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Global Phase

Global phase is one of the most important — and most frequently misunderstood — concepts in quantum mechanics. This page explains what it is, why it does not affect measurements, and why it still matters when working with multi-qubit systems or SU(2) representations.

What is Global Phase?

Given a quantum state |ψ⟩, the state e^(iφ)|ψ⟩ — where φ is any real number — is called a global phase shift of |ψ⟩.

The factor e^(iφ) is a complex number of magnitude 1. It scales every amplitude uniformly and does not change the relative weights of any superposition.

Why Global Phase is Unobservable

All measurable quantities in quantum mechanics are expectation values of the form:

⟨ψ|Â|ψ⟩

Under a global phase shift:

⟨e^(iφ)ψ| Â |e^(iφ)ψ⟩ = e^(-iφ) e^(iφ) ⟨ψ|Â|ψ⟩ = ⟨ψ|Â|ψ⟩

The phases cancel exactly. No measurement can distinguish |ψ⟩ from e^(iφ)|ψ⟩.

Global Phase on the Bloch Sphere

On the Bloch sphere, global phase is completely invisible. The map from spinors to Bloch vectors:

|ψ⟩  ──►  (x, y, z)

treats |ψ⟩ and e^(iφ)|ψ⟩ as identical — they land on the same point.

This is why the Bloch sphere has a well-defined geometric meaning: it factors out all global phase freedom and represents only the physically distinct states.

When Global Phase Matters

Global phase becomes physically meaningful as soon as it becomes relative phase. In a two-qubit system, a phase applied to only one component is no longer global — it is relative, and it affects measurement outcomes.

For example, in the Bell state:

|Φ⟩ = (|00⟩ + |11⟩) / √2

applying a phase to |00⟩ only changes the state to:

|Φ'⟩ = (e^(iφ)|00⟩ + |11⟩) / √2

which is a measurably different state.

Global Phase and SU(2)

In the SU(2) representation, global phase corresponds to the difference between SU(2) and U(1):

  • SU(2) matrices have determinant exactly 1.
  • A global phase e^(iφ) applied to a 2×2 unitary multiplies the determinant by e^(2iφ).
  • Removing global phase freedom is exactly what the det = 1 constraint imposes.

The double cover SU(2) → SO(3) is equivalent to saying: rotations of the Bloch sphere do not distinguish global phase, but spinor evolution in SU(2) tracks it.

Connection to the Ecosystem

In rqm-core:

  • Spinors are normalized to remove magnitude freedom, but global phase is preserved in the representation.
  • Functions that convert to Bloch vectors (bloch.state_to_bloch) implicitly factor out global phase.
  • Functions that work in SU(2) preserve the full phase structure.

Understanding global phase is important when interpreting the output of rqm-qiskit simulations: statevector outputs preserve global phase, but measurement probabilities and Bloch vector visualizations do not.

Summary

Global phase is unobservable in single-qubit measurements but becomes significant in multi-qubit entangled systems. The Bloch sphere representation factors out global phase; the SU(2) representation preserves it.