can also be connected to quantum complementarity. This question was addressed
recently [324, 325, 326, 327, 328, 329, 330, 331], where much of this work has
focused on single photons. This is somewhat special: for single photons, the
semiclassical degree of polarization is equivalent to the purity of the state.
Alternatively, the density matrix of a single-photon state contains only the
first-order moment (i.e., the dipole), so, from the perspective of
polarization, represents a classical state. However, the results have been
appropriately extended to arbitrary quantum states [332]; viz.,

$\mathbb{D}^{2}+\mathbb{V}^{2}=\mathbb{P}^{2}\,,$ (158)

where the intensity distinguishability $\mathbb{D}$ and the Stokes visibility
$\mathbb{V}$ are given by

$\mathbb{D}=\frac{\lvert\langle\hat{N}_{+}\rangle-\langle\hat{N}_{-}\rangle\rvert}{\langle\hat{N}_{+}\rangle+\langle\hat{N}_{-}\rangle}\,,\qquad\qquad\mathbb{V}=\frac{2\sqrt{\langle\hat{N}_{+}\rangle\langle\hat{N}_{-}\rangle}}{\langle\hat{N}_{+}\rangle+\langle\hat{N}_{-}\rangle}\lvert
g_{\pm}\rvert\,,$ (159)

and the associated degree of polarization here turns out to be

$\mathbb{P}=\left[1-\frac{4\langle\hat{N}_{+}\rangle\langle\hat{N}_{-}\rangle(1-\mid
g_{\pm}\mid^{2})}{(\langle\hat{N}_{+}\rangle+\langle\hat{N}_{-}\rangle)^{2}}\right]^{1/2}\,.$
(160)

In these expressions $g_{\pm}$ is a mode correlation coefficient defined as

$g_{\pm}=\frac{\langle\hat{a}_{+}^{\dagger}\hat{a}_{-}\rangle}{\sqrt{\langle\hat{N}_{+}\rangle\langle\hat{N}_{-}\rangle}}\,.$
(161)

This shows that all perfectly polarized quantum states [259]; i.e., states
with $\mathbb{P}=1$, obey strong complementarity. If the intensity
distinguishability (or path predictability for a single photon) is zero, we
obtain $\mathbb{V}=\mathbb{P}$; that is, the Stokes visibility is exactly
determined by the degree of polarization. This result unifies the
interpretations for the degree of polarization established in classical two-
way interferometry, where the degree of polarization has been connected to
intensity visibility and to polarization modulation [333]. Therefore, in
addition to its role as a complementarity measure, the degree of polarization
can be viewed as an intrinsic quantity that characterizes the ability of light
to exhibit intensity and polarization-state variation.

##  8 Unpolarized light

###  8.1 Higher-order unpolarized states

The constitution of unpolarized light has been investigated since the dawn of
modern optics. Indeed, Verdet [334] already offered a lucid characterization
of what was known as _natural_ light by using the projections of the intensity
onto the axes of a rotated Cartesian coordinate system. Unpolarized states are
those that remain invariant under any rotation of that coordinate system and
under any phase shift between its rectangular components.

In classical optics, the field components of unpolarized light are modeled by
zero-mean, uncorrelated, stationary Gaussian random processes [335]. The
previous invariance conditions thus determine the entire probabilistic
structure of the projected intensities [49]. In contrast, the standard theory
is limited to first-order moments, presenting unpolarized light as having a
zero-mean Stokes vector, which in geometrical terms means that the Stokes
vector reduces to the origin of the Poincaré sphere.

At the quantum level, the invariance requirement fixes once and for all the
structure of the density matrix, so it specifies the probability distribution
for and, as a result, all the moments of the Stokes variables, as we have
already discussed. However, one could think of extending this notion: when all
of the multipoles up to a given order (say $M$) vanish, the state lacks of
polarization information up to that order and hence will be called _$M$ th-
order unpolarized_. The classical picture matches the first-order theory,
corresponding to a certain set of quantum states that are only invariant to
first-order [259]; whereas, the quantum condition implies that all the
multipoles are identically null.

