Complexification of Information Geometry in view of quantum estimation theory

28/08/2013
Auteurs : Hiroshi Nagaoka
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Complexification of Information Geometry in view of quantum estimation theory

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Complexification of Information Geometry in view of quantum estimation theory Introduction • M : manifold with affine structure (flat connection) (θi ) ↓ TM : tangent bundle with a complex structure • M : manifold with a Hessian structure ((θi ), g, ψ) gij(θ) = ∂i∂jψ(θ) (∂ = ∂ ∂θi ) ↓ TM : tangent bundle with a K¨ahler structure with a K¨ahler potential ψ(θ) • M : manifold with affine structure (flat connection) (θi ) ↓ TM : tangent bundle with a complex structure • M : manifold with a Hessian structure ((θi ), g, ψ) gij(θ) = ∂i∂jψ(θ) (∂ = ∂ ∂θi ) ↓ TM : tangent bundle with a K¨ahler structure with a K¨ahler potential ψ(θ) As H. Shima pointed out in his book: and (dually flat structure) Introduction • M : manifold with affine structure (flat connection) (θi ) ↓ TM : tangent bundle with a complex structure • M : manifold with a Hessian structure ((θi ), g, ψ) gij(θ) = ∂i∂jψ(θ) (∂ = ∂ ∂θi ) ↓ TM : tangent bundle with a K¨ahler structure with a K¨ahler potential ψ(θ) • M : manifold with affine structure (flat connection) (θi ) ↓ TM : tangent bundle with a complex structure • M : manifold with a Hessian structure ((θi ), g, ψ) gij(θ) = ∂i∂jψ(θ) (∂ = ∂ ∂θi ) ↓ TM : tangent bundle with a K¨ahler structure with a K¨ahler potential ψ(θ) As H. Shima pointed out in his book: and (dually flat structure) Introduction (cont.) A similar situation will appear in the context of quantum estimation theory, where will be replaced with an(classical and quantum) exponential family • M : manifold with a Hessian structure ((θi ), g, ψ) gij(θ) = ∂i∂jψ(θ) (∂ = ∂ ∂θi ) ↓ TM : tangent bundle with a K¨ahler structure with a K¨ahler potential ψ(θ) M TM will be replaced with the complex projective space (the s pure states) and • M : manifold with affine structure (flat connection) (θi ) ↓ TM : tangent bundle with a complex structure • M : manifold with a Hessian structure ((θi ), g, ψ) gij(θ) = ∂i∂jψ(θ) (∂ = ∂ ∂θi ) ↓ TM : tangent bundle with a K¨ahler structure with a K¨ahler potential ψ(θ) M TM will be replaced with the complex projective space (the set of q will be replaced with the complex projective space (the set of quantum pure states) Introduction (cont.) A similar situation will appear in the context of quantum estimation theory, where will be replaced with an(classical and quantum) exponential family • M : manifold with a Hessian structure ((θi ), g, ψ) gij(θ) = ∂i∂jψ(θ) (∂ = ∂ ∂θi ) ↓ TM : tangent bundle with a K¨ahler structure with a K¨ahler potential ψ(θ) M TM will be replaced with the complex projective space (the s pure states) and • M : manifold with affine structure (flat connection) (θi ) ↓ TM : tangent bundle with a complex structure • M : manifold with a Hessian structure ((θi ), g, ψ) gij(θ) = ∂i∂jψ(θ) (∂ = ∂ ∂θi ) ↓ TM : tangent bundle with a K¨ahler structure with a K¨ahler potential ψ(θ) M TM will be replaced with the complex projective space (the set of q will be replaced with the complex projective space (the set of quantum pure states) Classical Exponential Families Let • X : a finite set, • P = P(X) := {p | p : X → (0, 1), X x∈X p(x) = 1}, • M = {pθ | θ ∈ Θ(⊂ Rn } (⊂ P), where pθ(x) = p0(x) exp h nX j=1 θi fi(x) − ψ(θ) i , ψ(θ) := log X x∈X p0(x) exp h nX j=1 θi fi(x) i . We assume {1, f1, . . . , fn} are linearly independent, which implies θ ￿→ p is injective. Let • X : a finite set, • P = P(X) := {p | p : X → (0, 1), X x∈X p(x) = 1}, • M = {pθ | θ ∈ Θ(⊂ Rn } (⊂ P), where pθ(x) = p0(x) exp h nX j=1 θi fi(x) − ψ(θ) i , ψ(θ) := log X x∈X p0(x) exp h nX j=1 θi fi(x) i . We assume {1, f1, . . . , fn} are linearly independent, which implies θ ￿→ pθ is injective. Classical Exponential Families Let • X : a finite set, • P = P(X) := {p | p : X → (0, 1), X x∈X p(x) = 1}, • M = {pθ | θ ∈ Θ(⊂ Rn } (⊂ P), where pθ(x) = p0(x) exp h nX j=1 θi fi(x) − ψ(θ) i , ψ(θ) := log X x∈X p0(x) exp h nX j=1 θi fi(x) i . We