相対エントロピーから情報と物理を組み立てる 第3部 第2章 §3.2.4 ★★★ 相手を周辺の積に取ると量子相互情報量と劣加法性が出る。純粋状態では部分系のエントロピーが等しく、条件付きエントロピーは負になりうる
部分トレースと量子相互情報量
2つの系 A A A B B B ρ ^ A B \hat\rho_{AB} ρ ^ A B ρ ^ A = Tr B ρ ^ A B \hat\rho_A=\operatorname{Tr}_B\hat\rho_{AB} ρ ^ A = Tr B ρ ^ A B ρ ^ B = Tr A ρ ^ A B \hat\rho_B=\operatorname{Tr}_A\hat\rho_{AB} ρ ^ B = Tr A ρ ^ A B S ( A ) ρ ≔ S ( ρ ^ A ) S(A)_\rho\coloneqq S(\hat\rho_A) S ( A ) ρ : = S ( ρ ^ A ) S ( B ) ρ ≔ S ( ρ ^ B ) S(B)_\rho\coloneqq S(\hat\rho_B) S ( B ) ρ : = S ( ρ ^ B ) S ( A B ) ρ ≔ S ( ρ ^ A B ) S(AB)_\rho\coloneqq S(\hat\rho_{AB}) S ( A B ) ρ : = S ( ρ ^ A B ) ρ \rho ρ
定義 3.2.8（量子相互情報量・条件付きエントロピー）
I ( A ; B ) ρ ≔ S ( A ) + S ( B ) − S ( A B ) , S ( A ∣ B ) ρ ≔ S ( A B ) − S ( B ) . (3.2.20) I(A;B)_\rho\coloneqq S(A)+S(B)-S(AB),\qquad S(A\,|\,B)_\rho\coloneqq S(AB)-S(B).\tag{3.2.20} I ( A ; B ) ρ : = S ( A ) + S ( B ) − S ( A B ) , S ( A ∣ B ) ρ : = S ( A B ) − S ( B ) . ( 3.2.20 )
古典の I ( X ; Y ) = D ( p X Y ∥ p X p Y ) I(X;Y)=D(p_{XY}\,\|\,p_Xp_Y) I ( X ; Y ) = D ( p X Y ∥ p X p Y )
定理 3.2.9（相互情報量は相対エントロピー ★★★）
I ( A ; B ) ρ = D ( ρ ^ A B ∥ ρ ^ A ⊗ ρ ^ B ) ≥ 0 , S ( A ∣ B ) ρ = − D ( ρ ^ A B ∥ 1 A ⊗ ρ ^ B ) . (3.2.21) I(A;B)_\rho=D(\hat\rho_{AB}\,\|\,\hat\rho_A\otimes\hat\rho_B)\ge0,\qquad S(A\,|\,B)_\rho=-D(\hat\rho_{AB}\,\|\,\mathbf 1_A\otimes\hat\rho_B).\tag{3.2.21} I ( A ; B ) ρ = D ( ρ ^ A B ∥ ρ ^ A ⊗ ρ ^ B ) ≥ 0 , S ( A ∣ B ) ρ = − D ( ρ ^ A B ∥ 1 A ⊗ ρ ^ B ) . ( 3.2.21 )
特に劣加法性 S ( A B ) ≤ S ( A ) + S ( B ) S(AB)\le S(A)+S(B) S ( A B ) ≤ S ( A ) + S ( B ) ρ ^ A B = ρ ^ A ⊗ ρ ^ B \hat\rho_{AB}=\hat\rho_A\otimes\hat\rho_B ρ ^ A B = ρ ^ A ⊗ ρ ^ B S ( A ∣ B ) ≤ ln d A S(A\,|\,B)\le\ln d_A S ( A ∣ B ) ≤ ln d A d A d_A d A A A A
証明 命題 3.2.5 (b) の証明と同じく ln ( ρ ^ A ⊗ ρ ^ B ) = ln ρ ^ A ⊗ 1 B + 1 A ⊗ ln ρ ^ B \ln(\hat\rho_A\otimes\hat\rho_B)=\ln\hat\rho_A\otimes\mathbf 1_B+\mathbf 1_A\otimes\ln\hat\rho_B ln ( ρ ^ A ⊗ ρ ^ B ) = ln ρ ^ A ⊗ 1 B + 1 A ⊗ ln ρ ^ B Tr ρ ^ A B ( ln ρ ^ A ⊗ 1 B ) = Tr ρ ^ A ln ρ ^ A = − S ( A ) \operatorname{Tr}\hat\rho_{AB}(\ln\hat\rho_A\otimes\mathbf 1_B)=\operatorname{Tr}\hat\rho_A\ln\hat\rho_A=-S(A) Tr ρ ^ A B ( ln ρ ^ A ⊗ 1 B ) = Tr ρ ^ A ln ρ ^ A = − S ( A ) B B B − S ( B ) -S(B) − S ( B ) D ( ρ ^ A B ∥ ρ ^ A ⊗ ρ ^ B ) = − S ( A B ) + S ( A ) + S ( B ) D(\hat\rho_{AB}\,\|\,\hat\rho_A\otimes\hat\rho_B)=-S(AB)+S(A)+S(B) D ( ρ ^ A B ∥ ρ ^ A ⊗ ρ ^ B ) = − S ( A B ) + S ( A ) + S ( B ) 1 A ⊗ ρ ^ B \mathbf 1_A\otimes\hat\rho_B 1 A ⊗ ρ ^ B D = − S ( A B ) + S ( B ) = − S ( A ∣ B ) D=-S(AB)+S(B)=-S(A\,|\,B) D = − S ( A B ) + S ( B ) = − S ( A ∣ B ) Tr ( 1 A ⊗ ρ ^ B ) = d A \operatorname{Tr}(\mathbf 1_A\otimes\hat\rho_B)=d_A Tr ( 1 A ⊗ ρ ^ B ) = d A (3.2.13) から− S ( A ∣ B ) ≥ − ln d A -S(A\,|\,B)\ge-\ln d_A − S ( A ∣ B ) ≥ − ln d A
古典の条件付きエントロピーは H ( X ∣ Y ) ≥ 0 H(X\,|\,Y)\ge0 H ( X ∣ Y ) ≥ 0 S ( A ∣ B ) S(A\,|\,B) S ( A ∣ B ) (3.2.21) で D D D 1 A ⊗ ρ ^ B \mathbf 1_A\otimes\hat\rho_B 1 A ⊗ ρ ^ B d A d_A d A 1 1 1 ln d A \ln d_A ln d A 0 0 0 S ( A ∣ B ) S(A\,|\,B) S ( A ∣ B )
Schmidt 分解と純粋状態の部分系
補題 3.2.10（Schmidt 分解）
A B A B A B ∣ Ψ ⟩ | \Psi \rangle ∣Ψ ⟩ A A A { ∣ u k ⟩ } \{| u_k \rangle \} { ∣ u k ⟩} B B B { ∣ v k ⟩ } \{| v_k \rangle \} { ∣ v k ⟩} λ k \lambda_k λ k ∑ k λ k = 1 \sum_k\lambda_k=1 ∑ k λ k = 1
∣ Ψ ⟩ = ∑ k λ k ∣ u k ⟩ ⊗ ∣ v k ⟩ (3.2.22) | \Psi \rangle =\sum_k\sqrt{\lambda_k}\,| u_k \rangle \otimes| v_k \rangle\tag{3.2.22} ∣Ψ ⟩ = k ∑ λ k ∣ u k ⟩ ⊗ ∣ v k ⟩ ( 3.2.22 )
