Categories for Quantum Theory

2019 ◽  
Author(s):  
Chris Heunen ◽  
Jamie Vicary
Keyword(s):  
Quantum Theory ◽  
Category Theory ◽  
Hopf Algebras ◽  
Quantum Teleportation ◽  
Graphical Calculus ◽  
Frobenius Structures ◽  
Nonstandard Models ◽  
Background Material ◽  
Quantum Phenomena ◽  
High Level

Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to understand many high-level quantum phenomena. Here, we lay the foundations for this categorical quantum mechanics, with an emphasis on the graphical calculus that makes computation intuitive. We describe superposition and entanglement using biproducts and dual objects, and show how quantum teleportation can be studied abstractly using these structures. We investigate monoids, Frobenius structures and Hopf algebras, showing how they can be used to model classical information and complementary observables. We describe the CP construction, a categorical tool to describe probabilistic quantum systems. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Previous knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text, we point out links with many other areas, such as representation theory, topology, quantum algebra, knot theory and probability theory, and present nonstandard models including sets and relations. All results are stated rigorously and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.

Complementarity

2019 ◽  
pp. 193-224
Author(s):  
Chris Heunen ◽  
Jamie Vicary
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Hopf Algebras ◽  
Quantum Information Theory ◽  
Graphical Calculus ◽  
Single Qubit ◽  
Frobenius Structures ◽  
Quantum Computations

Complementarity is a property of a pair of observables being ‘maximally distinct’ from each other and, in this chapter, we analyse this property in categorical terms as a pair of interacting Frobenius structures. Complementary observables play a central role in quantum information theory, and we will see how they can be used to understand the structure of the Deutsch—Jozsa algorithm. We show that complementarity is closely linked to the theory of Hopf algebras. We discuss how many-qubit gates can be modelled using only complementary Frobenius structures, such as controlled negation, controlled phase gates and arbitrary single qubit gates. This leads to the ZX calculus, a sound and complete way to handle quantum computations using only equations in the graphical calculus.

Frobenius Structures

2019 ◽  
pp. 147-192
Author(s):  
Chris Heunen ◽  
Jamie Vicary
Keyword(s):  
Quantum Teleportation ◽  
Operator Algebras ◽  
Coherence Property ◽  
Graphical Calculus ◽  
Classical Information ◽  
Frobenius Structures ◽  
State Transfer ◽  
Normal Form Theorem ◽  
Interaction Law ◽  
Model Measurement

A Frobenius structure is a monoid together with a comonoid, which satisfies an interaction law. Frobenius structures have a powerful graphical calculus and we prove a normal form theorem that makes them easy to work with. The Frobenius law itself is justified as a coherence property between daggers and closure of a category. We prove classification theorems for dagger Frobenius structures: in Hilb in terms of operator algebras and in Rel in terms of groupoids. Of special interest is the commutative case—as for Hilbert spaces this corresponds to a choice of basis—and provides a powerful tool to model classical information. We discuss phase gates and the state transfer protocol—as well as modules for Frobenius structures—and show how we can use these to model measurement, controlled operations and quantum teleportation.

scholarly journals Relativistic quantum mechanics

1932 ◽  
Vol 136 (829) ◽  
pp. 453-464 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Classical Theory ◽  
Correspondence Principle ◽  
Relativistic Quantum ◽  
Classical Mechanics ◽  
Hamiltonian Function ◽  
Relativistic Quantum Mechanics ◽  
Classical Electrodynamics ◽  
Quantum Phenomena

