Dirac Notation and Spectral
decomposition
Michele Mosca
review”: Dirac notation
t
ψ
For any vector ψ , we let
denote ψ , the
complex conjugate of ψ .
We denote by φ ψ  φ  ψ the inner
product between two vectors φ and ψ
ψ
defines a linear function that maps
φ  ψφ
(I.e.
ψ φ  ψ φ
… it maps any state
φ to the coefficient of its ψ component)
More Dirac notation
ψ ψ defines a linear operator that maps
ψ ψ φ ψ ψφ  ψφ ψ
This is a scalar so I
can move it to front
ψ (I.e.
ψ projects a state to its ψ component
Recall: this projection operator also corresponds to the
“density matrix” for ψ
)
More Dirac notation
More generally, we can also have operators
like θ ψ
θ ψ φ  θ ψφ  ψφ θ
Example of this Dirac notation
For example, the one qubit NOT gate
corresponds to the operator 0 1  1 0
e.g.
0
1  1 0  0

(sum of matrices
applied to ket vector)
 0 1 0 1 0 0
 0 10 1 00
 0 0 11
1
This is one more
notation to calculate
state from state and
operator
The NOT gate is a 1-qubit unitary operation.
Special unitaries:
Pauli Matrices in new notation
The NOT operation, is often called the X or
σX operation.
0 1 
X   X  NOT  0 1  1 0  

1
0


1 0 
Z   Z  signflip  0 0  1 1  

0  1
0  i 
Y   Y  i 0 1  i 1 0  

i
0


Recall: Special unitaries:
Pauli Matrices in new representation
Representation of
unitary operator
What is e
iHt
??
It helps to start with the spectral
decomposition theorem.
Spectral decomposition



Definition: an operator (or matrix) M is
t
t
“normal” if MM =M M
t
t
E.g. Unitary matrices U satisfy UU =U U=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
Spectral decomposition Theorem

Theorem: For any normal matrix M,
there is a unitary matrix P so that
M=PPt where  is a diagonal matrix.
The diagonal entries of  are the
eigenvalues.
 The columns of P encode the
eigenvectors.

Example: Spectral decomposition of
the NOT gate
X 0 1
X1  0
X 0 11 0
1 
1 
 1
 1
 1 0   2

0 1   2
2
2
X { 0 , 1 }      1  1  
 1



1
1 0 
 0  1 

 2
 2
2 
2 
1
1
1
1
 
0 
1
 
0 
1
2
2
2
2
X   
X  
X     
1 0 
X {  ,  }  

0

1


This is the middle matrix in
above decomposition
Spectral decomposition: matrix
from column vectors
 a11
a
21

P
 

an1
  ψ1
 a1n 

 a2 n 
  

 ann 
ψ2  ψn
a12
a22

an 2
Column vectors

Spectral decomposition:
eigenvalues on diagonal
 λ1

Λ



λ2
Eigenvalues on the
diagonal






λn 
Spectral decomposition: matrix
as row vectors
a

a
t

P 
 
 *
a1n
*
11
*
12
*
21
*
22
a
a

a2 n
*
 a   ψ1
 
ψ2
 a 

    
ψ
*
 ann   n
Adjont matrix = row vectors
*
n1
*
n2






Spectral decomposition: using row
and column vectors
PΛP
t
  ψ1
From theorem
ψ2
  λi ψi ψi
i

 λ1

ψn 



  λ1

0
 

0




λ2
0
λ2

0
  ψ1

ψ
 2
 

 ψ
λn   n







 0


th
 0  i row

 



 




i th colum n

 0

0
 0 
  λi 


 0
i


 λn 
0
0
1

0
Verifying eigenvectors and
eigenvalues
PΛP t ψ 2
  ψ1
  ψ1
Multiply on right by state vector Psi-2
ψ2
ψ2


 λ1

ψn 



 λ1

ψn 



λ2
λ2
  ψ1

  ψ2
 

 ψ
λn   n


 ψ2



  ψ1 ψ2

  ψ2 ψ2
 

 ψ ψ
λn   n 2






Verifying eigenvectors and
eigenvalues
  ψ1
ψ2

  ψ1
ψ2

 λ1
 
0


1 
λ
2
 
ψ n 

 


 0 
λn   

0
 λ2 
ψ n    λ2 ψ 2

 0 
useful
Why is spectral decomposition useful?
Because we can calculate f(A)
m-th power
Note that
So
ψ
i
ψi

m
 ψi ψi
recall
ψi ψ j  δij
m


m
  λi ψi ψi    λi ψi ψi
i
 i

Consider
f ( x)   am x
m
m
1
e.g. e   x m
m m!
x
Why is spectral decomposition
useful? Continue last slide
M


f M    am M   am   λi ψi ψi 
m
m
 i

m
m

m
  am  λi ψi ψi    am λi  ψi ψi
m
i
i  m

  f λi  ψi ψi
m
i
= f( i)
Now f(M) will be in matrix
notation
f ( M )   am M m
m

f ( PΛP )   am PΛP
t
m

 λ1


 P  am 
 m




m

t m

m t
  am PΛ P  P  am Λ  P
m
 m

m
t
m

am λ1


m
 t


  P  P 
m 
λn  





 Pt
m
am λn 

m

Same thing in matrix notation
 am λ1m
m
f ( PΛP t )  P 




 f  λ1 

 Pt
 P 



f  λn 
  ψ1
ψ2



 Pt
m
m am λn 
 f  λ1 
ψ n 


 ψ1

  ψ2
 
f  λn  
 ψn






Same thing in matrix notation
 f  λ1 
f ( PΛP t )  P 


  ψ1
ψ2

  f λi  ψi ψi
i

 Pt

f  λn 
 f  λ1 

ψ n 


 ψ1

  ψ2
 
f λn  
 ψn






Important formula in
matrix notation
f ( PP )   f i  i  i
t
i
“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 { 0 , 1 }
If we measure   (0 0  1 1 ) we get
2
with
probability
b
α
b
We knew it from beginning
but now we can generalize
Using new notation this can be described like this:

We have the projection operators P0  0 0
and P1  1 1 satisfying P0  P1  I
We consider the projection operator or
“observable” M  0P0  1P1  P1
 Note that 0 and 1 are the eigenvalues


When we measure this observable M, the
probability of getting the eigenvalue b is
2
Pr( b)  Φ Pb Φ  α b and we are in that
case left with the state
Pb 
b

b  b
p(b) b
Polar
Decomposition
Left polar decomposition
Right polar decomposition
This is for square
matrices
Gram-Schmidt
Orthogonalization
Hilbert Space:
Orthogonality:
Norm:
Orthonormal basis: