On Variational Definition of Quantum Entropy Roman V. Belavkin School of Science and Technology Middlesex University, London NW4 4BT, UK
1
September 24, 2014
1
This work was supported by EPSRC grant EP/H031936/1.
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
1 / 27
Viacheslav (Slava) Belavkin (1946–2012)
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
2 / 27
Viacheslav (Slava) Belavkin (1946–2012) Google: Belavkin equation Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
2 / 27
Motivation Crash Course in Quantum Probability Quantum Information and von Neumann Entropy Proper Quantum Entropy Variational Problems
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
3 / 27
Motivation
Motivation Crash Course in Quantum Probability Quantum Information and von Neumann Entropy Proper Quantum Entropy Variational Problems
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
4 / 27
Motivation
Motivation Quantum probability (QP) is a non-commutative generalization of classical probability (CP): Classical Probability ⊂ Quantum Probability
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
5 / 27
Motivation
Motivation Quantum probability (QP) is a non-commutative generalization of classical probability (CP): Classical Probability ⊂ Quantum Probability QP by von Neumann (1932) preceded Kolmogorov (1933).
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
5 / 27
Motivation
Motivation Quantum probability (QP) is a non-commutative generalization of classical probability (CP): Classical Probability ⊂ Quantum Probability QP by von Neumann (1932) preceded Kolmogorov (1933). Traditional definitions generalize classical: X H[p] := − (ln pi ) pi ⇐ S[p] := −tr {(ln p) p}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
5 / 27
Motivation
Motivation Quantum probability (QP) is a non-commutative generalization of classical probability (CP): Classical Probability ⊂ Quantum Probability QP by von Neumann (1932) preceded Kolmogorov (1933). Traditional definitions generalize classical: X H[p] := − (ln pi ) pi ⇐ S[p] := −tr {(ln p) p}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
5 / 27
Motivation
Motivation Quantum probability (QP) is a non-commutative generalization of classical probability (CP): Classical Probability ⊂ Quantum Probability QP by von Neumann (1932) preceded Kolmogorov (1933). Traditional definitions generalize classical: X H[p] := − (ln pi ) pi ⇐ S[p] := −tr {(ln p) p} X DKL [p, q] := (ln pi − ln qi ) pi ⇐ DAU [p, q] := tr {(ln p − ln q) p} (Kullback & Leibler, 1951; Araki, 1975; Umegaki, 1962)
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
5 / 27
Motivation
Motivation Quantum probability (QP) is a non-commutative generalization of classical probability (CP): Classical Probability ⊂ Quantum Probability QP by von Neumann (1932) preceded Kolmogorov (1933). Traditional definitions generalize classical: X H[p] := − (ln pi ) pi ⇐ S[p] := −tr {(ln p) p} X DKL [p, q] := (ln pi − ln qi ) pi ⇐ DAU [p, q] := tr {(ln p − ln q) p} (Kullback & Leibler, 1951; Araki, 1975; Umegaki, 1962) For any f : Y → R sup{f (y) : y is classical} ≤ sup f (y)
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
5 / 27
Motivation
Three definitions of entropy 1
As expected self-information: f (ω) = DKL [δω , P ] = − ln dP (ω): Z H(Ω) := H[P ] = −E{ln P } = − ln dP (ω) dP (ω) Ω
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
6 / 27
Motivation
Three definitions of entropy 1
As expected self-information: f (ω) = DKL [δω , P ] = − ln dP (ω): Z H(Ω) := H[P ] = −E{ln P } = − ln dP (ω) dP (ω) Ω
2
As negative KL-divergence DKL [P, ν]: Z dP (ω) H[P/ν] := − ln dP (ω) = ln ν(Ω) − DKL [P, Q] dν(ω) Ω
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
6 / 27
Motivation
Three definitions of entropy 1
As expected self-information: f (ω) = DKL [δω , P ] = − ln dP (ω): Z H(Ω) := H[P ] = −E{ln P } = − ln dP (ω) dP (ω) Ω
2
As negative KL-divergence DKL [P, ν]: Z dP (ω) H[P/ν] := − ln dP (ω) = ln ν(Ω) − DKL [P, Q] dν(ω) Ω
3
As information potential: maximum of IS (A.B) = DKL [P (A ∩ B), P (A) ⊗ P (B)]:
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
6 / 27
Motivation
Three definitions of entropy 1
As expected self-information: f (ω) = DKL [δω , P ] = − ln dP (ω): Z H(Ω) := H[P ] = −E{ln P } = − ln dP (ω) dP (ω) Ω
2
As negative KL-divergence DKL [P, ν]: Z dP (ω) H[P/ν] := − ln dP (ω) = ln ν(Ω) − DKL [P, Q] dν(ω) Ω
3
As information potential: maximum of IS (A.B) = DKL [P (A ∩ B), P (A) ⊗ P (B)]:
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
6 / 27
Motivation
Three definitions of entropy 1
As expected self-information: f (ω) = DKL [δω , P ] = − ln dP (ω): Z H(Ω) := H[P ] = −E{ln P } = − ln dP (ω) dP (ω) Ω
2
As negative KL-divergence DKL [P, ν]: Z dP (ω) H[P/ν] := − ln dP (ω) = ln ν(Ω) − DKL [P, Q] dν(ω) Ω
3
As information potential: maximum of IS (A.B) = DKL [P (A ∩ B), P (A) ⊗ P (B)]: Z H(B) := IS (B, B) = sup IS (A, B) : dP (a ∩ b) = dP (b) A
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
6 / 27
Crash Course in Quantum Probability
Motivation Crash Course in Quantum Probability Quantum Information and von Neumann Entropy Proper Quantum Entropy Variational Problems
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
7 / 27
