Polarization Nonclassicality in Quantum Light

• Polarization nonclassicality is defined by the failure of quantum light states to be described by classical polarization models, evidenced by negativities or singularities in Stokes-space and SU(2) quasiprobability distributions.
• Experimental reconstructions using photon-number–resolving detectors reveal thin negative lobes and quantifiable negativity volumes, clearly distinguishing weak coherent states from classical benchmarks.
• This phenomenon links to advanced quantum features such as entanglement conversion, higher-order and nonlinear Stokes criteria, and enhanced dynamical speed, providing actionable insights for quantum communication and metrology.

Searching arXiv for recent and foundational papers on polarization nonclassicality to ground the article. Polarization nonclassicality denotes the failure of a quantum state of light to admit a classical probabilistic description in terms of polarization variables, most commonly the Stokes observables or their SU(2)-coherent-state analogues. In the literature, it is characterized through several inequivalent but intersecting frameworks: negativity or singularity of a polarization quasiprobability distribution in Stokes space, negativities of a two-mode Glauber–Sudarshan $P$-representation or of SU(2)-coherent-state quasiprobabilities, complex joint quasiprobabilities for noncommuting polarization observables, higher-order and nonlinear Stokes criteria, distance- and majorization-based orderings of polarization fluctuations, and operational conversions of polarization quantumness into entanglement or dynamical speedup (Spasibko et al., 2015). A recurring conclusion is that polarization nonclassicality is not exhausted by first-moment Stokes analysis and, in several settings, persists even for states usually regarded as classical-like.

1. Definitions in Stokes space and SU(2) phase space

A central construction is the polarization quasiprobability distribution (PQD) $W(\vec S)$ over the three Stokes variables $S_1,S_2,S_3$, defined by the three-dimensional Fourier transform of the polarization characteristic function,

$$\chi(\vec\lambda)=\mathrm{Tr}\!\left[\hat\rho\,e^{i\vec\lambda\cdot\hat{\vec S}}\right], \qquad W(\vec S)=\frac{1}{(2\pi)^3}\int d^3\lambda\,e^{-i\vec\lambda\cdot \vec S}\chi(\vec\lambda).$$

In component form,

$$W(S_1,S_2,S_3)=\frac{1}{(2\pi)^3}\int d\lambda_1\,d\lambda_2\,d\lambda_3\, e^{-i(\lambda_1S_1+\lambda_2S_2+\lambda_3S_3)} \mathrm{Tr}\!\left[\hat\rho\,e^{i(\lambda_1\hat S_1+\lambda_2\hat S_2+\lambda_3\hat S_3)}\right].$$

An equivalent parametrization uses spherical angles $(\alpha,\beta)$ on the Poincaré sphere and the rotated Stokes component

$$S_{\alpha\beta}=S_1\cos\alpha+S_2\sin\alpha\cos\beta+S_3\sin\alpha\sin\beta$$

to rewrite the transform in terms of $\chi(\lambda,\alpha,\beta)$ (Spasibko et al., 2015).

Within a different but related language, polarization is represented by the quantum Stokes operators

$$\hat S_0=\hat a_H^\dagger\hat a_H+\hat a_V^\dagger\hat a_V,\quad \hat S_1=\hat a_H^\dagger\hat a_H-\hat a_V^\dagger\hat a_V,$$

$$\hat S_2=\hat a_H^\dagger\hat a_V+\hat a_V^\dagger\hat a_H,\quad \hat S_3=i(\hat a_V^\dagger\hat a_H-\hat a_H^\dagger\hat a_V),$$

which satisfy the SU(2) commutation relations

$W(\vec S)$0

From the Glauber–Sudarshan perspective, a two-mode polarization state is nonclassical if the joint $W(\vec S)$1 fails to be a bona fide probability; in the SU(2) formulation, any state whose $W(\vec S)$2-function in the space of SU(2) coherent states exhibits negativities is polarization-nonclassical (Ares et al., 2024).

These definitions establish two important reference classes. In Stokes-space tomography, the classical benchmark is a nonnegative Kolmogorov probability density over $W(\vec S)$3. In the SU(2)-coherent-state formulation, the classical benchmark is the convex set generated by angular-momentum coherent states $W(\vec S)$4. A plausible implication is that “polarization nonclassicality” is not a single scalar property but a family of failures of different classical reference models.

2. Negativity, singularity, and hidden-variable obstruction

The operational meaning of the PQD is direct: it reproduces the correct one-dimensional marginal probability distributions for Stokes measurements and therefore represents the natural choice for the probability distribution in classical hidden-variable models (Spasibko et al., 2015). If $W(\vec S)$5 were a set of preassigned classical variables, one would require $W(\vec S)$6 everywhere. Hence any observed region with $W(\vec S)$7 rules out such a model and serves as a direct operational signature of polarization nonclassicality (Spasibko et al., 2015).

A distinctive feature of polarization, absent from the standard quadrature Wigner function, is that the PQD demonstrates negativity for all quantum states. The stated origin is the discrete nature of the Stokes variables (Spasibko et al., 2015). Because each $W(\vec S)$8 has an integer spectrum, the one-dimensional marginals must be sums of Dirac $W(\vec S)$9-peaks at integer values, whereas rotationally symmetric two- and three-dimensional reconstructions can remain continuous. Reconciling these properties forces negative regions or derivative singularities in the full distribution (Chekhova et al., 2013).

The single-photon example makes this explicit. For the linearly polarized single-photon state $S_1,S_2,S_3$0, the $S_1,S_2,S_3$1-marginal is

$S_1,S_2,S_3$2

with

$S_1,S_2,S_3$3

for $S_1,S_2,S_3$4. In particular, $S_1,S_2,S_3$5 for $S_1,S_2,S_3$6 (Chekhova et al., 2013).

The same section of the literature also emphasizes a subtlety. In low-photon-number polarization tomography, even coherent states display PQD negativity because of the discreteness of the Stokes spectra (Chekhova et al., 2013). This does not weaken the hidden-variable conclusion stated above; rather, it identifies the specific classicality notion being excluded, namely any classical probability theory over Stokes space. This suggests that polarization nonclassicality, in the PQD sense, is stricter than classifications based only on quadrature squeezing or sub-Poissonian counting statistics.

3. Experimental reconstruction and direct observation

Direct observation of PQD negativity required photon-number resolving detection. Earlier experiments employed photon-number averaging detectors and therefore did not reveal the intrinsic negative regions (Spasibko et al., 2015). The experimental reconstruction reported for a weak horizontally polarized coherent state used a polarization tomography stage with a wave-plate sequence selecting a basis defined by $S_1,S_2,S_3$7 on the Poincaré sphere, a polarizing beam splitter, and photon-number-resolving transition-edge sensors at the two outputs. For each setting, the TES pair yielded $S_1,S_2,S_3$8, from which the conditional probability $S_1,S_2,S_3$9 was formed. The characteristic function was then estimated as

$$\chi(\vec\lambda)=\mathrm{Tr}\!\left[\hat\rho\,e^{i\vec\lambda\cdot\hat{\vec S}}\right], \qquad W(\vec S)=\frac{1}{(2\pi)^3}\int d^3\lambda\,e^{-i\vec\lambda\cdot \vec S}\chi(\vec\lambda).$$0

and inverted numerically on a finite grid to obtain $$\chi(\vec\lambda)=\mathrm{Tr}\!\left[\hat\rho\,e^{i\vec\lambda\cdot\hat{\vec S}}\right], \qquad W(\vec S)=\frac{1}{(2\pi)^3}\int d^3\lambda\,e^{-i\vec\lambda\cdot \vec S}\chi(\vec\lambda).$$1 (Spasibko et al., 2015).

