6,865 research outputs found
Graphical Methods in Device-Independent Quantum Cryptography
We introduce a framework for graphical security proofs in device-independent
quantum cryptography using the methods of categorical quantum mechanics. We are
optimistic that this approach will make some of the highly complex proofs in
quantum cryptography more accessible, facilitate the discovery of new proofs,
and enable automated proof verification. As an example of our framework, we
reprove a previous result from device-independent quantum cryptography: any
linear randomness expansion protocol can be converted into an unbounded
randomness expansion protocol. We give a graphical proof of this result, and
implement part of it in the Globular proof assistant.Comment: Publishable version. Diagrams have been polished, minor revisions to
the text, and an appendix added with supplementary proof
Informal proof, formal proof, formalism
Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened
Recommended from our members
Leveling transparency via situated intermediary learning objectives (SILOs)
When designers set out to create a mathematics learning activity, they have a fair sense of its objectives: students will understand a concept and master relevant procedural skills. In reform-oriented activities, students first engage in concrete situations, wherein they achieve situated, intermediary learning objectives (SILOs), and only then they rearticulate their solutions formally. We define SILOs as heuristics learners devise to accommodate contingencies in an evolving problem space, e.g., monitoring and repairing manipulable structures so that they model with fidelity a source situation. Students achieve SILOs through problem-solving with media, instructors orient toward SILOs via discursive solicitation, and designers articulate SILOs via analyzing implementation data. We describe the emergence of three SILOs in developing the activity Giant Steps for Algebra. Whereas the notion of SILOs emerged spontaneously as a framework to organize a system of practice, i.e. our collaborative design, it aligns with phenomenological theory of knowledge as instrumented action
Lower and Upper Conditioning in Quantum Bayesian Theory
Updating a probability distribution in the light of new evidence is a very
basic operation in Bayesian probability theory. It is also known as state
revision or simply as conditioning. This paper recalls how locally updating a
joint state can equivalently be described via inference using the channel
extracted from the state (via disintegration). This paper also investigates the
quantum analogues of conditioning, and in particular the analogues of this
equivalence between updating a joint state and inference. The main finding is
that in order to obtain a similar equivalence, we have to distinguish two forms
of quantum conditioning, which we call lower and upper conditioning. They are
known from the literature, but the common framework in which we describe them
and the equivalence result are new.Comment: In Proceedings QPL 2018, arXiv:1901.0947
- …