A global phase does not change a qubit’s Bloch-sphere point because it multiplies the entire state vector by the same unit-magnitude factor. The Bloch sphere represents the physical state after that overall phase redundancy is removed. A relative phase between the qubit’s two basis-state amplitudes is different: it affects the state and determines its position around the sphere.
What does global phase mean for a qubit?
A pure qubit state is a normalized vector, usually written as |ψ⟩ = α|0⟩ + β|1⟩, where |α|² + |β|² = 1. The complex numbers α and β are the amplitudes of the computational-basis states |0⟩ and |1⟩.
A global phase is a common factor of unit magnitude applied to the whole vector: |ψ′⟩ = eiγ|ψ⟩. It changes the mathematical representative, but not the physical single-qubit state. The National Academies of Sciences, Engineering, and Medicine puts it plainly: “the global phase α has no physical significance whatsoever,” and a single-qubit state can be specified by two real parameters. The statement appears in Box 2.3 of Quantum Computing: Progress and Prospects (2019).
How the Bloch-sphere angles separate global and relative phase
A normalized pure qubit can be written with an explicit overall phase as:
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|ψ⟩ = eiα(cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩).
Removing the shared factor eiα leaves the same physical state. The remaining angles locate the state on the unit Bloch sphere: θ sets its polar position, while φ sets its azimuth. This is why the sphere describes the state with two real parameters rather than recording every feature of the complex vector. The Introduction to Quantum Information Science’s Bloch-sphere explanation likewise factors out the overall phase and treats vectors differing only by global phase as physically indistinguishable.
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Global phase: shared by both amplitudes
Multiplying α and β by the same eiγ changes neither their relative phase nor their magnitudes. For example, |ψ⟩ and −|ψ⟩ differ by the global factor −1, so they represent the same single-qubit state.
Relative phase: changes the point
The factor eiφ in the parameterization applies to the |1⟩ amplitude relative to |0⟩. It is not a shared factor, so it generally changes the state’s azimuth. For instance, (|0⟩ + |1⟩)/√2 and (|0⟩ − |1⟩)/√2 have different relative phases and correspond to different Bloch-sphere points.
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Two quick checks that the shared phase cancels
The pure-state density operator
Represent a pure state by ρ = |ψ⟩⟨ψ|. If |ψ′⟩ = eiγ|ψ⟩, then:
ρ′ = |ψ′⟩⟨ψ′| = eiγe−iγ|ψ⟩⟨ψ| = ρ.
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The phase and its complex conjugate cancel, leaving the same state representation. Density matrices are also useful beyond pure states, including descriptions of noise and subsystems of entangled systems; IBM Quantum Learning introduces those broader uses.
Bloch-vector components
For a normalized pure qubit, the Bloch vector can be written as (2 Re(α*β), 2 Im(α*β), |α|² − |β|²). Under a shared phase, α*β stays unchanged because the phase on α is conjugated and cancels the phase on β. The magnitude squares are unchanged as well, so all three components remain the same.
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Why measurement probabilities alone are not the whole explanation
In a computational-basis measurement, outcome 0 has probability |α|² and outcome 1 has probability |β|²; a common phase leaves both probabilities unchanged. Microsoft Learn’s qubit overview describes these probability rules and notes the irrelevance of an overall sign. But unchanged probabilities for this particular measurement are only a quick check. The more general reason is that states differing by a global phase represent the same physical state, as explained in the Bloch-sphere treatment in Introduction to Quantum Information Science.
What the Bloch sphere does—and does not—represent
The familiar unit-sphere surface represents pure states of a single qubit. The sphere is not a literal drawing of every component of the complex state vector; it encodes the physical state modulo its unobservable global phase.
Mixed states require a broader picture. Density matrices can represent mixtures and reduced states, such as a qubit considered on its own when it is entangled with another system. In the single-qubit case, mixed states are represented inside the Bloch ball rather than only on its surface. For systems of multiple qubits, one qubit’s sphere cannot encode the full joint state; Microsoft Learn notes the limits of the Bloch-sphere representation for multi-qubit states.
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