The representation of a qubit on a Bloch sphere is a geometric visualization that helps us understand the state of a qubit. While the coefficients in the mathematical description of a qubit's state may be complex numbers, the Bloch sphere representation can still be used to depict the qubit's state by considering the real and imaginary components separately.
Here's how it works:
Complex Number Representation: A qubit state is typically described using a complex superposition of basis states, such as |0⟩ and |1⟩. The general form of a qubit state is |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex probability amplitudes. These amplitudes can be written as α = r e^(iθ), where r is the magnitude and θ is the phase angle.
Magnitude and Phase: The magnitude of the complex probability amplitudes, |α|^2 and |β|^2, represents the probabilities of measuring the qubit in the corresponding basis states. These probabilities sum up to 1, ensuring normalization.
Bloch Sphere Representation: On the Bloch sphere, the north and south poles represent the basis states |0⟩ and |1⟩, respectively. The equator of the sphere corresponds to a superposition of both basis states. The Bloch vector, which starts at the center of the sphere and extends to a point on the surface, represents the qubit state.
Real and Imaginary Components: To represent the complex probability amplitudes on the Bloch sphere, we can split the complex numbers into their real and imaginary components. For example, α = r e^(iθ) can be written as α = a + ib, where a = r cos(θ) and b = r sin(θ). The real component (a) corresponds to the x-coordinate of the Bloch vector, while the imaginary component (b) corresponds to the y-coordinate.
Bloch Vector Length: The length of the Bloch vector is determined by the magnitude of the probability amplitudes, |α| and |β|. The larger the magnitude, the longer the vector. However, the phase angle (θ) affects the orientation of the Bloch vector around the z-axis.
By considering the real and imaginary components separately, the Bloch sphere representation can effectively depict the qubit state, even when complex probability amplitudes are involved. It provides a geometric visualization of the qubit's superposition and helps in analyzing its properties and transformations.