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Complete measurements, while providing maximal information gain, results in destruction of the shared entanglement.
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The concurrence of a general Bell diagonal state, ρ B D \rho^{BD} , with probabilities, p 1 , p 2 , p 3 , p 4 p_{1},p_{2},p_{3},p_{4} in descending order is given by C ( ρ B D ) = max { p 1 − p 2 − p 3 − p 4 , 0 } C(\rho^{BD})=\max\{p_{1}-p_{2}-p_{3}-p_{4},0\} . The usual Horodecki protocol yields a fidelity of 2 + C ( ρ B D ) 3 \frac{2+C(\rho^{BD})}{3} , which is the maximal attainable fidelity for a given concurrence [ 2 ]
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The Bell basis consists of four maximally entangled states, given by | B i A B ⟩ = 1 2 ( | 00 ⟩ − ( − 1 ) i | 11 ⟩ ) , i = 1 , 2 , \displaystyle|B_{i}^{AB}\rangle=\frac{1}{\sqrt{2}}(|00\rangle-(-1)^{i}|11\rangle),\,\,i=1,2, | B i A B ⟩ = 1 2 ( | 01 ⟩ − ( − 1 ) i | 10 ⟩ ) , i = 3 , 4 , \displaystyle|B_{i}^{AB}\rangle=\frac{1}{\sqrt{2}}(|01\rangle-(-1)^{i}|10\rangle),\,\,i=3,4, where { | 0 ⟩ , | 1 ⟩ } \{|0\rangle,|1\rangle\} is the computational basis
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