## Physics from Computer Science -- a position statement (2006)

### BibTeX

@MISC{Abramsky06physicsfrom,

author = {Samson Abramsky and Bob Coecke},

title = {Physics from Computer Science -- a position statement },

year = {2006}

}

### OpenURL

### Abstract

In this statement we provide some examples of transdisciplinary journeys, from one field to another, and back. In particular, the quantum informatic endeavor is not just a matter of feeding physical theory into the general field of natural computation, but also one of using high-level methods developed in Computer Science to improve on the quantum physical formalism itself, and the understanding thereof. We highlight a seemingly contradictory phenomenon: passing to an abstract, categorical quantum informatic formalism leads directly to a simple and elegant graphical formulation of quantum theory itself, which for example makes the design of some important quantum informatic protocols completely transparent. It turns out that essentially all of the quantum informatic machinery can be recovered from this graphical calculus. But in turn, this graphical formalism provides a bridge between methods of logic and computer science, and some of the most exciting developments in the mathematics of the past two decades: namely those arising from the Jones polynomial invariant of knots and links, the Temperley-Lieb Algebra and related structures.

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Citation Context |

66 | Dagger compact closed categories and completely positive maps
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Citation Context ...osition vs. adjoint [6, 7], global phase and elimination thereof, vectorial vs. projective formalism [17]. Peter Selinger recovered mixed state, complete positivity and Jamiolkowski map-state duality =-=[43]-=-. Recently, in collaboration with Dusko Pavlovic and Eric Paquette, 5decoherence, generalized measurements and Naimark’s theorem have been recovered [20, 19], and Ross Duncan has exposed some foundat... |

65 | Retracing some paths in process algebra
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Citation Context ... the ordering relations on the two rows of dots. Moreover, the multiplication of the algebra can be described as a form of CutElimination, using the methods developed in the “Geometry of Interaction” =-=[26, 1, 3]-=-. We give a brief indication of the ideas. A diagram joining up a row of n dots with a row of m dots is formalized as a fixed-point free involution on [n] op ✁ [m], where [n] is the linear order 1 < 2... |

57 |
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50 |
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Citation Context ...decoherence, generalized measurements and Naimark’s theorem have been recovered [20, 19], and Ross Duncan has exposed some foundational structures for Measurement-Based quantum computation [23] — cf. =-=[39, 40, 21]-=-. This high-level of abstraction also comes with an intuitive and simple graphical calculus/notation. This “strongly compact closed graphical calculus” can be seen as a very substantial 2-dimensional ... |

40 |
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Citation Context ...e compositionality result expressed in picture (3), in the concrete setting of Hilbert spaces. In [32] Louis Kauffman relies on very similar topological ideas to derive the teleportation protocol. In =-=[11]-=- John Baez discusses structures close to strong compact closure and compares these to models of topological quantum field theories. 5 CATEGORICAL ALGEBRA A purely algebraic category-theoretic version ... |

40 |
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Citation Context ...cally at this high level of abstraction and generality. Moreover, we were able to derive the correctness of protocols such as quantum teleportation, entanglement swapping and logic-gate teleportation =-=[12, 27, 48]-=- in a transparent and very conceptual fashion. Also, while at this level of abstraction there is no underlying field of complex numbers, there is still an intrinsic notion of ‘scalar’, and we could st... |

36 |
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Citation Context ...cally at this high level of abstraction and generality. Moreover, we were able to derive the correctness of protocols such as quantum teleportation, entanglement swapping and logic-gate teleportation =-=[12, 27, 48]-=- in a transparent and very conceptual fashion. Also, while at this level of abstraction there is no underlying field of complex numbers, there is still an intrinsic notion of ‘scalar’, and we could st... |

28 | Quantum entanglement and topological entanglement
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Citation Context ... have braiding, which carries 3-dimensional geometric information. Does this information have some computational significance? Some ideas in this direction have been explored by Kauffman and Lomonaco =-=[33]-=-, but no clear understanding has yet been achieved. • Beyond this, we have the general setting of TQFT (Topological Quantum Field Theories) [46, 10] and related notions. This may be relevant to Quantu... |

25 |
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Citation Context ...ar logic [42], a resource sensitive logic developed in the late eighties. A key distinction between classical and quantum computation is indeed the inability to copy and delete unknown quantum states =-=[37, 47]-=-, and the ability to take such an inability into account was exactly the conceptual core of linear logic. At a more refined level, we also relied on the study of a particular categorical structure cal... |

