Foundations Of Quantum Theory Research Articles

On Entropic Uncertainty Relations for Measurements of Energy and Its “Complement”

AbstractHeisenberg's uncertainty principle in application to energy and time is a powerful heuristics. This statement plays an important role in foundations of quantum theory and statistical physics. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is well known that the case of energy and time principally differs from more familiar examples of two non‐commuting observables. Since quantum theory was originated, many approaches to energy–time uncertainties have been proposed. Entropic way to formulate the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, uncertainty relations of the “energy–time” type are obtained in terms of Rényi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state‐dependent and state‐independent formulations are of interest. Some of the derived state‐independent bounds are similar to the results obtained within a more general approach on the basis of sandwiched relative entropies. The developed method allows us to address the case of detection inefficiencies.

Nonlocality of all quantum networks

The multipartite correlations derived from local measurements on some composite quantum systems are inconsistent with those reproduced classically. This inconsistency is known as quantum nonlocality and shows a milestone in the foundations of quantum theory. Still, it is NP hard to decide a nonlocal quantum state. We investigate an extended question: how to characterize the nonlocal properties of quantum states that are distributed and measured in networks. We first prove the generic tripartite nonlocality of chain-shaped quantum networks using semiquantum nonlocal games. We then introduce a new approach to prove the generic activated nonlocality as a result of entanglement swapping for all bipartite entangled states. The result is further applied to show the multipartite nonlocality and activated nonlocality for all nontrivial quantum networks consisting of any entangled states. Our results provide the nonlocality witnesses and quantum superiorities of all connected quantum networks or nontrivial hybrid networks in contrast to classical networks.

Foundation of Quantum Mechanics: Once Again

Brukner and Dakic (Quantum theory and beyond: is entanglement special? 2009. https://arxiv.org/abs/0911.0695 ) proposed a very simple axiom system as a foundation for quantum theory. It implies the qubit and quantum entanglement. Because this axiom system aims at the core of our understanding of nature, it must be brought to the forum of the philosophy of nature. For philosophical reasons, a completely denied champion of quantum theory, imaginarity i, is added into this axiom system. In relation to Bell’s inequality, this leads to a deeper ‘philosophical’ understanding of quantum nature based on qubits and entanglement. Both opens a way as well as one can get to the fundamental Schrodinger equation of quantum mechanics with the help of a complex valued Brownian motion.

Ontological Clarity via Canonical Presentation: Electromagnetism and the Aharonov-Bohm Effect.

Quantum physics demands some radical revision of our fundamental beliefs about physical reality. We know that because there are certain verified physical phenomena—two-slit interference, the disappearance of interference upon monitoring, violations of Bell’s inequality—that have no classical analogs. But the exact nature of that revision has been under dispute since the foundation of quantum theory. I offer a method of clarifying what the commitments of a clearly formulated physical theory are, and apply it to a discussion of some options available to account for another non-classical phenomenon: the Aharonov–Bohm effect.

Macroscopic quantum states: Measures, fragility, and implementations

Large-scale quantum effects have always played an important role in the foundations of quantum theory. With recent experimental progress and the aspiration for quantum enhanced applications, the interest in macroscopic quantum effects has been reinforced. In this review, measures aiming to quantify various aspects of macroscopic quantumness are critically analyzed and discussed. Recent results on the difficulties and prospects to create, maintain, and detect macroscopic quantum states are surveyed. The role of macroscopic quantum states in foundational questions as well as practical applications is outlined. Finally, past and ongoing experimental advances aiming to generate and observe macroscopic quantum states are presented.16 MoreReceived 19 June 2017DOI:https://doi.org/10.1103/RevModPhys.90.025004© 2018 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum computationQuantum measurementsQuantum metrologyQuantum states of lightQuantum Information

Modular Spacetime and Metastring Theory

In this talk we review our recent work on metastring theory and its habitat, a new form of quantum spacetime, called modular spacetime. We emphasize that the geometry underlying modular spacetime, i.e. the background geometry of metastring theory is also the geometry underlying any quantum theory as formulated in terms of Aharonov’s modular variables. Thus the metastring sheds light on the foundations of quantum theory, and it represents a new formulation of string theory and quantum gravity based on the principle of relative locality.

Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory

The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness.

Anti Chsh—Refutation of the Chsh Inequality

Today, quantum-mechanical concepts i.e. such as non-locality refer to some mathematical foundations, especially to Bell’s inequality and the Chsh inequality. Experimental data, analyzed while using Bell’s inequality or the Chsh inequality, favor a quantum mechanical description of nature over local hidden variable theories (often referred to as local realism). In general, the use of mathematically inconsistent methods can imply a waste of money, time and effort on this account. Under some certain conditions (the assumption of independence) Bell’s theorem and the Chsh inequality are already refuted. The purpose of this publication is to explore the terra incognita, the interior logic that may lie beyond Bell’s original theorem and the Chsh inequality and to refute both Bell’s original theorem and the Chsh inequality under any circumstances by the proof that we can derive a logical contradiction out of Bell’s inequalities. Thus far, if you accept Bell’s theorem or the Chsh inequality as correct, then you must accept too that +0 = +1, which is a logical contradiction. Bell’s theorem and the Chsh inequality are refuted in general. In this insight, it appears to be necessary to revisit the very foundations of quantum theory and of physics as such.

Who made quantum theory popular with physicists and beyond?

My paper considers the impact of the early Solvay meetings from a perspective of popularization, i.e. looking at audiences besides science. In asking “Who made quantum theory popular?” I investigate whether this unusual congress introduced a kind of mechanism — institutional, disciplinary, popularizing or other — that had been recognized and applied also elsewhere. My interpretation of the early Solvay conferences, on the one hand, dispenses with linear stories often found with older accounts, e.g. that the Solvay meetings Nos. 1, 2 and 5 supposedly laid the foundations of quantum theory altogether. On the other hand, I attempt at dispelling the “three miracles” Peter Galison (2007) has found in the history of the first Solvay meeting. He was right to stress that the meetings were not so much spectacular turning points in the history of physics but rather “served as sites for powerful reviews of the field” and were “catalysts for intellectual and social networks”. However, neither the “precise balance between two philantropic forces” in Ernest Solvay’s motivation, nor Lorentz’s brilliant guidance of the discourse, nor the “presence and prior contributions” of Einstein, I would like to be willing to qualify in any way as miracles. There were good reasons, I argue, to give Lorentz the role he filled, and to invite Einstein; also Nernst may have made it clear enough to Solvay, what role would give him the most appreciation. My thesis is that the success of the first Solvay meeting did not much rely on miracles, but was a successful unfolding of a meticulously devised model — call it the Solvay model, or probably more accurately, the Nernst model. This model can be characterized as an externally-funded, agenda-setting, high-profile and international meeting of scientists concerned with one novel scientific problem or field, where select pre-circulated papers of some participants were discussed with more invited discussants, and later its papers and discussions were published for reaching a broader scientific audience as well as informing a more general audience, and all this with the aim to have follow-up meetings as well as a grant giving institution promoting research in the chosen field. For such a model, it obviously needed a Nernst, a Solvay and an interested scientific community, or to put it more generally, an organizer driven by a scientific problem he had encountered and which he wanted to dominate, a Philantropy that can offer funds and institutional resources, and also some disciplinary space, within which a new scientific subfield can be established. I test this model with two cases, which may be called Solvay 1.5 and Solvay 16.5, relating to high-profile meetings at Gottingen in 1913 and San Francisco in 1977.

Information Theoretic Characterization of Physical Theories with Projective State Space

Probabilistic theories are a natural framework to investigate the foundations of quantum theory and possible alternative or deeper theories. In a generic probabilistic theory, states of a physical system are represented as vectors of outcomes probabilities and state spaces are convex cones. In this picture the physics of a given theory is related to the geometric shape of the cone of states. In quantum theory, for instance, the shape of the cone of states corresponds to a projective space over complex numbers. In this paper we investigate geometric constraints on the state space of a generic theory imposed by the following information theoretic requirements: every non completely mixed state of a system is perfectly distinguishable from some other state in a single shot measurement; information capacity of physical systems is conserved under making mixtures of states. These assumptions guarantee that a generic physical system satisfies a natural principle asserting that the more a state of the system is mixed the less information can be stored in the system using that state as logical value. We show that all theories satisfying the above assumptions are such that the shape of their cones of states is that of a projective space over a generic field of numbers. Remarkably, these theories constitute generalizations of quantum theory where superposition principle holds with coefficients pertaining to a generic field of numbers in place of complex numbers. If the field of numbers is trivial and contains only one element we obtain classical theory. This result tells that superposition principle is quite common among probabilistic theories while its absence gives evidence of either classical theory or an implausible theory.

Entropic uncertainty relations for multiple measurements

We present the entropic uncertainty relations for multiple measurement settings in quantum mechanics. Those uncertainty relations are obtained for both cases with and without the presence of quantum memory. They take concise forms which can be proven in a unified method and easy to calculate. Our results recover the well known entropic uncertainty relations for two observables, which show the uncertainties about the outcomes of two incompatible measurements. Those uncertainty relations are applicable in both foundations of quantum theory and the security of many quantum cryptographic protocols.

Probability structures in subspace lattice approach to foundations of quantum theory

Noncommutative measure and probability theory develops parallel to classical probability theory. In our paper, we summarize recent results on the structure of subspaces affiliated with von Neumann algebras and noncommutative measures defined on them. We show the interplay of ideas from classical and noncommutative measure theories based on general inner product spaces and bring new examples in this area.

Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory (Extended Abstract)

The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality relation for a pair of arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the probabilistic interpretation of quantum theory to define the probability of equality relation for a pair of arbitrary observables. Applications of this new interpretation to measurement theory are discussed briefly.

Realism and Antirealism in Informational Foundations of Quantum Theory

Zeilinger-Brukner's informational foundations of quantum theory, a theory based on Zeilinger's foundational principle for quantum mechanics that an elementary system carried one bit of information, explains seemingly unintuitive quantum behavior with simple theoretical framework. It is based on the notion that distinction between reality and information cannot be made, therefore they are the same. As the critics of informational foundations of quantum theory show, this antirealistic move captures the theory in tautology, where information only refers to itself, while the relationships outside the information with the help of which the nature of information would be defined are lost and the questions "Whose information? Information about what?" cannot be answered. The critic's solution is a return to realism, where the observer's effects on the information are neglected. We show that radical antirealism of informational foundations of quantum theory is not necessary and that the return to realism is not the only way forward. A comprehensive approach that exceeds mere realism and antirealism is also possible: we can consider both sources of the constraints on the information, those coming from the observer and those coming from the observed system/nature/reality. The information is always the observer's information about the observed. Such a comprehensive philosophical approach can still support the theoretical framework of informational foundations of quantum theory: If we take that one bit is the smallest amount of information in the form of which the observed reality can be grasped by the observer, we can say that an elementary system (grasped and defined as such by the observer) correlates to one bit of information. Our approach thus explains all the features of the quantum behavior explained by informational foundations of quantum theory: the wave function and its collapse, entanglement, complementarity and quantum randomness. However, it does so in a more comprehensive and intuitive way. The presented approach is close to Husserl's explanation of the relationship between reality and the knowledge we have about it, and to Bohr's personal explanation of quantum mechanics, the complexity of which has often been missed and simplified to mere antirealism. Our approach thus reconnects phenomenology with contemporary philosophy of science and introduces the comprehensive approach that exceeds mere realism and antirealism to the field of quantum theories with informational foundations, where such an approach has not been taken before. Quanta 2014; 3: 32–42.

