Comment on “Quantum Superimposing Multiple Anti-Cloning Machine”

Abstract

Recently, Li et al. (Int. J. Theor. Phys. 46, 2599, 2007) has constructed the quantum superimposing multiple anti-cloning machine, moreover established the sufficient and necessary condition of this machine exists. In the proofs given by Li et al. (Int. J. Theor. Phys. 46, 2599, 2007), claimed that the following key fact to hold : Fact For an arbitrary unknown state |ψ〉 belongs to n-dimensional Hilbert space, there exists an antiunitary operator K such that K|ψ〉=|ψ⊥〉 here the state |ψ⊥〉 is an orthogonal complement state of |ψ〉, that is, it satisfies the following two conditions 〈ψ|K|ψ〉=〈ψ|ψ⊥〉=0 and 〈ψ|ψ〉=〈ψ⊥|ψ⊥〉=1 In this Comment, we would like to point out that (a). In 1-dimensional Hilbert space, for an arbitrary unknown state |ψ〉, the antiunitary operator K and the orthogonal complement state both do not exist in general. (b). In 3-dimensional Hilbert space, for an arbitrary unknown state |ψ〉, the antiunitary operator K do not exist in general, there are uncountably many orthogonal complement states that can be constructed through the skew-symmetric operator, but is not unitary one. Which shows that above Fact given by Li et al. [1] is incorrect in general for both 1 and 3-dimensional Hilbert space