assume {1, f1, . . . , fn} are linearly independent, which implies θ ￿→ p is injective. Let • X : a finite set, • P = P(X) := {p | p : X → (0, 1), X x∈X p(x) = 1}, • M = {pθ | θ ∈ Θ(⊂ Rn } (⊂ P), where pθ(x) = p0(x) exp h nX j=1 θi fi(x) − ψ(θ) i , ψ(θ) := log X x∈X p0(x) exp h nX j=1 θi fi(x) i . We assume {1, f1, . . . , fn} are linearly independent, which implies θ ￿→ pθ is injective. Classical Exponential Families Let • X : a finite set, • P = P(X) := {p | p : X → (0, 1), X x∈X p(x) = 1}, • M = {pθ | θ ∈ Θ(⊂ Rn } (⊂ P), where pθ(x) = p0(x) exp h nX j=1 θi fi(x) − ψ(θ) i , ψ(θ) := log X x∈X p0(x) exp h nX j=1 θi fi(x) i . We assume {1, f1, . . . , fn} are linearly independent, which implies θ ￿→ p is injective. Let • X : a finite set, • P = P(X) := {p | p : X → (0, 1), X x∈X p(x) = 1}, • M = {pθ | θ ∈ Θ(⊂ Rn } (⊂ P), where pθ(x) = p0(x) exp h nX j=1 θi fi(x) − ψ(θ) i , ψ(θ) := log X x∈X p0(x) exp h nX j=1 θi fi(x) i . We assume {1, f1, . . . , fn} are linearly independent, which implies θ ￿→ pθ is injective. Geometrical Structure of Exponential Family • Fisher information metric: gij = Eθ[∂i log pθ∂j log pθ] = ∂i∂jψ(θ) ( ⇒ Cram´er-Rao inequality : V (estimator) ≥ [gij]−1 ) • e-, m-connections: affine coordinates flat connection θi −→ ∇(e) ηi := Eθ[fi] −→ ∇(m) • Duality: Xg(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (m) X Z) ⇒ (M, g, ∇(e) , ∇(m) ) is dually flat Geometrical Structure of Exponential Family • Fisher information metric: gij = Eθ[∂i log pθ∂j log pθ] = ∂i∂jψ(θ) ( ⇒ Cram´er-Rao inequality : V (estimator) ≥ [gij]−1 ) • e-, m-connections: affine coordinates flat connection θi −→ ∇(e) ηi := Eθ[fi] −→ ∇(m) • Duality: Xg(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (m) X Z) ⇒ (M, g, ∇(e) , ∇(m) ) is dually flat ⇒ (M, g, ∇(e) , ∇(m) ) is dually flat • ˆη := (f1, . . . , fn) is an estimator achieving the Cram´er-Rao bound (: an efficient estimator). • P itself is an exponential family. ⇒ (M, g, ∇(e) , ∇(m) ) is dually flat • ˆη := (f1, . . . , fn) is an estimator achieving the Cram´er-Rao bound (: an efficient estimator). • P itself is an exponential family. Quantum State Space Let H ∼= Cd be a Hilbert space with an inner product ￿· | ·￿, and define L(H) := {A | A : H linear −→ H} = {linear operators}, Lh(H) := {A ∈ L(H) | A = A∗ } = {hermitian operators}, ¯S := {ρ | ρ ∈ Lh(H) | ρ ≥ 0, Tr[ρ] = 1} = {quantum states} = n[ r=1 Sr, where Sr := {ρ ∈ S | rank ρ = r}. Let H ∼= Cd be a Hilbert space with an inner product ￿· | ·￿, and define L(H) := {A | A : H linear −→ H} = {linear operators}, Lh(H) := {A ∈ L(H) | A = A∗ } = {hermitian operators}, ¯S := {ρ | ρ ∈ Lh(H) | ρ ≥ 0, Tr[ρ] = 1} = {quantum states} = n[ r=1 Sr, where Sr := {ρ ∈ S | rank ρ = r}. We mainly treat S1 and Sd in the sequel. Quantum State Space Let H ∼= Cd be a Hilbert space with an inner product ￿· | ·￿, and define L(H) := {A | A : H linear −→ H} = {linear operators}, Lh(H) := {A ∈ L(H) | A = A∗ } = {hermitian operators}, ¯S := {ρ | ρ ∈ Lh(H) | ρ ≥ 0, Tr[ρ] = 1} = {quantum states} = n[ r=1 Sr, where Sr := {ρ ∈ S | rank ρ = r}. Let H ∼= Cd be a Hilbert space with an inner product ￿· | ·￿, and define L(H) := {A | A : H linear −→ H} = {linear operators}, Lh(H) := {A ∈ L(H) | A = A∗ } = {hermitian operators}, ¯S := {ρ | ρ ∈ Lh(H) | ρ ≥ 0, Tr[ρ] = 1} = {quantum states} = n[ r=1 Sr, where Sr := {ρ ∈ S | rank ρ = r}. We mainly treat S1 and Sd in the sequel. Quantum State Space Let H ∼= Cd be a Hilbert space with an inner product ￿· | ·￿, and define L(H) := {A | A : H linear −→ H} = {linear operators}, Lh(H) := {A ∈ L(H) | A = A∗ } = {hermitian operators}, ¯S := {ρ | ρ ∈ Lh(H) | ρ ≥ 0, Tr[ρ] = 1} = {quantum states} = n[ r=1 Sr, where Sr := {ρ ∈ S | rank ρ = r}. Let H ∼= Cd be a Hilbert space with an inner product ￿· | ·￿, and define L(H) := {A | A : H linear −→ H} = {linear operators}, Lh(H) := {A ∈ L(H) | A = A∗ } = {hermitian operators}, ¯S := {ρ | ρ ∈ Lh(H) | ρ ≥ 0, Tr[ρ] = 1} = {quantum states} = n[ r=1 Sr, where Sr := {ρ ∈ S | rank ρ = r}. We mainly treat S1 and Sd in the sequel. SLD Fisher Metric Given a manifold M = {ρθ | θ = (θi ) ∈ Θ} ⊂ ¯S, let • Lθ,i ∈ Lh(H) s.t. ∂ ∂θi ρθ = 1 2 (ρθLθ,i + Lθ,iρθ) : Symmetric