と書くことができる。このとき ρ ^ A = ∑ k λ k ∣ u k ⟩ ⟨ u k ∣ \hat\rho_A=\sum_k\lambda_k| u_k \rangle \langle u_k | ρ ^ A = ∑ k λ k ∣ u k ⟩ ⟨ u k ∣ ρ ^ B = ∑ k λ k ∣ v k ⟩ ⟨ v k ∣ \hat\rho_B=\sum_k\lambda_k| v_k \rangle \langle v_k | ρ ^ B = ∑ k λ k ∣ v k ⟩ ⟨ v k ∣ S ( A ) = S ( B ) S(A)=S(B) S ( A ) = S ( B )
証明 基底で ∣ Ψ ⟩ = ∑ a , b c a b ∣ a ⟩ ∣ b ⟩ | \Psi \rangle =\sum_{a,b}c_{ab}| a \rangle | b \rangle ∣Ψ ⟩ = ∑ a , b c ab ∣ a ⟩ ∣ b ⟩ c c c c = ∑ k λ k u ( k ) v ( k ) T c=\sum_k\sqrt{\lambda_k}\,u^{(k)}v^{(k)\mathsf T} c = ∑ k λ k u ( k ) v ( k ) T u ( k ) u^{(k)} u ( k ) v ( k ) v^{(k)} v ( k ) (3.2.22) を得る。∑ k λ k = ⟨ Ψ ∣ Ψ ⟩ = 1 \sum_k\lambda_k=\langle \Psi | \Psi \rangle =1 ∑ k λ k = ⟨ Ψ∣Ψ ⟩ = 1 Tr B \operatorname{Tr}_B Tr B ⟨ v k ∣ v l ⟩ = δ k l \langle v_k | v_l \rangle =\delta_{kl} ⟨ v k ∣ v l ⟩ = δ k l ρ ^ A \hat\rho_A ρ ^ A ρ ^ B \hat\rho_B ρ ^ B λ k \lambda_k λ k
Schmidt 分解はエンタングルメントを測る基本の道具で、第3部第4章（第3部第4章 ）で詳しく使う。
純粋状態 ∣ Ψ ⟩ | \Psi \rangle ∣Ψ ⟩ S ( A B ) = 0 S(AB)=0 S ( A B ) = 0 I ( A ; B ) = 2 S ( A ) I(A;B)=2S(A) I ( A ; B ) = 2 S ( A ) S ( A ∣ B ) = − S ( B ) = − S ( A ) S(A\,|\,B)=-S(B)=-S(A) S ( A ∣ B ) = − S ( B ) = − S ( A ) ∣ Ψ ⟩ | \Psi \rangle ∣Ψ ⟩ λ k \lambda_k λ k S ( A ) > 0 S(A)>0 S ( A ) > 0 H ( X , Y ) ≥ H ( Y ) H(X,Y)\ge H(Y) H ( X , Y ) ≥ H ( Y ) 0 0 0 S ( A ∣ B ) < 0 S(A\,|\,B)<0 S ( A ∣ B ) < 0
Araki–Lieb の不等式
定理 3.2.11（Araki–Lieb の不等式 ★★）
∣ S ( A ) − S ( B ) ∣ ≤ S ( A B ) \lvert S(A)-S(B)\rvert \le S(AB) ∣ S ( A ) − S ( B )∣ ≤ S ( A B ) − ln d A ≤ S ( A ∣ B ) ≤ ln d A -\ln d_A\le S(A\,|\,B)\le\ln d_A − ln d A ≤ S ( A ∣ B ) ≤ ln d A I ( A ; B ) ≤ 2 min { S ( A ) , S ( B ) } I(A;B)\le2\min\{S(A),S(B)\} I ( A ; B ) ≤ 2 min { S ( A ) , S ( B )}
