Projective unitary equivalence and fusion frames

Abstract

Two finite sequences of vectors $\varPhi =(v_j)$ and $\varPsi =(w_j)$ in inner product spaces are unitarily equivalent if and only if their respective inner products (Gramian matrices) are equal (Corollary 2.1, §3.4).

Appendices

Notes

The characterisation of SICs up to projective unitary equivalence by their triple products was given by [AFF11]. This work was adapted to the general case (which includes MUBs) by [CW16] (see Theorems 8.1 and 8.2). The results of this chapter allow projective objects such as spherical (t, t)-designs and frames viewed as fusion frames to be classified (up to projective unitary equivalence) and their projective symmetries to be determined (see §9.3).

There is ongoing interest in fusion frames, e.g., see [BE15], the Fusion frame page of the Frame Research Centre, and www.fusionframe.org.

Tight signed frames were introduced in [PW02], where their relationship to the question of scaling to a tight frame (as presented here) was studied. The scaling question was also addressed in [KOPT13], who gave geometric descriptions of when a frame can be scaled to a tight frame.

Exercises

8.1.

Let $(\phi _j)_{j=1}^n$ be a sequence of vectors in ${\mathscr {H}}$, and $c_j\in \mathbb {F}$ be scalars.

(a) Show that there exists a representation of the form

$$\begin{aligned} f = \sum _j c_j\,\langle f,\phi _j\rangle \,\phi _j, \qquad \forall f\in {\mathscr {H}}, \end{aligned}$$

(8.29)

if and only if

$$\begin{aligned} \Vert f\Vert ^2 = \sum _j c_j\, |\langle f,\phi _j\rangle |^2, \qquad \forall f\in {\mathscr {H}}. \end{aligned}$$

(8.30)

(b) Suppose that (8.29) holds. Show that there is a unique choice for $(c_j)$ which minimises $\sum _j |c_j|^2$, and that this satisfies $c_j\in \mathbb {R}$, $\forall j$. Prove the analogue of (2.9), i.e.,

$$ \sum _j c_j\, \Vert \phi _j\Vert ^2 = \dim ({\mathscr {H}}). $$

8.2.

Let $(f_j)_{j=1}^n$ be vectors in ${\mathscr {H}}$, and $\sigma =(\sigma _j)$ , $\sigma _j\in \{\pm 1\}$. We say that $(f_j)$ is a signed frame with signature $\sigma$ for ${\mathscr {H}}$ if there exist (signed frame bounds) $A, B>0$ with

$$\begin{aligned} A\Vert f\Vert ^2\le \sum _j \sigma _j|\langle f, f_j\rangle |^2 \le B\Vert f\Vert ^2, \forall f\in {\mathscr {H}}. \end{aligned}$$

(8.31)

The signed frame operator $S;{\mathscr {H}}\rightarrow {\mathscr {H}}$ of a vector, signature pair $(f_j)$, $(\sigma _j)$ is given by

$$ Sf:= \sum _j \sigma _j \langle f, f_j\rangle f_j, \qquad \forall f\in {\mathscr {H}}. $$

(a) Show that the frame operator S of a signed frame with bounds A, B is invertible, with $(1/B)I_{\mathscr {H}}\le S^{-1}\le (1/A)I_{\mathscr {H}}$.

(b) For a signed frame $(f_j)$ with signature $\sigma$ and frame operator S, define the dual signed frame to be $(\tilde{f}_j)$ with signature $\sigma$, where $\tilde{f}_j:=S^{-1}f_j$. Show that the dual signed frame is a signed frame with frame operator $S^{-1}$, and one has the expansion

$$ f=\sum _j \sigma _j\langle f,\tilde{f}_j\rangle f_j =\sum _j \sigma _j\langle f, f_j\rangle \tilde{f}_j, \qquad \forall f\in {\mathscr {H}}. $$

Define the canonical tight signed frame to be $(f_j^\mathrm{can})$ with signature $\sigma$, where $f_j^\mathrm{can}:=S^{-1/2}f_j$, and show that this is a tight signed frame.

8.3.

Show that there is a tight signed frame of n vectors for $\mathbb {F}^d$ with signature $\sigma$ if and only if $\sigma$ takes the value $+1$ at least d times.

8.4.

Show that the Hadamard product satisfies

$$ (aa^*)\circ (bb^*) = (a\circ b)(a\circ b)^*, \qquad \forall a, b\in \mathbb {F}^d. $$

8.5.

Here we consider the space $\varPi _{r, s}^\circ (\mathbb {C}^d)$ of Exer. 6.17, which has dimension

$$n={d+r-1\atopwithdelims ()r}{d+s-1\atopwithdelims ()s}. $$

Let $v_1,\ldots , v_n\in \mathbb {C}^d$. Show that the following are equivalent

(a) The polynomials $p_j:z\mapsto \langle z,v_j\rangle ^r\langle v_j, z\rangle ^s$ are a basis for $\varPi _{r, s}^\circ (\mathbb {C}^d)$.

(b) The point evaluations $\delta _j:f\mapsto f(v_j)$ are a basis for dual space $\varPi _{r, s}^\circ (\mathbb {C}^d)'$.

(c) The $n\times n$ positive semidefinite matrix $A=[\langle v_j,v_k\rangle ^r\overline{\langle v_j, v_k\rangle }^s]$ is invertible.

Remark: Since $\varPi _{r, s}^\circ (\mathbb {C}^d)$ has a basis of ridge polynomials $z\mapsto \langle z,v\rangle ^r\langle v, z\rangle ^s$, it follows that A is invertible for some choices of $(v_j)$ .