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International Journal of Theoretical Physics

A Categorical Approach to T1 Separation and the Product of State Property Systems

Ann Van der Voorde

Published Mar 1, 2000• 11 citations
Publisher
DOI
10.1023/a:1003651602331

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An original presentation of Categorical Quantum Physics, in the line of Abramsky and Coecke, tries to introduce only objects and assumptions that are clearly relevant to Physics and does not assume compact closure. Adjoint arrows, tensor products and biproducts are the ingredients of this presentation. Tensor products are defined, up to a unitary arrow, by a universal property related to transformations of composite systems, not by assuming a monoidal structure. Entangled states of a tensor product define mixed states on the components of the tensor product. Coproducts that fit the adjoint structure are shown to be defined up to a unitary arrow and to provide biproducts. An abstract no-cloning result is proved.
Recently, Braunstein et al. [1] introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying graphs. We provide further results on the multipartite separability of Laplacian matrices of graphs. In particular, we identify complete bipartite graphs whose normalized Laplacian matrix is multipartite entangled under any vertex labeling. Furthermore, we give conditions on the vertex degrees such that there is a vertex labeling under which the normalized Laplacian matrix is entangled. These results address an open question raised in [1]. Finally, we extend some of the results in [1,2] to the multipartite case and show that the Laplacian matrix of any product of graphs (strong, Cartesian, tensor, categorical, etc.) is multipartite separable.
Abstract The tensor product of a monoidal category allows us to consider multiplications on its objects, leading to an abstract notion of monoid. This chapter investigates monoids and their relation to dual objects, as well as comonoids, which axiomatize copying and deleting operations. These play a major role in the categorical description of classical information, since classical information can be copied and deleted, while quantum information cannot. We prove categorical no-deleting and no-cloning theorems, showing that if these structures are able to copy and delete every state of the system, then the category collapses. We also characterize when a tensor product is a categorical product.