A NOTE ON LOCALLY UNEXTENDIBLE NON-MAXIMALLY ENTANGLED BASIS

We study the locally unextendible non-maximally entangled basis (LUNMEB) in H-d circle times H-d. We point out that there exists an error in the proof of the main result of LUNMEB [Quant. Inf. Comput. 12, 0271(2012)], which claims that there are at most d orthogonal vectors in a LUNMEB, constructed from a given non-maximally entangled state. We show that both the proof and the main result are not correct in general. We present a counter example for d = 4, in which five orthogonal vectors from a specific non-maximally entangled state are constructed. Besides, we completely solve the problem of LUNMEB for the case of d = 2.