Abstract
We describe a general procedure for associating a minimal informationally-complete quantum measurement (or MIC) and a set of linearly independent post-measurement quantum states with a purely probabilistic representation of the Born Rule. Such representations are motivated by QBism, where the Born Rule is understood as a consistency condition between probabilities assigned to the outcomes of one experiment in terms of the probabilities assigned to the outcomes of other experiments. In this setting, the difference between quantum and classical physics is the way their physical assumptions augment bare probability theory: Classical physics corresponds to a trivial augmentation---one just applies the Law of Total Probability (LTP) between the scenarios---while quantum theory makes use of the Born Rule expressed in one or another of the forms of our general procedure. To mark the irreducible difference between quantum and classical, one should seek the representations that minimize the disparity between the expressions. We prove that the representation of the Born Rule obtained from a symmetric informationally-complete measurement (or SIC) minimizes this distinction in at least two senses---the first to do with unitarily invariant distance measures between the rules, and the second to do with available volume in a reference probability simplex (roughly speaking a new kind of uncertainty principle). Both of these arise from a significant majorization result. This work complements recent studies in quantum computation where the deviation of the Born Rule from the LTP is measured in terms of negativity of Wigner functions.
Highlights
Quantum information theory represents a change of perspective
The distinction between quantum and classical arises naturally in the quantum interpretive project of QBism [12,13], 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 quantum experiments
Suppose our agent has a preferred reference process consisting of a measurement to which they ascribe the minimal informationally complete POVMs (MICs) {Hi} and, upon obtaining outcome i, the preparation of a state σi, drawn from a linearly independent set of postmeasurement states {σi}. (See Fig. 1.) In their choice of this reference process, they require linearly independent postmeasurement states so that the inner products trD jσi will uniquely characterize the operators D j
Summary
Informationally complete POVMs furnish a convenient way to bypass the language of quantum states, making quantum theory analogous to classical stochastic process theory, in which one puts probabilities in and gets probabilities out. The invertibility of is ensured by the linear independence of the MIC and postmeasurement sets This implies that the coefficients of ρ in the σi basis may be written as an application of the matrix on the vector of probabilities, ρ=. Prior work has given special attention to the reference procedure where the measurement and postmeasurement states are the same SIC [12,30,31] In this case we denote by SIC and Eq (5) takes the simple form. The inverse of a columnstochastic matrix is generally a column-quasistochastic matrix; in our case, inspection of Eq (3) reveals that −1 is column stochastic What would it mean if could equal I? Let p denote the column-quasistochastic matrix associated with a MIC and a proportional postmeasurement set. Det p det SIC with equality if and only if the MIC is a SIC
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