above numerical results, we can infer that, for different $\left(\theta,\phi\right)$ cases, there are different evolution curves of $\mathcal{E}\left(\rho_{aiso}\right)$ over $p$. When $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ is the Sn=1 state, $\mathcal{E}\left(\rho_{aiso}\right)$ remains at zero in the whole $p$ range. When $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ is not the Sn=1 state, $\mathcal{E}\left(\rho_{aiso}\right)$ will change with the $p$-value. For each $\left(\theta,\phi\right)$ case, there will be a $p$-value range satisfying $\mathcal{E}\left(\rho_{aiso}\right)=0$. That is to say, only when $p$-value exceeds a certain threshold, it is possible to observe $\mathcal{N}\left(\rho_{aiso}\right)>0$ for a given $\left(\theta,\phi\right)$ case. This can be seen from Fig.2, which depicts the feasibility regions satisfying $\mathcal{E}\left(\rho_{aiso}\right)>0$ in $\left(\theta,\phi,p\right)$ space.

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Figure 1: $\mathcal{E}\left(\rho_{aiso}\right)$ versus $p$ for eleven ($\theta$, $\phi$) cases. There are only five variation curves. For $p=1$, $\mathcal{E}\left(\rho_{aiso}\right)$ values are$\ 0$, $0.433013$, $0.5$, $0.932626$, $1$ in sequence.

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Figure 2: Three-dimensional feasibility region of $\mathcal{E}\left(\rho_{aiso}\right)>0$ showing entanglement in $(\theta,\phi,p)$ space. The blank region is that satisfying $\mathcal{E}\left(\rho_{aiso}\right)=0$.

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As expected, we further verify that the maximum entanglement value ($\mathcal{E}^{\max}\left(\rho_{aiso}\right)=1$) is positioned at $\left(\theta,\phi,p\right)\simeq\left(0.955317,0.785398,1\right)$, which corresponds exactly to the maximum entangled state $\left|\Phi_{3}^{+}\right\rangle$, i.e. $\mathcal{E}(\left|\Phi_{3}^{+}\right\rangle)=1$.

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IV Wigner negativity of AITTSs

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In this section, we shall analyze the DWFs and study Wigner negativities for the AITTSs. Regarding the foundations of this section, one can refer to two relevant works from Delfose’s group100 and Meyer’s group101 .

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IV.1 Discrete Wigner function

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Similar to the qubit Pauli operators $\sigma_{x}=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)$ and $\sigma_{z}=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right)$, one can introduce the qutrit Pauli operators

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$$X=\sum_{k=0}^{2}\left|k+1\right\rangle\left\langle k\right|=\left(\begin{array}[]{ccc}0&0&1\\ 1&0&0\\ 0&1&0\end{array}\right),$$ (6)

and

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$$Z=\sum_{k=0}^{2}\omega^{k}\left|k\right\rangle\left\langle k\right|=\left(\begin{array}[]{ccc}1&0&0\\ 0&\omega&0\\ 0&0&\omega^{2}\end{array}\right),$$ (7)

with $\omega=e^{\frac{2\pi i}{3}}$. They obey $X\left|k\right\rangle=\left|\left(k+1\right)\text{ mod }3\right\rangle$, $Z\left|k\right\rangle=\omega^{k}\left|k\right\rangle$, and $Z^{z}X^{x}=\omega^{xz}X^{x}Z^{z}$ for $x,z,k\in\mathbb{Z}_{3}$.

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For a qutrit, the Wiger operator in phase point $\left(u_{x},u_{z}\right)$ is defined as

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$$A_{\left(u_{x},u_{z}\right)}=\frac{1}{3}\sum_{v_{x}=0}^{2}\sum_{v_{z}=0}^{2}\omega^{u_{z}v_{x}-u_{x}v_{z}}D_{\left(v_{x},v_{z}\right)},$$ (8)

with the Heisenberg-Weyl displacement operator $D_{\left(x,z\right)}=\omega^{\frac{1}{2}xz}X^{x}Z^{z}$. Therefore, we rewrite $A_{\left(u_{x},u_{z}\right)}$ as $A_{\left(x,z\right)}$ with the following matrix

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$$A_{(x,z)}=\frac{1}{3}\allowbreak\left(\begin{array}[]{ccc}1+\omega^{-x}+\omega^{-2x}&\omega^{2z-x+2}+\omega^{2z-2x+4}+\omega^{2z}&\omega^{z}+\omega^{z-2x+2}+\omega^{z-x+\frac{5}{2}}\\ \omega^{z}+\omega^{z-2x+1}+\omega^{z-x+\frac{1}{2}}&1+\omega^{2-2x}+\omega^{-x+1}&\omega^{2z-x+3}+\omega^{2z-2x+3}+\omega^{2z}\\ \omega^{2z-x+1}+\omega^{2z-2x+2}+\omega^{2z}&\omega^{z}+\omega^{z-2x+3}+\omega^{z-x+\frac{3}{2}}&1+\omega^{1-2x}+\omega^{-x+2}\end{array}\right).$$ (9)

For the AITTSs with the density matrix $\rho_{aiso}$, the corresponding DWF can be calculated by

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$$W_{\left(x_{1},z_{1};x_{2},z_{2}\right)}\left(\rho_{aiso}\right)=\frac{1}{3^{2}}\mathrm{Tr}[(A_{\left(x_{1},z_{1}\right)}\otimes A_{\left(x_{2},z_{2}\right)})\rho_{aiso}].$$ (10)

For every two-qutrit state, there are eighty-one phase points due to $\left(x_{1},z_{1};x_{2},z_{2}\right)\in\mathbb{Z}_{3}^{4}$. In this paper, we will arrange them as follows