Crash Course in Quantum Probability
Classical ⊂ Quantum Algebra of subsets (events): Ω, A(Ω) ⊆ 2Ω
Roman Belavkin (Middlesex University)
Algebra of subspaces (q-events): H, A(H) ⊆ {E subspace of H}
Variational Quantum Entropy
September 24, 2014
8 / 27
Crash Course in Quantum Probability
Classical ⊂ Quantum Algebra of subsets (events): Ω, A(Ω) ⊆ 2Ω
H, A(H) ⊆ {E subspace of H}
Distributive lattice (Boolean)
Roman Belavkin (Middlesex University)
Algebra of subspaces (q-events):
Non-distributive (non-Boolean)
Variational Quantum Entropy
September 24, 2014
8 / 27
Crash Course in Quantum Probability
Classical ⊂ Quantum Algebra of subsets (events): Ω, A(Ω) ⊆ 2Ω
Algebra of subspaces (q-events): H, A(H) ⊆ {E subspace of H}
Distributive lattice (Boolean)
Non-distributive (non-Boolean)
Linear algebra of indicators f-ns:
Algebra of ortho-projectors: IE : H → H
IE : Ω → {0, 1}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
8 / 27
Crash Course in Quantum Probability
Classical ⊂ Quantum Algebra of subsets (events): Ω, A(Ω) ⊆ 2Ω
Algebra of subspaces (q-events): H, A(H) ⊆ {E subspace of H}
Distributive lattice (Boolean)
Non-distributive (non-Boolean)
Linear algebra of indicators f-ns:
Algebra of ortho-projectors: IE : H → H
IE : Ω → {0, 1} Commutative
Non-Commutative IA IB 6= IB IA
IA · IB = IB · IA
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
8 / 27
Crash Course in Quantum Probability
Classical ⊂ Quantum Algebra of subsets (events): Ω, A(Ω) ⊆ 2Ω
Algebra of subspaces (q-events): H, A(H) ⊆ {E subspace of H}
Distributive lattice (Boolean)
Non-distributive (non-Boolean)
Linear algebra of indicators f-ns:
Algebra of ortho-projectors: IE : H → H
IE : Ω → {0, 1} Commutative
Non-Commutative
IA · IB = IB · IA P Partition of unity I = Ω I{ω}
IA IB 6= IB IA P Not unique I = {ei } Iei
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
8 / 27
Crash Course in Quantum Probability
Classical ⊂ Quantum Algebra of subsets (events): Ω, A(Ω) ⊆ 2Ω
Algebra of subspaces (q-events): H, A(H) ⊆ {E subspace of H}
Distributive lattice (Boolean)
Non-distributive (non-Boolean)
Linear algebra of indicators f-ns:
Algebra of ortho-projectors: IE : H → H
IE : Ω → {0, 1} Commutative
Non-Commutative
IA · IB = IB · IA P Partition of unity I = Ω I{ω}
IA IB 6= IB IA P Not unique I = {ei } Iei
Remark Fixing Ω is equivalent to fixing a basis {ei } ∈ H Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
8 / 27
Crash Course in Quantum Probability
Algebra of random variables: X = {x : Ω → R , A-measurable}
Roman Belavkin (Middlesex University)
∗-Algebra of observables: X = {x : H → H , x = x∗ }
Variational Quantum Entropy
September 24, 2014
9 / 27
Crash Course in Quantum Probability
Algebra of random variables: X = {x : Ω → R , A-measurable} Commutative
Roman Belavkin (Middlesex University)
∗-Algebra of observables: X = {x : H → H , x = x∗ } Non-commutative
Variational Quantum Entropy
September 24, 2014
9 / 27
Crash Course in Quantum Probability
Algebra of random variables: X = {x : Ω → R , A-measurable}
∗-Algebra of observables: X = {x : H → H , x = x∗ }
Commutative Non-commutative Duality w.r.t. h·, ·i : X × Y → R (C) Z X hx, yi = x(ω)y(ω) = x(ω) dy(ω) , hx, yi = tr {xy}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
9 / 27
Crash Course in Quantum Probability
Algebra of random variables: X = {x : Ω → R , A-measurable}
∗-Algebra of observables: X = {x : H → H , x = x∗ }
Commutative Non-commutative Duality w.r.t. h·, ·i : X × Y → R (C) Z X hx, yi = x(ω)y(ω) = x(ω) dy(ω) , hx, yi = tr {xy} y(x) = hx, yi — regular signed measures (charges).
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
9 / 27
Crash Course in Quantum Probability
Algebra of random variables: X = {x : Ω → R , A-measurable}
∗-Algebra of observables: X = {x : H → H , x = x∗ }
Commutative Non-commutative Duality w.r.t. h·, ·i : X × Y → R (C) Z X hx, yi = x(ω)y(ω) = x(ω) dy(ω) , hx, yi = tr {xy} y(x) = hx, yi — regular signed measures (charges). Pointed convex cone of non-negative elements: X+ = {x : z ∗ z = x , ∃ z ∈ X} Y+ = {y : hx, yi ≥ 0 , ∀ x ∈ X+ }
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
9 / 27
Crash Course in Quantum Probability
Algebra of random variables: X = {x : Ω → R , A-measurable}
∗-Algebra of observables: X = {x : H → H , x = x∗ }
Commutative Non-commutative Duality w.r.t. h·, ·i : X × Y → R (C) Z X hx, yi = x(ω)y(ω) = x(ω) dy(ω) , hx, yi = tr {xy} y(x) = hx, yi — regular signed measures (charges). Pointed convex cone of non-negative elements: X+ = {x : z ∗ z = x , ∃ z ∈ X} Y+ = {y : hx, yi ≥ 0 , ∀ x ∈ X+ } Probabilities or states: P(X) = {y ∈ Y+ : h1, yi = 1}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
9 / 27
Crash Course in Quantum Probability
Algebra of random variables: X = {x : Ω → R , A-measurable}
∗-Algebra of observables: X = {x : H → H , x = x∗ }
Commutative Non-commutative Duality w.r.t. h·, ·i : X × Y → R (C) Z X hx, yi = x(ω)y(ω) = x(ω) dy(ω) , hx, yi = tr {xy} y(x) = hx, yi — regular signed measures (charges). Pointed convex cone of non-negative elements: X+ = {x : z ∗ z = x , ∃ z ∈ X} Y+ = {y : hx, yi ≥ 0 , ∀ x ∈ X+ } Probabilities or states: P(X) = {y ∈ Y+ : h1, yi = 1} (Y, ≤) is a lattice Roman Belavkin (Middlesex University)
(Y, ≤) is not a lattice Variational Quantum Entropy
