Quantum instrument

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