(Institute for Theoretical Physics, University of Cologne)
Quasiprobability representations, such as the Wigner function, play an important role in various research areas, such as quantum optics and quantum computation. The inevitable appearance of negativity in such representations is often regarded as a signature of nonclassicality, which has profound implications for quantum computation. However, little is known about the minimal negativity that is necessary in general quasiprobability representations. In this work I present an overview of quasiprobability representations for finite dimensional quantum systems and initiate a systematic study of the degree of negativity in these representations. I show that those representations with minimal negativity are in one-to-one correspondence with the elusive symmetric informationally complete measurements. In addition, most of them are automatically covariant with respect to the Heisenberg-Weyl groups. An interesting tradeoff between negativity and symmetry in quasiprobability representations is also revealed In the course of study. This work offers valuable insight on the distinction between quantum theory and classical probability theory in terms of negativity. Besides the foundational significance, it may have an impact on the current study on the theoretical foundations of quantum computation.
Reference: PRL 117, 120404 (2016)