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Scan for outdated or missing drivers - takes under a minuteDriver Scan →Clear out junk files and repair common Windows errorsFree Scan →In continuous-variable quantum physics, a Gaussian state is a quantum state whose phase-space representation, such as its Wigner function or characteristic function, has a Gaussian shape. Its essential information can often be captured by two things: its average quadrature values and a covariance matrix describing their uncertainty and correlations. This compact description makes Gaussian states useful in quantum optics and quantum information—but it does not make them classical, nor does it cover every capability quantum technologies may need.
What does “Gaussian” mean for a quantum state?
Continuous-variable systems are described using observables that can take continuous values. In quantum optics, these include two field quadratures, often treated as position-like and momentum-like variables. A phase-space representation gives a way to describe and visualize the state using those variables.
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For a Gaussian state, the Wigner function or characteristic function in that representation has a Gaussian form. The state is therefore characterized by its first moments—the average values of the quadratures—and its covariance matrix, which records their variances and correlations. The covariance matrix must obey the quantum uncertainty principle: a Gaussian-shaped function that violates the required uncertainty condition is not a physical quantum state. See Weedbrook et al.’s review of Gaussian quantum information and Adesso, Ragy, and Lee’s tutorial on continuous-variable quantum information.
Picture a bell-shaped region in phase space. Its center indicates the state’s average quadrature values, while its width and orientation convey the uncertainty and correlations. This is a visualization of quadrature statistics—not a literal cloud of particles. Also, the Wigner function is a quasiprobability representation; calling its shape Gaussian does not mean the state is simply an ordinary classical probability distribution.
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How do common Gaussian states differ?
Vacuum, coherent, squeezed, thermal, and two-mode squeezed states are standard examples. They differ in their displacement, quadrature noise, purity or thermal mixing, and whether correlations link multiple modes.
| State | What distinguishes it |
|---|---|
| Vacuum | The zero-excitation reference state for a mode. In the cited continuous-variable quantum key distribution review, its phase-space center is the origin. |
| Coherent | A displaced vacuum state, with its phase-space center shifted from the origin. It is often used to describe a laser-like field; the cited review notes that coherent states can be generated readily with a laser. |
| Squeezed | Its uncertainty is redistributed between conjugate quadratures: one has reduced noise and the other increased noise. |
| Thermal | A mixed state associated with thermal occupation. |
| Two-mode squeezed vacuum (EPR state) | A correlated two-mode state used in continuous-variable quantum information and communication. |
These labels describe different properties rather than interchangeable versions of one state. For example, displacement concerns the center in phase space, squeezing concerns quadrature uncertainty, thermal occupation is associated with mixing, and two-mode squeezing introduces correlations between modes. The standard examples and their continuous-variable context are discussed in the 2024 review of continuous-variable quantum key distribution.
Why are Gaussian states useful?
The first- and second-moment description can make calculations more manageable than tracking a state point by point across phase space. Gaussian transformations preserve Gaussian states, so many calculations can be carried out within a compact mathematical framework. Common quantum-optical states can also be prepared and handled with optical techniques, making the framework relevant to both theory and experiments.
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Researchers use Gaussian states and operations in continuous-variable approaches to quantum communication, cryptography, teleportation, sensing, and information processing. These are research areas and protocol families, not evidence that every proposed application is a mature consumer technology or will deliver a particular performance. What can be achieved depends on the task and on the operations and measurements available. For an overview, see Weedbrook et al. and Adesso, Ragy, and Lee.
What can Gaussian methods not do on their own?
Gaussian states can exhibit quantum properties, including entanglement and steering. But staying entirely within Gaussian states, operations, and measurements imposes restrictions: Gaussian methods are not sufficient for every proposed quantum capability. Some tasks require non-Gaussian resources. In particular, the non-Gaussian tutorial by Walschaers states that Wigner negativity is necessary for Bell-inequality violation and quantum computational advantage. That is a specific theoretical boundary, not a reason to dismiss Gaussian states, which remain valuable within their domain. See Walschaers, “Non-Gaussian Quantum States and Where to Find Them” (2021).
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.The practical takeaway
Gaussian states are quantum states with Gaussian-shaped phase-space representations, summarized in the continuous-variable framework by means and covariances. That structure makes them comparatively tractable and relevant to quantum optics and information research. Their usefulness has limits: some capabilities require non-Gaussian resources, so the right framework depends on the problem being studied.
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