Bosonic and fermionic Gaussian states from Kähler structures

Gaussian quantum states play an important role in almost all areas of quantum theory, ranging from quantum information and condensed matter physics to quantum field theory and quantum gravity. They form a family of states that is at the same time extremely versatile as it captures many physical phenomena (Bose-Einstein condensation, super conductivity,  super fluidity), but also sufficiently simple to allow analytical formulas for important theoretical concepts (entanglement entropy, squeezing, circuit complexity, logarithmic negativity).

In this talk, I will present a unifying mathematical framework to describe bosonic and fermionic Gaussian states as triple of mathematical structures on a classical phase space, namely a positive-definite metric, a symplectic form and (most importantly) a linear complex structure. I will illustrate how this framework incorporates and relates to various existing formalisms (covariance matrices, wave functions, squeezing formulas) and discuss how this framework has been recently used in practice for a wide range of applications (entanglement production, typical entanglement, circuit complexity, variational methods, entanglement extraction). If time permits, I will comment on future applications and possible generalizations.