---
title: Polarization Nonclassicality in Quantum Light
url: https://www.emergentmind.com/topics/polarization-nonclassicality
type: topic
---

# Polarization Nonclassicality in Quantum Light

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 [1508.03510]. 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 \(\chi(\lambda,\alpha,\beta)\) [1508.03510].

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
\[
[\hat S_i,\hat S_j]=2\,i\,\epsilon_{ijk}\,\hat S_k.
\]
From the Glauber–Sudarshan perspective, a two-mode polarization state is nonclassical if the joint \(P(\alpha_H,\alpha_V)\) fails to be a bona fide probability; in the SU(2) formulation, any state whose \(P\)-function in the space of SU(2) coherent states exhibits negativities is polarization-nonclassical [2407.07477].

These definitions establish two important reference classes. In Stokes-space tomography, the classical benchmark is a nonnegative Kolmogorov probability density over \((S_1,S_2,S_3)\). In the SU(2)-coherent-state formulation, the classical benchmark is the convex set generated by angular-momentum coherent states \(\{|s\rangle\equiv|q\rangle^{\otimes N}\}\). 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 [1508.03510]. If \(\vec S\) were a set of preassigned classical variables, one would require \(W(\vec S)\ge 0\) everywhere. Hence any observed region with \(W(\vec S)<0\) rules out such a model and serves as a direct operational signature of polarization nonclassicality [1508.03510].

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 [1508.03510]. Because each \(\hat S_j\) has an integer spectrum, the one-dimensional marginals must be sums of Dirac \(\delta\)-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 [1305.5322].

The single-photon example makes this explicit. For the linearly polarized single-photon state \(|1\rangle_H\langle1|\otimes|0\rangle_V\langle0|\), the \((S_2,S_3)\)-marginal is
\[
W_{23}(S_2,S_3)=w_1(S_{23}),\qquad S_{23}=\sqrt{S_2^2+S_3^2},
\]
with
\[
w_m(S_{23})=\frac{1}{2\pi}
\begin{cases}
-\dfrac{S_{23}}{m\sqrt{m^2-S_{23}^2}}, & 0\le S_{23}<m,\\[1ex]
0,& S_{23}\ge m,
\end{cases}
\]
for \(m>0\). In particular, \(w_1(S_{23})<0\) for \(0<S_{23}<1\) [1305.5322].

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 [1305.5322]. 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 [1508.03510]. 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 \((\alpha,\beta)\) 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 \((n_H,n_V)\), from which the conditional probability \(P(n_H,n_V|\alpha,\beta)\) was formed. The characteristic function was then estimated as
\[
\chi(\lambda,\alpha,\beta)\simeq
\sum_{n_H,n_V}P(n_H,n_V|\alpha,\beta)\,e^{i\lambda(n_H-n_V)},
\]
and inverted numerically on a finite grid to obtain \(W(S_1,S_2,S_3)\) [1508.03510].

The reconstructed PQD for the weak coherent state exhibited a broad positive halo centered near the classical Stokes vector \((\langle S_1\rangle\approx1,\langle S_2\rangle=\langle S_3\rangle=0)\), surrounded by thin negative lobes along the \(S_1=1\) shell of radius \(\simeq1\). A cross-section \(W(S_1=1,S_2,S_3)\) showed clear regions with \(W<0\) for \(S_2^2+S_3^2\lesssim0.3\), and the integrated negative volume
\[
V_-=\int_{W(\vec S)<0}|W(\vec S)|\,dS_1\,dS_2\,dS_3
\]
was found to be on the order of \(10^{-3}\), significantly above the statistical uncertainty [1508.03510].

