ELEC9705 lecture 05 Operators, Coupling, Entaglement.pdf


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Unitary operators
Defined by the property U-1 = U †.
All Pauli matrices are unitary.
E.g.
y

0

i

i

0
0


y

y

i

i

0

0



0

i

y

i

0

i

i

1 0

0



1

0 1

1

y

y

Properties of unitary operators:
- Unitary operators conserve the norm of the vector they
operate on. Consider:

~
1

~
2

U

1

U

2



~ ~
2

2 U U

1

1

2

1

1

2

1

this means that the norms of the vectors before
and after operating with U must be the same
- The eigenvalues of a unitary operator satisfy:

1

ei

-The imaginary exponential of a Hermitian operator
(meaning A = A† ) is unitary:

T

iA

e

T

1

e

iA

e

iA†

iA



(e ) T