September 24, 2014
9 / 27
Crash Course in Quantum Probability
Algebra of random variables: X = {x : Ω → R , A-measurable}
∗-Algebra of observables: X = {x : H → H , x = x∗ }
Commutative Non-commutative Duality w.r.t. h·, ·i : X × Y → R (C) Z X hx, yi = x(ω)y(ω) = x(ω) dy(ω) , hx, yi = tr {xy} y(x) = hx, yi — regular signed measures (charges). Pointed convex cone of non-negative elements: X+ = {x : z ∗ z = x , ∃ z ∈ X} Y+ = {y : hx, yi ≥ 0 , ∀ x ∈ X+ } Probabilities or states: P(X) = {y ∈ Y+ : h1, yi = 1} (Y, ≤) is a lattice P(X) is a simplex Roman Belavkin (Middlesex University)
(Y, ≤) is not a lattice P(X) is not a simplex Variational Quantum Entropy
September 24, 2014
9 / 27
Crash Course in Quantum Probability
Modular Structure on the Dual Space Y
Involution hx, y ∗ i = hx∗ , yi∗ and transposition (x∗ y)∗ = y ∗ x ,
Roman Belavkin (Middlesex University)
(xy)0 = y 0 x0
Variational Quantum Entropy
September 24, 2014
10 / 27
Crash Course in Quantum Probability
Modular Structure on the Dual Space Y
Involution hx, y ∗ i = hx∗ , yi∗ and transposition (x∗ y)∗ = y ∗ x ,
(xy)0 = y 0 x0
h·, ·i is regular w.r.t. left (right) multiplication and transposition: ∀ z ∈ X ∃ z0 ∈ Y :
Roman Belavkin (Middlesex University)
hzx, yi = hx, z 0 yi
Variational Quantum Entropy
September 24, 2014
10 / 27
Crash Course in Quantum Probability
Modular Structure on the Dual Space Y
Involution hx, y ∗ i = hx∗ , yi∗ and transposition (x∗ y)∗ = y ∗ x ,
(xy)0 = y 0 x0
h·, ·i is regular w.r.t. left (right) multiplication and transposition: ∀ z ∈ X ∃ z0 ∈ Y :
hzx, yi = hx, z 0 yi
hx, yz ∗0∗ i = hx∗ , z ∗0 y ∗ i∗ = hz ∗ x∗ , y ∗ i∗ = hxz, yi
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
10 / 27
Crash Course in Quantum Probability
Modular Structure on the Dual Space Y
Involution hx, y ∗ i = hx∗ , yi∗ and transposition (x∗ y)∗ = y ∗ x ,
(xy)0 = y 0 x0
h·, ·i is regular w.r.t. left (right) multiplication and transposition: ∀ z ∈ X ∃ z0 ∈ Y :
hzx, yi = hx, z 0 yi
hx, yz ∗0∗ i = hx∗ , z ∗0 y ∗ i∗ = hz ∗ x∗ , y ∗ i∗ = hxz, yi Y is a left (right) ∗-module over X 0 ⊆ Y .
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
10 / 27
Crash Course in Quantum Probability
Exponents and Logarithms Defined by the power series ex :=
∞ X xn n=0
Roman Belavkin (Middlesex University)
n!
,
ln y :=
∞ X (−1)n−1 n=1
Variational Quantum Entropy
n
(y − 1)n
September 24, 2014
11 / 27
Crash Course in Quantum Probability
Exponents and Logarithms Defined by the power series ex :=
∞ X xn n=0
n!
,
ln y :=
∞ X (−1)n−1 n=1
n
(y − 1)n
Homomorphism (X, +) → (X, ·) for xz = zx and yz = zy: ex+z = ex ez
Roman Belavkin (Middlesex University)
and
ln(yz) = ln y + ln z
Variational Quantum Entropy
September 24, 2014
11 / 27
Crash Course in Quantum Probability
Exponents and Logarithms Defined by the power series ex :=
∞ X xn n=0
n!
,
ln y :=
∞ X (−1)n−1 n=1
n
(y − 1)n
Homomorphism (X, +) → (X, ·) for xz = zx and yz = zy: ex+z = ex ez
and
ln(yz) = ln y + ln z
Homomorphism (X, ⊕) → (X, ⊗): ex⊕z = ex ⊗ ez
Roman Belavkin (Middlesex University)
and
ln(y ⊗ z) = ln y ⊕ ln z
Variational Quantum Entropy
September 24, 2014
11 / 27
Crash Course in Quantum Probability
Exponents and Logarithms Defined by the power series ex :=
∞ X xn n=0
n!
,
ln y :=
∞ X (−1)n−1 n=1
n
(y − 1)n
Homomorphism (X, +) → (X, ·) for xz = zx and yz = zy: ex+z = ex ez
and
ln(yz) = ln y + ln z
Homomorphism (X, ⊕) → (X, ⊗): ex⊕z = ex ⊗ ez
and
ln(y ⊗ z) = ln y ⊕ ln z
Because X 0 ⊆ Y , we can consider exp : X → Y
Roman Belavkin (Middlesex University)
and
ln : Int(Y+ ) ⊂ Y → X
Variational Quantum Entropy
September 24, 2014
11 / 27
Crash Course in Quantum Probability
Information Distance Definition (I : Y × Y → R ∪ {∞}) I[y, z] := hln y − ln z, yi − h1, y − zi
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
12 / 27
Crash Course in Quantum Probability
Information Distance Definition (I : Y × Y → R ∪ {∞}) I[y, z] := hln y − ln z, yi − h1, y − zi
Theorem (Generalized Law of Cosines) D[w, y] :=
R1 0
(1 − t)h∇2w D[y + t(w − y), y](w − y)2 i dx
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
12 / 27
Crash Course in Quantum Probability
Information Distance Definition (I : Y × Y → R ∪ {∞}) I[y, z] := hln y − ln z, yi − h1, y − zi
Theorem (Generalized Law of Cosines) D[w, y] :=
R1 0
(1 − t)h∇2w D[y + t(w − y), y](w − y)2 i dx
D[w, z] = D[w, y] + D[y, z] − h∇w D[y, z], y − wi
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
12 / 27
Crash Course in Quantum Probability
Information Distance Definition (I : Y × Y → R ∪ {∞}) I[y, z] := hln y − ln z, yi − h1, y − zi
Theorem (Generalized Law of Cosines) D[w, y] :=
R1 0
(1 − t)h∇2w D[y + t(w − y), y](w − y)2 i dx
D[w, z] = D[w, y] + D[y, z] − h∇w D[y, z], y − wi
Example (Bregman divergence (generalized)) DF [y, z] := inf{F (y) − F (z) − hx, y − zi : x ∈ ∂F (z)}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
12 / 27
Crash Course in Quantum Probability
Information Distance Definition (I : Y × Y → R ∪ {∞}) I[y, z] := hln y − ln z, yi − h1, y − zi
Theorem (Generalized Law of Cosines) D[w, y] :=
R1 0
(1 − t)h∇2w D[y + t(w − y), y](w − y)2 i dx
D[w, z] = D[w, y] + D[y, z] − h∇w D[y, z], y − wi
Example (Bregman divergence (generalized)) DF [y, z] := inf{F (y) − F (z) − hx, y − zi : x ∈ ∂F (z)} F (y) = hln y − 1, yi
