# A characterisation of projective unitary equivalence of
finite frames

## Tuan-Yow Chien and Shayne Waldron

## Abstract:

Many applications of finite tight frames, e.g., the use of
SICs and MUBs in quantum information theory and
harmonic frames for the analysis of signals subject to erasures,
depend only on the vectors up to projective unitary equivalence.
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 (Gramian 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 apply this characterisation to
SICs, MUBs and harmonic frames.
We also extend our results to the projective similarity of vectors.

**Keywords:**
Projective unitary equivalence,
Gram matrix (Gramian),
harmonic frame,
equiangular tight frame,
SIC-POVM (symmetric informationally complex positive operator valued measure),
MUBs (mutually orthogonal bases),
triple products,
Bargmann invariants,
frame graph,
cycle space,
chordal graph,
projective symmetry group

**Math Review Classification:**
Primary 05C50, 14N05, 14N20, 15A83;
Secondary 15A04, 42C15, 81P15, 81P45

**Length:** 21 pages

**Last Updated:** 17 February 2015

## Availability: