2005 q 0035 spectral decomposition

8/13/2019 2005 q 0035 Spectral Decomposition

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Dirac Notation and Spectraldecomposition

Michele Mosca


2/27

review:Dirac notation

For any vector , we let denote , thecomplex conjugate of .

t

We denote by the innerproduct between two vectors and

defines a linear function that maps

(I.e. it maps any stateto the coefficientof its component)


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More Dirac notation

defines a linear operatorthat maps

Recall:this projection operatoralso corresponds to the

density matrixfor )

(I.e. projects a state to its component

This is a scalar so I

can moveitto front


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More Dirac notation

More generally, we can also have operatorslike


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Example of this Dirac notation

For example, the one qubit NOT gatecorresponds to the operator

e.g.

0110

1

1100

001010

001010

00110

The NOT gate is a 1-qubit unitary operation.

This is one more

notation to calculate

state from state and

operator

(sum of m atr icesapplied to ket vector)


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Special unitaries:Pauli Matrices in new notation

The NOT operation, is often called the X orXoperation.

01100110NOTX X

10

011100signflipZ

Z

0

00110

i

iiiY Y


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Recall:Special unitaries:

Pauli Matrices in new representation

Representation of

unitary operator


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What is ??iHte

It helps to start with the spectral

decomposition theorem.


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Spectral decomposition Definition: an operator (or matrix) M is

normal if MMt=MtM

E.g. Unitary matrices U satisfy UUt=UtU=I

E.g. Density matrices(since they satisfy=t; i.e. Hermitian) are also normal

Remember:Unitary matrix operators and density

matrices are normal so can be decomposed


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Spectral decomposition Theorem

Theorem: For any normalmatrix M,there is a unitary matrix P so that

M=PPtwhere is a diagonalmatrix.

The diagonal entries of are the

eigenvalues. The columns of Pencode the

eigenvectors.


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Example: Spectral decompositionofthe NOT gate

10

01

12

10

2

11

2

10

2

1

2

1

2

12

1

2

1

10

01

2

1

2

12

1

2

1

01

10

01100110

},{

}1,0{

X

XXX

X

XXX

This is the middle matrix in

above decomposition


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Spectral decomposition: matrixfrom column vectors

nnnnn

n

n

aaa

aaa

aaa

P

21

21

22221

11211

Column vectors


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Spectral decomposition: matrixas row vectors

nnnnn

n

n

aaa

aaa

aaa

P

2

1

**2

*1

*2

*22

*12

*

1

*

21

*

11

t

Adjont matrix = row vectors

S t l d iti i


15/27

Spectral decomposition: using rowand column vectors

i

iii

nn

n

PP

2

1

2

1

21

t

columni

rowi

th

th

i

i

n

00

010

00

00

0

00

00

2

1

From theorem


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Verifying eigenvectors andeigenvalues

2

22

21

2

1

21

22

1

2

1

21

2

nn

n

nn

n

PP

t Multiply on right by state vector Psi-2


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Wh i t l d iti f l?


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Why is spectral decomposition useful?Because we can calculate f(A)

iim

ii

i

ii

m

i

m

i

iii

ijji

m

m

mxaxf )( mm

xx

me

!

1

Note that

So

recall

Considere.g.

m-th power

Wh i t l d iti


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Why is spectral decompositionuseful? Continue last slide

ii

ii

i

i

i

m

m

im

m i

ii

m

im

m m

m

i

iiim

m

m

f

aa

aMaMf

M

= f( i)


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Now f(M) will be in matrixnotation

tt

tttt

P

a

a

PPaP

PaPPPaPPaPPf

MaMf

m

n

m

m

m

m m

m

n

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

11

)(

)(


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Same thing in matrix notation

nn

n

n

m

n

m

m

m

m

m

f

f

P

f

f

P

P

a

a

PPPf

2

11

21

1

1

)(

t

tt


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Same thing in matrix notation

iii i

nn

n

n

f

f

f

P

f

f

PPPf

2

11

21

1

)( tt

t t


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mportant ormu a nmatrix notation

iii

ifPPf )( t


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Using new notation this can be described like this:

We have the projection operators

and satisfying IPP 10

110 PP1P0M

2)Pr( bbPb

bb)b(p

P

b

bb

When we measure this observable M, the

probability of getting the eigenvalue b isand we are in that

case left with the state

00P0

11P1

We consider the projection operator or

observable Note that 0 and 1 are the eigenvalues

P l


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Polar

DecompositionLeft polar decomposition

Right polar decomposition

This is for square

matrices


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Gram-Schmidt

OrthogonalizationHilbert Space:

Orthogonality:

Norm:

Orthonormal basis:

2005 q 0035 spectral decomposition

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