On Entropy for Quantum Compound Systems Noboru Watanabe Department - PowerPoint PPT Presentation

On Entropy for Quantum Compound Systems Noboru Watanabe Department of Information Sciences Tokyo University of Science Noda City, Chiba 278-8510, Japan E-mail : watanabe@is.noda.tus.ac.jp 51 th Symposium on Mathematical Physics, 16-18 of June, 2019, N. Copernicus Univ.,Torun, Poland

Introduction 1. Information Dynamics (M. Ohya) 2. Complexity (C) 3. Channel 4. Transmitted Complexity (T) 5. Other mutual entropy type complexities (T) How to construct compound states

Quantum Physics Quantum Probability Operator Algebra Quantum Entropy Information Dynamics Chaos Dynamical Entropy Information Genetics Quantum Information Quantum Communication

1. Information Dynamics [M. Ohya] Information dynamics is an attempt to provide a new view for the study of complex systems and their chaotic behavior Information Dynamics = Synthesis of dynamics of state change and complexity of state Two Complxities ( ) ( ) ϕ ;Λ ∗ S T ϕ C S (1)Complexity of a state (2) complexity of a dynamics describing the system describing the state change Complexity of state Transmitted complexity Examples of Complexity of state Examples of Transmitted Complexity entropy, fractal dimension, chaos degree, Lyapunov exponent, ergodicity, etc. dynamical entropy, computational complexity, etc.

1. Information Dynamics [M. Ohya] Information Dynamics ( ) ( ) ( ) ⇔ α α Λ ϕ ϕ Λ * * , , , ; , , , ; ; C , T ; ; R S S A S S A S S whe re is a certain relation among above quantitie R s. Therefore, in Information Dynamics we have to α α (i) mathematically determine , , , ; , , , , S S A S A S Λ * (ii) choose and , R ( ) ( ) ϕ ϕ Λ * (iii) define C , T ; . S S One can apply several fields including (1) Recognition of Chaos, (2) Quantum SAT Algorithm, (3) From DNA (Amino Acid Sequences) to Life Science (4) Quantum Information Communication

1. Information Dynamics [M. Ohya] Property of Two Complexities in ID [M. Ohya] C and T should satisfy the following properties : ( ) ( ) ( ) ϕ ≥ ϕ Λ ≥ ϕ ∈ ⊂ * 1 C 0 and T ; 0 for any S S S. S ( ) → 2 For any bijection : j ex e x (the set of all extremal elements of ), S S S ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) • j ϕ = ϕ j ϕ Λ = ϕ Λ * * S S C j C S , T j ; T S ; ( )     ϕ ∈ ⊂ ϕ ≡ ϕ ↑ ∈ = ↑ ϕ ≡ ϕ ↑ ∈ = ↑    3 For any pu t , , S S, A S S A A S S A ( ) ( ) ( ) ( ) ( ) ( )  ⊗ • ϕ ≤ ϕ + ϕ ϕ ⊗ ϕ = ϕ + ϕ  C C C , C C C S S S S S S S ( ) ( ) ( ) ≤ ϕ Λ ≤ ϕ * 4 0 T ; C S S ( ) ( ) ( ) ϕ = ϕ = 5 T ; id C , i d identity channel S S

1. Shannon’s Type Inequality r Compound states ( ) ( ) Λ S p ( ) * S p I p Λ * ; ( ) I p Λ * ; Mutual Entropy (Information) Shannon’s Type inequalities ( ) ( ) ( ) ≤ Λ = ⊗ Λ ≤ * * 0 I p ; S r , p p S p

Introduction 1. Information Dynamics (M. Ohya) 2. Complexity (C) 3. Channel 4. Transmitted Complexity (T) 5. Other mutual entropy type complexities (T) How to construct compound states

2. Complexity (C) (Entropy Type ) 1) Shannon Entropy [Shannon] ( ) ( ) ≡ − ∑ ⇔ S C p S p p log p 1 k k k Entrop of finite decomposition (straight extension of 2) Shannon entropy) for Gaussian measure     ∑ ( ) ( ) ( ) ( )  ( ) µ ⇔ µ = − µ µ ∈ S C S sup A log A ; ,   A P B 2 k k 2      ∈ A A k ( ) where is the set of all finite partitions of . P B B 2 2 3) Differential entropy [Gibbs] µ µ d d ( ) ( ) = − ∫ R µ ⇔ µ S C S log dm 3 dm dm 2

2. A. Entropy Type Complexity 4) Von Neumann Entropy [von Neumann] ( ) ( ) ρ ⇔ ρ ≡ − ρ ρ S C S tr log 4 5) S- mixing entropy [M. Ohya 1985] (CNT entropy 1987) ( )  µ ; µ ∈ ≠ ∅ inf{ H ( ) D ( )} D ( ) , S S  ϕ ϕ ( ) ρ ⇔ ϕ =  S C S ( ) S ( ) 5 ∞ = ∅ D ( ) .  S  ϕ

