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

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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

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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

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Information Dynamics

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

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Information Dynamics = Synthesis of dynamics of state change and complexity of state

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 (1)Complexity of a state describing the system Complexity of state (2) complexity of a dynamics describing the state change Transmitted complexity Examples of Complexity of state entropy, fractal dimension, ergodicity, etc.

Two Complxities

( )

C ϕ

S

( )

S

T ϕ

∗

;Λ

Examples of Transmitted Complexity chaos degree, Lyapunov exponent, dynamical entropy, computational complexity, etc.

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( )

* *

whe , , , ; , , , ; ; re is a certain relation among above quantitie ; s. , ; C T R R α α ϕ ϕ Λ Λ ⇔ Information Dynamics S S

S S

A S A S

( )

* *

(i) , , , ; , , , , (ii) , (iii) , ; . R C T α α ϕ ϕ Λ Λ Therefore, in Information Dynamics we have to mathematically determine choose and define S S

S S

A S A S

One can apply several fields including

1. Information Dynamics [M. Ohya]

(1) Recognition of Chaos, (2) Quantum SAT Algorithm, (3) From DNA (Amino Acid Sequences) to Life Science (4) Quantum Information Communication

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( ) ( )

( )

( ) ( )

( )

* * *

1 and for any 2 For any : (the set of all

f

), , 3 For any pu ; ; ;

j j

C C T j ex e T T x T j C j C ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ∈ ⊂ →

∈

≥ Λ ≥ ⊂ = Λ = Λ and should satisfy the following properties : bijection extremal elements

S S S S S S

S S S. S S S S,   

( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( )

*

t , , , 4 ; 5 , ; C C C C C T i id C C d C T ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ

⊗

≤ + ⊗ = ≤ ≡ ↑ ∈ = ↑ ≡ ↑ = ∈ = ↑

=

≤ + Λ identity channel

S S S S S S S S S S S

A S S A A S S A



    

Property of Two Complexities in ID [M. Ohya]

1. Information Dynamics [M. Ohya]

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1. Shannon’s Type Inequality

Mutual Entropy (Information)

( )

S p

( )

*

S p Λ

( )

*

; I p Λ

( ) ( )

( )

* *

, ; r I S p S p p p Λ = ⊗ ≤ ≤ Λ

Shannon’s Type inequalities

r

( )

*

; I p Λ

Compound states

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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

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2. Complexity (C) (Entropy Type )

1) Shannon Entropy [Shannon]

( ) ( )

1

log

S k k k

C p S p p p ⇔ ≡ −∑

2)

( ) ( ) ( ) ( ) ( ) ( )

2 2 2 2

sup log ; , where is the set of all finite partitions of .

k

S k k A

C S A A µ µ µ µ

∈

    ⇔ = − ∈      

∑

Entrop of finite decomposition (straight extension of Shannon entropy) for Gaussian measure

A

A P B P B B



( ) ( )

2

3

log

S

d d C S dm dm dm µ µ µ µ ⇔ = −∫R Differential entropy [Gibbs]

3)

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4) Von Neumann Entropy [von Neumann]

( ) ( )

4

log

S

C S tr ρ ρ ρ ρ ⇔ ≡ −

5) S-mixing entropy [M. Ohya 1985] (CNT entropy 1987)

2. A. Entropy Type Complexity

( )

( ) ( )

5

inf{ ( ) ( )} ( ) , ( ) ( ) .

S

H D D C S D

ϕ ϕ ϕ

µ µ ρ ϕ  ; ∈ ≠ ∅  ⇔ =  ∞ = ∅  

S

S S S

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5) S -mixing entropy [Ohya]

( )

( ) ; , A A S

a C*-system,

( ); ⊂ S A S

a weak* compact convex subset of

( ) A S

; ϕ ∈S

It is decomposed such as

d ϕ ω µ = ∫

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.

( );

Mϕ S

the set of all such measures for ϕ ∈S

{

( ) ( ) { } { } ex 1 ( )

k k k k k k k

D M and s t

ϕ ϕ

µ µ ϕ µ µ µ δ ϕ

+

 ≡ ∈ ; ∃ ⊂ ⊂ . . = , = ,  

∑ ∑

R S S S

where is Dirac measure concentrated on

( )

δ ϕ

{ }.

ϕ

For a measure ,

( )

Dϕ µ ∈ S ( ) log

k k k

H µ µ µ = −∑

S-mixing entropy of a general state w.r.t. is defined by

ϕ ∈S

S

( )

( ) ( )

5

inf{ ( ) ( )} ( ) , ( ) ( ) .

S

H D D C S D

ϕ ϕ ϕ

µ µ ρ ϕ  ; ∈ ≠ ∅  ⇔ =  ∞ = ∅  

S

S S S

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Properties of S-mixing entropy Theorem [Ohya]

If and (i.e., for any ) with a unitary

perator , for any state , given by with a density operator ρ , the

following facts hold: 1.

2. If

is an α-invariant faithful state and every eigenvalue of ρ is non- degenerate, then where is the set of all α-invariant faithful states.

3. , then , where is the set of all KMS states.

ϕ

( ) = B A H ( )

t t

Ad U α =

( )

t t t

A U AU α

∗

=

A∈A

t

U ( ) tr ϕ ρ ⋅ = ⋅

( )

( ) log ; v.N. entropy S tr S ϕ ρ ρ ρ = − =

ϕ

( )( )

( )

I

S S

α ϕ

ϕ = ,

( )

I α ( ) K ϕ α ∈

( )( )

K

S

α ϕ =

( )

K α

5) S -mixing entropy [Ohya]

Theorem [Ohya] For any , we have

( ) K ϕ α ∈

1. 2.

(1) This - mixing entropy gives a measure of the uncertainty observed from the reference system. (2) This entropy can be applied to characterize normal states and quantum Markov chains in von Neumann algebras.

( )

( ) ( )

K I

S S

α α

ϕ ϕ ≤

( )( )

( )

K

S S

α ϕ

ϕ ≤

S

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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

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Communication Process

Quantum Input System Quantum Output System Classical Output System Classical Input System

Λ

*

Ξ

*

˜ Ξ

* *

Γ

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3. Channel

1) Transition Probability Matrix

{ } ( )

* 1 1

: , , , ; , 1

n n m n n i i i

p p p p i p

=

  Λ ∆ → ∆ ∆ = = ≥ ∀ =    

∑



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

* 1

1|1 1| 2 1| 3 1| 2 |1 2 | 2 2 | 3 2 | 3|1 3| 2 3| 3 3| , | 1, 1,2, , |1 | 2 | 3 |

m j

p p p p n p p p p n p p p p n p j k k n p m p m p m p m n

=

        Λ = = =        

∑

         

A. Classical Channel

( ) ( ) ( ) ( )

[ ] ( )( ) ( ) ( ) ( )

( )

{ }

1

1 2 1 2 1 2 1 1 2 1 2 1 2

: ; a map from to is defined by the : 0,1 such as , , , ; , , , where is a linear transformation from to , Gaussian channel s t sf a i i

G G G G x x

Q x Q d x x Q Q Q y Ax y Q x Q A µ λ µ λ λ λ µ → × → Γ ≡ ≡ ≡ ∈ + ∈ Γ ∈ ∈

∫

P P P P

H

H B H H B H H

( )

( ) ( )

2 1 1 1 2

es the following conditions: (1) , for exch fixed (2) , is a measurable function on , for each fixed

G

x x Q Q λ λ

∈

∈

∈

P H H B B

2) Gaussian channels

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B. Quantum Channel

( )

; the set of v.N. alg. on 1,2 A H

k k k =

( ) ( )

* 1 2

: ; S S A A

Λ

→ Quantum Channel

*

(1) satisfying the Λ affine property

( ) ( ) ( )

1

1 S A

k k k k k k k k k k

λ λ λ ϕ λ ϕ ϕ

∗ ∗

  = ≥ ⇒ Λ = Λ ,∀ ∈    

∑ ∑ ∑

is called a linear Channel (2) Predual map

f

satisfying the

∗

Λ Λ completely positivity

( )

* 1 2 , 1

0, , , A A

n j j k k k k j k

B A A B n B A

∗ =

Λ ≥ ∀ ∈ ∀ ∈ ∀ ∈

∑

N

( )

is called a completely positive CP channel

( ) ( )

; the set of all normal state on 1,2 S A A

k k k =

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B. Quantum Channel

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B. Quantum Channel

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B. Quantum Channel

Open System Dynamics

* 2 1 j j j

H b b

=

= ∑

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B. Quantum Channel

Open System Dynamics

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B. Quantum Channel

Open System Dynamics

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as

B. Quantum Channel

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B. Quantum Channel

0 0 θ θ αθ αθ −β θ −β θ

*

π beam splitter

Attenuation channel and beam Splitter [Ohya, 1983]

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

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κ κ θ θ αθ βκ αθ βκ βθ ακ βθ ακ + + − + − +

*

π generalized beam splitter

the CP channel . The mathemetical formulations of beam splitting are π

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

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B. Quantum Channel

SLIDE 27

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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

SLIDE 29

B. Transmitted Complexity (T)

1) Mutual Entropy [Shannon]

( ) ( )

( )

* 1 ,

; log ,

ij S ij i j i j

r T p I p r S r p q p q

∗

;Λ ⇔ Λ ≡ = ⊗

∑

( ) ( ) ( ) ( )

1 2

1 * 2 1 1 12 1 2 12 12 1 2 12 1 2 1 2 1 2 1

with respect to and is defined by the such as I ; | lo ( ) [ ] g

S

T S d d d d d µ µ λ µ λ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ

×

Γ ; ⇔ = ⊗ ⊗ ⊗ ⊗ ⊗ = ∞

∫

Kullback -Leibler infor Mutual entropy information GKY mation

H H



( ) ( ) ( ) ( )

2 1 2 12 12 1 2 1 2 2 3 1 2 1 1

where is the Radon-Nikodym derivative (1) , (2)

f

w.r.t.

S S S

d C T C d µ µ µ µ µ µ µ µ λ µ µ µ     < = ⊗  ⊗ ∞ ⊗ ; + Theorem [OW] 

2)

( )

{ }

1 1 1

min ,

S S S

T p C p C p

∗ ∗

≤ ;Λ ≤ Λ

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B. Transmitted Complexity (T)

3) Ohya Mutual Entropy for density operator [Ohya] In general, there does not exist the joint states in the quantum systems [Urbanik] How to define the quantum mutual entropy?

( ) ( )

;

ne dimensional orthogonal decomposition of ρ

Schatten decomposition is not uniqe in general

n n n E n n n n

E E E ρ λ ρ σ λ

∗

= = ⊗ Λ

∑ ∑

Quantum System Schatten decomposition of Remark Ohya Compound state  

( )

{ }

Classical mutual entropy constructed by .

ij

r r ⇐ = Classical System Joint prob measure (by Jamiolkowski isometric channel)

n n n E n n n n n

id E E E E λ σ λ

∗ ∗

  ⊗ Λ ⊗ → = ⊗ Λ    

∑ ∑

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B. Transmitted Complexity (T)

3) Ohya Mutual Entropy for density operator [Ohya]

( ) ( ) ( )

{ }

* 3

sup ,

S E E

T I S ρ ρ σ ρ ρ

∗ ∗

;Λ ⇔ ;Λ ≡ ⊗ Λ

(Ohya compound state) (Trivial compound state)

E n n n n

E E σ λ σ ρ ρ

∗ ∗

= ⊗ Λ = ⊗ Λ

∑

Two compound states

( )

{ }

* 3 4 4

min ,

S S S

T C C ρ ρ ρ

∗

≤ ;Λ ≤ Λ

( ) ( ) ( )

* *

* 1 1 2 , * * ,

in the sense of Accardi and Ohya [AO] : (nonlinear lifting)

E E n n n E n

E E ρ σ λ

Λ Λ

→ ⊗ → = ⊗ Λ

∑

Lifting S S E H H H E

Theorem [Ohya] (1) (2) If the system is classical, then the quantum mutual entropy equals to the classical mutual entropy. (3)

( ) ( ) ( )

3 4

von Neumann entropy

S S

T id C ρ ρ ; = Relative Entropy [Umegaki]

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B. Transmitted Complexity (T)

3) Ohya Mutual Entropy for density operator [Ohya]

( ) ( ) ( )

{ }

* 3

sup ,

S E E

T I S ρ ρ σ ρ ρ

∗ ∗

;Λ ⇔ ;Λ ≡ ⊗ Λ

(Ohya compound state) (Trivial compound state)

E n n n n

E E σ λ σ ρ ρ

∗ ∗

= ⊗ Λ = ⊗ Λ

∑

Two compound states

( )

{ }

* 3 4 4

min ,

S S S

T C C ρ ρ ρ

∗

≤ ;Λ ≤ Λ

( )

* * * * *

(1) If is , then ; . (2) If is , then ; 0. (3) If is faithful stationary state and all eigenvales of are not degenerate, and is , the I S I ρ ρ ρ ρ ρ Λ Λ = Λ Λ = Λ deterministic chaio T tic er heorem [ godi O y c h a]

( ) ( )

* *

n ; . I S ρ ρ Λ = Λ

Relative Entropy [Umegaki]

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S ρ,σ

( )≡

trρ logρ − logσ

( )

s ρ

( )≤ s σ

( )

∞

therwise

( )

  

support projection of σ

It was extended by Araki (for von Neumann alg.) and by Uhlmann (for *-algebra).

B. Transmitted Complexity (T)

Relative Entropy for density operator [Umegaki]

5) 6)

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B. Transmitted Complexity (T)

4) Ohya Mutual Entropy for general C*- systems [Ohya]

( ) { }

4

( ) limsup ( ) ( )

S S S

T I I F S

ε µ ϕ ε

ρ ϕ ϕ µ

∗ ∗ ∗ →

;Λ ⇔ ;Λ = ;Λ ; ∈

(Ohya compound state) (Trivial compound state)

S

d

µ

ω ω µ ϕ ϕ

∗ ∗

Φ = ⊗ Λ Φ = ⊗ Λ

∫

Two compound states for general C*-systems

S

( )

{ }

* 4 5 5

min ,

S S S

T C C ϕ ϕ ϕ

∗

≤ ;Λ ≤ Λ

( ) ( ) ( )

4 5

S - mixing entropy

S S

T id C ϕ ϕ ; = Relative Entropy [Araki & Uhlmann]

( ) ϕ ∈ ⊂ S A S

and

( ) ( ),

∗

Λ : → A A S S

For

( ) ( ) I S

µ µ

ϕ

∗

;Λ = Φ ,Φ

S S

where

{ }

( ) ( ) ( ) ( ) ( ) ( ) ( ) D S H S F M if S

ϕ ε ϕ ϕ

µ ϕ µ ϕ ε ϕ  ∈ ; ≤ ≤ + < +∞  =  = +∞  

S S S

SLIDE 35

B. Transmitted Complexity (T)

4) Ohya Mutual Entropy for general C*- systems [Ohya] Ohya quntum mutual entropy

Levitin and Holevo’s (LH for short) semi-quantum mutual entropy Thus the Ohya’s q. mutual entropy contaions the LH s-q. mutual entropy as a special case.

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Entropy and mutual entropy type complexities of Classical and Quantum Systems

← ↑ ← Semi -quantum ME Ohya ME Shannon's ME Ohya ME ME for GQ (GKY) S  

← ↑ ← Semi -classical E v.N. E Shannon's E E Ohya

mixing E for G

(D,Finite P.) QS S  

( )

1 S

T p

∗

;Λ

( )

1 S

C p

( )

2 S

T µ

∗

;Λ

( ) ( )

2 3

,

S S

C C µ µ

( )

3 S

T ρ

∗

;Λ

( )

4 S

C ρ

( )

5 S

C ϕ

( )

4 S

T ϕ

∗

;Λ

Entropy type complexities of Classical and Quantum Systems Mutual Entropy type complexities of Classical and Quantum Systems

S-mixing entropy of quantum channel Generalized by Prof. Mukhamedov and NW S-mixing Renyi entropy Generalized by Prof. Mukhamedov, Ohmura and NW

SLIDE 37

(3) Information Transmission of Gaussian Communication Process

Information Transmission of Gaussian Communication Process

Input state
Gaussian Channel
Entropy of Finite partition
Differential Entropy
Mutual Entropy by GKY
Inequality

Information Transmission of General Quantum Communication Process

Input state
Quantum channel
C*-mixing Entropy
State Decomposition
Quantum Mutual Entropy
Inequality

( )

1 G

P µ ∈

λ

( ) ( ) ( ) ( ) ( )

2

sup log ;

k

k S k A

S C A A µ µ µ µ

∈

    ⇔ ≡ − ∈ Ω      

∑

A A

A P

 



( ) ( )

3

log

S

d d S dm dm C dm µ µ µ µ ⇔ ≡ −∫R

( ) ( ) ( )

1 2 2 2 1 1 1

; | ;

S

I S T µ λ µ µ µ µ λ ⇔ = ⊗ ( ) ( )

2 1 2 1

;

S S

T C µ λ µ ≤ = +∞

( ) ( )

2 1 3 1

;

S S

T C µ λ µ ≥

( )

1

S A ϕ ∈

*

Λ

( ) ( ) ( )

{ }

4

inf ; ( )

S S

S H C v D S

ϕ

ϕ ν ϕ ⇔ = ∈

{ }

4

( ) limsup ( ) ( ) ( )

S

T I I F ε

µ ϕ ε

ϕ ϕ ϕ µ

∗ ∗ → ∗ ⇔

;Λ = ;Λ ; ∈ ;Λ

S S

S

d ϕ ω ν = ∫

( )

* 4 5

;