A complementary experimental development replaces linear Stokes analysis by click-counting nonlinear Stokes observables. In that approach, one defines click operators
\[
\hat\pi=1-{:}\exp[-(\eta/N)\hat n]{:},
\qquad
\hat\Pi_k={:}\binom{N}{k}\hat\pi^k(1-\hat\pi)^{N-k}{:},
\]
and constructs nonlinear Stokes operators
\[
\hat S_{\rm NL}
= N\hat\pi_A-N\hat\pi_B
=2N\,{:}\exp\!\biggl(-\frac{\eta}{2N}\hat S_0\biggr)
\sinh\!\biggl(\frac{\eta}{2N}\,\mathbf e\!\cdot\!\hat{\mathbf S}\biggr){:},
\]
\[
\hat S_{0,{\rm NL}}
= N\hat\pi_A+N\hat\pi_B
=2N\,{:}\Bigl[1-\exp\!\bigl(-\tfrac{\eta}{2N}\hat S_0\bigr)
\cosh\!\bigl(\tfrac{\eta}{2N}\,\mathbf e\!\cdot\!\hat{\mathbf S}\bigr)\Bigr]{:}.
\]
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 \(\langle{:}(\Delta S_{\rm NL})^2{:}\rangle\) became significantly negative, up to \(\simeq-0.02\) in normalized units, with statistical significance \(>15\sigma\), and the minimal eigenvalue of the higher-order moment matrix remained strictly below zero up to the eighth moment [2204.07083].

## 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 \(\{|q\rangle^{\otimes N}\}\) for particle entanglement and SU(2) coherent states for polarization. Accordingly, the quasiprobability-for-quantum-coherence decomposition
\[
\rho=\sum_i P_i\,|c_i\rangle\langle c_i|
\]
has identical coefficients in the two formalisms, so any negative \(P_i\) simultaneously witnesses polarization nonclassicality and entanglement [2407.07477].

This equivalence is constructive. One solves the separability eigenvalue equation
\[
\bigl(\rho_{q^{\otimes(N-1)}}\bigr)\,|q\rangle=g\,|q\rangle,
\]
uses the corresponding eigenvalues \(g_i\) and Gram matrix \(G_{ij}=\langle c_i|c_j\rangle\), and obtains the optimal quasiprobability weights from
\[
\mathbf P=G^{-1}\mathbf g.
\]
The total amount of nonclassicality may then be quantified by
\[
\mathcal N=-\sum_{i:P_i<0}P_i,
\]
while a basis-independent witness can be written as
\[
W=g_{\max}\,\mathbb1^{\otimes N}-|D_{k,N-k}\rangle\langle D_{k,N-k}|,
\]
with \(\langle W\rangle<0\) if and only if the state is nonclassical or entangled. For the two-photon Dicke state \(|1,1\rangle\), the maximal classical overlap is \(g_{\max}=1/2\), and the experiment reported \(\langle W\rangle\approx -0.35\), in agreement with \(\mathcal N>0\) [2407.07477].

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 \(q=a+a^\dagger\) and subsequent beam-splitter recombination, the resulting state acquires entanglement negativity
\[
\mathcal N(\rho)=\frac{\|\rho^{T_H}\|_1-1}{2}
=\frac{1}{2+8\xi_R^2},
\]
maximal at \(1/2\) for \(\xi_R=0\), while the classical polarization coherence matrix yields a Schmidt number
\[
K=\bigl[1-\tfrac12\sin^2(\Delta\phi)\sin^2(2\theta)\bigr]^{-1},
\qquad 1\le K\le 2.
\]
The paper characterizes the nonclassicality of the precursor state through Wigner negativity and Mandel \(Q<0\) [2310.04065].

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 \(QSL_{\rm cl}\), whereas unrestricted evolution obeys the usual quantum speed limit \(QSL\). A process displays a nonclassical speedup precisely if \(QSL>QSL_{\rm cl}\), quantified for example by \(Q\equiv QSL/QSL_{\rm cl}>1\). For the cross-Kerr Hamiltonian \(\hat H_{\rm int}=\epsilon\,\hat n_+\hat n_-\), the classical bound scales as
\[
QSL_{\rm cl}=\epsilon\,\sqrt{N}(N-1)/2\sim O(N^{3/2}),
\]
whereas the unrestricted bound scales as
\[
QSL=\epsilon\,[N^2-(N\bmod 2)]/(4\hbar)\sim O(N^2),
\]
so the speedup ratio
\[
Q(N)=\frac{N^2-(N\bmod 2)}{2\sqrt{N}(N-1)}
\]
satisfies \(Q(N)>1\) for all \(N>1\) and asymptotically grows as \(O(\sqrt N)\) [2603.23124].

