Glauber–Sudarshan States in Quantum Optics

• Glauber–Sudarshan states are a representation of quantum states using the P-function over coherent states, offering a clear link between classical and quantum descriptions.
• They reveal nonclassical behavior through the negativity or singularity of the P-function, which is key to identifying and engineering nonclassical states.
• Extensions to finite-dimensional systems, group-theoretical formulations, and quantum field applications demonstrate their versatility in quantum process tomography and state reconstruction.

Glauber–Sudarshan states constitute a central concept in quantum optics and the theory of quantum state representations, embodying the decomposition of quantum states as distributions over “most classical” basis states—coherent states. Their mathematical structure, physical significance, connection to phase-space quasi-probabilities, and extensions to finite-dimensional and group-theoretic settings have given them a foundational role in both theoretical and applied quantum physics.

1. Mathematical Foundations and Phase-Space Representation

The defining structure of a Glauber–Sudarshan (GS) state is the P-representation of the density operator $\rho$ as an (in general distributional) integral over coherent states: $\rho = \int d^2a\, P(a) |a\rangle \langle a|$ where $|a\rangle$ are canonical coherent states parameterized by $a\in\mathbb{C}$ and $P(a)$ is the Glauber–Sudarshan P-function (Moya-Cessa, 2013).

Coherent states $|a\rangle$ themselves are eigenstates of the field annihilation operator, saturate the Heisenberg uncertainty relation, and exhibit Poissonian photon number statistics (Gazeau, 2018). The P-function, when a regular non-negative function, can be interpreted as a classical probability distribution; for nonclassical states, $P(a)$ may become negative, highly singular, or even distributional (involving derivatives of the Dirac delta function) (Sperling, 2016).

In phase-space, the GS P-function underlies other quasiprobability representations—such as Wigner and Husimi functions—via convolution with Gaussian or other smoothing kernels, which “regularize” singularities or, in the case of the Kirkwood-Riháczek function, via specific phase-space transformations (Moya-Cessa, 2013). Explicitly,

$$K(\beta, \beta^*) = \int d^2a\, P(a)\, e^{\beta^2 - |a|^2} e^{\sqrt{2}(X a^* - i Y a)}$$

demonstrates the K-R function as a filtered (Gaussian-exponential) image of the P-function.

2. Physical Interpretation, Classicality, and Nonclassicality

The conventional boundary between “classical” and “nonclassical” quantum states is established by the positivity and regularity of the P-function: states with $P(a)\geq 0$ and $P(a)$ a regular function are called classical; nonclassicality is associated with negativity or increased singularity (beyond delta-type) in $P(a)$ (Damanet et al., 2017, Sperling, 2016). However, this interpretation is non-universal.

Recent research demonstrates that even states with a regular, positive P-function (e.g., coherent states) may display nonclassical features when probed with appropriate measurements—every quantum state except the maximally mixed state can show statistical entanglement in the joint outcomes of tailored measurements (Luis, 2016). This situational manifestation of nonclassicality reveals that the traditional GS-positivity criterion captures only a subset of quantum resources.

Regularization of the P-function via filter functions (e.g., Gaussian or nonclassicality filters) leads to smoother distributions, and operationally any quantum state can be approximated to arbitrary accuracy by one with a regular (smooth) P-function, provided the filtering preserves completely positivity and trace (i.e., the filter’s Fourier transform is a probability density) (Zartab et al., 2022).

3. Generalizations: Finite Dimensions, Spin, and Group-Theoretical Frameworks

The GS formalism is generalized to finite-dimensional systems, notably spin-$j$ and multi-qubit systems, by replacing canonical coherent states with spin coherent states, forming an analogous decomposition: $$\rho = \frac{2j+1}{4\pi} \int P_{\rho}(\Omega) |\Omega\rangle\langle\Omega|\, d\Omega$$ where $|\Omega\rangle$ span the sphere $S^2$ (Denis et al., 2023, Giraud et al., 2011).

In these finite-dimensional cases, the set of GS-positive states coincides with the set of symmetric separable (i.e., classically correlated) states, and the geometry of classical states can be characterized explicitly. For spin-1, classicality in the GS sense corresponds to the positivity of a certain matrix built from the state’s moment tensors, with the set of classical states forming a union of ellipsoids in generalized Bloch-vector space (Giraud et al., 2011). For higher spins, the intersection between absolute GS-positivity and Wigner positivity becomes more subtle (Denis et al., 2023).

Discrete variant constructions further enable the definition of GS-like distributions on finite phase-space “lattices” via group-theoretical operator bases, unifying classical, quantum, and quantum-information settings (including two-qubit and ququart systems) (Marchiolli et al., 2019).