証明 ρ ^ A B = ∑ i μ i ∣ w i ⟩ ⟨ w i ∣ \hat\rho_{AB}=\sum_i\mu_i| w_i \rangle \langle w_i | ρ ^ A B = ∑ i μ i ∣ w i ⟩ ⟨ w i ∣ C C C ∣ i ⟩ | i \rangle ∣ i ⟩ ∣ Ψ ⟩ A B C = ∑ i μ i ∣ w i ⟩ ∣ i ⟩ | \Psi \rangle _{ABC}=\sum_i\sqrt{\mu_i}| w_i \rangle | i \rangle ∣Ψ ⟩ A B C = ∑ i μ i ∣ w i ⟩ ∣ i ⟩ Tr C ∣ Ψ ⟩ ⟨ Ψ ∣ = ρ ^ A B \operatorname{Tr}_C| \Psi \rangle \langle \Psi |=\hat\rho_{AB} Tr C ∣Ψ ⟩ ⟨ Ψ∣ = ρ ^ A B 純粋化 ）である。補題 3.2.10 を ( A B ∣ C ) (AB|C) ( A B ∣ C ) ( A ∣ B C ) (A|BC) ( A ∣ B C ) ( B ∣ A C ) (B|AC) ( B ∣ A C ) S ( C ) = S ( A B ) S(C)=S(AB) S ( C ) = S ( A B ) S ( B C ) = S ( A ) S(BC)=S(A) S ( B C ) = S ( A ) S ( A C ) = S ( B ) S(AC)=S(B) S ( A C ) = S ( B ) B C BC B C S ( A ) = S ( B C ) ≤ S ( B ) + S ( C ) = S ( B ) + S ( A B ) S(A)=S(BC)\le S(B)+S(C)=S(B)+S(AB) S ( A ) = S ( B C ) ≤ S ( B ) + S ( C ) = S ( B ) + S ( A B ) A A A B B B S ( B ) − S ( A ) ≤ S ( A B ) S(B)-S(A)\le S(AB) S ( B ) − S ( A ) ≤ S ( A B )
S ( A ∣ B ) = S ( A B ) − S ( B ) ≥ − S ( A ) ≥ − ln d A S(A\,|\,B)=S(AB)-S(B)\ge-S(A)\ge-\ln d_A S ( A ∣ B ) = S ( A B ) − S ( B ) ≥ − S ( A ) ≥ − ln d A I ( A ; B ) = S ( A ) + S ( B ) − S ( A B ) ≤ S ( A ) + S ( B ) − ∣ S ( A ) − S ( B ) ∣ = 2 min { S ( A ) , S ( B ) } I(A;B)=S(A)+S(B)-S(AB)\le S(A)+S(B)-\lvert S(A)-S(B)\rvert =2\min\{S(A),S(B)\} I ( A ; B ) = S ( A ) + S ( B ) − S ( A B ) ≤ S ( A ) + S ( B ) − ∣ S ( A ) − S ( B )∣ = 2 min { S ( A ) , S ( B )}
古典では I ( X ; Y ) ≤ min { H ( X ) , H ( Y ) } I(X;Y)\le\min\{H(X),H(Y)\} I ( X ; Y ) ≤ min { H ( X ) , H ( Y )} 2 ln d 2\ln d 2 ln d 1 1 H. Araki and E. H. Lieb (1970), Communications in Mathematical Physics 18 , 160–170。
例題 3.2.4 Bell 状態と古典的に相関した状態 ★★★
オリジナル 2つの量子ビットの状態
∣ Φ + ⟩ = 1 2 ( ∣ 00 ⟩ + ∣ 11 ⟩ ) , ρ ^ c l = 1 2 ( ∣ 00 ⟩ ⟨ 00 ∣ + ∣ 11 ⟩ ⟨ 11 ∣ ) | \Phi^+ \rangle =\frac{1}{\sqrt2}(| 00 \rangle +| 11 \rangle ),\qquad \hat\rho_{\mathrm{cl}}=\frac12\bigl(| 00 \rangle \langle 00 |+| 11 \rangle \langle 11 |\bigr) ∣ Φ + ⟩ = 2 1 ( ∣00 ⟩ + ∣11 ⟩) , ρ ^ cl = 2 1 ( ∣00 ⟩ ⟨ 00∣ + ∣11 ⟩ ⟨ 11∣ )
