## Quantum information-flow, concretely, abstractly (2004)

Venue: | PROC. QPL 2004 |

Citations: | 11 - 5 self |

### BibTeX

@MISC{Coecke04quantuminformation-flow,,

author = {Bob Coecke},

title = { Quantum information-flow, concretely, abstractly},

year = {2004}

}

### OpenURL

### Abstract

These ‘lecture notes ’ are based on joint work with Samson Abramsky. I will survey and informally discuss the results of [3, 4, 5, 12, 13] in a pedestrian not too technical way. These include: • ‘The logic of entanglement’, that is, the identification and abstract axiomatization of the ‘quantum information-flow ’ which enables protocols such as quantum teleportation. 1 To this means we defined strongly compact closed categories which abstractly capture the behavioral properties of quantum entanglement. • ‘Postulates for an abstract quantum formalism ’ in which classical informationflow (e.g. token exchange) is part of the formalism. As an example, we provided a purely formal description of quantum teleportation and proved correctness in abstract generality. 2 In this formalism types reflect kinds, contra the essentially typeless von Neumann formalism [25]. Hence even concretely this formalism manifestly improves on the usual one. • ‘A high-level approach to quantum informatics’. 3 Indeed, the above discussed work can be conceived as aiming to solve: von Neumann quantum formalism � high-level language low-level language. I also provide a brief discussion on how classical and quantum uncertainty can be mixed in the above formalism (cf. density matrices). 4

### Citations

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Citation Context ...al terms this non-existence means no-cloning [30] and no-deleting [25]. In categorical terms it means that ⊠ is not a categorical product. 6 In logical terms this means that we are doing linear logic =-=[16, 22, 23, 27]-=- as opposed to classical logic. In linear logic we are not allowed to copy and delete assumptions, that is, A ∧ B ⇒ A and A ⇒ A ∧ A are not valid. Compact closure and information-flow. Crucial in the ... |

488 | The mathematics of sentence structure
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Citation Context ...al terms this non-existence means no-cloning [30] and no-deleting [25]. In categorical terms it means that ⊠ is not a categorical product. 6 In logical terms this means that we are doing linear logic =-=[16, 22, 23, 27]-=- as opposed to classical logic. In linear logic we are not allowed to copy and delete assumptions, that is, A ∧ B ⇒ A and A ⇒ A ∧ A are not valid. Compact closure and information-flow. Crucial in the ... |

478 |
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Citation Context ...provide a purely formal description of quantum teleportation and prove correctness in abstract generality. In this formalism types reflect kinds, contra the essentially typeless von Neumann formalism =-=[29]-=-. Hence even concretely this formalism manifestly improves on the usual one. • ‘Towards a high-level approach to quantum informatics’ [2]. Indeed, the above discussed work can be conceived as aiming t... |

460 | Teleporting an unknown quantum state via dual classic and Einstein-Podolsky-Rosen
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Citation Context ...ssion. The corresponding mathematical formalism was considered to have reached its maturity in von Neumann’s book [29]. However! The quantum teleportation protocol. The quantum teleportation protocol =-=[10]-=- involves three qubits a, b and c and two spatial regions A (for “Alice”) and B (for “Bob”). Qubit a is in a state |φ〉 and located in A. Qubits b and c form an ‘EPR-pair’, that is, their joint state i... |

296 |
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Citation Context ...o one which has enough types to reflect this! 2 )What is the true essence of quantum? John von Neumann himself was the first to look for this, teaming up with the ‘king of lattices’ Garrett Birkhoff =-=[11]-=-. It is fair to say that as an attempt to understand ‘the whole of quantum mechanics’ this particular ‘quantum logic’ program has failed. While it provided a much better understanding of quantum super... |

169 | Traced monoidal categories - Joyal, Street, et al. - 1996 |

162 | A categorical semantics of quantum protocols
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(Show Context)
Citation Context ....ox.ac.uk:8080/BobCoecke/Home en.html bob.coecke@comlab.ox.ac.uk Abstract These lecture notes survey joint work with Samson Abramsky. I will somewhat informally discuss the main results of the papers =-=[2, 4, 5, 6, 13, 14]-=- in a pedestrian not too technical way. These include: • ‘The logic of entanglement’ [14], that is, the identification and abstract axiomatization of the ‘quantum information-flow’ which enables proto... |

158 |
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Citation Context ...anics’ this particular ‘quantum logic’ program has failed. While it provided a much better understanding of quantum superposition and the superselection rules (for a survey try to get hold of Piron’s =-=[26]-=- and Varadarajan’s [28] books), it failed at teaching us anything about quantum entanglement, and definitely didn’t teach us anything on how quantum and classical information interact. So lattices don... |

149 |
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Citation Context ...als ∆A : A → A ⊠ A to exist. More precisely, we don’t want them to exist, and this will be guaranteed by a piece of structure we shall introduce. In physical terms this non-existence means no-cloning =-=[30]-=- and no-deleting [25]. In categorical terms it means that ⊠ is not a categorical product. 6 In logical terms this means that we are doing linear logic [16, 22, 23, 27] as opposed to classical logic. I... |

123 |
Exact categories
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Citation Context ...is of the quantum information-flow was H∗ 1 ⊗ H2 ≃ H1 → H2. In categorical terms, making sense of H1 → H2 requires the category to be closed. 7 To give sense to the ∗ we require it to be ∗-autonomous =-=[9]-=-, 8 and finally, requiring H∗ 1 ⊗ H2 ≃ H1 →H2 implies that the category is compact closed [20]. In logical terms this means that we have the multiplicative fragment of linear logic, with negation, and... |

120 |
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Citation Context ...eory. Although this style is usually conceived (even by category theoreticians) as of an abstract∞ nature, in our particular case, it’s bull’s-eye for understanding the quantum information-flow. 9 In =-=[21]-=- a category C is defined to be compact closed iff for each object A three additional pieces of data are specified, an object denoted A∗ , a morphism ηA : I → A∗ ⊠ A called unit and a morphism ǫA : A ⊠... |

107 | Linear logic, *-autonomous categories, and cofree coalgebras
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Citation Context ...al terms this non-existence means no-cloning [30] and no-deleting [25]. In categorical terms it means that ⊠ is not a categorical product. 6 In logical terms this means that we are doing linear logic =-=[16, 22, 23, 27]-=- as opposed to classical logic. In linear logic we are not allowed to copy and delete assumptions, that is, A ∧ B ⇒ A and A ⇒ A ∧ A are not valid. Compact closure and information-flow. Crucial in the ... |

99 |
The Geometry of Quantum Theory
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Citation Context ...‘quantum logic’ program has failed. While it provided a much better understanding of quantum superposition and the superselection rules (for a survey try to get hold of Piron’s [26] and Varadarajan’s =-=[28]-=- books), it failed at teaching us anything about quantum entanglement, and definitely didn’t teach us anything on how quantum and classical information interact. So lattices don’t seem to be capable o... |

71 | Theory of Categories - Mitchell - 1965 |

67 | Retracing some paths in process algebra, in
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Citation Context ... but we do want to mention that much of the inspiration for [13, 14] emerged from [4] where we studied the physical realization of ‘abstract traces’ [19], which generalize traditional feedback traces =-=[1, 8]-=-. It turns out that both on (Rel, ×) and (FdVecK, ⊗), due to compact closure, the trace also admits a feedback-loop type interpretation, but a linear ‘only-use-once’ one. Please consult [6] for more d... |

44 |
Quantum teleportation is a universal computational primitive
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Citation Context ...the line changes its temporal direction every time it passes a projector box (see [13]). Using the same line of thought it is also easy to reconstruct other protocols such as logic-gate teleportation =-=[17]-=- and entanglement swapping [31], and, the quantum information-flow interpretation also extends to multipartite projectors. We refer the reader to [13, 14] for details on this. Then we asked: “Are thes... |

43 | Quantum quandaries: a category-theoretic perspective - Baez - 2004 |

42 |
Event-ready-detectors” Bell experiment via entanglement swapping
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Citation Context ...irection every time it passes a projector box (see [13]). Using the same line of thought it is also easy to reconstruct other protocols such as logic-gate teleportation [17] and entanglement swapping =-=[31]-=-, and, the quantum information-flow interpretation also extends to multipartite projectors. We refer the reader to [13, 14] for details on this. Then we asked: “Are these information-flow features spe... |

29 |
Deductive systems and categories i
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Citation Context |

29 |
Impossibility of deleting an unknown quantum state”, arXiv: quant-ph/9911090, Nature
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Citation Context ... exist. More precisely, we don’t want them to exist, and this will be guaranteed by a piece of structure we shall introduce. In physical terms this non-existence means no-cloning [30] and no-deleting =-=[25]-=-. In categorical terms it means that ⊠ is not a categorical product. 6 In logical terms this means that we are doing linear logic [16, 22, 23, 27] as opposed to classical logic. In linear logic we are... |

28 | Nuclear and trace ideal in tensored *- categories
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Citation Context ... ) ∗ provides transposition while ( )∗ provides complex conjugation. Let us end this paragraph by saying that most things discussed above extend to infinite dimensional settings when using ideas from =-=[3]-=-. 10A note on categorical traces. This paragraph slightly diverges from our story line, but we do want to mention that much of the inspiration for [13, 14] emerged from [4] where we studied the physi... |

24 |
An abstract approach to coherence
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Citation Context ...e of H1 → H2 requires the category to be closed. 7 To give sense to the ∗ we require it to be ∗-autonomous [9], 8 and finally, requiring H∗ 1 ⊗ H2 ≃ H1 →H2 implies that the category is compact closed =-=[20]-=-. In logical terms this means that we have the multiplicative fragment of linear logic, with negation, and where conjunction is self-dual, that is, it coincides with disjunction — indeed, you read thi... |

16 | Physical traces: Quantum vs. classical information processing
- Abramsky, Coecke
- 2003
(Show Context)
Citation Context ....ox.ac.uk:8080/BobCoecke/Home en.html bob.coecke@comlab.ox.ac.uk Abstract These lecture notes survey joint work with Samson Abramsky. I will somewhat informally discuss the main results of the papers =-=[2, 4, 5, 6, 13, 14]-=- in a pedestrian not too technical way. These include: • ‘The logic of entanglement’ [14], that is, the identification and abstract axiomatization of the ‘quantum information-flow’ which enables proto... |

15 | Abstract physical traces. Theory and Applications of Categories 14(6):111–124 - Abramsky, Coecke - 2005 |

15 |
Feedback and generalized logic
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- 1976
(Show Context)
Citation Context ... but we do want to mention that much of the inspiration for [13, 14] emerged from [4] where we studied the physical realization of ‘abstract traces’ [19], which generalize traditional feedback traces =-=[1, 8]-=-. It turns out that both on (Rel, ×) and (FdVecK, ⊗), due to compact closure, the trace also admits a feedback-loop type interpretation, but a linear ‘only-use-once’ one. Please consult [6] for more d... |

12 | Pure operations and measurements - Hellwig, Kraus - 1969 |

12 | Operational quantum logic: An overview - Coecke, Moore, et al. - 2000 |

10 |
Quantum logic in intuitionistic perspective
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Citation Context ...and ǫV :: ei ⊗ ēj ↦→ δij . i=1 4Compare this to ‘superposition’ in lattice theoretic terms: an atomic lattice has superposition states if the join of two atoms has additional atoms below it (e.g. cf. =-=[12]-=-). 5A categorical isomorphism is a morphism f : A → B with an inverse f −1 : B → A, that is, f ◦ f −1 = 1A and f −1 ◦ f = 1B. A natural isomorphism is a strong notion of categorical isomorphism. For v... |

4 |
The Logic of entanglement. An invitation
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(Show Context)
Citation Context ....ox.ac.uk:8080/BobCoecke/Home en.html bob.coecke@comlab.ox.ac.uk Abstract These lecture notes survey joint work with Samson Abramsky. I will somewhat informally discuss the main results of the papers =-=[2, 4, 5, 6, 13, 14]-=- in a pedestrian not too technical way. These include: • ‘The logic of entanglement’ [14], that is, the identification and abstract axiomatization of the ‘quantum information-flow’ which enables proto... |

4 |
Abstract physical traces, Theory and
- Abramsky, Coecke
(Show Context)
Citation Context ...1 ⊠ A2 → (⊞iI) ⊠ A2. For epistemic density matrices this means that the classical data and the quantum data are truly distinct entities. 12 A discussion of normalization of projectors can be found in =-=[6]-=-. 13Using Lemma 7.6 of [5] one verifies that ω∗ : ⊞iI → A ∗ is given by ω∗ = [si • (π ∗ i ◦ uI)]i where si := πi ◦ φ and uI : I ≃ I ∗ (a natural transformation which exists by compact closure). Hence... |

3 |
2004a) The logic of entanglement. arXiv:quant-ph/0402014
- Coecke
(Show Context)
Citation Context ...inite dimensional settings when using ideas from [3]. 10A note on categorical traces. This paragraph slightly diverges from our story line, but we do want to mention that much of the inspiration for =-=[13, 14]-=- emerged from [4] where we studied the physical realization of ‘abstract traces’ [19], which generalize traditional feedback traces [1, 8]. It turns out that both on (Rel, ×) and (FdVecK, ⊗), due to c... |

2 |
High-Level Methods for Quantum Computation and Quantum Information. Pleanary talk at 19th Annual
- Abramsky
(Show Context)
Citation Context |

1 |
Operations and measurents. Volume II
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(Show Context)
Citation Context ...ts an external part of the system (ontic). The other concepts that can be derived from basic quantum mechanics by ‘acting on part of a bigger system’ (non-isolated dynamics, generalized measurements, =-=[18]-=- etc.) can also be defined abstractly, e.g. generalized measurements as 〈fi〉 i=n i=1 : A → ⊞iA with ∑i=n i=1 f † i ◦ fi = 1A , while abstract analogous of theorems such as Naimark’s can be proven. Of ... |