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 .