Unknown Quantum States and Operations, a Bayesian View

The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. In this paper, we motivate and review two results that generalize de Finetti's theorem to the quantum mechanical setting: Namely a de Finetti theorem for quantum states and a de Finetti theorem for quantum operations. The quantum-state theorem, in a closely analogous fashion to the original de Finetti theorem, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an "unknown quantum state" in quantum-state tomography. Similarly, the quantum-operation theorem gives an operational definition of an "unknown quantum operation" in quantum-process tomography. These results are especially important for a Bayesian interpretation of quantum mechanics, where quantum states and (at least some) quantum operations are taken to be states of belief rather than states of nature.Comment: 37 pages, 3 figures, to appear in "Quantum Estimation Theory," edited by M.G.A. Paris and J. Rehacek (Springer-Verlag, Berlin, 2004

Negativity Bounds for Weyl-Heisenberg Quasiprobability Representations

The appearance of negative terms in quasiprobability representations of quantum theory is known to be inevitable, and, due to its equivalence with the onset of contextuality, of central interest in quantum computation and information. Until recently, however, nothing has been known about how much negativity is necessary in a quasiprobability representation. Zhu proved that the upper and lower bounds with respect to one type of negativity measure are saturated by quasiprobability representations which are in one-to-one correspondence with the elusive symmetric informationally complete quantum measurements (SICs). We define a family of negativity measures which includes Zhu's as a special case and consider another member of the family which we call "sum negativity." We prove a sufficient condition for local maxima in sum negativity and find exact global maxima in dimensions $3$ and $4$. Notably, we find that Zhu's result on the SICs does not generally extend to sum negativity, although the analogous result does hold in dimension $4$. Finally, the Hoggar lines in dimension $8$ make an appearance in a conjecture on sum negativity.Comment: 21 pages. v2: journal version, added reference