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Glauber Sudarshan P FunctionEdit

The Glauber-Sudarshan P function, also known as the P representation, is a foundational tool in the phase-space formulation of quantum optics. It provides a diagonal expansion of the density operator in the basis of coherent states, encoding quantum states as a (quasi) probability distribution P(α) over the complex plane. Named after Roy J. Glauber and E. C. Sudarshan, the P function plays a central role in distinguishing classical-like light from genuinely quantum states of the electromagnetic field. If P(α) behaves like a true probability density, the state is effectively a classical mixture of coherent states; if P becomes highly singular or takes negative values, the state exhibits nonclassical features with no direct classical analogue. In this sense, the P representation sharpens the boundary between classical optics and quantum phenomena.

From a practical standpoint, the P function is most useful when one needs to relate quantum expectations to classical averages. Coherent states coherent state model the closest thing to classical light, such as stabilized laser beams, and the P representation expresses any state as an ensemble of these states with weights given by P. This makes it possible to compute expectations of normally ordered operators by integrating over P(α). The linkage between quantum statistics and classical-like phase-space pictures rests on the resolution of the identity in the coherent-state basis, often written as ∫ d^2α/π |α><α| = I, with ρ = ∫ P(α) |α><α| d^2α defining the diagonal representation of the density operator ρ.

Foundations

  • Diagonal representation: For a quantum state described by a density operator ρ, there exists a function P(α) such that ρ = ∫ P(α) |α><α| d^2α, where |α> denotes a coherent state. The integration measure d^2α is over the complex plane, reflecting the two real degrees of freedom (amplitude and phase) of the field mode. See coherent states.

  • Classical vs quantum mixtures: If P(α) is a positive, well-behaved probability density, the state can be interpreted as a statistical mixture of coherent states, i.e., a classical-like scene in phase space. When P(α) fails to be a true function—being singular, or taking negative values—the state cannot be written as a classical mixture of coherent states, signaling nonclassicality. See nonclassical state.

  • Normally ordered moments: For normally ordered operators, expectation values can be computed as classical averages with respect to P. In practical terms, this connects quantum optical measurements to phase-space statistics through the P representation and its relatives. See normal ordering.

Properties

  • Quasi-probability nature: The P function is a quasi-probability distribution. It shares some properties with classical probability distributions, but its support can be singular or include negative values, reflecting genuine quantum features. See quasi-probability distribution.

  • Singularity and nonclassicality: For many quantum states with nonclassical features, P(α) is more singular than a function (e.g., derivatives of delta functions) and thus not a regular function in the mathematical sense. This contrasts with classical light, where P is a bona fide positive function. See squeezed states and nonclassical states.

  • Relation to classical limits: The P representation highlights the classical limit of quantum optics: when P becomes a positive probability density, the quantum state admits a classical stochastic interpretation. See classical limit.

  • Accessibility and reconstruction: Directly measuring P(α) is generally not feasible because it is a property of the quantum state rather than an observable. Experimental work often reconstructs P(α) (or regularized versions) from measured quadrature data via tomographic methods. See homodyne tomography.

Connections to other representations

  • Wigner and Q representations: The P function is one member of a family of phase-space representations. The Wigner function Wigner function provides a symmetric-ordering view of the state and can be negative even for some Gaussian states, while the Q function Q function corresponds to anti-normal ordering and is always positive but more coarse-grained. The P function is typically the most delicate in terms of singularities but offers the most direct diagonal coherent-state interpretation. See phase-space methods.

  • Regularized and alternative representations: Because of its often-singular nature, researchers use regularized versions of the P representation (sometimes called a regularized P function) and related constructs such as the positive-P representation to enable numerical simulation and more robust interpretation. See positive-P representation.

  • Measurement and reconstruction methods: In practice, one infers phase-space properties from measurements such as homodyne detection, which yields marginal distributions from which one can reconstruct P or its regularized forms. See quantum state tomography.

Applications and limitations

  • State characterization in quantum optics: The P representation remains a standard diagnostic for determining whether a light field state can be viewed as a classical mixture of coherent states or if it exhibits intrinsic quantum features. See quantum optics.

  • Classical-quantum boundary and nonclassicality criteria: The presence of nonpositive or singular P signals nonclassicality, which is closely tied to phenomena such as photon bunching, antibunching, and squeezing. See nonclassical light and photon statistics.

  • Controversies and debates (from a practical, right-of-center vantage): The usefulness of the P function is sometimes debated in terms of interpretability and measurability. Critics point out that P can be ill-behaved or not a true probability density for many interesting quantum states, which can hinder intuitive understanding. Proponents emphasize that P provides the cleanest link between quantum states and classical mixtures of coherent states, offering a sharp criterion for classicality and a straightforward route to compute normally ordered observables. Some critics have targeted the broader project of labeling nonclassicality in absolute terms, arguing that such labels can be overbroad or culturally loaded; defenders counter that scientific criteria should be judged by predictive power, experimental viability, and internal consistency, not by fashionable trends. In any case, the field often uses a suite of representations—P, Wigner, Q, and their regularized variants—to cross-check conclusions and to accommodate both theoretical clarity and experimental practicality. See quantum-state representations and phase-space formulation.

See also