Minimal determination of a pure qutrit state and four-measurement protocol for pure qudit state

Abstract

Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. Peres conjectured that two unbiased bases may be sufficient to determine a pure state up to some finite ambiguities. Here we find that the Peres conjecture is correct in the case d = 3 where a pure state is determined up to at most six candidates states and is incorrect in the case of d = 4 for which a counterexample is constructed. We observe that the target state can be picked out from candidates states by an adaptive two-outcome measurement. We thus provided a minimal qutrit tomography protocol constituted of three measurements in contrast to the four measurements required in previous non-adaptive method. We also reduce a five-measurement protocol for a pure qudit state into three full-dimensional measurements and one two-outcome measurement protocol, which exhibits robustness to the usual white noise thus also applies to nearly pure state.