I show how classical and quantum physics approach the problem of randomness and probability. Contrary to popular opinions, neither we can prove that classical mechanics is a deterministic theory, nor that quantum mechanics is a nondeterministic one. In other words it is not possible to show that randomness in classical mechanics has a purely epistemic character and that of quantum mechanics an ontic one. Nevertheless, recent developments of quantum theory and increasing experimental possibilities to check its predictions call for returning to the problem of comparing possibilities given by classical and quantum physics to accommodate and prove the existence of a `genuine randomness'. Recent results concerning `amplification of randomness' show that, in certain sense, quantum physics is in fact ‘more random’ that classical and outperforms it in producing a `truly random process'.
I show how classical and quantum physics approach the problem of randomness and probability. Contrary to popular opinions, neither we can prove that classical mechanics is a deterministic theory, nor that quantum mechanics is a nondeterministic one. In other words it is not possible to show that randomness in classical mechanics has a purely epistemic character and that of quantum mechanics an ontic one. Nevertheless, recent developments of quantum theory and increasing experimental possibilities to check its predictions call for returning to the problem of comparing possibilities given by classical and quantum physics to accommodate and prove the existence of a genuine randomness'. Recent results concerning amplification of randomness' show that, in certain sense, quantum physics is in fact ‘more random’ that classical and outperforms it in producing a `truly random process'.
Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive non-relativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical statistical mechanics: an "ontic extension" defines a nonseparable (global) random variable that generates physical correlations, and an "epistemic restriction" constrains allowed phase space distributions. The ontic extension and epistemic restriction, with strength on the order of Planck's constant, imply quantum entanglement and uncertainty relations. This framework suggests that the wave function is epistemic, yet it does not provide an ontic dynamics for individual systems.
Abstract Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive non-relativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical statistical mechanics: an “ontic extension” defines a nonseparable (global) random variable that generates physical correlations, and an “epistemic restriction” constrains allowed phase space distributions. The ontic extension and epistemic restriction, with strength on the order of Planck’s constant, imply quantum entanglement and uncertainty relations. This framework suggests that the wave function is epistemic, yet it does not provide an ontic dynamics for individual systems.
The description of physical systems obtained in the classical limit of quantum mechanics is not consistent with the one given by classical statistical mechanics. In qantum mechanics the complete description of a physical system is given by specifying the wave function, which evolves completely deterministically. Probability comes into play only when a measurement is made. Classical statistical mechanics, on the contrary, concerns systems whose description is not complete and where probability enters as a consequence of a lack of knowledge. We show that Nelson's stochastic mechanics, whose predictions are identical with those of quantum mechanics, provides nevertheless a unified description of the deterministic and random aspects of quantum mechanics which eliminates the inconsistency outlined above and gives a simple and satisfactory solution to the measurement problem.
Quantum mechanics was developed as a response to the inadequacy of classical physics in explaining certain physical phenomena. While it has proved immensely successful, it also presents several features that severely challenge our classicality based intuition. Randomness in quantum theory is one such and is the central theme of this dissertation. Randomness is a notion we have an intuitive grasp on since it appears to abound in nature. It a icts weather systems and nancial markets and is explicitly used in sport and gambling. It is used in a wide range of scienti c applications such as the simulation of genetic drift, population dynamics and molecular motion in fluids. Randomness (or the lack of it) is also central to philosophical concerns such as the existence of free will and anthropocentric notions of ethics and morality. The conception of randomness has evolved dramatically along with physical theory. While all randomness in classical theory can be fully attributed to a lack of knowledge of the observer, quantum theory qualitatively departs by allowing the existence of objective or intrinsic randomness. It is now known that intrinsic randomness is a generic feature of hypothetical theories larger than quantum theory called the non-signalling theories. They are usually studied with regards to a potential future completion of quantum mechanics or from the perspective of recognizing new physical principles describing nature. While several aspects have been studied to date, there has been little work in globally characterizing and quantifying randomness in quantum and non-signalling theories and the relationship between them. This dissertation is an attempt to ll this gap. Beginning with the unavoidable assumption of a weak source of randomness in the universe, we characterize upper bounds on quantum and non-signalling randomness. We develop a simple symmetry argument that helps identify maximal randomness in quantum theory and demonstrate its use in several explicit examples. Furthermore, we show that maximal randomness is forbidden within general non-signalling theories and constitutes a quantitative departure from quantum theory. We next address (what was) an open question about randomness ampli cation. It is known that a single source of randomness cannot be ampli ed using classical resources alone. We show that using quantum resources on the other hand allows a full ampli cation of the weakest sources of randomness to maximal randomness even in the presence of supra-quantum adversaries. The signi cance of this result spans practical cryptographic scenarios as well as foundational concerns. It demonstrates that conditional on the smallest set of assumptions, the existence of the weakest randomness in the universe guarantees the existence of maximal randomness. The next question we address is the quanti cation of intrinsic randomness in non-signalling correlations. While this is intractable in general, we identify cases where this can be quanti ed. We nd that in these cases all observed randomness is intrinsic even relaxing the measurement independence assumption. We nally turn to the study of the only known resource that allows generating certi able intrinsic randomness in the laboratory i.e. entanglement. We address noisy quantum systems and calculate their entanglement dynamics under decoherence. We identify exact results for several realistic noise models and provide tight bounds in some other cases. We conclude by putting our results into perspective, pointing out some drawbacks and future avenues of work in addressing these concerns.
