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For a pure state described by the state vector |χ, the density operator is ρ=|χ χ|. Its trace is Tr(ρ)= χ|χ. The operator ρ2=|χ χ|χ χ|, while its trace is Tr(ρ2)= χ|χ χ|χ=[Tr(ρ)]2. Hence, the ratio Tr(ρ2)/[Tr(ρ)]2= 1, for nonzero values of Tr(ρ). For a mixed state, there is no single state vector describing the system, and Tr(ρ2)/[Tr(ρ)]2<1. The transition from a pure state to a mixed state is caused by decoherence.
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For a pure state described by the state vector |χ, the density operator is ρ=|χ χ|. Its trace is Tr(ρ)= χ|χ. The operator ρ2=|χ χ|χ χ|, while its trace is Tr(ρ2)= χ|χ χ|χ=[Tr(ρ)]2. Hence, the ratio Tr(ρ2)/[Tr(ρ)]2= 1, for nonzero values of Tr(ρ). For a mixed state, there is no single state vector describing the system, and Tr(ρ2)/[Tr(ρ)]2<1. The transition from a pure state to a mixed state is caused by decoherence.
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