This paper explores the relation between the concept of symmetry and its formalisms. The standard view among philosophers and physicists is that symmetry is completely formalized by mathematical groups. For some mathematicians however, the groupoid is a competing and more general formalism. An analysis of symmetry which justifies this extension has not been adequately spelled out. After a brief explication of how groups, equivalence, * Previous versions of this paper were presented at the University of Pittsburgh and at theÉcole Normale Supérieure (Paris). Thanks to the members of those audiences for their comments and questions and especially to John Earman and Gordon Belot. Guay's work was supported by a postdoctoral grant form the Fonds de Recherche sur la Société et la Culture du Québec (Canada). 1 and symmetries classes are related, we show that, while it's true in some instances that groups are too restrictive, there are other instances for which the standard extension to groupoids is too unrestrictive. The connection between groups and equivalence classes, when generalized to groupoids, suggests a middle ground between the two.
In support of a recent conjecture by Nielsen (1999), we prove that the phenomena of 'incomparable entanglement'-whereby, neither member of a pair of pure entangled states can be transformed into the other via local operations and classical communication (LOCC)-is a generic feature when the states at issue live in an infinite-dimensional Hilbert space.
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