Open Problems:35 - Revision history

Krzysztof Onak: updated header

2013-03-07T01:52:27Z

updated header

Krzysztof Onak at 05:27, 16 November 2012

2012-11-16T05:27:58Z

Krzysztof Onak: 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..."

2012-11-13T01:40:18Z

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..."

New page

{{DISPLAYTITLE:Problem 35: Maximal Complex Equiangular Tight Frames}}
{{Infobox
|label1 = Proposed by
|data1 = Joel Tropp
|label2 = Source
|data2 = [[Workshops:Kanpur_2009|Kanpur 2009]]
|label3 = Short link
|data3 = http://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.

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-{{DISPLAYTITLE:Problem 35: Maximal Complex Equiangular Tight Frames}}
+{{Header
-{{Infobox
+|title=Maximal Complex Equiangular Tight Frames
-|label1 = Proposed by
+|source=kanpur09
-|data1 = Joel Tropp
+|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