The reconstructed PQD for the weak coherent state exhibited a broad positive halo centered near the classical Stokes vector $$\chi(\vec\lambda)=\mathrm{Tr}\!\left[\hat\rho\,e^{i\vec\lambda\cdot\hat{\vec S}}\right], \qquad W(\vec S)=\frac{1}{(2\pi)^3}\int d^3\lambda\,e^{-i\vec\lambda\cdot \vec S}\chi(\vec\lambda).$$2, surrounded by thin negative lobes along the $$\chi(\vec\lambda)=\mathrm{Tr}\!\left[\hat\rho\,e^{i\vec\lambda\cdot\hat{\vec S}}\right], \qquad W(\vec S)=\frac{1}{(2\pi)^3}\int d^3\lambda\,e^{-i\vec\lambda\cdot \vec S}\chi(\vec\lambda).$$3 shell of radius $$\chi(\vec\lambda)=\mathrm{Tr}\!\left[\hat\rho\,e^{i\vec\lambda\cdot\hat{\vec S}}\right], \qquad W(\vec S)=\frac{1}{(2\pi)^3}\int d^3\lambda\,e^{-i\vec\lambda\cdot \vec S}\chi(\vec\lambda).$$4. A cross-section $$\chi(\vec\lambda)=\mathrm{Tr}\!\left[\hat\rho\,e^{i\vec\lambda\cdot\hat{\vec S}}\right], \qquad W(\vec S)=\frac{1}{(2\pi)^3}\int d^3\lambda\,e^{-i\vec\lambda\cdot \vec S}\chi(\vec\lambda).$$5 showed clear regions with $$\chi(\vec\lambda)=\mathrm{Tr}\!\left[\hat\rho\,e^{i\vec\lambda\cdot\hat{\vec S}}\right], \qquad W(\vec S)=\frac{1}{(2\pi)^3}\int d^3\lambda\,e^{-i\vec\lambda\cdot \vec S}\chi(\vec\lambda).$$6 for $$\chi(\vec\lambda)=\mathrm{Tr}\!\left[\hat\rho\,e^{i\vec\lambda\cdot\hat{\vec S}}\right], \qquad W(\vec S)=\frac{1}{(2\pi)^3}\int d^3\lambda\,e^{-i\vec\lambda\cdot \vec S}\chi(\vec\lambda).$$7, and the integrated negative volume

$$\chi(\vec\lambda)=\mathrm{Tr}\!\left[\hat\rho\,e^{i\vec\lambda\cdot\hat{\vec S}}\right], \qquad W(\vec S)=\frac{1}{(2\pi)^3}\int d^3\lambda\,e^{-i\vec\lambda\cdot \vec S}\chi(\vec\lambda).$$8

was found to be on the order of $$\chi(\vec\lambda)=\mathrm{Tr}\!\left[\hat\rho\,e^{i\vec\lambda\cdot\hat{\vec S}}\right], \qquad W(\vec S)=\frac{1}{(2\pi)^3}\int d^3\lambda\,e^{-i\vec\lambda\cdot \vec S}\chi(\vec\lambda).$$9, significantly above the statistical uncertainty (Spasibko et al., 2015).

A complementary experimental development replaces linear Stokes analysis by click-counting nonlinear Stokes observables. In that approach, one defines click operators

$$W(S_1,S_2,S_3)=\frac{1}{(2\pi)^3}\int d\lambda_1\,d\lambda_2\,d\lambda_3\, e^{-i(\lambda_1S_1+\lambda_2S_2+\lambda_3S_3)} \mathrm{Tr}\!\left[\hat\rho\,e^{i(\lambda_1\hat S_1+\lambda_2\hat S_2+\lambda_3\hat S_3)}\right].$$0

and constructs nonlinear Stokes operators

$$W(S_1,S_2,S_3)=\frac{1}{(2\pi)^3}\int d\lambda_1\,d\lambda_2\,d\lambda_3\, e^{-i(\lambda_1S_1+\lambda_2S_2+\lambda_3S_3)} \mathrm{Tr}\!\left[\hat\rho\,e^{i(\lambda_1\hat S_1+\lambda_2\hat S_2+\lambda_3\hat S_3)}\right].$$1

$$W(S_1,S_2,S_3)=\frac{1}{(2\pi)^3}\int d\lambda_1\,d\lambda_2\,d\lambda_3\, e^{-i(\lambda_1S_1+\lambda_2S_2+\lambda_3S_3)} \mathrm{Tr}\!\left[\hat\rho\,e^{i(\lambda_1\hat S_1+\lambda_2\hat S_2+\lambda_3\hat S_3)}\right].$$2