A quantum-information-theoretic complement to a general-relativistic implementation of a beyond-Turing computer

There exists a growing literature on the so-called physical Church-Turing thesis in a relativistic spacetime setting. The physical Church-Turing thesis is the conjecture that no computing device that is physically realizable (even in principle) can exceed the computational barriers of a Turing machine. By suggesting a concrete implementation of a beyond-Turing computer in a spacetime setting, Istv\'an N\'emeti and Gyula D\'avid (2006) have shown how an appreciation of the physical Church-Turing thesis necessitates the confluence of mathematical, computational, physical, and indeed cosmological ideas. In this essay, I will honour Istv\'an's seventieth birthday, as well as his longstanding interest in, and his seminal contributions to, this field going back to as early as 1987 by modestly proposing how the concrete implementation in N\'emeti and D\'avid (2006) might be complemented by a quantum-information-theoretic communication protocol between the computing device and the logician who sets the beyond-Turing computer a task such as determining the consistency of Zermelo-Fraenkel set theory. This suggests that even the foundations of quantum theory and, ultimately, quantum gravity may play an important role in determining the validity of the physical Church-Turing thesis.

Physics, inconsistency, and quasi-truth

In this work, the first of a series, we study the nature of informal inconsistency in physics, focusing mainly on the foundations of quantum theory, and appealing to the concept of quasi-truth. We defend a pluralistic view of the philosophy of science, grounded on the existence of inconsistencies and on quasi-truth. Here, we treat only the ‘classical aspects’ of the subject, leaving for a forthcoming paper the ‘non-classical’ part.

ψ-Epistemic models are exponentially bad at explaining the distinguishability of quantum states.

The status of the quantum state is perhaps the most controversial issue in the foundations of quantum theory. Is it an epistemic state (state of knowledge) or an ontic state (state of reality)? In realist models of quantum theory, the epistemic view asserts that nonorthogonal quantum states correspond to overlapping probability measures over the true ontic states. This naturally accounts for a large number of otherwise puzzling quantum phenomena. For example, the indistinguishability of nonorthogonal states is explained by the fact that the ontic state sometimes lies in the overlap region, in which case there is nothing in reality that could distinguish the two states. For this to work, the amount of overlap of the probability measures should be comparable to the indistinguishability of the quantum states. In this Letter, I exhibit a family of states for which the ratio of these two quantities must be ≤2de-cd in Hilbert spaces of dimension d that are divisible by 4. This implies that, for large Hilbert space dimension, the epistemic explanation of indistinguishability becomes implausible at an exponential rate as the Hilbert space dimension increases.

Quantifying contextuality.

Contextuality is central to both the foundations of quantum theory and to the novel information processing tasks. Despite some recent proposals, it still faces a fundamental problem: how to quantify its presence? In this work, we provide a universal framework for quantifying contextuality. We conduct two complementary approaches: (i)the bottom-up approach, where we introduce a communication game, which grasps the phenomenon of contextuality in a quantitative manner; (ii)the top-down approach, where we just postulate two measures, relative entropy of contextuality and contextuality cost, analogous to existent measures of nonlocality (a special case of contextuality). We then match the two approaches by showing that the measure emerging from the communication scenario turns out to be equal to the relative entropy of contextuality. Our framework allows for the quantitative, resource-type comparison of completely different games. We give analytical formulas for the proposed measures for some contextual systems, showing in particular that the Peres-Mermin game is by order of magnitude more contextual than that of Klyachko etal. Furthermore, we explore properties of these measures such as monotonicity or additivity.

QUANTUM SYSTEMS BASED UPON GALOIS FIELDS — FROM SUB-QUANTUM TO SUPER-QUANTUM CORRELATIONS

In this paper, we describe our recent work on discrete quantum theory based on Galois fields. In particular, we discuss how discrete quantum theory sheds new light on the foundations of quantum theory and we review an explicit model of super-quantum correlations we have constructed in this context. We also discuss the larger questions of the origins and foundations of quantum theory, as well as the relevance of super-quantum theory for the quantum theory of gravity.

Physics in Discrete Spaces: On Fundamental Interactions

This contribution is the third of a series of articles devoted to the physics of discrete spaces. After the building of space-time [1] and the foundation of quantum theory [2] one studies here how the three fundamental interactions could emerge from the model of discrete space-time that we have put forward in previous contributions. The gauge interactions are recovered. We also propose an original interpretation of gravitational interactions.