Conal representation of quantum states and non-trace-preserving quantum operations

Abstract

We represent generalized density matrices of a d-complex dimensional quantum system as a subcone of a real pointed cone of revolution in ${R}^{{d}^{2}},$ or indeed a Minkowskian cone in ${E}^{{1,d}^{2}\ensuremath{-}1}.$ Generalized pure states correspond to certain future-directed lightlike vectors of ${E}^{{1,d}^{2}\ensuremath{-}1}.$ This extension of the generalized Bloch sphere enables us to cater for non-trace-preserving quantum operations, and in particular 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 formulas 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 trade-off in the case of two equiprobable pure states. Thus we recover Fuchs and Peres's formula in an elegant manner.