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Chapter 4 Projective unitary equivalences of frames
One natural question to consider when studying frames is when two frames are alike. A classic notion to describe likeness is the idea of unitary equivalence. Unitary transformations are de-sirable as they preserve the inner product relations and hence the geometry of the transformed vectors. The unitary operators form a group as well.
When studying projective objects like lines, one hopes to be able to leverage the machinery of vector spaces. This can be accomplished by considering the lines as vectors of a fixed length (customarily unit length) and consider vectors scaled by arbitrary constants (of unit modulus) as equivalent. The analogue is captured in the following definition.
Definition
Let := (fj)j2J and := (gj)j2J be finite sequences of vectors in a Hilbert space H. Then and are projectively unitarily equivalent if there exists a unitary operator U 2 U(H) and a sequence (cj)j2J with jcjj = 1 such that cjU(fj) = gj; for all j 2 J: and are unitarily equivalent if and only if (cf. Corollary 2.1.9) where C is the diagonal matrix with complex diagonal entries cj of unit modulus.
Prima facie, solving (4.0.2) looks promising, but for complex inner product spaces, the only thing to work with is hwj; wki = cjckhvj; vki; 8j; k:
From a survey of the literature, it appears that projective unitary equivalence has only been previously calculated in a few situations using situation specific approaches.
These situations are: H = Rd by considering all possible cj = 1 in (4.0.2) or by using two– graphs in the case of equiangular lines (Chapter 11 of [GR01]), for a specific construction of MUBs (using Weyl-Heisenberg or Clifford groups) via symplectic spreads, Hadamard matrices, and for the equal–norm tight frames of n vectors in C2 robust to n 2 erasures (Section 5 of [CCKS97], Section 3 of [GR09], Section 3 of [Ben07], for SIC-POVMs Theorem 3 of [AFF11], and Section 2 of [HP04]).
Following in the footsteps of [MACR01], a general method is developed for determining whether any two finite sequence of vectors are projectively unitarily equivalent.
The graph theory definitions and basic results in this Chapter can be found in a standard textbook on graph theory, e.g., Chapter 1 of [Die10]. The key result in this chapter is Theorem 4.2.2 which shows that a sequence of vectors (vj) is determined up to projective unitary equivalence by its m–products for all m. While these results apply to arbitrary finite sequences of vectors, it is sometimes convenient to refer to them as frames on the space that they span.
The proof of Theorem 4.2.2 identifies certain subsets of the m–products which classify the sequences. These depend on which of the m–products are nonzero, a fact conveniently encap-sulated by the notion of a frame graph.
Definition
The frame graph (cf. Section 3 of [AN13]) of a sequence of vectors (vj)j2J is the graph with vertices fvjgj2J (or the indices j themselves) and an edge between vj and vk, j 6= k () hvj; vki =6 0:
Remark
Projectively unitarily equivalent frames have the same frame graph.
Complete frame graphs
A generic sequence of vectors is unlikely to have any pairs of orthogonal vectors most of the time. Hence, its corresponding frame graphs is often a complete graph. Appleby et al showed that d2 equiangular vectors in Cd are characterised up to projective unitary equivalence by their triple products (3–products) in [AFF11]. This section extends their proof to cover frames with complete graphs, however these results do not extend to a general sequence of vectors.
Definition
The angles of a sequence of vectors = (vj)j2J are the jk 2 R=(2 Z) defined by hvj; vki = jhvj; vkijei jk hvj; vki 6= 0:
Since hvj; vki = hvk; vji, these angles satisfy the relation
Remark
A sequence of vectors might only have few angles, e.g., an orthogonal basis has no angles.
Lemma
Let = (vj)j2J and = (wj)j2J be finite sequences of vectors in Hilbert spaces, with angles jk and jk0. Then and are projectively unitarily equivalent if and only if
- Their Gramians have entries with equal moduli, i.e.,
- Their angles are “gauge equivalent”, i.e., there exist j 2 R=(2 Z)
Since the frame graph of is complete, the projective unitary equivalence class of (equiv-alently G, S or gr(S)) is characterised by the triple products of (Theorem 4.1.4). It is only necessary to consider triple products with distinct indices. If an index is repeated twice or thrice, by (4.1.1), the triple products depends only on . These triple products are also independent of the ordering of the indices.
Example 4.1.7 (Equiangular lines in Cd).
Let be a sequence of n equiangular unit vectors (lines) in Cd, with C > 0. Then is determined by its triple products (up to projective unitary equivalence). When n = d2 (a SIC-POVM), this is the result in Theorem 3 and its remarks in [AFF11].
Characterisation of projective unitary equivalence
A sequence of n vectors (vj)j2J is determined up to projective unitary equivalence by its m– products for all 1 m n. Verifying that two projectively unitarily equivalent frames induce the same m–products is a standard fare calculation. The crux is in proving the con-verse. The strategy will be to construct a sequence of vectors (wj)j2J with given m–products (vj1 ; : : : ; vjm ). It amounts to reconstructing all Gramians for a given set of m–products.
Example
A sequence of n vectors (vj)j2J can be constructed in different ways for a given a Gram matrix G. If V is a frame of n vectors spanning a dimension d space, then G is positive semidefinite of rank d and unitarily diagonalisable by using only the m–products of = (vj)j2J . The frame graph of and the modulus of each entry of G are known by (4.2.1). It remains to determine the arguments of the (nonzero) inner products. These correspond to edges of the frame graph. Without loss of generality, assume the frame graph of is connected. Otherwise apply the method to each connected component of .
Let T be a spanning tree of with root vertex r. While fanning out from the root r (breadth-first search), multiply the vertices v 2 n frg by unit scalars so that the arguments of the inner products corresponding to the edges of take arbitrarily assigned values. Continue in this fashion until you have reached every vertex.
The remaining undefined entries of the Gram matrix G are given by the edges of which are not in T . Since T is a spanning tree, adding each such edge to T gives an m–cycle. The corresponding nonzero m–product has all inner products already determined, except the one corresponding to the edge added. It is therefore uniquely determined by the m–product.
Corollary
A frame is projectively unitarily equivalent to a real frame if and only if its m–products are all real.
Reconstruction from the m–products
Theorem 4.2.2 shows that only a small subset of the m–products is required to determine a sequence of vectors up to projective unitary equivalence.
Definition
A determining set is a subset of the m–products (or the corresponding indices) that can deter-mine all m–products.
Corollary
Let be the frame graph of a sequence of n vectors . For each connected component j of let Tj be a spanning tree. Then is determined up to projective unitary equivalence by the following m–products
The 2–products
(ii) The m–products, 3 m n, used to obtain j from Tj by completing m–cycles (these have indices in j), as detailed in the proof of Theorem 4.2.2.
Remark
If M is the number of edges of which are not in any Tj, then it is sufficient to know all of the 2–products and M of the m–products, 3 m n.
Remark
The m–products from (i) and (ii) are a determining set.
Although Example 4.2.6 shows the m–products of (ii) may not be a minimal set of m-products needed to characterise a sequence of vectors, it is a minimal subset from which the m–products can be determined using only the proof of Theorem 4.2.2.
Abstract
Acknowledgements
Symbols
1 Introduction
2 Background
2.1 Frame theory
2.2 SIC-POVM existence problem (Zauner’s conjecture)
2.3 Weyl-Heisenberg group
2.4 Clifford group
2.5 Three dimensional SIC-POVMs
3 Reverse engineering numerical SIC-POVMs
3.1 Galois theory fundamentals
3.2 SIC-POVM field structure
3.3 Methodology
3.4 Improvements
3.5 Results
4 Projective unitary equivalences of frames
4.1 Complete frame graphs
4.2 Characterisation of projective unitary equivalence
4.3 Reconstruction from the m–products
4.4 Similarity and m–products for vector spaces
4.5 Projectively equivalent harmonic frames
5 Projective symmetry groups
5.1 Introduction
5.2 Tight frames and the complement of a frame
5.3 Projective invariants
5.4 The algorithm
5.5 The extended projective symmetry group
5.6 Group frames, nice error bases, SIC-POVMs and MUBs
5.7 Harmonic frames
6 Nice error frames
6.1 Background
6.2 Nice error frames and canonical abstract error groups
6.3 Calculations
6.4 SIC-POVMs from nonabelian group in 6 dimensions
6.5 Equivalence to Heisenberg SIC-POVMs
Bibliography
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Equiangular lines, projective symmetries and nice error frames