The problem of the determinism of Quantum Mechanics has been a main one during the 20th century. At the same time, in the context of Logic and Set Theory, the importance of ancient paradoxes as well as the appearance of many new ones, has shed light on and deeply influenced the foundations of Mathematics and somehow of Physics. But, strangely, concerning Physics, a paradox which we call the Memory Paradox has remained yet undiscovered, despite its simplicity and remarkable consequences, mostly in Physics and surprisingly in classical Physics that appear to be non deterministic, contrary to the general belief since Newton, Laplace, etc.. The non determinism of Quantum Physics follows without any supplementary hypothesis. This paper extends a previous one (arXiv: 1203.2945v1 [physics.gen-ph] 13 Mar 2012).
Quantum nonlocality has been a long-standing and fascinating problem in the foundations of quantum theory since its proof by the Bell inequality. It seems to contradict our intuitive picture of space-time and thus urges us to reconsider its foundation. With the advent of quantum information science, it has attracted renewed interest from a fundamental point of view as well as from a quantum information-theoretic one. Another fundamental question on the nature of quantum theory is how we understand what quantum states represent. (Quantum states hereafter refer to pure states.) This is also a long-standing problem since the early days of quantum theory. Roughly speaking, there are two conflicting views on this matter: one is ontology and the other is epistemology. Ontological models of quantum theory are models in which we assume some underlying reality for physical systems and require that all the predictions of quantum theory should be reproduced. The ontological models are divided into two categories, depending on which views the models are based on, ontology or epistemology. In the models based on ontology, quantum states represent the underlying reality in some sense, and different quantum states correspond to different realities. In the models based on epistemology, quantum states do not necessarily correspond to the underlying realities, and even different quantum states can be consistent with a common underlying reality. The former type of ontological models of quantum theory is called 1⁄4-ontic. The latter is called 1⁄4-epistemic. Thus, it might be rather confusing that ontological models are classified further into 1⁄4-ontic models and 1⁄4-epistemic ones. We use these terms here in accordance with the recent literature on quantum foundations. (For details of these notions in the ontological models of quantum theory, see the paper by Harrigan and Spekkens.) Quite recently, the question just mentioned has been answered by Pusey et al., and one of the above models has been refuted. It was shown that 1⁄4-epistemic models are inconsistent with the predictions of quantum theory. In other words, epistemological views on quantum states turned out to be false as long as we assume some underlying reality for physical systems. Therefore, only 1⁄4-ontic models of quantum theory are valid for describing quantum systems. This significantly advances our understanding on the nature of quantum states. It is expected that various conclusions will be drawn from this fact. In this paper, we will discuss a connection with nonlocality. The above two fundamental questions, nonlocality and the nature of quantum states, are intimately connected, and the connection was first implicitly recognised by Einstein in the early days of quantum theory. In a private correspondence with Schrodinger, Einstein constructed a fascinating argument that relates the above two. (That argument is distinctly different from the well-known EPR argument.) Recently, it was revisited and reconstructed in accordance with more rigorous notions in quantum foundations. Nonlocality is usually shown by violations of Bell inequalities. (Bell inequalities hereafter include all the subsequent developments such as the CHSH inequality.) Even if it is not clear whether the models of quantum theory are 1⁄4-ontic or 1⁄4-epistemic, we can prove nonlocality just by showing the violation of Bell inequalities. However, the above argument originated by Einstein implies that if we know the models are 1⁄4-ontic in advance (as we do now), we can prove nonlocality in a simpler way without Bell inequalities. (This argument had been known in quantum foundations before the refutation of 1⁄4-epistemic models.) Nonlocality is rigorously defined as a violation of the notion of local causality defined by Bell. Showing the violation of Bell inequalities is one way of demonstrating the violation of local causality. Since the simpler proof initiated by Einstein directly demonstrates the violation of local causality, it is a more straightforward proof of nonlocality from a conceptual point of view. In the following, we will see that this simple argument of nonlocality without Bell inequalities can be used to show multipartite nonlocality as well. Before moving onto multipartite cases, first let us define the necessary notions for presenting our result. First of all, we assume that there exists a complete specification of the properties of a physical system, which we refer to as the ontic state of that system. The ontic states represent the underlying realities and are not necessarily equal to ordinary quantum states. (In other words, the ontic states correspond to hidden variables.) Ontological models of quantum theory are models that assume ontic states and reproduce all the predictions of quantum theory. An important question is how ontic states correspond to quantum states in quantum theory. In order to frame the question accurately, we need the notion of an epistemic state, which represents the relation between them. Suppose we prepare a physical system in a quantum state 1⁄4. Although the system is uniquely specified as 1⁄4 in quantum theory, there still remains a possibility that it allows a more detailed specification in the ontic state space, in which case many different ontic states turn out to be consistent with 1⁄4. The relation between ontic states and the preparation procedure, P1⁄4, is described by a conditional probability distribution over the ontic state space, pð jP1⁄4Þ, which we refer to as an epistemic state. In general, the preparation P can be a preparation of a physical system in a mixed sate. Unless there exists a one-to-one correspondence between ontic states and quantum states, the probability distribution spreads over . (See Fig. 1.) The question of whether an ontological model is 1⁄4-ontic or 1⁄4-epistemic is concerned with the behaviour of the epistemic states for two distinct preparations of quantum states. Consider two distinct preparations of quantum states, Journal of the Physical Society of Japan 82 (2013) 055003