The steady development of the quantum theory that has taken place during the present century was made possible only by continual reference to the Correspondence Principle of Bohr, according to which, classical theory can give valuable information about quantum phenomena in spite of the essential differences in the fundamental ideas of the two theories. A masterful advance was made by Heisenberg in 1925, who showed how equations of classical physics could be taken over in a formal way and made to apply to quantities of importance in quantum theory, thereby establishing the Correspondence Principle on a quantitative basis and laying the foundations of the new Quantum Mechanics. Heisenberg’s scheme was found to fit wonderfully well with the Hamiltonian theory of classical mechanics and enabled one to apply to quantum theory all the information that classical theory supplies, in so far as this information is consistent with the Hamiltonian form. Thus one was able to build up a satisfactory quantum mechanics for dealing with any dynamical system composed of interacting particles, provided the interaction could be expressed by means of an energy term to be added to the Hamiltonian function. This does not exhaust the sphere of usefulness of the classical theory. Classical electrodynamics, in its accurate (restricted) relativistic form, teaches us that the idea of an interaction energy between particles is only an approxi­mation and should be replaced by the idea of each particle emitting waves which travel outward with a finite velocity and influence the other particles in passing over them. We must find a way of taking over this new information into the quantum theory and must set up a relativistic quantum mechanics, before we can dispense with the Correspondence Principle.

Philosophy of Quantum Mechanics: Dynamical Collapse Theories

2021 ◽  
Author(s):  
Angelo Bassi
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
New Technologies ◽  
Stochastic Dynamics ◽  
Laws Of Nature ◽  
Nonlinear Effects ◽  
Foundation Of Quantum Mechanics ◽  
Quantum Phenomena ◽  
The One ◽  
Collapse Theories

Quantum Mechanics is one of the most successful theories of nature. It accounts for all known properties of matter and light, and it does so with an unprecedented level of accuracy. On top of this, it generated many new technologies that now are part of daily life. In many ways, it can be said that we live in a quantum world. Yet, quantum theory is subject to an intense debate about its meaning as a theory of nature, which started from the very beginning and has never ended. The essence was captured by Schrödinger with the cat paradox: why do cats behave classically instead of being quantum like the one imagined by Schrödinger? Answering this question digs deep into the foundation of quantum mechanics. A possible answer is Dynamical Collapse Theories. The fundamental assumption is that the Schrödinger equation, which is supposed to govern all quantum phenomena (at the non-relativistic level) is only approximately correct. It is an approximation of a nonlinear and stochastic dynamics, according to which the wave functions of microscopic objects can be in a superposition of different states because the nonlinear effects are negligible, while those of macroscopic objects are always very well localized in space because the nonlinear effects dominate for increasingly massive systems. Then, microscopic systems behave quantum mechanically, while macroscopic ones such as Schrödinger’s cat behave classically simply because the (newly postulated) laws of nature say so. By changing the dynamics, collapse theories make predictions that are different from quantum-mechanical predictions. Then it becomes interesting to test the various collapse models that have been proposed. Experimental effort is increasing worldwide, so far limiting values of the theory’s parameters quantifying the collapse, since no collapse signal was detected, but possibly in the future finding such a signal and opening up a window beyond quantum theory.

Elements of ∞-Category Theory

2022 ◽  
Author(s):  
Emily Riehl ◽  
Dominic Verity
Keyword(s):  
First Principles ◽  
Category Theory ◽  
Higher Dimensional ◽  
Background Material ◽  
Model Independent ◽  
The Universe ◽  
Formal Category

The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

scholarly journals Reality, Indeterminacy, Probability, and Information in Quantum Theory

Entropy ◽  
2020 ◽  
Vol 22 (7) ◽  
pp. 747
Author(s):  
Arkady Plotnitsky
Keyword(s):  
Information Theory ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Information ◽  
Quantum Correlations ◽  
Quantum Information Theory ◽  
Quantum Phenomena

Following the view of several leading quantum-information theorists, this paper argues that quantum phenomena, including those exhibiting quantum correlations (one of their most enigmatic features), and quantum mechanics may be best understood in quantum-informational terms. It also argues that this understanding is implicit already in the work of some among the founding figures of quantum mechanics, in particular W. Heisenberg and N. Bohr, half a century before quantum information theory emerged and confirmed, and gave a deeper meaning to, to their insights. These insights, I further argue, still help this understanding, which is the main reason for considering them here. My argument is grounded in a particular interpretation of quantum phenomena and quantum mechanics, in part arising from these insights as well. This interpretation is based on the concept of reality without realism, RWR (which places the reality considered beyond representation or even conception), introduced by this author previously, in turn, following Heisenberg and Bohr, and in response to quantum information theory.

scholarly journals The de Broglie–Bohm Quantum Theory and Its Application to Quantum Cosmology

