Estimation of group action with energy constraint arXiv:1209.3463v3 - - PowerPoint PPT Presentation

Estimation of group action with energy constraint

Masahito Hayashi

Graduate School of Mathematics, Nagoya University Centre for Quantum Technologies, National University of Singapore

arXiv:1209.3463v3

• Summary of estimation in group

covariant family

• Estimation of group action

with average energy restriction

• Practical construction of asymptotically
• ptimal estimator
• Application to uncertainty relation

(Robertson type)

Contents

, U(1), SU(2), and SO(3) 

Given a projective unitary representation of on .

Estimation of group action

( , ) M   E H G f

:Our operation :error function

Average error when the true parameter is

1 1

ˆ ˆ ˆ ( , ) ( , ) ( , ) R g g R e g g R e gg

 

 

* ,

ˆ ˆ ( ) : ( , )Tr ( ) ( ) ( )

R g G R g g

M dg f g f g    D E D E g

Bayesian: Input state measurement

 M

estimate



g ( ) f g

Unknown Unitary to be estimated

, ,

( ) : ( ) ( )

R R g G

M M dg



   D D D D

,

( ): max ( )

R R g g

M M  D D D D

Mini-max: prior: 

A POVM taking values in is called covariant if ：Set of POVMs taking the values in

Group covariant measurement

:Hilbert space

G :group H

*

( ) ( ) ( ) ( ) f g M B f g M gB  f

:projective unitary representation

M G

( ) G M

cov( )

G M G

：Set of covariant POVMs taking values inG is included in

cov( )

G M ( ) M G M

*

( ) ( ): ( ) ( ) ( )

T B

M B M B f g Tf g dg    

Holevo 1979

Group-action-version of quantum Hunt-Stein theorem

Invariant probability measure exists for when is compact. Then, the following equations hold.

G

cov cov

, , ( ) , ( ) :pure, ( ) , :pure, ( )

min ( , ) min ( , ) min ( , ) min ( , )

R R M G M G R M G R M G

M M M M

     

   

   

  

M M M M M M

D D D D D D G 

The following relation holds even when is not compact.

G

cov

, ( ) :pure, ( )

min ( , ) min ( , )

R R M G M G

M M

 

 

 



M M M M

D D D D

Fourier transform and inverse Fourier transform on group

：Set of irreducible unitary representation of

ˆ G G

2 2 ˆ

ˆ ( ): ( )

G

L G L

 

  U

2(

) L



U

：Set of HS operators on



U

2 2 ˆ

: ( ) ( ) L G L G  F

：Fourier transform

*

( [ ]) : ( ) ( ) ( )

G

d f g g dg

  

   



F

• 1

2 2

ˆ : ( ) ( ) L G L G  F

：Inverse Fourier transform

• 1

ˆ

[ ]( ): Tr ( )

G

A g d f g A

   

  F

Energy constraint

Tr H E  



cov 2 2 ,

, ( ) :pure, ( ), Tr Tr ( ), 1

min ( , ) min ( , ) min ( )

H E

R R M G M G H E H E R X L G X

M M D X

   

 

     

 

M M M M

D D D D



1 1 2

ˆ ˆ ( ): ( , )| [ ]( )| ( )

R G

D X R e g X g d g 

 

  F

2 2 ,

ˆ ˆ ( ) : { ( ) | }

H E

L G X L G X H X E   

Our target is Optimization with Energy constraint via inverse Fourier transform

Example: G  

 

  

 

2 cov 2 2

2 ( ) ( ( )) 2 1 2 2 2 ( ) 2 2 ( )

min min ( , ) | Tr min | [ ]( ) | | ( ) | 2 2 1 min 4

R M L L L

D M Q E d g d g g E Q P E E

  

            

    

               

 

M S

F

     

 

2

( , ) ( ) , R g g g g   ( ) ,

ig

f g e



  

2

H Q 

Minimum is attained with

2 2

4

( )

E

e E



 



 ˆ ( ) G  

Mathieu Function

2

2 cos2 P q Q 

Periodic differential operator

Minimum eigenvalue Eigen function space

2 p,even((

, ]) 2 2 L   

2 p,odd((

, ]) 2 2 L   

2 a,even((

, ]) 2 2 L   

2 a,odd((

, ]) 2 2 L   

0( )

a q

2( )

b q

1( )

a q

1( )

b q ce ( , ) q 

2

se ( , ) q 

1

se ( , ) q 

1

ce ( , ) q 

Estimation of U(1)

 

 

2 cov 2 p,even

ˆ (U(1)) ( (U(1))) 2 (( , ]) 2 3 2

min min ( , ) | Tr min cos (2 / ) max 1 4 1 1 as 8 128 7 2 1 2 as 16

R M L L s

D M H E I Q P E sa s sE E E E E E E

   

     

    

                    

M S

 

( , ) 1 cos( ), R g g g g    ( ) ,

ikg

f g k e k 

2 k

H k k k  

ce ( , ) q 

Optimal input is constructed by

Graphs

Estimation of SU(2)

 

2 cov 2 p,odd

ˆ (SU(2)) ( (SU(2))) 2 (( , ]) 2 3 11 2 3 2

min min ( , ) | Tr 1 min cos 2 4 (8 / ) 1 max 1 ( ) 4 4 9 7 3 as 32 2 2 5 1 as 3 6 3

R M L L s

D M H E Q I P E sb s s E E E E E E E

   

     

    

                             

M S

 

1 1 2

1 ( , ) 1 ( ), 2 R g g gg 



 

