Alternative method for deriving nonlocal multipartite product states

We can only perform finite rounds of measurements in protocols with local operations and classical communication (LOCC). In this paper, we propose a set of product states that require infinite rounds of measurements in order to distinguish this given set of states perfectly by LOCC. Therefore, we can conclude that these sets of states are locally indistinguishable. More accurately, given any multipartite LOCC indistinguishable set where every local system cannot start with a nontrivial measurement, then after appending these states with arbitrarily choosing two nonorthogonal states, we obtain another LOCC indistinguishable set. It can be seen that some parties can perform some nontrivial measurements. Hence, these sets are quite different from those constructed before. This result broadens the knowledge of nonlocality without entanglement to a certain extent.

Figure 1

This is a sketch map to show the distinguishing protocol of Theorem t5. Here we use $A_1,A_2,...,A_m$ to denote the first $m$ parties and $B_1$ to denote the last partite. There exists an infinite sequence of outcomes (indicated by the red dotted line) performed by the last partite. Moreover, the postmeasurement states along these outcomes are pictured by the gray rectangle.

Figure 2

This is a sketch map to show the distinguishing protocol of Theorem t6 with $n=3$. Here we use $A_1,$ $A_2,\cdots ,$ $A_m$ to denote the first $m$ parties and $B_1,B_2,B_3$ to denote the last three parties. The significant point is that there exists an infinite sequence of outcomes (indicated by the red dotted line) performed by the last three parties, and these parties send a set with the $\mathcal{P}(m,3)$ property to another set with the same property. Moreover, the postmeasurement states along these outcomes are pictured by the gray rectangle.