The physical Church-Turing thesis and the principles of quantum theory

Notoriously, quantum computation shatters complexity theory, but is innocuous to computability theory. Yet several works have shown how quantum theory as it stands could breach the physical Church-Turing thesis. We draw a clear line as to when this is the case, in a way that is inspired by Gandy. Gandy formulates postulates about physics, such as homogeneity of space and time, bounded density and velocity of information --- and proves that the physical Church-Turing thesis is a consequence of these postulates. We provide a quantum version of the theorem. Thus this approach exhibits a formal non-trivial interplay between theoretical physics symmetries and computability assumptions.Comment: 14 pages, LaTe

Free fall and cellular automata

Three reasonable hypotheses lead to the thesis that physical phenomena can be described and simulated with cellular automata. In this work, we attempt to describe the motion of a particle upon which a constant force is applied, with a cellular automaton, in Newtonian physics, in Special Relativity, and in General Relativity. The results are very different for these three theories.Comment: In Proceedings DCM 2015, arXiv:1603.0053

Discrete Geodesics and Cellular Automata

International audienceThis paper proposes a dynamical notion of discrete geodesics, understood as straightest trajectories in discretized curved spacetime. The notion is generic, as it is formulated in terms of a general deviation function, but readily specializes to metric spaces such as discretized pseudo-riemannian manifolds. It is effective: an algorithm for computing these geodesics naturally follows, which allows numerical validation—as shown by computing the perihelion shift of a Mercury-like planet. It is consistent, in the continuum limit, with the standard notion of timelike geodesics in a pseudo-riemannian manifold. Whether the algorithm fits within the framework of cellular automata is discussed at length

Computational irreducibility and compatibilism: towards a formalization

If our actions are determined by the laws of nature, can we meaningfully claim to possess free will? Compatibilists argue that the answer is yes, and that free will is compatible with complete determinism. Previously, it has been suggested that the notion of computational irreducibility can shed light on this relation: it implies that there cannot in general be "shortcuts" to the decisions of agents, explaining why deterministic agents often appear to have free will. In this paper, we introduce a variant of computational irreducibility that intends to capture more accurately aspects of actual (as opposed to apparent) free will: computational sourcehood, i.e. the phenomenon that the successful prediction of a process' outputs must typically involve an almost-exact representation of the relevant features of that process, regardless of the time it takes to arrive at the prediction. We conjecture that many processes have this property, and we study different possibilities for how to formalize this conjecture in terms of universal Turing machines. While we are not able to settle the conjecture, we give several results and constructions that shed light on the quest for its correct formulation.Comment: 15 pages, 1 figur

Thermodynamic costs of Turing Machines

Turing Machines (TMs) are the canonical model of computation in computer science and physics. We combine techniques from algorithmic information theory and stochastic thermodynamics to analyze the thermodynamic costs of TMs. We consider two different ways of realizing a given TM with a physical process. The first realization is designed to be thermodynamically reversible when fed with random input bits. The second realization is designed to generate less heat, up to an additive constant, than any realization that is computable (i.e., consistent with the physical Church-Turing thesis). We consider three different thermodynamic costs: the heat generated when the TM is run on each input (which we refer to as the "heat function"), the minimum heat generated when a TM is run with an input that results in some desired output (which we refer to as the "thermodynamic complexity" of the output, in analogy to the Kolmogorov complexity), and the expected heat on the input distribution that minimizes entropy production. For universal TMs, we show for both realizations that the thermodynamic complexity of any desired output is bounded by a constant (unlike the conventional Kolmogorov complexity), while the expected amount of generated heat is infinite. We also show that any computable realization faces a fundamental tradeoff between heat generation, the Kolmogorov complexity of its heat function, and the Kolmogorov complexity of its input-output map. We demonstrate this tradeoff by analyzing the thermodynamics of erasing a long string.Comment: Physical Review Research, 202

