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Density Matrices .612512020 Iti > =/ ! )
Notice : we
. 1%7 : a particular state assume that
. pi : corresponding probability 12h. ) is normalized,
so Hill Yi > =L .
. for observable a ,the mean value is :
CA > = ¥,Pi
ATM→ it's a scalar .
omensmy( Quadratic forms . )
"
weights" L4il: conjugate Transpose of Itis
--
K
- Then, density matrix is defined as :
→ e .. E:p .ni, ±×±
- lil' " '
ran "
o - → a linear combinationa rank - one matrixI of composite States .
→. Can rewrite : LA > =
tripA ) .
←- -
Hi : eigenvector . eigenvector→ Look at it
in a"
math"
↳ ifwe write it as : X"
,
way : then CHI A 14 ;) ⇒ X*AXmm
-
e I
⇒ Xx : Ax,
so Xt'Ax =
XIII=④II = d
- -
⇒ so,
we basically sum up the eigenvalues .
← sum of the⇒ Edi= trace
. diagonal entries .
--
→ Look at it A : physics operator g⇒ PA : mean
in a "physics"
p : densityway :
Hermitian :
DensityMatrices eigenvectors can be
taken to be a
• Example : 2.4 .Ortho norad set
.
Pure state : f scalar >
-
dm= E. HIA
KEN>
P a
✓= L 41 A ? 147%147ith diagonal entry -
-Identity .
= LIVIA 147 .
. A general density matrix is a convex combination
of pure state .
• Convex combination : a linear combination of pointswhere all coefficients are non - negative and
sum to 1 .
Positive Definite ,Positive Semidefinite , etc
.
Recall : 147 : similar to vectorsC ' ' - '
If :)-
x" A x
mm244A147 i similar to quadratic forms : IAx
- -
(PD) d
Positive Definite
'TQLX ) = WAX 70 tf x to
Pus. Semidefinite :
QLX ) = X* AX 70
?(PSD)
-
Note : PD : for Hermitian matrices .
→ a function denpends on
( ND ) Qu ) X.similarly :
Negative Definite . ↳ x' ' Ax < 0
Neg .
- SemidefiniteL NSD ) X* AX SO
Also :
Indefinite
LID ).
What's x' Ax ?
X*AX : Quadratic forms .
x.*
xz=oMaximizing ? {x. xx ,
=L.
Suppose : A -- A
*,
A E Mn.
→ di Eda E . . - Edu eigvals
- -
T nxn X , Xz . . . Xn eigvec .Hermitian- - -
↳ For Hermitian Matrices ,Pick : arbitrary X .um eigenvectors can alwaysX =L
,X
, t Azxz t - . ' t An Xn.
be taken as an
Kil'
- orthonormal set .
= did xxx = ElExit ) L djXj ) = El dit '
. ( so it's a basis
T I T itj ,xixfo for RYE "
)dx*Ey=xEdie = E =
T put A back : A ( tix , -142kt - - tan Xn) . weighted= Edi A- Xi
average ofSo, for QCX ) > O t Xto ,
- - = Edit Xineed :Rio, fi .
di .
similarly ⇒
PSI: QLX) 70
didViNI: QCXKO
RicoNSD i QCXIEO di EO
- -
LDi some di> O
,some djCO
.
Spectral Decomposition :
x*AX -- Id , I 'd
,t t.dz/-dat.---ldntdn
- o 't o I o 't
Max & Min of X*AX Qu ) =x*AX = -214-1 di.
X =L , X , t 42×2 t . . . t An XnMax X
'AX = An ( put all weights on dn
.
)xxx -
- I Talked, ,
X = Xn . A ,E X
,E . - - Edn
.
th =/.
by : Imax -
min X* AX = A , ( put all weights on di
. )×*× "
realized by taking × -- × '
PSD
min X*AX = ? dz QCXI 30Xxx -
- I t , 30I realized by taking X - - X'
min ×* Ax 30
. . .So What ? ⇒ Hermitian matrices : physics observable .
X* AX 70
In mixed state , ( PSD )• Density matrices are positive semidefinite
?4itAHi> 30 .
a If p is separable ,then the partial transpose are PSD .