Physics 618, Spring 2020

March 20,

2020

Second Test

Last time : projective report's

Some aspects of QIM.

Born Rule :

State p pos . trace class

Trip ) = 1

observable 0 self. ag :

}°k

ECR

toE

Pade ) = Try (PPOCE ) )

Symmetry in QM.

1- 1

map

s, :S - S

sz : O - O

Preserves prob 's .

Try ) Ps .¥±,' =Tr pBCE )

Such pairs form a

groupAut ( QM ) .

Reduce this to a map

s : Spine - $ Pure

preserves overlaps

Stone = { Rank one projectors

p=nsy÷, use}

Tr (Py Pq ) = Kuhnle414,7<4.14

.)

SPM = PH

If fl=¢Nt 'then SPFCIRN

o(l%kk)=@sdU¥y'

PLL = { set of lines

through origin }= { set of rank one

projectors }Aut ( QM ) = Isomety group

of ERN for

D= Fubini . Study metric

Example : Lf=¢2 I Qbit

→→

g= at b. a

11,

Ispan all 2×2

complex matrices.

p >o < ⇒ ( 4, py )=ot4

p+=p < ⇒ a ,J'

real.

positivity ? e.v. 's ?

Is'

~ (&

is ;) ✓

£ . F Hemmitian In I real

Tr& . I ) = o

tie ~ (''

. a)E.'s)2= £2 = Is

Atb .F e.v. 's a±lb→l

⇒ as Rl

Trip ) =L ⇒ a= tz

p=I(I+E.8 ) Ruel

s=€#tcp3External points

flare.±£(I+aF ) iii. ,

p- I (1+12) £31

p2=p Ip =L

⇒ e - Eooo)p=

14 ><

4¥ ,for some

nonzero

Tech

Yk '

Auto ) uesuk )

End (eit¥as±zA=( sinoosohsinosingeoso ) &U¥since )

145<41 = tzftn. F)

§3ttssnHopf libration{ tee

'

IHHH }

Else ) is a U ( l ) tensor

phase ambiguity in if

Overly

trpnpn.

= tz(HA .ae )= cos 202

O = angle between A

avge¥e-Aut ( QM ) = Isomety goop of 5= OC3)

Next time :

.

Wigner 's theorem.

Awkward to describe symmetriesin terms of isometrics of the

Fubini Study metric on PH.

Replaces this by thegroup

of

unityand

antiveninoperators on Ll,

This linearized=

the description .

A positive operator is a-fortiori

HermitAZO then A = BTB for some

B

B+B>o (4,13+134)=1113411230

( 4 A 4) >- o

(4.tF4zgAthtt4.Dzo@1.A4 )*= ( 4 , At )

iAy , 4) = ( 4 , Atul )( eitej ,

A Ceitej ) )= ( eitej ,

At Ceitej ) )⇒ (es ; Aei )+@i , Aej ) = Faint

Do the same thing with

E,

+ FT '

ej Combine thetwo

eqs

( ei , Aej ) = @,At

Wigner's Theorem :

U ( fl ) =

groupof unitary ops

�1�- linear

ops U sit .

key 11=11411VitekSet of anteingop 's .

@ - anti linear : A (14+42)=4 ,+AkZEE A (z4)=z*A(y )

*anti "

omt=y_ : aisE- antilinean

and Ham = 11411 fnallyak.

Aotffl ) =

groupof unitary

and anti - unitaryuiuzeufk ) operators .

U

,°Uz#l )

AFUZanti unitary

U,°Az.iaz unitaryAct on the set of pure states

P - uput Palmasum >

P→aP@←

aat-=+

= I

Thispreserves overlap fonetion

of R ,R ) at Pip

→ Tr up,uYu/Ru+-

=TrRPz

itI.

1→UH→Autffl) → AUHQM

)→l11 k 11

groys ofIsom of

unitary F- S.

anti unitary

Tw°point÷O # is surject 've

.

�2� kernel is U ( l )

{ u=ei91 } R¥= Ker ( it )

.