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$$\begin{array}[]{ccccccccc}W_{0000}&W_{0100}&W_{0200}&W_{0001}&W_{0101}&W_{0201}&W_{0002}&W_{0102}&W_{0202}\\ W_{1000}&W_{1100}&W_{1200}&W_{1001}&W_{1101}&W_{1201}&W_{1002}&W_{1102}&W_{1202}\\ W_{2000}&W_{2100}&W_{2200}&W_{2001}&W_{2101}&W_{2201}&W_{2002}&W_{2102}&W_{2202}\\ W_{0010}&W_{0110}&W_{0210}&W_{0011}&W_{0111}&W_{0211}&W_{0012}&W_{0112}&W_{0212}\\ W_{1010}&W_{1110}&W_{1210}&W_{1011}&W_{1111}&W_{1211}&W_{1012}&W_{1112}&W_{1212}\\ W_{2010}&W_{2110}&W_{2210}&W_{2011}&W_{2111}&W_{2211}&W_{2012}&W_{2112}&W_{2212}\\ W_{0020}&W_{0120}&W_{0220}&W_{0021}&W_{0121}&W_{0221}&W_{0022}&W_{0122}&W_{0222}\\ W_{1020}&W_{1120}&W_{1220}&W_{1021}&W_{1121}&W_{1221}&W_{1022}&W_{1122}&W_{1222}\\ W_{2020}&W_{2120}&W_{2220}&W_{2021}&W_{2121}&W_{2221}&W_{2022}&W_{2122}&W_{2222}\end{array}.$$ (11)

Note that $W_{x_{1}z_{1}x_{2}z_{2}}\equiv W_{\left(x_{1},z_{1};x_{2},z_{2}\right)}$. According to Eqs.(10) and (11), we can obtain the DWF values and plot the DWF figures for $\rho_{aiso}$.

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In Fig.3, we plot the DWFs for $\left|S_{1}^{(1)}\right\rangle$, $\left|S_{1}^{(2)}\right\rangle$, and $\left|S_{1}^{(3)}\right\rangle$. Their DWFs all have nine points with value $1/9$ and seventy-two points with value $0$, i.e. {$\frac{1}{9}\rightarrow 9$, $0\rightarrow 72$}. Moreover, all of their values are non-negative.

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In Fig.4, we plot the DWFs for $\left|S_{2}^{(1)}\right\rangle$, $\left|S_{2}^{(2)}\right\rangle$, and $\left|S_{2}^{(3)}\right\rangle$. Their DWFs all have three points with value $5/81$, six points with value $-5/162$, twenty-four points with value $1/162$, twenty-four points with value $-1/81$, twelve points with value $7/162$, six points with value $13/162$, and six points with value $2/81$, i.e. {$\frac{5}{81}\rightarrow 3$, $-\frac{5}{162}\rightarrow 6$, $\frac{1}{162}\rightarrow 24$, $-\frac{1}{81}\rightarrow 24$, $\frac{7}{162}\rightarrow 12$, $\frac{13}{162}\rightarrow 6$, $\frac{2}{81}\rightarrow 6$}. For these three states, their DWFs may be negative.

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In Fig.5, we plot the DWFs for $\left|S_{3}^{(1)}\right\rangle$ (see left sub-figure) and $\rho_{noise}=1_{9}/9$ (see right sub-figure). For $\left|S_{3}^{(1)}\right\rangle$, its DWF has eight points with value $5/81$, thirty-six points with value $2/81$, thirty-six points with value $-1/81$, and one point with value $7/162$, i.e. {$\frac{5}{81}\rightarrow 8$, $\frac{2}{81}\rightarrow 36$, $-\frac{1}{81}\rightarrow 36$, $\frac{7}{162}\rightarrow 1$}. That is, the DWF of $\left|\Phi_{3}^{+}\right\rangle$ may be negative. For $\rho_{noise}$, its DWF has thirty-six points with value $0$, thirty-six points with value $1/54$, and nine points with value $1/27$, i.e. {$0\rightarrow 36$, $\frac{1}{54}\rightarrow 36$, $\frac{1}{27}\rightarrow 9$}. That is, the DWF of $\rho_{noise}$ is non-negative.

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Figure 3: DWFs for $\left|S_{1}^{(1)}\right\rangle$(Left), $\left|S_{1}^{(2)}\right\rangle$(Middle); $\left|S_{1}^{(3)}\right\rangle$(Right). They have same 81 values but different distributions.

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Figure 4: DWFs for $\left|S_{2}^{(1)}\right\rangle$ (Left); $\left|S_{2}^{(2)}\right\rangle$ (Middle); $\left|S_{2}^{(3)}\right\rangle$ (Right). They have same 81 values but different distributions.

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Figure 5: DWFs for $\left|\Phi_{3}^{+}\right\rangle=\left|S_{3}^{(1)}\right\rangle$ (Left) and $\rho_{noise}=1_{9}/9$ (Right).

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As examples, we only plot the DWFs for $\left|S_{1}^{(1)}\right\rangle$, $\left|S_{1}^{(2)}\right\rangle$, $\left|S_{1}^{(3)}\right\rangle$, $\left|S_{2}^{(1)}\right\rangle$, $\left|S_{2}^{(2)}\right\rangle$, $\left|S_{2}^{(3)}\right\rangle$, $\left|S_{3}^{(1)}\right\rangle$ and $\rho_{noise}$ in this work. The DWFs, for $\left|S_{2}^{(4)}\right\rangle$, $\left|S_{2}^{(5)}\right\rangle$, $\left|S_{2}^{(6)}\right\rangle$, $\left|S_{3}^{(2)}\right\rangle$ and other $\rho_{aiso}$s, are not plotted here.

