Quantum Entropy Functions
Abstract¶ Some Quantum entropy functions and their properties are given.
Keywords: Entropy Relative Entropy Rényi Entropies. ¶ Von Neumann Entropy ¶ Let ρ \rho ρ ρ \rho ρ
S ( ρ ) = − tr ( ρ log ( ρ ) ) S(\rho) = -\textrm{tr}(\rho \log (\rho)) S ( ρ ) = − tr ( ρ log ( ρ )) It can be thought of as a measure of how far from a pure state a given state ρ \rho ρ
S ( ρ ) ≥ 0 S(\rho) \geq 0 S ( ρ ) ≥ 0 S ( ρ ) = 0 S(\rho)=0 S ( ρ ) = 0 ρ \rho ρ max ρ S ( ρ ) = log d \max_{\rho} S(\rho)= \log{d} max ρ S ( ρ ) = log d d d d ρ \rho ρ S ( U ρ U † ) = S ( ρ ) S(U \rho U^{\dagger}) = S(\rho) S ( U ρ U † ) = S ( ρ ) U U † = U † U = I UU^{\dagger} = U^{\dagger}U = \mathbb{I} U U † = U † U = I If ∑ i p i = 1 \sum_{i} p_i = 1 ∑ i p i = 1 S ( ∑ i p i ρ i ) ≥ ∑ i p i S ( ρ i ) . S \bigg( \sum_{i} p_{i} \rho_i \bigg) \geq \sum_{i} p_{i} S(\rho_{i}). S ( i ∑ p i ρ i ) ≥ i ∑ p i S ( ρ i ) . S ( ρ ⊗ σ ) = S ( ρ ) ⊗ S ( σ ) S(\rho \otimes \sigma) = S(\rho) \otimes S(\sigma) S ( ρ ⊗ σ ) = S ( ρ ) ⊗ S ( σ ) S ( ρ A B ) ≤ S ( ρ A ) + S ( ρ B ) S(\rho_{AB}) \leq S(\rho_{A}) + S(\rho_{B}) S ( ρ A B ) ≤ S ( ρ A ) + S ( ρ B ) ρ A = tr B ( ρ A B ) \rho_{A} = \textrm{tr}_{B}(\rho_{AB}) ρ A = tr B ( ρ A B ) S ( ρ A B ) ≥ ∣ S ( ρ A ) − S ( ρ B ) ∣ S(\rho_{AB}) \geq \vert S(\rho_{A}) - S(\rho_{B}) \vert S ( ρ A B ) ≥ ∣ S ( ρ A ) − S ( ρ B ) ∣ Quantum Relative Entropy ¶ Let ρ , σ \rho, \sigma ρ , σ
S ( ρ ∣ ∣ σ ) = tr ( ρ log ( ρ ) − ρ log ( σ ) ) . S( \rho \vert \vert \sigma) = \textrm{tr} \big( \rho \log(\rho) - \rho \log(\sigma)). S ( ρ ∣∣ σ ) = tr ( ρ log ( ρ ) − ρ log ( σ )) . if supp ( ρ ) ⊆ supp ( σ ) \textrm{supp}(\rho) \subseteq \textrm{supp}(\sigma) supp ( ρ ) ⊆ supp ( σ ) + ∞ + \infty + ∞
It can be thought of as a measure of how similar two states ρ \rho ρ σ \sigma σ
