A process theoretic triptych: two roads to the emergence of classicality, reconstructing quantum theory from diagrams, looking for post-quantum theories
This thesis asks what can be learnt about quantum theory by investigating it from the perspective of process theories. This is based on the diagrammatic compositional structure of Categorical Quantum Mechanics, leading to a very general framework to describe alternate theories of nature. In particular this framework is well suited to understanding the relationship between different theories. In the first part of the thesis we investigate the relationship between quantum and classical theory, showing how an abstract description of decoherence in terms of leaking information leads to emergent classicality. Moreover, this process theoretic notion of a `leak' allows us to capture the distinction between quantum and classical theory in a particularly simple way, highlighting how the quantum and classical worlds diverge. In the second part we look at how to reconstruct quantum theory from diagrammatic principles showing that i) the existence of a classical interface with the theory plus ii) standard notions of composition and iii) a time symmetric form of purification are sufficient to reconstruct the standard quantum formalism. Thereby demonstrating that the standard tools of Categorical Quantum Mechanics come very close to capturing the essence of quantum theory. In the third part we abstract the key features of this emergence of classicality to define a notion of `hyperdecoherence' whereby some post-quantum theory might appear quantum due to an uncontrolled interaction with an environment. We prove a no-go theorem which states that any operational post-quantum theory must violate the purification principle, and so must radically challenge our understanding of how information behaves. To summarise, we use the framework of process theories to gain a better understanding of quantum theory, its sub-theories, and its potential super-theories.
This thesis asks what can be learnt about quantum theory by investigating it from the perspective of process theories. This is based on the diagrammatic compositional structure of Categorical Quantum Mechanics, leading to a very general framework to describe alternate theories of nature. In particular this framework is well suited to understanding the relationship between different theories. In the first part of the thesis we investigate the relationship between quantum and classical theory, showing how an abstract description of decoherence in terms of leaking information leads to emergent classicality. Moreover, this process theoretic notion of a leak' allows us to capture the distinction between quantum and classical theory in a particularly simple way, highlighting how the quantum and classical worlds diverge. In the second part we look at how to reconstruct quantum theory from diagrammatic principles showing that i) the existence of a classical interface with the theory plus ii) standard notions of composition and iii) a time symmetric form of purification are sufficient to reconstruct the standard quantum formalism. Thereby demonstrating that the standard tools of Categorical Quantum Mechanics come very close to capturing the essence of quantum theory. In the third part we abstract the key features of this emergence of classicality to define a notion of hyperdecoherence' whereby some post-quantum theory might appear quantum due to an uncontrolled interaction with an environment. We prove a no-go theorem which states that any operational post-quantum theory must violate the purification principle, and so must radically challenge our understanding of how information behaves. To summarise, we use the framework of process theories to gain a better understanding of quantum theory, its sub-theories, and its potential super-theories.
A reconstruction of quantum theory refers to both a mathematical and a conceptual paradigm that allows one to derive the usual formulation of quantum theory from a set of primitive assumptions. The motivation for doing so is a discomfort with the usual formulation of quantum theory, a discomfort that started with its originator John von Neumann. We present a reconstruction of finite-dimensional quantum theory where all of the postulates are stated in diagrammatic terms, making them intuitive. Equivalently, they are stated in category-theoretic terms, making them mathematically appealing. Again equivalently, they are stated in process-theoretic terms, establishing that the conceptual backbone of quantum theory concerns the manner in which systems and processes compose. Aside from the diagrammatic form, the key novel aspect of this reconstruction is the introduction of a new postulate, symmetric purification. Unlike the ordinary purification postulate, symmetric purification applies equally well to classical theory as well as quantum theory. Therefore we first reconstruct the full process theoretic description of quantum theory, consisting of composite classical-quantum systems and their interactions, before restricting ourselves to just the ‘fully quantum’ systems as the final step. We propose two novel alternative manners of doing so, ‘no-leaking’ (roughly that information gain causes disturbance) and ‘purity of cups’ (roughly the existence of entangled states). Interestingly, these turn out to be equivalent in any process theory with cups & caps. Additionally, we show how the standard purification postulate can be seen as an immediate consequence of the symmetric purification postulate and purity of cups. Other tangential results concern the specific frameworks of generalised probabilistic theories (GPTs) and process theories (a.k.a. CQM). Firstly, we provide a diagrammatic presentation of GPTs, which, henceforth, can be subsumed under process theories. Secondly, we argue that the ‘sharp dagger’ is indeed the right choice of a dagger structure as this sharpness is vital to the reconstruction.
