August 2015
Beginner
270 pages
8h 21m
English
Content preview from Introduction to Quantum Physics and Information Processing
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58 Introduction to Quantum Physics and Information Processing
Definition 3.1. The tensor product of two matrices A of dimensions m × n
and B of any dimensions is given by
A ⊗ B =
A
11
B A
12
B . . . A
1n
B
A
21
B A
22
B . . . A
2n
B
.
.
.
.
.
.
.
.
.
A
m1
B A
m2
B . . . A
mn
B
. (3.31)
Thus we have
|00i =
"
1
0
#
⊗
"
1
0
#
=
1
0
0
0
; |01i =
"
1
0
#
⊗
"
0
1
#
=
0
1
0
0
;
|10i =
"
0
1
#
⊗
"
1
0
#
=
0
0
1
0
; |11i =
"
0
1
#
⊗
"
0
1
#
=
0
0
0
1
.
(3.32)
Thus we have the natural basis for the 4-dimensional vector space from those
of two 2-dimensional spaces.
Example 3.5.1. Direct products: to express σ
x
on a 2-qubit state as a matrix,
we take the direct product of two σ
x
s acting on each single
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