Quantum theory is a probabilistic calculus that enables the calculation of the probabilities of the possible outcomes of a measurement performed on a physical system. But what is the relationship between this probabilistic calculus and probability theory itself? Is quantum theory compatible with probability theory? If so, does it extend or generalize probability theory? In this paper, we answer these questions, and precisely determine the relationship between quantum theory and probability theory, by explicitly deriving both theories from first principles. In both cases, the derivation depends upon identifying and harnessing the appropriate symmetries that are operative in each domain. We prove, for example, that quantum theory is compatible with probability theory by explicitly deriving quantum theory on the assumption that probability theory is generally valid.Quantum theory is a probabilistic calculus that enables the calculation of the probabilities of the possible outcomes of a measurement performed on a physical system. But what is the relationship between this probabilistic calculus and probability theory itself? Is quantum theory compatible with probability theory? If so, does it extend or generalize probability theory? In this paper, we answer these questions, and precisely determine the relationship between quantum theory and probability theory, by explicitly deriving both theories from first principles. In both cases, the derivation depends upon identifying and harnessing the appropriate symmetries that are operative in each domain. We prove, for example, that quantum theory is compatible with probability theory by explicitly deriving quantum theory on the assumption that probability theory is generally valid.

SymmetryApr 27, 201129 citations

Theory

Naresh Chandra, R. Ghosh

Springer series on atomic, optical, and plasma physicsMar 15, 20120 citations

Theory

Vera M. Schäfer

Springer thesesJan 1, 20200 citations

Convex operational theories and non-classical features of quantum theory

Leevi Leppäjärvi

Convex operational theories form a class of physical theories that are built on the operational mixing of states of the system resulting in convex state spaces. Following the operational approach to describe physical experiments, the other operational concepts, such as measurements and state transformations, rise from the properties of the state space. In addition to quantum theory, the convex operational theories include but are not restricted to classical theories and quantum theory of processes. In the light of recent deep interest in quantum information theory, convex operational theories serve as means to consider information-theoretic principles in a more abstract framework. This allows to compare different types of theories against each other and further study the nature of these principles. Some of these principles can then even be used for different axiomatizations of quantum theory. This thesis serves to introduce the mathematical concepts related to convex operational theories and then use them to construct this class of theories in the ordered vector space formalism. We use the constructed class of theories to consider the most important aspects of the theories with applications in physical theories such as quantum theory. We study some of the most important non-classical properties of quantum theory in the more abstract framework of convex operational theories including original research on one of these features.Convex operational theories form a class of physical theories that are built on the operational mixing of states of the system resulting in convex state spaces. Following the operational approach to describe physical experiments, the other operational concepts, such as measurements and state transformations, rise from the properties of the state space. In addition to quantum theory, the convex operational theories include but are not restricted to classical theories and quantum theory of processes. In the light of recent deep interest in quantum information theory, convex operational theories serve as means to consider information-theoretic principles in a more abstract framework. This allows to compare different types of theories against each other and further study the nature of these principles. Some of these principles can then even be used for different axiomatizations of quantum theory. This thesis serves to introduce the mathematical concepts related to convex operational theories and then use them to construct this class of theories in the ordered vector space formalism. We use the constructed class of theories to consider the most important aspects of the theories with applications in physical theories such as quantum theory. We study some of the most important non-classical properties of quantum theory in the more abstract framework of convex operational theories including original research on one of these features.

Aug 8, 20170 citations

Theory

Richard Hydomako

Springer thesesDec 13, 20120 citations

General Quantum Theory

Koen Thas

Inspired by classical ("actual") Quantum Theory over $\mathbb{C}$ and Modal Quantum Theory (MQT), which is a model of Quantum Theory over certain finite fields, we introduce General Quantum Theory as a Quantum Theory -- in the København interpretation -- over general division rings with involution, in which the inner product "is" a $(σ,1)$ -Hermitian form $φ$ . This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical Quantum Theory over $\mathbb{C}$ and in some MQTs, hold for any General Quantum Theory. On the other hand, in many General Quantum Theories, a geometrical object which we call "quantum kernel" arises, which is invariant under the unitary group $\mathbf{U}(V,φ)$ , and which carries the geometry of a so-called polar space. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic $0$ , every General Quantum Theory over an algebraically closed field behaves like classical Quantum Theory over $\mathbb{C}$ at many levels, and that all such theories share one model, which we pin down as the "minimal model," which is countable and defined over $\overline{\mathbb{Q}}$ . Moreover, to make the analogy with classical Quantum Theory even more striking, we show that Born's rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for General Quantum Theories in characteristic $0$ which allows one to extend any such theory over algebraically closed fields (such as classical complex Quantum Theory) to larger theories in which a quantum kernel is present.Inspired by classical ("actual") Quantum Theory over $\mathbb{C}$ and Modal Quantum Theory (MQT), which is a model of Quantum Theory over certain finite fields, we introduce General Quantum Theory as a Quantum Theory -- in the København interpretation -- over general division rings with involution, in which the inner product "is" a $(σ,1)$-Hermitian form $φ$. This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical Quantum Theory over $\mathbb{C}$ and in some MQTs, hold for any General Quantum Theory. On the other hand, in many General Quantum Theories, a geometrical object which we call "quantum kernel" arises, which is invariant under the unitary group $\mathbf{U}(V,φ)$, and which carries the geometry of a so-called polar space. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic $0$, every General Quantum Theory over an algebraically closed field behaves like classical Quantum Theory over $\mathbb{C}$ at many levels, and that all such theories share one model, which we pin down as the "minimal model," which is countable and defined over $\overline{\mathbb{Q}}$. Moreover, to make the analogy with classical Quantum Theory even more striking, we show that Born's rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for General Quantum Theories in characteristic $0$ which allows one to extend any such theory over algebraically closed fields (such as classical complex Quantum Theory) to larger theories in which a quantum kernel is present.

arXiv:1712.04669Dec 13, 20170 citations

Inspired by classical ("actual") Quantum Theory over

\mathbb{C}

and Modal Quantum Theory (MQT), which is a model of Quantum Theory over certain finite fields, we introduce General Quantum Theory as a Quantum Theory -- in the København interpretation -- over general division rings with involution, in which the inner product "is" a

(σ,1)

-Hermitian form

φ

. This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical Quantum Theory over

\mathbb{C}

and in some MQTs, hold for any General Quantum Theory. On the other hand, in many General Quantum Theories, a geometrical object which we call "quantum kernel" arises, which is invariant under the unitary group

\mathbf{U}(V,φ)

, and which carries the geometry of a so-called polar space. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic

0

, every General Quantum Theory over an algebraically closed field behaves like classical Quantum Theory over

\mathbb{C}

at many levels, and that all such theories share one model, which we pin down as the "minimal model," which is countable and defined over

\overline{\mathbb{Q}}

. Moreover, to make the analogy with classical Quantum Theory even more striking, we show that Born's rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for General Quantum Theories in characteristic

0

which allows one to extend any such theory over algebraically closed fields (such as classical complex Quantum Theory) to larger theories in which a quantum kernel is present.