## 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,
\[
\hat\sigma_H\equiv \hat S_{HV},\qquad
\hat\sigma_D\equiv \hat S_{PM},
\]
with \(\hat\sigma_H\hat\sigma_D=i\hat\sigma_R\), the experimentally observed joint probabilities can be expressed as a convolution of an underlying complex Dirac distribution \(\rho(s_D,s_H)\) with resolution, transmission, and dynamical-correlation parameters \((\varepsilon,\tau,\nu)\). The inversion formula
\[
\rho(s_D,s_H)
=\sum_{m,f=\pm1}
\Bigl(\tfrac14+\tfrac{m s_D}{4\varepsilon}
+\tfrac{f s_H}{4\tau}
+\tfrac{i m f}{4\nu}\Bigr)\,
P_{\rm exp}(m,f)
\]
reconstructs a complex joint quasiprobability whose imaginary part yields the nonclassical correlation
\[
\mathrm{Im}\langle \hat\sigma_H\hat\sigma_D\rangle=\langle \hat\sigma_R\rangle=\nu.
\]
For an elliptically polarized input state, the real parts gave \(\langle\hat\sigma_D\rangle\approx0.36\) and \(\langle\hat\sigma_H\rangle\approx0.84\), while the imaginary parts yielded \(\langle\hat\sigma_H\hat\sigma_D\rangle=i\,0.50\), independent of the measurement strength. No negative probabilities appeared in \(\mathrm{Re}\,\rho\); here the nonclassicality appears purely as a non-zero imaginary off-diagonal in the Dirac distribution [1606.00148].

Higher-order optical-polarization proposes a different extension. Instead of requiring a fixed ratio \(A_\perp/A\), one demands a nonrandom \(n\)th power,
\[
\left(\frac{A_\perp}{A}\right)^n=p_n,
\]
or, quantum mechanically,
\[
\hat b^n|\psi\rangle=p_n\,\hat a^n|\psi\rangle,
\]
with \(n>1\) 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 [1303.3952]. The same source emphasizes that the polarization order \(n\) 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 \(\hat H_j\) define a degree of hidden optical polarization
\[
H(t)=\frac{\sqrt{h_1(t)^2+h_2(t)^2+h_3(t)^2}}{h_0(t)},
\]
with \(0\le H(t)\le 1\) classically. Therefore \(H(t)>1\) is a nonclassicality witness. In degenerate parametric amplification, the threshold condition was given as
\[
t>t_0=\frac1{2k}\tanh^{-1}
\Bigl[\frac{1+|p_h|^2}{|p_h\sin\Delta_h|}\Bigr],
\]
beyond which the degree of hidden optical polarization exceeds unity [1007.2610].

The inadequacy of first-moment measures also appears in degree-of-polarization theory. The straightforward quantum analogue
\[
P_1=\frac{\sqrt{\langle\hat S_1\rangle^2+\langle\hat S_2\rangle^2+\langle\hat S_3\rangle^2}}{\langle\hat S_0\rangle}
\]
assigns \(P_1=0\) to amplitude-coherent, phase-randomized states and to hidden-polarized states, even though higher-order correlations reveal anisotropic polarization structure. A “second-generalized” intensity,
\[
n_{\hat\varepsilon}^{(2)}=\mathrm{Tr}\!\left[\hat\rho\,\hat N_{\hat\varepsilon}\,\hat V_{\hat\varepsilon_\perp}\right],
\]
leads to
\[
P_2=\frac{n_{\hat\varepsilon,\max}^{(2)}-n_{\hat\varepsilon,\min}^{(2)}}{n_{\hat\varepsilon,\max}^{(2)}+n_{\hat\varepsilon,\min}^{(2)}},
\]
which restores sensitivity to these hidden structures and explicitly depends on mean photon number [1301.3616].