24 | Abstract scalars, loops, and free traced and strongly compact closed categories
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Citation Context ...sed graphical calculus” can be seen as a very substantial 2-dimensional extension of Dirac’s bra-ket notation [22], and relying on category-theoretic results on free constructions of these categories =-=[3, 28, 34, 43]-=- one can show that an equational statement is derivable in the graphical calculus if and only if it is derivable from categorical algebra. An informal introduction to this calculus is provided in [15,... |

23 | Quantum measurements without sums
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(Show Context)
Citation Context ...positivity and Jamiolkowski map-state duality [43]. Recently, in collaboration with Dusko Pavlovic and Eric Paquette, 5decoherence, generalized measurements and Naimark’s theorem have been recovered =-=[20, 19]-=-, and Ross Duncan has exposed some foundational structures for Measurement-Based quantum computation [23] — cf. [39, 40, 21]. This high-level of abstraction also comes with an intuitive and simple gra... |

21 | A categorical quantum logic
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(Show Context)
Citation Context ...ly and categorytheoretically, uses the full power of (3) since in this case g will be the teleported gate. A logical approach to proving such facts, using normalization of proof-nets, is developed in =-=[8]-=-. An alternative axiomatization, which uses coalgebraic structure instead of biproducts, was recently developed in [20]. 6 FROM CUT-ELIMINATION TO KNOT THEORY VIA QUANTUM MECHANICS We shall also menti... |

21 |
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Citation Context ...s direction have been explored by Kauffman and Lomonaco [33], but no clear understanding has yet been achieved. • Beyond this, we have the general setting of TQFT (Topological Quantum Field Theories) =-=[46, 10]-=- and related notions. This may be relevant to Quantum Informatic concerns in (at least) two ways: 1. A novel and promising paradigm of Topological Quantum Computing has recently been proposed [24]. 2.... |

20 | De-linearizing linearity: projective quantum axiomatics from strong compact closure
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- 2006
(Show Context)
Citation Context ...ex numbers, there is still an intrinsic notion of ‘scalar’, and we could still make sense of transposition vs. adjoint [6, 7], global phase and elimination thereof, vectorial vs. projective formalism =-=[17]-=-. Peter Selinger recovered mixed state, complete positivity and Jamiolkowski map-state duality [43]. Recently, in collaboration with Dusko Pavlovic and Eric Paquette, 5decoherence, generalized measur... |

15 | Abstract physical traces. Theory and Applications of Categories - Abramsky, Coecke - 2005 |

13 | Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead
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Citation Context ...omatics. Other advantages can be found in [6, 2, 5]. 4 QUANTUM NON-LOGIC VS. QUANTUM HYPER-LOGIC The term quantum logic is usually understood in connection with the 1936 Birkhoff-von Neumann proposal =-=[13, 41]-=- to consider the (closed) linear subspaces of a Hilbert space ordered by inclusion as the formal expression of the logical distinction between quantum and classical physics. While in classical logic w... |

12 | Introducing categories to the practicing physicist
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- 2006
(Show Context)
Citation Context ... 43] one can show that an equational statement is derivable in the graphical calculus if and only if it is derivable from categorical algebra. An informal introduction to this calculus is provided in =-=[15, 16, 18]-=-. In the graphical calculus we depict physical processes by boxes, and we label the inputs and outputs of these boxes by types which tell on which kind of system these boxes act cf. one qubit, several... |

12 |
The Principles of Quantum Mechanics (third edition
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Citation Context ...comes with an intuitive and simple graphical calculus/notation. This “strongly compact closed graphical calculus” can be seen as a very substantial 2-dimensional extension of Dirac’s bra-ket notation =-=[22]-=-, and relying on category-theoretic results on free constructions of these categories [3, 28, 34, 43] one can show that an equational statement is derivable in the graphical calculus if and only if it... |

11 | Types for quantum computing
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Citation Context ...aquette, 5decoherence, generalized measurements and Naimark’s theorem have been recovered [20, 19], and Ross Duncan has exposed some foundational structures for Measurement-Based quantum computation =-=[23]-=- — cf. [39, 40, 21]. This high-level of abstraction also comes with an intuitive and simple graphical calculus/notation. This “strongly compact closed graphical calculus” can be seen as a very substan... |

