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71
Interacting quantum observables
 of Lecture Notes in Computer Science
, 2008
"... Abstract. We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum account on the quantum d ..."
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Cited by 23 (13 self)
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Abstract. We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum account on the quantum data encoded in complex phases, and prove a normal form theorem for it. Together these enable us to describe all observables of finite dimensional Hilbert space quantum mechanics. The resulting equations suffice to perform computations with elementary quantum gates, translate between distinct quantum computational models, establish the equivalence of entangled quantum states, and simulate quantum algorithms such as the quantum Fourier transform. All these computations moreover happen within an intuitive diagrammatic calculus. 1
Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Cited by 19 (4 self)
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.
The local GromovWitten theory of curves
, 2008
"... We study the equivariant GromovWitten theory of a rank 2 vector bundle N over a nonsingular curve X of genus g: (i) We define a TQFT using the GromovWitten partition functions. The full theory is determined in the TQFT formalism from a few exact calculations. We use a reconstruction result proven ..."
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Cited by 12 (3 self)
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We study the equivariant GromovWitten theory of a rank 2 vector bundle N over a nonsingular curve X of genus g: (i) We define a TQFT using the GromovWitten partition functions. The full theory is determined in the TQFT formalism from a few exact calculations. We use a reconstruction result proven jointly with C. Faber and A. Okounkov in the appendix.
A new description of orthogonal bases
 Math. Structures in Comp. Sci
"... We show that an orthogonal basis for a finitedimensional Hilbert space can be equivalently characterised as a commutative †Frobenius monoid in the category FdHilb, which has finitedimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal st ..."
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Cited by 12 (6 self)
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We show that an orthogonal basis for a finitedimensional Hilbert space can be equivalently characterised as a commutative †Frobenius monoid in the category FdHilb, which has finitedimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative †Frobenius monoid is special. Hence orthogonal and orthonormal bases can be axiomatised in terms of composition of operations and tensor product only, without any explicit reference to the underlying vector spaces. This axiomatisation moreover admits an operational interpretation, as the comultiplication copies the basis vectors and the counit uniformly deletes them. That is, we rely on the distinct ability to clone and delete classical data as compared to quantum data to capture basis vectors. For this reason our result has important implications for categorical quantum mechanics. 1
Statesum construction of twodimensional openclosed TQFTs
 In preparation
"... We present a state sum construction of twodimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional twodimensional cobordisms to those of openclosed cobordisms, i.e. smoo ..."
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Cited by 11 (5 self)
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We present a state sum construction of twodimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional twodimensional cobordisms to those of openclosed cobordisms, i.e. smooth compact oriented 2manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a twodimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an openclosed TQFT with a finite set of Dbranes using the example of the groupoid algebra of a finite groupoid.
When the theories meet: Khovanov homology as Hochschild homology of links, arXiv:math.GT/0509334
"... ABSTRACT. We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2,n)torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we ..."
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Cited by 10 (3 self)
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ABSTRACT. We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2,n)torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of KhovanovRozansky, sl(n), homology and their deformations we conjecture the connection. The best framework to explore our ideas is to use a comultiplicationfree version of Khovanov homology for graphs developed by L. HelmeGuizon and Y. Rong and extended here to Mreduced case, and in the case of a polygon to noncommutative algebras. In this framework we prove that for any unital algebra A the Hochschild homology of A is isomorphic to graph homology over A of a polygon. We expect that this
Quantum and classical structures in nondeterministic computation
 Proceedings of Quanum Interaction 2009, Lecture
"... Abstract. In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum theories, classical structures over a space correspon ..."
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Cited by 9 (2 self)
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Abstract. In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum theories, classical structures over a space correspond to its orthonormal bases. In the present paper, we show that classical structures in the category of relations correspond to direct sums of abelian groups. Although relations are, of course, not an interesting model of quantum computation, this result has some interesting computational interpretations. If relations are viewed as denotations of nondeterministic programs, it uncovers a wide variety of nonstandard quantum structures in this familiar area of classical computation. Ironically, it also opens up a version of what in philosophy of quantum mechanics would be called an onticepistemic gap, as it provides no interface to these nonstandard quantum structures. 1