Unextendible product bases from tile structures and their local entanglement-assisted distinguishability

We completely characterize the condition when a tile structure provides an unextendible product basis (UPB) and construct UPBs of different large sizes in ${\mathbb{C}}^m\otimes {\mathbb{C}}^n$. In particular, we show that there exists a UPB of size $(mn-4\lfloor \frac{m-1}{2}\rfloor )$ in ${\mathbb{C}}^m\otimes {\mathbb{C}}^n$ for any $n\ge m\ge 3$, which solves an open problem [S. Halder et al., Phys. Rev. A 99, 062329 (2019)]. As an application, we show that this class of UPBs can be perfectly distinguished by local operations and classical communications assisted with a $\lceil \frac{m}{2}\rceil \otimes \lceil \frac{m}{2}\rceil$ maximally entangled state.

Figure 1

Tile structure of system $A,B$ in ${\mathbb{C}}^4\otimes {\mathbb{C}}^4$. A tile structure is a rectangle paved by some disjoint tiles. It gives a complete orthogonal product basis (COPB). We can obtain a UPB by deleting some states and adding a special state of COPB. More details are given in Example t1.

Figure 2

Tile structure in ${\mathbb{C}}^4\otimes {\mathbb{C}}^4$.

Figure 3

A U-tile structure with ($2m-3$)-tiles in ${\mathbb{C}}^m\otimes {\mathbb{C}}^n$ when $m$ is even.

Figure 4

A U-tile structure with $(2m-1)$-tiles in ${\mathbb{C}}^m\otimes {\mathbb{C}}^n$ when $m$ is odd.

Figure 5

U-tile structures with 5,6,7,8-tiles in ${\mathbb{C}}^4\otimes {\mathbb{C}}^4$.

Figure 6

U-tile structures with 5,6,7-tiles in ${\mathbb{C}}^5\otimes {\mathbb{C}}^5$ based on U-tile structures with 5,6,7-tiles in ${\mathbb{C}}^4\otimes {\mathbb{C}}^4$.

Figure 7

U-tile structures with 8,9,10-tiles in ${\mathbb{C}}^5\otimes {\mathbb{C}}^5$ based on the U-tile structure with 8-tiles in ${\mathbb{C}}^4\otimes {\mathbb{C}}^4$.

Figure 8

U-tile structures with $(2m-1),\left(2m\right)$-tiles in ${\mathbb{C}}^m\otimes {\mathbb{C}}^m$ based on the U-tile structure with $2(m-1)$-tiles in ${\mathbb{C}}^{m-1}\otimes {\mathbb{C}}^{m-1}$, where $m\ge 6$ is even.

Figure 9

U-tile structures with $(2m-1),\left(2m\right)$-tiles in ${\mathbb{C}}^m\otimes {\mathbb{C}}^m$ based on the U-tile structure with $2(m-1)$-tiles in ${\mathbb{C}}^{m-1}\otimes {\mathbb{C}}^{m-1}$, where $m\ge 6$ is odd.

Figure 10

A U-tile structure with 5-tiles in ${\mathbb{C}}^m\otimes {\mathbb{C}}^n$.