Difference between revisions of "Open Problems:35"
(Created page with "{{DISPLAYTITLE:Problem 35: Maximal Complex Equiangular Tight Frames}} {{Infobox |label1 = Proposed by |data1 = Joel Tropp |label2 = Source |data2 = [[Workshops:Kanpur_2009|Kan...") |
m (updated header) |
||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | {{ | + | {{Header |
| − | + | |source=kanpur09 | |
| − | | | + | |who=Joel Tropp |
| − | | | ||
| − | |||
| − | |||
| − | |||
| − | |||
}} | }} | ||
Consider a system of unit vectors $\{ x_k : k = 1, 2, \dots, N \}$ in $\mathbb{C}^d$. It can be shown that the maximum inner product among these vectors satisfies the Welch bound | Consider a system of unit vectors $\{ x_k : k = 1, 2, \dots, N \}$ in $\mathbb{C}^d$. It can be shown that the maximum inner product among these vectors satisfies the Welch bound | ||
Latest revision as of 01:52, 7 March 2013
| Suggested by | Joel Tropp |
|---|---|
| Source | Kanpur 2009 |
| Short link | https://sublinear.info/35 |
Consider a system of unit vectors $\{ x_k : k = 1, 2, \dots, N \}$ in $\mathbb{C}^d$. It can be shown that the maximum inner product among these vectors satisfies the Welch bound $$ \max_{i \neq j} |\langle x_i, x_j \rangle | \geq \sqrt{\frac{N-d}{d(N - 1)}}. $$ Miraculously, when this bound is attained, the modulus of the inner product between every pair of vectors is identical. Such a configuration is referred to as an equiangular tight frame (ETF).
It can be shown that the cardinality $N$ of an ETF is $\mathbb{C}^d$ must satisfy the bound $N \leq d^2$. When this bound is attained, the ETF is referred to as a maximal ETF. In other words, a maximal ETF is a system of $d^2$ unit vectors in $\mathbb{C}^d$ whose pairwise inner products share the modulus $(d+1)^{-1/2}$.
A striking geometric fact about maximal ETFs is that each one corresponds with a regular simplex consisting of $d^2$ points embedded in the set of rank-one, trace-one, complex, Hermitian matrices with dimension $d$. This correspondence is achieved by mapping each vector $x$ in the ETF to the matrix $xx^*$. Researchers believe that there is a maximal ETF for every dimension $d$. This question, so far, has resisted all efforts at solution.
