Law of Cosines and Shannon-Pythagorean Theorem for Quantum Information

28/08/2013
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Law of Cosines and Shannon-Pythagorean Theorem for Quantum Information

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Law of Cosines and Shannon-Pythagorean Theorem for Quantum Information Roman V. Belavkin School of Science and Technology Middlesex University, London NW4 4BT, UK 1 August 29, 2013 1 This work was supported by EPSRC grant EP/H031936/1. Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 1 / 16 Main Result: Shannon-Pythagorean Theorem w q p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 2 / 16 Main Result: Shannon-Pythagorean Theorem w q ⊗ q q ⊗ p w joint measure (state) q, p its marginals Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 2 / 16 Main Result: Shannon-Pythagorean Theorem w q ⊗ q q ⊗ p w joint measure (state) q, p its marginals w defines T : P → P transofrming q → T(q) = p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 2 / 16 Main Result: Shannon-Pythagorean Theorem w  q ⊗ q // << q ⊗ p w joint measure (state) q, p its marginals w defines T : P → P transofrming q → T(q) = p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 2 / 16 Main Result: Shannon-Pythagorean Theorem w  q ⊗ q I(p,q) // << q ⊗ p w joint measure (state) q, p its marginals w defines T : P → P transofrming q → T(q) = p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 2 / 16 Main Result: Shannon-Pythagorean Theorem w I(w,q⊗p)  q ⊗ q I(p,q) // << q ⊗ p w joint measure (state) q, p its marginals w defines T : P → P transofrming q → T(q) = p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 2 / 16 Main Result: Shannon-Pythagorean Theorem w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) << q ⊗ p w joint measure (state) q, p its marginals w defines T : P → P transofrming q → T(q) = p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 2 / 16 Duality: Observables and States Quantum Information Distance Law of Cosines and Shannon-Pythagorean Theorem Discussion Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 3 / 16 Duality: Observables and States Duality: Observables and States Quantum Information Distance Law of Cosines and Shannon-Pythagorean Theorem Discussion Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 4 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} X is a ∗-algebra with 1 ∈ X Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} X is a ∗-algebra with 1 ∈ X Y dual of X Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} X is a ∗-algebra with 1 ∈ X Involution (x∗z)∗ = z∗x Y dual of X Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} X is a ∗-algebra with 1 ∈ X Involution (x∗z)∗ = z∗x Y dual of X Involution x, y∗ = x∗, y ∗ Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} X is a ∗-algebra with 1 ∈ X Involution (x∗z)∗ = z∗x X+ := {x : z∗z = x, ∃ z ∈ X} Y dual of X Involution x, y∗ = x∗, y ∗ Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} X is a ∗-algebra with 1 ∈ X Involution (x∗z)∗ = z∗x X+ := {x : z∗z = x, ∃ z ∈ X} Y dual of X Involution x, y∗ = x∗, y ∗ Y+ := {y : x, y ≥ 0, ∀ x ∈ X+} Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} X is a ∗-algebra with 1 ∈ X Involution (x∗z)∗ = z∗x X+ := {x : z∗z = x, ∃ z ∈ X} Observables x = x∗ Y dual of X Involution x, y∗ = x∗, y ∗ Y+ := {y : x, y ≥ 0, ∀ x ∈ X+} Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} X is a ∗-algebra with 1 ∈ X Involution (x∗z)∗ = z∗x X+ := {x : z∗z = x, ∃ z ∈ X} Observables x = x∗ Y dual of X Involution x, y∗ = x∗, y ∗ Y+ := {y : x, y ≥ 0, ∀ x ∈ X+} States y ≥ 0, 1, y = 1 Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} X is a ∗-algebra with 1 ∈ X Involution (x∗z)∗ = z∗x X+ := {x : z∗z = x, ∃ z ∈ X} Observables x = x∗ Y dual of X Involution x, y∗ = x∗, y ∗ Y+ := {y : x, y ≥ 0, ∀ x ∈ X+} States y ≥ 0, 1, y = 1 The base of Y+ is the set of all states (statistical manifold): P(X) := {p ∈ Y+ : 1, p = 1} Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} X is a ∗-algebra with 1 ∈ X Involution (x∗z)∗ = z∗x X+ := {x : z∗z = x, ∃ z ∈ X} Observables x = x∗ Y dual of X Involution x, y∗ = x∗, y ∗ Y+ := {y : x, y ≥ 0, ∀ x ∈ X+} States y ≥ 0, 1, y = 1 The base of Y+ is the set of all states (statistical manifold): P(X) := {p ∈ Y+ : 1, p = 1} Transposition: ∀ z ∈ X ∃ z ∈ Y : zx, y = x, z y Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Duality: Observables ∈ X ← ·, · → Y States ·, · : X × Y → C x, y := xiyi , x, y := x dy , x, y := tr {xy} X is a ∗-algebra with 1 ∈ X Involution (x∗z)∗ = z∗x X+ := {x : z∗z = x, ∃ z ∈ X} Observables x = x∗ Y dual of X Involution x, y∗ = x∗, y ∗ Y+ := {y : x, y ≥ 0, ∀ x ∈ X+} States y ≥ 0, 1, y = 1 The base of Y+ is the set of all states (statistical manifold): P(X) := {p ∈ Y+ : 1, p = 1} Transposition: ∀ z ∈ X ∃ z ∈ Y : zx, y = x, z y Y is a left (resp. right) module over X ⊆ Y w.r.t. z y (resp. yz∗ ∗). Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 5 / 16 Duality: Observables and States Exponents and Logarithms Define by the power series ex := ∞ n=0 xn n! , ln y := ∞ n=1 (−1)n−1 n (y − 1)n Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 6 / 16 Duality: Observables and States Exponents and Logarithms Define by the power series ex := ∞ n=0 xn n! , ln y := ∞ n=1 (−1)n−1 n (y − 1)n Group homomorphisms for xz = zx and yz = zy: ex+z = ex ez and ln(yz) = ln y + ln z Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 6 / 16 Duality: Observables and States Exponents and Logarithms Define by the power series ex := ∞ n=0 xn n! , ln y := ∞ n=1 (−1)n−1 n (y − 1)n Group homomorphisms for xz = zx and yz = zy: ex+z = ex ez and ln(yz) = ln y + ln z Group homomorphisms for tesnor product ⊗ and Kronecker ⊕: ex⊕z = ex ⊗ ez and ln(y ⊗ z) = ln y ⊕ ln z Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 6 / 16 Duality: Observables and States Exponents and Logarithms Define by the power series ex := ∞ n=0 xn n! , ln y := ∞ n=1 (−1)n−1 n (y − 1)n Group homomorphisms for xz = zx and yz = zy: ex+z = ex ez and ln(yz) = ln y + ln z Group homomorphisms for tesnor product ⊗ and Kronecker ⊕: ex⊕z = ex ⊗ ez and ln(y ⊗ z) = ln y ⊕ ln z Because X ⊆ Y , we can consider exp : X → Y and ln : Y → X Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 6 / 16 Quantum Information Distance Duality: Observables and States Quantum Information Distance Law of Cosines and Shannon-Pythagorean Theorem Discussion Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 7 / 16 Quantum Information Distance Additivity Axiom (Khinchin, 1957) I(p1 ⊗ p2, q1 ⊗ q2) = I(p1, q1) + I(p2, q2) Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 8 / 16 Quantum Information Distance Additivity Axiom (Khinchin, 1957) I(p1 ⊗ p2, q1 ⊗ q2) = I(p1, q1) + I(p2, q2) Let F : Y → R ∪ {∞} and F∗ : X → R ∪ {∞} be dual cl. convex F∗ (x) := sup{ x, y − F(y)} F∗∗ = F Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 8 / 16 Quantum Information Distance Additivity Axiom (Khinchin, 1957) I(p1 ⊗ p2, q1 ⊗ q2) = I(p1, q1) + I(p2, q2) Let F : Y → R ∪ {∞} and F∗ : X → R ∪ {∞} be dual cl. convex F∗ (x) := sup{ x, y − F(y)} F∗∗ = F Sub-differentials ∂F : Y → 2X, ∂F∗ : X → 2Y are inverse of each other (Moreau, 1967; Rockafellar, 1974): ∂F(y) x ⇐⇒ y ∈ ∂F∗ (x) Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 8 / 16 Quantum Information Distance Additivity Axiom (Khinchin, 1957) I(p1 ⊗ p2, q1 ⊗ q2) = I(p1, q1) + I(p2, q2) Let F : Y → R ∪ {∞} and F∗ : X → R ∪ {∞} be dual cl. convex F∗ (x) := sup{ x, y − F(y)} F∗∗ = F Sub-differentials ∂F : Y → 2X, ∂F∗ : X → 2Y are inverse of each other (Moreau, 1967; Rockafellar, 1974): ∂F(y) x ⇐⇒ y ∈ ∂F∗ (x) Example F(y) = ln y − 1, y F∗ (x) = 1, ex Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 8 / 16 Quantum Information Distance Additivity Axiom (Khinchin, 1957) I(p1 ⊗ p2, q1 ⊗ q2) = I(p1, q1) + I(p2, q2) Let F : Y → R ∪ {∞} and F∗ : X → R ∪ {∞} be dual cl. convex F∗ (x) := sup{ x, y − F(y)} F∗∗ = F Sub-differentials ∂F : Y → 2X, ∂F∗ : X → 2Y are inverse of each other (Moreau, 1967; Rockafellar, 1974): ∂F(y) x ⇐⇒ y ∈ ∂F∗ (x) Example F(y) = ln y − 1, y F∗ (x) = 1, ex F(y) = ln y ex = F∗ (x) Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 8 / 16 Quantum Information Distance Additive Quantum Information Distance Definition (I : Y × Y → R ∪ {∞}) I(y, z) := ln y − ln z, y − 1, y − z Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 9 / 16 Quantum Information Distance Additive Quantum Information Distance Definition (I : Y × Y → R ∪ {∞}) I(y, z) := ln y − ln z, y − 1, y − z yI(y, z) = ln y − ln z Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 9 / 16 Quantum Information Distance Additive Quantum Information Distance Definition (I : Y × Y → R ∪ {∞}) I(y, z) := ln y − ln z, y − 1, y − z yI(y, z) = ln y − ln z 2 yI(y, z) = y−1 Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 9 / 16 Quantum Information Distance Additive Quantum Information Distance Definition (I : Y × Y → R ∪ {∞}) I(y, z) := ln y − ln z, y − 1, y − z yI(y, z) = ln y − ln z 2 yI(y, z) = y−1 I∗(x, z) := 1, exz Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 9 / 16 Quantum Information Distance Additive Quantum Information Distance Definition (I : Y × Y → R ∪ {∞}) I(y, z) := ln y − ln z, y − 1, y − z yI(y, z) = ln y − ln z 2 yI(y, z) = y−1 I∗(x, z) := 1, exz Non-commutativity ln(ex+z) = x + z iff xz = zx, so that Radon-Nikodym derivative y/z: y/z := exp(ln y − ln z) (Araki, 1975; Umegaki, 1962) Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 9 / 16 Quantum Information Distance Additive Quantum Information Distance Definition (I : Y × Y → R ∪ {∞}) I(y, z) := ln y − ln z, y − 1, y − z yI(y, z) = ln y − ln z 2 yI(y, z) = y−1 I∗(x, z) := 1, exz Non-commutativity ln(ex+z) = x + z iff xz = zx, so that Radon-Nikodym derivative y/z: y/z := exp(ln y − ln z) (Araki, 1975; Umegaki, 1962) y/z := y1/2z−1y1/2 z−1/2yz−1/2 (Belavkin & Staszewski, 1984) Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 9 / 16 Quantum Information Distance Additive