Lgarithmic Derivatives, or SLDs of M • gij := Re Tr [ρθLθ,iLθ,j] . ⇒ g = [gij] defines a Riemannian metric on M. In particular, every Sr becomes a Riemannian space with g. SLD Fisher Metric Given a manifold M = {ρθ | θ = (θi ) ∈ Θ} ⊂ ¯S, let • Lθ,i ∈ Lh(H) s.t. ∂ ∂θi ρθ = 1 2 (ρθLθ,i + Lθ,iρθ) : Symmetric Lgarithmic Derivatives, or SLDs of M • gij := Re Tr [ρθLθ,iLθ,j] . ⇒ g = [gij] defines a Riemannian metric on M. In particular, every Sr becomes a Riemannian space with g. SLD Fisher Metric Given a manifold M = {ρθ | θ = (θi ) ∈ Θ} ⊂ ¯S, let • Lθ,i ∈ Lh(H) s.t. ∂ ∂θi ρθ = 1 2 (ρθLθ,i + Lθ,iρθ) : Symmetric Lgarithmic Derivatives, or SLDs of M • gij := Re Tr [ρθLθ,iLθ,j] . ⇒ g = [gij] defines a Riemannian metric on M. In particular, every Sr becomes a Riemannian space with g. SLD Fisher Metric (cont.) • The metric g is a quantum version of the classical Fisher metric, and is called the SLD metric. • A quantum version of Cram´er-Rao inequality: V (estimator) ≥ [gij]−1 . (Helstrom, 1967) • The minimum monotone metric. (Petz, 1996) • Every Sr becomes a Riemannian space with the SLD metric. How about the e-, m-connections and the dualistic structure? SLD Fisher Metric (cont.) • The metric g is a quantum version of the classical Fisher metric, and is called the SLD metric. • A quantum version of Cram´er-Rao inequality: V (estimator) ≥ [gij]−1 . (Helstrom, 1967) • The minimum monotone metric. (Petz, 1996) • Every Sr becomes a Riemannian space with the SLD metric. How about the e-, m-connections and the dualistic structure? SLD Fisher Metric (cont.) • The metric g is a quantum version of the classical Fisher metric, and is called the SLD metric. • A quantum version of Cram´er-Rao inequality: V (estimator) ≥ [gij]−1 . (Helstrom, 1967) • The minimum monotone metric. (Petz, 1996) • Every Sr becomes a Riemannian space with the SLD metric. How about the e-, m-connections and the dualistic structure? SLD Fisher Metric (cont.) • The metric g is a quantum version of the classical Fisher metric, and is called the SLD metric. • A quantum version of Cram´er-Rao inequality: V (estimator) ≥ [gij]−1 . (Helstrom, 1967) • The minimum monotone metric. (Petz, 1996) • Every Sr becomes a Riemannian space with the SLD metric. How about the e-, m-connections and the dualistic structure? r=d: faithful states • Sd = © ρ ∈ ¯S ρ > 0 ™ = {faithful states}. • Since Sd is an open subset in the affine space {A A = A∗ and TrA = 1}, the m-connection ∇(m) on Sd is defined as the natural flat con- nection. • The e-connection ∇(e) is defined as the dual of ∇(m) w.r.t. g: Xg(Y, Z) = g(∇(e) XY, Z) + g(Y, ∇(m) XZ) • R(e) = 0 (curvature), T(e) ￿= 0 (torsion), so (Sd, g, ∇(e) , ∇(m) ) is not dually flat. r=d: faithful states • Sd = © ρ ∈ ¯S ρ > 0 ™ = {faithful states}. • Since Sd is an open subset in the affine space {A A = A∗ and TrA = 1}, the m-connection ∇(m) on Sd is defined as the natural flat con- nection. • The e-connection ∇(e) is defined as the dual of ∇(m) w.r.t. g: Xg(Y, Z) = g(∇(e) XY, Z) + g(Y, ∇(m) XZ) • R(e) = 0 (curvature), T(e) ￿= 0 (torsion), so (Sd, g, ∇(e) , ∇(m) ) is not dually flat. r=d: faithful states • Sd = © ρ ∈ ¯S ρ > 0 ™ = {faithful states}. • Since Sd is an open subset in the affine space {A A = A∗ and TrA = 1}, the m-connection ∇(m) on Sd is defined as the natural flat con- nection. • The e-connection ∇(e) is defined as the dual of ∇(m) w.r.t. g: Xg(Y, Z) = g(∇(e) XY, Z) + g(Y, ∇(m) XZ) • R(e) = 0 (curvature), T(e) ￿= 0 (torsion), so (Sd, g, ∇(e) , ∇(m) ) is not dually flat. r=d: faithful states • Sd = © ρ ∈ ¯S ρ > 0 ™ = {faithful states}. • Since Sd is an open subset in the affine space {A A = A∗ and TrA = 1}, the m-connection ∇(m) on Sd is defined as the natural flat con- nection. • The e-connection ∇(e) is defined as the dual of ∇(m) w.r.t. g: Xg(Y, Z) = g(∇(e) XY, Z) + g(Y, ∇(m) XZ) • R(e) = 0 (curvature), T(e) ￿= 0 (torsion), so (Sd, g, ∇(e) , ∇(m) ) is not dually flat. r=1: pure states • S1 = {|ξ￿￿ξ| | ξ ∈ H, ￿ξ￿ = 1} = {pure states}. • S1 ∼= P(H) := (H \ {0})/ ∼ (complex projective space), where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c ξ2. • The SLD metric g on S1 coincides with the well-known