ar X iv : q ua nt - p h / 06 11 18 4 v 1 1 7 N ov 2 00 6 Comment on ’ Teleportation of two quNit entanglement : Exploiting local resources ’

Recently, in Ref.[1] N. Ba An proposed a teleportation protocol, in which the new interesting idea of exploiting local resources to do teleportation [2] is generalized to the case of arbitrary quNits. After carefully rededucting the formulae in Ref.[1], however, we find some of them are wrong. As a consequence, the unitary operations which need to be performed to reconstruct the unknown bipartite entangled state are also incorrect. This means that in terms of N. Ba An’s protocol the teleportation of two-quNit entanglement can not be successfully realized. To solve this problem, in this comment we present two different efficient methods to modify some details of N. Ba An’s protocol: (1) we correct the wrong formulae and accordingly offer right unitary operations; (2) we redefine some definitions such that the so-called wrong formulae and unitary operations all maintain their original forms. After our modifications, we believe, the revised N. Ba An’s protocol can work successfully. Method 1: For convenience, throughout this method all the definitions, such as the two-qutrit(quNit) maximally entangled (ME) pairs {|Φmn〉}, the control change gate, the rotated basis, etc, are all same as those in the Ref.[1]. Along the line of N. Ba An’s work [1], we also first consider the case of qutrits. In this case, the whole teleportation process is as follows: (i) Alice has two qutrits 1 and 2 in an entangled state |Ψ〉12 = (α|00〉+ β|11〉+ γ|22〉)12. Besides, she also shares a single two-qutrit maximally entangled pair in the state |Φ00〉34 with Bob (say, Alice has the qutrit 3 while Bob the qutrit 4). (ii) Alice performs a joint measurement on the 1-3 system in the basis {|Φmn〉13} and measures qutrit 2 in a rotated basis. (iii) After her measurements, Alice tells Bob her two measurement results via classical channel. (iv) Bob introduces an ancillary qutrit 5 and performs the control change gate C45 on the 4-5 system. (v) Conditioned on the classical message from Alice, Bob performs an appropriate unitary operation U on the 4-5 system to reconstruct the state |Ψ〉. In Ref[1], the total 1-2-3-4 system state in terms of {|Φmn〉13} is written as its formula 2, that is,