Inadequacy of Modal Logic in Quantum Settings

We test the principles of classical modal logic in fully quantum settings. Modal logic models our reasoning in multi-agent problems, and allows us to solve puzzles like the muddy children paradox. The Frauchiger-Renner thought experiment highlighted fundamental problems in applying classical reasoning when quantum agents are involved; we take it as a guiding example to test the axioms of classical modal logic. In doing so, we find a problem in the original formulation of the Frauchiger-Renner theorem: a missing assumption about unitarity of evolution is necessary to derive a contradiction and prove the theorem. Adding this assumption clarifies how different interpretations of quantum theory fit in, i.e., which properties they violate. Finally, we show how most of the axioms of classical modal logic break down in quantum settings, and attempt to generalize them. Namely, we introduce constructions of trust and context, which highlight the importance of an exact structure of trust relations between agents. We propose a challenge to the community: to find conditions for the validity of trust relations, strong enough to exorcise the paradox and weak enough to still recover classical logic.Comment: In Proceedings QPL 2018, arXiv:1901.0947

Thermalisation and entropy in Heisenberg spin chains

Mestrado em Engenharia FísicaEste projecto tem como objectivo estudar o comportamento de medidas de entropia quântica em pequenos sistemas quânticos, de forma a se obter uma intuição sobre o assunto que veja a ser útil para um futuro projecto dedicado ao desenvolvimento de uma termodinâmica para pequenos sistemas quânticos baseada em entropias quânticas. Mostramos como modelar sistemas quânticos que interagem entre si. Introduzimos a noção de entropia quântica e discutimos o significado físico de algumas medidas de entropia, bem como as relações entre si. Apresentamos uma abordagem do ponto de vista da informação quântica ao problema da termalização e equilíbrio. De seguida introduzimos e discutimos os sistemas modelo estudados, cadeias de spin de Heisenberg. Para contribuir para esta linha de trabalho, desenvolvemos e executamos simulações numéricas nestas cadeias de spin, de forma a estudar o comportamento de várias medidas de entropia à medida que pequenos subsistemas termalizavam. Entre outras coisas, concluímos que as diferentes medidas de entropia apresentam diferentes tempos de saturação. Apenas a medida de entropia mais lenta constitui um indicador adequado acerca do estado de termalização do sistema. ABSTRACT: The goal of this project is to study the behaviour of quantum entropy measures in small quantum systems, in order to get an intuition for the subject that may help to orientate a future project dedicated to the development of a thermodynamic theory of small quantum systems based on quantum entropies. We show how to model composite (in particular interacting) quantum systems. We introduce the notion of quantum entropy and discuss the physical meaning of some entropy measures and the relations between them. We present a quantum-information framework to the phenomena of thermalisation and equilibration and recall state-of-the-art results in this topic. Then we introduce the toy systems of study, Heisenberg spin chains, from the most basic principles of spin and exchange interaction. We discuss their physical meaning and how they behave under the action of the XXZ Hamiltonian. To contribute to that body of work we finally develop and perform numerical simulations in those spin chains in order to study the behaviour of several entropy measures as small systems thermalised. Amongst other things, we find that distinct entropy measures saturate at different times. Only the slowest of these measures is appropriate to indicate whether a system has equilibrated

The role of quantum information in thermodynamics --- a topical review

This topical review article gives an overview of the interplay between quantum information theory and thermodynamics of quantum systems. We focus on several trending topics including the foundations of statistical mechanics, resource theories, entanglement in thermodynamic settings, fluctuation theorems and thermal machines. This is not a comprehensive review of the diverse field of quantum thermodynamics; rather, it is a convenient entry point for the thermo-curious information theorist. Furthermore this review should facilitate the unification and understanding of different interdisciplinary approaches emerging in research groups around the world.Comment: published version. 34 pages, 6 figure

Toys can't play: physical agents in Spekkens' theory

Information is physical, and for a physical theory to be universal, it should model observers as physical systems, with concrete memories where they store the information acquired through experiments and reasoning. Here we address these issues in Spekkens' toy theory, a non-contextual epistemically restricted model that partially mimics the behaviour of quantum mechanics. We propose a way to model physical implementations of agents, memories, measurements, conditional actions and information processing. We find that the actions of toy agents are severely limited: although there are non-orthogonal states in the theory, there is no way for physical agents to consciously prepare them. Their memories are also constrained: agents cannot forget in which of two arbitrary states a system is. Finally, we formalize the process of making inferences about other agents' experiments and model multi-agent experiments like Wigner's friend. Unlike quantum theory or box world, in the toy theory there are no inconsistencies when physical agents reason about each other's knowledge.Comment: 18 + 19 page

Fitch's knowability axioms are incompatible with quantum theory

How can we consistently model the knowledge of the natural world provided by physical theories? Philosophers frequently use epistemic logic to model reasoning and knowledge abstractly, and to formally study the ramifications of epistemic assumptions. One famous example is Fitch's paradox, which begins with minimal knowledge axioms and derives the counter-intuitive result that "every agent knows every true statement." Accounting for knowledge that arises from physical theories complicates matters further. For example, quantum mechanics allows observers to model other agents as quantum systems themselves, and to make predictions about measurements performed on each others' memories. Moreover, complex thought experiments in which agents' memories are modelled as quantum systems show that multi-agent reasoning chains can yield paradoxical results. Here, we bridge the gap between quantum paradoxes and foundational problems in epistemic logic, by relating the assumptions behind the recent Frauchiger-Renner quantum thought experiment and the axioms for knowledge used in Fitch's knowability paradox. Our results indicate that agents' knowledge of quantum systems must violate at least one of the following assumptions: it cannot be distributive over conjunction, have a kind of internal continuity, and yield a constructive interpretation all at once. Indeed, knowledge provided by quantum mechanics apparently contradicts traditional notions of how knowledge behaves; for instance, it may not be possible to universally assign objective truth values to claims about agent knowledge. We discuss the relations of this work to results in quantum contextuality and explore possible modifications to standard epistemic logic that could make it consistent with quantum theory.Comment: 22 + 7 page

Operational locality in global theories

Within a global physical theory, a notion of locality allows us to find and justify information-processing primitives, like non-signalling between distant agents. Here, we propose exploring the opposite direction: to take agents as the basic building blocks through which we test a physical theory, and recover operational notions of locality from signalling conditions. First, we introduce an operational model for the effective state spaces of individual agents, as well as the range of their actions. We then formulate natural secrecy conditions between agents and identify the aspects of locality relevant for signalling. We discuss the possibility of taking commutation of transformations as a primitive of physical theories, as well as applications to quantum theory and generalized probability frameworks. This ‘it from bit’ approach establishes an operational connection between local actions and local observations, and gives a global interpretation to concepts like discarding a subsystem or composing local functions.ISSN:1364-503XISSN:1471-296

Operational locality in global theories

Within a global physical theory, a notion of locality allows us to find and justify information-processing primitives, like non-signalling between distant agents. Here, we propose exploring the opposite direction: to take agents as the basic building blocks through which we test a physical theory, and recover operational notions of locality from signalling conditions. First, we introduce an operational model for the effective state spaces of individual agents, as well as the range of their actions. We then formulate natural secrecy conditions between agents and identify the aspects of locality relevant for signalling. We discuss the possibility of taking commutation of transformations as a primitive of physical theories, as well as applications to quantum theory and generalized probability frameworks. This ‘it from bit’ approach establishes an operational connection between local actions and local observations, and gives a global interpretation to concepts like discarding a subsystem or composing local functions.ISSN:1364-503XISSN:1471-296

Composable security in relativistic quantum cryptography

Relativistic protocols have been proposed to overcome certain impossibility results in classical and quantum cryptography. In such a setting, one takes the location of honest players into account, and uses the signalling limit given by the speed of light to constraint the abilities of dishonest agents. However, composing such protocols with each other to construct new cryptographic resources is known to be insecure in some cases. To make general statements about such constructions, a composable framework for modelling cryptographic security in Minkowski space is required. Here, we introduce a framework for performing such a modular security analysis of classical and quantum cryptographic schemes in Minkowski space. As an application, we show that (1) fair and unbiased coin flipping can be constructed from a simple resource called channel with delay; (2) biased coin flipping, bit commitment and channel with delay through any classical, quantum or post-quantum relativistic protocols are all impossible without further setup assumptions; (3) it is impossible to securely increase the delay of a channel, given several short-delay channels as ingredients. Results(1) and (3) imply in particular the non-composability of existing relativistic bit commitment and coin flipping protocols

The thermodynamic meaning of negative entropy

Landauer's erasure principle exposes an intrinsic relation between thermodynamics and information theory: the erasure of information stored in a system, S, requires an amount of work proportional to the entropy of that system. This entropy, H(S|O), depends on the information that a given observer, O, has about S, and the work necessary to erase a system may therefore vary for different observers. Here, we consider a general setting where the information held by the observer may be quantum-mechanical, and show that an amount of work proportional to H(S|O) is still sufficient to erase S. Since the entropy H(S|O) can now become negative, erasing a system can result in a net gain of work (and a corresponding cooling of the environment).Comment: Added clarification on non-cyclic erasure and reversible computation (Appendix E). For a new version of all technical proofs see the Supplementary Information of the journal version (free access