S S

T C ϕ ϕ ≤ Λ ≤

SLIDE 38

New Treatment of Information Transmission of Gaussian Communication Process

Input state
Gaussian Channel
Entropy Func. Str. Eq.
Mutual Ent. Func. Str. Eq.
Inequality
Input state
Quantum channel
V.N. Entropy
Schatten Decomposition
Quantum Mutual Entropy
Inequality

( )

1 G

P µ ∈

λ ( )

1

ρ ∈ H S

*

Λ ( ) ( )

4

log

S

S tr C ρ ρ ρ ρ = − ⇔

( ) ( ) ( )

* 3 * *

; s ; up ,

E E S

T I S ρ σ ρ ρ ρ Λ = ⊗ Λ Λ ⇔

n n n n n n n

E x x ρ λ λ = =

∑ ∑

( )

* 3 4

;

S S

T C ρ ρ ≤ Λ ≤

( ) ( ) ( ) ( ) ( ) ( )

* * 1 1 1 1 1 * * 1 1 1 1 1

log .

S SE SE

S tr C tr tr µ µ µ µ µ µ Ξ Ξ = −     Ξ Ξ    ⇔   ( ) ( ) ( ) ( ) ( ) ( )

* * * 1 1 1 1 1 * * * 1 1 1 1 1

; sup , ;

SE E S SE E

I S t T r µ µ µ λ σ µ µ µ λ   Ξ ⊗ Π Ξ =       Ξ ⊗ Π Ξ     ⇔   

( ) ( )

1 1

;

S S SE SE

T C µ λ µ ≤ ≤

New Treatment of Information Transmission of Gaussian Communication Process Information Transmission of Quantum Communication Process

SLIDE 39

SLIDE 40

( )( ) ( )( )

* ' 1*,

is hold for any : A I I ϕ ϕ ϕ

+

=

∈

Θ (1) Linearity condition (Linear Approximation) (2 Treatment [Ohya and Watanabe] ) Trace preserving condition

ϕ

( )

* ϕ

Θ

' 2*,

A

+ ' 1*,

A

+

( )

I ϕ

( )( )

*

I ϕ Θ

( )( ) ( )( )

*

I I ϕ ϕ Θ =

( )

*

I A A trR α α = −

( )

*

1 A A trR I = −

( )

1 I ϕ =

( )

I ϕ α =

(Ohya & NW) (Makiwara & NW)

SLIDE 41

( ) ( )

* * ' 1*,

( ) ' ( )is hold for any ' satisfying ( ) : '( ) A I I , I I ϕ ϕ ϕ ϕ ϕ ϕ

+

Θ = Θ =

∈

Weak trace pres (1) Linearity condit erving condi ion (Linear Approximat A new treatment [Wa io t n) (2) anabe] tion

' 1*,

A

+ ' 2*,

A

+

ϕ ' ϕ

( )

* ϕ

Θ

( )

* ϕ

Θ ( ) '( ) I I ϕ ϕ =

( ) ( )

* *

( ) ' ( ) I I ϕ ϕ Θ = Θ

( )

*

I A A trR α α = −

( )

*

1 A A trR I = −

( )

1 I ϕ =

( )

I ϕ α =

(NW) (NW)

SLIDE 42

SLIDE 43

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

SLIDE 44

1) Ohya mutual entropy w.r.t. [Ohya,1983, 1989]

( ) ( ) ( )

{ }

* * * 4

; sup , ;

Araki ex

T I S d d ϕ ϕ ω ϕ µ ϕ ω µ

∗

;Λ ⇔ Λ ≡ Λ Λ =

∫ ∫

S

S S

*

, ϕ Λ

2) Lindblad-Nielsen’s entropy w.r.t.

( ) ( ) ( )

( )

* * * 5

; ,

S

L N e

T I S S S ϕ ρ ρ ρ ρ

∗ −

;Λ ⇔ Λ ≡ Λ + − Λ

3) Coherent information w.r.t.

*

, ρ Λ

*

, ρ Λ

( ) ( ) ( ) ( )

* * * 6

; ,

S

C e

T I S S ϕ ρ ρ ρ

∗

;Λ ⇔ Λ ≡ Λ − Λ

In the above two cases, the entropy exchange w.r.t. is defined as follows: When is given by

( )

*

,

e

S ρ Λ

*

, ρ Λ

*

Λ

( )

*

, ,

e

S S W ρ Λ ≡

( )

* * *

,

i i i i i i

A A A A I ρ ρ   Λ ≡ =    

∑ ∑

( )

*

,

ij ij i j

W W W trA A ρ ≡ ≡

5. Other Mutual Entropy Type Complexities

SLIDE 45

A. Comparison among these quantum mutual entropy type complexities

Theorem [OW] For the attenuation channel

( )

n n j j i i j i

A Q VW z z n V y y

= =

      ≡ = ⊗ ⊗              

∑ ∑

ne can obtain the following results for any input states

( )

{ }

( )

( ) ( )

( )

* * 3 4 4 * 5 * 6

1 min , , 2 ; , 3 ; 0.

S S S S S

T C C T S T ρ ρ ρ ρ ρ ρ ≤ ;Λ ≤ Λ Λ = Λ =

( )

* * *

0 0 ,

n n n

tr V V A A ρ ρ ρ

=

Λ ≡ ⊗ = ∑

K

2

,

n n n

E ρ λ = ∑ ,

n

E n n =

Coherent Entropy Ohya Mutual Entropy Lindblad Entropy

5. Other Mutual Entropy Type Complexities

Coherent Entropy Ohya Mutual Entropy Lindblad Entropy

SLIDE 46

Calculation of Mutual Entropy-type Complexity for Attenuation Channel

* *

For the attenuation channel and the input state (1 ) , there exists a unitary operator such that (1 ) , U UWU ρ λ λ θ θ λ λ βθ βθ Λ = + − = + − − − Lemma

( )

( ) ( )

{ } (

)

2 2

* 1 *

1 1 1 1 4 1 1 exp 2

where 0,1 .

