This paper develops an operator-theoretic framework for characterising a phenomenon we call phase-free recurrence: the exact return of a quantum state to itself under unitary time evolution, with no global phase change whatsoever.Starting from the unitary time-evolution operator , we construct a Hermitian operator via symmetrisation, yielding the cosine operator . Applying the real sign function to this operator defines a new operator , which we prove to be an involution — a self-inverse linear operator — whenever none of the cosines of the energy eigenvalues vanish. The involutory structure of induces a canonical decomposition of the Hilbert space into three mutually orthogonal subspaces corresponding to eigenvalues +1, −1, and 0. We study the kernel subspace in detail, characterising the discrete periodic times at which each energy component enters it.Using this structure we derive a precise algebraic criterion for phase-free recurrence. For a two-level system with energy eigenvalues and , we prove that phase-free recurrence occurs if and only if the ratio is rational. This result is generalised to arbitrary n-level systems, where it takes the form of a simultaneous proportionality condition on the integers labelling the kernel-recurrence times of each energy component.The framework is purely algebraic, relying only on the spectral theorem and elementary properties of involutions, and applies to any quantum system with a discrete non-degenerate spectrum.
Alim Mohammed (Sun,) studied this question.