Many applications of finite tight frames (e.g., the use of SICs and mutually unbiased bases (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 result is equivalent to finding a rank-one matrix completion of a certain matrix. We give an algorithm to recover the sequence of vectors (up to projective unitary equivalence) from a small subset of these projective invariants and apply it to SICs, MUBs, and harmonic frames. We also extend our results to the projective similarity of vectors.
The existence of maximal sets of equiangular lines (SIC-POVMs) is of interest to the mathematics and physics communities due to their connection to quantum information theory, quantum cryptography, spherical 2-designs and cubature rules. This thesis looks at a way to recover analytic SIC-POVMs from numerical SIC-POVMs, including the discovery of a new analytic SIC-POVM in 17 dimensions. It also develops a way to comprehensively find nice error bases (used for finding SIC-POVMs). Finally, a general theory to classify sequences of vectors up to projective unitary equivalence is developed and applied to study the projective symmetry groups of sequences of vectors.
Bargmann invariants, a class of unitary-invariant quantities arising from the overlaps of quantum state vectors, provide a profound and unifying framework for understanding the relative geometry of the projective Hilbert space. This survey offers a comprehensive overview of their theoretical characterization and practical applications, with particular emphasis on recent progress in determining the full structure of their admissible set. The core of this review demonstrates how these invariants serve as a powerful tool for characterizing the intrinsic geometry of the space of quantum states, leading to applications in determining local unitary equivalence and constructing a complete set of polynomial invariants for mixed states. On the operational side, we review the cycle-test quantum circuit for the direct estimation of Bargmann invariants without full state tomography, and demonstrate their utility in witnessing quantum imaginary, discriminating local unitary equivalence, and detecting entanglement via partial-transpose moments---with explicit complete invariant sets provided for two-qubit systems. By connecting fundamental geometric classification with experimentally feasible estimation protocols,this survey establishes Bargmann invariants as indispensable probes of the relational, noncommutative, and geometric structure of quantum states, and identifies key open problems for multipartite high-dimensional systems and for quantum resource theories.
In this brief report, we consider the equivalence between two sets of
m+1 bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree
m matrix polynomials are unitarily equivalent; i.e.
UAiV†=Bi for
0≤i≤m where
U and
V are unitary and
(Ai,Bi) are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices
U and
V.
In this brief report, we consider the equivalence between two sets of
m+1 bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree
m matrix polynomials are unitarily equivalent; i.e.
UAiV†=Bi for
0≤i≤m where
U and
V are unitary and
(Ai,Bi) are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices
U and
V.
Let
H be a finite-dimensional complex Hilbert space, and let
w(P,M) be a normalized probability weight assigned to an outcome projection
P as it occurs in a projective measurement
M. For fixed
P, let
GP≅U(P⊥) be the group of unitaries acting identically on
ranP and arbitrarily on
P⊥. We classify the
GP-orbits of projective measurements containing
P: two measurements lie in the same orbit exactly when the multisets of ranks of their complementary outcomes agree. The orbit with profile
λ=(r1,…,rk) is a compact homogeneous space of real dimension
(d−rankP)2−∑jrj2. On maximal rank-one measurements there is one orbit, so context independence is equivalent to
GP-invariance, with equality of the corresponding uniform defects; invariance under two-level complementary unitaries already suffices. For arbitrary projective measurements, every context containing an outcome
P=I coarsens to the unique binary context
{P,I−P}. Consequently, refinement consistency alone is equivalent to context independence. The finite orbit space carries a natural rank-profile refinement graph, whose diameter is
d−rankP−1 when
P=I. We prove a stability theorem that compares the binary-coarsening path with a shortest path in this graph followed by one complementary unitary. In dimension at least three, Gleason's theorem converts the maximal-context invariance condition and the all-context refinement condition, on their respective domains, into the Born form
Tr(ρP). The results are structural characterizations, not independent physical derivations of context independence.