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
([email protected] )*= ( 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 )
.
Top Related