Positivization and regularization of quantum phase space distributions

Abstract

It is shown that, to any quasiprobability distribution corresponding to a given density operator, one can associate a class of positive distributions in an extended space. In situations where the quasiprobability distributions are singular, these distributions are shown to provide regularized versions thereof. The positive distribution introduced by Drummond and Gardiner (1981) in the context of the Glauber-Sudarshan P-representation is shown to arise as a special case. One can thus associate a positive distribution to the Wigner function such that its moments give the averages of Weyl-ordered operators with respect to the given density operator. It is also found that one can associate to the Glauber-Sudarshan P-function, a distribution in the extended space which, though not positive, interestingly, involves the Wigner function (1932). A measurement scheme which directly yields these positive distributions is presented.