dorsal/arxiv
View SchemaA categorical semantics of quantum protocols
| Authors | Samson Abramsky, Bob Coecke |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0402130 |
| URL | https://arxiv.org/abs/quant-ph/0402130 |
| Journal | Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004) |
Abstract
We study quantum information and computation from a novel point of view. Our approach is based on recasting the standard axiomatic presentation of quantum mechanics, due to von Neumann, at a more abstract level, of compact closed categories with biproducts. We show how the essential structures found in key quantum information protocols such as teleportation, logic-gate teleportation, and entanglement-swapping can be captured at this abstract level. Moreover, from the combination of the --apparently purely qualitative-- structures of compact closure and biproducts there emerge `scalars` and a `Born rule'. This abstract and structural point of view opens up new possibilities for describing and reasoning about quantum systems. It also shows the degrees of axiomatic freedom: we can show what requirements are placed on the (semi)ring of scalars C(I,I), where C is the category and I is the tensor unit, in order to perform various protocols such as teleportation. Our formalism captures both the information-flow aspect of the protocols (see quant-ph/0402014), and the branching due to quantum indeterminism. This contrasts with the standard accounts, in which the classical information flows are `outside' the usual quantum-mechanical formalism.
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"abstract": "We study quantum information and computation from a novel point of view. Our\napproach is based on recasting the standard axiomatic presentation of quantum\nmechanics, due to von Neumann, at a more abstract level, of compact closed\ncategories with biproducts. We show how the essential structures found in key\nquantum information protocols such as teleportation, logic-gate teleportation,\nand entanglement-swapping can be captured at this abstract level. Moreover,\nfrom the combination of the --apparently purely qualitative-- structures of\ncompact closure and biproducts there emerge `scalars` and a `Born rule\u0027. This\nabstract and structural point of view opens up new possibilities for describing\nand reasoning about quantum systems. It also shows the degrees of axiomatic\nfreedom: we can show what requirements are placed on the (semi)ring of scalars\nC(I,I), where C is the category and I is the tensor unit, in order to perform\nvarious protocols such as teleportation. Our formalism captures both the\ninformation-flow aspect of the protocols (see quant-ph/0402014), and the\nbranching due to quantum indeterminism. This contrasts with the standard\naccounts, in which the classical information flows are `outside\u0027 the usual\nquantum-mechanical formalism.",
"arxiv_id": "quant-ph/0402130",
"authors": [
"Samson Abramsky",
"Bob Coecke"
],
"categories": [
"quant-ph",
"cs.LO",
"math-ph",
"math.CT",
"math.MP"
],
"journal_ref": "Proceedings of the 19th IEEE conference on Logic in Computer\n Science (LiCS\u002704). IEEE Computer Science Press (2004)",
"title": "A categorical semantics of quantum protocols",
"url": "https://arxiv.org/abs/quant-ph/0402130"
},
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