In consequence, using the previous notion of higher-order polarization
degrees, a state is $M$th-order unpolarized when $\mathbb{P}_{M}^{(S)}=0$,
which obviously implies $\mathcal{A}_{M}^{(S)}=0$; i.e., all the multipoles up
to order $M$ vanish. We will denote these states as
$\hat{\varrho}_{\mathrm{unpol},M}^{(S)}$. In more physical terms, the
condition of $M$th-order unpolarization amounts to imposing that the moments
$\langle\hat{\mathbf{S}}_{\mathbf{n}}^{\ell}\rangle$ be independent of the
direction $\mathbf{n}$ for $\ell=1,\ldots,M$ (i.e., that they be isotropic).
Therefore, all of the moments up to order $M$ do not show any directional
structure, while higher-order moments do. Because of this, $M$th-order
unpolarized states do carry polarization information when one inspects higher-
order moments. As we have said, this is referred to as hidden polarization,
according to the terminology coined by Klyshko [55, 56, 57], although we will
say that such states display higher-order polarization [336].

We illustrate this point with a few examples, starting with single-photon
states ($S=1/2$). The multipole expansion of an arbitrary single-photon state
reads

$\hat{\varrho}^{(1/2)}=\varrho_{00}^{(1/2)}\,\hat{T}_{00}^{(1/2)}+\sum_{q}\varrho_{1q}^{(1/2)}\,\hat{T}_{1q}^{(1/2)}\,.$
(162)

Since the state only has dipolar components, the quantum and classical
descriptions coincide and these states can only be first-order unpolarized.
Positivity constrains the possible values of the dipole to the range
$0\leq\mathcal{A}_{1}^{(1/2)}\leq 1/2$. The condition
$\mathcal{A}_{1}^{(1/2)}=0$ immediately fixes the unpolarized states; viz.,

$\varrho_{\mathrm{unpol},1}^{(1/2)}=\tfrac{1}{2}\;\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}_{2}\,.$ (163)

We stress, though, that like all quantum objects, these states can only be
considered as elements of an ensemble [337].

Figure 12: Second-order degree of polarization as a function of purity, for
the first-order unpolarized states (166).

For two-photon states, there are first-order (classical) and second-order
(quantum) unpolarized states. The general condition for first-order
unpolarization is that the dipolar term vanishes; i.e.,

$\hat{\varrho}_{\mathrm{unpol},1}^{(1)}=\varrho_{00}^{(1)}\,\hat{T}_{00}^{(1)}+\sum_{q}\varrho_{2q}^{(1)}\,\hat{T}_{2q}^{(1)}\,,$
(164)

with the extra constraint of positivity. Let us assume that the density matrix
can be diagonalized via an SU(2) transformation (which is true for a broad
class of axially symmetric states [338]), so that it can be expressed as

$\hat{\varrho}_{\mathrm{diag}}^{(1)}=\mathrm{diag}(\lambda_{1},\lambda_{2},\lambda_{3})=\frac{1}{\sqrt{3}}\hat{T}_{00}^{(1)}+\frac{\lambda_{1}-\lambda_{3}}{\sqrt{2}}\;\hat{T}_{10}^{(1)}+\frac{1-3\lambda_{2}}{\sqrt{6}}\;\hat{T}_{20}^{(1)}\,,$
(165)

where $\mathrm{diag}(d_{1},\ldots,d_{n})$ represents a diagonal matrix whose
diagonal elements are those in the argument. The state is first-order
unpolarized when $\lambda_{1}=\lambda_{3}$. Since
$\mathop{\mathrm{Tr}}\nolimits(\hat{\varrho}_{d})=1$, we can write

$\hat{\varrho}^{(1)}_{\mathrm{unpol},1}=\mathrm{diag}(\lambda,1-2\lambda,\lambda)$
(166)

and positivity enforces $0\leq\lambda\leq 1/2$; i.e.,
$0\leq\mathcal{A}_{2}^{(1)}\leq 2/3$. Both the purity
$\mathcal{P}=\mathop{\mathrm{Tr}}\nolimits[(\hat{\varrho}_{\mathrm{unpol},1}^{(1)})^{2}]$
and the second-order degree $\mathbb{P}_{2}$ depend on $\lambda$

$\mathcal{P}^{(1)}=6\lambda^{2}-4\lambda+1\,,\qquad\qquad\qquad\mathbb{P}_{2}^{(1)}=\sqrt{(3\lambda-1)^{2}}\,,$
(167)

while $\mathbb{P}_{1}^{(1)}=0$, as anticipated. This can be concisely recast
as

$\mathbb{P}_{2}^{(1)}=\sqrt{\tfrac{1}{2}[3\mathcal{P}^{(1)}-1]}\,,$ (168)

which is plotted in Fig. 12. The maximum degree $\mathbb{P}_{2}^{(1)}$ is
attained for the pure states

