. 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.

DOI: 10.26421/qic13.11-12-9

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A note on locally unextendible non-maximally entangled basis : comment on 'Quant. Inf. Comput. 12, 0271(2012)'

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We study the locally unextendible non-maximally entangled basis (LUNMEB) in

H^{d}\bigotimes H^{d}

We study the locally unextendible non-maximally entangled basis (LUNMEB) in

H^{d}\bigotimes 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

We introduce the concept of the locally unextendible non-maximally entangled basis (LUNMEB) in

H^d \bigotimes H^d

. It is shown that such a basis consists of

d

orthogonal vectors for a non-maximally entangled state. However, there can be a maximum of

(d-1)^2

orthogonal vectors for non-maximally entangled state if it is maximally entangled in

(d-1)

dimensional subspace. Such a basis plays an important role in determining the number of classical bits that one can send in a superdense coding protocol using a non-maximally entangled state as a resource. By constructing appropriate POVM operators, we find that the number of classical bits one can transmit using a non-maximally entangled state as a resource is

(1+p_0\frac{d}{d-1})\log d

, where

p_0

is the smallest Schmidt coefficient. However, when the state is maximally entangled in its subspace then one can send up to $2\log (d-1) $ bits. We also find that for

d= 3

, former may be more suitable for the superdense coding.

{d}^{2}

vectors which have no additional maximally entangled vectors orthogonal to all of them. We prove that UMEBs do not exist for

d=2

and give explicit constructions for a six-member UMEB with

d=3

and a 12-member UMEB with

d=4