Exploiting the higher-order statistics of random-coefficient pure states for quantum information processing

Quantum mechanics and hence quantum information processing methods widely use two types of states, namely (deterministic-coefficient) pure states and statistical mixtures. Density operators can be associated with them. We here address a third type of states, whose ket coefficients are random variables, as opposed to the deterministic coefficients of usual pure states. We therefore call them Random-Coefficient Pure States, or RCPS. We define physical setups that yield RCPS. We analyze the properties of RCPS and show that they contain much richer information than the density operator and mean of observables that we associate with them , because that operator only exploits the second-order statistics of the random state coefficients, whereas their higher-order statistics contain additional information. That information can be accessed in practice with the multiple-preparation procedure that we propose for RCPS, by using second-order and higher-order statistics of associated random probabilities of measurement outcomes. Exploiting these higher-order statistics yields a very general approach to advanced quantum information processing. We illustrate its relevance with a generic quantum parameter estimation problem related to quantum process tomography , especially considering its blind/unsupervised version. We show that this problem cannot be solved by using only the density operator ρ of an RCPS and the associated mean value Tr( ρ ) of the operator  corresponding to the considered physical quantity. We solve it by exploiting a fourth-order statistical parameter of state coefficients, in addition to second-order statistics. Numerical tests validate this result and show that the proposed method yields accurate parameter estimation for the considered number of state preparations.