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IV.2 Wigner negativity

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For any quantum state, the Wigner function is normalized. As expected, we can definitely verify

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$$\sum_{x_{1},z_{1};x_{2},z_{2}\in\mathbb{Z}_{3}^{4}}W_{\left(x_{1},z_{1};x_{2},z_{2}\right)}\left(\rho_{aiso}\right)=1$$ (12)

for any $\rho_{aiso}$. The amount of the Wigner negativity is just minus the sum of all negative values among the DWF. Hence, the Wigner negativity of $\rho_{aiso}$ can be calculated by

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$$\mathcal{N}\left(\rho_{aiso}\right)=\frac{1}{2}[\sum_{x_{1},z_{1};x_{2},z_{2}\in\mathbb{Z}_{3}^{4}}\left|W_{\left(x_{1},z_{1};x_{2},z_{2}\right)}\left(\rho_{aiso}\right)\right|-1].$$ (13)

After making numerical calculation, we easily obtain $\mathcal{N}(\left|S_{1}^{(1)}\right\rangle)=\mathcal{N}(\left|S_{1}^{(2)}\right\rangle)=\mathcal{N}(\left|S_{1}^{(3)}\right\rangle)=0$, $\mathcal{N}(\left|S_{2}^{(4)}\right\rangle)=\mathcal{N}(\left|S_{2}^{(5)}\right\rangle)=\mathcal{N}(\left|S_{2}^{(6)}\right\rangle)\simeq 0.416975$, $\mathcal{N}(\left|S_{2}^{(1)}\right\rangle)=\mathcal{N}(\left|S_{2}^{(2)}\right\rangle)=\mathcal{N}(\left|S_{2}^{(3)}\right\rangle)=\frac{13}{27}\simeq 0.481481$, $\mathcal{N}(\left|S_{3}^{(2)}\right\rangle)\simeq 0.421011$, $\mathcal{N}(\left|S_{3}^{(1)}\right\rangle)=\frac{4}{9}\simeq 0.444444$, and $\mathcal{N}\left(\rho_{noise}\right)=0$. Some of these values can be validated from our plotted DWFs.

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Figure 6 depicts the variation of Wigner negativity $\mathcal{N}\left(\rho_{aiso}\right)$ versus $p$ for eleven $(\theta,\phi)$ cases. There are five curves in this figure. Each curve is illustrated as follows:

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(wL1) The first curve corresponds to the cases of $\left(\theta,\phi\right)=(\pi/2,0)$, $(\pi/2,\pi/2)$, $\left(0,\phi\right)$. It satisfy $\mathcal{N}\left(\rho_{aiso}\right)\equiv 0$ for any $p\in[0,1]$.

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(wL2) The second curve corresponds to the cases of $\left(\theta,\phi\right)=(\pi/2,\pi/6)$, $(\pi/6,0)$, $(\pi/6,\pi/2)$. It is a piecewise function line, satisfying $\mathcal{E}\left(\rho_{aiso}\right)=0$ in the interval of $0\leq p\lesssim 0.315949$, $\mathcal{N}\left(\rho_{aiso}\right)\simeq 0.234449p-0.0740741$ in the interval of $0.315949\lesssim p\leq 0.535898$, and $\mathcal{N}\left(\rho_{aiso}\right)=0.787346p-0.37037$ in the interval of $0.535898\lesssim p\leq 1$.

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(wL3) The third curve corresponds to the cases of $\left(\theta,\phi\right)=(\pi/2,\pi/4)$, $(\pi/4,0)$, $(\pi/4,\pi/2)$. In this curve, there are three typical points $\left(p,\mathcal{N}\right)=(0.285714,0)$, $(0.5,1/18)$, $(1,23/27)$. This curve is a piecewise function line, satisfying $\mathcal{N}\left(\rho_{aiso}\right)=0$ in the interval of $0\leq p\lesssim 0.285714$, $\mathcal{N}\left(\rho_{aiso}\right)\simeq 0.259259p-0.0740741$ in the interval of $0.285714\lesssim p\leq 0.5$, and $\mathcal{N}\left(\rho_{aiso}\right)=$ $23p/27-10/27$ in the interval of $0.5\lesssim p\leq 1$.

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(wL4) The fourth curve corresponds to the case of $\left(\theta,\phi\right)=(\arccos(1/\sqrt{3}),\pi/6)$. It is also a piecewise function line, satisfying $\mathcal{N}\left(\rho_{aiso}\right)=0$ in the interval of $0\leq p\lesssim 0.463361$, $\mathcal{N}\left(\rho_{aiso}\right)\simeq 0.319725p-0.148148$ in the interval of $0.463361\lesssim p\lesssim 0.500194$, $\mathcal{N}\left(\rho_{aiso}\right)\simeq 0.615907p-0.296296$ in the interval of $0.500194\lesssim p\lesssim 0.61731$, and $\mathcal{N}\left(\rho_{aiso}\right)\simeq$ $0.0787166p^{2}+0.753744p-0.411381$ in the interval of $0.61731\lesssim p\leq 1$.

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(wL5) The fifth curve corresponds to the case of $\left(\theta,\phi\right)=(\arccos(1/\sqrt{3}),\pi/4)$. It is also a piecewise function line, satisfying $\mathcal{N}\left(\rho_{aiso}\right)=0$ in the interval of $0\leq p\leq 0.5$ and $\mathcal{N}\left(\rho_{aiso}\right)=8p/9-4/9$ in the interval of $0.5\leq p\leq 1$.