Quantum Information Distance Definition (I : Y × Y → R ∪ {∞}) I(y, z) := ln y − ln z, y − 1, y − z yI(y, z) = ln y − ln z 2 yI(y, z) = y−1 I∗(x, z) := 1, exz Non-commutativity ln(ex+z) = x + z iff xz = zx, so that Radon-Nikodym derivative y/z: y/z := exp(ln y − ln z) (Araki, 1975; Umegaki, 1962) y/z := y1/2z−1y1/2 z−1/2yz−1/2 (Belavkin & Staszewski, 1984) I(y, z) := I∗∗(y, z) = sup{ x, y − I∗(x, z)}, where I∗ (x, z) := 1, z1/2 ex z1/2 or I∗ (x, z) := 1, ex/2 zex/2 Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 9 / 16 Law of Cosines and Shannon-Pythagorean Theorem Duality: Observables and States Quantum Information Distance Law of Cosines and Shannon-Pythagorean Theorem Discussion Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 10 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Logarithmic Law of Cosines) For any w, y, z in Y with finite distances I(w, z) = I(w, y) + I(y, z) − ln y − ln z, y − w w z y Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 11 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Logarithmic Law of Cosines) For any w, y, z in Y with finite distances I(w, z) = I(w, y) + I(y, z) − ln y − ln z, y − w w z y Proof. First order Taylor expansion of I(·, z) at y: I(w, z) = I(y, z) + wI(y, z), w − y + R1(y, w) where wI(y, z) = ln y − ln z and the remainder R1(y, w) = I(w, y) Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 11 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Logarithmic Law of Cosines) For any w, y, z in Y with finite distances I(w, z) = I(w, y) + I(y, z) − ln y − ln z, y − w w z y Proof. First order Taylor expansion of I(·, z) at y: I(w, z) = I(y, z) + wI(y, z), w − y + R1(y, w) where wI(y, z) = ln y − ln z and the remainder R1(y, w) = I(w, y) Corollary (Log-Pythagorean Theorem) If ln y − ln z, y − w = 0, then I(w, z) = I(w, y) + I(y, z) Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 11 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Inequality for Information) I(y, z) ≥ 1, (y − z)2 2 max{ y ∞, z ∞} Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 12 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Inequality for Information) I(y, z) ≥ 1, (y − z)2 2 max{ y ∞, z ∞} Proof. Recall that I(y, z) is the remainder R1(z, y) in Taylor expansion: I(y, w) = I(z, w) + yI(z, w), y − z + R1(z, y) Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 12 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Inequality for Information) I(y, z) ≥ 1, (y − z)2 2 max{ y ∞, z ∞} Proof. Recall that I(y, z) is the remainder R1(z, y) in Taylor expansion: I(y, w) = I(z, w) + yI(z, w), y − z + R1(z, y) R1(z, y) = 1 0 (1 − t) 1, 2 yI(z + t(y − z), w)(y − z)2 dt Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 12 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Inequality for Information) I(y, z) ≥ 1, (y − z)2 2 max{ y ∞, z ∞} Proof. Recall that I(y, z) is the remainder R1(z, y) in Taylor expansion: I(y, w) = I(z, w) + yI(z, w), y − z + R1(z, y) R1(z, y) = 1 0 (1 − t) 1, 2 yI(z + t(y − z), w)(y − z)2 dt = 1 2 1, 2 yI(ξ, w)(y − z)2 for some ξ ∈ [z, y) Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 12 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Inequality for Information) I(y, z) ≥ 1, (y − z)2 2 max{ y ∞, z ∞} Proof. Recall that I(y, z) is the remainder R1(z, y) in Taylor expansion: I(y, w) = I(z, w) + yI(z, w), y − z + R1(z, y) R1(z, y) = 1 0 (1 − t) 1, 2 yI(z + t(y − z), w)(y − z)2 dt = 1 2 1, 2 yI(ξ, w)(y − z)2 for some ξ ∈ [z, y) Corollary (Stratonovich, 1975) I(p, q) + I(q, p) ≥ 1, (p − q)2 for all p, q ∈ P(X). Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 12 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Shannon-Pythagorean) w ∈ P(A ⊗ B), q ∈ P(A), p ∈ P(B), A ⊆ B w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 13 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Shannon-Pythagorean) w ∈ P(A ⊗ B), q ∈ P(A), p ∈ P(B), A ⊆ B If q = 1, w B, p = 1, w A, then I(w, q ⊗ q) = I(w, q ⊗ p) + I(p, q) w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 13 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Shannon-Pythagorean) w ∈ P(A ⊗ B), q ∈ P(A), p ∈ P(B), A ⊆ B If q = 1, w B, p = 1, w A, then I(w, q ⊗ q) = I(w, q ⊗ p) + I(p, q) w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 13 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Shannon-Pythagorean) w ∈ P(A ⊗ B), q ∈ P(A), p ∈ P(B), A ⊆ B If q = 1, w B, p = 1, w A, then I(w, q ⊗ q) = I(w, q ⊗ p) + I(p, q) w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Proof. I(w, q⊗q) = I(w, q⊗p)+I(q ⊗ p, q ⊗ q) I(p,q) − ln q ⊗ p − ln q ⊗ q, q ⊗ p − w 0 Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 13 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Shannon-Pythagorean) w ∈ P(A ⊗ B), q ∈ P(A), p ∈ P(B), A ⊆ B If q = 1, w B, p = 1, w A, then I(w, q ⊗ q) = I(w, q ⊗ p) + I(p, q) w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Proof. I(w, q⊗q) = I(w, q⊗p)+I(q ⊗ p, q ⊗ q) I(p,q) − ln q ⊗ p − ln q ⊗ q, q ⊗ p − w 0 ln q ⊗ p − ln q ⊗ q = 1A ⊗ (ln p − ln q) Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 13 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Shannon-Pythagorean) w ∈ P(A ⊗ B), q ∈ P(A), p ∈ P(B), A ⊆ B If q = 1, w B, p = 1, w A, then I(w, q ⊗ q) = I(w, q ⊗ p) + I(p, q) w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Proof. I(w, q⊗q) = I(w, q⊗p)+I(q ⊗ p, q ⊗ q) I(p,q) − ln q ⊗ p − ln q ⊗ q, q ⊗ p − w 0 ln q ⊗ p − ln q ⊗ q = 1A ⊗ (ln p − ln q) B b → 1A ⊗ b ∈ A ⊗ B, where B = ln p − ln q Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 13 / 16 Law of Cosines and Shannon-Pythagorean Theorem Theorem (Shannon-Pythagorean) w ∈ P(A ⊗ B), q ∈ P(A), p ∈ P(B), A ⊆ B If q = 1, w B, p = 1, w A, then I(w, q ⊗ q) = I(w, q ⊗ p) + I(p, q) w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Proof. I(w, q⊗q) = I(w, q⊗p)+I(q ⊗ p, q ⊗ q) I(p,q) − ln q ⊗ p − ln q ⊗ q, q ⊗ p − w 0 ln q ⊗ p − ln q ⊗ q = 1A ⊗ (ln p − ln q) B b → 1A ⊗ b ∈ A ⊗ B, where B = ln p − ln q b, p = 1A ⊗ b, w = 1A ⊗ b, z ⊗ p , as p = 1, w A = 1, z ⊗ p A Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 13 / 16 Discussion Duality: Observables and States Quantum Information Distance Law of Cosines and Shannon-Pythagorean Theorem Discussion Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 14 / 16 Discussion Applications to Optimisation of Dynamical Systems w ∈ P(A ⊗ B) defines a channel (Markov morphism or operation): T : P(A) q → Tq = p ∈ P(B) w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 15 / 16 Discussion Applications to Optimisation of Dynamical Systems w ∈ P(A ⊗ B) defines a channel (Markov morphism or operation): T : P(A) q → Tq = p ∈ P(B) I(p(t), q) = I(Ttq, q) divergence