In the mathematical physics of quantum mechanics, Liouville space, also known as line space, is the space of operators on Hilbert space. Liouville space is itself a Hilbert space under the Hilbert-Schmidt inner product.[1][2]

Abstractly, Liouville space is equivalent (isometrically isomorphic) to the tensor product of a Hilbert space with its dual.[1][3] A common computational technique to organize computations in Liouville space is vectorization.[2]

Liouville space underlies the density operator formalism and is a common computation technique in the study of open quantum systems.[2][3]


  1. ^ a b "Hilbert space". Uni Hamburg. Archived from the original on 29 Oct 2014. Retrieved 24 April 2022.
  2. ^ a b c Gyamfi, Jerryman A. (16 October 2020). "Fundamentals of quantum mechanics in Liouville space". European Journal of Physics. 41 (6): 063002. arXiv:2003.11472. Bibcode:2020EJPh...41f3002G. doi:10.1088/1361-6404/ab9fdd.
  3. ^ a b Janos Polonyi; Rachid, Ines (2021). "Elementary open quantum states". Symmetry. 13 (9): 1624. arXiv:2106.01443v2. Bibcode:2021Symm...13.1624P. doi:10.3390/sym13091624.