A characterisation of projective unitary equivalence of finite frames

It is well known that two finite sequences of vectors in inner product spaces are unitarily equivalent if and only if their respective inner products (Gram matrices) are equal. Here we present a corresponding result for the projective unitary equivalence of two sequences of vectors (lines) in inner product spaces, i.e., that a finite number of (Bargmann) projective (unitary) invariants are equal. This is based on an algorithm to recover the sequence of vectors (up to projective unitary equivalence) from a small subset of these projective invariants. We consider the implications for the characterisation of SICs, MUBs and harmonic frames up to projective unitary equivalence. We also extend our results to projective similarity of vectors.