## A new description of orthogonal bases

Venue: | Math. Structures in Comp. Sci |

Citations: | 11 - 6 self |

### BibTeX

@ARTICLE{Coecke_anew,

author = {Bob Coecke and Dusko Pavlovic and Jamie Vicary},

title = {A new description of orthogonal bases},

journal = {Math. Structures in Comp. Sci},

year = {}

}

### OpenURL

### Abstract

We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative †-Frobenius monoid in the category FdHilb, which has finite-dimensional 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

### Citations

209 |
C ∗ -algebras and operator theory
- Murphy
- 1990
(Show Context)
Citation Context ...s in [14]. Corollary 4.5. The copyable elements for any commutative †-Frobenius monoid on H in FdHilb form a basis for H. Proof. By the spectral theorem for finite-dimensional commutative C*-algebras =-=[11]-=-, the involution-preserving homomorphisms from a finite-dimensional commutative C*-algebra to the complex numbers form a basis for the dual of the underlying vector space, in our case H ≃ FdHilb(C,H) ... |

145 | The geometry of tensor calculus - JOYAL, STREET - 1991 |

137 | A single quantum cannot be cloned - Wootters, Zurek - 1982 |

77 |
Frobenius algebras and 2-d topological quantum field theories
- Kock
(Show Context)
Citation Context ...bers, and morphisms given by isomorphisms of sets that preserve the functions into the real numbers. This is interesting from the perspective of unitary 2-dimensional topological quantum field theory =-=[9]-=-, since such things are given by commutative †-Frobenius monoids in FdHilb, and the natural notion of homomorphism is one that preserves all of the Frobenius structure. If we only require that our hom... |

75 |
Cartesian bicategories
- Carboni, Walters
- 1987
(Show Context)
Citation Context ... Within this context one then aims to maximise the expressiveness additional structure 2�� � � �� �� ratio. Additional structure on which we rely in this paper is that of internal Frobenius algebras =-=[3, 10]-=-, more specifically, internal commutative †- Frobenius monoids [6]. Relative to the quantum universe which is modelled by the symmetric monoidal †-category these commutative †-Frobenius monoids model ... |

51 | Communication by EPR Devices - Dieks - 1982 |

22 | R.: Interacting quantum observables
- Coecke, Duncan
- 2008
(Show Context)
Citation Context ...anics, for example, for describing the flow of classical information in quantum informatic protocols [6], for defining complementarity and special quantum logic gates in quantum computational schemes =-=[4]-=- and for constructing discrete models for quantum reasoning [5]. In this paper, we establish that every commutative special †-Frobenius monoids arises from an orthonormal basis, and that dropping the ... |

12 |
Coecke (2004): A categorical semantics of quantum protocols
- Abramsky, B
(Show Context)
Citation Context ...m mechanics emerged from the observation that the subtle details of important, experimentally-established quantum informatic protocols can already be specified at an abstract category-theoretic level =-=[1]-=-. The background structure is that of a symmetric monoidal †-category [1, 13], a symmetric monoidal category together with a identity-on-objects involutive endofunctor which coherently preserves the s... |

7 |
Toy quantum categories
- Coecke, Edwards
- 2008
(Show Context)
Citation Context ...ation in quantum informatic protocols [6], for defining complementarity and special quantum logic gates in quantum computational schemes [4] and for constructing discrete models for quantum reasoning =-=[5]-=-. In this paper, we establish that every commutative special †-Frobenius monoids arises from an orthonormal basis, and that dropping the specialty condition gives an orthogonal basis. The plan of the ... |

6 |
2007) Dagger compact categories and completely positive maps
- Selinger
(Show Context)
Citation Context ...tant, experimentally-established quantum informatic protocols can already be specified at an abstract category-theoretic level [1]. The background structure is that of a symmetric monoidal †-category =-=[1, 13]-=-, a symmetric monoidal category together with a identity-on-objects involutive endofunctor which coherently preserves the symmetric monoidal structure. Within this context one then aims to maximise th... |

5 |
2007) Quantum measurements without sums. In: Mathematics of Quantum Computing and
- Coecke, Pavlovic
(Show Context)
Citation Context ... any orthonormal basis {|φi〉}i in a finite dimensional Hilbert space H we can always define the linear maps δ : H → H ⊗ H :: |φi〉 ↦→ |φi〉 ⊗ |φi〉 (1) and ǫ : H → C :: |φi〉 ↦→ 1. (2) It was observed in =-=[6]-=- that the triple (H,δ,ǫ) is a so-called commutative special †-Frobenius comonoid in the category FdHilb of finite dimensional Hilbert spaces and linear maps with the tensor product as monoidal structu... |

5 | Braunstein (2000) Impossibility of deleting an unknown quantum state. Nature 404 - Pati, L |

2 |
Categorical formulation of C*-algebras
- Vicary
(Show Context)
Citation Context ...s monoid in FdHilb is a C*-algebra, in particular, it can be given a C*-algebra norm. Remark 4.4. Note that in the above we did not assume the †-Frobenius monoid to be commutative. More on this is in =-=[14]-=-. Corollary 4.5. The copyable elements for any commutative †-Frobenius monoid on H in FdHilb form a basis for H. Proof. By the spectral theorem for finite-dimensional commutative C*-algebras [11], the... |

1 |
2000) A note on strongly separable algebras. In: Boletín de la Academia Nacional de Ciencias
- Aguiar
(Show Context)
Citation Context ...he fact that a special commutative Frobenius algebra is necessarily strongly separable, and since the ground field is of characteristic 0, it is therefore necessarily finitedimensional and semisimple =-=[2]-=-. Such an algebra is canonically isomorphic to a finite cartesian product of the complex numbers, up to permutation, and the basis elements are given by the number 1 in each of the complex factors. In... |