Roman Belavkin (Middlesex University)
⇒
DF [y, z] = I[y, z]
Variational Quantum Entropy
September 24, 2014
12 / 27
Crash Course in Quantum Probability
Information Distance Definition (I : Y × Y → R ∪ {∞}) I[y, z] := hln y − ln z, yi − h1, y − zi
Theorem (Generalized Law of Cosines) D[w, y] :=
R1 0
(1 − t)h∇2w D[y + t(w − y), y](w − y)2 i dx
D[w, z] = D[w, y] + D[y, z] − h∇w D[y, z], y − wi
Example (Bregman divergence (generalized)) DF [y, z] := inf{F (y) − F (z) − hx, y − zi : x ∈ ∂F (z)} F (y) = hln y − 1, yi ∇F (y) = ln y and
Roman Belavkin (Middlesex University)
⇒
∇F ∗ (x)
DF [y, z] = I[y, z] = ex
Variational Quantum Entropy
September 24, 2014
12 / 27
Crash Course in Quantum Probability
Information Distance Definition (I : Y × Y → R ∪ {∞}) I[y, z] := hln y − ln z, yi − h1, y − zi
Theorem (Generalized Law of Cosines) D[w, y] :=
R1 0
(1 − t)h∇2w D[y + t(w − y), y](w − y)2 i dx
D[w, z] = D[w, y] + D[y, z] − h∇w D[y, z], y − wi
Example (Bregman divergence (generalized)) DF [y, z] := inf{F (y) − F (z) − hx, y − zi : x ∈ ∂F (z)} F (y) = hln y − 1, yi ∇F (y) = ln y and
⇒
∇F ∗ (x)
DF [y, z] = I[y, z] = ex
Additivity axiom (Khinchin, 1957): I[p1 ⊗ p2 , q1 ⊗ q2 ] = I[p1 , q1 ] + I[p2 , q2 ] Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
12 / 27
Quantum Information and von Neumann Entropy
Motivation Crash Course in Quantum Probability Quantum Information and von Neumann Entropy Proper Quantum Entropy Variational Problems
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
13 / 27
Quantum Information and von Neumann Entropy
Compound States H = HA ⊗ H B , X = A ⊗ B
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
14 / 27
Quantum Information and von Neumann Entropy
Compound States H = HA ⊗ H B , X = A ⊗ B Let q ∈ P(A), p ∈ P(B)
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
14 / 27
Quantum Information and von Neumann Entropy
Compound States H = HA ⊗ H B , X = A ⊗ B Let q ∈ P(A), p ∈ P(B) Product states q ⊗ p ∈ P(A ⊗ B)
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
14 / 27
Quantum Information and von Neumann Entropy
Compound States H = HA ⊗ H B , X = A ⊗ B Let q ∈ P(A), p ∈ P(B) Product states q ⊗ p ∈ P(A ⊗ B) Separable states ws ∈ co cl [P(A) ⊗ P(B)]
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
14 / 27
Quantum Information and von Neumann Entropy
Compound States H = HA ⊗ H B , X = A ⊗ B Let q ∈ P(A), p ∈ P(B) Product states q ⊗ p ∈ P(A ⊗ B) Separable states ws ∈ co cl [P(A) ⊗ P(B)] Non-separable states w ∈ P(A ⊗ B) \ co cl [P(A) ⊗ P(B)]: P(A ⊗ B) \ co cl [P(A) ⊗ P(B)] 6= ∅
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
14 / 27
Quantum Information and von Neumann Entropy
Compound States H = HA ⊗ H B , X = A ⊗ B Let q ∈ P(A), p ∈ P(B) Product states q ⊗ p ∈ P(A ⊗ B) Separable states ws ∈ co cl [P(A) ⊗ P(B)] Non-separable states w ∈ P(A ⊗ B) \ co cl [P(A) ⊗ P(B)]: P(A ⊗ B) \ co cl [P(A) ⊗ P(B)] 6= ∅ w defines a channel Λ : P(A) → P(B), q 7→ Λq = p.
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
14 / 27
Quantum Information and von Neumann Entropy
Compound States H = HA ⊗ H B , X = A ⊗ B Let q ∈ P(A), p ∈ P(B) Product states q ⊗ p ∈ P(A ⊗ B) Separable states ws ∈ co cl [P(A) ⊗ P(B)] Non-separable states w ∈ P(A ⊗ B) \ co cl [P(A) ⊗ P(B)]: P(A ⊗ B) \ co cl [P(A) ⊗ P(B)] 6= ∅ w defines a channel Λ : P(A) → P(B), q 7→ Λq = p. q and p are reduced states (partial traces) of w: q = h1A ⊗ 1B , wiB ,
Roman Belavkin (Middlesex University)
h1A ⊗ 1B , wiA = p
Variational Quantum Entropy
September 24, 2014
14 / 27
Quantum Information and von Neumann Entropy
Entanglement Definition (Entanglements) of q ∈ P(A) and p ∈ P(B) associated with w ∈ P(A ⊗ B) are CP maps: π : A → P(B) ⊂ B 0 :
Roman Belavkin (Middlesex University)
π(a) = ha ⊗ 1B , wiA
Variational Quantum Entropy
September 24, 2014
15 / 27
Quantum Information and von Neumann Entropy
Entanglement Definition (Entanglements) of q ∈ P(A) and p ∈ P(B) associated with w ∈ P(A ⊗ B) are CP maps: π : A → P(B) ⊂ B 0 : 0
0
π : B → P(A) ⊂ A :
Roman Belavkin (Middlesex University)
π(a) = ha ⊗ 1B , wiA π 0 (b) = h1A ⊗ b, wiB
Variational Quantum Entropy
September 24, 2014
15 / 27
Quantum Information and von Neumann Entropy
Entanglement Definition (Entanglements) of q ∈ P(A) and p ∈ P(B) associated with w ∈ P(A ⊗ B) are CP maps: π : A → P(B) ⊂ B 0 : 0
0
π : B → P(A) ⊂ A :
π(a) = ha ⊗ 1B , wiA π 0 (b) = h1A ⊗ b, wiB
hb, π(a)iB = ha ⊗ b, wi = ha, π 0 (b)iA
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
15 / 27
Quantum Information and von Neumann Entropy
Entanglement Definition (Entanglements) of q ∈ P(A) and p ∈ P(B) associated with w ∈ P(A ⊗ B) are CP maps: π : A → P(B) ⊂ B 0 : 0
0
π : B → P(A) ⊂ A :
π(a) = ha ⊗ 1B , wiA π 0 (b) = h1A ⊗ b, wiB
hb, π(a)iB = ha ⊗ b, wi = ha, π 0 (b)iA π(1A ) = p ∈ P(B), π 0 (1B ) = q ∈ P(A)
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
15 / 27
Quantum Information and von Neumann Entropy
Entanglement Definition (Entanglements) of q ∈ P(A) and p ∈ P(B) associated with w ∈ P(A ⊗ B) are CP maps: π : A → P(B) ⊂ B 0 : 0
0
π : B → P(A) ⊂ A :
π(a) = ha ⊗ 1B , wiA π 0 (b) = h1A ⊗ b, wiB
hb, π(a)iB = ha ⊗ b, wi = ha, π 0 (b)iA π(1A ) = p ∈ P(B), π 0 (1B ) = q ∈ P(A) If w is separable, then transposed π(a) and π 0 (b) are also CP.
