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Two types of states are widely used in quantum mechanics, namely(deterministic-coefficient) pure states and statistical mixtures. A densityoperator can be associated with each of them. We here address a third type ofstates, that we previously introduced in a more restricted framework. Thesestates generalize pure ones by replacing each of their deterministic ketcoefficients by a random variable. We therefore call them Random-CoefficientPure States, or RCPS. We analyze their properties and their relationships withboth types of usual states. We show that RCPS contain much richer informationthan the density operator and mean of observables that we associate with them.This occurs because the latter operator only exploits the second-orderstatistics of the random state coefficients, whereas their higher-orderstatistics contain additional information. That information can be accessed inpractice with the multiple-preparation procedure that we propose for RCPS, byusing second-order and higher-order statistics of associated randomprobabilities of measurement outcomes. Exploiting these higher-order statisticsopens the way to a very general approach for performing advanced quantuminformation processing tasks. We illustrate the relevance of this approach witha generic example, dealing with the estimation of parameters of a quantumprocess and thus related to quantum process tomography. This parameterestimation is performed in the non-blind (i.e. supervised) or blind (i.e.unsupervised) mode. We show that this problem cannot be solved by using onlythe density operator \rho of an RCPS and the associated mean value Tr(\rho A)of the operator A that corresponds to the considered physical quantity. Wesucceed in solving this problem by exploiting a fourth-order statisticalparameter of state coefficients, in addition to second-order statistics.Numerical tests validate this result.

cornell university

Beyond the density operator and Tr(\rho A): Exploiting the higher-order  statistics of random-coefficient pure states for quantum information  processing