Using a spectrally decorrelated type-II phase-matched waveguide inside a Sagnac interferometer and eight-time-bin quasi-photon-number-resolving detection, nonlinear polarization squeezing was certified up to eighth order. The normally ordered variance $$W(S_1,S_2,S_3)=\frac{1}{(2\pi)^3}\int d\lambda_1\,d\lambda_2\,d\lambda_3\, e^{-i(\lambda_1S_1+\lambda_2S_2+\lambda_3S_3)} \mathrm{Tr}\!\left[\hat\rho\,e^{i(\lambda_1\hat S_1+\lambda_2\hat S_2+\lambda_3\hat S_3)}\right].$$3 became significantly negative, up to $$W(S_1,S_2,S_3)=\frac{1}{(2\pi)^3}\int d\lambda_1\,d\lambda_2\,d\lambda_3\, e^{-i(\lambda_1S_1+\lambda_2S_2+\lambda_3S_3)} \mathrm{Tr}\!\left[\hat\rho\,e^{i(\lambda_1\hat S_1+\lambda_2\hat S_2+\lambda_3\hat S_3)}\right].$$4 in normalized units, with statistical significance $$W(S_1,S_2,S_3)=\frac{1}{(2\pi)^3}\int d\lambda_1\,d\lambda_2\,d\lambda_3\, e^{-i(\lambda_1S_1+\lambda_2S_2+\lambda_3S_3)} \mathrm{Tr}\!\left[\hat\rho\,e^{i(\lambda_1\hat S_1+\lambda_2\hat S_2+\lambda_3\hat S_3)}\right].$$5, and the minimal eigenvalue of the higher-order moment matrix remained strictly below zero up to the eighth moment (Prasannan et al., 2022).

4. Relations to entanglement, coherence conversion, and dynamical resources

A major recent result is the strict equivalence between nonclassical polarization and the entanglement of indistinguishable photons. The key observation is that the classical reference set is the same in both descriptions: fully separable symmetric product states $$W(S_1,S_2,S_3)=\frac{1}{(2\pi)^3}\int d\lambda_1\,d\lambda_2\,d\lambda_3\, e^{-i(\lambda_1S_1+\lambda_2S_2+\lambda_3S_3)} \mathrm{Tr}\!\left[\hat\rho\,e^{i(\lambda_1\hat S_1+\lambda_2\hat S_2+\lambda_3\hat S_3)}\right].$$6 for particle entanglement and SU(2) coherent states for polarization. Accordingly, the quasiprobability-for-quantum-coherence decomposition

$$W(S_1,S_2,S_3)=\frac{1}{(2\pi)^3}\int d\lambda_1\,d\lambda_2\,d\lambda_3\, e^{-i(\lambda_1S_1+\lambda_2S_2+\lambda_3S_3)} \mathrm{Tr}\!\left[\hat\rho\,e^{i(\lambda_1\hat S_1+\lambda_2\hat S_2+\lambda_3\hat S_3)}\right].$$7

has identical coefficients in the two formalisms, so any negative $$W(S_1,S_2,S_3)=\frac{1}{(2\pi)^3}\int d\lambda_1\,d\lambda_2\,d\lambda_3\, e^{-i(\lambda_1S_1+\lambda_2S_2+\lambda_3S_3)} \mathrm{Tr}\!\left[\hat\rho\,e^{i(\lambda_1\hat S_1+\lambda_2\hat S_2+\lambda_3\hat S_3)}\right].$$8 simultaneously witnesses polarization nonclassicality and entanglement (Ares et al., 2024).

