The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement

  title={The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement},
  author={Teiko Heinosaari and M{\'a}rio Ziman},
Introduction 1. Hilbert space refresher 2. States and effects 3. Observables 4. Operations and channels 5. Measurement models and instruments 6. Entanglement Bibliography Index. 
Quantum Entanglement: Spooky Action at a Distance
Quantum entanglement is an important resource in quantum information technologies. Here, we study and characterize in a precise mathematical language some of the weird and nonintuitive features ofExpand
Complete quantum measurements break entanglement
Abstract A complete measurement of a quantum observable (POVM) is a measurement of the maximally refined version of the POVM. Complete measurements give information on multiplicities of measurementExpand
Effectuses in Categorical Quantum Foundations
This thesis develops the theory of effectuses as a categorical axiomatic approach to quantum theory. It provides a comprehensive introduction to effectus theory and reveals its connections withExpand
Complete quantum measurements break entanglement
Complete measurement of a quantum observable (POVM) is a measurement of the maximally refined version of the POVM. Complete measurements give information on multiplicities of measurement outcomes andExpand
Relation between the state distinction power and disturbance in quantum measurements
The measurement of an informative observable strongly disturbs a quantum state. We examine the so-called information-disturbance relation by introducing order relations based on the state distinctionExpand
Fuzzy Approach for Toffoli Gate in Quantum Computation With Mixed States
In the framework of quantum computation with mixed states, a fuzzy representation based on continuous t -norms for Toffoli gate is introduced. In this representation, the incidence of nonseparabilityExpand
Quantum instruments: I. Extreme instruments
The structure of completely positive quantum instruments is analyzed thoroughly. We characterize the extreme points of the convex set of instruments. The characterization holds also for nondiscreteExpand
Holistic type extension for classical propositional logic in quantum computation
A holistic extension of classical propositional logic is introduced in the framework of quantum computation with mixed states. The concepts of tautology and contradiction are investigated in thisExpand
A Theory of Entanglement
  • S. Gudder
  • Computer Science, Mathematics
  • Quanta
  • 2020
The basis of a theory of entanglement for quantum states is presented, which begins with a classical theory of entangled discrete measures, then treats quantum mechanics and discusses the statistics of bounded operators on a Hilbert space in terms of context coefficients. Expand


Optimal measurements in quantum mechanics
Abstract Four common optimality criteria for measurements are formulated using relations in the set of observables, and their connections are clarified. As case studies, 1–0 observables, localizationExpand
Quantum Information Theory and Quantum Statistics
Prerequisites from Quantum Mechanics.- Information and its Measures.- Entanglement.- More About Information Quantities.- Quantum Compression.- Channels and Their Capacity.- Hypothesis Testing.-Expand
The Role of Entanglement in Quantum Measurement and Information Processing
The significance of the quantum feature of entanglement between physical systems is investigated in the context of quantum measurements. It is shown that, while there are measurement couplings thatExpand
"No Information Without Disturbance": Quantum Limitations of Measurement
In this contribution I review rigorous formulations of a variety of limitations of measurability in quantum mechanics. To this end I begin with a brief presentation of the conceptual tools of modernExpand
Quantum entanglement
Entanglement is a fundamental resource in quantum information theory. It allows performing new kinds of communication, such as quantum teleportation and quantum dense coding. It is an essentialExpand
Entanglement-Assisted Local Manipulation of Pure Quantum States
We demonstrate that local transformations on a composite quantum system can be enhanced in the presence of certain entangled states. These extra states act much like catalysts in a chemical reaction:Expand
The Theory of Symmetry Actions in Quantum Mechanics: with an Application to the Galilei Group
A Synopsis of Quantum Mechanics.- The Automorphism Group of Quantum Mechanics.- The Symmetry Actions and Their Representations.- The Galilei Groups.- Galilei Invariant Elementary Particles.- GalileiExpand
Fuzzy observables in quantum mechanics
The formalism of covariant conditional expectations is described as leading to an operational definition of generalized observables in quantum mechanics, wide enough to account for the fuzzinessExpand
Quantum computation and quantum information
  • T. Paul
  • Mathematics, Computer Science
  • Mathematical Structures in Computer Science
  • 2007
This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers dealExpand
On the Problem of Hidden Variables in Quantum Mechanics
The demonstrations of von Neumann and others, that quantum mechanics does not permit a hidden variable interpretation, are reconsidered. It is shown that their essential axioms are unreasonable. ItExpand