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]}

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^{a}^{b}"Hilbert space". Uni Hamburg. Archived from the original on 29 Oct 2014. Retrieved 24 April 2022. - ^
^{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. - ^
^{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.