Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition

In a multipartite scenario quantum entanglement manifests its most dramatic form when the state is genuinely entangled. Such a state is more beneficial for information theoretic applications if it contains distillable entanglement in every bipartition. It is, therefore, of significant operational interest to identify subspaces of multipartite quantum systems that contain such properties a priori. In this article we introduce the notion of unextendible biseparable bases (UBB) that provides an adequate method to construct genuinely entangled subspaces (GES). We provide explicit construction of two types of UBBs—party symmetric and party asymmetric—for every 3-qudit quantum system, with local dimension $d\ge 3$. Furthermore, we show that the GES resulting from the symmetric construction is indeed a bidistillable subspace, i.e., all the states supported on it contain distillable entanglement across every bipartition.

Figure 1

The block structure for the t-OPB (1). An arbitrary subblock made of $n$ basic blocks can contain at most $n$ mutually orthogonal states. Each subblock ${\mathcal{B}}_l$ is made of four basic blocks, $\forall l\in \{1\cdots 6\}$. In the right figure all six subblocks and the three body-diagonal blocks are shown.

Figure 2

Tile structure of the UPB, ${\mathcal{U}}_{BB}^{AB|C}$ in AB|C cut. This particular tile structure is a bit similar to that of ${\mathbb{C}}^3\otimes {\mathbb{C}}^3$ tile UPB [22]—for that look into four blue tiles and the central (white) tile.

Figure 3

The block structure for ${\left({\mathbb{C}}^4\right)}^{\otimes 3}$. The states of the t-OPB (5) are placed in the corresponding Blocks.

Figure 4

The block structure for ${\left({\mathbb{C}}^5\right)}^{\otimes 3}$. The states of the t-OPB (6) are placed in the corresponding blocks.