[ ] For the attenuation channel and the input state 0 0 (1 ) , the entropy exchange is obtained by log log ,

j j

e j j j

j

OW S trW W

µ λ λ β θ

ρ λ λ θ θ ρ µ µ

=

= + − − − − −

=

Λ = + − ,Λ = − = −∑ Theorem

SLIDE 47

Calculation of Mutual Entropy-type Complexity for Attenuation Channel

( ) ( )

* * *

[ ] For the attenuation channel and the input state (1 ) , (1) if > then the cohent entropy ; 0 is holds, (2) if < then the cohent entropy ; 0 is holds, (3) if < then t

C C

OW I I ρ λ λ θ θ α β ρ α β ρ α β Λ = + − Λ > Λ < Theorem

( )

*

he cohent entropy ; 0 is holds.

C

I ρ Λ =

( )

* * *

Lindblad- [ ] F Niels

r the att

en’s entro enuation channel and the input py Lindblad-Nielsen’s entr state (1 ) , (1) if > then the ; is holds, (2) if < then the

py

;

LN LN

OW I S I ρ λ λ θ θ α β ρ ρ α β ρ Λ = + − Λ > Λ Theorem

( )

*

Lindblad-Nielsen’s is holds, (3) if < entropy then the ; is holds. LN S I S ρ α β ρ ρ < Λ =

SLIDE 48

Compound State (signal is transmitted) Let us consider construct of the compound state. If the initial state ρ and the quantum channel Λ* (signal is transmitted from the initial system to the final system) are given, compound states Φ should satisfy the following marginal conditions: (1) tr2 Φ = ρ, (Marginal condition 1) (2) tr1 Φ = Λ* ρ (Marginal condition 2)

5. Compound States

SLIDE 49

Compound State (signal is transmitted)

5. Compound States

Λ

*ρ =

λ n

n

∑

Λ

*En

ρ = λ n

n

∑

E n

Φ

2

; CP channel tr Φ

H

1

; CP channel tr Φ

H * CP channel

Λ

* *

(1) (2)

E n n n n

E E σ ρ ρ σ λ = ⊗ Λ = ⊗ Λ

∑

Ohya compound st Trivial compound stat ate e

SLIDE 50

Compound State (signal is transmitted)

5. Compound States

Λ

*ρ =

λ n

n

∑

Λ

*En

ρ = λ n

n

∑

E n

E

σ

* CP channel

Λ

( ) ( ) ( )

* *

* * * * 1 1 2 , ,

(1) (2) in the sense of Accardi and Ohya [AO] , :

E n n n n E E E

E E σ ρ ρ σ λ σ ρ

Λ Λ

= ⊗ Λ = ⊗ Λ = → ⊗

∑

Li Trivial compound state Ohya compound state fting S S E E H H H

*

* ,

nonlinear liftings

E Λ

E

*

* ,

Liftings

E Λ

E

SLIDE 51

5. Compound States

( )(

)

( )( )(

)

( )(

)

( )(

)(

)

* * *

0 0 (1 ) 1 0 0 (1 ) 1 1 2 if and = , then there exists a 2 3

*

For the attenuation channel and the input state a b a b c d c d a b a b c d c d ρ λ λ θ θ ρ θ θ ρ θ θ ρ λ λ αθ αθ ρ αθ αθ ρ αθ αθ λ β Λ = + − = + + + − + + Λ = + − ′ ′ ′ ′ = Λ + + ′ ′ ′ ′ + − Λ + + = Theorem [NW]

( ) ( ) ( )

* * *

compound state satisfying ; , = ; .

LN

I S I ρ ρ ρ ρ Φ Λ = Φ ⊗ Λ Λ

SLIDE 52

5. Compound States

( ) ( )

( )

( ) ( ) ( )

( )

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 1 , , 1 1 2exp 1 2exp 1 2 2 , , , , 1 2 1 4 1 1 exp 1 2 1 exp 2 1 2 1 4 1 1 exp 1 2 1 exp 2 b c t t a b c t d ab ab b cd cd t c t τ θ τ θ τ τ λ λ λ θ τ λ θ λ λ λ θ λ θ = =     + − + + − +         = = = = = = − − + − − − − =   − −     − − − − − − − =   − −    

SLIDE 53

5. Compound States

( ) ( )

( )

( ) ( ) ( )

( )

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 1 , , 1 1 2exp 1 2exp 1 2 2 , , , , 1 2 1 4 1 1 exp 1 2 1 exp 2 1 2 1 4 1 1 exp 1 2 1 exp 2 b c t t a b c t d a b a b b c d c d t c t τ αθ τ αθ τ τ λ λ λ αθ τ λ αθ λ λ λ αθ λ αθ ′ ′ = =     ′ ′ ′ ′ + − + + − +         ′ ′ ′ ′ ′ ′ = = ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ = = = = − − + − − − − ′ =   − −     − − − − − − − ′ =   − −    

SLIDE 54

5. Compound States

Λ

*ρ =

λ n

n

∑

Λ

*En

ρ = λ n

n

∑

E n

2

; CP channel tr Φ

H

1

; CP channel tr Φ

H * CP channel

Λ

SLIDE 55

5. Compound States

( )

*

0 0 (1 ) , 1 if and =1, then there exists a compound state satisfying 2 , .

