Abstract
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.
Highlights
Graphical methods have long been used in the study of physics and computation
Our graphical proof of unbounded expansion was based on two central steps: one was the application of the Spot-Check Lemma (Lemma 4.2), and the other was the principle of causality
Causality is an elementary step in symbolic proofs for quantum information, but in the case of our graphical proof it is an important manipulation
Summary
Graphical methods have long been used in the study of physics and computation. In physics, this can be traced back at least as far as Penrose’s use of diagrams [1]. Pictures are regularly used as a conceptual aid in discussions of quantum cryptography, but it would be beneficial for both accessibility and rigor if proofs themselves could be expressed as pictures For this reason, we believe that the field of quantum cryptography can benefit from the abstract methods of categorical quantum mechanics [8]. One of the recent major achievements of quantum cryptography is the development of device-independent security proofs [18]. In such proofs, the adversary and the quantum devices are untrusted, and allowed to exhibit uncharacterized behavior. The recent paper [38] addresses graphical proofs of quantum cryptography, albeit with a different focus (device-dependent quantum key distribution). The graphical concepts in the present paper and in [38] were developed independently, and we expect that there will be a useful synergy between the two papers
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