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Figure 6: $\mathcal{N}\left(\rho_{aiso}\right)$ versus $p$ for eleven ($\theta$, $\phi$) cases. There are only five variation curves. For $p=1$, $\mathcal{N}\left(\rho_{aiso}\right)$ values are$\ 0$, $0.416975$, $0.421011$, $4/9$, and $13/27\$in order from small to large.

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Figure 7: Three-dimensional feasibility region with $\mathcal{N}\left(\rho_{aiso}\right)>0$ showing Wigner negativity in $(\theta,\phi,p)$ space. The blank region is that satisfying $\mathcal{N}\left(\rho_{aiso}\right)=0$.

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For different $\left(\theta,\phi\right)$ cases, the evolution curves of $\mathcal{N}\left(\rho_{aiso}\right)$ over $p$ are different. In most cases, $\mathcal{N}\left(\rho_{aiso}\right)$ value is not equal to $p\mathcal{N}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\left\langle\psi_{\left(\theta,\phi\right)}\right|\right)+(1-p)\mathcal{N}\left(\rho_{noise}\right)$, except that $\mathcal{N}\left(\rho_{aiso}\right)\equiv 0$ when $\left|\psi_{\left(\theta,\phi\right)}\right\rangle=\left|00\right\rangle$, $\left|11\right\rangle$, $\left|22\right\rangle$. Due to the effects of the noise, the Wigner negativity only occurs when the $p$-value exceeds a certain value. This can be seen from Fig.7, which plot the feasibility regions in $\left(\theta,\phi,p\right)$ space, satisfying $\mathcal{N}\left(\rho_{aiso}\right)>0$. That is to say, only when $p$-value exceeds a certain threshold, it is possible to observe $\mathcal{N}\left(\rho_{aiso}\right)>0$ for a given $\left(\theta,\phi\right)$ case.

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In addition, we find that $\mathcal{N}^{\max}\left(\rho_{aiso}\right)=\frac{13}{27}$ is found for $\rho_{aiso}\rightarrow\left|\psi_{\left(\theta,\phi\right)}\right\rangle=\left|S_{2}^{(i)}\right\rangle$ ($i=1,2,3$). It is worth to note that states with stronger entanglement do not necessarily have greater Wigner negativity. For instance, although $\mathcal{E}(\left|S_{3}^{(1)}\right\rangle)$ is greater than $\mathcal{E}(\left|S_{2}^{(i)}\right\rangle)$ ($i=1,2,3$), $\mathcal{N}(\left|S_{3}^{(1)}\right\rangle)$ is less than $\mathcal{N}(\left|S_{2}^{(i)}\right\rangle)$ ($i=1,2,3$).

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V Bell nonlocality of AITTSs

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As Meyer et al. pointed out101 , Wigner negativity is necessary for nonlocality in qudit systems. In this section, we study Bell nonlocality for the AITTSs by virtue of the CGLMP inequalities. We consider two separated observers, Alice and Bob. Alice can conduct two different measurements ($A_{1}$ and $A_{2}$) with respective three outcomes, i.e., $A_{1}=j$ ($j=0,1,2$) and $A_{2}=k$ ($k=0,1,2$). Similarly, Bob can conduct two different measurements ($B_{1}$ and $B_{2}$) with respective three outcomes, i.e., $B_{1}=l$ ($l=0,1,2$) and $B_{2}=m$ ($m=0,1,2$). In order to avoid the eigenstates degenerate, we further assume that their eigenstates satisfy

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$$\left|j\right\rangle_{A_{1}} = \frac{1}{\sqrt{3}}\sum_{n=0}^{2}\omega^{n\left(j+\alpha_{1}\right)}\left|n\right\rangle_{A},$$ (14)

$$\left|k\right\rangle_{A_{2}} = \frac{1}{\sqrt{3}}\sum_{n=0}^{2}\omega^{n(k+\alpha_{2})}\left|n\right\rangle_{A},$$ (15)

$$\left|l\right\rangle_{B_{1}} = \frac{1}{\sqrt{3}}\sum_{n=0}^{2}\omega^{n(-l+\beta_{1})}\left|n\right\rangle_{B},$$ (16)

$$\left|m\right\rangle_{B_{2}} = \frac{1}{\sqrt{3}}\sum_{n=0}^{2}\omega^{n(-m+\beta_{2})}\left|n\right\rangle_{B},$$ (17)

which lead to $\Pi_{A_{1}}^{\left(j\right)}=\left|j\right\rangle_{A_{1}}\left\langle j\right|$, $\Pi_{A_{2}}^{\left(k\right)}=\left|k\right\rangle_{A_{2}}\left\langle k\right|$, $\Pi_{B_{1}}^{\left(l\right)}=\left|l\right\rangle_{B_{1}}\left\langle l\right|$, and $\Pi_{B_{2}}^{\left(m\right)}=\left|m\right\rangle_{B_{2}}\left\langle m\right|$ ($j,k,l,m\in\mathbb{Z}_{3}$), respectively.

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Now, we let Alice and Bob share the AITTSs. The joint probabilities can be calculated as

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$$P\left(A_{1}=j,B_{1}=l\right) = \mathrm{Tr}[(\Pi_{A_{1}}^{\left(j\right)}\otimes\Pi_{B_{1}}^{\left(l\right)})\rho_{aiso}],$$ (18)

$$P\left(A_{1}=j,B_{2}=m\right) = \mathrm{Tr}[(\Pi_{A_{1}}^{\left(j\right)}\otimes\Pi_{B_{2}}^{\left(m\right)})\rho_{aiso}],$$ (19)

$$P\left(A_{2}=k,B_{1}=l\right) = \mathrm{Tr}[(\Pi_{A_{2}}^{\left(k\right)}\otimes\Pi_{B_{1}}^{\left(l\right)})\rho_{aiso}],$$ (20)

$$P\left(A_{2}=k,B_{2}=m\right) = \mathrm{Tr}[(\Pi_{A_{2}}^{\left(k\right)}\otimes\Pi_{B_{2}}^{\left(m\right)})\rho_{aiso}].$$ (21)