Fubini-Study metric on P(H) (up to constant). r=1: pure states • S1 = {|ξ￿￿ξ| | ξ ∈ H, ￿ξ￿ = 1} = {pure states}. • S1 ∼= P(H) := (H \ {0})/ ∼ (complex projective space), where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c ξ2. • The SLD metric g on S1 coincides with the well-known Fubini-Study metric on P(H) (up to constant). r=1: pure states • S1 = {|ξ￿￿ξ| | ξ ∈ H, ￿ξ￿ = 1} = {pure states}. • S1 ∼= P(H) := (H \ {0})/ ∼ (complex projective space), where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c ξ2. • The SLD metric g on S1 coincides with the well-known Fubini-Study metric on P(H) (up to constant). as a complex manifold• S1 ∼= P(H) := (H \ {0})/ ∼ (complex projective space), where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c ξ2. • The SLD metric g on S1 coincides with the well-known Fubini-Study metric on P(H) (up to constant). z ￿−→ ρz S1 ∼= P(H) Rn Cn • A (1, 1)-tensor field J satisfying J2 = −1 (almost complex structure) is canonicaly defined by J µ ∂ ∂xj ∂ = ∂ ∂yi , J µ ∂ ∂yj ∂ = − ∂ ∂xi for an arbitrary holomorphic (complex analytic) coordinate system (zj ) = (xj + √ −1yj ). • g(JX, JY ) = g(X, Y ). • A diffirential 2-form ω is defined by ω(X, Y ) = g(X, JY ). • g (or (J, g, ω)) is a K¨ahler metric in the sense that ω is a symplectic form: dω = 0, or equivalently that there is a funtion called a K¨ahler potential f satisfying ω = √ −1 2 ∂ ¯∂f. as a complex manifold• S1 ∼= P(H) := (H \ {0})/ ∼ (complex projective space), where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c ξ2. • The SLD metric g on S1 coincides with the well-known Fubini-Study metric on P(H) (up to constant). z ￿−→ ρz S1 ∼= P(H) Rn Cn • A (1, 1)-tensor field J satisfying J2 = −1 (almost complex structure) is canonicaly defined by J µ ∂ ∂xj ∂ = ∂ ∂yi , J µ ∂ ∂yj ∂ = − ∂ ∂xi for an arbitrary holomorphic (complex analytic) coordinate system (zj ) = (xj + √ −1yj ). • g(JX, JY ) = g(X, Y ). • A diffirential 2-form ω is defined by ω(X, Y ) = g(X, JY ). • g (or (J, g, ω)) is a K¨ahler metric in the sense that ω is a symplectic form: dω = 0, or equivalently that there is a funtion called a K¨ahler potential f satisfying ω = √ −1 2 ∂ ¯∂f. as a complex manifold• S1 ∼= P(H) := (H \ {0})/ ∼ (complex projective space), where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c ξ2. • The SLD metric g on S1 coincides with the well-known Fubini-Study metric on P(H) (up to constant). z ￿−→ ρz S1 ∼= P(H) Rn Cn • A (1, 1)-tensor field J satisfying J2 = −1 (almost complex structure) is canonicaly defined by J µ ∂ ∂xj ∂ = ∂ ∂yi , J µ ∂ ∂yj ∂ = − ∂ ∂xi for an arbitrary holomorphic (complex analytic) coordinate system (zj ) = (xj + √ −1yj ). • g(JX, JY ) = g(X, Y ). • A diffirential 2-form ω is defined by ω(X, Y ) = g(X, JY ). • g (or (J, g, ω)) is a K¨ahler metric in the sense that ω is a symplectic form: dω = 0, or equivalently that there is a funtion called a K¨ahler potential f satisfying ω = √ −1 2 ∂ ¯∂f. as a complex manifold• S1 ∼= P(H) := (H \ {0})/ ∼ (complex projective space), where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c ξ2. • The SLD metric g on S1 coincides with the well-known Fubini-Study metric on P(H) (up to constant). z ￿−→ ρz S1 ∼= P(H) Rn Cn • A (1, 1)-tensor field J satisfying J2 = −1 (almost complex structure) is canonicaly defined by J µ ∂ ∂xj ∂ = ∂ ∂yi , J µ ∂ ∂yj ∂ = − ∂ ∂xi for an arbitrary holomorphic (complex analytic) coordinate system (zj ) = (xj + √ −1yj ). • g(JX, JY ) = g(X, Y ). • A diffirential 2-form ω is defined by ω(X, Y ) = g(X, JY ). • g (or (J, g, ω)) is a K¨ahler metric in the sense that ω is a symplectic form: dω = 0, or equivalently that there is a funtion called a K¨ahler potential f satisfying ω = √ −1 2 ∂ ¯∂f. Kahler potential Let ajk = g µ ∂ ∂xj , ∂ ∂xk ∂ = g µ ∂ ∂yj , ∂ ∂yk ∂ , bjk = g µ ∂ ∂yj , ∂ ∂xk ∂ = −g µ ∂ ∂xj , ∂ ∂yk ∂ . Then f is a K¨ahler potential iff ajk = 1 4 µ ∂2 f ∂xj∂xk + ∂2 f ∂yj∂yk ∂ , and bjk = 1 4 µ ∂2 f ∂xj∂yk − ∂2 f ∂yj∂xk ∂ . Kahler potential Let ajk = g µ ∂ ∂xj , ∂ ∂xk ∂ = g µ ∂ ∂yj , ∂ ∂yk ∂ , bjk = g µ ∂ ∂yj , ∂ ∂xk ∂ = −g µ ∂ ∂xj , ∂ ∂yk ∂ . Then f is a K¨ahler potential iff ajk = 1 4 µ ∂2 f ∂xj∂xk + ∂2 f ∂yj∂yk ∂ , and bjk = 1 4 µ ∂2 f ∂xj∂yk − ∂2 f ∂yj∂xk ∂ . Quasi-Classical Exponential Family (QCEF) M = {ρθ | θ ∈ Rn } ⊂ ¯S is called a quasi-classical exponential family when it is represented as ρθ = exp " 1 2 ≥X i θi Fi − ψ(θ) ¥ # ρ0 exp " 1 2 ≥X i θi Fi − ψ(θ) ¥ # where {F1, . . . , Fn} ⊂ Lh(H), [Fi, Fj] := FiFj − FjFi = 0 (commutative), {ρ0, F1ρ0, . . . , Fnρ0} are linearly independent, ψ(θ) = log Tr h ρ0 exp £X j θi Fj §i . Properties of QCEFs • e-, m-connections are defined by affine coordinates flat connection θi −→ ∇(e) ηi := Tr[ρθFi] −→ ∇(m) • (M, g, ∇(e) , ∇(m) ) is dually flat, where g is the SLD metric. • Suppose M ⊂ Sd. Then M is e-autoparallel in Sd, and (g, ∇(e) , ∇(m) ) on M is induced from (Sd, g, ∇(e) , ∇(m) ). • (F1 . . . , Fn) is an estimator for the coordi- nates (η1, . . . , ηn) achieving the SLD Cram´er- Rao bound. Properties of QCEFs • e-, m-connections are defined by affine coordinates flat connection θi −→ ∇(e) ηi := Tr[ρθFi] −→ ∇(m) • (M, g, ∇(e) , ∇(m) ) is dually flat, where g is the SLD metric. • Suppose M ⊂ Sd. Then M is e-autoparallel in Sd, and (g, ∇(e) , ∇(m) ) on M is induced from (Sd, g, ∇(e) , ∇(m) ). • (F1 . . . , Fn) is an estimator for the coordi- nates (η1, . . . , ηn) achieving the SLD Cram´er- Rao bound. Properties of QCEFs • e-, m-connections are defined by affine coordinates flat connection θi −→ ∇(e) ηi := Tr[ρθFi] −→ ∇(m) • (M, g, ∇(e) , ∇(m) ) is dually flat, where g is the SLD metric. • Suppose M ⊂ Sd. Then M is e-autoparallel in Sd, and (g, ∇(e) , ∇(m) ) on M is induced from (Sd, g, ∇(e) , ∇(m) ). • (F1 . . . , Fn) is an estimator for the coordi- nates (η1, . . . , ηn) achieving the SLD Cram´er- Rao bound. Properties of QCEFs • e-, m-connections are defined by affine coordinates flat connection θi −→ ∇(e) ηi := Tr[ρθFi] −→ ∇(m) • (M, g, ∇(e) , ∇(m) ) is dually flat, where g is the SLD metric. • Suppose M ⊂ Sd. Then M is e-autoparallel in Sd, and (g, ∇(e) , ∇(m) ) on M is induced from (Sd, g, ∇(e) , ∇(m) ). • (F1 . . . , Fn) is an estimator for the coordi- nates (η1, . . . , ηn) achieving the SLD Cram´er- Rao bound. Properties of QCEFs (cont.) nates (η1, . . . , ηn) achieving the SLD Cram´er- Rao bound. • Since {Fi} are commutative, there exist an or- thonormal basis {|x￿}x∈X (eigenvectors) with X = {1, 2, · · · , d = dim H} and functions (eigen- values) fi : X → R (i = 1, . . . , n) such that Fi = X x∈X fi(x) |x￿￿x|. Then we have: pθ(x):= ￿x|ρθ|x￿ = p0(x) exp[ X i θi fi(x) − ψ(θ)] (: a classical exponential family) and M = {ρθ} ∼= {pθ} w.r.t. (g, ∇(e) , ∇(m) ). Properties of QCEFs (cont.) nates (η1, . . . , ηn) achieving the SLD Cram´er- Rao bound. • Since {Fi} are commutative, there exist an or- thonormal basis {|x￿}x∈X (eigenvectors) with X = {1, 2, · · · , d = dim H} and functions (eigen- values) fi : X → R (i = 1, . . . , n) such that Fi = X x∈X fi(x) |x￿￿x|. Then we have: pθ(x):= ￿x|ρθ|x￿ = p0(x) exp[ X i θi fi(x) − ψ(θ)] (: a classical exponential family) and M = {ρθ} ∼= {pθ} w.r.t. (g, ∇(e) , ∇(m) ). Complexification of a pure state QCEF Let M = {ρθ} be a quasi-classical exp. family: ρθ = exp " 1 2 ≥X i θi Fi − ψ(θ) ¥ # ρ0 exp " 1 2 ≥X i θi Fi − ψ(θ) ¥ # (with the same assumption on {Fi} as before), and suppose that M ⊂ S1(H) ∼= P(H). For z = (z1 , . . . , zn )∈ Cn , zi = θi + √ −1 yj , θi , yi : real, let ρz := exp " 1 2 ≥X i zi Fi − ψ(θ) ¥ # ρ0 exp " 1 2 ≥X i ¯ziFi − ψ(θ) ¥ # = UyρθU∗ y where Uy := exp "√ −1 2 X i yi Fi # : unitary. Complexification of a pure state QCEF Let M = {ρθ} be a quasi-classical exp. family: ρθ = exp " 1 2 ≥X i θi Fi − ψ(θ) ¥ # ρ0 exp " 1 2 ≥X i θi Fi − ψ(θ) ¥ # (with the same assumption on {Fi} as before), and suppose that M ⊂ S1(H) ∼= P(H). For z = (z1 , . . . , zn )∈ Cn , zi = θi + √ −1 yj , θi , yi : real, let ρz := exp " 1 2 ≥X i zi Fi − ψ(θ) ¥ # ρ0 exp " 1 2 ≥X i ¯ziFi − ψ(θ) ¥ # = UyρθU∗ y where Uy := exp "√ −1 2 X i yi Fi # : unitary. Complexification of a pure state QCEF Let M = {ρθ} be a quasi-classical exp. family: ρθ = exp " 1 2 ≥X i θi Fi − ψ(θ) ¥ # ρ0 exp " 1 2 ≥X i θi Fi − ψ(θ) ¥ # (with the same assumption on {Fi} as before), and suppose that M ⊂ S1(H) ∼= P(H). For z = (z1 , . . . , zn )∈ Cn , zi = θi + √ −1 yj , θi , yi : real, let ρz := exp " 1 2 ≥X i zi Fi − ψ(θ) ¥ # ρ0 exp " 1 2 ≥X i ¯ziFi − ψ(θ) ¥ # = UyρθU∗ y where Uy := exp "√ −1 2 X i yi Fi # : unitary. Complexification of pure state QCEF (cont.) Letting V be a nbd of Rn in Cn for which V ￿ z ￿→ ρz is injective, define ˜M := {ρz | z ∈ V } (⊃ M = {ρθ | θ ∈ Rn }). Then, ˜M is a complex (holomorphic) submanifold of S1 with a holomorphic coordinate system (zi ), and hence is K¨ahler w.r.t. gM = (Fubini-Study)|M . • 4ψ(θ) gives a K¨ahler potential on ˜M: ωM := ω|M = 2 √ −1 ∂ ¯∂ ψ. i y M • S1 = {|ξ￿￿ξ| | ξ ∈ H, ￿ξ￿ = 1} = {pur • S1 ∼= P(H) := (H \ {0})/ ∼ (complex where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c ξ2. • The SLD metric g on S1 coincides wi well-known Fubini-Study metric on P (up to constant). S1 ∼= P(H) Rn • S1 = P(H) := (H \ {0})/ ∼ (com where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c • The SLD metric g on S1 coincide well-known Fubini-Study metric (up to constant). z ￿−→ ρz S1 ∼= P(H) Rn Cn • S1 = {|ξ￿￿ξ| | ξ • S1 ∼= P(H) := ( where ξ1 ∼ ξ2 • The SLD metr well-known Fu (up to constan z ￿−→ ρz S1 ∼= P(H) Rn Complexification of pure state QCEF (cont.) Letting V be a nbd of Rn in Cn for which V ￿ z ￿→ ρz is injective, define ˜M := {ρz | z ∈ V } (⊃ M = {ρθ | θ ∈ Rn }). Then, ˜M is a complex (holomorphic) submanifold of S1 with a holomorphic coordinate system (zi ), and hence is K¨ahler w.r.t. gM = (Fubini-Study)|M . • 4ψ(θ) gives a K¨ahler potential on ˜M: ωM := ω|M = 2 √ −1 ∂ ¯∂ ψ. i y M • S1 = {|ξ￿￿ξ| | ξ ∈ H, ￿ξ￿ = 1} = {pur • S1 ∼= P(H) := (H \ {0})/ ∼ (complex where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c ξ2. • The SLD metric g on S1 coincides wi well-known Fubini-Study metric on P (up to constant). S1 ∼= P(H) Rn • S1 = P(H) := (H \ {0})/ ∼ (com where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c • The SLD metric g on S1 coincide well-known Fubini-Study metric (up to constant). z ￿−→ ρz S1 ∼= P(H) Rn Cn • S1 = {|ξ￿￿ξ| | ξ • S1 ∼= P(H) := ( where ξ1 ∼ ξ2 • The SLD metr well-known Fu (up to constan z ￿−→ ρz S1 ∼= P(H) Rn Complexification of pure state QCEF (cont.) Letting V be a nbd of Rn in Cn for which V ￿ z ￿→ ρz is injective, define ˜M := {ρz | z ∈ V } (⊃ M = {ρθ | θ ∈ Rn }). Then, ˜M is a complex (holomorphic) submanifold of S1 with a holomorphic coordinate system (zi ), and hence is K¨ahler w.r.t. gM = (Fubini-Study)|M . • 4ψ(θ) gives a K¨ahler potential on ˜M: ωM := ω|M = 2 √ −1 ∂ ¯∂ ψ. i y MV M • S1 = {|ξ￿￿ξ| | ξ ∈ H, ￿ξ￿ = 1} = {pur • S1 ∼= P(H) := (H \ {0})/ ∼ (complex where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c ξ2. • The SLD metric g on S1 coincides wi well-known Fubini-Study metric on P (up to constant). S1 ∼= P(H) Rn • S1 = {|ξ￿￿ξ| | ξ ∈ H, ￿ξ￿ = • S1 ∼= P(H) := (H \ {0})/ ∼ where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C • The SLD metric g on S1 c well-known Fubini-Study (up to constant). z ￿−→ ρz S1 ∼= P(H) Rn • S1 = P(H) := (H \ {0})/ ∼ (com where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c • The SLD metric g on S1 coincide well-known Fubini-Study metric (up to constant). z ￿−→ ρz S1 ∼= P(H) Rn Cn • S1 = {|ξ￿￿ξ| | ξ • S1 ∼= P(H) := ( where ξ1 ∼ ξ2 • The SLD metr well-known Fu (up to constan z ￿−→ ρz S1 ∼= P(H) Rn Complexification of pure state QCEF (cont.) Letting V be a nbd of Rn in Cn for which V ￿ z ￿→ ρz is injective, define ˜M := {ρz | z ∈ V } (⊃ M = {ρθ | θ ∈ Rn }). Then, ˜M is a complex (holomorphic) submanifold of S1 with a holomorphic coordinate system (zi ), and hence is K¨ahler w.r.t. gM = (Fubini-Study)|M . • 4ψ(θ) gives a K¨ahler potential on ˜M: ωM := ω|M = 2 √ −1 ∂ ¯∂ ψ. i y MV M • S1 = {|ξ￿￿ξ| | ξ ∈ H, ￿ξ￿ = 1} = {pur • S1 ∼= P(H) := (H \ {0})/ ∼ (complex where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c ξ2. • The SLD metric g on S1 coincides wi well-known Fubini-Study metric on P (up to constant). S1 ∼= P(H) Rn • S1 = {|ξ￿￿ξ| | ξ ∈ H, ￿ξ￿ = • S1 ∼= P(H) := (H \ {0})/ ∼ where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C • The SLD metric g on S1 c well-known Fubini-Study (up to constant). z ￿−→ ρz S1 ∼= P(H) Rn • S1 = P(H) := (H \ {0})/ ∼ (com where ξ1 ∼ ξ2 def ⇐⇒ ∃c ∈ C, ξ1 = c • The SLD metric g on S1 coincide