Reply to ”Comment on quantum secret sharing based on reusable Greenbergr-Horne-Zeilinger states as secure carriers”

In a recent comment, it has been shown that in a quantum secret sharing protocol proposed in [S. Bagherinezhad, V. Karimipour, Phys. Rev. A, 67, 044302, (2003)], one of the receivers can cheat by splitting the entanglement of the carrier and intercepting the secret, without being detected. In this reply we show that a simple modification of the protocol prevents the receivers from this kind of cheating. PACS Numbers: 03.67.Dd, 03.65.Ud. email:[email protected] To set up the context and the notations, it is appropriate to first review briefly the protocol itself [1] and the basic feature of the attack or cheating suggested in [2]. 1 The basic steps of the protocol and the cheating First we need the concept of a reusable secure carrier [3], . A Bell state like |φ〉ab = 1 √ 2 (|00〉+ |11〉)ab, shared between Alice(a) and Bob(b) can be used as a reusable secure carrier between two parties as follows. Alice entangles a qubit |q〉1 by the action of a CNOT gate Ca1 (acting on the qubit 1 and controlled by a), which produces a state like 1 √ 2 (|00q〉+ |11q〉)ab1. At the destination Bob disentangles the qubit by a CNOT operation Cb1, leaving the carrier in its original state for reusing. During the transmission the qubit has been disguised in a highly mixed state. Any of the Bell states |φ〉ab = 1 √ 2 (|00〉 ± |11〉)ab, |ψ〉ab = 1 √ 2 (|01〉 ± |10〉)ab (1) can be used as a carrier. For three parties [1], a carrier shared between Alice(a), Bob(b) and Charlie(c) can be a GHZ state like |GHZ〉 := 1 √ 2 (|000〉+ |111〉)abc, (2) or an even parity state like |E〉 := 1 2 (|000〉+ |110〉+ |101〉+ |011〉)abc. (3) Throughout [1], the comment [2] and the present reply the subscripts a, b and c are used for the quibts shared by, or the local operators acted by, Alice, Bob and Charlie respectively, while the subscripts 1 and 2 are used for the qubits sent to Bob and Charlie respectively. It was shown in [1] that by suitable local operations, Alice can send a qubit q to Bob and Charlie, by entangling it to the above carriers (hence hiding it from Eavesdroppers). In order to share the secret between Bob and Charlie, half of the bits (the bits in the odd rounds) were sent to Bob and Charlie, as states of the form |qq〉12 which they could read without the help