Let X be a countable set describing the outcomes of a quantum measurement, and let { E x } x ∈ ∈ X denote a collection of trace-non-increasing completely positive maps, such that the sum of all E x is trace-preserving, i.e. tr ⁡ ⁡ ( ∑ ∑ x E x ( ρ ρ ) ) = tr ⁡ ⁡ ( ρ ρ ) for all positive operators ρ ρ .