| Gaussian state | |
|---|---|
| Name | Gaussian state |
| Field | Quantum optics |
| Introduced | 1960s |
| Applications | Quantum information science, Quantum key distribution, Quantum metrology |
Gaussian state
A Gaussian state is a class of quantum state of continuous-variable systems whose Wigner function (or other quasiprobability distributions) is a multivariate Gaussian in phase space. Gaussian states are central in Quantum optics and Continuous-variable quantum information because they are fully characterised by first and second statistical moments, which simplifies analysis of dynamics, measurement and entanglement for systems such as modes of the electromagnetic field or collective motional modes of trapped ions. Their mathematical simplicity and experimental accessibility make them a cornerstone for quantum communication, sensing, and simulation.
A Gaussian state is defined for bosonic modes described by canonical operators satisfying the canonical commutation relations (CCR). Physically relevant systems include optical modes produced by lasers and parametric devices in laboratories such as Max Planck Institute for Quantum Optics or Bell Labs experiments. The defining property is that the state’s characteristic function or Wigner quasiprobability distribution is Gaussian, so all higher-order moments are determined by the covariance matrix and the displacement vector. This property links Gaussian states to classical Gaussian statistics while retaining quantum features like squeezing and entanglement, making them practical resources in quantum communication protocols (e.g., protocols studied at École Normale Supérieure and MIT).
Mathematically, a Gaussian state for n bosonic modes is specified by a displacement vector d_i = ⟨R_i⟩ and a real symmetric covariance matrix σ_{ij} = ½⟨{R_i,R_j}⟩−⟨R_i⟩⟨R_j⟩, where R = (q_1,p_1,...,q_n,p_n)^T are quadrature operators. The Wigner function W(R) = (2π)^{-n} (det σ)^{-1/2} exp[−½(R−d)^T σ^{-1} (R−d)] is a multivariate Gaussian. The physicality of σ requires the Robertson–Schrödinger uncertainty relation, equivalent to σ + iΩ/2 ≥ 0 with symplectic form Ω. Transformations preserving Gaussianity correspond to symplectic linear maps implemented by quadratic Hamiltonians and represented by Williamson's theorem and Bogoliubov transformation techniques. The Husimi Q function and characteristic functions provide alternative Gaussian phase-space descriptions.
Prominent Gaussian examples include: - Coherent states (Glauber states) produced by displacement operators acting on the vacuum; minimal uncertainty states extensively used in laser physics and described in Roy J. Glauber’s work. - Squeezed states where one quadrature variance is reduced below the vacuum level at the expense of the conjugate quadrature; generated via parametric down-conversion and essential for gravitational wave detectors like LIGO. - Thermal (Gibbs) states of harmonic oscillators characterized by a diagonal covariance matrix proportional to mean occupation number; relevant for cavities studied at NIST and in cryogenic experiments at IBM research.
Each of these examples is Gaussian and serves distinct roles: coherent states for communication, squeezed states for metrology, and thermal states for decoherence modelling.
Gaussian states are routinely generated using optical components and nonlinear media. Techniques include coherent displacement via beam splitter and coherent sources, squeezing via optical parametric amplifiers and four-wave mixing in nonlinear crystals (e.g., periodically poled Lithium niobate), and thermal states produced by attenuated blackbody radiation or engineered reservoirs in microwave circuits at Yale University and Caltech. Detection typically uses homodyne and heterodyne detection schemes with photodiodes and local oscillators, enabling full reconstruction of displacement and covariance via quantum tomography protocols implemented in experiments by groups such as those at University of Vienna and IQOQI.
Gaussian states remain Gaussian under dynamics generated by quadratic Hamiltonians and under a broad class of noise models known as Gaussian channels. Gaussian channels include attenuation, amplification, phase-space displacement, and additive classical noise, formally represented by completely positive trace-preserving maps characterized by matrices acting on d and σ. Important channel classes are the quantum-limited amplifier, lossy bosonic channel, and Gaussian thermal channel studied in quantum Shannon theory and by researchers at Telefónica and academic groups investigating channel capacities and additivity. Tools such as symplectic diagonalisation and the Williamson decomposition facilitate analysis of evolution and steady states under Lindblad master equations with quadratic jump operators.
Entanglement in Gaussian states is fully characterised via covariance matrices and criteria like the Peres–Horodecki criterion implemented as the positive partial transpose (PPT) test for Gaussian states (Simon’s criterion). Quantitative measures include the logarithmic negativity, entanglement of formation (computed in some Gaussian cases), and mutual information derived from von Neumann entropy of Gaussian density matrices using symplectic eigenvalues. Multipartite correlations and Gaussian cluster states underpin continuous-variable measurement-based quantum computation proposals developed in groups at University of Tokyo and Université Paris-Saclay.
Gaussian states underpin many practical continuous-variable quantum information tasks: quantum key distribution (CV-QKD) protocols like those analysed by Anthony Leverrier and collaborators, quantum error correction proposals for bosonic codes, and quantum sensing enhancements (squeezed-light metrology in LIGO Collaboration). Gaussian optics forms the basis of linear optical quantum computing architectures and interfaces to non-Gaussian resources required for universal quantum computation, such as photon-number-resolving detectors and cubic phase gates. Industrial and academic applications span optical communications research at Bell Labs and quantum hardware development at companies like Xanadu (company) and Rigetti Computing working on hybrid continuous-variable platforms.
Category:Quantum states Category:Quantum optics Category:Continuous-variable quantum information