This equivalence is constructive. One solves the separability eigenvalue equation

$$W(S_1,S_2,S_3)=\frac{1}{(2\pi)^3}\int d\lambda_1\,d\lambda_2\,d\lambda_3\, e^{-i(\lambda_1S_1+\lambda_2S_2+\lambda_3S_3)} \mathrm{Tr}\!\left[\hat\rho\,e^{i(\lambda_1\hat S_1+\lambda_2\hat S_2+\lambda_3\hat S_3)}\right].$$9

uses the corresponding eigenvalues $(\alpha,\beta)$0 and Gram matrix $(\alpha,\beta)$1, and obtains the optimal quasiprobability weights from

$(\alpha,\beta)$2

The total amount of nonclassicality may then be quantified by

$(\alpha,\beta)$3

while a basis-independent witness can be written as

$(\alpha,\beta)$4

with $(\alpha,\beta)$5 if and only if the state is nonclassical or entangled. For the two-photon Dicke state $(\alpha,\beta)$6, the maximal classical overlap is $(\alpha,\beta)$7, and the experiment reported $(\alpha,\beta)$8, in agreement with $(\alpha,\beta)$9 (Ares et al., 2024).

A different conversion paradigm appears when nonclassicality of polarization superpositions is activated by beam-splitter mixing with vacuum. Two orthogonal polarizations are modeled as a bipartite infinite-dimensional system of coherent-product states; after a conditional quadrature operation $$S_{\alpha\beta}=S_1\cos\alpha+S_2\sin\alpha\cos\beta+S_3\sin\alpha\sin\beta$$0 and subsequent beam-splitter recombination, the resulting state acquires entanglement negativity

$$S_{\alpha\beta}=S_1\cos\alpha+S_2\sin\alpha\cos\beta+S_3\sin\alpha\sin\beta$$1

maximal at $$S_{\alpha\beta}=S_1\cos\alpha+S_2\sin\alpha\cos\beta+S_3\sin\alpha\sin\beta$$2 for $$S_{\alpha\beta}=S_1\cos\alpha+S_2\sin\alpha\cos\beta+S_3\sin\alpha\sin\beta$$3, while the classical polarization coherence matrix yields a Schmidt number

$$S_{\alpha\beta}=S_1\cos\alpha+S_2\sin\alpha\cos\beta+S_3\sin\alpha\sin\beta$$4

The paper characterizes the nonclassicality of the precursor state through Wigner negativity and Mandel $$S_{\alpha\beta}=S_1\cos\alpha+S_2\sin\alpha\cos\beta+S_3\sin\alpha\sin\beta$$5 (Li et al., 2023).

Polarization nonclassicality has also been identified as a dynamical resource. Restricting dynamics to the manifold of angular-momentum coherent states defines a classical speed limit $$S_{\alpha\beta}=S_1\cos\alpha+S_2\sin\alpha\cos\beta+S_3\sin\alpha\sin\beta$$6, whereas unrestricted evolution obeys the usual quantum speed limit $$S_{\alpha\beta}=S_1\cos\alpha+S_2\sin\alpha\cos\beta+S_3\sin\alpha\sin\beta$$7. A process displays a nonclassical speedup precisely if $$S_{\alpha\beta}=S_1\cos\alpha+S_2\sin\alpha\cos\beta+S_3\sin\alpha\sin\beta$$8, quantified for example by $$S_{\alpha\beta}=S_1\cos\alpha+S_2\sin\alpha\cos\beta+S_3\sin\alpha\sin\beta$$9. For the cross-Kerr Hamiltonian $\chi(\lambda,\alpha,\beta)$0, the classical bound scales as

$\chi(\lambda,\alpha,\beta)$1

whereas the unrestricted bound scales as

$\chi(\lambda,\alpha,\beta)$2

so the speedup ratio

$\chi(\lambda,\alpha,\beta)$3

satisfies $\chi(\lambda,\alpha,\beta)$4 for all $\chi(\lambda,\alpha,\beta)$5 and asymptotically grows as $\chi(\lambda,\alpha,\beta)$6 (Aßbrock et al., 24 Mar 2026).

