Simultaneous Estimation of Dimension, States and Measurements: Computation of representative density matrices and POVMs

Abstract: This investigation continues a program aiming at obtaining effective quantum models to describe measurement statistics. In [Stark, arXiv:1209.5737 (2012)], we have described how the Gram matrix associated to the prepared states and the performed POVM elements can be estimated via convex relaxation of a rank minimization problem. This Gram matrix determines the density matrices and the POVM elements uniquely up to simultaneous rotations (with respect to the Hilbert-Schmidt inner product) in the vector space of Hermitian matrices. However, when the description of the experiment needs to be connected to textbook quantum mechanics, explicit expressions for the states and the POVM elements in terms of positive semidefinite matrices are required. In this paper, we describe a heuristic algorithm that takes the state-meaurement Gram matrix as input and searches explicit realizations of this Gram matrix in terms of quantum mechanically valid density matrices and POVM elements.