## Discrete Quantum Causal Dynamics (2003)

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Venue: | International Journal of Theoretical Physics |

Citations: | 9 - 4 self |

### BibTeX

@ARTICLE{Blute03discretequantum,

author = {Richard F. Blute and Ivan T. Ivanov and Prakash Panangaden},

title = {Discrete Quantum Causal Dynamics},

journal = {International Journal of Theoretical Physics},

year = {2003},

volume = {2003}

}

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### Abstract

We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolution, driven both by unitary operators and measurements, is coded in a precise mathematical structure in such a way that the crucial properties of causality, covariance and entanglement are faithfully represented. We show how our framework may be expressed using the language of (poly)categories and functors. Remarkably, important physical consequences - such as covariance - follow directly from the functoriality of our axioms. We establish strong links between the physical picture we propose and linear logic. Specifically we show that the rened logical connectives of linear logic can be used to describe the entanglements of subsystems in a precise way. Furthermore, we show that there is a precise correspondence between the evolution of a given system and deductions in a certain formal logical system based on the rules of linear logic. This framework generalizes and enriches both causal posets and the histories approach to quantum mechanics. 1

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Citation Context ... finite.Discrete Quantum Causal Dynamics 2031 outgoing edges. The process at a vertex could be an intervention 5 corresponding to a positive operator-valued measure (POVM) (Nielsen and Chuang, 2000; =-=Peres, 1995-=-) or a unitary transformation. Or instead of an external or unitary action there could be several quantum subsystems that come together and then split apart, possibly in a different way. We will consi... |

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Citation Context ...ection. 3. DYNAMICS ON GRAPHS 3.1. Measurements and Interventions For standard material on density matrices, positive operator-valued measures (POVMs), and completely positive operators, we refer to (=-=Davies, 1976-=-; Nielsen and Chuang, 2000; Preskill, ...). The measurement of a property of a quantum system involves interaction with a classical apparatus. When a classical apparatus measures an observable of a qu... |

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Citation Context ...tive slice. More precisely, let V ={v1,...,vp} be the set of vertices to the past of S. Let I ={e1,...,eq} 6This ordering is well-known in computation theory where it is called the Egli–Milner order (=-=Plotkin, 1976-=-).Discrete Quantum Causal Dynamics 2037 be the set of initial edges to the past of S. Constructing a sequence of slices by incrementally incorporating the vertices of V in a manner similar to what we... |

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Citation Context ...ation for histories described by more general graphs as proposed here. 1.1.2. Causal Sets Causal sets form the basis of an approach to quantum gravity mainly advocated by R. Sorkin and collaborators (=-=Bombelli et al., 1987-=-; Sorkin, 1991), where the basic idea is to take the notion of causality as the primitive. In classical relativity, the structure of the space-time manifold together with a metric of Lorentzian signat... |

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Citation Context ...ally invariant, i.e. determined by only the conformal equivalence class of the metric and hence more primitive than the metric. Various proposals for quantum gravity—for example, the twistor program (=-=Penrose and MacCallum, 1972-=-)—have taken as their point of departure the idea that the causal structure is more fundamental than the metric structure. In the causal sets approach, one takes the point of view that, at the smalles... |

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Citation Context ...approximates a manifold as one “sprinkles” more and more points into the causal set in a uniform fashion. Applications and extensions of these ideas can be found in papers such as (Markopoulou, 2000; =-=Markopoulou and Smolin, 1997-=-; Raptis, 2000), although this list is by no means exhaustive. For us, a finite causal set is the kinematic framework on which we describe evolution and information flow. 1.1.3. Quantum Causal Histori... |

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Citation Context ...f this is that one approximates a manifold as one “sprinkles” more and more points into the causal set in a uniform fashion. Applications and extensions of these ideas can be found in papers such as (=-=Markopoulou, 2000-=-; Markopoulou and Smolin, 1997; Raptis, 2000), although this list is by no means exhaustive. For us, a finite causal set is the kinematic framework on which we describe evolution and information flow.... |

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Citation Context ... “sprinkles” more and more points into the causal set in a uniform fashion. Applications and extensions of these ideas can be found in papers such as (Markopoulou, 2000; Markopoulou and Smolin, 1997; =-=Raptis, 2000-=-), although this list is by no means exhaustive. For us, a finite causal set is the kinematic framework on which we describe evolution and information flow. 1.1.3. Quantum Causal Histories The notion ... |

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Citation Context ...have to be mutually noninterfering. Sets of histories obeying this condition are selected with the use of a special bilinear form on histories— the decoherence functional (Gell-Mann and Hartle, 1993; =-=Griffiths, 1996-=-; Omnés, 1994). A particular history is mathematically represented as a linearly ordered sequence of projection operators in the Hilbert space of the quantum mechanicalDiscrete Quantum Causal Dynamic... |

8 |
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Citation Context ...ly separated entangled quantum systems. This issue is even more pressing for quantum cosmology considerations. An application of the histories approach to quantum field theory on a curved space-time (=-=Blencowe, 1991-=-) must assume the existence of a globally hyperbolic manifold, and thus via the associated foliation, a linear ordering of the histories of the quantum field. Our proposal for describing the evolution... |

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