Quantum theories with alternative measurement postulates

Two measurements of an observable always yield identical outcomes when implemented in quick succession on a quantum system. In the standard formulation of non-relativistic quantum theory, this phenomenon is accounted for by the collapse of the state induced by a non-destructive measurement. In the first part of this thesis, we investigate the role of the collapse in quantum theory in a systematic way. We introduce the framework of Alternative-Measurement Theories (AMTs) defined as no-signalling foils of quantum mechanics sharing with it all standard postulates, except for the update rule assigning post-measurement states. By embedding quantum mechanics in a set of structurally similar theories, we are able to identify properties that depend on the collapse and show which of them are unique to the Lüders projection rule. We show that update rules can affect the ontological status of quantum states, or not allow for non-local correlations. The operational equivalence of different experimental strategies to measure product observables is found to rely on the chosen update rule, as well as the feasibility of protocols such as local tomography. The comparison with individual AMTs, such as "passive quantum theory" characterised by measurements which cause no state update, allows us to identify operational assumptions from which the Lüders rule can be derived. We show that the repeatability of measurement outcomes is insufficient; a stronger assumption on degenerate observables proves sufficient, at least for non-composite systems. In the second part, we focus on support uncertainty relations in spaces of prime dimensions. Tao derived a state-independent inequality which holds for the support sizes of a pure qudit state in two bases related by a discrete Fourier transform. We generalise Tao’s uncertainty relation to complete sets of (d+1) mutually unbiased bases in prime-dimensional spaces and investigate the sharpness of the obtained lower bounds.