Genuinely entangled subspaces and strongly nonlocal unextendible biseparable bases in four-partite systems

Abstract

A set of orthogonal pure states is an unextendible biseparable basis (UBB), which means that its complementary subspace contains only genuinely entangled states. UBBs thus serve as an effective tool for constructing genuinely entangled subspaces. If every state within such a subspace exhibits distillable entanglement across all bipartitions, it becomes particularly advantageous for applications in quantum information. In this paper, we mainly conduct research on the 4-qudit quantum systems, where the local dimension $d$ is not less than 3. We present an approach for constructing UBB and prove that the UBB established in this way is strongly nonlocal. We build several genuinely entangled subspaces and demonstrate the distillability of the genuinely entangled subspaces across all bipartitions. In addition, we also describe the specific orthonormal basis for some genuinely entangled subspaces. These results will not only contribute to the development of quantum nonlocality theory, but also provide a crucial theoretical foundation for practical quantum information processing tasks.

Figures (3)

• Figure 1: This is the structure of set $\mathcal{U}$ under bipartition $X_{1}|X_{234}$. The regions denoted as $\mathcal{C}_{j}$ and $\mathcal{D}_{j}$ with $j=1,\ldots,8$ correspond to the subsets of Eq. (\ref{['ye4']}), respectively. Each colored area is marked with a red number in the left lower corner. The union of the regions with the same color and red number $j$ corresponds the subsets $\mathcal{U}_{j}^{-}$ of set $\mathcal{U}$.
• Figure 2: Divide the matrix $M$ into blocks based on $|111\rangle$ and $\mathcal{P}_{j}^{(234)}$$(\mathcal{P}=\mathcal{C},\mathcal{D}$ and $j=1,5,6,7)$. The off-diagonal subblocks are all zero matrix. In figure, $\mathcal{P}_{j}$ represents the subblock $M_{\mathcal{P}_{j}}$.
• Figure 3: This is the structure of set $\mathcal{U}^{d}$ under bipartition $X_{1}|X_{234}$. In figure, $i$ expresses the dimension of $1,\ldots,d-2$. The regions denoted as $\mathcal{C}_{j}^{d,1}$ and $\mathcal{D}_{j}^{d,1}$ with $j=1,\ldots,8$ correspond to the subsets of Eq. (\ref{['yel']}), respectively. Each colored area is marked with a red number in the left lower corner. The union of the regions with the same color and red number $j$ correspond the subsets $\mathcal{U}_{j}^{-}$ of set $\mathcal{U}^{d}$.