Random access test as an identifier of nonclassicality

Random access codes are an intriguing class of communication tasks that reveal an operational and quantitative difference between classical and quantum information processing. We formulate a natural generalization of random access codes and call them random access tests, defined for any finite collection of measurements in an arbitrary finite dimensional general probabilistic theory. These tests can be used to examine collective properties of collections of measurements. We show that the violation of a classical bound in a random access test is a signature of either measurement incompatibility or super information storability. The polygon theories are exhaustively analyzed and a critical difference between even and odd polygon theories is revealed.