Further, we use the CGLMP inequality

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$$\mathcal{I}_{3} \equiv [P\left(A_{1}=B_{1}\right)+P\left(B_{1}=A_{2}+1\right)$$ (22)

$$+P\left(A_{2}=B_{2}\right)+P\left(B_{2}=A_{1}\right)]$$

$$-[P\left(A_{1}=B_{1}-1\right)+P\left(B_{1}=A_{2}\right)$$

$$+P\left(A_{2}=B_{2}-1\right)+P\left(B_{2}=A_{1}-1\right)]$$

$$\leq 2.$$

to study the Bell nonlocality for $\rho_{aiso}$ by observing $\mathcal{I}_{3}\left(\rho_{aiso}\right)>2$ and setting $(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})=(0,1/2,1/4,-1/4)$.

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As our references, we give $\mathcal{I}_{3}(\left|S_{2}^{(1)}\right\rangle)=\mathcal{I}_{3}(\left|S_{2}^{(2)}\right\rangle)=\mathcal{I}_{3}(\left|S_{2}^{(3)}\right\rangle)=0$, $\mathcal{I}_{3}(\left|S_{2}^{(4)}\right\rangle)=\mathcal{I}_{3}(\left|S_{2}^{(6)}\right\rangle)\simeq 1$, $\mathcal{I}_{3}(\left|S_{2}^{(1)}\right\rangle)=\mathcal{I}_{3}(\left|S_{2}^{(3)}\right\rangle)\simeq 1.1547$, $\mathcal{I}_{3}(\left|S_{2}^{(5)}\right\rangle)\simeq 1.73205$, $\mathcal{I}_{3}(\left|S_{2}^{(2)}\right\rangle)\simeq 2$, $\mathcal{I}_{3}(\left|S_{3}^{(1)}\right\rangle)\simeq 2.84399$, $\mathcal{I}_{3}(\left|S_{3}^{(1)}\right\rangle)\simeq 2.87293$, and $\mathcal{I}_{3}\left(\rho_{noise}\right)=0$. Surprisedly, for those Sn-2 states, we find $\mathcal{I}_{3}(\left|S_{2}^{(1)}\right\rangle)=\mathcal{I}_{3}(\left|S_{2}^{(3)}\right\rangle)\neq\mathcal{I}_{3}(\left|S_{2}^{(2)}\right\rangle)$ and $\mathcal{I}_{3}(\left|S_{2}^{(4)}\right\rangle)=\mathcal{I}_{3}(\left|S_{2}^{(6)}\right\rangle)\neq\mathcal{I}_{3}(\left|S_{2}^{(5)}\right\rangle)$. These results are different from those in studying $\mathcal{E}$ and $\mathcal{N}$.

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Figure 8: $\mathcal{I}_{3}\left(\rho_{aiso}\right)$ versus $p$ for eleven ($\theta$, $\phi$) cases. For $p=1$, $\mathcal{I}_{3}$ values are$\ 0$, $1$, $1.1547$, $1.73205$, $2$, $2.84399$, $2.87293$ in sequence. Each line is a straight line.

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Figure 9: Three-dimensional feasibility region with $\mathcal{I}_{3}\left(\rho_{aiso}\right)>2$ showing Bell nonlocality in $(\theta,\phi,p)$ space. The blank region is that satisfying $\mathcal{I}_{3}\left(\rho_{aiso}\right)\leq 2$.

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Fig.8 depicts $\mathcal{I}_{3}\left(\rho_{aiso}\right)$s as functions of $p$ for eleven $(\theta,\phi)$cases. There are seven lines in this figure. Each line is illustrated as follows:

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(bL1) The first line corresponds to the cases of $\left(\theta,\phi\right)=(\pi/2,0)$, $(\pi/2,\pi/2)$, $\left(0,\phi\right)$. It satisfy $\mathcal{I}_{3}\left(\rho_{aiso}\right)\equiv 0$ for any $p\in[0,1]$.

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(bL2) The second line corresponds to the cases of $\left(\theta,\phi\right)=(\pi/2,\pi/6)$, $(\pi/6,\pi/2)$. It is a straight line piecewise satisfy $\mathcal{I}_{3}\left(\rho_{aiso}\right)=p$ in the whole $p$ range.

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(bL3) The third curve corresponds to the cases of $\left(\theta,\phi\right)=(\pi/2,\pi/4)$, $(\pi/4,\pi/2)$. It is a straight line piecewise satisfy $\mathcal{I}_{3}\left(\rho_{aiso}\right)=1.1547p$ in the whole $p$ range.

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(bL4) The fourth line corresponds to the cases of $\left(\theta,\phi\right)=(\pi/6,0)$. It is a straight line piecewise satisfy $\mathcal{I}_{3}\left(\rho_{aiso}\right)=1.73205p$ in the whole $p$ range.

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(bL5) The fifth curve corresponds to the cases of $\left(\theta,\phi\right)=(\pi/4,0)$. It is a straight line piecewise satisfy $\mathcal{I}_{3}\left(\rho_{aiso}\right)=2p$ in the whole $p$range.

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(bL6) The sixth curve corresponds to the case of $\left(\theta,\phi\right)=(\arccos(1/\sqrt{3}),\pi/6)$. It is a straight line piecewise satisfy $\mathcal{I}_{3}\left(\rho_{aiso}\right)=2.84399p$ in the whole $p$ range.

