# Find the qutrit analogue of certain qubit and ququart formulas of Li and Qiao for testing separability

In eqs. (33), (43)-(46), (56) of their paper, "Separable Decompositions of Bipartite Mixed States" https://arxiv.org/abs/1708.05336, Li and Qiao present a number of formulas pertinent to testing the separability of certain states. In particular, in eq. (54), they state that $$\begin{equation} \Pi_{\mu=1}^3 \mathcal{E}_{\mu}(A) \leq \frac{1}{27}, \hspace{.1in} \Pi_{\nu=1}^3 \mathcal{E}_{\nu}(B) \leq (\frac{2}{27})^2, \end{equation}$$ is required for separability of certain $$2 \times 4$$ states, where $$A$$ correponds to qubits and $$B$$ to ququarts. There doesn't seem to be an explicit formula for qutrits, say $$C$$, however (although some peripheral discussions). Some manipulations of mine--of which I am not fully convinced--lead me to suspect the value $$\frac{1}{128}$$, but I would like to be more confident of its correctness, or if not, the proper alternative.

After eq. (33), Li and Qiao note that "$$\mathcal{E}_{\mu} (A)$$ and $$\mathcal{E}_{\nu} (B)$$ are the means of the squares of the components along the unit directions $$\vec{u}_{\mu} \in \mathcal{S}_l^{(A)}$$ and $$\vec{v}_{\nu} \in \mathcal{S}_l^{(B)}$$, respectively, and $$\tau_{\mu}$$ are the singular values of $$\mathcal{T}$$".

Also, preceding this they write: "[W]e define the average of the squares of the components along the directions $$\vec{u}_{\mu} \in \mathcal{S}^{(A)}_{l}$$ and $$\vec{v}_{\nu} \in \mathcal{S}^{(B)}_{l}$$ as follows $$\begin{equation} \mathcal{E}_{\mu}(A) \equiv \sum_{i=1}^L p_i |\vec{u}_{\mu} \cdot \vec{r}_i|^2 \; ,\; \mathcal{E}_{\nu}(B) \equiv \sum_{i=1}^L p_i |\vec{v}_{\nu} \cdot \vec{s}_i|^2 \; ." \end{equation}$$

Another background passage reads: "Let $$\mathcal{T} = (\vec{u}_1,\cdots,\vec{u}_{N^2-1}) \Lambda_{\tau} (\vec{v}_1, \cdots, \vec{v}_{M^2-1})^{\mathrm{T}}$$ be the singular value decomposition of the correlation matrix $$\mathcal{T}$$ and $$\Lambda_{\tau}$$ has rank $$l$$, then we have $$\begin{equation} \mathcal{T} = \sum_{\mu =1}^l \tau_{\mu} \vec{u}_{\mu}\vec{v}_{\mu}^{\,\mathrm{T}} \; . \label{Singular-tau-def} \end{equation}$$ For the $$l$$ nonzero values of $$\tau_{\mu}$$, the corresponding singular vectors $$\{\vec{u}_1,\cdots,\vec{u}_l\}$$ and $$\{\vec{v}_1,\cdots,\vec{v}_l\}$$ span two $$l$$-dimensional subspaces in Bloch vector space: $$\mathcal{S}_l^{(A)} \equiv \mathrm{span}\{\vec{u}_1,\cdots,\vec{u}_l\} \subseteq \mathcal{S}_{N^2-1}$$ and $$\mathcal{S}_l^{(B)} \equiv \mathrm{span} \{\vec{v}_1,\cdots,\vec{v}_l\} \subseteq \mathcal{S}_{M^2-1}$$.".

In addition, "the corresponding decomposition of $$\rho_{AB}$$ can be read from equations (63) and (64), where $$p_1=p_2=p_3=p_4=1/4$$, and the Bloch vectors for $$\rho_i^{(A,B)}$$ are \begin{align} \vec{r}_1 = \begin{pmatrix} \alpha_1 \\ -\alpha_2 \\ -\alpha_3 \end{pmatrix}\; , \; \vec{r}_2 = \begin{pmatrix} -\alpha_1 \\ -\alpha_2 \\ \alpha_3 \end{pmatrix}\; , \; \vec{r}_3 = \begin{pmatrix} -\alpha_1 \\ \alpha_2 \\ -\alpha_3 \end{pmatrix}\; , \; \vec{r}_4 = \begin{pmatrix} \alpha_1 \\ \alpha_2 \\ \alpha_3 \end{pmatrix} \; , \label{Example-I-detail-1}\\ \vec{s}_1 = \begin{pmatrix} \beta_1 \\ -\beta_2 \\ -\beta_3 \end{pmatrix}\; , \; \vec{s}_2 = \begin{pmatrix} -\beta_1 \\ -\beta_2 \\ \beta_3 \end{pmatrix}\; , \; \vec{s}_3 = \begin{pmatrix} -\beta_1 \\ \beta_2 \\ -\beta_3 \end{pmatrix}\; , \; \vec{s}_4 = \begin{pmatrix} \beta_1 \\ \beta_2 \\ \beta_3 \end{pmatrix} \; . \label{Example-I-detail-2}". \end{align}

Clearly, some command of the authors' detailed (and rather challenging, it seems to me) argument would seem to be needed to successfully address the question.