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We consider quantum-memory assisted protocols for discriminating quantum channels.
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This is the delayed measurement principle
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The orthogonality condition is 0 = ( I ⊗ Ψ T ) C 0 ( I ⊗ Ψ ∗ Ψ T ) C 1 ( I ⊗ Ψ ∗ ) = H 0 † H 0 H 1 † H 1 0=(I\otimes\Psi^{T})C_{0}(I\otimes\Psi^{*}\Psi^{T})C_{1}(I\otimes\Psi^{*})=H_{0}^{\dagger}H_{0}H_{1}^{\dagger}H_{1} , with H i = C i 1 2 ( I ⊗ Ψ ∗ ) H_{i}=C_{i}^{\frac{1}{2}}(I\otimes\Psi^{*}) , which holds iff H 0 H 1 † = 0 H_{0}H_{1}^{\dagger}=0
Cited in the paper.
This distance is referred to in the literature as cb-norm distance, since it is induced by the norm of complete boundedness [ 17 ] (cb-norm for short, also defined diamond norm in Ref. [ 18 ] )
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In the degenerate case, if Θ ( U ) + Θ ( V ) > π \Theta(U)+\Theta(V)>\pi , it is always possible to find T T such that Θ ( U T V T † ) = π \Theta(UTVT^{\dagger})=\pi
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