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International Journal of Quantum Information

THE RELATION BETWEEN LOCAL UNITARY TRANSFORMATION INVARIANT AND PERFECT QUANTUM TELEPORTATION

Xin-Wei Zha, Haiyang Song, Kuanfang Ren

Published Dec 1, 2010• 8 citations
Publisher

Abstract

We introduce a concept of local unitary (LU) transformation invariant operators which permits us to easily obtain arbitrary multi-qubit states LU transformation invariant. The general relation between the LU transformation invariant and perfect quantum teleportation is examined. The optimal match of measurement basis is given.
DOI
10.1142/s0219749910006915

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Physical Review A•Mar 16, 2009•22 citations
We proposed a concept of LU transformation invariant operators. By using this operator, arbitrary multi-qubit states LU transformation invariant and SLOCC invariant could be easily obtained. And we find that presences two kinds of invariant operators and corresponding invariants. One kind of operators yields LU invariants and the other operators results in SLOCC invariants. For three-qubit states, all independence LU transformation invariant are obtained. Furthermore, by this system method, arbitrary multi-qubit states invariants can be given.
A scheme of teleportation of an arbitrary three-qubit state is presented by using the generalized Bell-basis measurement and the Non-Bell-Pair quantum channel.Teleportation can be perfectly realized if Alice makes Bell-state measurement on her particles and Bob carries out appropriate unitary transformation(Toffoli transformation)on his particle.The unknown three-particle entangled state can be teleported perfectly,and the successful possibilities and fidelities of the two schemes both reach unity.Based on the transformation operator,the collapses states can easily obtained and Bob's unitary transformation can also easily given.
We present a simple and precise protocol for standard quantum teleportation of N-qubit state, considering the most general resource q-channel and Bell states. We find condition on these states for perfect teleportation and give explicitly the unitary transformation required to be done by Bob for achieving perfect teleportation. We discuss connection of our simple theory with the complicated related work on this subject and with character matrix, transformation, judgment and kernel operators defined in this context. We also prove that the magic basis discussed by Hill and Wootters [Phys. Rev. Lett. 78 (1997) 5022] does not exist for entangled 2N-qubit states with N > 1 but magic partial bases, similar to those discussed recently by Prakash and Maurya [Optics Commun. 284 (2011) 5024] do exist. We give explicitly all magic partial bases for N = 2.
In this paper, we make a few corrections to a paper by Zha and Ren [Phys. Rev. A 77, 014306 (2008)]. The Zha-Ren protocol for teleportation in principle is equivalent to Rigolin's protocol [G. RigolinPhys. Rev. A 71, 032303 (2005)] and the associated Comment [F. G. DengPhys. Rev. A 72, 036301 (2005)]. We feel that the transformation operator is not well suited as a criterion for the faithful teleportation, but can be used as a means to transform an arbitrary four-qubit entangled state into a tensor product state of two Bell states. We give all the transformation operators.