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# A Categorical Semantics of Quantum Protocols

logic in computer science, (2004): 415-425

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Abstract

We present a concise and natural encoding of the spi-calculus into the more basic 驴-calculus and establish its correctness with respect to a formal notion of testing.This is particularly relevant for security protocols modelled in spi since the tests ...

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Introduction

- Quantum information and computation is concerned with the use of quantum-mechanical systems to carry out computational and information-processing tasks [20].
- Teleportation is the most basic of a family of quantum protocols, and already illustrates the basic ideas, in particular the use of preparations of entangled states as carriers for information flow, performing measurements to propagate information, using classical information to control branching behaviour to ensure the required behaviour despite quantum indeterminacy, and performing local data transformations using unitary operations
- It should be born in mind that the information required to specify Õ is an arbitrary pair of complex numbers« ¬μ satisfying « 3⁄4 · ¬ 3⁄4 1⁄2, so achieving this information transfer with just two classical bits is no mean feat! Teleportation is the most basic of a family of quantum protocols, and already illustrates the basic ideas, in particular the use of preparations of entangled states as carriers for information flow, performing measurements to propagate information, using classical information to control branching behaviour to ensure the required behaviour despite quantum indeterminacy, and performing local data transformations using unitary operations

Highlights

- Quantum information and computation is concerned with the use of quantum-mechanical systems to carry out computational and information-processing tasks [20]
- Transfer with just two classical bits is no mean feat! Teleportation is the most basic of a family of quantum protocols, and already illustrates the basic ideas, in particular the use of preparations of entangled states as carriers for information flow, performing measurements to propagate information, using classical information to control branching behaviour to ensure the required behaviour despite quantum indeterminacy, and performing local data transformations using unitary operations
- ̄ In order to handle protocols and quantum computations more systematically, it would be desirable to have an effective syntax, whose design should be guided by the categorical semantics

Conclusion

- ̄ The information flow level of analysis using only the compact-closed structure allows some very elegant and convenient ‘qualitative’ reasoning, while adding biproducts allows very fine-grained modelling and analysis.
- The interplay between these two levels merits further investigation.
- A significant step towards the infinite dimensional case is provided by the previous work on nuclear ideals in tensored £-categories [1].
- One of the main intended models of nuclear ideals is given by the category of all Hilbert spaces and bounded linear maps

Reference

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