10 | Quantum information-flow, concretely, and axiomatically
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(Show Context)
Citation Context ... 43] one can show that an equational statement is derivable in the graphical calculus if and only if it is derivable from categorical algebra. An informal introduction to this calculus is provided in =-=[15, 16, 18]-=-. In the graphical calculus we depict physical processes by boxes, and we label the inputs and outputs of these boxes by types which tell on which kind of system these boxes act cf. one qubit, several... |

9 | Kindergarten quantum mechanics — lecture notes
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(Show Context)
Citation Context ... 43] one can show that an equational statement is derivable in the graphical calculus if and only if it is derivable from categorical algebra. An informal introduction to this calculus is provided in =-=[15, 16, 18]-=-. In the graphical calculus we depict physical processes by boxes, and we label the inputs and outputs of these boxes by types which tell on which kind of system these boxes act cf. one qubit, several... |

8 |
High-level methods in quantum computation and information. EPSRC research grant EP/C500032/1
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(Show Context)
Citation Context ...ank the black line: = — we called this line the quantum information flow [16]. The intuitive graphical calculus is an important benefit of the categorical axiomatics. Other advantages can be found in =-=[6, 2, 5]-=-. 4 QUANTUM NON-LOGIC VS. QUANTUM HYPER-LOGIC The term quantum logic is usually understood in connection with the 1936 Birkhoff-von Neumann proposal [13, 41] to consider the (closed) linear subspaces ... |

8 |
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(Show Context)
Citation Context ...decoherence, generalized measurements and Naimark’s theorem have been recovered [20, 19], and Ross Duncan has exposed some foundational structures for Measurement-Based quantum computation [23] — cf. =-=[39, 40, 21]-=-. This high-level of abstraction also comes with an intuitive and simple graphical calculus/notation. This “strongly compact closed graphical calculus” can be seen as a very substantial 2-dimensional ... |

8 | Teleportation topology
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- 2005
(Show Context)
Citation Context ...n [14] Braunstein, D’Adriano, Milburn and Sacchi extend Dirac notation to obtain results similar to the compositionality result expressed in picture (3), in the concrete setting of Hilbert spaces. In =-=[32]-=- Louis Kauffman relies on very similar topological ideas to derive the teleportation protocol. In [11] John Baez discusses structures close to strong compact closure and compares these to models of to... |

4 |
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(Show Context)
Citation Context ...he interesting phenomenon of ‘apparant reversal of the causal order’ which is the source of many quite mystical interpretations of quantum teleportation in terms of ‘traveling backward in time’ — cf. =-=[35]-=-. Indeed, while on the left, physically, we first prepare the state labeled g and then apply the costate labeled f, the global effect is as if we first applied f itself first, and only then g. Derivat... |

3 |
Generalized measurements and Naimark’s theorem without sums. Draft paper
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(Show Context)
Citation Context ...positivity and Jamiolkowski map-state duality [43]. Recently, in collaboration with Dusko Pavlovic and Eric Paquette, 5 decoherence, generalized measurements and Naimark’s theorem have been recovered =-=[20, 19]-=-, and Ross Duncan has exposed some foundational structures for Measurement-Based quantum computation [23] — cf. [39, 40, 21]. This high-level of abstraction also comes with an intuitive and simple gra... |

2 |
Topological quantum computation. arxiv:quant-ph/0101025
- Freedman, Kitaev, et al.
(Show Context)
Citation Context ...[46, 10] and related notions. This may be relevant to Quantum Informatic concerns in (at least) two ways: 1. A novel and promising paradigm of Topological Quantum Computing has recently been proposed =-=[24]-=-. 2. As the issues arising from distributed quantum computing, quantum security protocols etc. are investigated, the interactions between quantum informatics and spatio-temporal structure will inevita... |

1 |
Temperley-Lieb algebras and geometry of interaction
- Abramsky
- 2006
(Show Context)
Citation Context ... g ⊙ f). There seems some potential here for a non-trivial interaction between geometrical and computational/logical ideas, at a foundational level. Further details will appear in a forthcoming paper =-=[4]-=-. 7 CONCLUDING REMARKS We see an exciting agenda for future research at the interface between Computer Science and Physics. This seems to be the right context for addressing many issues which are fund... |

1 |
Universal teleportation with a twist
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(Show Context)
Citation Context ...cal communication is now implicit in the fact that the index i is both present in the costate (= measurement-branch) and the correction, and hence needs to be send from Alice to Bob. Related work. In =-=[14]-=- Braunstein, D’Adriano, Milburn and Sacchi extend Dirac notation to obtain results similar to the compositionality result expressed in picture (3), in the concrete setting of Hilbert spaces. In [32] L... |