One-way local distinguishability of generalized Bell states in arbitrary dimension

In this paper, we study the distinguishability of arbitrary dimensional generalized Bell states under one-way local operations and classical communication (LOCC). We introduce an admissible solutions set and a nonadmissible solutions set for each set of generalized Bell states. Any element that exists in the admissible solution set but not in the nonadmissible one implies the one-way LOCC distinguishablity of a given set. This criterion is very convenient in the following two aspects: both solution sets can be calculated easily and, once the criterion can be applied successfully for a given set, it immediately presents an explicit distinguishing strategy. Furthermore, Fan's results in Phys. Rev. Lett. 92, 177905 (2004) can also be deduced as special cases of our results. For the case no such element exists, we disprove a conjecture proposed by Yang et al. in Quantum Inf. Process. 17, 29 (2018) by presenting a counterexample.

Figure 1

Here we fix $N=2000$. And we calculate out the successful proportion $p_l\left(d\right)=N_T(d,l)/N$ for each dimension $100\le d\le 2000$ and three $l{}^{\prime }\mathrm{s}$: $l=r\lceil \sqrt{d}\rceil$ with $r=2,3,4$. The points $[d,p_l\left(d\right)]$ are plotted by red (top line), green (middle line), and pink (bottom line) color according to $l=r\lceil \sqrt{d}\rceil$ with $r=2,3,4$, respectively.

Figure 2

Here we use the same data $p_l\left(d\right)$ obtained in Fig. 1. In order to study whether the successful proportion depends on the parity or not, we plot the dots $[d,p_l\left(d\right)]$ with red color for even $d$ and green color for odd $d$. It is difficult to distinguish these two lines. Therefore, we could conclude that $p_l\left(d\right)$ may not depend on the parity.