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Mn } on H such that ni=1 Mi† Mi = IH . 3) where ΠnX is a one-dimensional orthogonal projection on HX for every n with sum IX , for X = A, B, C. 3) have the property that n=1 X X and so Πn Πm = 0(m = n) for X = A, B, C. Similar to [4], we introduce the following. 1 Let ρ ∈ D(HABC ). 3) such that Π (ρ) = ρ, then ρ is said to be completely classical correlated (shortly, CCC). Otherwise, ρ is said to be quantum correlated (QC). 3), a state ρ ∈ D(HABC ) is changed to dA dC dB ΠiA ⊗ ΠjB ⊗ ΠkC ρ ΠiA ⊗ ΠjB ⊗ ΠkC .

The above algorithm can be illustrated by Fig. 3. 2 Orthogonal Duality Quantum Computer N4 Algorithms for Unsorted Database Search A novel duality quantum algorithm called the N4 algorithm was designed in [7], which finds a marked state in a fixed manner in N/4 steps. Here, we give an OGDQC version of the N4 algorithm. By stating from the initial state 1 |ψ = √ N N−1 |i |0 = |ψ0 |0 , i=0 where 1 |ψ0 = √ |0 + |1 + · · · + |τ + · · · + |N − 1 , N replacing Hadamard gate H with the unitary gate ⎡ V =⎣ N 2N−4 N−4 2N−4 N−4 2N−4 − N 2N−4 ⎤ ⎦ := V00 V10 V01 , V11 Int J Theor Phys (2013) 52:1751–1767 1765 Fig.

A 328, 6–10 (2004) 4. : Revisiting the security of quantum dialogue and bidirectional quantum secure direct communication. Sci. China Ser. G 51(5), 559–566 (2008) 5. : Classical correlation in quantum dialogue. Int. J. Quantum Inf. 325 (2008). doi:10. 1142/S021974990800344X 6. : Bidirectional quantum secure communication based on a shared private Bell state. Opt. Commun. 282, 2460–2463 (2009) 7. : Two quantum dialogue protocols without information leakage. Opt. Commun. 283, 2288–2293 (2010) 8. : Bidirectional quantum secure communication scheme based on Bell states and auxiliary particles.

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