S ( ρ ∣ ∣ σ ) ≥ 0 S( \rho \vert \vert \sigma) \geq 0 S ( ρ ∣∣ σ ) ≥ 0 S ( ρ ∣ ∣ σ ) ≠ S ( σ ∣ ∣ ρ ) S( \rho \vert \vert \sigma) \neq S( \sigma \vert \vert \rho) S ( ρ ∣∣ σ ) = S ( σ ∣∣ ρ ) If E \mathcal{E} E S ( ρ ∣ ∣ σ ) ≥ S ( E ( ρ ) ∣ ∣ E ( σ ) ) . S( \rho \vert \vert \sigma) \geq S( \mathcal{E}(\rho) \vert \vert \mathcal{E}(\sigma)). S ( ρ ∣∣ σ ) ≥ S ( E ( ρ ) ∣∣ E ( σ )) . S ( ρ ∣ ∣ I / d ) = log ( d ) − S ( ρ ) S( \rho \vert \vert \mathbb{I}/d) = \log(d) - S(\rho) S ( ρ ∣∣ I / d ) = log ( d ) − S ( ρ ) If ∑ i p i = 1 \sum_{i} p_i = 1 ∑ i p i = 1 S ( ∑ i p i ρ i ∣ ∣ ∑ i p i σ i ) ≤ ∑ i p i S ( ρ i ∣ ∣ σ i ) S \bigg( \sum_{i} p_i \rho_i \vert \vert \sum_i p_i \sigma_i \bigg) \leq \sum_i p_i S(\rho_i \vert \vert \sigma_i) S ( i ∑ p i ρ i ∣∣ i ∑ p i σ i ) ≤ i ∑ p i S ( ρ i ∣∣ σ i ) Quantum α \alpha α ¶ Let ρ \rho ρ α \alpha α ρ \rho ρ
S α = 1 1 − α log ( tr ( ρ α ) ) , α ∈ ( 0 , 1 ) ∩ ( 1 , ∞ ) . S_{\alpha} = \frac{1}{1-\alpha} \log(\textrm{tr}(\rho^{\alpha})), ~ \alpha \in (0,1) \cap (1, \infty). S α = 1 − α 1 log ( tr ( ρ α )) , α ∈ ( 0 , 1 ) ∩ ( 1 , ∞ ) . These are a set of functions that generalise entropy functions (such as the Von Neumann Entropy α \alpha α
In the limit of α → 1 \alpha \rightarrow 1 α → 1 α \alpha α Von Neumann Entropy
S ( ρ ) = lim α → 1 S α ( ρ ) . S(\rho) = \lim_{\alpha \rightarrow 1} S_{\alpha}(\rho). S ( ρ ) = α → 1 lim S α ( ρ ) . S α ( ρ ) ≥ 0 S_\alpha(\rho) \geq 0 S α ( ρ ) ≥ 0 S α ( ρ ) = 0 S_\alpha(\rho)=0 S α ( ρ ) = 0 ρ \rho ρ max ρ S α ( ρ ) = log d \max_{\rho} S_\alpha(\rho)= \log{d} max ρ S α ( ρ ) = log d d d d ρ \rho ρ S α ( U ρ U † ) = S α ( ρ ) S_\alpha(U \rho U^{\dagger}) = S_\alpha(\rho) S α ( U ρ U † ) = S α ( ρ ) U U † = U † U = I UU^{\dagger} = U^{\dagger}U = \mathbb{I} U U † = U † U = I If ∑ i p i = 1 \sum_{i} p_i = 1 ∑ i p i = 1 S α ( ∑ i p i ρ i ) ≥ ∑ i p i S α ( ρ i ) . \begin{align*}
S_\alpha \bigg( \sum_{i} p_{i} \rho_i \bigg) \geq \sum_{i} p_{i} S_\alpha(\rho_{i}).