を比べる。
どちらも ρ ^ A = ρ ^ B = 1 / 2 \hat\rho_A=\hat\rho_B=\mathbf 1/2 ρ ^ A = ρ ^ B = 1 /2
それぞれの S ( A B ) S(AB) S ( A B ) I ( A ; B ) I(A;B) I ( A ; B ) S ( A ∣ B ) S(A\,|\,B) S ( A ∣ B )
ρ ^ c l = Δ B ( ∣ Φ + ⟩ ⟨ Φ + ∣ ) \hat\rho_{\mathrm{cl}}=\Delta_{\mathcal B}(| \Phi^+ \rangle \langle \Phi^+ |) ρ ^ cl = Δ B ( ∣ Φ + ⟩ ⟨ Φ + ∣ ) B \mathcal B B { ∣ 00 ⟩ , ∣ 01 ⟩ , ∣ 10 ⟩ , ∣ 11 ⟩ } \{| 00 \rangle ,| 01 \rangle ,| 10 \rangle ,| 11 \rangle \} { ∣00 ⟩ , ∣01 ⟩ , ∣10 ⟩ , ∣11 ⟩} 3.2.6 で S ( A B ) S(AB) S ( A B )
解答 (1) ∣ Φ + ⟩ | \Phi^+ \rangle ∣ Φ + ⟩ λ 1 = λ 2 = 1 / 2 \lambda_1=\lambda_2=1/2 λ 1 = λ 2 = 1/2 3.2.10 から ρ ^ A = ρ ^ B = 1 2 ( ∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣ ) = 1 / 2 \hat\rho_A=\hat\rho_B=\frac12(| 0 \rangle \langle 0 |+| 1 \rangle \langle 1 |)=\mathbf 1/2 ρ ^ A = ρ ^ B = 2 1 ( ∣0 ⟩ ⟨ 0∣ + ∣1 ⟩ ⟨ 1∣ ) = 1 /2 ρ ^ c l \hat\rho_{\mathrm{cl}} ρ ^ cl Tr B \operatorname{Tr}_B Tr B 1 2 ( ∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣ ) \frac12(| 0 \rangle \langle 0 |+| 1 \rangle \langle 1 |) 2 1 ( ∣0 ⟩ ⟨ 0∣ + ∣1 ⟩ ⟨ 1∣ )
(2) ∣ Φ + ⟩ | \Phi^+ \rangle ∣ Φ + ⟩ S ( A B ) = 0 S(AB)=0 S ( A B ) = 0 S ( A ) = S ( B ) = ln 2 S(A)=S(B)=\ln2 S ( A ) = S ( B ) = ln 2 I ( A ; B ) = 2 ln 2 = 1.3863 I(A;B)=2\ln2=1.3863 I ( A ; B ) = 2 ln 2 = 1.3863 S ( A ∣ B ) = − ln 2 = − 0.6931 S(A\,|\,B)=-\ln2=-0.6931 S ( A ∣ B ) = − ln 2 = − 0.6931 ρ ^ c l \hat\rho_{\mathrm{cl}} ρ ^ cl ( 1 / 2 , 1 / 2 , 0 , 0 ) (1/2,1/2,0,0) ( 1/2 , 1/2 , 0 , 0 ) S ( A B ) = ln 2 S(AB)=\ln2 S ( A B ) = ln 2 I ( A ; B ) = ln 2 I(A;B)=\ln2 I ( A ; B ) = ln 2 S ( A ∣ B ) = 0 S(A\,|\,B)=0 S ( A ∣ B ) = 0 2 min { S ( A ) , S ( B ) } = 2 ln 2 2\min\{S(A),S(B)\}=2\ln2 2 min { S ( A ) , S ( B )} = 2 ln 2