## 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 \(N\), one considers the SU(2)-invariant distribution induced by
\[
p_m(\theta,\phi)=|\langle J,m|\psi\rangle_{\mathbf n}|^2,
\]
or, in smoothed form, the SU(2)-\(Q\) function
\[
Q(\Omega)=\frac{N+1}{4\pi}\,\langle N,\Omega|\rho|N,\Omega\rangle.
\]
If \(p\succ q\), then every Schur-concave measure—Shannon entropy, Rényi entropies, confidence-interval widths, purity—orders \(p\) as more concentrated and therefore more polarized than \(q\). Within each fixed-\(N\) sector, the numerically observed chain is
\[
\text{SU(2) coherent}\succ \text{squeezed}\succ N00N\succ \text{unpolarized-pure},
\]
equivalently \(H_C<H_S<H_{N00N}<H_H\) for Shannon entropies, and coherent states majorize every other \(N\)-photon state [1610.00408].

Distance-based measures exhibit a different behavior under de-Gaussification. For two-mode thermal states and their photon-added counterparts, the first-moment degree \(P_1\) always decreases when photons are added, while \(P_2\) and the Hilbert–Schmidt degree \(P_{\rm HS}\) behave non-monotonically. By contrast, the Bures and relative-entropy degrees increase monotonically under photon addition:
\[
P_{\rm B}^{\rm(add)}>P_{\rm B}^{\rm(th)},\qquad
P_{\rm RE}^{\rm(add)}>P_{\rm RE}^{\rm(th)}.
\]
The authors interpret this as a consistent link between nonclassicality and increased distance from the set of unpolarized states [1809.10547].

Basis dependence is another persistent issue. The unified quasiprobability/entanglement treatment reports basis-independence of the witness \(\langle W(\theta)\rangle\) under polarization rotations \(U(\theta)\), in contrast with coherence measures such as the \(\ell_2\)-norm of off-diagonal matrix elements, which fluctuate strongly with basis [2407.07477]. By contrast, higher-order optical-polarization explicitly states that the order \(n\) may change under basis rotation [1303.3952]. 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 [1402.4997]. 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 [1508.03510]. The highlighting procedure—embedding the unknown mode \(\hat\rho_H\) into \(\hat\rho_H\otimes|\alpha_0\rangle_V\langle\alpha_0|\) with \(|\alpha_0|\gg 1\)—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 [1305.5322].

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
\[
d_\rho=|P^{(3)}(t)|^2-|P^{(2)}(t)|^2
\]
with the classical bounds obtained from control experiments on eigenstates \(|i\rangle\):
\[
\min_i d_i \le |P^{(3)}(t)|^2-|P^{(2)}(t)|^2 \le \max_i d_i.
\]
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 \(X\) and \(Y\), the experiment reconstructed deviations \(\delta\approx 0.007\), observed \(S(x_{ij},y_{ij})=0.719\)–\(0.750>0\), and found \(C_{\min}^{\rm prep}=0.069\)–\(0.071>0\), thereby violating preparation-noncontextual and bounded-ontological-distinctness constraints under the assumption that the measurements form a tomographically complete set [2601.16952]. The same states and measurements are exactly those of the \(2\!\to\!1\) quantum random access code, with success probability \(\tfrac12(1+1/\sqrt2)\approx0.8536\) [2601.16952].

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
\[
\zeta_j=\frac{V_j}{|\langle \hat S_{j\perp}\rangle|_{\max}},
\]
with reported minima
\[
\zeta_{1,\min}=\frac1{\sqrt2\,[1-\min(r_H,r_V)^2]},\qquad
\zeta_{2,\min}=\zeta_{3,\min}=1-\min(r_H,r_V)^2.
\]
The maximum degree of polarization squeezing reported was up to \(65\%\) along \(\hat S_1\) and \(50\%\) along \(\hat S_2,\hat S_3\) [1603.08477].

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.

Source: https://www.emergentmind.com/topics/polarization-nonclassicality