\end{align*} S α ( i ∑ p i ρ i ) ≥ i ∑ p i S α ( ρ i ) . S α ( ρ ⊗ σ ) = S α ( ρ ) ⊗ S α ( σ ) S_\alpha(\rho \otimes \sigma) = S_\alpha(\rho) \otimes S_\alpha(\sigma) S α ( ρ ⊗ σ ) = S α ( ρ ) ⊗ S α ( σ ) S α ( ρ ) − S 0 ( ρ B ) ≤ S α ( ρ A B ) ≤ S α ( ρ A ) + S 0 ( ρ B ) S_\alpha(\rho) - S_0(\rho_{B}) \leq S_\alpha(\rho_{AB}) \leq S_\alpha(\rho_{A}) + S_0(\rho_{B}) S α ( ρ ) − S 0 ( ρ B ) ≤ S α ( ρ A B ) ≤ S α ( ρ A ) + S 0 ( ρ B ) S α 1 ( ρ ) ≥ S α 2 ( ρ ) S_{\alpha_1}(\rho) \geq S_{\alpha_2}(\rho) S α 1 ( ρ ) ≥ S α 2 ( ρ ) 1 < α 1 ≤ α 2 1 < \alpha_1 \leq \alpha_2 1 < α 1 ≤ α 2 Quantum Relative α \alpha α ¶ Let ρ , σ \rho, \sigma ρ , σ α \alpha α
S α ( ρ ∣ ∣ σ ) = 1 1 − α log [ tr ( ρ α σ 1 − α ) ] S_{\alpha}(\rho \vert \vert \sigma) = \frac{1}{1-\alpha} \log \big[ \textrm{tr} \big( \rho^{\alpha} \sigma^{1-\alpha} \big) \big] S α ( ρ ∣∣ σ ) = 1 − α 1 log [ tr ( ρ α σ 1 − α ) ] The following entropy functions are limits of the quantum relative α \alpha α
S α ( ρ ∣ ∣ σ ) ≥ 0 S_{\alpha}(\rho \vert \vert \sigma) \geq 0 S α ( ρ ∣∣ σ ) ≥ 0 S α ( ρ ∣ ∣ σ ) = 0 S_{\alpha}(\rho \vert \vert \sigma)=0 S α ( ρ ∣∣ σ ) = 0 ρ = σ \rho=\sigma ρ = σ If E \mathcal{E} E S α ( ρ ∣ ∣ σ ) ≥ S α ( E ( ρ ) ∣ ∣ E ( σ ) ) . S_{\alpha}(\rho \vert \vert \sigma) \geq S_{\alpha}( \mathcal{E}(\rho) \vert \vert \mathcal{E}(\sigma)). S α ( ρ ∣∣ σ ) ≥ S α ( E ( ρ ) ∣∣ E ( σ )) . Max Relative Entropy ¶ Let ρ , σ \rho, \sigma ρ , σ ρ ≥ 0 , tr ( ρ ) = 1 \rho \geq 0, ~\textrm{tr}(\rho)=1 ρ ≥ 0 , tr ( ρ ) = 1 σ ≥ 0 \sigma \geq 0 σ ≥ 0 max relative entropy
D max ( ρ ∣ ∣ σ ) = log [ min { λ : ρ ≤ λ σ } ] D_{\textrm{max}}(\rho \vert \vert \sigma) = \log \big[ \textrm{min} \{ \lambda~:~\rho \leq \lambda \sigma \} \big] D max ( ρ ∣∣ σ ) = log [ min { λ : ρ ≤ λσ } ] or
D max ( ρ ∣ ∣ σ ) = log [ μ m a x ( σ 1 2 ρ σ 1 2 ) ] D_{\textrm{max}}(\rho \vert \vert \sigma) = \log \big[ \mu_{max} (\sigma^{\frac{1}{2}} \rho \sigma^{\frac{1}{2}} ) \big] D max ( ρ ∣∣ σ ) = log [ μ ma x ( σ 2 1 ρ σ 2 1 ) ] where μ m a x ( A ) \mu_{max}(A) μ ma x ( A ) A A A supp ( ρ ) ⊆ supp ( σ ) \textrm{supp}(\rho) \subseteq \textrm{supp}(\sigma) supp ( ρ ) ⊆ supp ( σ ) + ∞ + \infty + ∞
