Quantum measurement with minimal state alteration

Abstract

We construct a positive-operator valued measurement (POVM) that only minimally changes the state of the quantum system. The problem is delimited to a 2-element POVM and 2-dimensional Hilbert space. The elements are given by M₁ = √αA^† and M2 = U √I − αAA^†. If the measurement outcome is M1, then the system is unaltered. However, if the outcome is M2, the state is altered. We focus on the unitary operator U that will minimize ‖ρ_f − ρ₀‖. The minimum of this function is zero for a = 0 , which is trivial, and nonzero, otherwise. We conclude that there are no unitary operators that will leave a state completely unaltered. However, we were able to find a unitary operator that will minimize state alteration.