5. Beyond first moments: sequential quasiprobabilities, higher-order structure, and hidden polarization

Several works show that first-moment Stokes criteria are incomplete. In sequential measurements of two noncommuting polarization observables,

$\chi(\lambda,\alpha,\beta)$7

with $\chi(\lambda,\alpha,\beta)$8, the experimentally observed joint probabilities can be expressed as a convolution of an underlying complex Dirac distribution $\chi(\lambda,\alpha,\beta)$9 with resolution, transmission, and dynamical-correlation parameters $$\hat S_0=\hat a_H^\dagger\hat a_H+\hat a_V^\dagger\hat a_V,\quad \hat S_1=\hat a_H^\dagger\hat a_H-\hat a_V^\dagger\hat a_V,$$0. The inversion formula

$$\hat S_0=\hat a_H^\dagger\hat a_H+\hat a_V^\dagger\hat a_V,\quad \hat S_1=\hat a_H^\dagger\hat a_H-\hat a_V^\dagger\hat a_V,$$1

reconstructs a complex joint quasiprobability whose imaginary part yields the nonclassical correlation

$$\hat S_0=\hat a_H^\dagger\hat a_H+\hat a_V^\dagger\hat a_V,\quad \hat S_1=\hat a_H^\dagger\hat a_H-\hat a_V^\dagger\hat a_V,$$2

For an elliptically polarized input state, the real parts gave $$\hat S_0=\hat a_H^\dagger\hat a_H+\hat a_V^\dagger\hat a_V,\quad \hat S_1=\hat a_H^\dagger\hat a_H-\hat a_V^\dagger\hat a_V,$$3 and $$\hat S_0=\hat a_H^\dagger\hat a_H+\hat a_V^\dagger\hat a_V,\quad \hat S_1=\hat a_H^\dagger\hat a_H-\hat a_V^\dagger\hat a_V,$$4, while the imaginary parts yielded $$\hat S_0=\hat a_H^\dagger\hat a_H+\hat a_V^\dagger\hat a_V,\quad \hat S_1=\hat a_H^\dagger\hat a_H-\hat a_V^\dagger\hat a_V,$$5, independent of the measurement strength. No negative probabilities appeared in $$\hat S_0=\hat a_H^\dagger\hat a_H+\hat a_V^\dagger\hat a_V,\quad \hat S_1=\hat a_H^\dagger\hat a_H-\hat a_V^\dagger\hat a_V,$$6; here the nonclassicality appears purely as a non-zero imaginary off-diagonal in the Dirac distribution (Suzuki et al., 2016).

Higher-order optical-polarization proposes a different extension. Instead of requiring a fixed ratio $$\hat S_0=\hat a_H^\dagger\hat a_H+\hat a_V^\dagger\hat a_V,\quad \hat S_1=\hat a_H^\dagger\hat a_H-\hat a_V^\dagger\hat a_V,$$7, one demands a nonrandom $$\hat S_0=\hat a_H^\dagger\hat a_H+\hat a_V^\dagger\hat a_V,\quad \hat S_1=\hat a_H^\dagger\hat a_H-\hat a_V^\dagger\hat a_V,$$8th power,

$$\hat S_0=\hat a_H^\dagger\hat a_H+\hat a_V^\dagger\hat a_V,\quad \hat S_1=\hat a_H^\dagger\hat a_H-\hat a_V^\dagger\hat a_V,$$9

or, quantum mechanically,

$$\hat S_2=\hat a_H^\dagger\hat a_V+\hat a_V^\dagger\hat a_H,\quad \hat S_3=i(\hat a_V^\dagger\hat a_H-\hat a_H^\dagger\hat a_V),$$0

with $$\hat S_2=\hat a_H^\dagger\hat a_V+\hat a_V^\dagger\hat a_H,\quad \hat S_3=i(\hat a_V^\dagger\hat a_H-\hat a_H^\dagger\hat a_V),$$1 minimal. This criterion captures polarization structures carried by higher-order correlations, including Schrödinger-cat and entangled coherent states that are invisible to ordinary Stokes theory (Singh et al., 2013). The same source emphasizes that the polarization order $$\hat S_2=\hat a_H^\dagger\hat a_V+\hat a_V^\dagger\hat a_H,\quad \hat S_3=i(\hat a_V^\dagger\hat a_H-\hat a_H^\dagger\hat a_V),$$2 can change under SU(2) basis rotation, so higher-order polarization is highly basis-sensitive.

