Quantum States as Vectors
In Quantum States and Probabilities,
we saw that a quantum state can be expressed as
a combination of the 
and
states as follows:
The parameters 
and 
are the amplitudes whose squares are the probabilities
of collapsing to 0 or 1, respectively.
We can express the quantum state more compactly.
Since a qubit is a blended state of just
the two quantum states,
and ,
we can represent it as a 
vector of
two rows and one column
(see Working with Matrices and Vectors):
In this vector, the top element represents the amplitude
for the pentagon qubelets, and
the bottom represents the amplitude for the triangle qubelets
in the qubit.
The vector for the idealized quantum state
will have 
and 
:
Pictorially, this vector represents the following qubelets in the qubit:
The triangle qubelet is drawn
with a dotted outline as it’s not active in this qubit.
Likewise, the vector for the idealized quantum state
will have 
and 
:
In this case, this vector corresponds to the following qubelets in the qubitshown.
The pentagon
qubelet is drawn with a dotted outline since it’s
not active in the qubit.
Consider a qubit with
seven pentagon qubelets
and three inverted triangle qubelets,
shown in the following figure:
You can calculate its quantum state as described in Quantum States and Probabilities to get:
Rewrite this equation as a vector:
Sometimes, you’ll find it convenient
to write the previous vector in terms of the
vectors for the idealized quantum states
and ,
as shown here:
In general, the quantum state of a single qubit in terms of vectors is:
The squares of the amplitudes and
still add up to 1 for the vector
to be a valid representation of a quantum state:
