statement of “mere” engineering and find a physical principle? In this paper, we prove some strong results in this regard in the context of finite dimensional Hilbert spaces. In particular, we find the unique form of the quantum mechanical Born Rule that makes it resemble the classical Law of Total Probability (LTP) as closely as possible in at least two senses. Both come from a significant majorization result which may be of general interest for resource theory. This way of tackling the distinction between quantum and classical arises naturally in the quantum interpretive project of QBism [@Fuchs:2013; @Fuchs:2016], where the Born Rule is seen as an empirically motivated constraint that one adds to probability theory when using it in the context of alternative (complementary) quantum experiments. We expect the techniques developed here to give an alternative way to explore the paradigm of negativity and to be of use for a range of practical problems.

The standard procedure in quantum theory for generating probabilities starts with an observer, or agent, assigning a quantum state $\rho$ to a system. When the agent plans to measure the system, she represents the outcomes of her measurement with a positive operator-valued measure (POVM) $\{D_j\}$. Assigning 