Universe ◽  
2021 ◽  
Vol 7 (5) ◽  
pp. 134
Author(s):  
Nelson Pinto-Neto
Keyword(s):  
Quantum Theory ◽  
Configuration Space ◽  
Physical System ◽  
Quantum Cosmology ◽  
Dynamical Theory ◽  
Cosmological Models ◽  
Cosmological Perturbations ◽  
Alternative Description ◽  
Quantum Phenomena ◽  
De Broglie Bohm

We review the de Broglie–Bohm quantum theory. It is an alternative description of quantum phenomena in accordance with all the quantum experiments already performed. Essentially, it is a dynamical theory about objectively real trajectories in the configuration space of the physical system under investigation. Hence, it is not necessarily probabilistic, and it dispenses with the collapse postulate, making it suitable to be applied to cosmology. The emerging cosmological models are usually free of singularities, with a bounce connecting a contracting era with an expanding phase, which we are now observing. A theory of cosmological perturbations can also be constructed under this framework, which can be successfully confronted with current observations, and can complement inflation or even be an alternative to it.

scholarly journals New Quantum Theory Explains All the Mysterious Quantum Phenomena

2016 ◽  
Vol 07 (15) ◽  
pp. 2135-2154
Author(s):  
Narendra Swarup Agarwal
Keyword(s):  
Quantum Theory ◽  
Quantum Phenomena

scholarly journals Nature Has No Elementary Particles and Makes No Measurements or Predictions: Quantum Measurement and Quantum Theory, from Bohr to Bell and from Bell to Bohr

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1197
Author(s):  
Arkady Plotnitsky
Keyword(s):  
Field Theory ◽  
Quantum Theory ◽  
Quantum Measurement ◽  
Physical Reality ◽  
Quantum Field ◽  
Quantum Object ◽  
Classical Physics ◽  
Quantum Phenomenon ◽  
Quantum Objects ◽  
Quantum Phenomena

This article reconsiders the concept of physical reality in quantum theory and the concept of quantum measurement, following Bohr, whose analysis of quantum measurement led him to his concept of a (quantum) “phenomenon,” referring to “the observations obtained under the specified circumstances,” in the interaction between quantum objects and measuring instruments. This situation makes the terms “observation” and “measurement,” as conventionally understood, inapplicable. These terms are remnants of classical physics or still earlier history, from which classical physics inherited it. As defined here, a quantum measurement does not measure any preexisting property of the ultimate constitution of the reality responsible for quantum phenomena. An act of measurement establishes a quantum phenomenon by an interaction between the instrument and the quantum object or in the present view the ultimate constitution of the reality responsible for quantum phenomena and, at the time of measurement, also quantum objects. In the view advanced in this article, in contrast to that of Bohr, quantum objects, such as electrons or photons, are assumed to exist only at the time of measurement and not independently, a view that redefines the concept of quantum object as well. This redefinition becomes especially important in high-energy quantum regimes and quantum field theory and allows this article to define a new concept of quantum field. The article also considers, now following Bohr, the quantum measurement as the entanglement between quantum objects and measurement instruments. The argument of the article is grounded in the concept “reality without realism” (RWR), as underlying quantum measurement thus understood, and the view, the RWR view, of quantum theory defined by this concept. The RWR view places a stratum of physical reality thus designated, here the reality ultimately responsible for quantum phenomena, beyond representation or knowledge, or even conception, and defines the corresponding set of interpretations quantum mechanics or quantum field theory, such as the one assumed in this article, in which, again, not only quantum phenomena but also quantum objects are (idealizations) defined by measurement. As such, the article also offers a broadly conceived response to J. Bell’s argument “against ‘measurement’”.

scholarly journals Any modification of the Born rule leads to a violation of the purification and local tomography principles

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 104 ◽  
Author(s):  
Thomas D. Galley ◽  
Lluis Masanes
Keyword(s):  
Quantum Theory ◽  
Born Rule ◽  
Mixed States ◽  
Many Worlds ◽  
Local Tomography ◽  
Local Measurements ◽  
State Tomography ◽  
Quantum Phenomena ◽  
The Many

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.

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