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(bL7) The seventh curve corresponds to the case of $\left(\theta,\phi\right)=(\arccos(1/\sqrt{3}),\pi/4)$. It is a straight line piecewise satisfy $\mathcal{I}_{3}\left(\rho_{aiso}\right)=2.87293p$ in the whole $p$ range.

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It is worth noting that we have $\mathcal{I}_{3}\left(\rho_{aiso}\right)=p\mathcal{I}_{3}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$ for arbitrarily determined $\left(\theta,\phi\right)$ case. That is, $\mathcal{I}_{3}\left(\rho_{aiso}\right)$ is a linear function of $p$ ($\in[0,1]$) with slope value $\mathcal{I}_{3}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$. Interestingly, we can also draw the conclusion $\mathcal{I}_{3}\left(\rho_{aiso}\right)=p\mathcal{I}_{3}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)+(1-p)\mathcal{I}_{3}\left(\rho_{noise}\right)$.

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Theoretically, Bell nonlocality can be witnessed by observing $\mathcal{I}_{3}\left(\rho_{aiso}\right)>2$, i.e. the violation of Eq.(22). From above numerical results, we immediately know that there is the possibility of $\mathcal{I}_{3}\left(\rho_{aiso}\right)>2$ if and only if $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ is the Sn=3 state. In Fig.9, we plot the feasibility regions in $\left(\theta,\phi,p\right)$ space showing Bell nonlocality. Only in the range of $0.686141\lesssim p\leq 1$, we can choose proper $\left(\theta,\phi\right)$ values to satisfy $\mathcal{I}_{3}\left(\rho_{aiso}\right)>2$. Moreover, the optional $\left(\theta,\phi\right)$ region area will decrease as $p$ decreases until $p\approx 0.686141$. In other words, no matter how you choose $\left(\theta,\phi\right)$ if $0\leq p\lesssim 0.686141$, it is impossible to ensure that this inequality $\mathcal{I}_{3}\left(\rho_{aiso}\right)>2$ holds true. Here, I would like to remind everyone that $\mathcal{I}_{3}\left(\rho_{aiso}\right)$ values may be less than zero for some parameter $\left(\theta,\phi\right)$ regions, but $\mathcal{I}_{3}\left(\rho_{aiso}\right)$ values are absolutely impossible to be less than $-2$ for any parameter $\left(\theta,\phi\right)$ regions.

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In addition, we find that the maximum value of $\mathcal{I}_{3}\left(\rho_{aiso}\right)$ is $\mathcal{I}_{3}^{\max}=2.91485$, which is positioned at $\left(\theta,\phi,p\right)\simeq\left(0.906006,0.67002,1\right)\$or $\left(2.23559,3.81161,1\right)$. This is an astonishing result for us because of $\mathcal{I}_{3}^{\max}\left(\rho_{aiso}\right)\geq\mathcal{I}_{3}^{\max}\left(\rho_{iso}\right)=\mathcal{I}_{3}(\left|\Phi_{3}^{+}\right\rangle)\simeq 2.87293$. That is to say, the maximally entangled state is not the maximally Bell-nonlocality state. In fact, there are a lot of $\rho_{aiso}$s, whose $\mathcal{I}_{3}$ values are greater than $\mathcal{I}_{3}(\left|\Phi_{3}^{+}\right\rangle)$. As shown in Fig.10, two feasibility regions satisfying $\mathcal{I}_{3}\left(\rho_{aiso}\right)>\mathcal{I}_{3}(\left|\Phi_{3}^{+}\right\rangle)$ are distributed in the interval of $0.985618\lesssim p\leq 1$, together with proper $\left(\theta,\phi\right)$ values satisfying $0.8112\lesssim\theta\lesssim 1.002$, $0.5388\lesssim\phi\lesssim 0.7996$, or $2.141\lesssim\theta\lesssim 2.330$, $3.681\lesssim\phi\lesssim 3.941$.

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Figure 10: Two feasibility regions satisfying $\mathcal{I}_{3}\left(\rho_{aiso}\right)>2.87293$ in $(\theta,\phi,p)$ space.

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VI Conclusions and discussions

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In summary, we introduced the AITTSs, and then explored their properties, including entanglement (see index $\mathcal{E}$), Wigner negativity (see index $\mathcal{N}$) and Bell nonlocality (see index $\mathcal{I}_{3}$). For all these properties, we tried our best to obtain their respective analytical and numerical results.

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As one extremal case of $\rho_{aiso}$ with $p=0$, we always had $\mathcal{E}\left(\rho_{noise}\right)\equiv 0$, $\mathcal{N}\left(\rho_{noise}\right)\equiv 0$, and $\mathcal{I}_{3}\left(\rho_{noise}\right)\equiv 0$. As another extremal case of $\rho_{aiso}$ with $p=1$, we know that $\mathcal{E}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$, $\mathcal{N}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$, and $\mathcal{I}_{3}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$ are determined by $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ in itself. So, in the end of this paper, we specialize in analyzing the influence of parameters $\left(\theta,\phi\right)$ on these three properties of $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$. Figure 11 depicts the contours for $\mathcal{E}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$, $\mathcal{N}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$, and $\mathcal{I}_{3}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$ in $\left(\theta,\phi\right)$ plains. From Fig.11(a), we see that $\mathcal{E}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$ is a periodic function of $\theta$ with period $\pi$ and a periodic function of $\phi$ with period $\pi/4$. Compared with $\mathcal{E}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$, the periodic features of $\mathcal{N}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$ are gone, as shown in Fig.11(b). By observing $\mathcal{I}_{3}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$ in Fig.11(c), we find that $\mathcal{I}_{3}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$ values are in the range of $[-2,2.91485]$. Specially, when $\left(\theta,\phi,p\right)=\left(\pi/4,\pi,1\right)$, the minimal $\mathcal{I}_{3}^{\min}=-2$ is found, corresponding to $\left|\psi_{\left(\theta,\phi\right)}\right\rangle=\frac{1}{\sqrt{2}}(\left|22\right\rangle-\left|00\right\rangle)$.

