Dirac Notation and Spectral decomposition

Michele Mosca

Dirac notationFor 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 notationFor 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

10011100signflipZ Z

0

00110i

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

1001

1210

211

210

21

21

21

21

21

1001

21

21

21

21

0110

01100110

},{

}1,0{

X

XXX

X

XXX

Spectral decomposition

n

nnnn

n

n

aaa

aaaaaa

P

ψψψ

21

21

22221

11211

Spectral decomposition

nλ

λλ

2

1

Λ

Spectral decomposition

nnnnn

n

n

aaa

aaaaaa

P

ψ

ψψ

2

1

**2

*1

*2

*22

*12

*1

*21

*11

t

Spectral decomposition

iiii

nn

n

PP

ψψλ

ψ

ψψ

λ

λλ

ψψψ

2

1

2

1

21

Λ t

columni

rowi

th

th

ii

n

00

01000

0000000

2

1

λ

λ

λλ

Verifying eigenvectors and eigenvalues

2

22

21

2

1

21

22

1

2

1

21

2Λ

ψψ

ψψψψ

λ

λλ

ψψψ

ψ

ψ

ψψ

λ

λλ

ψψψ

ψ

nn

n

nn

n

PP

t

Verifying eigenvectors and eigenvalues

222

21

2

1

21

0

0

0

10

ψλλ

ψψψ

λ

λλ

ψψψ

n

n

n

Why is spectral decomposition useful?

iim

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

Pa

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

Pf

fP

Pa

a

PPPf

ψ

ψψ

λ

λψψψ

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

Pf

fPPPf

ψψλ

ψ

ψψ

λ

λψψψ

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

b2ΦΦ)Pr( bbPb α

bb)b(p

Pb

bb