*

For the attenuation channel and the input state S S ρ λ λ θ θ λ α ρ ρ ρ Λ = + − = Φ Φ ⊗ Λ = Theorem [NW]

How to construct the following compound states? 1 Separable compound state Entangled compoun P d ) s 2) ta roblem te

SLIDE 56

5. Compound States

' ' ' ' E k k k k k k k k

x x x x λ λ     Φ = ⊗ ⊗        

∑ ∑

Entangled compound state

E n n n n

E E ω λ = ⊗

∑

Separable compound state

n n nE

ρ λ = ∑ the Schatten decomposition of For a given initial state ρ, ρ is given by

SLIDE 57

5. Compound States

E n n n n

E E ω λ = ⊗

∑

Separable compound state

*

( )

E n n n n

E E σ λ = ⊗ Λ

∑

Ohya compound state Separable Jamiolkowski isomorphism channel (CP channel)

( )(

) ( )

* * * *

is given by and

n n n n n n

I V I V I I V V ρ ρ ⊗ ⊗ = ⊗ Λ Λ =

∑ ∑

SLIDE 58

Compound State (two states are transmitted)

* CP channel

Λ

ρ = λ n

n

∑

E n Λ

*ρ =

λ n

n

∑

Λ

*En

Separable State

E

σ

5. Compound States

Jamiolkowski isomorphism channel (CP channel)

ρ = λ n

n

∑

E n

E

ω

SLIDE 59

Compound State (two states are transmitted)

* CP channel

Λ

ρ = λ n

n

∑

E n Λ

*ρ =

λ n

n

∑

Λ

*En

Separable State

E

σ

5. Compound States

E

σ

Compound State (one state is transmitted)

SLIDE 60

5. Compound States

' ' ' ' E k k k k k k k k

x x x x λ λ     Φ = ⊗ ⊗        

∑ ∑

Entangled compound state

( )

' * ' ' ' E k k n k k k k n n k k

I V x x x x I V λ λ     Ψ = ⊗ ⊗ ⊗ ⊗        

∑ ∑ ∑

Entangled compound state

Jamiolkowski isomorphism channel (CP channel)

( )(

) ( )

* * * *

is given by and

n n n n n n

I V I V I I V V ρ ρ ⊗ ⊗ = ⊗ Λ Λ =

∑ ∑

SLIDE 61

Compound State (two states are transmitted)

* CP channel

Λ

ρ = λ n

n

∑

E n Λ

*ρ =

λ n

n

∑

Λ

*En

Entangled State

E

Ψ

5. Compound States

Jamiolkowski isomorphism channel (CP channel)

E

Φ

SLIDE 62

Compound State (two states are transmitted)

* CP channel

Λ

*ρ =

λ n

n

∑

Λ

*En

Entangled State

E

Ψ

5. Compound States

ρ = λ n

n

∑

E n

Compound State (one state is transmitted)

SLIDE 63

5. Compound States

( )

' * ' * ' '

. . . , under t

k n k k k k n n E k k E k k k k

Let be a compound state w r t the initial state the quantum CP channel and a S E I V x x chatten decomposition of defined by x x I V λ λ λ ρ ρ Ψ     Ψ = ⊗ ⊗ ⊗ ⊗        Λ = 

∑ ∑ ∑ ∑

Theorem [NW]

( )(

) ( ) ( )

( ) ( )

( )

2 1

* * * * * *

two marginal he condition and . One can obtain as follows , . : states Upper bo is S un given by , d 2

E E E n n n n n n

I V I V I I V V tr S tr S S ρ ρ ρ ρ ρ ρ ρ Λ Λ ⊗ ⊗ = ⊗ = Λ Ψ = Ψ = Ψ ⊗ ≤ Λ

∑ ∑

H H

SLIDE 64

5. Compound States

( )

*

[NW] . . . . w.r.t.

E k k k E k k k

Let be a compound state w r t the initial state the quantum CP channel and a Schatten decomposition of d E I V x ef x ined by ρ λ λ ρ Λ = Ψ Ψ = ⊗ ⊗

∑

1) Pure entangled compound stat Corolla e ry

( ) ( )(

) ( ) ( )

( )

2 1

* ' ' ' ' * * * * *

under the condition and . One can ob two marginal states tain as follows Up , . : S , is giv per bound en by

k k k E k E E k

x x I V I V I V I I V tr S tr S V λ ρ ρ ρ ρ ρ     ⊗ ⊗         ⊗ ⊗ = ⊗ Ψ = Ψ = Ψ ⊗ Λ Λ = Λ

∑ ∑

H H

( )

*

2S ρ ρ Λ ≤

SLIDE 65

5. Compound States

( )

*

. . . . w.r.t.

k n E k k n k E

Let be a compound state w r t the initial state the quantum CP channel and a Schatten decomposition of defined by E I V x ρ ρ λ λ ⊗ = Ψ = Λ Ψ

∑ ∑

2) Mixed entangled compound sta Corollary e [NW] t

( )

( )(

) ( ) ( )

2 1

* ' ' ' * ' * * *

is given b two mar under the condition and . One can obtain as follows , ginal y states

k k k k k n k k n n n m m m n n E n n E n n

x x x I V V y x I V I V I I V V tr S tr λ ρ ρ µ ρ     ⊗ ⊗ ⊗         = ⊗ ⊗ = ⊗ Ψ = Ψ = Λ Λ =

∑ ∑ ∑ ∑ ∑

H H

( )

( ) ( ) (

)

( )

' * *

Upper . ) , ' : boun 2 d S ,

n n m E m

S a y y m n n S ρ ρ ρ ρ Λ ⊥ ∀ ∀ ⇒ Ψ ≤ Λ ≠ ⊗

SLIDE 66

5. Compound States

( )

*

. . . w [NW] .r.t. .

k k k k k k E k E

Let be a compound state w r t the initial state the quantum CP channel and a Schatten decomposition of define E I d x b V y x λ λ ρ ρ Λ = Ψ  Ψ = ⊗ ⊗

∑ ∑

Lem 2) Mixed entangled compound a a st m te

( ) ( )(

) ( ) ( )

( ) ( )

2 1

' * ' ' ' * * * * * *

under the condition and . One can obtain as follows two marginal st i , ates Upper s given bo . : by S d n , u

k E k k E E k

x x I V I V I V I I r r S V V t S t ρ λ ρ ρ ρ ρ ρ Λ    ⊗ ⊗         ⊗ ⊗ Λ = Λ = ⊗ Ψ = Ψ = Ψ Λ ⊗

∑

H H

( )

2S ρ ≤

SLIDE 67

5. Compound States

[ ] ( ) ( )

2 1

, , * , ,

. 1 . 0,1 , , a rg

E E E E E E E

is the compound state given above For any let be a compound state defin One can obtain as follows tr S tr two m in ed al state b S y s

µ µ µ µ

µ µ ρ µ σ Ψ Ψ Ψ = + − Ψ Ψ = Ψ ∈ = Λ Theorem [NW]

H H

( ) ( )

* ,

. : S , 2 .

