Efficient discrimination schemes for unextendible product bases with strong quantum nonlocality

TL;DR

The paper addresses the problem of locally discriminating strongly nonlocal unextendible product bases (UPBs) in multipartite systems with minimal entanglement resources, focusing on $3\otimes3\otimes3$ and general $d\otimes d\otimes d$. It introduces three entanglement-allocation LOCC schemes for the $3\otimes3\otimes3$ UPB and generalizes the approach to higher dimensions, presenting multiple resource configurations that avoid full-state teleportation when possible. The main contributions are Theorems 1–3 for the 3-party case, showing perfect LOCC discrimination with limited entanglement, and Theorems 4–6 for $d\otimes d\otimes d$ that compare teleportation-based and teleportation-free strategies, including a cost analysis. Overall, the results demonstrate resource-efficient UPB discrimination and elucidate the operational role of maximally entangled states in local quantum state discrimination.

Abstract

Entanglement is a central resource in quantum information science; therefore, it is important to design local discrimination protocols that minimize resource consumption. In this paper, we propose three entanglement-allocation schemes for the local discrimination of particular unextendible product bases (UPB) exhibiting strong quantum nonlocality in a \(3 \otimes 3 \otimes 3\) system. By exploiting the structural features of these UPB and the operational advantages of maximally entangled states, we further extend our protocols to strongly nonlocal UPB in \(d \otimes d \otimes d\) systems. In particular, we show that these UPB can be perfectly distinguished with only two maximally entangled states. Moreover, a resource-cost analysis indicates that our protocols, which avoid quantum teleportation whenever possible, can reduce the entanglement consumption. These results not only facilitate resource-efficient quantum information processing, but also provide further insight into the operational role of maximally entangled states.

Figures (5)

• Figure 1: With the shared EPR state $|\phi^+(2)\rangle_{ab}$, the state in Eq. (\ref{['e3.2']}) can be represented on a $2d_A\times 2d_Bd_C$ grid. The lavender-shaded region indicates the support of $M_{11}$.
• Figure 2: The lightly shaded regions represent the measurement outcomes in the $M_{21}$, $M_{22}$, and $M_{23}$ directions for Bob, respectively.
• Figure 3: The measurement result for direction $M_{31}$ in Alice's subsystem corresponds to the pale green region.
• Figure 4: Layered cubic structure of the set in Eq. (\ref{['e4.1']}) within a $d\otimes d\otimes d$ tripartite system.
• Figure 5: T4, T5, and T6 are the curves of entanglement resource consumed by discrimination protocols in Theorems 4, 5, 6 corresponding to even dimension $d$, respectively.