Dirac Notation and Spectral decomposition

Michele Mosca

Dirac notation

For any vector , we let denote , the complex conjugate of .

ψ ψ

ψ

tψ

We denote by the inner product between two vectors and

ψφψφ φ

defines a linear function that maps

φψφ

ψ

ψ

φψφψ (I.e. … it maps any state to the coefficient of its

component) φ ψ

More Dirac notation

defines a linear operator that maps

ψφψφψψφψψ

ψψ

(Aside: this projection operator also corresponds to the

“density matrix” for ) ψ

θφψφψθφψθ

More generally, we can also have operators like ψθ

(I.e. projects a state to its component) ψ

More Dirac notation

For example, the one qubit NOT gate corresponds to the operator

e.g.

0110

1

1100

001010

001010

00110

The NOT gate is a 1-qubit unitary operation.

Special unitaries: Pauli Matrices

The NOT operation, is often called the X or σX operation.

01

100110NOTX X

10

011100signflipZ Z

0

00110

i

iiiY Y

Special unitaries: Pauli Matrices

What is ?? iHte

It helps to start with the spectral decomposition theorem.

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

Spectral decomposition

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

M=PPt where is a diagonal matrix. The diagonal entries of are the

eigenvalues. The columns of P encode the eigenvectors.

e.g. 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

Spectral decomposition

n

nnnn

n

n

aaa

aaa

aaa

P

ψψψ

21

21

22221

11211

Spectral decomposition

nλ

λ

λ

2

1

Λ

Spectral decomposition

nnnnn

n

n

aaa

aaa

aaa

P

ψ

ψψ

2

1

**2

*1

*2

*22

*12

*1

*21

*11

t

Spectral decomposition

iiii

nn

n

PP

ψψλ

ψ

ψψ

λ

λ

λ

ψψψ

2

1

2

1

21

Λ t

columni

rowi

th

th

ii

n

00

010

00

00

0

00

00

2

1

λ

λ

λ

λ

Verifying eigenvectors and eigenvalues

2

22

21

2

1

21

22

1

2

1

21

2Λ

ψψ

ψψψψ

λ

λ

λ

ψψψ

ψ

ψ

ψψ

λ

λ

λ

ψψψ

ψ

nn

n

nn

n

PP

t

Verifying eigenvectors and eigenvalues

222

21

2

1

21

0

0

0

10

ψλλ

ψψψ

λ

λ

λ

ψψψ

n

n

n

Why is spectral decomposition useful?

ii

m

ii ψψψψ

iii

mi

m

iiii ψψλψψλ

ijji δψψ

m

mmxaxf )( m

m

x xm

e !

1

Note that

So

recall

Consider e.g.

Why is spectral decomposition useful?

ii

ii

ii

im

mim

m iii

mim

m m

m

iiiim

mm

f

aa

aMaMf

ψψλ

ψψλψψλ

ψψλ

Same thing in matrix notation

tt

tttt

P

a

a

PPaP

PaPPPaPPaPPf

MaMf

mn

mm

m

mm

mn

m

mm

m

mm

m

mm

m

mm

m

mm

λ

λ

λ

λ

11

ΛΛΛ)Λ(

)(

Same thing in matrix notation

nn

n

n

mn

mm

m

mm

f

f

P

f

f

P

P

a

a

PPPf

ψ

ψψ

λ

λ

ψψψ

λ

λ

λ

λ

2

11

21

1

1

)Λ(

t

tt

Same thing in matrix notation

iii

i

nn

n

n

f

f

f

P

f

f

PPPf

ψψλ

ψ

ψψ

λ

λ

ψψψ

λ

λ

2

11

21

1

)Λ( tt

“Von Neumann measurement in the computational basis”

Suppose we have a universal set of quantum gates, and the ability to measure each qubit in the basis

If we measure we get with probability

}1,0{

2

bαb)10( 10

In section 2.2.5, this is described as follows

00P0 11P1

We have the projection operatorsand satisfying

We consider the projection operator or “observable”

Note that 0 and 1 are the eigenvalues When we measure this observable M, the

probability of getting the eigenvalue is and we

are in that case left with the state

IPP 10

110 PP1P0M

b2

ΦΦ)Pr( bbPb αbb

)b(p

P

b

bb