E

Upper bound S

µ

ρ ρ ρ µ ρ Ψ ≤ − Λ ⊗

SLIDE 68

5. Compound States

( ) ( )

' * ' ' ' * * ' ' , '

One can define a linear CP channel depending on the Schatten decomposition of from to as follows: where ,

E k k k k k k E n n n n nn nn n n k k

x x x x E E W W λ ρ ω λ λ Ξ     Φ = ⊗ ⊗         = ⊗ Ξ

=
∑

∑ ∑ ∑

Theorem [NW]

' ' ' ' * ' ' , '

is given by satisfying .

nn nn n n n n nn nn n n

W W x x x x W W I I = ⊗ = ⊗

∑

SLIDE 69

5. Compound States

( ) ( )

' * ' ' ' * ' ' , '

One can also define a linear CP channel depending on the Schatten decomposition of from to as follows:

E n n n n E k k k k k k nn nn n n k k

E E x x x x w w λ λ ρ ω λ Ξ = ⊗     Φ = ⊗ ⊗         Ξ

=
∑

∑ ∑ ∑

Theorem [NW]  

* ' ' ' * ' ' , '

wh , is given by satisfying ere .

nn nn k k k n n k nn nn n n

w w x x x x w w I I λ   = ⊗ ⊗     = ⊗

∑ ∑

SLIDE 70

Ξ

*

State 0 State θ θ Signal 0 Information Source Signal 1

( )

1- ρ λ λ θ θ = + Input state

5. Compound States

λ 1 λ −

SLIDE 71

* α

Λ

( )

1- ρ λ λ θ θ = + Input state

( )(

)

( )( )(

)

1 a b a b c d c d ρ ρ ρ θ θ ρ θ θ = + + + − + + Schatten decompotition of

( )

*

1- ρ λ λ αθ αθ Λ = + Output state

( )(

) ( )(

)(

)

* * * *

1 a b a b c d c d ρ ρ ρ αθ αθ ρ αθ αθ Λ ′ ′ ′ ′ Λ = Λ + + ′ ′ ′ ′ + − Λ + + Schatten decompotition of

5. Compound States

SLIDE 72

* α

Λ

( )(

)

( )( )(

)

1 a b a b c d c d ρ ρ ρ θ θ ρ θ θ = + + + − + + Schatten decompotition of input state

( )(

) ( )(

)(

)

* * * *

1 a b a b c d c d ρ ρ ρ αθ αθ ρ αθ αθ Λ ′ ′ ′ ′ Λ = Λ + + ′ ′ ′ ′ + − Λ + + Schatten decompotition of output state

( ) ( )

(

( ) ( ))

( )

(

( )

( ))

( )

(

( ) ( ) ( ))

* * * * * * *

1 1 1 1 1 1 a b a b c d c d a b c d c d a b a b c d c d a b ρ ρ ρ θ ρ αθ ρ θ ρ αθ ρ θ ρ αθ ρ θ ρ αθ ρ θ ρ αθ ρ θ ρ αθ Φ Λ ′ ′ Φ = + ⊗ Λ + ′ ′ + − + ⊗ − Λ + ′ ′ + ⊗ − Λ + ′ ′ + − + ⊗ Λ + ′ ′ + + ⊗ − Λ + ′ ′ + − + ⊗ Λ + Compound state with respect to and

2

tr Φ

H

1

tr Φ

H

5. Compound States

SLIDE 73

Communication Process

* α

Λ

( )(

)

( )( )(

)

1 a b a b c d c d ρ ρ ρ θ θ ρ θ θ = + + + − + + Schatten decompotition of input state

( )(

) ( )(

)(

)

* * * *

1 a b a b c d c d ρ ρ ρ αθ αθ ρ αθ αθ Λ ′ ′ ′ ′ Λ = Λ + + ′ ′ ′ ′ + − Λ + + Schatten decompotition of output state

( ) ( )

( ) ( ) ( ) ( )

( )

( ) ( )

2 * * 1 2 2 2

, 1 1 1 1 4 1 1 exp 1 4 1 1 exp 2 log 1

k k k k

S S S t t t t

α α

ρ ρ ρ ρ η µ µ λ λ θ λ λ α θ η

=

Φ ⊗ Λ = Λ + −   = + − − − − − − − − −     = − ≤ ≤

∑

SLIDE 74

* α

Λ

( )(

)

( )( )(

)

1 a b a b c d c d ρ ρ ρ θ θ ρ θ θ = + + + − + + Schatten decompotition of input state

( )(

) ( )(

)(

)

* * * *

1 a b a b c d c d ρ ρ ρ αθ αθ ρ αθ αθ Λ ′ ′ ′ ′ Λ = Λ + + ′ ′ ′ ′ + − Λ + + Schatten decompotition of output state

( ) ( )

( )

( ) ( )

( ) ( ) ( ) ( )

( )

* * * 2 * 1 2 2

; , 1 1 1 1 4 1 1 exp 1 2

LN e k k k k

I S S S S S

α α α α

ρ ρ ρ ρ ρ ρ η ν ν λ λ α θ

=

Λ ≡ Λ + − Λ = Λ + −   = + − − − − − −    

∑

5. Compound States

SLIDE 75

( ) ( )

( ) ( ) ( ) ( )

( )

* *

2 * * 1 2 2 2 2

2 2 , 1 1 1 1 4 1 1 exp 1 4 1 1 exp 2 there exists a positive number 2log 2 1 (1) if the input state satifying then ; ,

k k k k * * LN

S S S I S

α α

α α α α

α β ρ ρ ρ ρ η µ µ λ λ θ λ λ α θ ω ρ ω θ ρ ρ

= Λ Λ

= = Φ ⊗ Λ = Λ + −   = + − − − − − − − − −     = + < Λ > Φ ⊗ Λ

∑

Case 1)