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
15 / 27
Quantum Information and von Neumann Entropy
Entanglement Definition (Entanglements) of q ∈ P(A) and p ∈ P(B) associated with w ∈ P(A ⊗ B) are CP maps: π : A → P(B) ⊂ B 0 : 0
0
π : B → P(A) ⊂ A :
π(a) = ha ⊗ 1B , wiA π 0 (b) = h1A ⊗ b, wiB
hb, π(a)iB = ha ⊗ b, wi = ha, π 0 (b)iA π(1A ) = p ∈ P(B), π 0 (1B ) = q ∈ P(A) If w is separable, then transposed π(a) and π 0 (b) are also CP. Decomposition (Belavkin & Ohya, 2002): π(a) = p1/2 Π(a)p1/2 ,
Roman Belavkin (Middlesex University)
π 0 (b) = q 1/2 Π0 (b)q 1/2
Variational Quantum Entropy
September 24, 2014
15 / 27
Quantum Information and von Neumann Entropy
Entanglement Definition (Entanglements) of q ∈ P(A) and p ∈ P(B) associated with w ∈ P(A ⊗ B) are CP maps: π : A → P(B) ⊂ B 0 : 0
0
π : B → P(A) ⊂ A :
π(a) = ha ⊗ 1B , wiA π 0 (b) = h1A ⊗ b, wiB
hb, π(a)iB = ha ⊗ b, wi = ha, π 0 (b)iA π(1A ) = p ∈ P(B), π 0 (1B ) = q ∈ P(A) If w is separable, then transposed π(a) and π 0 (b) are also CP. Decomposition (Belavkin & Ohya, 2002): π(a) = p1/2 Π(a)p1/2 ,
π 0 (b) = q 1/2 Π0 (b)q 1/2
Standard entanglement: π(a) = p1/2 ap1/2 (i.e. Π(a) = a). Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
15 / 27
Quantum Information and von Neumann Entropy
Quantum Mutual Information Definition (Stratonovich, 1965) IS (A, B) := I[w, q ⊗ p]
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
16 / 27
Quantum Information and von Neumann Entropy
Quantum Mutual Information Definition (Stratonovich, 1965) IS (A, B) := I[w, q ⊗ p] =
hln w − ln q ⊗ p, wi
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
16 / 27
Quantum Information and von Neumann Entropy
Quantum Mutual Information Definition (Stratonovich, 1965) IS (A, B) := I[w, q ⊗ p] =
hln w − ln q ⊗ p, wi
=
S(A) + S(B) − S(A ⊗ B)
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
S(A) := S[q] = −hln q, qi
September 24, 2014
16 / 27
Quantum Information and von Neumann Entropy
Quantum Mutual Information Definition (Stratonovich, 1965) IS (A, B) := I[w, q ⊗ p] =
hln w − ln q ⊗ p, wi
=
S(A) + S(B) − S(A ⊗ B)
=
S(A) − S(A | B)
Roman Belavkin (Middlesex University)
S(A) := S[q] = −hln q, qi S(A | B) := S(A) − IS (A, B)
Variational Quantum Entropy
September 24, 2014
16 / 27
Quantum Information and von Neumann Entropy
Quantum Mutual Information Definition (Stratonovich, 1965) IS (A, B) := I[w, q ⊗ p] =
hln w − ln q ⊗ p, wi
=
S(A) + S(B) − S(A ⊗ B)
=
S(A) − S(A | B)
S(A | B) := S(A) − IS (A, B)
=
S(B) − S(B | A)
S(B | A) := S(B) − IS (A, B)
Roman Belavkin (Middlesex University)
S(A) := S[q] = −hln q, qi
Variational Quantum Entropy
September 24, 2014
16 / 27
Quantum Information and von Neumann Entropy
Quantum Mutual Information Definition (Stratonovich, 1965) IS (A, B) := I[w, q ⊗ p] =
hln w − ln q ⊗ p, wi
=
S(A) + S(B) − S(A ⊗ B)
=
S(A) − S(A | B)
S(A | B) := S(A) − IS (A, B)
=
S(B) − S(B | A)
S(B | A) := S(B) − IS (A, B)
S(A) := S[q] = −hln q, qi
Shannon’s inequality is false: 0 ≤ IS (A, B) min[S(A), S(B)]
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
16 / 27
Quantum Information and von Neumann Entropy
Quantum Mutual Information Definition (Stratonovich, 1965) IS (A, B) := I[w, q ⊗ p] =
hln w − ln q ⊗ p, wi
=
S(A) + S(B) − S(A ⊗ B)
=
S(A) − S(A | B)
S(A | B) := S(A) − IS (A, B)
=
S(B) − S(B | A)
S(B | A) := S(B) − IS (A, B)
S(A) := S[q] = −hln q, qi
Shannon’s inequality is false: 0 ≤ IS (A, B) min[S(A), S(B)] Negative conditional entropy 0 S(A | B) Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
16 / 27
Proper Quantum Entropy
Motivation Crash Course in Quantum Probability Quantum Information and von Neumann Entropy Proper Quantum Entropy Variational Problems
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
17 / 27
Proper Quantum Entropy
Proper Quantum Entropy Definition (Belavkin & Ohya, 2002) H(B) := IS (B, B) =
sup
{I[w, q ⊗ p] : h1, wiA = p}
w∈P(A⊗B)
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
18 / 27
Proper Quantum Entropy
Proper Quantum Entropy Definition (Belavkin & Ohya, 2002) H(B) := IS (B, B) =
sup
{I[w, q ⊗ p] : h1, wiA = p}
w∈P(A⊗B)
Achieved by standard entanglement π(a) = p1/2 ap1/2 .
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
18 / 27
Proper Quantum Entropy
Proper Quantum Entropy Definition (Belavkin & Ohya, 2002) H(B) := IS (B, B) =
sup
{I[w, q ⊗ p] : h1, wiA = p}
w∈P(A⊗B)
Achieved by standard entanglement π(a) = p1/2 ap1/2 . Shannon’s inequality: 0 ≤ IS (A, B) ≤ min[H(A), H(B)]
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
18 / 27
Proper Quantum Entropy
Proper Quantum Entropy Definition (Belavkin & Ohya, 2002) H(B) := IS (B, B) =
sup
{I[w, q ⊗ p] : h1, wiA = p}
w∈P(A⊗B)
Achieved by standard entanglement π(a) = p1/2 ap1/2 . Shannon’s inequality: 0 ≤ IS (A, B) ≤ min[H(A), H(B)] Proper quantum conditional entropy 0 ≤ H(A | B) := H(A) − IS (A, B)
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
18 / 27
Proper Quantum Entropy
Proper Quantum Entropy Definition (Belavkin & Ohya, 2002) H(B) := IS (B, B) =
sup
{I[w, q ⊗ p] : h1, wiA = p}
w∈P(A⊗B)
Achieved by standard entanglement π(a) = p1/2 ap1/2 . Shannon’s inequality: 0 ≤ IS (A, B) ≤ min[H(A), H(B)] Proper quantum conditional entropy 0 ≤ H(A | B) := H(A) − IS (A, B) S(B) ≤ H(B) Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
18 / 27
Proper Quantum Entropy
Decomposition of Quantum Information Lemma w ∈ P(A ⊗ B), h1, wiB = q, h1, wiA = p.
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
19 / 27
Proper Quantum Entropy
Decomposition of Quantum Information Lemma w ∈ P(A ⊗ B), h1, wiB = q, h1, wiA = p. Separable projection ws ∈ cl co [P(A) ⊗ P(B)]: I[w, ws ] = inf{I[w, r] : r ∈ co cl [P(A) ⊗ P(B)]} h1, ws iB = q,
Roman Belavkin (Middlesex University)
h1, ws iA = p
Variational Quantum Entropy
September 24, 2014
19 / 27
Proper Quantum Entropy
Decomposition of Quantum Information Lemma w ∈ P(A ⊗ B), h1, wiB = q, h1, wiA = p. Separable projection ws ∈ cl co [P(A) ⊗ P(B)]: I[w, ws ] = inf{I[w, r] : r ∈ co cl [P(A) ⊗ P(B)]} h1, ws iB = q,
h1, ws iA = p
hln ws − ln q ⊗ p, ws − wi = 0
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
19 / 27
Proper Quantum Entropy
Decomposition of Quantum Information Lemma w ∈ P(A ⊗ B), h1, wiB = q, h1, wiA = p. Separable projection ws ∈ cl co [P(A) ⊗ P(B)]: I[w, ws ] = inf{I[w, r] : r ∈ co cl [P(A) ⊗ P(B)]} h1, ws iB = q,
h1, ws iA = p
hln ws − ln q ⊗ p, ws − wi = 0
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
19 / 27
Proper Quantum Entropy
Decomposition of Quantum Information Lemma w ∈ P(A ⊗ B), h1, wiB = q, h1, wiA = p. Separable projection ws ∈ cl co [P(A) ⊗ P(B)]: I[w, ws ] = inf{I[w, r] : r ∈ co cl [P(A) ⊗ P(B)]} h1, ws iB = q,
h1, ws iA = p
hln ws − ln q ⊗ p, ws − wi = 0 Then IS (A, B) = IC (A, B) + IQ (A, B) | {z } | {z } I[ws ,q⊗p]
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
I[w,ws ]
September 24, 2014
19 / 27
Proper Quantum Entropy
Decomposition of Quantum Information Lemma w ∈ P(A ⊗ B), h1, wiB = q, h1, wiA = p. Separable projection ws ∈ cl co [P(A) ⊗ P(B)]: I[w, ws ] = inf{I[w, r] : r ∈ co cl [P(A) ⊗ P(B)]} h1, ws iB = q,
h1, ws iA = p
hln ws − ln q ⊗ p, ws − wi = 0 Then IS (A, B) = IC (A, B) + IQ (A, B) | {z } | {z } I[ws ,q⊗p]
I[w,ws ]
Remark 0 ≤ IC (A, B) ≤ min[S(A), S(B)] Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
19 / 27
Proper Quantum Entropy
Cross-Entropy
Definition (von Neumann Cross-Entropy) S[p, q] := −hln q, pi
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
20 / 27
Proper Quantum Entropy
Cross-Entropy
Definition (von Neumann Cross-Entropy) S[p, q] := −hln q, pi =
Roman Belavkin (Middlesex University)
S[p] + I[p, q]
Variational Quantum Entropy
September 24, 2014
20 / 27
Proper Quantum Entropy
Cross-Entropy
Definition (von Neumann Cross-Entropy) S[p, q] := −hln q, pi =
S[p] + I[p, q]
Definition (Proper Cross-Entropy) H[p, q] := H[p] + I[p, q]
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
20 / 27
Proper Quantum Entropy
Cross-Entropy
Definition (von Neumann Cross-Entropy) S[p, q] := −hln q, pi =
S[p] + I[p, q]
Definition (Proper Cross-Entropy) H[p, q] := H[p] + I[p, q] ≥
Roman Belavkin (Middlesex University)
S[p, q]
Variational Quantum Entropy
September 24, 2014
20 / 27
Proper Quantum Entropy
Cross-Information If A ⊆ B, then q ∈ P(A) is also q ∈ P(B).