$|\Psi_{\mathrm{unpol,1}}^{(1)}\rangle=\frac{1}{\sqrt{2}}\sin\beta\left(e^{i\alpha}|1,1\rangle-e^{-i\alpha}|1,-1\rangle\right)+\cos\beta|1,0\rangle\,,$
(169)

which are transformed versions of the state $|1,0\rangle$ under SU(2)
transformations. These states have served as the conduit to experimentally
verify the existence of hidden polarization [199, 294]; for a more detailed
analysis of these states, the reader is referred to Ref. [339].

###  8.2 Majorana representation

The SU(2) coherent states correspond as nearly as possible to a classical spin
vector pointing in a particular direction. It is irresistible to ask which
pure states are, in a sense, _the opposite_ of SU(2) coherent states and
therefore the most quantum ones. This idea has been recently pursued [340,
341, 342] as those states that _point nowhere_ ; i.e., the average Stokes
vector vanishes and the fluctuations up to order $M$ are isotropic: they have
been dubbed _anticoherent states_.

To investigate this point it is advantageous to use the Majorana stellar
representation [343], which allows us to uniquely depict a spin state state
living in $\mathcal{H}_{S}$ by $2S$ points on the unit sphere [344]. Several
decades after its conception, this representation has recently attracted a
great deal of attention in several fields [345, 346, 347, 348, 349, 350, 351,
352, 353, 354, 355, 356, 357, 358].

The Majorana representation is a direct generalization of the Bloch sphere
from spin-$1/2$ particles to spin-$S$ particles, making use of the coherent-
state representation of the wave function
$\Psi(\mathbf{n})=\langle\mathbf{n}|\Psi\rangle$ (to lighten notation, we will
skip the label $S$, henceforth restricting to a fixed $S$). Once we insert the
identity in terms of the states $|S,m\rangle$ and take into account the
overlap $\langle\mathbf{n}|S,m\rangle$, the wavefunction can be written as

$\Psi(\mathbf{n})=\frac{1}{(1+\lvert\zeta\rvert^{2})^{S}}\sum_{m=-S}^{S}\sqrt{\frac{(2S)!}{(S-m)!(S+m)!}}\Psi_{m}\,\zeta^{S+m}\,,$
(170)

where $\zeta=\tan(\theta/2)e^{i\phi}$ is the stereographic projection mapping
a point on the unit sphere with angle $(\theta,\phi)$ to the point
$\zeta\in\mathbb{C}$. Apart from an unessential positive factor, this wave
function is a polynomial of order $2S$; thus, $|\Psi\rangle$ is determined by
the set $\\{\zeta_{i}\\}$ of the $2S$ complex zeros of $\Psi(\mathbf{n})$
[344]

$\Psi(\mathbf{n})=\frac{Z^{\ast}_{2S}}{(1+|\zeta|^{2})^{S}}(\zeta-\zeta_{1})\ldots(\zeta-\zeta_{2S})\,,$
(171)

with the resulting set suitably completed by points at infinity if the degree
of $\Psi(\mathbf{n})$ is less than $2S$. Here $Z^{\ast}_{2S}$ is the final
component of the vector of coefficients of $|\Psi\rangle$ in the angular
momentum basis. The corresponding configuration of points on the unit sphere
is called the Majorana constellation associated with the state $|\Psi\rangle$.

There is a complementary way to look at the Majorana polynomial that helps us
gain further insights. Any pure state $|\Psi\rangle\in\mathcal{H}_{S}$ can be
factorized in terms of the bosonic operators $\hat{a}_{\pm}$ as

$|\Psi\rangle=\frac{1}{\sqrt{\cal{N}}}\prod_{m=1}^{2S}[\cos({\theta_{m}}/{2})\hat{a}_{+}^{\dagger}+e^{i\phi_{m}}\sin({\theta_{m}}/{2})\hat{a}_{-}^{\dagger}]|0_{+},0_{-}\rangle\,,$
(172)

where $\cal{N}$ is a normalization factor, $|0_{+},0_{-}\rangle$ is the two-
mode vacuum and the angles $\theta_{m}$ and $\phi_{m}$ satisfy the natural
constraints $0\leq\theta_{m}\leq\pi$ and $0\leq\phi_{m}<2\pi$. Thus, each
factor in (172) can be visualized as a point on the unit sphere. Since the
operators $\hat{a}^{\dagger}_{+}$ and $\hat{a}^{\dagger}_{-}$ create an
excitation in right- and left-hand circularly polarized modes, respectively,
each of the factors in (172) can also be naively thought of as creating an
_excitation component_ of a polarization state corresponding to its position
on the sphere [344].

