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

Chen, B., Halqem Nizamidin, Shao-Ming Fei

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

We study the locally unextendible non-maximally entangled basis (LUNMEB) in H d ⊗ 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 ve orthogonal vectors from a specic nonmaximally entangled state are constructed. Besides, we completely solve the problem of LUNMEB for the case of d = 2.

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We study the locally unextendible non-maximally entangled basis (LUNMEB) in H d ⊗ 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 ve orthogonal vectors from a specic nonmaximally entangled state are constructed. Besides, we completely solve the problem of LUNMEB for the case of d = 2.
We study the locally unextendible non-maximally entangled basis (LUNMEB) in Hd⨂HdH^{d}\bigotimes H^{d}Hd⨂Hd. 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 ddd 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=4d=4d=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=2d=2d=2.
We study the locally unextendible non-maximally entangled basis (LUNMEB) in Hd⨂HdH^{d}\bigotimes H^{d}Hd⨂Hd. 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 Hd⨂HdH^d \bigotimes H^dHd⨂Hd. It is shown that such a basis consists of ddd orthogonal vectors for a non-maximally entangled state. However, there can be a maximum of (d−1)2(d-1)^2(d−1)2 orthogonal vectors for non-maximally entangled state if it is maximally entangled in (d−1)(d-1)(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+p0dd−1)log⁡d(1+p_0\frac{d}{d-1})\log d(1+p0​d−1d​)logd, where p0p_0p0​ 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=3d= 3d=3, former may be more suitable for the superdense coding.
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.
We introduce the notion of the unextendible maximally entangled basis (UMEB), a set of orthonormal maximally entangled states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ consisting of fewer than d2{d}^{2}d2 vectors which have no additional maximally entangled vectors orthogonal to all of them. We prove that UMEBs do not exist for d=2d=2d=2 and give explicit constructions for a six-member UMEB with d=3d=3d=3 and a 12-member UMEB with d=4d=4d=4.
We solve the unextendible maximally entangled basis (UMEB) problem in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}(d\ensuremath{\ne}{d}^{\ensuremath{'}}).$ The results show that a UMEB will always exist. In addition, there might be two sets of UMEBs with different numbers. The main difficulty is to prove the unextendibility of the set of states. We give an explicit construction of UMEBs by considering the Schmidt number of the complementary space of the states we construct.