D max ( ρ ∣ ∣ σ ) ≥ 0 D_{\textrm{max}}( \rho \vert \vert \sigma) \geq 0 D max ( ρ ∣∣ σ ) ≥ 0 D max ( ρ ∣ ∣ σ ) = 0 D_{\textrm{max}}(\rho \vert \vert \sigma)=0 D max ( ρ ∣∣ σ ) = 0 ρ = σ \rho=\sigma ρ = σ If E \mathcal{E} E D max ( ρ ∣ ∣ σ ) ≥ D max ( E ( ρ ) ∣ ∣ E ( σ ) ) . D_{\textrm{max}}( \rho \vert \vert \sigma) \geq D_{\textrm{max}}( \mathcal{E}(\rho) \vert \vert \mathcal{E}(\sigma)). D max ( ρ ∣∣ σ ) ≥ D max ( E ( ρ ) ∣∣ E ( σ )) . If ρ = ∑ i n p i ρ i \rho = \sum_{i}^{n} p_i \rho_i ρ = ∑ i n p i ρ i σ = ∑ i n q i σ 1 \sigma = \sum_i^{n} q_i \sigma_1 σ = ∑ i n q i σ 1 D max ( ρ ∣ ∣ σ ) ≤ m a x i D max ( ρ i ∣ ∣ σ i ) D_{\textrm{max}}(\rho \vert \vert \sigma) \leq max_{i} D_{\textrm{max}}(\rho_i \vert \vert \sigma_i) D max ( ρ ∣∣ σ ) ≤ ma x i D max ( ρ i ∣∣ σ i ) D max ( ρ ∣ ∣ σ ) ≥ S ( ρ ∣ ∣ σ ) D_{\textrm{max}}(\rho \vert \vert \sigma) \geq S(\rho \vert \vert \sigma) D max ( ρ ∣∣ σ ) ≥ S ( ρ ∣∣ σ ) D max ( U ρ U † ∣ ∣ U σ U † ) = D max ( ρ ∣ ∣ σ ) D_{\textrm{max}}(U \rho U^{\dagger} \vert \vert U \sigma U^{\dagger}) = D_{\textrm{max}}(\rho \vert \vert \sigma) D max ( U ρ U † ∣∣ U σ U † ) = D max ( ρ ∣∣ σ ) D max ( ρ ∣ ∣ σ ) ≤ − log μ m i n ( σ ) D_{\textrm{max}}(\rho \vert \vert \sigma) \leq - \log{\mu_{min}(\sigma)} D max ( ρ ∣∣ σ ) ≤ − log μ min ( σ ) D max ( ρ 1 ⊗ ρ 2 ∣ ∣ σ 1 ⊗ σ 2 ) = D max ( ρ 1 ∣ ∣ σ 2 ) + D max ( ρ 2 ∣ ∣ σ 2 ) D_{\textrm{max}}(\rho_1 \otimes \rho_2 \vert \vert \sigma_1 \otimes \sigma_2) = D_{\textrm{max}}(\rho_1 \vert \vert \sigma_2) + D_{\textrm{max}}(\rho_2 \vert \vert \sigma_2) D max ( ρ 1 ⊗ ρ 2 ∣∣ σ 1 ⊗ σ 2 ) = D max ( ρ 1 ∣∣ σ 2 ) + D max ( ρ 2 ∣∣ σ 2 ) D max ( A : B ) = D max ( ρ A B ∣ ∣ ρ A ⊗ ρ B ) D_{\textrm{max}}(A:B) = D_{\textrm{max}}(\rho_{AB} \vert \vert \rho_A \otimes \rho_B) D max ( A : B ) = D max ( ρ A B ∣∣ ρ A ⊗ ρ B ) A mutual information like quantity Min Relative Entropy ¶ Let ρ , σ \rho, \sigma ρ , σ ρ ≥ 0 , tr ( ρ ) = 1 \rho \geq 0, ~\textrm{tr}(\rho)=1 ρ ≥ 0 , tr ( ρ ) = 1 σ ≥ 0 \sigma \geq 0 σ ≥ 0 min relative entropy
