Huhhh. Continuing the thing from yesterday, I think the point is that if your kernel has orthonormal eigenvectors v1, ... vm all with eigenvalues 1, then it's like taking a state that's a bunch of fermions tensored together (v11|1> + v12|2> + ... + v1n|n>) ⊗ ... ⊗ (vm1|1> + vm2|2> + ... + vmn|n>) and observing it and seeing which basis state pops out. And if your kernel has orthonormal eigenvectors with eigenvalues that aren't all 1, then it behaves like a statistical mixture of these fermionic tensor products, where an eigenvalue being p tell you to independently flip a p-biased coin and keep the particle if it comes up heads.