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Figure 11: Contourplots of (a) $\mathcal{E}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$, (b) $\mathcal{N}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$, and (c) $\mathcal{I}_{3}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)$ in $\left(\theta,\phi\right)$ plain space. It is necessary to remind in sub-figure (c) that (i) the red dashed lines correspond to $\mathcal{I}_{3}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)=2$; (ii) For some $\left(\theta,\phi\right)$ regions, $\mathcal{I}_{3}$ values are less than 0.

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If $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ was the Sn=1 state, then we always have $\mathcal{E}\left(\rho_{aiso}\right)\equiv 0$, $\mathcal{N}\left(\rho_{aiso}\right)\equiv 0$, and $\mathcal{I}_{3}\left(\rho_{aiso}\right)\equiv 0$. While $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ was not the Sn=1 states, then we found that (i) $\mathcal{E}\left(\rho_{aiso}\right)$ was not a linear function of $p$; (ii) $\mathcal{N}\left(\rho_{aiso}\right)$ was also not a linear function of $p$; (iii) but, $\mathcal{I}_{3}\left(\rho_{aiso}\right)$ was a linear function of $p$. Although $\rho_{aiso}$ was the mixture between $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ and $\rho_{noise}$ (see Eq.(2)), we concluded that (i) The equality of $\mathcal{E}\left(\rho_{aiso}\right)=p\mathcal{E}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)+(1-p)\mathcal{E}\left(\rho_{noise}\right)$ may not necessarily hold true; (ii) The equality of $\mathcal{N}\left(\rho_{aiso}\right)=p\mathcal{N}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)+(1-p)\mathcal{N}\left(\rho_{noise}\right)$ may not necessarily hold true; (iii) But the equality of $\mathcal{I}_{3}\left(\rho_{aiso}\right)=p\mathcal{I}_{3}\left(\left|\psi_{\left(\theta,\phi\right)}\right\rangle\right)+(1-p)\mathcal{I}_{3}\left(\rho_{noise}\right)$ may always hold true.

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Finally, we summarize several key results as follows:

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(1) There is no decisive relationship between these three properties. If a quantum state has the highest entanglement, its negativity may not necessarily be the highest, and its Bell non-locality may not necessarily be the strongest. This point can be verified from $\mathcal{E}$, $\mathcal{N}$, and $\mathcal{I}_{3}$ of state $\left|\Phi_{3}^{+}\right\rangle$. The maximally entangled state is not the maximally Wigner-negativity state and the maximally Bell-nonlocality state.

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(2) A quantum pure state with a larger Schmidt number does not necessarily have a greater Wigner negativity. This point can be verified from $\mathcal{N}(\left|S_{2}^{(1)}\right\rangle)>\mathcal{N}(\left|S_{3}^{(1)}\right\rangle)$.

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(3) It is the effects of the noise that three properties will exhibit only when $p$ exceeds a certain threshold. This point can be seen from our numerical results. Of course, the optimal properties of $\rho_{aiso}$ are those of $\left|\psi_{\left(\theta,\phi\right)}\right\rangle$ corresponding to $p=1$, without the effects of the noise.

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Appendix

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Appendix A: Pauli matrices related with qutrit system

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The bases are set as $\left|0\right\rangle=\left(\begin{array}[]{ccc}1&0&0\end{array}\right)^{T}$, $\left|1\right\rangle=\left(\begin{array}[]{ccc}0&1&0\end{array}\right)^{T}$, $\left|2\right\rangle=\left(\begin{array}[]{ccc}0&0&1\end{array}\right)^{T}$, $\left\langle 0\right|=\left(\begin{array}[]{ccc}1&0&0\end{array}\right)$, $\left\langle 1\right|=\left(\begin{array}[]{ccc}0&1&0\end{array}\right)$, and $\left\langle 2\right|=\left(\begin{array}[]{ccc}0&0&1\end{array}\right)$, for qutrit system with dimensionality $d=3$ and $\omega=e^{i\frac{2\pi}{3}}$. Related Pauli matrices are created via repeated matrix multiplication $X^{x}Z^{z}$ ($x$, $z\in\mathbb{Z}_{3}$)56 as

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$$X^{0}Z^{0} =\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right),X^{0}Z^{1}=\left(\begin{array}[]{ccc}1&0&0\\ 0&\omega&0\\ 0&0&\omega^{2}\end{array}\right),$$ (A.1)

$$X^{0}Z^{2} =\left(\begin{array}[]{ccc}1&0&0\\ 0&\omega^{2}&0\\ 0&0&\omega\end{array}\right),X^{1}Z^{0}=\left(\begin{array}[]{ccc}0&0&1\\ 1&0&0\\ 0&1&0\end{array}\right),$$ (A.2)

$$X^{1}Z^{1} =\left(\begin{array}[]{ccc}0&0&\omega^{2}\\ 1&0&0\\ 0&\omega&0\end{array}\right),X^{1}Z^{2}=\left(\begin{array}[]{ccc}0&0&\omega\\ 1&0&0\\ 0&\omega^{2}&0\end{array}\right),$$ (A.3)

$$X^{2}Z^{0} =\left(\begin{array}[]{ccc}0&1&0\\ 0&0&1\\ 1&0&0\end{array}\right),X^{2}Z^{1}=\left(\begin{array}[]{ccc}0&\omega&0\\ 0&0&\omega^{2}\\ 1&0&0\end{array}\right),$$ (A.4)

$$X^{2}Z^{2} =\left(\begin{array}[]{ccc}0&\omega^{2}&0\\ 0&0&\omega\\ 1&0&0\end{array}\right).$$ (A.5)

According to Eq.(8) and using above matrices, we can obtain the matrix in Eq.(9).

