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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
