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

Estimation of group action with energy constraint arXiv:1209.3463v3 Masahito Hayashi Graduate School of Mathematics, Nagoya University Centre for Quantum Technologies, National University of Singapore

Contents • Summary of estimation in group covariant family • Estimation of group action  , U(1), SU(2), and SO(3) with average energy restriction • Practical construction of asymptotically optimal estimator • Application to uncertainty relation (Robertson type)

Estimation of group action H f G Given a projective unitary representation of on . Unknown estimate measurement Input state Unitary to be estimated   ( ) f g g M   E ( , ) M :Our operation     1 1 ˆ ˆ ˆ ( , ) ( , ) ( , ) R g g R e g g R e gg :error function g Average error when the true parameter is    * ˆ ˆ D D E E ( ) : ( , )Tr ( ) ( ) ( ) G R g g M dg f g f g , R g   prior:   D D D D ( ) : ( ) ( ) M M dg Bayesian:  , , R R g G  D D D D ( ): max ( ) M M Mini-max: , R R g g

Group covariant measurement H :Hilbert space Holevo 1979 G :group f :projective unitary representation M G A POVM taking values in is called covariant if  * ( ) ( ) ( ) ( ) f g M B f g M gB M ( ) G G ： Set of POVMs taking the values in M cov ( ) G ： Set of covariant POVMs taking values in G  M M ( ) cov ( ) G M G is included in     * ( ) ( ): ( ) ( ) ( ) M B M B f g Tf g dg T B

Group-action-version of quantum Hunt-Stein theorem  G Invariant probability measure exists for G when is compact. Then, the following equations hold.    D D D D min ( , ) min ( , ) M M  , R R     M M M M , ( ) , ( ) M G M G   D min ( , ) M R   M :pure, ( ) M G cov   D min ( , ) M  , R   M :pure, ( ) M G cov G The following relation holds even when is not compact.    D D D D min ( , ) min ( , ) M M R R     M M M M , ( ) :pure, ( ) M G M G cov

Fourier transform and inverse Fourier transform on group ˆ G ： Set of irreducible unitary representation of G ˆ   2 2 U ( ): ( ) L G L    ˆ G 2 ( U U ) L ： Set of HS operators on   2 ˆ  2 F : ( ) ( ) L G L G ： Fourier transform      * F ( [ ]) : ( ) ( ) ( ) d f g g dg    G ˆ  -1 2 2 F : ( ) ( ) L G L G ： Inverse Fourier transform   -1 F [ ]( ): Tr ( ) A g d f g A    ˆ   G

Optimization with Energy constraint via inverse Fourier transform Energy constraint   Tr H E       1 1 2 ˆ ˆ F ( ): ( , )| [ ]( )| ( ) D X R e g X g d g R G Our target is    D D D D min ( , ) min ( , ) M M R R     M M M M , ( ) :pure, ( ), M G M G cov     Tr Tr H E H E  min ( ) D X R  2  2  ( ), 1 X L G X , H E ˆ ˆ    2 2 ( ) : { ( ) | } L G X L G X H X E , H E

Example: G   ˆ G   ( )          ig 2 2 ( ) , f g e ( , ) ( ) , H Q R g g g g      2 min min ( , ) | Tr D M Q E R    2  M  ( ) S M ( ( )) L cov       d g d   2          1 2 2 2   F min | [ ]( ) | | ( ) | g g E         2 2 2    ( ) L   1        2 2 min Q P E 4   2 E  ( ) L  2  Minimum is attained with    2 ( ) 4 e E E

Mathieu Function Periodic differential operator  2 2 cos2 P q Q Minimum Eigen function space eigenvalue    0 ( ) a q  ce ( , ) 2 q p,even (( , ]) L 0 2 2     2 se ( , ) 2 ( ) q p,odd (( , ]) b q L 2 2 2    1 ( ) ce ( , ) a q q  2 a,even (( , ]) L 1 2 2    se ( , ) q  1 ( ) 2 b q a,odd (( , ]) L 1 2 2

Estimation of U(1)      ( , ) 1 cos( ), R g g g g  ikg ( ) , f g k e k   2 H k k k   k    min min ( , ) | Tr D M H E R    ˆ 2 M (U(1)) S M ( (U(1))) L cov          2 min cos I Q P E   2    (( , ]) L p,even Optimal input is (2 / ) sa s     ce ( , ) 0 q max 1 constructed by sE 0  4 0 s  1 1    as E  2 8 128  E E   3   7 2 2 E    1 2 0 E as E  16

Graphs

Estimation of SU(2) 1      k k     1    ( , ) 1 ( ), R g g gg  1  H I 1 2 k    2 2 0 k 2 2    ˆ 2 2 p,odd (( , ]) L (SU(2)) L Reduce to      min min ( , ) | Tr D M H E R  ˆ   2 M (SU(2)) S M ( L (SU(2))) cov   1 Q         2   min cos 2 I P E   4   2    (( , ]) L p,odd (8 / ) 1 sb s     2 max 1 ( ) s E Optimal input is  4 4 s 0 constructed by   3 9 7 3     as E  se ( , ) q 11 2 32 2 E E  2   3   2 5 2 E   1 0  E as E  3 6 3

Graphs

Factor system of Chiribella 2011 projective unitary representation Factor system    ( , ') 1 i g g : ( ) ( ') ( ') e f g f g f gg e   ( , ') i g g L : { } , ' g g ˆ[ ] ： Set of projective irreducible representation G L with the factor system L      1 1 2 ˆ ˆ ˆ F ( ) : ( , ) | [ ]( ) | ( ) D X R e g X g dg L R G

Estimation of SO(3)    1 k k          1 ( , ) (3 ( )), R g g gg  1  H I 1 k   2  2 2 0 k 2       ˆ 2 2 2 a,odd (( , ]) p,odd (( , ]) L L (SO(3)) L Reduce to or      min min ( , ) | Tr D M H E R  ˆ   2 M (SO(3)) S M ( (SO(3))) L cov  1        2 min { | cos | | | | } I Q P E  4   2 L  a,odd  Integer case   1         2 min { | cos | | | | } I Q P E  4   2 L p,odd   Half integer case

Integer case 1        2 min { | cos | | | | } I Q P E 4   2 L a,odd (2 / ) 1 sa s     1 max 1 ( ) s E  4 4 0 s  9 81    E   2 8 128 E E   3  E E    0 E   2 4 2  ce ( , ) q Optimal input is constructed by 1

Half integer case 1        2 min { | cos | | | | } I Q P E 4   2 L p,odd (2 / ) 1 sb s     2 max 1 ( ) s E  4 4 0 s  9 81    E   2 8 128 E E   1 3 1 3 5 3 3       1 ( ) 2 ( ) 2 E E E   4 4 4 3 48 3  se ( , ) q Optimal input is constructed by 2

Graphs Κ SO � 3 � � E � & Κ SO � 3 � , � � 1 � � E � 1.4 1.2 1.0 0.8 0.6 0.4 0.2 E 0 5 10 15 20 Thick line expresses the projective case, and Normal line expresses the representation case

G   Non-compact Example: 2 f : Heisenberg representation X   L  2 (  2 ( ) ) L multiplicity Minimize  x yi    2 2 1 2 F ( )| [ ]( )| x y X dxdy 2  2 under    2 2 ( ) X Q P I X E 1 Minimum value: 2 E

How to derive minimum      2 2 2 2 F : ( ) ( ) ( ) L L L Fourier transform 1 1        1 1 F F F F = F F F F = ( ) , ( ) Q I P Q P I P Q 2 1 1 2 2 2   F  1 [ ] X Via , minimizing problem is equivalent with    2 2 Q Q Minimize 1 2 1 1        2 2 ( ) ( ) P Q P Q E under 2 1 1 2 2 2 By choosing suitable coordinate, minimizing problem is equivalent with Minimum value    2 2 Minimize Q Q Uncertainty 1 2 12 E relation     2 2 P P E under 1 1

Practical realization of asymptotically optimal estimator     G  2 H , k H k k k U(1)   k k   Assume that satisfies 2 | | 0 k  F [ ] is even function k MLE      U M 1      U M 2  n       U M n This method attains the optimal performance.