The Mathematics of Qubits: Bloch Sphere and State Vectors
A qubit is the basic unit of quantum information. Unlike a classical bit (0 or 1), a qubit can exist in a superposition of both states simultaneously.
Mathematically, a qubit is represented as a state vector in a 2-dimensional complex Hilbert space.
1️⃣ Qubit State Vector
A general qubit state can be written as:
∣
π
⟩
=
πΌ
∣
0
⟩
+
π½
∣
1
⟩
∣Ο⟩=Ξ±∣0⟩+Ξ²∣1⟩
Where:
∣
0
⟩
=
[
1
0
]
∣0⟩=[
1
0
]
∣
1
⟩
=
[
0
1
]
∣1⟩=[
0
1
]
πΌ
,
π½
∈
πΆ
Ξ±,Ξ²∈C
Normalization condition:
∣
πΌ
∣
2
+
∣
π½
∣
2
=
1
∣Ξ±∣
2
+∣Ξ²∣
2
=1
∣
πΌ
∣
2
∣Ξ±∣
2
is the probability of measuring the qubit in state
∣
0
⟩
∣0⟩
∣
π½
∣
2
∣Ξ²∣
2
is the probability of measuring the qubit in state
∣
1
⟩
∣1⟩
2️⃣ Global Phase Irrelevance
Multiplying a state vector by a global phase
π
π
π
e
iΟ
does not change measurement outcomes:
∣
π
⟩
∼
π
π
π
∣
π
⟩
∣Ο⟩∼e
iΟ
∣Ο⟩
This allows us to describe any qubit using only two real parameters.
3️⃣ Parametrization via ΞΈ and Ο
A qubit can be expressed as:
∣
π
⟩
=
cos
π
2
∣
0
⟩
+
π
π
π
sin
π
2
∣
1
⟩
∣Ο⟩=cos
2
ΞΈ
∣0⟩+e
iΟ
sin
2
ΞΈ
∣1⟩
Where:
0
≤
π
≤
π
0≤ΞΈ≤Ο
0
≤
π
<
2
π
0≤Ο<2Ο
Here:
π
ΞΈ → angle from the north pole (z-axis)
π
Ο → rotation around the z-axis (azimuthal angle)
This is the basis of the Bloch sphere representation.
4️⃣ The Bloch Sphere
The Bloch sphere is a geometric representation of a single qubit as a point on a unit sphere.
The north pole corresponds to
∣
0
⟩
∣0⟩
The south pole corresponds to
∣
1
⟩
∣1⟩
Any other point represents a superposition
Coordinates of the qubit on the Bloch sphere:
π₯
=
sin
π
cos
π
,
π¦
=
sin
π
sin
π
,
π§
=
cos
π
x=sinΞΈcosΟ,y=sinΞΈsinΟ,z=cosΞΈ
Visualization:
|0⟩
●
|
|
|
●
|1⟩
Points on the surface represent pure states, inside the sphere represent mixed states.
5️⃣ Quantum Gates as Rotations
Quantum gates correspond to rotations of the Bloch vector:
Pauli-X gate: rotation 180° around x-axis
Pauli-Y gate: rotation 180° around y-axis
Pauli-Z gate: rotation 180° around z-axis
Hadamard gate: rotates |0⟩ to an equal superposition (45° rotation)
Mathematically:
∣
π
new
⟩
=
π
∣
π
⟩
∣Ο
new
⟩=U∣Ο⟩
Where
π
U is a unitary 2x2 matrix (
π
†
π
=
πΌ
U
†
U=I).
6️⃣ Measurement in Computational Basis
When measuring in the
{
∣
0
⟩
,
∣
1
⟩
}
{∣0⟩,∣1⟩} basis:
Probability of |0⟩:
π
(
0
)
=
∣
⟨
0
∣
π
⟩
∣
2
=
cos
2
(
π
/
2
)
P(0)=∣⟨0∣Ο⟩∣
2
=cos
2
(ΞΈ/2)
Probability of |1⟩:
π
(
1
)
=
∣
⟨
1
∣
π
⟩
∣
2
=
sin
2
(
π
/
2
)
P(1)=∣⟨1∣Ο⟩∣
2
=sin
2
(ΞΈ/2)
After measurement, the qubit collapses to the observed state.
7️⃣ Key Takeaways
Concept Bloch Sphere Interpretation
Qubit state vector Point on unit sphere in β³
0⟩,
Superposition Any point on sphere surface
ΞΈ Polar angle (0 → Ο)
Ο Azimuthal angle (0 → 2Ο)
Quantum gate Rotation of vector around an axis
Measurement Projects vector onto z-axis (
✅ Summary
A single qubit is fully described by two angles ΞΈ and Ο, up to a global phase.
The Bloch sphere provides an intuitive way to visualize qubit states, superpositions, and gate operations.
Quantum gates = rotations on the Bloch sphere, measurement = collapse along z-axis.
This geometric perspective is essential for understanding quantum algorithms, entanglement, and quantum error correction.
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