in t ∈ N0 w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 15 / 16 Discussion Applications to Optimisation of Dynamical Systems w ∈ P(A ⊗ B) defines a channel (Markov morphism or operation): T : P(A) q → Tq = p ∈ P(B) I(p(t), q) = I(Ttq, q) divergence in t ∈ N0 I(w, q ⊗ p) capacity of T w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 15 / 16 Discussion Applications to Optimisation of Dynamical Systems w ∈ P(A ⊗ B) defines a channel (Markov morphism or operation): T : P(A) q → Tq = p ∈ P(B) I(p(t), q) = I(Ttq, q) divergence in t ∈ N0 I(w, q ⊗ p) capacity of T I(w, q ⊗ q) hypotenuse of T w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 15 / 16 Discussion Applications to Optimisation of Dynamical Systems w ∈ P(A ⊗ B) defines a channel (Markov morphism or operation): T : P(A) q → Tq = p ∈ P(B) I(p(t), q) = I(Ttq, q) divergence in t ∈ N0 I(w, q ⊗ p) capacity of T I(w, q ⊗ q) hypotenuse of T w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Information-Theoretic Variational Problems Type I Maximize Ep{u} = u, p subject to I(p, q) ≤ λ Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 15 / 16 Discussion Applications to Optimisation of Dynamical Systems w ∈ P(A ⊗ B) defines a channel (Markov morphism or operation): T : P(A) q → Tq = p ∈ P(B) I(p(t), q) = I(Ttq, q) divergence in t ∈ N0 I(w, q ⊗ p) capacity of T I(w, q ⊗ q) hypotenuse of T w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Information-Theoretic Variational Problems Type I Maximize Ep{u} = u, p subject to I(p, q) ≤ λ Type III Maximize Ew{v} = v, w subject to I(w, q ⊗ p)} ≤ γ Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 15 / 16 Discussion Applications to Optimisation of Dynamical Systems w ∈ P(A ⊗ B) defines a channel (Markov morphism or operation): T : P(A) q → Tq = p ∈ P(B) I(p(t), q) = I(Ttq, q) divergence in t ∈ N0 I(w, q ⊗ p) capacity of T I(w, q ⊗ q) hypotenuse of T w I(w,q⊗p)  q ⊗ q I(p,q) // I(w,q⊗q) :: q ⊗ p Information-Theoretic Variational Problems Type I Maximize Ep{u} = u, p subject to I(p, q) ≤ λ Type III Maximize Ew{v} = v, w subject to I(w, q ⊗ p)} ≤ γ I+III=IV I(w, q ⊗ q) ≤ γ + λ Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 15 / 16 References Duality: Observables and States Quantum Information Distance Law of Cosines and Shannon-Pythagorean Theorem Discussion Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 16 / 16 Discussion Araki, H. (1975). Relative entropy of states of von Neumann algebras. Publications of the Research Institute for Mathematical Sciences, 11(3), 809–833. Belavkin, V. P., & Staszewski, P. (1984). Relative entropy in C∗-algebraic statistical mechanics. Reports in Mathematical Physics, 20, 373–384. Khinchin, A. I. (1957). Mathematical foundations of information theory. New York: Dover. Moreau, J.-J. (1967). Functionelles convexes. Paris: Coll´ege de France. Rockafellar, R. T. (1974). Conjugate duality and optimization (Vol. 16). PA: Society for Industrial and Applied Mathematics. Stratonovich, R. L. (1975). Information theory. Moscow, USSR: Sovetskoe Radio. (In Russian) Umegaki, H. (1962). Conditional expectation in an operator algebra. IV. entropy and information. Kodai Mathematical Seminar Reports, 14(2), 59–85. Roman Belavkin (Middlesex University) Shannon-Pythagorean Theorem for Quantum Information August 29, 2013 16 / 16