D min ( ρ ∣ ∣ σ ) = − log [ Tr ( Π ρ σ ) ] D_{\textrm{min}}(\rho \vert \vert \sigma) = - \log \big[ \textrm{Tr} \big( \Pi_{\rho} \sigma \big) \big] D min ( ρ ∣∣ σ ) = − log [ Tr ( Π ρ σ ) ] where Π ρ \Pi_{\rho} Π ρ ρ \rho ρ supp ( ρ ) ⊆ supp ( σ ) \textrm{supp}(\rho) \subseteq \textrm{supp}(\sigma) supp ( ρ ) ⊆ supp ( σ )
Note that
D min ( ρ ∣ ∣ σ ) = lim α → 0 + S α ( ρ ∣ ∣ σ ) D_{\textrm{min}}( \rho \vert \vert \sigma) = \lim_{\alpha \rightarrow 0^+} S_{\alpha}(\rho \vert \vert \sigma) D min ( ρ ∣∣ σ ) = α → 0 + lim S α ( ρ ∣∣ σ ) D min ( ρ ∣ ∣ σ ) ≥ 0 D_{\textrm{min}}( \rho \vert \vert \sigma) \geq 0 D min ( ρ ∣∣ σ ) ≥ 0 D min ( ρ ∣ ∣ σ ) = 0 D_{\textrm{min}}(\rho \vert \vert \sigma)=0 D min ( ρ ∣∣ σ ) = 0 ρ = σ \rho=\sigma ρ = σ supp ( ρ ) = supp ( σ ) \textrm{supp}(\rho)=\textrm{supp}(\sigma) supp ( ρ ) = supp ( σ ) D min ( ρ ∣ ∣ σ ) = 0 D_{\textrm{min}}( \rho \vert \vert \sigma)=0 D min ( ρ ∣∣ σ ) = 0 ρ \rho ρ σ \sigma σ D min ( ρ ∣ ∣ σ ) ≤ D max ( ρ ∣ ∣ σ ) D_{\textrm{min}}( \rho \vert \vert \sigma) \leq D_{\textrm{max}}( \rho \vert \vert \sigma) D min ( ρ ∣∣ σ ) ≤ D max ( ρ ∣∣ σ ) If E \mathcal{E} E D min ( ρ ∣ ∣ σ ) ≥ D min ( E ( ρ ) ∣ ∣ E ( σ ) ) . D_{\textrm{min}}( \rho \vert \vert \sigma) \geq D_{\textrm{min}}( \mathcal{E}(\rho) \vert \vert \mathcal{E}(\sigma)). D min ( ρ ∣∣ σ ) ≥ D min ( E ( ρ ) ∣∣ E ( σ )) . If ρ = ∑ i n p i ρ i \rho = \sum_{i}^{n} p_i \rho_i ρ = ∑ i n p i ρ i σ = ∑ i n p i σ 1 \sigma = \sum_i^{n} p_i \sigma_1 σ = ∑ i n p i σ 1 D min ( ρ ∣ ∣ σ ) ≤ ∑ i p i D min ( ρ i ∣ ∣ σ i ) D_{\textrm{min}}(\rho \vert \vert \sigma) \leq \sum_i p_i D_{\textrm{min}}(\rho_i \vert \vert \sigma_i) D min ( ρ ∣∣ σ ) ≤ i ∑ p i D min ( ρ i ∣∣ σ i ) D min ( ρ ∣ ∣ σ ) ≤ S ( ρ ∣ ∣ σ ) D_{\textrm{min}}(\rho \vert \vert \sigma) \leq S(\rho \vert \vert \sigma) D min ( ρ ∣∣ σ ) ≤ S ( ρ ∣∣ σ ) D min ( U ρ U † ∣ ∣ U σ U † ) = D min ( ρ ∣ ∣ σ ) D_{\textrm{min}}(U \rho U^{\dagger} \vert \vert U \sigma U^{\dagger}) = D_{\textrm{min}}(\rho \vert \vert \sigma) D min ( U ρ U † ∣∣ U σ U † ) = D min ( ρ ∣∣ σ ) D min ( ρ ∣ ∣ σ ) ≤ − log μ m i n ( σ ) D_{\textrm{min}}(\rho \vert \vert \sigma) \leq - \log{\mu_{min}(\sigma)} D min ( ρ ∣∣ σ ) ≤ − log μ min ( σ ) D min ( A : B ) = D min ( ρ A B ∣ ∣ ρ A ⊗ ρ B ) D_{\textrm{min}}(A:B) = D_{\textrm{min}}(\rho_{AB} \vert \vert \rho_A \otimes \rho_B) D min ( A : B ) = D min ( ρ A B ∣∣ ρ A ⊗ ρ B ) A mutual information like quantity Quantum Sandwiched α \alpha α ¶ Let ρ , σ \rho, \sigma ρ , σ α \alpha α
D α ( ρ ∣ ∣ σ ) = 1 1 − α log [ tr ( σ 1 − α 2 α ρ σ 1 − α 2 α ) α ] . D_{\alpha}( \rho \vert \vert \sigma) = \frac{1}{1-\alpha} \log { \bigg[ \textrm{tr} \big( \sigma^{\frac{1-\alpha}{2 \alpha}} \rho \sigma^{\frac{1-\alpha}{2 \alpha}} \big)^{\alpha} \bigg] } . D α ( ρ ∣∣ σ ) = 1 − α 1 log [ tr ( σ 2 α 1 − α ρ σ 2 α 1 − α ) α ] . if supp ( ρ ) ⊆ supp ( σ ) \textrm{supp}(\rho) \subseteq \textrm{supp}(\sigma) supp ( ρ ) ⊆ supp ( σ ) + ∞ + \infty + ∞
D α ( ρ ∣ ∣ σ ) ≥ 0 D_\alpha(\rho \vert \vert \sigma) \geq 0 D α ( ρ ∣∣ σ ) ≥ 0 D α ( ρ ∣ ∣ σ ) D_\alpha(\rho \vert \vert \sigma) D α ( ρ ∣∣ σ ) ρ = σ \rho=\sigma ρ = σ If E \mathcal{E} E α ∈ [ 1 2 , 1 ) \alpha \in [\frac{1}{2}, 1) α ∈ [ 2 1 , 1 ) ( 1 , ∞ ) (1, \infty) ( 1 , ∞ ) D α ( ρ ∣ ∣ σ ) ≥ D α ( E ( ρ ) ∣ ∣ E ( σ ) ) . D_{\alpha}( \rho \vert \vert \sigma) \geq D_{\alpha}( \mathcal{E}(\rho) \vert \vert \mathcal{E}(\sigma)). D α ( ρ ∣∣ σ ) ≥ D α ( E ( ρ ) ∣∣ E ( σ )) . D α ( ρ ∣ ∣ σ ) ≥ D β ( ρ ∣ ∣ σ ) D_{\alpha}(\rho \vert \vert \sigma) \geq D_{\beta}(\rho \vert \vert \sigma) D α ( ρ ∣∣ σ ) ≥ D β ( ρ ∣∣ σ ) α ≥ β \alpha \geq \beta α ≥ β The following entropy functions are limits of the quantum sandwiched α \alpha α
Min Relative Entropy ¶ Let ρ , σ \rho, \sigma ρ , σ