An SU(2) rotation simply corresponds to a rigid rotation of the Majorana
constellation and, consequently, it does not affect the degree of
polarization: states with the same constellation, irrespective of their
relative orientations, have the same polarization invariants.

For SU(2) coherent states, the Majorana constellation collapses to a single
point on the unit sphere. Intuitively, one would guess that states with the
most isotropic polarization moments would have the most symmetric
constellations that are possible. We further develop this idea in the
subsequent section.

###  8.3 Extremal polarizaton states

To formalize the idea of the most quantum states we will use the cumulative
distribution $\mathcal{A}_{M}$. The idea is to identify which states minimize
$\mathcal{A}^{(S)}_{M}$ for each order $M$. We shall be considering only pure
states, which we expand as
$|\Psi\rangle=\sum_{m=-S}^{S}\Psi_{m}\,|S,m\rangle$. We then have that

$\mathcal{A}_{M}=\sum_{K=1}^{M}\sum_{q=-K}^{K}\frac{2K+1}{2S+1}\left|\sum_{m,m^{\prime}=-S}^{S}C_{Sm,Kq}^{Sm^{\prime}}\Psi_{m^{\prime}}\Psi_{m}^{\ast}\right|^{2}\,.$
(173)

As we have discussed, SU(2) coherent states maximize $\mathcal{A}_{M}$ for all
orders $M$.

The strategy to minimize $\mathcal{A}_{M}$ is simple to state: starting from a
set of unknown normalized state amplitudes in Eq. (173), which we write as
$\Psi_{m}=a_{m}+ib_{m}$ ($a_{m},b_{m}\in\mathbb{R}$), we try to obtain
$\mathcal{A}_{M}=0$ for the highest possible $M$. This yields a system of
polynomial equations of degree two for $a_{m}$ and $b_{m}$, which we solve
using Gröbner bases [359] implemented in the computer algebra system magma
[360]. Since the orientation of the constellation is irrelevant, one can
reduce the number of variables by fixing one of the points to be at, say, the
north pole and another to lie in the $S_{2}$-$S_{3}$ plane. In this way, we
get exact algebraic expressions and we can detect when there is no feasible
solution. Table 1 lists the resulting states (which, in some cases, are not
unique) for selected values of $S$. We also indicate the solutions’ associated
Majorana constellations. For completeness, in Fig. 13 we plot the
constellations as well as the $Q$-functions for some of these states. A
complete list can be found in [361] and a more detailed discussion can be
found in [362]. The resulting states systematize the notion of anticoherent
states (of note is the fierce objection of Roy Glauber, the father of coherent
states, to such a term) they have been called the Kings of Quantumness to
avoid a disputable denomination.