To date, there has been no experimental evidence that invalidates quantum theory. Yet it may only be an effective description of the world, in the same way that classical physics is an effective description of the quantum world. We ask whether there exists an operationally defined theory superseding quantum theory, but which reduces to it via a decoherence-like mechanism. We prove that no such post-quantum theory exists if it is demanded that it satisfy two natural physical principles: causality and purification . Causality formalizes the statement that information propagates from present to future, and purification that each state of incomplete information arises in an essentially unique way due to lack of information about an environment. Hence, our result can be viewed either as evidence that the fundamental theory of Nature is quantum or as showing in a rigorous manner that any post-quantum theory must abandon causality, purification or both.
To date, there has been no experimental evidence that invalidates quantum theory. Yet it may only be an effective description of the world, in the same way that classical physics is an effective description of the quantum world. We ask whether there exists an operationally defined theory superseding quantum theory, but which reduces to it via a decoherence-like mechanism. We prove that no such post-quantum theory exists if it is demanded that it satisfy two natural physical principles: causality and purification. Causality formalizes the statement that information propagates from present to future, and purification that each state of incomplete information arises in an essentially unique way due to lack of information about an environment. Hence, our result can be viewed either as evidence that the fundamental theory of Nature is quantum or as showing in a rigorous manner that any post-quantum theory must abandon causality, purification or both.
The first half of this thesis is dedicated to the study of generalizations of probability theory which include quantum theory and many of its alternatives. This research program seeks to understand quantum theory by exploring it ‘from the outside’, where it sits within a landscape of all operationally defined theories. We approach this problem with a new perspective: generalized decoherence. Quantum theory recovers classical theory via decoherence, where a ‘classical limit’ is achieved by a quantum system interacting with its environment, losing its quantum coherence. Typically it is the emergence of the classical world from the quantum that is studied. We reverse this paradigm and ask - how does the existence of a classical limit define the structure of quantum theory, or any other post-classical theory? We derive the existence of entangled states as a necessary feature of any theory that allows for the emergence of the classical world in this way. We then use our framework for generalized decoherence to explore the properties of theories that can ‘hyperdecohere’ to quantum theory. We find that any such theory must be very exotic compared to quantum theory, violating either the postulate of tomographic locality, reversibility, causality or purity. In the second half of this thesis we turn our attention to the second law of thermodynamics. We present a framework for deriving the second law under general constraints and explore two pertinent examples, constraining the fluctuations in work and constraining the size of the thermal bath and deriving the second law in these cases. Bounding fluctuations results in a unified free energy that contains the single-shot and Helmholtz free energies as limiting cases. Bounding the size of the thermal bath results in a finite-bath second law that is more general that previous attempts and draws connections between non-asymptotic thermodynamics and second-order information theory.
This is the second part of a three-part overview, in which we derive the category-theoretic backbone of quantum theory from a process ontology, treating quantum theory as a theory of systems, processes and their interactions. In this part, we focus on classical–quantum interaction. Classical and quantum systems are treated as distinct types, of which the respective behavioral properties are specified in terms of processes and their compositions. In particular, classicality is witnessed by ‘spiders’ which fuse together whenever they connect. We define mixedness and show that pure processes are extremal in the space of all processes, and we define entanglement and show that quantum theory indeed exhibits entanglement. We discuss the classification of tripartite qubit entanglement and show that both the GHZ-state and the W-state come from spider-like families of processes, which differ only in how they behave when they are connected by two or more wires. We define measurements and provide fully comprehensive descriptions of several quantum protocols involving classical data flow. Finally, we give a notion of ‘genuine quantumness’, from which special processes called ‘phase spiders’ arise, and get a first glimpse of quantum nonlocality.
We introduce the notion of a leak for general process theories and identify quantum theory as a theory with minimal leakage, while classical theory has maximal leakage. We provide a construction that adjoins leaks to theories, an instance of which describes the emergence of classical theory by adjoining decoherence leaks to quantum theory. Finally, we show that defining a notion of purity for processes in general process theories has to make reference to the leaks of that theory, a feature missing in standard definitions; hence, we propose a refined definition and study the resulting notion of purity for quantum, classical and intermediate theories.