On a universal descriptor for quantum state information: a non-commutative extension of the Born rule

We introduce the universal Q function, a one-parameter class of probability representations that encode quantum state information relative to an arbitrary collection of self-adjoint observables. These are always positive probability distributions on Rⁿ for any choice of quantum state, making them pertinent for analysis and visualization of quantum information with respect to a collection of (possibly non-commuting) self-adjoint operators. The class is formulated using the framework of continuous measurements and serves as a natural extension of the Born rule into the non-commutative setting. Intuitively, the new representation can be viewed as the probabilistic landscape generated by the accumulation of information obtained by continuous monitoring, which in turn squeezes the support of the probability towards the structure of minimal uncertainty states. In great generality, the framework identifies the natural smearing kernel that relates the Wigner distribution to its Husimi counterpart, which need not be Gaussian as in the kernel that yields the classical Husimi function in phase space. The class is formulated most naturally when the self-adjoint observables generate unitary representations of Lie groups, just as in the case of a single self-adjoint operator. We illustrate our framework by deriving the universal Q function for various examples and show that it recovers the Born rule and all Husimi functions in the literature, without any explicit reference to coherent states.