well-known Fubini-Study metric (up to constant). z ￿−→ ρz S1 ∼= P(H) Rn Cn • S1 = {|ξ￿￿ξ| | ξ • S1 ∼= P(H) := ( where ξ1 ∼ ξ2 • The SLD metr well-known Fu (up to constan z ￿−→ ρz S1 ∼= P(H) Rn Complexification of pure state QCEF (cont.) • ˜M is a complex (holomorphic) submanifold of S1 with a holomorphic coordinate system (zi ), and hence is K¨ahler w.r.t. g ˜M = (Fubini-Study)| ˜M . • When n = d − 1, ˜M is open in S1. • 4ψ(θ) gives a K¨ahler potential on ˜M: ω ˜M := ω| ˜M = 2 √ −1 ∂ ¯∂ ψ. • ( ˜M, ηi, yi ) forms a Darboux coordinate system: n i Complexification of pure state QCEF (cont.) • ˜M is a complex (holomorphic) submanifold of S1 with a holomorphic coordinate system (zi ), and hence is K¨ahler w.r.t. g ˜M = (Fubini-Study)| ˜M . • When n = d − 1, ˜M is open in S1. • 4ψ(θ) gives a K¨ahler potential on ˜M: ω ˜M := ω| ˜M = 2 √ −1 ∂ ¯∂ ψ. • ( ˜M, ηi, yi ) forms a Darboux coordinate system: n i Complexification of pure state QCEF (cont.) • ˜M is a complex (holomorphic) submanifold of S1 with a holomorphic coordinate system (zi ), and hence is K¨ahler w.r.t. g ˜M = (Fubini-Study)| ˜M . • When n = d − 1, ˜M is open in S1. • 4ψ(θ) gives a K¨ahler potential on ˜M: ω ˜M := ω| ˜M = 2 √ −1 ∂ ¯∂ ψ. • ( ˜M, ηi, yi ) forms a Darboux coordinate system: n i Similar to the case of Shima's observation on M and TM Complexification of pure state QCEF (cont.) • 4ψ(θ) gives a K¨ahler potential on M: ω ˜M := ω| ˜M = 2 √ −1 ∂ ¯∂ ψ. • ( ˜M, ηi, yi ) forms a Darboux coordinate system: ω ˜M = n i=1 dηi ∧ dyi . • Letting (m) be the flat connection with affine coordinates (ηi; yi ) and (e) be its dual w.r.t. g ˜M , (e) ◦ J = J ◦ (m) and (e) ω ˜M = (m) ω ˜M = 0. Complexification of pure state QCEF (cont.) • 4ψ(θ) gives a K¨ahler potential on M: ω ˜M := ω| ˜M = 2 √ −1 ∂ ¯∂ ψ. • ( ˜M, ηi, yi ) forms a Darboux coordinate system: ω ˜M = n i=1 dηi ∧ dyi . • Letting (m) be the flat connection with affine coordinates (ηi; yi ) and (e) be its dual w.r.t. g ˜M , (e) ◦ J = J ◦ (m) and (e) ω ˜M = (m) ω ˜M = 0. Complexification of pure state QCEF (cont.) • 4ψ(θ) gives a K¨ahler potential on M: ω ˜M := ω| ˜M = 2 √ −1 ∂ ¯∂ ψ. • ( ˜M, ηi, yi ) forms a Darboux coordinate system: ω ˜M = n i=1 dηi ∧ dyi . • Letting (m) be the flat connection with affine coordinates (ηi; yi ) and (e) be its dual w.r.t. g ˜M , (e) ◦ J = J ◦ (m) and (e) ω ˜M = (m) ω ˜M = 0. duality ⇐⇒ ∀X, Y, Z, Xg(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (m) X Z) ∇(e) ◦J = J◦∇(m) ⇔ ∀X, Y, ∇ (e) X J(Y ) = J(∇ (m) X Y ) ∇(e) ω = ∇(m) ω = 0 ∇(e) ω = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (e) X Z) (m) (m) (m) Relation to parallel displacement = ∇(m) ω = 0 Z, Xω(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (e) X Z) Z, Xω(Y, Z) = g(∇ (m) X Y, Z) + g(Y, ∇ (m) X Z) X￿ X X￿ e −→ X￿ X e −→ X￿ m −→ X￿ X m −→ X￿ Y ￿ Y Y ￿ 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (e) X Z) = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (m) X Y, Z) + g(Y, ∇ (m) X Z) X X￿ X X￿ X e −→ X￿ X e −→ X￿ X e −→ X￿ X m −→ X￿ X m −→ X￿ X m −→ X￿ Y Y ￿ Y Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ ⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (m) X Y, Z) + g(Y, ∇ (m) X Z) X X￿ X X￿ → X￿ X e −→ X￿ X e −→ X￿ → X￿ X m −→ X￿ X m −→ X￿ Y Y ￿ Y Y ￿ → Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ m → Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ X X￿ X X￿ X e −→ X￿ X e −→ X￿ X e −→ X￿ X m −→ X￿ X m −→ X￿ X m −→ X￿ Y Y ￿ Y Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ ∇(e) ω = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g ∇(m) ω = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g X X￿ X X￿ X e −→ X￿ X e −→ X￿ X e −→ X m −→ X￿ X m −→ X￿ X m −→ Y Y ￿ Y Y ￿ X e −→ X￿ X X m −→ X￿ X Y Y e −→ Y ￿ Y Y m −→ Y ￿ Y Y Y ￿ Y Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ g(X, Y ) = g(X￿ , Y ￿ ) ω(X, Y ) = ω(X￿ , Y ￿ ) X e −→ X￿ iff J(X) m −→ J(X￿ ), and X m −→ X￿ iff J(X) e −→ J(X￿ ). duality ⇐⇒ ∀X, Y, Z, Xg(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (m) X Z) ∇(e) ◦J = J◦∇(m) ⇔ ∀X, Y, ∇ (e) X J(Y ) = J(∇ (m) X Y ) ∇(e) ω = ∇(m) ω = 0 ∇(e) ω = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (e) X Z) (m) (m) (m) Relation