(3) ∣ Φ + ⟩ ⟨ Φ + ∣ = 1 2 ( ∣ 00 ⟩ ⟨ 00 ∣ + ∣ 00 ⟩ ⟨ 11 ∣ + ∣ 11 ⟩ ⟨ 00 ∣ + ∣ 11 ⟩ ⟨ 11 ∣ ) | \Phi^+ \rangle \langle \Phi^+ |=\frac12(| 00 \rangle \langle 00 |+| 00 \rangle \langle 11 |+| 11 \rangle \langle 00 |+| 11 \rangle \langle 11 |) ∣ Φ + ⟩ ⟨ Φ + ∣ = 2 1 ( ∣00 ⟩ ⟨ 00∣ + ∣00 ⟩ ⟨ 11∣ + ∣11 ⟩ ⟨ 00∣ + ∣11 ⟩ ⟨ 11∣ ) ρ ^ c l \hat\rho_{\mathrm{cl}} ρ ^ cl 3.2.6 から S ( ρ ^ c l ) − S ( ∣ Φ + ⟩ ⟨ Φ + ∣ ) = D ( ∣ Φ + ⟩ ⟨ Φ + ∣ ∥ ρ ^ c l ) = ln 2 S(\hat\rho_{\mathrm{cl}})-S(| \Phi^+ \rangle \langle \Phi^+ |)=D(| \Phi^+ \rangle \langle \Phi^+ |\,\|\,\hat\rho_{\mathrm{cl}})=\ln2 S ( ρ ^ cl ) − S ( ∣ Φ + ⟩ ⟨ Φ + ∣ ) = D ( ∣ Φ + ⟩ ⟨ Φ + ∣ ∥ ρ ^ cl ) = ln 2 ∣ 00 ⟩ | 00 \rangle ∣00 ⟩ ∣ 11 ⟩ | 11 \rangle ∣11 ⟩ ln 2 \ln2 ln 2 S ( A B ) S(AB) S ( A B ) I ( A ; B ) I(A;B) I ( A ; B ) S ( A ∣ B ) S(A\,|\,B) S ( A ∣ B )
∣ Φ + ⟩ | \Phi^+ \rangle ∣ Φ + ⟩ ρ ^ p = p ∣ Φ + ⟩ ⟨ Φ + ∣ + ( 1 − p ) 1 / 4 \hat\rho_p=p| \Phi^+ \rangle \langle \Phi^+ |+(1-p)\mathbf 1/4 ρ ^ p = p ∣ Φ + ⟩ ⟨ Φ + ∣ + ( 1 − p ) 1 /4 ( 1 + 3 p ) / 4 (1+3p)/4 ( 1 + 3 p ) /4 ( 1 − p ) / 4 (1-p)/4 ( 1 − p ) /4 ρ ^ A = ρ ^ B = 1 / 2 \hat\rho_A=\hat\rho_B=\mathbf 1/2 ρ ^ A = ρ ^ B = 1 /2 p p p 3.2.2 のように、S ( A ∣ B ) S(A\,|\,B) S ( A ∣ B ) p ≈ 0.748 p\approx0.748 p ≈ 0.748 p > 1 / 3 p>1/3 p > 1/3 S ( A ∣ B ) < 0 S(A\,|\,B)<0 S ( A ∣ B ) < 0
図 3.2.2 Werner 状態 ρ ^ p \hat\rho_p ρ ^ p S ( A ) = S ( B ) = ln 2 S(A)=S(B)=\ln2 S ( A ) = S ( B ) = ln 2 p p p S ( A ∣ B ) = S ( A B ) − ln 2 S(A\,|\,B)=S(AB)-\ln2 S ( A ∣ B ) = S ( A B ) − ln 2 I ( A ; B ) = 2 ln 2 − S ( A B ) I(A;B)=2\ln2-S(AB) I ( A ; B ) = 2 ln 2 − S ( A B ) S ( A ∣ B ) S(A\,|\,B) S ( A ∣ B ) p = 0.748 p=0.748 p = 0.748
1 H. Araki and E. H. Lieb (1970), Communications in Mathematical Physics 18 , 160–170。
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