Physics:Gaussian state
In quantum mechanics and quantum information theory, a Gaussian state (also called a quasi-free state in quantum field theory) is a specific type of mixed quantum state of either multiple bosons or multiple fermions.
Bosonic Gaussian states
Gaussian states are named as such because, at least in the bosonic case, they behave like a quantum analog of the classical multivariate Gaussian distribution. A bosonic Gaussian state is uniquely determined by quantum analogs of the mean and covariance, which are equivalent to its one-point and two-point correlation functions. The Wigner quasiprobability distribution of a bosonic Gaussian state is always a classical multivariate Gaussian. For a single mode, a bosonic Gaussian state is the same as a squeezed coherent state, and general Gaussian states can be seen as a generalization of squeezed coherent states to multiple modes. The thermal state at any temperature of a Hamiltonian which is quadratic in the bosonic creation and annihilation operators is a bosonic Gaussian state, and this characterizes the class of bosonic Gaussian states.[1][2]
Fermionic Gaussian states
Gaussian states can also be defined in the fermionic setting. Similarly to the bosonic setting, the thermal state at any temperature of a Hamiltonian which is quadratic in the fermionic creation and annihilation operators is a fermionic Gaussian state, and any fermionic Gaussian state can be written in this way. A fermionic Gaussian state is in general determined by just its two-point correlation function, or covariance matrix.[1][3]
Geometric structure
In both the bosonic and fermionic cases, the set of Gaussian states which are pure has a rich geometric structure. Both can be equipped with natural Riemannian metrics which are in fact Kähler. They turn out to be the Hermitian symmetric spaces DIIIn in the bosonic case and CIn in the fermionic space, using Cartan's classification. While this space is noncompact in the bosonic case, it is compact in the fermionic case and thus there is additional structure: it is also an algebraic variety, as can be shown using results from complex geometry such as the Kodaira embedding theorem and Serre's GAGA theorem.[1]
References
- ↑ 1.0 1.1 1.2 Hackl, Lucas; Bianchi, Eugenio (2021-09-22). "Bosonic and fermionic Gaussian states from Kähler structures". SciPost Physics Core 4 (3). doi:10.21468/SciPostPhysCore.4.3.025. ISSN 2666-9366.
- ↑ Holevo, Alexander S. (2011-05-05). Probabilistic and Statistical Aspects of Quantum Theory. Pisa: Springer Science & Business Media. ISBN 978-88-7642-378-9.
- ↑ Surace, Jacopo; Tagliacozzo, Luca (2022-05-16). "Fermionic Gaussian states: an introduction to numerical approaches". SciPost Physics Lecture Notes. doi:10.21468/SciPostPhysLectNotes.54. ISSN 2590-1990.
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