A related strand concerns hidden optical-polarized states (HOPS), where the nonrandom parameters are the ratio of real amplitudes and the sum of phases rather than the phase difference. The corresponding hidden-Stokes operators $$\hat S_2=\hat a_H^\dagger\hat a_V+\hat a_V^\dagger\hat a_H,\quad \hat S_3=i(\hat a_V^\dagger\hat a_H-\hat a_H^\dagger\hat a_V),$$3 define a degree of hidden optical polarization

$$\hat S_2=\hat a_H^\dagger\hat a_V+\hat a_V^\dagger\hat a_H,\quad \hat S_3=i(\hat a_V^\dagger\hat a_H-\hat a_H^\dagger\hat a_V),$$4

with $$\hat S_2=\hat a_H^\dagger\hat a_V+\hat a_V^\dagger\hat a_H,\quad \hat S_3=i(\hat a_V^\dagger\hat a_H-\hat a_H^\dagger\hat a_V),$$5 classically. Therefore $$\hat S_2=\hat a_H^\dagger\hat a_V+\hat a_V^\dagger\hat a_H,\quad \hat S_3=i(\hat a_V^\dagger\hat a_H-\hat a_H^\dagger\hat a_V),$$6 is a nonclassicality witness. In degenerate parametric amplification, the threshold condition was given as

$$\hat S_2=\hat a_H^\dagger\hat a_V+\hat a_V^\dagger\hat a_H,\quad \hat S_3=i(\hat a_V^\dagger\hat a_H-\hat a_H^\dagger\hat a_V),$$7

beyond which the degree of hidden optical polarization exceeds unity (Gupta et al., 2010).

The inadequacy of first-moment measures also appears in degree-of-polarization theory. The straightforward quantum analogue

$$\hat S_2=\hat a_H^\dagger\hat a_V+\hat a_V^\dagger\hat a_H,\quad \hat S_3=i(\hat a_V^\dagger\hat a_H-\hat a_H^\dagger\hat a_V),$$8

assigns $$\hat S_2=\hat a_H^\dagger\hat a_V+\hat a_V^\dagger\hat a_H,\quad \hat S_3=i(\hat a_V^\dagger\hat a_H-\hat a_H^\dagger\hat a_V),$$9 to amplitude-coherent, phase-randomized states and to hidden-polarized states, even though higher-order correlations reveal anisotropic polarization structure. A “second-generalized” intensity,

$W(\vec S)$00

leads to

$W(\vec S)$01

which restores sensitivity to these hidden structures and explicitly depends on mean photon number (Singh et al., 2013).

6. Alternative orderings, basis dependence, and measurement-context controversies

Because different definitions encode different classical references, the ordering of states by “degree of polarization” need not be unique. Majorization offers a measure-independent partial order on polarization distributions. For a fixed total photon number $W(\vec S)$02, one considers the SU(2)-invariant distribution induced by

$W(\vec S)$03

or, in smoothed form, the SU(2)-$W(\vec S)$04 function

$W(\vec S)$05

If $W(\vec S)$06, then every Schur-concave measure—Shannon entropy, Rényi entropies, confidence-interval widths, purity—orders $W(\vec S)$07 as more concentrated and therefore more polarized than $W(\vec S)$08. Within each fixed-$W(\vec S)$09 sector, the numerically observed chain is

$W(\vec S)$10

equivalently $W(\vec S)$11 for Shannon entropies, and coherent states majorize every other $W(\vec S)$12-photon state (Luis et al., 2016).

Distance-based measures exhibit a different behavior under de-Gaussification. For two-mode thermal states and their photon-added counterparts, the first-moment degree $W(\vec S)$13 always decreases when photons are added, while $W(\vec S)$14 and the Hilbert–Schmidt degree $W(\vec S)$15 behave non-monotonically. By contrast, the Bures and relative-entropy degrees increase monotonically under photon addition: $W(\vec S)$16 The authors interpret this as a consistent link between nonclassicality and increased distance from the set of unpolarized states (Ghiu et al., 2018).