D min ( ρ ∣ ∣ σ ) = − log [ Tr ( ρ σ ) 2 ] D_{\textrm{min}}(\rho \vert \vert \sigma) = - \log \big[ \textrm{Tr} \big( \sqrt{\rho} \sqrt{\sigma} \big)^{2} \big] D min ( ρ ∣∣ σ ) = − log [ Tr ( ρ σ ) 2 ] D min ( ρ ∣ ∣ σ ) ≥ 0 D_{\textrm{min}}( \rho \vert \vert \sigma) \geq 0 D min ( ρ ∣∣ σ ) ≥ 0 D min ( ρ ∣ ∣ σ ) = 0 D_{\textrm{min}}(\rho \vert \vert \sigma)=0 D min ( ρ ∣∣ σ ) = 0 ρ = σ \rho=\sigma ρ = σ If E \mathcal{E} E D min ( ρ ∣ ∣ σ ) ≥ D min ( E ( ρ ) ∣ ∣ E ( σ ) ) . D_{\textrm{min}}( \rho \vert \vert \sigma) \geq D_{\textrm{min}}( \mathcal{E}(\rho) \vert \vert \mathcal{E}(\sigma)). D min ( ρ ∣∣ σ ) ≥ D min ( E ( ρ ) ∣∣ E ( σ )) . Hypothesis Testing Relative Entropy ¶ Let ρ , σ \rho, \sigma ρ , σ
D H ϵ ( ρ ∣ ∣ σ ) ≔ − log inf 0 ≤ Q ≤ I , tr [ Q ρ ] ≥ 1 − ϵ tr [ Q σ ] D_{H}^{\epsilon}(\rho \vert \vert \sigma) \coloneqq - \log \inf_{ \substack{0 \leq Q \leq \mathbb{I}, \\ \textrm{tr}[ Q \rho ] \geq 1 - \epsilon}} \textrm{tr}\big[Q \sigma \big] D H ϵ ( ρ ∣∣ σ ) : = − log 0 ≤ Q ≤ I , tr [ Qρ ] ≥ 1 − ϵ inf tr [ Q σ ] D H ϵ ( ρ ∣ ∣ σ ) ≥ 0 D_{H}^{\epsilon}(\rho \vert \vert \sigma) \geq 0 D H ϵ ( ρ ∣∣ σ ) ≥ 0 D H ϵ ( ρ ∣ ∣ σ ) = 0 D_{H}^{\epsilon}(\rho \vert \vert \sigma) = 0 D H ϵ ( ρ ∣∣ σ ) = 0 ρ = σ \rho = \sigma ρ = σ If E \mathcal{E} E D H ϵ ( ρ ∣ ∣ σ ) ≥ D H ϵ ( E ( ρ ) ∣ ∣ E ( σ ) ) . D_{H}^{\epsilon}(\rho \vert \vert \sigma) \geq D_{H}^{\epsilon}( \mathcal{E}(\rho) \vert \vert \mathcal{E}(\sigma)). D H ϵ ( ρ ∣∣ σ ) ≥ D H ϵ ( E ( ρ ) ∣∣ E ( σ )) . Let H b ( ⋅ ) H_b(\cdot) H b ( ⋅ ) binary entropy function then D H ϵ ( ρ ∣ ∣ σ ) ≤ ( S ( ρ ∣ ∣ σ ) + H b ( ϵ ) ) 1 − ϵ . D_{H}^{\epsilon}(\rho \vert \vert \sigma) \leq \frac{ \bigg( S(\rho \vert \vert \sigma ) + H_b(\epsilon) \bigg)}{1 - \epsilon}. D H ϵ ( ρ ∣∣ σ ) ≤ 1 − ϵ ( S ( ρ ∣∣ σ ) + H b ( ϵ ) ) . For any ϵ ∈ ( 0 , 1 ) \epsilon \in (0,1) ϵ ∈ ( 0 , 1 ) lim n → ∞ 1 n D H ϵ ( ρ ⊗ n ∣ ∣ σ ⊗ n ) = S ( ρ ∣ ∣ σ ) . \lim_{n \rightarrow \infty} \frac{1}{n} D_{H}^{\epsilon}(\rho^{\otimes n} \vert \vert \sigma^{\otimes n}) = S(\rho \vert \vert \sigma). n → ∞ lim n 1 D H ϵ ( ρ ⊗ n ∣∣ σ ⊗ n ) = S ( ρ ∣∣ σ ) . This is the Quantum Stein’s Lemma
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