2

1 2 2

k k

k k H I

 

         Optimal input is constructed by

2

se ( , ) q 

Reduce to

2 p,odd((

, ]) L   

2

ˆ (SU(2)) L

Graphs

Factor system of projective unitary representation

( , ') 1

: ( ) ( ') ( ')

i g g

e f g f g f gg

 

 ˆ[ ] G L

Factor system

( , ') , '

: { }

i g g g g

e   L

：Set of projective irreducible representation

with the factor system L

1 1 2

ˆ ˆ ˆ ( ) : ( , ) | [ ]( ) | ( )

R G

D X R e g X g dg 

 

 

L

F

Chiribella 2011

Estimation of SO(3)

 

2 cov 2 a,odd 2 p,odd

ˆ (SO(3)) ( (SO(3))) 2 2

min min ( , ) | Tr 1 min { | cos | | | | } 4 1 min { | cos | | | | } 4

R M L L L

D M H E I Q P E I Q P E

  

         

   

                

M S

 

1 1

1 ( , ) (3 ( )), 2 R g g gg 



 

2

1 2 2

k k

k k H I

 

        

Integer case Half integer case

Reduce to or

2 a,odd((

, ]) L   

2

ˆ (SO(3)) L

2 p,odd((

, ]) L   

2 a,odd

2 1 2

1 min { | cos | | | | } 4 (2 / ) 1 max 1 ( ) 4 4 9 81 8 128 3 2 4 2

L s

I Q P E sa s s E E E E E E E



   

 

                    

Integer case

Optimal input is constructed by

1

ce ( , ) q 

2 p,odd

2 2 2 1 3 2 2

1 min { | cos | | | | } 4 (2 / ) 1 max 1 ( ) 4 4 9 81 8 128 1 3 5 3 3 1 ( ) ( ) 4 4 4 3 48 3

L s

I Q P E sb s s E E E E E E E



   

 

                      

Half integer case

Optimal input is constructed by

2

se ( , ) q 

Graphs

5 10 15 20 E 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ΚSO 3E & ΚSO 3,1E

Thick line expresses the projective case, and Normal line expresses the representation case

Non-compact Example:

2

G   f : Heisenberg representation

2 2

( ) X Q P I X E   

2

2 2 1 2

( )| [ ]( )| 2 x yi x y X dxdy



 



F

 2(

) L  

2(

) L 

multiplicity Minimize

X 

under Minimum value:

1 2E

How to derive minimum

2 2 2 2

: ( ) ( ) ( ) L L L   F   

Fourier transform

1 1 2

1 ( ) 2 P I P Q



   F F F F =

1 2 1

1 ( ) , 2 Q I P Q



  F F F F =

Via , minimizing problem is equivalent with

2 2 1 2

Q Q   

2 2 2 1 1 2

1 1 ( ) ( ) 2 2 P Q P Q E       

Minimize under

1[

] X 



 F

By choosing suitable coordinate, minimizing problem is equivalent with

2 2 1 2

Q Q   

2 2 1 1

P P E    

Minimize under Minimum value

12E

Uncertainty relation

1

U M     

Practical realization of asymptotically optimal estimator

2

| |

k

k k  



[ ]  F

Assume that satisfies

 U(1) G 

2

,

k

k H k k k     H

is even function

2

U M     

n

U M     

 n



MLE This method attains the optimal performance.

1

U M     

Practical realization of asymptotically optimal estimator

Assume that the support of contains both of integer rep. and half integer rep.

 SU(2) G 

 

  H H H H

2

U M     

n

U M     

 n



MLE This method attains the optimal performance.

1

U M     

Practical realization of asymptotically optimal estimator

Assume that the support of contains

• nly integer rep. or half integer rep.

 SO(3) G 

 

  H H H H

2

U M     

n

U M     

 n



MLE This method attains the optimal performance.

1

U M     

Implication of these optimal estimators

When we consider the energy constraint, entangled input state and measurement with entangled basis do not enhance the quality of estimation.

2

U M     

n

U M     

 n



MLE This method attains the optimal performance.

2 ce ( , ) 2

E

s  

Uncertainty relation on

The minimum is realized by

2 2 2

(cos ,sin ) : cos sin Q Q Q Q

  

    

 

2

2 2 (( , ]) 2

min (cos ,sin )| (2 / ) max1 ( ) 4

p

L s

Q Q P E sa s sE

       

     

2 ((

, ])

p

L   

2

2 ( ) : argmax1 ( ) 4

E s

sa s s sE



  

2 2 1

(U(1)) ( ) L L S  

Uncertainty relation on

3 2 2

: ,

j j

Q Q

  

  



 

/2

( ) : |

j

it j t

d e g P dt



 





2(SU(2))

L

3 1 2 3

( ( ), ( ), ( ), ( )) g x g x g x g x g S  

3 2 2 1

:

j j

P P

  

  



 

 

2

2 2 (SU(2)) 2 2

min | 8 max1 ( ( 1/ 4) ( ) / 16)

L s

Q P E s E sb s

   

          

Function realizing the minimum is given by using



2

8 se ( , ) 4

E

s  

2 3

( ) L S 

• We have proposed a method with

Inverse Fourier transform as a unified approach for estimation of group action

• Using this method, we have derived the
• ptimal estimator with energy constraint

in several groups.

• We have shown that entanglement of

input and output cannot improve under energy constraint.

• We have applied it to uncertainty relation.

Conclusion

• MH arXiv:1209.3463
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References