Basis dependence is another persistent issue. The unified quasiprobability/entanglement treatment reports basis-independence of the witness $W(\vec S)$17 under polarization rotations $W(\vec S)$18, in contrast with coherence measures such as the $W(\vec S)$19-norm of off-diagonal matrix elements, which fluctuate strongly with basis (Ares et al., 2024). By contrast, higher-order optical-polarization explicitly states that the order $W(\vec S)$20 may change under basis rotation (Singh et al., 2013). This is not necessarily contradictory: it indicates that basis-independent resource witnesses and basis-sensitive structural descriptors answer different questions.

A final controversy concerns whether “classical-like” states can still display polarization nonclassicality. The Bohmian trajectory analysis reports that even Glauber and SU(2) coherent states exhibit trajectory topologies with vortices and saddles in polarization configuration space, unlike the single classical ellipse of electrodynamics. The authors treat the winding numbers and singularity patterns as topological indicators of polarization nonclassicality (Luis et al., 2014). This suggests that classicality with respect to quadrature amplitudes does not imply classicality of polarization dynamics.

7. Applications and broader operational significance

Polarization nonclassicality has been linked to several operational tasks. In quantum communication and metrology, the observation of PQD negativity for a weak coherent state implies that no model in which the three Stokes parameters are preassigned classical values can reproduce the measured statistics, thereby extending nonclassicality certification into the full Poincaré-sphere description of polarization (Spasibko et al., 2015). The highlighting procedure—embedding the unknown mode $W(\vec S)$21 into $W(\vec S)$22 with $W(\vec S)$23—maps polarization tomography onto Wigner-function tomography while preserving immunity to common phase fluctuations in the light path, and is proposed for bright states such as squeezed Fock states (Chekhova et al., 2013).

In nonlinear spectroscopy, “polarization nonclassicality” is used in a different sense: violation of a non-disturbance condition expressed through induced polarization intensities. The witness compares

$W(\vec S)$24

with the classical bounds obtained from control experiments on eigenstates $W(\vec S)$25: $W(\vec S)$26 Any violation signals nonclassicality rooted in coherence that cannot be reproduced by a convex mixture of classical pathways (1808.04427). This usage is conceptually distinct from Stokes-space nonclassicality but retains the same structural logic: experimentally accessible polarization observables violate a classical mixture model.

Prepare-and-measure experiments with polarization qubits connect nonclassical polarization statistics to contextuality-based advantages. Using four equatorial preparations and two binary measurements $W(\vec S)$27 and $W(\vec S)$28, the experiment reconstructed deviations $W(\vec S)$29, observed $W(\vec S)$30–$W(\vec S)$31, and found $W(\vec S)$32–$W(\vec S)$33, thereby violating preparation-noncontextual and bounded-ontological-distinctness constraints under the assumption that the measurements form a tomographically complete set (Gama et al., 23 Jan 2026). The same states and measurements are exactly those of the $W(\vec S)$34 quantum random access code, with success probability $W(\vec S)$35 (Gama et al., 23 Jan 2026).

Finally, polarization squeezing remains a practical route to polarization nonclassicality. Mixing a two-mode squeezed vacuum with a strong coherent beam at a linear polarization beam splitter can generate output light for which all three Stokes components are squeezed according to

$W(\vec S)$36

with reported minima

$W(\vec S)$37

The maximum degree of polarization squeezing reported was up to $W(\vec S)$38 along $W(\vec S)$39 and $W(\vec S)$40 along $W(\vec S)$41 (Shukla et al., 2016).

Taken together, these developments show that polarization nonclassicality is best understood as a structured family of obstructions to classical polarization models: nonpositive Stokes-space quasiprobabilities, negative or singular SU(2) phase-space representations, complex joint probabilities for incompatible Stokes observables, higher-order or nonlinear moment violations, failure of classical majorization hierarchies, convertibility into entanglement, and excess dynamical speed beyond angular-momentum coherent-state evolution. This suggests that the topic is less a single criterion than a unifying lens on how quantum coherence manifests in the polarization degree of freedom.