Quantum theory can be formulated using a small number of mathematical postulates. These postulates describe how quantum systems interact and evolve as well as describing measurements and probabilities of measurement outcomes. The measurement postulates are logically independent from the other postulates, which are dynamical and compositional in nature. In this thesis we study all theories which have the same dynamical and compositional postulates as quantum theory but different measurement postulates. In the first part we introduce the necessary tools for this task: the operational approach to physical theories (general probabilistic theories) and the representation theory of the unitary group. Following this we introduce a framework which is used to describe theories with modified measurement postulates and we classify all possible alternative measurement postulates using representation theory. We then study informational properties of single systems described by these theories and compare them to quantum systems. Finally we study properties of bi-partite systems in these theories. We show that all bi-partite systems in these theories violate two properties which are met by quantum systems: purification and local tomography.
Quantum theory can be formulated using a small number of mathematical postulates. These postulates describe how quantum systems interact and evolve as well as describing measurements and probabilities of measurement outcomes. The measurement postulates are logically independent from the other postulates, which are dynamical and compositional in nature. In this thesis we study all theories which have the same dynamical and compositional postulates as quantum theory but different measurement postulates. In the first part we introduce the necessary tools for this task: the operational approach to physical theories (general probabilistic theories) and the representation theory of the unitary group. Following this we introduce a framework which is used to describe theories with modified measurement postulates and we classify all possible alternative measurement postulates using representation theory. We then study informational properties of single systems described by these theories and compare them to quantum systems. Finally we study properties of bi-partite systems in these theories. We show that all bi-partite systems in these theories violate two properties which are met by quantum systems: purification and local tomography.
Using the existing classification of all alternatives to the measurement postulates of quantum theory we study the properties of bi-partite systems in these alternative theories. We prove that in all these theories the purification principle is violated, meaning that some mixed states are not the reduction of a pure state in a larger system. This allows us to derive the measurement postulates of quantum theory from the structure of pure states and reversible dynamics, and the requirement that the purification principle holds. The violation of the purification principle implies that there is some irreducible classicality in these theories, which appears like an important clue for the problem of deriving the Born rule within the many-worlds interpretation. We also prove that in all such modifications the task of state tomography with local measurements is impossible, and present a simple toy theory displaying all these exotic non-quantum phenomena. This toy model shows that, contrarily to previous claims, it is possible to modify the Born rule without violating the no-signalling principle. Finally, we argue that the quantum measurement postulates are the most non-classical amongst all alternatives.
The standard postulates of quantum theory can be divided into two groups: the first one characterizes the structure and dynamics of pure states, while the second one specifies the structure of measurements and the corresponding probabilities. In this work we keep the first group of postulates and characterize all alternatives to the second group that give rise to finite-dimensional sets of mixed states. We prove a correspondence between all these alternatives and a class of representations of the unitary group. Some features of these probabilistic theories are identical to quantum theory, but there are important differences in others. For example, some theories have three perfectly distinguishable states in a two-dimensional Hilbert space. Others have exotic properties such as lack of bit symmetry, the violation of no simultaneous encoding (a property similar to information causality) and the existence of maximal measurements without phase groups. We also analyze which of these properties single out the Born rule.
The standard postulates of quantum theory can be divided into two groups: the first one characterizes the structure and dynamics of pure states, while the second one specifies the structure of measurements and the corresponding probabilities. In this work we keep the first group of postulates and characterize all alternatives to the second group that give rise to finite-dimensional sets of mixed states. We prove a correspondence between all these alternatives and a class of representations of the unitary group. Some features of these probabilistic theories are identical to quantum theory, but there are important differences in others. For example, some theories have three perfectly distinguishable states in a two-dimensional Hilbert space. Others have exotic properties such as lack of "bit symmetry", the violation of "no simultaneous encoding" (a property similar to information causality) and the existence of maximal measurements without phase groups. We also analyze which of these properties single out the Born rule.
Understanding the core content of quantum mechanics requires us to disentangle the hidden logical relationships between the postulates of this theory. Here we show that the mathematical structure of quantum measurements, the formula for assigning outcome probabilities (Born's rule) and the post-measurement state-update rule, can be deduced from the other quantum postulates, often referred to as "unitary quantum mechanics", and the assumption that ensembles on finite-dimensional Hilbert spaces are characterized by finitely many parameters. This is achieved by taking an operational approach to physical theories, and using the fact that the manner in which a physical system is partitioned into subsystems is a subjective choice of the observer, and hence should not affect the predictions of the theory. In contrast to other approaches, our result does not assume that measurements are related to operators or bases, it does not rely on the universality of quantum mechanics, and it is independent of the interpretation of probability.
Understanding the core content of quantum mechanics requires us to disentangle the hidden logical relationships between the postulates of this theory. Here we show that the mathematical structure of quantum measurements, the formula for assigning outcome probabilities (Born's rule) and the post-measurement state-update rule, can be deduced from the other quantum postulates, often referred to as unitary quantum and the assumption that ensembles on finite-dimensional Hilbert spaces are characterised by finitely many parameters. This is achieved by taking an operational approach to physical theories, and using the fact that the manner in which a physical system is partitioned into subsystems is a subjective choice of the observer, and hence should not affect the predictions of the theory. In contrast to other approaches, our result does not assume that measurements are related to operators or bases, it does not rely on the universality of quantum mechanics, and it is independent of the interpretation of probability.
Quantum theory combines density matrices, Born probabilities, tensor-product composites, positive-operator-valued measures (POVMs), and quantum channels. In a finite-dimensional causal operational theory, we prove that two postulates suffice: local input-output statistics identify channels, and every state admits an equivalent-system purification, unique up to reversible dynamics. The full complex quantum formalism follows; every consistent probability rule is realized as a POVM, so measurement no-restriction is derived rather than assumed.