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2
is the probability of
0
. To properly conserve
probability, a mathematical constraint known as the normalization condition must
apply to qubits:
j
>
;|
β
is the probability of
1
j
>
2
2
jα
jþjβ
j¼
1
:
(4.9)
This is an equation for a sphere and suggests a point on the surface of a sphere
[
4
,
5
]. For equal proportions of true and false probabilities, a state vector could be
denoted as
a
0
>¼ η½
11
,where
η ¼
1/
2 achieves vector normalization, that is,
√
2)
2
+(1/
2)
2
(1/
¼
1. This means 50 % chance of observing a
0
j
>
and 50 %
√
√
chance of observing a
1
.
The probability sphere for a given qubit portrays only the relative phase of
j
>
α
relative to
may have their own independent phases, but
often it is the phase difference that is important to quantum calculations.
Note that
a
β
. In general both
α
and
β
within the x-z plane
about the y-axis. Relative phase shift involves another angle
j
>
is thought of as rotating through an angle
θ
φ
in the x-y plane
about the z-axis. The result is that
a
can have any direction. In particular,
negative signs are permitted, for instance, either +
>
j
are possible.
This is considered a phase reversal. Note that the sphere can be misleading since it
does not show the phase for an ideal
0
j
ψ >
or
ψ >
j
j
>
or
1
j
>
; but both +
1
j
>
and
j
1
>
would
read out as a one with 100 % because phase does not affect the probability.
Suppose now that two qubits are available in a quantum mechanical system.
Thus there are four possible states. These states can be expressed using a direct
product, to be taken as below. A direct product is sometimes indicted with
, for
example
a
j
>
j
b
>:
>
0
ψ >¼
a
>
b
>¼½
a
1
b
>
a
2
b
;
a
1
a
2
b
1
b
2
j
a
>¼
j
b
>¼
(4.10)
In this case a direct product gives the state vector as
2
4
3
5
a
1
b
1
a
1
b
2
a
2
b
1
a
2
b
2
jψ >¼
(4.11)
For illustration, assume each qubit is prepared to have probability levels of 50 %.
Then
a
0
0
>¼ η½
11
and
b
>¼ η½
11
. Upon readout, there is a 25 % chance of any
given combination
00
j
>
,
01
j
>
,
10
j
>
,
11
j
>
. This may be expressed as
2
4
3
5
:
1
1
1
1
2
jψ >¼ η
(4.12)
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