## COMPACTLY ACCESSIBLE CATEGORIES AND QUANTUM KEY DISTRIBUTION

Citations: | 2 - 2 self |

### BibTeX

@MISC{Heunen_compactlyaccessible,

author = {Chris Heunen},

title = {COMPACTLY ACCESSIBLE CATEGORIES AND QUANTUM KEY DISTRIBUTION},

year = {}

}

### OpenURL

### Abstract

Abstract. Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finite-dimensional, they cannot accomodate (co)limit-based constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large numbers. To overcome this limitation, we introduce the notion of a compactly accessible category, relying on the extra structure of a factorisation system. This notion allows for infinite dimension while retaining key properties of compact categories: the main technical result is that the choice-of-duals functor on the compact

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Citation Context ... Figure 1 will be used in Section 7 as an example that can be modeled by compactly accessible categories. As such, we need to distinguish between correctness 1 Such a protocol, like Diffie-Hellmann’s =-=[DH76]-=-, regulates key distribution, but gives no guarantee about authenticity of the two parties involved.�� � �� �� 4 C. HEUNEN and security. A quantum key distribution protocol is correct if both parties... |

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Citation Context ...ently secure by the laws of nature, i.e. not depending on any unsolved or computationally unfeasible mathematical problems. The most well-known protocol in this family is that of Bennett and Brassard =-=[BB84]-=-, which essentially relies on Bell’s inequality and the law of large numbers to provide secure keys. 1 There are several improvements upon this protocol. Especially Ekert [Eke91] developed a very nice... |

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(Show Context)
Citation Context ...xample 3.2 is a compactly accessible category precisely if every element is a directed supremum of the compact elements below it, i.e. when the Abelian group is in fact ordered by an algebraic domain =-=[AJ94]-=-. 5.4. Properties of compactly accessible categories. Compactly accessible categories inherit some of the pleasant properties from compact categories, and others only partly. The choice-of-duals-funct... |

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Citation Context ...t, go to step 2. Alice uses cj, j ∈ {1, . . . , 3n}\I as her key bits. Bob uses 1 − c ′ j , j ∈ {1, . . . , 3n}\I as his key bits. Figure 1: A quantum protocol to obtain a 2n-bits shared secret key =-=[Eke91]-=-. Section 2 first introduces the quantum key distribution protocol mentioned above. We recall the necessary details of compact categories in Section 3. Subsequently, Section 4 builds up to the notion ... |

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(Show Context)
Citation Context ...he presented material also indicates some directions for future research. First, a categorical version of the Bell inequalities would lend a definiteness to the categorical approach to quantum theory =-=[AC04]-=-. Secondly, the notion of complete positivity [Sel07] could be extended to compactly accessible categories. The usual formulation of a completely positive morphism in a dagger compact category relies ... |

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Citation Context ...tting in mind, Barr gave a construction to embed a category with certain minimal properties fully into a complete and cocomplete category that is *-autonomous, a notion closely related to compactness =-=[Bar79]-=-. However, as we will see, the important category of Hilbert spaces and bounded maps, that is the traditional model of quantum physics, is neither complete nor cocomplete. Another proposal revolves ar... |

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Citation Context ...ory of the monad on Set given by P(M × −) for an arbitrary commutative monoid M instead of the trivial monoid. It can also be generalized to the category of relations on an arbitrary regular category =-=[CKS84]-=-. In both generalized categories, every object is compact. At first sight, one might expect that the category Sup of complete lattices and suppreserving functions is compact, but it is not [Bar79, pag... |

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Citation Context ...restrictions and hence is essentially for free. Several connections to related research present themselves. First, a compact category has a canonical trace [Abr05]. Although the nuclear ideal setting =-=[ABP99]-=- seems ideal to study this phenomenon, perhaps the trace class morphisms can also be characterized by a colimit property, analogous to the passage from compact categories to compactly accessible ones.... |

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Citation Context ...categories as models of quantum computation is that from very few axioms, surprisingly many consequences ensue that were postulates explicitly in the traditional Hilbert space formalism, e.g. scalars =-=[Abr05]-=-. Moreover, the connection to linear logic provides quantum computation with a resource sensitive type theory of its own [Dun06]. Much of the structure of compact categories is due to a seemingly ingr... |

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Citation Context ...PACTLY ACCESSIBLE CATEGORIES AND QUANTUM KEY DISTRIBUTION 21 6.2. Classical structures and measurements. Finally, we need to model measurements in our semantics. These can be dealt with categorically =-=[CP06]-=- — the following definitions recall the necessary notions, adapted to the compactly accessible setting. Definition 6.7. An object C in a dagger compactly accessible category C is said to be a classica... |

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(Show Context)
Citation Context ...ure a quantum object. Any infinite-dimensional Hilbert space is a quantum object in Hilb, since it cannot carry any classical structure as that would entail finite-dimensionality [Koc03]. We refer to =-=[CD08]-=- for a way to select quantum objects representing qubits. The type of a (demolition) measurement now is X → C, for a classical structure C. As in the traditional Hilbert space formalism, we first defi... |

15 |
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(Show Context)
Citation Context ... colimit property, analogous to the passage from compact categories to compactly accessible ones. Secondly, compactly accessible categories can be seen as a ‘technical implementation’ of shape theory =-=[Blu06]-=-, with the benefit of actually having concrete structure. One could look for the initial or terminal such implementation. Thirdly, the store of qubit-pairs in Section 7 strongly resembles Fock space [... |

13 |
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(Show Context)
Citation Context ...es, model a large fragment of linear logic. It is also known that Barr’s free construction of a *-autonomous category provides a model of full linear logic when one starts with an accessible category =-=[Bar90]-=-. Thus compactly presentable categories qualify as likely candidates to model linear logic, perhaps with unusual properties [Dun06]. Fourthly, locally presentable categories are known to be precisely ... |

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(Show Context)
Citation Context ...tes explicitly in the traditional Hilbert space formalism, e.g. scalars [Abr05]. Moreover, the connection to linear logic provides quantum computation with a resource sensitive type theory of its own =-=[Dun06]-=-. Much of the structure of compact categories is due to a seemingly ingrained ‘finitedimensionality’. This feature is most apparent in the prime example, the category of vector spaces and linear maps.... |

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1 | Conformal field theory as a nuclear functor
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(Show Context)
Citation Context ...n system. This approach provides a proper category in which to model quantum protocols, and hence is automatically compositional — as opposed to ideals that typically do not include all identity maps =-=[BPP07]-=-. Physically, directed colimits provide the intuition of ‘time’. The main result, that justifies our definition of compactly accessible category, is Theorem 5.11. It shows that the choice-of-duals fun... |

1 |
Series A 24(3):309–311
- Hellman
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(Show Context)
Citation Context ... Figure 1 will be used in Section 7 as an example that can be modeled by compactly accessible categories. As such, we need to distinguish between correctness 1 Such a protocol, like Diffie-Hellmann’s =-=[DH76]-=-, regulates key distribution, but gives no guarantee about authenticity of the two parties involved.�� � �� �� 4 CHRIS HEUNEN and security. A quantum key distribution protocol is correct if both part... |

1 |
Shape theory for nuclear ideals. Theory and Application of Categories
- Blute
- 2006
(Show Context)
Citation Context ... colimit property, analogous to the passage from compact categories to compactly accessible ones. Secondly, compactly accessible categories can be seen as a ‘technical implementation’ of shape theory =-=[Blu06]-=-, with the benefit of actually having concrete structure. One could look for the initial or terminal such implementation. Thirdly, the store of qubit-pairs in Section 7 strongly resembles Fock space [... |