Elementary quantum cloning machines.pdf

a r X i v : q u a n t - p h / 0 6 0 2 0 3 5 v 1 3 F e b 2 0 0 6 Elementary quantum cloning machines V.N.Dumachev February 1, 2008 Abstract The task of reception of a copy of an arbitrary quantum state with use of a minimum quantity of quantum operations is considered. 1 Introduction It is known, that an arbitrary quantum state cannot be copied perfectly [1]. However V.Buzek and M.Hillery in work [2] offered a universal quantum cloning machine (UQCM), allowing to create 2 identical qubits from 1 qubit. Two output qubits are a copy of each other, but they are not a copy of an initial quantum state, and are similar to it only in 5/6 ～= 0.83. Universality of the UQCM is that it clones any quantum state with identical accuracy. In the present work some variants of not universal QCM are considered. Their lack is that they clone qubits non-uniformly depending on their state. In a final section the calculation scheme of UQCM from the general principles is submitted which includes known results [3,4] as special cases. It is shown, how one can choose optimum by quantity of used quantum operations UQCM from given ones. 1.1 Brief theoretical data Any quantum state |ψ0〉 = α |0〉+ β |1〉 (1) is represented by a point on Bloch sphere and changes by means of rotation operator [5]: R (θ, ?) = ( cos θ ?ie?i? sin θ ?iei? sin θ cos θ ) . For simplicity we shall work with equatorial qubits, then ? = π/2 and R (θ) |0〉 = cos θ |0〉+ sin θ |1〉 , R (θ) |1〉 = ? sin θ |0〉+ cos θ |1〉 . 1 Pauli matrices can be written down as projectors I? = σ0 = |0〉 〈0|+ |1〉 〈1| , σ1 = |1〉 〈0|+ |0〉 〈1| , σ2 = i (|1〉 〈0| ? |0〉 〈1|) , σ3 = |0〉 〈0| ? |1〉 〈1| . Action of Pauli matrixes on an input |ψ0〉 is given by expression |ψi〉 = σi |ψ0〉 . (2) We shall note at once, that scalar product 〈ψ0| ψ2〉 = 0. It means, that for any |ψ0〉 with the help of the universal operator NOT = ?iσ2 it is possible to create an orthogonal state |ψ2〉. For corresponding (2) matrixes of density we shell receive ρini = |ψi〉 〈ψi| . (3) Si