Appendix B: Matrices for $\Pi_{A_{1}}^{\left(j\right)}$, $\Pi_{A_{2}}^{\left(k\right)}$, $\Pi_{B_{1}}^{\left(l\right)}$, and $\Pi_{B_{2}}^{\left(m\right)}$

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(1) Matrix of $\Pi_{A_{1}}^{\left(j\right)}$ is

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$$\Pi_{A_{1}}^{\left(j\right)}=\frac{1}{3}\allowbreak\left(\begin{array}[]{ccc}1&\omega^{-j-\alpha_{1}}&\omega^{-2j-2\alpha_{1}}\\ \omega^{j+\alpha_{1}}&1&\omega^{-j-\alpha_{1}}\\ \omega^{2j+2\alpha_{1}}&\omega^{j+\alpha_{1}}&1\end{array}\right),$$ (B.1)

(2) Matrix of $\Pi_{A_{2}}^{\left(k\right)}$ is

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$$\Pi_{A_{2}}^{\left(k\right)}=\frac{1}{3}\allowbreak\left(\begin{array}[]{ccc}1&\omega^{-k-\alpha_{2}}&\omega^{-2k-2\alpha_{2}}\\ \omega^{k+\alpha_{2}}&1&\omega^{-k-\alpha_{2}}\\ \omega^{2k+2\alpha_{2}}&\omega^{k+\alpha_{2}}&1\end{array}\right),$$ (B.2)

(3) Matrix of $\Pi_{B_{1}}^{\left(l\right)}$ is

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$$\Pi_{B_{1}}^{\left(l\right)}=\frac{1}{3}\left(\begin{array}[]{ccc}1&\omega^{l-\beta_{1}}&\omega^{2l-2\beta_{1}}\\ \omega^{\beta_{1}-l}&1&\omega^{l-\beta_{1}}\\ \omega^{2\beta_{1}-2l}&\omega^{\beta_{1}-l}&1\end{array}\right),$$ (B.3)

(4) Matrix of $\Pi_{B_{2}}^{\left(m\right)}$ is

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$$\Pi_{B_{2}}^{\left(m\right)}=\frac{1}{3}\left(\begin{array}[]{ccc}1&\omega^{m-\beta_{2}}&\omega^{2m-2\beta_{2}}\\ \omega^{\beta_{2}-m}&1&\omega^{m-\beta_{2}}\\ \omega^{2\beta_{2}-2m}&\omega^{\beta_{2}-m}&1\end{array}\right).$$ (B.4)

Appendix C: Details of each term in Eq.(22)

(1) For $P\left(A_{1}=B_{1}\right)$, we have

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$$P\left(A_{1}=B_{1}\right) =P\left(A_{1}=0,B_{1}=0\right)$$

$$+P\left(A_{1}=1,B_{1}=1\right)$$

$$+P\left(A_{1}=2,B_{1}=2\right),$$ (C.1)

(2) For $P\left(B_{1}=A_{2}+1\right)$, we have

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$$P\left(B_{1}=A_{2}+1\right) =P\left(A_{2}=0,B_{1}=1\right)$$

$$+P\left(A_{2}=1,B_{1}=2\right)$$

$$+P\left(A_{2}=2,B_{1}=0\right),$$ (C.2)

(3) For $P\left(A_{2}=B_{2}\right)$, we have

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$$P\left(A_{2}=B_{2}\right) =P\left(A_{2}=0,B_{2}=0\right)$$

$$+P\left(A_{2}=1,B_{2}=1\right)$$

$$+P\left(A_{2}=2,B_{2}=2\right),$$ (C.3)

(4) For $P\left(B_{2}=A_{1}\right)$, we have

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$$P\left(B_{2}=A_{1}\right) =P\left(A_{1}=0,B_{2}=0\right)$$

$$+P\left(A_{1}=1,B_{2}=1\right)$$

$$+P\left(A_{1}=2,B_{2}=2\right),$$ (C.4)

(5) For $P\left(A_{1}=B_{1}-1\right)$, we have

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$$P\left(A_{1}=B_{1}-1\right) =P\left(A_{1}=2,B_{1}=0\right)$$

$$+P\left(A_{1}=0,B_{1}=1\right)$$

$$+P\left(A_{1}=1,B_{1}=2\right),$$ (C.5)

(6) For $P\left(B_{1}=A_{2}\right)$, we have

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$$P\left(B_{1}=A_{2}\right) =P\left(A_{2}=0,B_{1}=0\right)$$

$$+P\left(A_{2}=1,B_{1}=1\right)$$

$$+P\left(A_{2}=2,B_{1}=2\right),$$ (C.6)

(7) For $P\left(A_{2}=B_{2}-1\right)$, we have

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$$P\left(A_{2}=B_{2}-1\right) =P\left(A_{2}=2,B_{2}=0\right)$$

$$+P\left(A_{2}=0,B_{2}=1\right)$$

$$+P\left(A_{2}=1,B_{2}=2\right),$$ (C.7)

(8) For $P\left(B_{2}=A_{1}-1\right)$, we have

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$$P\left(B_{2}=A_{1}-1\right) =P\left(A_{1}=0,B_{2}=2\right)$$

$$+P\left(A_{1}=1,B_{2}=0\right)$$

$$+P\left(A_{1}=2,B_{2}=1\right).$$ (C.8)