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Husimi Q function

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description Husimi Q function Overview

The Husimi Q function is a quasiprobability distribution used in quantum mechanics to represent quantum states in phase space. It is constructed by smoothing the Wigner function, which produces a non-negative distribution that resembles a fuzzy density. Because it is always non-negative, it is often used in quantum optics to visualize states and calculate expectation values.

help Husimi Q function FAQ

Why is the Husimi Q function never negative?

It is defined from the expectation value of the density operator in a coherent state, conventionally written as Q(α) = ⟨α|ρ|α⟩/π. Because a valid density operator is positive semidefinite, that expectation value cannot be negative.

How is the Husimi Q function related to the Wigner function?

The Q function can be obtained by smoothing the Wigner function with a Gaussian corresponding to coherent-state uncertainty. This removes negative fine structure but also discards some phase-space detail.

Can the Husimi Q function reveal nonclassical states?

Yes, but not through negative values, since Q is non-negative by construction. Nonclassicality can instead appear through shapes, zeros or statistical properties, while Wigner-function negativity provides a more direct signature for many states.

How is the Husimi Q function measured in quantum optics?

It is closely connected to heterodyne detection, which jointly samples two field quadratures with added vacuum noise. Repeated heterodyne outcomes can therefore reconstruct the state's Q distribution in optical phase space.

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