Open linear maps and Gibbs varieties

This talk explores the idea that a phase transition of a system occurs at a given state if an arbitrarily small local change of this state forces the global state to change abruptly. Surprisingly, this –apparently too general– idea has a practical meaning in the setting of finite-dimensional quantum systems, as the collection of partial traces is not an open map. This lack of openness contrasts with classical mechanics.

We propose to use Gibbs manifolds and Gibbs varieties as tools to characterize the openness and we discuss examples and prior results. Related discontinuity phenomena of the maximum-entropy inference were associated with phase transitions in the work of Chen et al. arXiv:1406.5046 [quant-ph] before.