( ) ( ) ( ) ( ) ( )

( )

1 2 1 2 1 2 2 2 2 2

for ; , for

r

; , for 0< <

r

1 3 1 exp exp 1 2 2 where 1 1 2 1 1 exp 1 exp 2

* * LN * * LN k k

I S I S

α α α α

ρ λ λ λ ρ ρ ρ λ λ λ ρ ρ ρ λ λ λ λ θ θ λ θ θ < < Λ = Φ ⊗ Λ = Λ < Φ ⊗ Λ < <       − + −             = − −       − − − −            

5. Compound States

SLIDE 76

* *

2

0 0 (1 ) , if is given by 0 1 then there exists a positive number . (2) if the input state satifying then ;

* * LN

For the attenuation channel and the input state I

α α

ρ λ λ θ θ α ω ρ ω θ ρ

Λ Λ

Λ = + − Λ ≤ ≤ = Theorem [NW]

( ) ( ) ( ) ( )

*

2

, for 1 or 0 (3) if the input state satifying 0 < then ; , for 0 1

* * * * LN

S I S

α

α α α α

ρ ρ λ ρ θ ω ρ ρ ρ λ

Λ

Λ = Φ ⊗Λ = ≤ Λ > Φ ⊗Λ ≤ ≤

5. Compound States

SLIDE 77

( ) ( )

( ) ( ) ( ) ( )

( )

*

2 2 2 * * 1 2 2 2 3 3

1 2 , 3 3 , 1 1 1 1 4 1 1 exp 1 4 1 1 exp 2 3 7 33 33 5 33 27 34 33 586 1 there exists a positive number 3log 2 9 17 4 243 729 3 (1) if the input

k k k k

S S S

α

α α

α β ρ ρ ρ ρ η µ µ λ λ θ λ λ α θ ω

= Λ

= = Φ ⊗ Λ = Λ + −   = + − − − − − − − − −       − −   = + + + +    

∑

Case 2)

( ) ( ) ( ) ( ) ( ) ( )

( )

*

2 1 2 1 2 1 2 2 2 2

state satifying then ; , for ; , for

r

; , for 0< <

r

1 4 2 exp exp 1 3 3 where 1 1 2 1 exp 1 exp

* * LN * * LN * * LN k k

I S I S I S

α

α α α α α α

ρ ω θ ρ ρ ρ λ λ λ ρ ρ ρ λ λ λ ρ ρ ρ λ λ λ λ θ θ λ θ

Λ <

Λ > Φ ⊗ Λ < < Λ = Φ ⊗ Λ = Λ < Φ ⊗ Λ < <     − + −         = − − − − −

( )

2

2 θ         −    

5. Compound States

SLIDE 78

* *

2

0 0 (1 ) , if is given by 0 1 then there exists a positive number . (2) if the input state satifying then ;

* * LN

For the attenuation channel and the input state I

α α

ρ λ λ θ θ α ω ρ ω θ ρ

Λ Λ

Λ = + − Λ ≤ ≤ = Theorem [NW]

( ) ( ) ( ) ( )

*

2

, for 1 or 0 (3) if the input state satifying 0 < then ; , for 0 1

* * * * LN

S I S

α

α α α α

ρ ρ λ ρ θ ω ρ ρ ρ λ

Λ

Λ = Φ ⊗Λ = ≤ Λ > Φ ⊗Λ ≤ ≤

5. Compound States

SLIDE 79

5. Compound States

* *

2

0 0 (1 ) , if is given by 0 1 then there exists a positive number . (1) if the input state satifying then ;

* * LN

For the attenuation channel and the input state I

α α

ρ λ λ θ θ α ω ρ ω θ ρ

Λ Λ

Λ = + − Λ ≤ ≤ < Theorem [NW]

( ) ( ) ( ) ( ) ( ) ( )

( )

1 2 1 2 1 2 2 2 2 2 2 2 2

, for ; , for

r

; , for 0< <

r

1 exp 1 exp 1 1 where 1 1 2 1 exp 1 exp

* * * * LN * * LN k k

S I S I S

α α α α α α

ρ ρ λ λ λ ρ ρ ρ λ λ λ ρ ρ ρ λ λ λ λ α θ α θ λ θ α θ Λ > Φ ⊗Λ < < Λ = Φ ⊗Λ = Λ < Φ ⊗Λ < <   − + + − −   = − −   − − − −    

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5. Compound States

* *

2

0 0 (1 ) , if is given by 0 1 then there exists a positive number . (2) if the input state satifying then ;

* * LN

For the attenuation channel and the input state I

α α

ρ λ λ θ θ α ω ρ ω θ ρ

Λ Λ

Λ = + − Λ ≤ ≤ = Theorem [NW]

( ) ( ) ( ) ( )

*

2

, for 1 or 0 (3) if the input state satifying 0 < then ; , for 0 1

* * * * LN

S I S

α

α α α α

ρ ρ λ ρ θ ω ρ ρ ρ λ

Λ

Λ = Φ ⊗Λ = ≤ Λ > Φ ⊗Λ ≤ ≤

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1) We explained the quantum channels associated with the open system dynamics and the quantum communication processes. 2) The Ohya mutual entropy is treated for purely quantum systems, and semi-quantum mutual entropy are special case of the Ohya mutual entropy. 3) The Ohya mutual entrpy is one of the most suitable transmitted complexity for discussing the efficiency of information transmission in quantum communication systems 4) We briefly reviewed the mean entropy and the mean mutual entropy for general quantum systems. 5) The lower bound of the mean entropy for the open system dynamics is obtained. For a given assumption, the mean entropy and the mean mutual entropy for the open system dynamics are calculated. 6) We discuss how to construct the compound states. We will show the relation between the Lindblad entropy and the relative entropy with respect to the entangled compound state and trivial compound state consisted of the input state and the output state.

Conclusion

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Thank you very much!

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