  We represent generalized density matrices of a $d$-complex dimensional
quantum system as a subcone of a real pointed cone of revolution in
$\mathbb{R}^{d^2}$, or indeed a Minkowskian cone in $\mathbb{E}^{1,d^2-1}$.
Generalized pure states correspond to certain future-directed light-like
vectors of $\mathbb{E}^{1,d^2-1}$. This extension of the Generalized Bloch
Sphere enables us to cater for non-trace-preserving quantum operations, and in
particluar to view the per-outcome effects of generalized measurements. We show
that these consist of the product of an orthogonal transform about the axis of
the cone of revolution and a positive real linear transform. We give detailed
formulae for the one qubit case and express the post-measurement states in
terms of the initial state vectors and measurement vectors. We apply these
results in order to find the information gain versus disturbance tradeoff in
the case of two equiprobable pure states. Thus we recover Fuchs and Peres'
formula in an elegant manner.