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
21 / 27
Proper Quantum Entropy
Cross-Information If A ⊆ B, then q ∈ P(A) is also q ∈ P(B). q ⊗ q ∈ P(A ⊗ B)
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
21 / 27
Proper Quantum Entropy
Cross-Information If A ⊆ B, then q ∈ P(A) is also q ∈ P(B). q ⊗ q ∈ P(A ⊗ B)
Definition (Cross-Information) I[w, q ⊗ q] := hln w − ln q ⊗ q, wi
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
21 / 27
Proper Quantum Entropy
Cross-Information If A ⊆ B, then q ∈ P(A) is also q ∈ P(B). q ⊗ q ∈ P(A ⊗ B)
Definition (Cross-Information) I[w, q ⊗ q] := hln w − ln q ⊗ q, wi
Theorem (Shannon-Pythagorean) I[w, q ⊗ q] = I[w, q ⊗ p] + I[p, q]
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
21 / 27
Proper Quantum Entropy
Cross-Information If A ⊆ B, then q ∈ P(A) is also q ∈ P(B). q ⊗ q ∈ P(A ⊗ B)
Definition (Cross-Information) I[w, q ⊗ q] := hln w − ln q ⊗ q, wi
Theorem (Shannon-Pythagorean) I[w,q⊗q]
I[w, q ⊗ q] = I[w, q ⊗ p] + I[p, q] q⊗q
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
:w
I[p,q]
I[w,q⊗p]
/q⊗p
September 24, 2014
21 / 27
Proper Quantum Entropy
Cross-Information If A ⊆ B, then q ∈ P(A) is also q ∈ P(B). q ⊗ q ∈ P(A ⊗ B)
Definition (Cross-Information) I[w, q ⊗ q] := hln w − ln q ⊗ q, wi
Theorem (Shannon-Pythagorean) I[w,q⊗q]
I[w, q ⊗ q] = I[w, q ⊗ p] + I[p, q] q⊗q
:w
I[p,q]
I[w,q⊗p]
/q⊗p
I[w, q ⊗ q] = S[p, q] − S(B | A) = H[p, q] − H(B | A)
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
21 / 27
Proper Quantum Entropy
Cross-Information If A ⊆ B, then q ∈ P(A) is also q ∈ P(B). q ⊗ q ∈ P(A ⊗ B)
Definition (Cross-Information) I[w, q ⊗ q] := hln w − ln q ⊗ q, wi
Theorem (Shannon-Pythagorean) I[w,q⊗q]
I[w, q ⊗ q] = I[w, q ⊗ p] + I[p, q] q⊗q
:w
I[p,q]
I[w,q⊗p]
/q⊗p
I[w, q ⊗ q] = S[p, q] − S(B | A) = H[p, q] − H(B | A) I[w, q ⊗ q] ≤ min{H[q], H[p]} + I[p, q] ≤ H[p, q] Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
21 / 27
Variational Problems
Motivation Crash Course in Quantum Probability Quantum Information and von Neumann Entropy Proper Quantum Entropy Variational Problems
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
22 / 27
Variational Problems
First Variational Problem Definition (Utility operator) u : H → H, u∗ = u, with u =
P
ui . uj
λi ui and ({ui }, .) such that ⇐⇒
λi ≤ λj
i.e. order ({λi }, ≤) ⇐⇒ ({ui }, .) preference relation.
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
23 / 27
Variational Problems
First Variational Problem Definition (Utility operator) u : H → H, u∗ = u, with u =
P
ui . uj
λi ui and ({ui }, .) such that ⇐⇒
λi ≤ λj
i.e. order ({λi }, ≤) ⇐⇒ ({ui }, .) preference relation.
Type I problem u(λ) = sup{hu, pi : I[p, q] ≤ λ}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
23 / 27
Variational Problems
First Variational Problem Definition (Utility operator) u : H → H, u∗ = u, with u =
P
ui . uj
λi ui and ({ui }, .) such that ⇐⇒
λi ≤ λj
i.e. order ({λi }, ≤) ⇐⇒ ({ui }, .) preference relation.
Type I problem u(λ) = sup{hu, pi : I[p, q] ≤ λ} −1
u
(υ) = inf{I[p, q] : hu, pi ≥ υ}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
23 / 27
Variational Problems
First Variational Problem Definition (Utility operator) u : H → H, u∗ = u, with u =
P
ui . uj
λi ui and ({ui }, .) such that ⇐⇒
λi ≤ λj
i.e. order ({λi }, ≤) ⇐⇒ ({ui }, .) preference relation.
Type I problem u(λ) = sup{hu, pi : I[p, q] ≤ λ} −1
u
(υ) = inf{I[p, q] : hu, pi ≥ υ}
Theorem (Solution) 1
1
pβ = ∇I ∗ [βu, q] = e 2 [βu−Ψ(β)] qe 2 [βu−Ψ(β)] , Roman Belavkin (Middlesex University)
Variational Quantum Entropy
β −1 = u0 (λ) September 24, 2014
23 / 27
Variational Problems
Third Variational Problem Type III problem H = HA ⊗ H B , u ∈ A ⊗ B
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
24 / 27
Variational Problems
Third Variational Problem Type III problem H = HA ⊗ H B , u ∈ A ⊗ B u(λ) = sup{hu, wi : I[w, q ⊗ p] ≤ λ}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
24 / 27
Variational Problems
Third Variational Problem Type III problem H = HA ⊗ H B , u ∈ A ⊗ B u(λ) = sup{hu, wi : I[w, q ⊗ p] ≤ λ} −1
u
(υ) = inf{I[w, q ⊗ p] : hu, wi ≥ υ}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
24 / 27
Variational Problems
Third Variational Problem Type III problem H = HA ⊗ H B , u ∈ A ⊗ B u(λ) = sup{hu, wi : I[w, q ⊗ p] ≤ λ} −1
u
(υ) = inf{I[w, q ⊗ p] : hu, wi ≥ υ}
Theorem (Solution) 1
1
wβ = ∇I ∗ [βu, q ⊗ p] = e 2 [βu−ΨA (β)⊕0B ] (q ⊗ p)e 2 [βu−ΨA (β)⊕0B ] β −1 = u0 (λ) ← I[wβ , q ⊗ p] = λ
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
24 / 27
Variational Problems
Third Variational Problem Type III problem H = HA ⊗ H B , u ∈ A ⊗ B u(λ) = sup{hu, wi : I[w, q ⊗ p] ≤ λ} −1
u
(υ) = inf{I[w, q ⊗ p] : hu, wi ≥ υ}
Theorem (Solution) 1
1
wβ = ∇I ∗ [βu, q ⊗ p] = e 2 [βu−ΨA (β)⊕0B ] (q ⊗ p)e 2 [βu−ΨA (β)⊕0B ] β −1 = u0 (λ) ← I[wβ , q ⊗ p] = λ D 1 E 1 eΨA (β) = e 2 βu (1A ⊗ p)e 2 βu B
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
24 / 27
Variational Problems
Third Variational Problem Type III problem H = HA ⊗ H B , u ∈ A ⊗ B u(λ) = sup{hu, wi : I[w, q ⊗ p] ≤ λ} −1
u
(υ) = inf{I[w, q ⊗ p] : hu, wi ≥ υ}
Theorem (Solution) 1
1
