We determine the set of the Bloch vectors for N -level systems, generalizing the familiar Bloch ball in 2-level systems. An origin of the structural difference from the Bloch ball in 2-level systems is clarified.
Bloch-vector spaces for N -level systems are investigated from the spherical-coordinate point of view in order to understand their geometrical aspects. We present a characterization of the space by using the spectra of (orthogonal) generators of SU(N ). As an application, we find a dual property of the space which provides an overall picture of the space. We also provide three classes of quantum-state representations based on actual measurements and discuss their state-spaces.
As a part of the construction of an information theory based on general probabilistic theories, we propose and investigate the several distinguishability measures and "entropies" in general probabilistic theories. As their applications, no-cloning theorems, information-disturbance theorems are reformulated, and a bound of the accessible informations is discussed in any general probabilistic theories, not resorting to quantum theory. We also propose the principle of equality for pure states which makes general probabilistic theories to be more realistic, and discuss the role of entropies as a measure of pureness.
We investigate a state discrimination problem in operationally the most general framework to use a probability, including both classical, quantum theories, and more. In this wide framework, introducing closely related family of ensembles (which we call a Helstrom family of ensembles) with the problem, we provide a geometrical method to find an optimal measurement for state discrimination by means of Bayesian strategy. We illustrate our method in 2-level quantum systems and in a probabilistic model with square-state space to reproduce e.g., the optimal success probabilities for binary state discrimination and N numbers of symmetric quantum states. The existences of families of ensembles in binary cases are shown both in classical and quantum theories in any generic cases.
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