Direct and Reverse Secret-Key Capacities of a Quantum Channel

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For simplicity, we include error correction and privacy amplification (which correspond to a classical channel) into the decoding POVM

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In the formulas for K ▶ ​ ( 𝒩 ) K_{\blacktriangleright}(\mathcal{N}) , K ◀ ​ ( 𝒩 ) K_{\blacktriangleleft}(\mathcal{N}) , K ▶ ⊗ ​ ( 𝒩 ) K_{\blacktriangleright}^{\otimes}(\mathcal{N}) and K ◀ ⊗ ​ ( 𝒩 ) K_{\blacktriangleleft}^{\otimes}(\mathcal{N}) we implicitly assume max ⁡ x = max ⁡ { 0 , x } \max x=\max\{0,x\}

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Detailed derivations will be presented elsewhere

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In the formulas for K ▶ ​ ( 𝒩 ) K_{\blacktriangleright}(\mathcal{N}) and K ▶ ⊗ ​ ( 𝒩 ) K_{\blacktriangleright}^{\otimes}(\mathcal{N}) , the information quantities refer to the classical-quantum state ω X ​ T ​ B n ​ E n = ∑ t , x p ( t | x ) p ( x ) | x ⟩ ⟨ x | X ⊗ | t ⟩ ⟨ t | T ⊗ ρ B n ​ E n ( x ) \omega_{XTB^{n}E^{n}}=\sum_{t,x}p(t|x)p(x)\left|x\right\rangle\left\langle x\right|_{X}\otimes\left|t\right\rangle\left\langle t\right|_{T}\otimes\rho_{B^{n}E^{n}}(x) , where ρ B n ​ E n ​ ( x ) \rho_{B^{n}E^{n}}(x) is Bob and Eve’s state conditioned to the outcome x x . A corresponding state ω Y ​ T ​ A n ​ E n \omega_{YTA^{n}E^{n}} must be considered for K ◀ ​ ( 𝒩 ) K_{\blacktriangleleft}(\mathcal{N}) and K ◀ ⊗ ​ ( 𝒩 ) K_{\blacktriangleleft}^{\otimes}(\mathcal{N})

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