wβ = ∇I ∗ [βu, q ⊗ p] = e 2 [βu−ΨA (β)⊕0B ] (q ⊗ p)e 2 [βu−ΨA (β)⊕0B ] β −1 = u0 (λ) ← I[wβ , q ⊗ p] = λ D 1 E 1 eΨA (β) = e 2 βu (1A ⊗ p)e 2 βu B D 1 E 1 [βu−Ψ (β)⊕0 ] A B 1B = e2 (q ⊗ 1B )e 2 [βu−ΨA (β)⊕0B ] Roman Belavkin (Middlesex University)
Variational Quantum Entropy
A
September 24, 2014
24 / 27
Variational Problems
Fourth Variational Problem Type III+I=IV problem: I[w, q ⊗ q] = I[w, q ⊗ p] + I[p, q] H = HA ⊗ H B , u ∈ A ⊗ B
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
25 / 27
Variational Problems
Fourth Variational Problem Type III+I=IV problem: I[w, q ⊗ q] = I[w, q ⊗ p] + I[p, q] H = HA ⊗ H B , u ∈ A ⊗ B u(λ) = sup{hu, wi : I[w, q ⊗ q] ≤ λ}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
25 / 27
Variational Problems
Fourth Variational Problem Type III+I=IV problem: I[w, q ⊗ q] = I[w, q ⊗ p] + I[p, q] H = HA ⊗ H B , u ∈ A ⊗ B u(λ) = sup{hu, wi : I[w, q ⊗ q] ≤ λ} −1
u
(υ) = inf{I[w, q ⊗ q] : hu, wi ≥ υ}
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
25 / 27
Variational Problems
Fourth Variational Problem Type III+I=IV problem: I[w, q ⊗ q] = I[w, q ⊗ p] + I[p, q] H = HA ⊗ H B , u ∈ A ⊗ B u(λ) = sup{hu, wi : I[w, q ⊗ q] ≤ λ} −1
u
(υ) = inf{I[w, q ⊗ q] : hu, wi ≥ υ}
Theorem (Solution) 1
1
wβ = ∇I ∗ [βu, q ⊗ p] = e 2 [βu−ΨA (β)⊕0B ] (q ⊗ p)e 2 [βu−ΨA (β)⊕0B ] β −1 = u0 (λ) ← I[wβ , q ⊗ q] = λ
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
25 / 27
Variational Problems
Fourth Variational Problem Type III+I=IV problem: I[w, q ⊗ q] = I[w, q ⊗ p] + I[p, q] H = HA ⊗ H B , u ∈ A ⊗ B u(λ) = sup{hu, wi : I[w, q ⊗ q] ≤ λ} −1
u
(υ) = inf{I[w, q ⊗ q] : hu, wi ≥ υ}
Theorem (Solution) 1
1
wβ = ∇I ∗ [βu, q ⊗ p] = e 2 [βu−ΨA (β)⊕0B ] (q ⊗ p)e 2 [βu−ΨA (β)⊕0B ] β −1 = u0 (λ) ← I[wβ , q ⊗ q] = λ
Theorem (Entropy bound) I[w, q ⊗ q] ≤ H(A) Roman Belavkin (Middlesex University)
⇐⇒
I[p, q] ≤ H(A | B)
Variational Quantum Entropy
September 24, 2014
25 / 27
Variational Problems
Summary
Different definitions of a function in classical probability (CP) may coincide, but they may differ in quantum probability (QP).
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
26 / 27
Variational Problems
Summary
Different definitions of a function in classical probability (CP) may coincide, but they may differ in quantum probability (QP). Entropy is one such example.
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
26 / 27
Variational Problems
Summary
Different definitions of a function in classical probability (CP) may coincide, but they may differ in quantum probability (QP). Entropy is one such example. von Neumann’s entropy S appears to be a natural generalization, but it only represents classical component of quantum information.
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
26 / 27
Variational Problems
Summary
Different definitions of a function in classical probability (CP) may coincide, but they may differ in quantum probability (QP). Entropy is one such example. von Neumann’s entropy S appears to be a natural generalization, but it only represents classical component of quantum information. Proper quantum entropy H is based on variational definition, and it has all the desired geometric properties.
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
26 / 27
Variational Problems
Summary
Different definitions of a function in classical probability (CP) may coincide, but they may differ in quantum probability (QP). Entropy is one such example. von Neumann’s entropy S appears to be a natural generalization, but it only represents classical component of quantum information. Proper quantum entropy H is based on variational definition, and it has all the desired geometric properties. Definitions of quantum cross-entropy and cross-information have been presented alongside the corresponding variational problems.
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
26 / 27
References
Motivation Crash Course in Quantum Probability Quantum Information and von Neumann Entropy Proper Quantum Entropy Variational Problems
Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
27 / 27
References
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., & Ohya, M. (2002, January). Entanglement, quantum entropy and mutual information. Royal Society of London Proceedings Series A, 458(209). Khinchin, A. I. (1957). Mathematical foundations of information theory. New York: Dover. Kolmogorov, A. N. (1933). Grundbegriffe der wahrscheinlichkeitsrechnung. Berlin: Julius Springer. (in German) Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. The Annals of Mathematical Statistics, 22(1), 79–86. Neumann, J. von. (1932). Mathematische Grundlagen der Quantenmechanik. (German) [Mathematical foundations of quantum mechanics]. Berlin: Springer-Verlag. Stratonovich, R. L. (1965). On mutual information and the capacity of quantum channels. Izvestia Vuzov: Radiophysics, 4, 15–24. (In Russian) Roman Belavkin (Middlesex University)
Variational Quantum Entropy
September 24, 2014
27 / 27
Variational Problems
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)
Variational Quantum Entropy
September 24, 2014
27 / 27
