1 Introduction

Quantum Teleportation is a process of transferring an arbitrary unknown qubit to a distant receiver with the help of shared entanglement and some classical information without any transfer of mass or energy. In their initial proposal, Bennett et al. [1] proposed quantum teleportation using maximally entangled two qubit states. Thereafter, this was realized in tripartite GHZ state [2], four partite GHZ state [3], SLOCC equivalent W-class state [4] and the cluster state [5], etc. The perfect teleportation of an arbitrary two-qubit state was proposed using quantum channels formed by the tensor product of two Bell states [6], tensor product of two orthogonal states [7], genuinely entangled five qubit state [8], five qubit cluster state [9] and six qubit genuinely entangled states [10], etc.

Yang et al. [11] had shown that perfect teleportation of a single qubit is possible using a GHZ-Like state. Tsai et al. [12] had shown that this state can also be used successfully to teleport a pure EPR state α|01〉+β|10〉. However, we show that a more general two qubit state can be teleported with this GHZ-Like state.

Apart from this GHZ-Like state, deterministic quantum teleportation of a single arbitrary qubit state through a tripartite state as quantum channel is possible through the GHZ state [2] and a SLOCC equivalent W-class states [4]. Of the three, GHZ-Like state is more robust from the application point of view. For a pure state |Ψ〉 AB , the entanglement between the two subsystems A and B are given by,

$$ E_{A|B}\bigl(|\varPsi\rangle_{AB}\bigr)=S(\rho_A) $$
(1)

where \(\rho_{A}=\operatorname{tr}_{B}(|\varPsi\rangle \langle\varPsi|)\) and S(ρ A ) is the von Neumann entropy. For the tripartite GHZ state and the GHZ-Like states, E 1|23=E 2|13=E 3|12=1 while for the other state E 3|12=1 and E 1|23=E 2|13<1. As a result, quantum teleportation of a single qubit is only possible through any bipartite cut of the first two states while it is possible through the 12|3 cut of the third system. Now if we consider these states for the teleportation of a two qubit state, it can be seen that only a pure EPR state of the form α|00〉+β|11〉 can be teleported by the GHZ state while it is not possible through the asymmetric W state in a deterministic manner. The GHZ-Like state on the other hand can be used for perfect teleportation of a more general two-qubit state.

2 Quantum Teleportation of a Two Qubit Entangled State via the GHZ-Like State

The most general form of a two qubit state is given by,

$$ |\chi\rangle_{12} =\alpha|00\rangle+\beta|01\rangle+ \gamma|10\rangle+\delta|11\rangle $$
(2)

with α, β, γ and δ being complex and |α|2+|β|2+|γ|2+|δ|2=1. Let us consider that, Alice had an unknown two qubit state, which is a special form of Eq. (2) with α=δ and β=γ,

$$ |\tau\rangle_{12} =\alpha\bigl(|00\rangle+|11\rangle\bigr)+\beta\bigl(|01\rangle+|10 \rangle\bigr) $$
(3)

where \(|\alpha|^{2}+|\beta|^{2}=\tfrac{1}{2}\). She wants to send this state to a distant receiver Bob. The quantum channel linking Alice and Bob is a GHZ-Like state given by,

$$ |\phi_G\rangle_{345}=\frac{1}{2}\bigl[|001\rangle+|010 \rangle+|100\rangle+|111\rangle\bigr] $$
(4)

with Alice owning qubit 3 and qubits 4 and 5 belong to Bob. The combined state of the five qubit system is given by,

$$\begin{aligned} |\varGamma\rangle_{12345} =& |\tau\rangle_{12} \otimes |\phi_G\rangle_{345} \\ =& \frac{1}{2}\alpha\bigl(|00001\rangle+|00010\rangle+|00100\rangle+|00111 \rangle+|11001\rangle+|11010\rangle \\ &{}+|11100\rangle+|11111\rangle\bigr) +\frac{1}{2}\beta\bigl(|01001\rangle+|01010 \rangle+|01100\rangle+|01111\rangle \\ &{}+|10001\rangle+|10010\rangle+|10100\rangle+|10111\rangle\bigr)_{12345} \end{aligned}$$
(5)

Thereafter, Alice performs a three-qubit von-Neumann measurement on her qubits 1, 2 and 3 in the following basis,

$$ \begin{aligned} \bigl|\xi^{\pm} \bigr\rangle =& \frac{1}{2} \bigl[|001 \rangle+|111\rangle\pm\bigl(|010\rangle+|100\rangle\bigr) \bigr] \\ \bigl|\eta^{\pm} \bigr\rangle =& \frac{1}{2} \bigl[|000\rangle+|110\rangle \pm\bigl(|011\rangle+|101\rangle\bigr) \bigr] \end{aligned} $$
(6)

Alice then sends her measurement results via two bits of classical information to Bob and Bob applies suitable recovery operations to reconstruct the two qubit unknown state. Alice’s measurement results and Bob’s recovery operations are listed in Table 1.

Table 1 Strategy for recovering the two qubit state at Bob’s end
Full size table

In Tsai et al.’s [12] proposal, after measurement in the {+,−} basis and Bell basis, Bob’s state becomes a product state and Bob has to use a specific entangled unitary operation U s to recover the EPR state. In contrast, in our case, Alice had to perform a single three qubit measurement and thereafter, Bob had to perform simple Pauli rotations to recover the state. Again, in [12], the classical communication required is 3 cbits while in our case it is 2 cbits.

3 Conclusion

Here we have shown that a more general form of two qubit state than that shown by Tsai et al. [12] can be teleported by using a GHZ-Like state. In [12], the measurement cost consist of a measurement in the diagonal basis followed by a measurement in the Bell basis. In our case, one has to perform a single three qubit measurement. The classical communication cost is also less in our case.