Bell diagonal state
Bell diagonal states are a class of bipartite qubit states that are frequently used in quantum information and quantum computation theory. == Definition == The Bell diagonal state is defined as the probabilistic mixture of Bell states: | ϕ + ⟩ = 1 2 ( | 0 ⟩ A ⊗ | 0 ⟩ B + | 1 ⟩ A ⊗ | 1 ⟩ B ) {\displaystyle |\phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B})} | ϕ − ⟩ = 1 2 ( | 0 ⟩ A ⊗ | 0 ⟩ B − | 1 ⟩ A ⊗ | 1 ⟩ B ) {\displaystyle |\phi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}-|1\rangle _{A}\otimes |1\rangle _{B})} | ψ + ⟩ = 1 2 ( | 0 ⟩ A ⊗ | 1 ⟩ B + | 1 ⟩ A ⊗ | 0 ⟩ B ) {\displaystyle |\psi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}+|1\rangle _{A}\otimes |0\rangle _{B})} | ψ − ⟩ = 1 2 ( | 0 ⟩ A ⊗ | 1 ⟩ B − | 1 ⟩ A ⊗ | 0 ⟩ B ) {\displaystyle |\psi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B})} In density operator form, a Bell diagonal state is defined as ϱ B e l l = p 1 | ϕ + ⟩ ⟨ ϕ + | + p 2 | ϕ − ⟩ ⟨ ϕ − | + p 3 | ψ + ⟩ ⟨ ψ + | + p 4 | ψ − ⟩ ⟨ ψ − | {\displaystyle \varrho ^{Bell}=p_{1}|\phi ^{+}\rangle \langle \phi ^{+}|+p_{2}|\phi ^{-}\rangle \langle \phi ^{-}|+p_{3}|\psi ^{+}\rangle \langle \psi ^{+}|+p_{4}|\psi ^{-}\rangle \langle \psi ^{-}|} where p 1 , p 2 , p 3 , p 4 {\displaystyle p_{1},p_{2},p_{3},p_{4}} is a probability distribution.