Quantum phase

Multiplying a quantum state by one global complex phase does not change any measurement probability; relative phases between components can affect interference.

The global phase is common to the whole state; relative phases control interference between state components.

The phase difference between nonzero amplitudes in a specified basis, for example between $a$ and $b$ in $a|1⟩+b|2⟩$. Unlike a global phase, it affects quantum interference when the two basis components are recombined. It is encoded in the off-diagonal density matrix elements and depends on the phase convention for the basis states.

A geometric phase is a quantum phase associated with the geometry of a path of state rays, after separating the dynamical phase. The Berry phase is its cyclic case governed by the adiabatic theorem; more general cyclic evolution can also have a geometric phase.

For an energy eigenstate with instantaneous energy eigenvalue $E(t)$, the dynamical phase isIt comes from the Schrödinger equation and depends on the time spent evolving. In an ideal cycle under the adiabatic theorem, a periodic choice of instantaneous energy eigenstate gives $|\Psi (T)⟩=e^{i(\delta +\gamma )}|\Psi (0)⟩$, with Berry phase $\gamma$.

A global phase multiplies an entire quantum state by one complex number of modulus one. It does not change any measurement probability, unlike a relative phase between components.

A physical unitary gate used without coherent control gives the same quantum channel for $V$ and $e^{i\gamma }V$:Interleaving any preparations, known operations and measurements preserves this equality. Equivalently, each fixed measurement branch with $k$ uses gains only the overall factor $e^{ik\gamma }$, which cancels in its probability. Thus unrestricted repeated uncontrolled uses do not determine the common phase of the eigenvalues. A supplied controlled unitary gate is a different resource: $|0⟩⟨0|\otimes I+|1⟩⟨1|\otimes V$ changes by a relative control phase when $V$ is replaced by $e^{i\gamma }V$. Such controlled access permits quantum phase estimation to measure that phase relative to the identity branch.