Les deux formes de la thèse de Church-Turing et l’épistémologie du calcul

La thèse de Church-Turing stipule que toute fonction calculable est calculable par une machine de Turing. En distinguant, à la suite de nombreux auteurs, une forme algorithmique de la thèse de Church-Turing portant sur les fonctions calculables par un algorithme d’une forme empirique de cette même thèse, portant sur les fonctions calculables par une machine, il devient possible de poser une nouvelle question : les limites empiriques du calcul sont-elles identiques aux limites des algorithmes ? Ou existe-t-il un moyen empirique d’effectuer un calcul qu’aucun algorithme ne permet d’effectuer ? Je montrerai ici la pertinence philosophique de cette question. Elle interroge la capacité de processus symboliques comme les calculs à simuler certains processus empiriques. Elle permet également d’étudier le statut épistémologique des calculs réalisés par des machines. S’il existait une fonction calculable par une machine sans être calculable par un algorithme, il existerait un problème mathématique qui serait soluble par un dispositif empirique, sans être soluble par aucune méthode mathématique a priori.Church-Turing’s thesis states that every computable function is computable by a Turing machine. By distinguishing, like many authors in the recent literature, between an algorithmic form of Church-Turing’s thesis on the functions computable by an algorithm, and an empirical form of the same on the functions computable by a machine, it becomes possible to ask a new question: Are the limits of empirical calculation identical to the limits of the algorithms? Or is there a way to empirically perform a calculation no algorithm can perform? I will show the philosophical relevance of this ques­tion. It questions the ability of symbolic processes as calculations to simulate certain empirical processes. It is also essential to the epistemological status of calculations performed by machines. If there were a function computable by a machine without being computable by an algorithm, there would be a mathematical problem, which would be solvable by an empirical device, but would be unsolvable by any a priori mathematical method

From quantum foundations to quantum information protocols and back

Physics has two main ambitions: to predict and to understand. Indeed, physics aims for the prediction of all natural phenomena. Prediction entails modeling the correlation between an action, the input, and what is subsequently observed, the output.Understanding, on the other hand, involves developing insightful principles and models that can explain the widest possible varietyof correlations present in nature. Remarkably, advances in both prediction and understanding foster our physical intuition and, as a consequence, novel and powerful applications are discovered. Quantum mechanics is a very successful physical theory both in terms of its predictive power as well as in its wide applicability. Nonetheless and despite many decades of development, we do not yet have a proper physical intuition of quantum phenomena. I believe that improvements in our understanding of quantum theory will yield better, and more innovative, protocols and vice versa.This dissertation aims at advancing our understanding and developing novel protocols. This is done through four approaches. The first one is to study quantum theory within a broad family of theories. In particular, we study quantum theory within the family of locally quantum theories. We found out that the principle that singles out quantum theory out of this family, thus connecting quantum local and nonlocal structure, is dynamical reversibility. This implies that the viability of large scale quantum computing can be based on concrete physical principles that can be experimentally tested at a local level without needing to test millions of qubits simultaneously. The second approach is to study quantum correlations from a black box perspective thus making as few assumptions as possible. The strategy is to study the completeness of quantum predictions by benchmarking them against alternative models. Three main results and applications come out of our study. Firstly, we prove that performing complete amplification of randomness starting from a source of arbitrarily weak randomness - a task that is impossible with classical resources - is indeed possible via nonlocality. This establishes in our opinion the strongest evidence for a truly random event in nature so far. Secondly, we prove that there exist finite events where quantum theory gives predictions as complete as any no-signaling theory can give, showing that the completeness of quantum theory is not an asymptotic property. Finally, we prove that maximally nonlocal theories can never be maximally random while quantum theory can, showing a trade-off between the nonlocality of a theory and its randomness capabilities. We also prove that quantum theory is not unique in this respect. The third approach we follow is to study quantum correlations in scenarios where some parties have a restriction on the available quantum degrees of freedom. The future progress of semi-device-independent quantum information depends crucially on our ability to bound the strength of these correlations. Here we provide a full characterization via a complete hierarchy of sets that approximate the target set from the outside. Each set can be in turn characterized using standard numerical techniques. One application of our work is certifying multidimensional entanglement device-independently.The fourth approach is to confront quantum theory with computer science principles. In particular, we establish two interesting implications for quantum theory results of raising the Church-Turing thesis to the level of postulate. Firstly, we show how different preparations of the same mixed state, indistinguishable according to the quantum postulates, become distinguishable when prepared computably. Secondly, we identify a new loophole for Bell-like experiments: if some parties in a Bell-like experiment use private pseudorandomness to choose their measurement inputs, the computational resources of an eavesdropper have to be limited to observe a proper violation of non locality.La física tiene dos finalidades: predecir y comprender. En efecto, la física aspira a poder predecir todos los fenómenos naturales. Predecir implica modelar correlaciones entre una acción y la reacción subsiguiente.Comprender, implica desarrollar leyes profundas que expliquen la más amplia gama de correlaciones presentes en la naturaleza. Avances tanto en la capacidad de predicción como en nuestra comprensión fomentan la intuición física y, como consecuencia, surgen nuevas y poderosas aplicaciones. La mecánica cuántica es una teoría física de enorme éxito por su capacidad de predicción y amplia aplicabilidad.Sin embargo, a pesar de décadas de gran desarrollo, no poseemos una intuición física satisfactoria de los fenómenos cuánticos.Creo que mejoras en nuestra comprensión de la teoría cuántica traerán consigo mejores y más innovadores protocolos y vice versa.Ésta tesis doctoral trata simultáneamente de avanzar nuestra comprensión y de desarrollar nuevos protocolos mediante cuatro enfoques distintos.El primero consiste en estudiar la mecánica cuántica como miembro de una familia de teorías: las teorías localmente cuánticas. Probamos que el principio que selecciona a la mecánica cuántica, conectando por tanto la estructura cuántica local y no local, es la reversibilidad de su dinámica.Ésto implica que la viabilidad de la computación cuántica a gran escala puede ser estudiada de manera local, comprobando experimentalmente ciertos principios físicos. El segundo enfoque consiste en estudiar las correlaciones cuánticas desde una perspectiva de 'caja negra', haciendo así el mínimo de asunciones físicas. La estrategia consiste en estudiar la completitud de las predicciones cuánticas, comparándolas con todos los modelos alternativos. Hemos obtenido tres grandes resultados. Primero, probamos que se puede amplificar completamente la aleatoriedad de una fuente de aleatoriedad arbitrariamente débil.Ésta tarea, imposible mediante recursos puramente clásicos, se vuelve factible gracias a la no localidad. Ésto establece a nuestro parecer la evidencia más fuerte de la existencia de eventos totalmente impredecibles en la naturaleza. Segundo, probamos que existen eventos finitos cuyas predicciones cuánticas son tan completas como permite el principio de 'no signaling'. Ésto prueba que la completitud de la mecánica cuántica no es una propiedad asintótica. Finalmente, probamos que las teorías máximamente no locales no pueden ser máximamente aleatorias, mientras que la mecánica cuántica lo es. Ésto muestra que hay una compensación entre la no localidad de una teoría y su capacidad para generar aleatoriedad. También probamos que la mecánica cuántica no es única en éste respecto. En tercer lugar, estudiamos las correlaciones cuánticas en escenarios dónde algunas partes tienen restricciones en el número de grados de libertad cuánticos accesibles. Éste escenario se denomina 'semi-device-independent'. Aquí encontramos una caracterización completa de éstas correlaciones mediante una jerarquía de conjuntos que aproximan al conjunto buscado desde fuera y que pueden ser caracterizados a su vez mediante técnicas numéricas estandar. Un aplicación de nuestro trabajo es la certificación de entrelazamiento multidimensional de manera 'device-independent'. El cuarto y último enfoque consiste en enfrentar a la mecánica cuántica con principios provenientes de la computación. En particular, establecemos dos implicaciones para la mecánica cuántica de elevar la tesis de Church-Turing al nivel de postulado. Primero, mostramos que diferentes preparaciones de un mismo estado mixto, indistinguibles de acuerdo a los axiomas cuánticos, devienen distinguibles cuando son preparados de manera computable. Segundo, identificamos un nuevo 'loophole' en experimentos de Bell: si algunas partes en un experimento de Bell usan pseudo aleatoriedad para escoger sus medidas, los recursos computacionales de un espía deben ser limitados a fin de observar verdaderamente la no localidad