While playing with what I called "quantum matching", the following problem arose: construct a map $F$ from the unit sphere $S_2$ in $R^3$ to itself such that $F(X)$ is orthogonal to $X$ plus has one of the properties:
Strong: if 3 vectors $X_1,X_2, X_3$ are independent then 3 vectors $F(X_1), F(X_2), F(X_3)$ are also independent.
Weak: if 3 vectors $X_1,X_2, X_3$ are orthogonal then 3 vectors $F(X_1), F(X_2), F(X_3)$ are independent.
It is not hard to prove that such maps with the Strong property don't exist. I proved the existence of such maps satisfying the Weak property, but my proof uses the Axiom of Choice and some version of the Continuum Hypothesis. Granted the maps I am after supposed to be bad, at least discontinuous.
Are things like Axiom of Choice really needed here?
The discussion and proofs are in http://arxiv.org/abs/quant-ph/0201022