Strongly symmetric spectral convex sets are Jordan algebra state spaces

Howard Barnum (No current affiliation), Joachim Hilgert (University of Paderborn)

We show that the finite-dimensional convex compact sets (“state spaces”) having the properties of spectrality and strong symmetry are precisely the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and the simplices. This improves upon a result of Barnum, Mueller, and Ududec (New J Phys 2014; arxiv:1403.4147), who characterized the same set of state spaces using the same two properties plus the assumption of no higher-order interference. Various natural ways to characterize complex quantum state spaces among the Jordan state spaces are known; these combine with our theorem to give simple characterizations of finite-dimensional quantum state space. 

A finite-dimensional convex compact set is called spectral if every element (“state”) is a convex combination of perfectly distinguishable pure (i.e. extremal) states. It is called strongly symmetric if every list of perfectly distinguishable pure states can be taken to any other list of the same length by a reversible transformation (affine map on the affine span of the convex set, taking the set onto itself). 

While our result is purely convex-geometric in nature, it has strong implications for work relating geometric properties of the state spaces of systems to physical and information processing characteristics of general probabilistic theories (GPTs). 

For instance, important aspects of quantum and classical thermodynamics, and also of query complexity, have been generalized to GPTs satisfying natural postulates. We will survey examples in which the assumed postulates either include or imply spectrality and strong symmetry, so that their results apply to a narrower class of theories than might have been hoped, already relatively close to complex quantum theory since their systems are Jordan algebraic.