Visualizing a Qubit: The Bloch Sphere, Explained
Early in any study of quantum computing, the Bloch sphere appears as a tool for visualizing the state of a single qubit. It represents that state as a point on the surface of a sphere:
This explanation covers both how the Bloch sphere works and the mathematical reasoning behind it.
From Four Dimensions to Three
A qubit in an arbitrary state is described by two complex coefficients, a and b. If these were real numbers, the state would have just two degrees of freedom and would be straightforward to plot. Because they are complex, there are four real parameters to account for. The Bloch sphere is a clever mapping that compresses this four-dimensional space into something we can visualize.
We start by expressing each complex coefficient in polar form, separating its magnitude and phase angle. Doing so for both coefficients and expanding the state gives:
Since a global phase has no measurable effect on a qubit, we can multiply the entire state by a factor that removes the phase from the first coefficient. This leaves us with only the relative phase between the two components. Calling this relative phase φ, the state simplifies to:
This reduces our four real parameters to three. Still, we have not used the normalization constraint: the squared magnitudes of the coefficients must sum to one:
Applying this constraint and using the Cartesian representation of complex numbers leads to an algebraic rearrangement that yields a familiar form. If we rename one of the variables to z, the equation becomes that of a unit sphere. This allows us to switch to spherical coordinates, where the state is expressed in terms of a polar angle (from the positive z axis) and an azimuthal angle (from the positive x axis):
Substituting these spherical coordinates back into the quantum state reduces it to just the two angles. At this point, the mapping appears straightforward: each point on the sphere corresponds to a unique quantum state. However, a subtle problem emerges when we check the endpoints. If we set the polar angle to zero, the state we derive is not the state we expect at the "north pole" of the sphere. Instead, it produces a state that is orthogonal to the intended one.
The states |0⟩ and |1⟩ are orthogonal by definition, and we see them as opposite poles on the Bloch sphere. But the raw mapping does not give us that. To fix this, we stretch the state space by using the half-angle θ/2 instead of θ in the state equation. With this substitution, setting θ=0 correctly yields the |0⟩ state.
Reading States on the Sphere
With the math in place, we can interpret quantum superpositions visually. Consider the important state:
Because the coefficient of |0⟩ is real, the polar angle is θ=π/2. The azimuthal angle is 0. In spherical coordinates, this places the state on the positive x axis. Other standard states like |+⟩, |−⟩, |0⟩, and |1⟩ can be located similarly:
A key insight is that measurement probabilities for a state like |+⟩ are 1/2 for both |0⟩ and |1⟩. The Bloch sphere makes this intuitive, since |+⟩ lies halfway between the poles on the equator. In fact, all points on the equator have these same measurement probabilities; what differentiates them is only the relative phase, encoded in the azimuthal angle φ.
As a final mapping example, take a state where the coefficient of |1⟩ is larger than that of |0⟩ and has a complex phase. Rewriting the coefficients using Euler's formula and factoring out a global phase, we can express the state in standard spherical form. Extracting the angles places the qubit at a point that is visibly biased upward on the sphere, indicating a higher probability of measuring |1⟩. The same geometric reasoning applies when measuring in other bases, such as the |+⟩/|−⟩ basis, where the sphere's orientation gives an immediate sense of the outcome probabilities.
Why Global Phases Don't Matter
The derivation relies on discarding a global phase, a step that deserves justification. For a qubit in superposition, the measurement probabilities in the standard basis depend on the squared magnitudes of the coefficients. If we multiply the entire state by a phase factor such as eiγ, both coefficients pick up that same factor. When we compute probabilities, the magnitude of each coefficient is squared, and the phase factor disappears entirely.
Because measurement is the only way to observe a quantum state, and this phase does not affect any measurement outcome in any basis, it is physically irrelevant and can be safely ignored.
