Glaubersudarshan P RepresentationEdit
The Glauber–Sudarshan P representation is a phase-space formulation used in quantum optics to describe states of the electromagnetic field. It expresses any quantum state through a diagonal expansion in the basis of coherent states, associating with each state a function P(α) over the complex plane. In practical terms, one writes the density operator ρ as an integral over coherent states |α⟩ with weight P(α): ρ = ∫ P(α) |α⟩⟨α| d^2α. Coherent states themselves are eigenstates of the annihilation operator, a|α⟩ = α|α⟩, and they form an overcomplete basis for the quantum harmonic oscillator that underpins most optical field descriptions. The completeness relation ∫ |α⟩⟨α| d^2α / π = I underlines how every quantum state can be represented in terms of coherent-state components.
The P representation sits at the intersection of quantum theory and a classical intuition for light. If P(α) behaves like a genuine probability distribution—nonnegative and not more singular than a function—the corresponding quantum state can be interpreted as a classical statistical mixture of coherent states. In this sense, such states are often labeled “classical-like.” Conversely, when P is highly singular (for example, involving derivatives of delta functions) or when it takes negative values, the state exhibits nonclassical features that lack a straightforward classical analogue. This dichotomy provides a convenient diagnostic tool for assessing the classicality of optical states and has become a standard reference point in discussions of quantum vs. classical light.
Overview and mathematical formulation
Coherent states and their role: Coherent states coherent state are central to the P representation. They minimize quantum uncertainties and resemble classical oscillations of the electromagnetic field. The P representation uses |α⟩ as the building blocks for expressing ρ, and the integral over α spans the entire phase space of complex amplitudes.
Normal ordering and expectation values: Observables built from normally ordered operators can be calculated as averages with respect to P. In particular, the expectation value of a normally ordered function f(a, a†) can be written as ⟨f(a, a†)⟩ = ∫ P(α) f(α, α*) d^2α. This links the quantum description to a classical-looking ensemble over coherent amplitudes.
Classicality criterion: If P is a nonnegative, well-behaved function, the state admits a classical interpretation as a statistical mixture of coherent states. If P is negative or more singular than a function, the state is deemed nonclassical, with quantum features that cannot be reproduced by any classical probability distribution over coherent states.
Singularities and practical limits: For many important nonclassical states—such as certain Fock states, squeezed states, or other highly nonclassical light—the P distribution is not a regular function; it can be a distribution in the mathematical sense (involving derivatives of delta functions). This singular character reflects genuine quantum properties and poses challenges for direct interpretation or numerical treatment.
Relation to other phase-space representations
Wigner function: The Wigner function Wigner function is another phase-space representation that provides a quasi-probability distribution. Unlike the P distribution, the Wigner function can take negative values, highlighting nonclassicality in a different way. It is often used for visualizing quantum states in a more symmetrical ordering framework.
Husimi Q function: The Husimi Q function Husimi Q function is a smoothed, always nonnegative representation associated with anti-normal ordering. While easier to visualize and measure in some contexts, it can obscure fine-grained quantum features that the P distribution makes explicit.
Positive P representation: To address the practical difficulties posed by singular P functions, the positive P representation positive P representation extends the formalism by introducing additional complex variables. This provides a framework in which all quantum states have a positive, well-behaved representation, enabling stochastic methods to simulate quantum dynamics beyond what the standard P representation can handle.
Applications, interpretation, and limits
Diagnostic and design tool: The P representation serves as a bridge between quantum theory and experimental practice. By testing whether a state’s P distribution is a genuine probability density, researchers can assess whether the light field behaves in a manner that can be explained by classical statistics or whether inherently quantum features are at play. This has implications for designing optical sources, detectors, and communication protocols, where classical simulability can influence expectations about performance and security.
Quantum optics and state reconstruction: In experiments, researchers often reconstruct quasi-probability distributions to characterize states of light. While direct measurement of P(α) is rarely straightforward due to singularities, the P framework guides the interpretation of data and complements other representations such as the Wigner function Wigner function and the Q function Husimi Q function.
Nonclassical light and quantum technologies: Nonclassical states—signaled by nonpositive or singular P distributions—underpin many quantum technologies, including quantum-enhanced metrology, secure communications, and certain quantum information protocols. The P representation makes explicit where quantum advantages arise: when a state cannot be written as a classical mixture of coherent states, its behavior cannot be fully captured by classical stochastic models alone.
Controversies and debates
Classicality versus quantum advantage: A fundamental debate in this area centers on how best to delineate the boundary between classical and quantum descriptions of light. Proponents of the P representation as a classicality criterion argue that a positive, well-behaved P corresponds to states that can be simulated by classical random processes. Critics point out that the boundary is subtle: even in some states with benign-looking P distributions, quantum features may manifest in other observables or in interference phenomena, challenging any overly simplistic classical interpretation.
Practicality of the P representation: The fact that the P distribution can be highly singular for many useful nonclassical states means that the representation is not always the most convenient for computation or tomography. This has motivated the development of alternative phase-space tools (Wigner, Q) and, more recently, the positive P representation, which trades some interpretational simplicity for computational robustness.
Interpretational nuances: While the P representation cleanly connects to a diagonal expansion in coherent states, some researchers caution against overreliance on a probabilistic intuition for quantum states. Quantum theory retains intrinsic nonlocality and contextuality in ways that may resist a purely classical probabilistic reading, even when a P-like decomposition exists for certain states.
See also