Generating random vectors using Mathematica and an inequality

Question

Consider $n$-dimensional vectors $c^i = \left(c^i_1, c^i_2 ...c^i_n \right)$ , whose entries are complex and generated at random. These vectors are normalized such that

$$|c^i|^2 = 1.$$

Using Mathematica, how do I construct such a set of $k$ vectors $\{ c^i\}$? Does it numerically satisfy the following relation in the limit of large-$n$,

$$|(c^i)^* c^j| <<1, \quad i \neq j ?$$

Note: As an explicit example, can this be seen numerically for $n \approx 1000$?

Answer 1

One way to generate some random vectors that satisfy your normalization rule:

v = RandomPoint[Sphere[1000],100] /. {x_?NumericQ :> x Exp[I RandomReal[{0, 2 Pi}]]}

v will be a set of 100 1000-d vectors where

Norm/@v

will yield all 1. Now we look at your test

test = Outer[Abs@*Dot, Conjugate[v], v, 1];

ListDensityPlot[test, PlotLegends -> Automatic,ScalingFunctions -> "Log"]

A slice: