Equiangular lines

The goal is to attack the following problem:

Problem Find as many 1-spaces in ℂn as possible, such that any two of them make the same angle.

Here "angle" is defined as follows: equip ℂn with the standard hermitian form. Given 1-spaces l, m in ℂn choose unit vectors u in l and v in m and define the inner product between l and m to be the absolute value of the inner product of u and v (which is independent of the choice of u, v), and the angle between l and m to be the arccos of their inner product. A collection of 1-spaces (or "lines", or projective points) as in the problem is called equiangular.

It is not hard to see that n2 is an upper bound on the size of an equiangular set of lines.

Conjecture 1 There exist n2 equiangular lines in ℂn.

Why should we believe this? Because numerically such lines have been found up to n=45, which up to machine precision are equiangular.
These examples are of the following shape. Let A be the matrix diag(1,z,...,zn–1) where z=exp(2πi/n) and let S be the shift matrix
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
(for n=4). Modulo scalar matrices, A and S commute. Their images in the projective unitary group PUn generate a group PH isomorphic to ℤn x ℤn. (In Un, they generate the Heisenberg group H which has the group of scalar matrices <μI | μn=1> as centre, the quotient by which is ℤn x ℤn.)

Conjecture 2 There exists an equiangular PH-orbit of size n2.

Fact A set of n2 equiangular lines li would maximise ∑i,j <li,lj>4.

This leads to the following numeric approach: optimise the function that on input of a line l computes the latter expression for the PH-orbit of l. Using numerical optimisation, a suitable l ("fiducial state") has been found for n up to 45, thus numerically supporting conjecture 2. See this page.

 

Low dimensions

Conjecture 2 has been verified in dimensions 2,3,...,15,19,24,35,48 in the papers below.

 

Normaliser

The normaliser of PH in PUn is not hard to compute. Of course it operates on the lines whose orbit is an equiangular set with size n2. Modulo PH the normaliser is isomorphic to SL2(ℤn). It was observed by Zauner that a suitable "fiducial state" is often an eigenvector of an element of order 3 in this normaliser---unfortunately the corresponding eigenvector has multiplicity larger than one.

 

Some references

SIC-POVMs: A new computer study
Numerical SICPOVM solutions
Equiangular tight frames from Paley tournaments
Symmetrically information complete quantum measurements
Some remarks on Heisenberg frames and sets of equiangular lines
On SIC-POVMs and MUBs in dimension 6
Quantum measurements and finite geometry
Joe Renes's dissertation
Markus Grassl's slides
Gerhard Zauner's dissertation


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ERC Starting grant for Daniel Dadush
26-9-2018

Daniel Dadush (CWI) has been awarded an ERC Starting Grant of 1.5 ME for his proposal ‘Towards a Quantitative Theory of Integer Programming’. With this grant, Dadush aims to revolutionize the understanding of integer programming (IP), the most popular method used today for finding optimal solutions to real-world optimization problems.

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Nikhil Bansal has been awarded a Vici grant of 1.5 ME. He is one of the 35 academics to receive this grant from NWO in 2018. Bansal aims to use his grant to develop new algorithmic methods to make discrete decisions in a continuous way. He expects this to lead to applications in the fields of logistics, bio-informatics, chip design and machine learning.

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NWO, following a shortlist provided by the DIAMANT board, has decided to award 3 PhD positions to young DIAMANT members: Dion Gijswijt (TU Delft), Jan Steffen Müller (RUG) and Arno Kret (UvA).
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19-12-2017

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