Graphical Calculus for Quantum Key Distribution (Extended Abstract)

Coecke, Bob; Wang, Quanlong; Wang, Baoshan; Wang, Yongjun; Zhang, Qiye

doi:10.1016/j.entcs.2011.01.034

Cited by 10 publications

(6 citation statements)

References 16 publications

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“…As these approaches are based on process calculi, they are categorical under the hood [62,65] even if not overtly so. There is also earlier work explicitly discussing category theory in the context of cryptography [12,13,25,26,40,42,43,47,66,67,76,77], but they concern stand-alone security of particular (kinds of) cryptographic protocols, rather than categorical aspects of composable security definitions.…”

Section: Related Workmentioning

confidence: 99%

Categorical composable cryptography

Broadbent,

Karvonen

2021

Preprint

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We formalize the simulation paradigm of cryptography in terms of category theory and show that protocols secure against abstract attacks form a symmetric monoidal category, thus giving an abstract model of composable security definitions in cryptography. Our model is able to incorporate computational security, set-up assumptions and various attack models such as colluding or independently acting subsets of adversaries in a modular, flexible fashion. We conclude by using string diagrams to rederive no-go results concerning the limits of bipartite and tripartite cryptography, ruling out e.g. composable commitments and broadcasting. On the way, we exhibit two categorical constructions of resource theories that might be of independent interest: one capturing resources shared among n parties and one capturing resource conversions that succeed asymptotically.

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Section: Related Workmentioning

confidence: 99%

Categorical composable cryptography

Broadbent,

Karvonen

2021

Preprint

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“…Categorical quantum mechanics (CQM) [34][35][36][37][38][39][40][41][42][43][44][45] is the study of quantum computation and quantum foundations using category theory, as well as the graphical language closely related to category theory. In CQM, dagger monoidal categories (DMC) are used as an axiomatic basis for quantum mechanics, providing a generalization of the usual axiomatization in terms of Hilbert spaces.…”

Section: Bit Commitment In Categorical Quantum Mechanicsmentioning

confidence: 99%

Impossibility of Quantum Bit Commitment, a Categorical Perspective

Sun

Wang

2020

Axioms

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Bit commitment is a cryptographic task in which Alice commits a bit to Bob such that she cannot change the value of the bit after her commitment and Bob cannot learn the value of the bit before Alice opens her commitment. According to the Mayers-Lo-Chau (MLC) no-go theorem, ideal bit commitment is impossible within quantum theory. In the information theoretic-reconstruction of quantum theory, the impossibility of quantum bit commitment is one of the three information-theoretic constraints that characterize quantum theory. In this paper, we first provide a very simple proof of the MLC no-go theorem and its quantitative generalization. Then, we formalize bit commitment in the theory of dagger monoidal categories. We show that in the setting of dagger monoidal categories, the impossibility of bit commitment is equivalent to the unitary equivalence of purification.Axioms 2020, 9, 28 2 of 12 with quantitative bounds on the degree of concealment and bindingness. D'Ariano et al. [27] provided a strengthened and explicit impossibility proof exhausting all conceivable protocols in which not only quantum information, but also classical information is exchanged between the two parties. However, the considerable length of the proof in [27] makes it still hard to follow. Chiribella et al. [28] simplified the proof in [27]. In the works of Cohn-Gordon [29] and Heunen and Kissinger [30], a clear and rigorous formalization of QBC is developed in the setting of categorical quantum mechanics. also provided a proof of the no-go theorem. While this proof is already simpler than all previous proofs, we find there is still room for simplification and extension.In Clifton et al.'s information theoretic-reconstruction of quantum theory [31], the impossibility of bit commitment is conceived as one of the three fundamental information-theoretic constraints that characterize quantum theory. In [31], the authors partially proved that the impossibility of bit commitment is equivalent to the existence of entangled, or nonlocal, states. This result was questioned by Heunen and Kissinger [30], who demonstrated that, in the categorical setting, the impossibility of bit commitment is not equivalent to the existence of entangled states. Which quantum feature is the one that is equivalent to the impossibility of bit commitment is left unanswered in [30].The contributions of this paper are ass follows.

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“…Effects correspond to positive semidefinite operators on Q with operator norm less than or equal to one. The effect Q → C given by ij w ij |ij → i w ii is denoted by the diagram below Q (11) and corresponds to taking the trace of a linear operator. If V is a classical register then…”

Section: The Graphical Language Of Categorical Quantum Mechanicsmentioning

confidence: 99%

“…To our knowledge the use of graphical methods to formalize quantum cryptography is fairly new, although the literature provides some useful beginnings. Graphical security proofs for quantum key distribution (one of the original problems in the field) have been presented in [10][11][12], although these are not yet at the level of security that has been proved through non-graphical means. Meanwhile, the literature has a number of formal treatments of cryptography that are not primarily based on graphical reasoning.…”

Section: Introductionmentioning

confidence: 99%

Graphical Methods in Device-Independent Quantum Cryptography

Breiner

Miller

Ross

2019

Quantum

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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.

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Graphical Calculus for Quantum Key Distribution (Extended Abstract)

Cited by 10 publications

References 16 publications

Categorical composable cryptography

Categorical composable cryptography

Impossibility of Quantum Bit Commitment, a Categorical Perspective

Graphical Methods in Device-Independent Quantum Cryptography

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