Path integral and quantization methods have extended the GS concept to general compact Lie groups, where the role of coherent states is played by group orbits through highest-weight vectors, and the GS quantization is formalized via integrals over these orbits (Yamashita, 2018). This generalization provides explicit geometric quantization and rigorous coherent-state path integrals on group manifolds.

4. Glauber–Sudarshan States in Quantum Field Theory and String Theory

The GS coherent state construction underpins the quantum-classical correspondence in quantum field theory and quantum optics, ensuring that averages of classical observables over ensembles match quantum expectation values after GS mapping (Werbos, 2013). For classical ensembles of definite energy, the mapped quantum density matrices are supported on the kernel of normal-ordered spectral operators, refining the correspondence between the classical and quantum spectra.

In string theory and quantum gravity, the GS construction has been employed to realize nontrivial spacetime backgrounds—notably four-dimensional de Sitter space—as coherent states (GS states) over supersymmetric solitonic vacua (Brahma et al., 2020, Brahma et al., 2020, Brahma et al., 2022). Here, the explicit construction requires quantum corrections, temporal variation of internal fluxes, and the understanding that de Sitter space is not a vacuum but a GS state encoding breaking of supersymmetry and controlling entropy and vacuum energy through expectation values.

Techniques leveraging resurgent summation and nodal diagrammatics show that non-perturbative quantum corrections can be consistently resummed in such states, with positive-definite cosmological constant and stability within appropriate domains (Brahma et al., 2022). The GS formalism has further been used to describe gravitational solitons, branes, and dualities, establishing a new quantum description of solitonic backgrounds as GS states with full resurgence structure (Chakravarty et al., 2023).

5. Operational Consequences and Applications

The GS decomposition underlies operational approaches such as quantum process tomography with coherent states, where the knowledge of process action on coherent state inputs, together with the P-function, allows for characterization of arbitrary quantum processes. Recently, methods have been developed to bypass the need for explicit P-function computation, expressing process tensors directly in terms of derivatives of process outputs on coherent states, thus improving numerical stability and eliminating regularization ambiguities (Rahimi-Keshari et al., 2010). Photon number (energy) cutoffs determine the accuracy-resource balance in such reconstructions.

Preparation of engineered nonclassical states with smooth but non-positive P-functions, via controlled “puncturing” of classical P-functions, opens pathways for experimental realization and fine-tuned quantum state engineering with nonclassical photon statistics or antibunching (Damanet et al., 2017).

In quantum information, the geometry of the set of GS-positive spin states (in terms of polytopes or ellipsoids) allows for precise criteria for absolute classicality and the distinction between different resource-theoretic concepts of nonclassicality and entanglement (Denis et al., 2023, Giraud et al., 2011).

6. Summary Table: Central Mathematical Structures

Representation Type	Mathematical Expression / Key Feature	Physical/Operational Role
P-function (GS)	$\rho = \int P(a) |a\rangle\langle a|$	Encodes classicality, allows quasiprobabilistic state description
Spin GS function	$$\rho = \frac{2j+1}{4\pi} \int P_\rho(\Omega)|\Omega\rangle\langle\Omega| d\Omega$$	Quantum-state decomposition for finite spins
Discrete GS function	$$P(\mu,\nu) = \mathrm{Tr}[\rho\, \hat{T}^{(+1)}(\mu,\nu)]$$	Phase-space representation for finite systems
Filtered GS distribution	$$P_\Omega(\alpha) = \int P(\alpha - \alpha_0) \Omega(\alpha_0)\, d^2\alpha_0$$	Regularization for practical/experimental uses
GS state in QFT	$\rho = \int P(a) |a\rangle\langle a|\, d^2a$	Ensures quantum-classical correspondence for averages
GS state in string theory	$|\Omega\rangle = D(\Omega)|0\rangle$	Coherent state over solitonic vacuum, encodes nontrivial backgrounds

7. Conceptual Significance and Ongoing Developments

Glauber–Sudarshan states anchor the interface of quantum optics, quantum information, and quantum field theory, both as representational tools and as organizing principles for nonclassicality and quantum resource theory. Contemporary research continues to refine their operational boundaries (e.g., the universality of nonclassicality beyond GS-positivity), their mathematical structure (including distribution-theoretic singularities and regularizations), geometric characterizations for finite systems, and their foundational role in emergent spacetime and quantum gravity scenarios.

This suggests ongoing developments will further link GS-state concepts to resource theories of nonclassicality, refined topologies of quantum state sets, and physically realizable schemes for regularizing and reconstructing quantum states and processes across quantum platforms.