@techreport{RR-09-04, abstract = "We describe a general framework in which we can precisely compare the structures of quantum-like theories which may initially be formulated in quite different mathematical terms. We then use this framework to compare two theories: quantum mechanics restricted to qubit stabiliser states and operations, and a toy theory proposed by Spekkens. We discover that viewed within our framework these theories are very similar, but differ in one key aspect - a four element group we term the phase group which emerges naturally within our framework. In the case of the stabiliser theory this group is Z4 while for Spekkens's theory the group is Z2 �Z2. We further show that the structure of this group is intimately involved in a key physical difference between the theories: whether or not they can be modelled by a local hidden variable theory. This is done by establishing a connection between the phase group, and an abstract notion of GHZ state correlations. We go on to formulate precisely how the stabiliser theory and toy theory are 'similar' by defining a notion of 'mutually unbiased qubit theory', noting that all such theories have four element phase groups. Since Z4 and Z2 �Z2 are the only such groups we conclude that the GHZ correlations in this type of theory can only take two forms, exactly those appearing in the stabiliser theory and those appearing in Spekkens's theory. The results point at a classification of local/non-local behaviours by finite Abelian groups, extending beyond qubits to any finitary theory whose observables are all mutually unbiased.", author = "Bob Coecke and Bill Edwards and Rob Spekkens", institution = "OUCL", number = "RR-09-04", pages = "22", title = "The Group Theoretic Origin of Non-Locality For Qubits", year = "2009", } Warning - the bibtex entry below may be invalid: Missing 'journal' field @article{cpv08-ndob, author = "Bob Coecke, Dusko Pavlovic and Jamie Vicary", title = "A new description of orthogonal bases", url = "http://arxiv.org/abs/0810.0812", year = "2008", } @techreport{RR-08-02, author = "Bob Coecke and Eric O Paquette and Dusko Pavlovic", institution = "OUCL", number = "RR-08-02", title = "Classical and Quantum Structures", year = "2008", } @techreport{RR-08-03, author = "Bob Coecke and Dusko Pavlovic", institution = "OUCL", number = "RR-08-03", title = "Scalar Inverses in Quantum Structuralism", year = "2008", } @techreport{RR-07-05, abstract = "Given any †-symmetric monoidal category C we construct a new category Mix(C), which, in the case that C is a †-compact category, is isomorphic to Selinger's CPM(C) [Sel]. Hence, if C is the category FdHilb of finite dimensional Hilbert spaces and linear maps we exactly obtain completely positive maps as morphisms. This means that mixedness of states and operations, within the categorical quantum axiomatics developed in [AC1, AC2, Sel, CPv, CPq], is a concept which exists independently of the quantum and classical structure. Moreover, since our construction does not require †-compactness, it can be applied to categories which have infinite dimensional Hilbert spaces as objects. Finally, in general Mix(C) is not a †-category, so does not admit a notion of positivity. This means that, in the abstract, the notion of 'complete positivity' can exist independently of a notion of 'positivity', which points at a very unfortunate terminology.", author = "Bob Coecke", institution = "Oxford University Computing Laboratory", month = "September", number = "RR-07-05", title = "Complete Positivity without Positivity and Without Compactness", year = "2007", } @techreport{RR-06-02, abstract = "Sums play a prominent role in the formalisms of quantum mechanics, be it for mixing and superposing states, or for composing state spaces. Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product. The corresponding axioms define classical spaces as objects that allow copying and deleting data. Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data. The sums turn out to be an implicit implementation of this capabilities. Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus. In category-theoretic terms, classical data types are dagger-compact Frobenius algebras, and quantum spectra underlying quantum measurements are Eilenberg-Moore coalgebras induced by these Frobenius algebras.", author = "Bob Coecke and Dusko Pavlovic", institution = "Oxford University Computing Laboratory", month = "July", number = "RR-06-02", title = "Quantum measurements without sums", year = "2006", } @techreport{RR-04-02, 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.", author = "Samson Abramsky and Bob Coecke", institution = "Oxford University Computing Laboratory", month = "February", number = "RR-04-02", title = "A Categorical semantics of Quantum Protocols", year = "2004", } @techreport{RR-03-12, abstract = "In SHORT: We expose the information flow capabilities of quantum entanglement. In LONG: This paper contains several components: (a) We prove a general characterization theorem on information flow through bipartite entanglement. This theorem will enable us to provide a unified view on protocols such as quantum teleportation, quantum logic gate teleportation and entanglement swapping. (b) We accomplish the extension of the above to multipartite entanglement which exposes the necessity of logical tools such as typing. Also the need for linear logic connectives and polarities arises naturally. (c) We expose a methodology emerging from our information flow based reasoning about entanglement which yields a two-way compilation scheme enabling design of computational and communicational protocols. This tool allows evident reconstruction of the above mentioned protocols of quantum information processing and also the design of new ones in terms of a classical travelling token-interpretation. We use this methodology to realize a passage from sequential to parallel composition for quantum logic gates. This mechanism also yields a fault-tolerant methodology to prepare multipartite entangled states. (d) At a more advanced level this methodology allows to accommodate classical functional programming features such as Currying, lambda-calculi, Abramsky style geometry of interaction in the sense of and other high level specification logics. (e) Finally, the information flow capabilities of entanglement exposed in this paper yield a canonical family of entanglement measures for multipartite systems. They also provide an interpretation in terms of information flow capabilities for non-local untary operations.", author = "Bob Coecke", institution = "Oxford University Computing Laboratory", month = "October", number = "RR-03-12", title = "The Logic of Entanglement. An invitation. (Version 0.9999)", year = "2003", }