5) S -mixing entropy [Ohya] ( ) , ( ) ; a C*-system, A S A ⊂ ( ); ( ) a weak* compact convex subset of S S A S A = ∫ ϕ ∈ S ϕ ω µ ; d It is decomposed such as S where μ is maximum measures pseudosupported on exS (the set of all extremal points) in S . The measure μ giving the above decomposition is not unique. ( ) ; the set of all such measures for ϕ ∈ S M ϕ S  { ∑ ∑ ≡ µ ∈ ; ∃ µ ⊂ + ϕ ⊂ . . µ = , µ = µ δ ϕ , D ( ) M ( ) { } R and { } ex s t 1 ( )  S S S ϕ ϕ k k k k k  k k { } . ( ) ϕ δ ϕ where is Dirac measure concentrated on = − ∑ ( ) H µ µ µ µ ∈ D ϕ ( ) log For a measure , S k k k ϕ ∈ S S -mixing entropy of a general state w.r.t. is defined by S ( )  µ ; µ ∈ ≠ ∅ inf{ H ( ) D ( )} D ( ) , S S  ϕ ϕ ( ) ρ ⇔ ϕ =  S C S ( ) S ( ) 5 ∞ = ∅ D ( ) .  S  ϕ

5) S -mixing entropy [Ohya] Theorem [Ohya] Properties of S -mixing entropy α = ∗ = B α = ( ) A U AU A ∈ A If and ( ) Ad U ( ) (i.e., for any ) with a unitary A H t t t t t ϕ ϕ ⋅ = ρ ⋅ operator , for any state , given by with a density operator ρ , the ( ) tr U t following facts hold: ( ) ϕ = − ρ ρ = ρ 1. S ( ) tr log S ; v.N. entropy ϕ 2. If is an α -invariant faithful state and every eigenvalue of ρ is non- α ϕ ( ) I α degenerate , then where is the set of all α -invariant faithful I ( ) ( ) = ϕ , S S ( ) states. α ϕ = ( ) ( ) ( ) ϕ ∈ α K K α 3. , then , where is the set of all KMS states. K ( ) S 0 ϕ ∈ α Theorem [Ohya] For any , we have K ( ) ( ) α 1. K ϕ ≤ α ϕ I ( ) S ( ) S ( ) α ϕ ( ) ( ) K ≤ ϕ 2. S S ( ) (1) This - mixing entropy gives a measure of the uncertainty observed from the S reference system. (2) This entropy can be applied to characterize normal states and quantum Markov chains in von Neumann algebras.

Introduction 1. Information Dynamics (M. Ohya) 2. Complexity (C) 3. Channel 4. Transmitted Complexity (T) 5. Other mutual entropy type complexities (T) How to construct compound states

Communication Process Classical Quantum Input Input System System Ξ * Λ * Γ * Classical Quantum Output Output ˜ System System Ξ *

3. Channel A. Classical Channel 1) Transition Probability Matrix   n ∑ { } ( ) Λ ∆ → ∆ ∆ = = ≥ ∀ = * : , p p ,  , p ; p 0 i , p 1   n m n 1 n i i   = i 1 ( ) ( ) ( ) ( )   p 1|1 p 1| 2 p 1| 3  p 1| n   ( ) ( ) ( ) ( ) p 2 |1 p 2 | 2 p 2 | 3  p 2 | n   m ∑ ( ) ( ) ( ) ( ) ( ) ( )   Λ = * = = p 3|1 p 3| 2 p 3| 3  p 3| n , p j k | 1, k 1,2,  , n   = j 1          ( ) ( ) ( ) ( ) p m |1 p m | 2 p m | 3  p m n |   2) Gaussian channels ( ) ( ) ( ) ( ) Γ 1 → 2 1 2 : P P ; a map from P to P is defined by the Gaussian channel G G G G [ ] ( )( ) ( ) ( ) ∫ λ × → Γ µ ≡ λ µ : 0,1 such as Q x Q d , x H B 1 2 1 1 H 1 ( ) ( ) { } λ ≡ µ x x ≡ ∈ + ∈ ∈ ∈ x Q , Q , Q y ; Ax y Q , x , Q , H H B 0 2 1 2 λ where is a linear transformation from A to , s t sf a i i es H H 1 2 the following conditions: ( ) ( ) λ • ∈ 2 ∈ (1) x , P for exch fixed x H G 1 ( ) ( ) λ • ∈ (2) , Q is a measurable function on , for each fixed Q H B B 1 1 2

B. Quantum Channel ( ) k k = ; the set of v.N. alg. on 1,2 A H k ( ) ( ) k k = ; the set of all normal state on 1,2 S A A k ( ) ( ) • Λ → * : ; Quantum Channel S A S A 1 2 Λ * (1) satisfying the affine property   ∑ ∑ ∑ ( ) ( ) ( ) ∗ ∗ λ = λ ≥ ⇒ Λ λ ϕ = λ Λ ϕ ,∀ ϕ ∈ 1 0 S A   k k k k k k k 1   k k k is called a linear Channel ∗ Λ Λ (2) Predual map of satisfying the completely positivity n ( ) ∑ ∗ Λ * ≥ ∀ ∈ ∀ ∈ ∀ ∈ B A A B 0, n N , B , A A A j j k k k 1 k 2 = j k , 1 ( ) is called a completely positive CP channel

B. Quantum Channel

B. Quantum Channel

B. Quantum Channel Open System Dynamics = ∑ * H b b 2 j j = j 1

B. Quantum Channel Open System Dynamics

B. Quantum Channel Open System Dynamics

B. Quantum Channel as

B. Quantum Channel Attenuation channel and beam Splitter [Ohya, 1983] 0 0 π beam splitter * 0 θ θ αθ αθ − β θ − β θ

B. Quantum Channel Noisy optical channel and generalized beam Splitter[Ohya NW,1984]