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A qubit’s global phase does not move its point on the Bloch sphere because it multiplies the entire state vector by the same unit-magnitude factor. The sphere represents the physical state after this shared phase has been removed. A relative phase between the qubit’s two basis-state amplitudes is different: it changes the state and determines its position around the sphere.
What global phase means for a qubit
A normalized pure qubit is commonly written as |ψ⟩ = α|0⟩ + β|1⟩, where |α|² + |β|² = 1. The complex numbers α and β are amplitudes for the computational-basis states |0⟩ and |1⟩.
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Multiplying the whole ket by eiγ gives |ψ′⟩ = eiγ|ψ⟩. This applies the same phase factor to both amplitudes. The vector’s written representation changes, but the physical state it represents does not. The National Academies of Sciences, Engineering, and Medicine puts it plainly in Quantum Computing: Progress and Prospects (2019): “It turns out that the global phase α has no physical significance whatsoever.” National Academies, Chapter 2, Box 2.3.
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After factoring out the shared phase, a pure qubit can be written in the conventional form:
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|ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩
Here, θ sets the polar angle and φ sets the azimuthal angle. The common phase has disappeared, while the relative phase between the |0⟩ and |1⟩ amplitudes remains. That relative phase generally changes the state, so it cannot simply be discarded. The Introduction to Quantum Information Science’s Bloch-sphere discussion likewise treats states identical up to global phase as physically indistinguishable.
Global phase: same point
If |ψ′⟩ = eiγ|ψ⟩, both amplitudes receive the same factor. For instance, |ψ⟩ and −|ψ⟩ differ by a global phase of π, yet represent the same point on the sphere.
Relative phase: potentially a different point
In (|0⟩ + |1⟩)/√2, the amplitudes have one relative phase; in (|0⟩ − |1⟩)/√2, the relative phase differs by π. These states occupy different positions on the Bloch sphere. The minus sign belongs to one amplitude relative to the other, not to the whole vector.
Two quick checks that the shared phase cancels
Density-operator check
A pure state’s density operator is ρ = |ψ⟩⟨ψ|. Replacing the state by |ψ′⟩ = eiγ|ψ⟩ gives:
ρ′ = |ψ′⟩⟨ψ′| = eiγe−iγ|ψ⟩⟨ψ| = ρ
The phase on the ket is canceled by its complex conjugate on the bra. Since the density operator is unchanged, so are the state’s observable predictions.
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Bloch-vector check
For amplitudes α and β, the Bloch vector can be written as (2 Re(α*β), 2 Im(α*β), |α|² − |β|²). Under a shared phase, α and β become eiγα and eiγβ. Their magnitude squares are unchanged, and the product of the conjugate of the first amplitude with the second remains α*β. Every Bloch-vector component therefore stays the same.
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The familiar unit-sphere surface represents pure states of a single qubit, with global-phase-equivalent vectors identified as one state. It is not a literal picture of every detail of the complex vector. A convention that makes one amplitude real and nonnegative, or one that writes an explicit global-phase factor, can describe the same state; the choice of representative does not alter the physics.
The surface picture should not be extended unqualified to every quantum state. Density matrices also describe mixed states, which are represented within the Bloch ball rather than only on its surface. They are also useful for describing a subsystem when another entangled system is ignored. A single-qubit Bloch sphere does not encode the complete joint state of a multi-qubit system. See Microsoft Learn’s qubit overview and IBM Quantum Learning’s introduction to density matrices.
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