to parallel displacement = ∇(m) ω = 0 Z, Xω(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (e) X Z) Z, Xω(Y, Z) = g(∇ (m) X Y, Z) + g(Y, ∇ (m) X Z) X￿ X X￿ e −→ X￿ X e −→ X￿ m −→ X￿ X m −→ X￿ Y ￿ Y Y ￿ 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (e) X Z) = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (m) X Y, Z) + g(Y, ∇ (m) X Z) X X￿ X X￿ X e −→ X￿ X e −→ X￿ X e −→ X￿ X m −→ X￿ X m −→ X￿ X m −→ X￿ Y Y ￿ Y Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ ⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (m) X Y, Z) + g(Y, ∇ (m) X Z) X X￿ X X￿ → X￿ X e −→ X￿ X e −→ X￿ → X￿ X m −→ X￿ X m −→ X￿ Y Y ￿ Y Y ￿ → Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ m → Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ X X￿ X X￿ X e −→ X￿ X e −→ X￿ X e −→ X￿ X m −→ X￿ X m −→ X￿ X m −→ X￿ Y Y ￿ Y Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ ∇(e) ω = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g ∇(m) ω = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g X X￿ X X￿ X e −→ X￿ X e −→ X￿ X e −→ X m −→ X￿ X m −→ X￿ X m −→ Y Y ￿ Y Y ￿ X e −→ X￿ X X m −→ X￿ X Y Y e −→ Y ￿ Y Y m −→ Y ￿ Y Y Y ￿ Y Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ g(X, Y ) = g(X￿ , Y ￿ ) ω(X, Y ) = ω(X￿ , Y ￿ ) X e −→ X￿ iff J(X) m −→ J(X￿ ), and X m −→ X￿ iff J(X) e −→ J(X￿ ). Relation to parallel displacement (cont.) duality ⇐⇒ ∀X, Y, Z, Xg(Y, Z) = g(∇X Y, Z) + g(Y, ∇X ∇(e) ◦J = J◦∇(m) ⇔ ∀X, Y, ∇ (e) X J(Y ) = J(∇ (m) X Y ) ∇(e) ω = ∇(m) ω = 0 ∇(e) ω = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇(m) ω = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (m) X Y, Z) + g(Y ￿ ￿ ω(X, Y ) = ω(X￿ , Y ￿ ) X e −→ X￿ iff J(X) m −→ J(X￿ ), and X m −→ X￿ iff J(X) e −→ J(X￿ ). X e −→ X￿ ↓ J ↓ J J(X) m −→ J(X￿ ) 1 (Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (m) X Z) Y, ∇ (e) X J(Y ) = J(∇ (m) X Y ) (m) ω = 0 0 ⇐⇒ ∀X, Y, Z Z) + ω(Y, ∇ (e) X Z) X m −→ X￿ ↓ J ↓ J J(X) e −→ J(X￿ ) Relation to parallel displacement (cont.) duality ⇐⇒ ∀X, Y, Z, Xg(Y, Z) = g(∇X Y, Z) + g(Y, ∇X ∇(e) ◦J = J◦∇(m) ⇔ ∀X, Y, ∇ (e) X J(Y ) = J(∇ (m) X Y ) ∇(e) ω = ∇(m) ω = 0 ∇(e) ω = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇(m) ω = 0 ⇐⇒ ∀X, Y, Z, Xω(Y, Z) = g(∇ (m) X Y, Z) + g(Y ￿ ￿ ω(X, Y ) = ω(X￿ , Y ￿ ) X e −→ X￿ iff J(X) m −→ J(X￿ ), and X m −→ X￿ iff J(X) e −→ J(X￿ ). X e −→ X￿ ↓ J ↓ J J(X) m −→ J(X￿ ) 1 (Y, Z) = g(∇ (e) X Y, Z) + g(Y, ∇ (m) X Z) Y, ∇ (e) X J(Y ) = J(∇ (m) X Y ) (m) ω = 0 0 ⇐⇒ ∀X, Y, Z Z) + ω(Y, ∇ (e) X Z) X m −→ X￿ ↓ J ↓ J J(X) e −→ J(X￿ ) Relation to parallel displacement (cont.)∇(e) ω = ∇(m) ω = 0 ∇(e) ω = ∇(m) ω = 0 ⇐⇒ ∀X, Y, Z Xω(Y, Z)= ω(∇ (e) X Y, Z) + ω(Y, ∇ (e) X Z) = ω(∇ (m) X Y, Z) + ω(Y, ∇ (m) X Z) X X￿ X X￿ X e −→ X￿ X e −→ X￿ X e −→ X￿ m ￿ m ￿ m ￿ ∇(e) ω = ∇(m) ω = 0 ∇(e) ω = ∇(m) ω = 0 ⇐⇒ ∀X, Y, Z Xω(Y, Z)= ω(∇ (e) X Y, Z) + ω(Y, ∇ (e) X Z) = ω(∇ (m) X Y, Z) + ω(Y, ∇ (m) X Z) X X￿ X X￿ X e −→ X￿ X e −→ X￿ X e −→ X￿ X m −→ X￿ X m −→ X￿ X m −→ X￿ Y Y ￿ Y Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ g(X, Y ) = g(X￿ , Y ￿ ) ω(X, Y ) = ω(X￿ , Y ￿ ) X e −→ X￿ iff J(X) m −→ J(X￿ ), and X m −→ X￿ iff J(X) e −→ J(X￿ ). ∇ (m) X Z) X m −→ X￿ ↓ J ↓ J J(X) e −→ J(X￿ ) X e −→ X￿ and Y e −→ Y ￿ X m −→ X￿ and Y m −→ Y ￿ ∇ (m) X Z) X m −→ X￿ ↓ J ↓ J J(X) e −→ J(X￿ ) X e −→ X￿ and Y e −→ Y ￿ X m −→ X￿ and Y m −→ Y ￿ Relation to parallel displacement (cont.)∇(e) ω = ∇(m) ω = 0 ∇(e) ω = ∇(m) ω = 0 ⇐⇒ ∀X, Y, Z Xω(Y, Z)= ω(∇ (e) X Y, Z) + ω(Y, ∇ (e) X Z) = ω(∇ (m) X Y, Z) + ω(Y, ∇ (m) X Z) X X￿ X X￿ X e −→ X￿ X e −→ X￿ X e −→ X￿ m ￿ m ￿ m ￿ ∇(e) ω = ∇(m) ω = 0 ∇(e) ω = ∇(m) ω = 0 ⇐⇒ ∀X, Y, Z Xω(Y, Z)= ω(∇ (e) X Y, Z) + ω(Y, ∇ (e) X Z) = ω(∇ (m) X Y, Z) + ω(Y, ∇ (m) X Z) X X￿ X X￿ X e −→ X￿ X e −→ X￿ X e −→ X￿ X m −→ X￿ X m −→ X￿ X m −→ X￿ Y Y ￿ Y Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y e −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ Y m −→ Y ￿ g(X, Y ) = g(X￿ , Y ￿ ) ω(X, Y ) = ω(X￿ , Y ￿ ) X e −→ X￿ iff J(X) m −→ J(X￿ ), and X m −→ X￿ iff J(X) e −→ J(X￿ ). ∇ (m) X Z) X m −→ X￿ ↓ J ↓ J J(X) e −→ J(X￿ ) X e −→ X￿ and Y e −→ Y ￿ X m −→ X￿ and Y m −→ Y ￿ ∇ (m) X Z) X m −→ X￿ ↓ J ↓ J J(X) e −→ J(X￿ ) X e −→ X￿ and Y e −→ Y ￿ X m −→ X￿ and Y m −→ Y ￿