Table 1: States for which $\mathcal{A}_{M}$ vanishes for the indicated values
of $S$. In the second column, we indicate the order $M$, which we conjecture
is the highest possible. We give the nonzero state components $\Psi_{m}$
($m=-S,\ldots,S$) and describe the corresponding Majorana constellation.
$S$ | $M$ | State | Constellation
---|---|---|---
1 | 1 | $\Psi_{0}=1$ | radial line
$\frac{3}{2}$ | 1 | $\Psi_{\scriptsize{\pm\frac{3}{2}}}=\scriptsize{\frac{1}{\sqrt{2}}}$ | equatorial triangle
2 | 2 | $\Psi_{-1}=\scriptsize{\frac{2}{\sqrt{3}}},\quad\Psi_{2}=\scriptsize{\sqrt{\frac{1}{3}}}$ | tetrahedron
$\frac{5}{2}$ | 1 | $\Psi_{\scriptsize{\pm\frac{3}{2}}}=\scriptsize{\frac{1}{\sqrt{2}}}$ | equatorial triangle + poles
3 | 3 | $\Psi_{\pm 2}=\scriptsize{\frac{1}{\sqrt{2}}}$ | octahedron
$\frac{7}{2}$ | 2 | $\Psi_{-\scriptsize{\frac{5}{2}}}=\Psi_{\scriptsize{\frac{1}{2}}}=\scriptsize{\sqrt{\frac{7}{18}}},\quad\Psi_{\scriptsize{\frac{7}{2}}}=\scriptsize{\sqrt{\frac{2}{9}}}$ | two triangles + pole
4 | 3 | $\Psi_{\pm 4}=\scriptsize{\sqrt{\frac{5}{24}}},\quad\Psi_{0}=\scriptsize{\sqrt{\frac{7}{12}}}$ | cube
$\frac{9}{2}$ | 2 | $\Psi_{\pm\scriptsize{\frac{9}{2}}}=\scriptsize{\frac{1}{\sqrt{6}}},\quad\Psi_{\pm\scriptsize{\frac{3}{2}}}=\scriptsize{\frac{1}{\sqrt{3}}}$ | three triangles
5 | 3 | $\Psi_{\pm 5}=\scriptsize{\frac{1}{\sqrt{5}}},\quad\Psi_{0}=\scriptsize{\frac{3}{\sqrt{5}}}$ | pentagonal prism
$\frac{11}{2}$ | 3 | $\Psi_{\pm\scriptsize{\frac{11}{2}}}=\scriptsize{\frac{\sqrt{17}}{12}},\quad\Psi_{\pm\scriptsize{\frac{5}{2}}}=i\scriptsize{\frac{\sqrt{55}}{12}}$ | pentagon + two triangles
6 | 5 | $\Psi_{\pm 5}=\pm\scriptsize{\frac{\sqrt{7}}{5}},\quad\Psi_{0}=-\scriptsize{\frac{\sqrt{11}}{5}}$ | icosahedron
7 | 4 | $\Psi_{\pm 6}=\scriptsize{\sqrt{\frac{854}{3645}}},\quad\Psi_{\pm 3}=\scriptsize{\sqrt{\frac{637}{13420}}+i\sqrt{\frac{512603}{9783180}}}$ | three squares + poles
|  | $\Psi_{0}=\scriptsize{\sqrt{\frac{12561757}{163053000}}}-\scriptsize{i\sqrt{\frac{512603}{2013000}}}$ |
10 | 5 | $\Psi_{\pm 10}=\scriptsize{\sqrt{\frac{187}{1875}}},\quad\Psi_{\pm 5}=\pm\scriptsize{\sqrt{\frac{209}{625}}},\quad\Psi_{0}=\scriptsize{\sqrt{\frac{247}{1875}}}$ | deformed dodecahedron

From a physical perspective one would expect these constellations to have
their points distributed as symmetrically as possible on the unit sphere. For
some values of $S$, such as 4, 6, 8, 12 and 20, one can guess a maximally
unpolarized constellation, in each case corresponding to the vertices of a
Platonic solid. For other numbers it is not easy to guess an optimal, exact
constellation, but solving the system of polynomial equations, as described
before, yields exact algebraic expressions for the coefficients $\Psi_{m}$,
from which one can easily compute the points of the Majorana constellation
with arbitrary numerical precision.

The problem of distributing $N$ points on a sphere in the _most symmetric_
fashion has a long history and has many different solutions depending on the
cost function one tries to optimize [363, 364]. Here, we shall only discuss a
few of the formulations: spherical $t$-designs [365], the Thomson problem
[366, 367, 368, 369] and the Queens of quantumness [370].

Spherical $t$-designs are configurations of $N$ points on a sphere such that
the average value of any polynomial of degree at most $t$ is the same when
evaluated over the $N$ points as over the entire sphere. Thus, the $N$ points
can be seen to give a representative average value of all polynomials up to
degree $t$. It has been conjectured that a state is unpolarized to order $t$
if and only if its Majorana constellation is a spherical $t$-design [341].
However, while the statement is true for some $t$-designs, such as those
represented by the Platonic solids, the general conjecture has been disproven
[342].

Figure 13: Density plots of the SU(2) $Q$-functions for the optimal states in
Table I for the cases $S=5/2,3,7/2,9/2,5$, and $7$ (from left to right, blue
indicates the zero values and red maximal ones). On top, we sketch the
Majorana constellation for each of them.

It is clear that there should be some connection between the number of points
$N$ and the maximal degree $t$ for which a spherical $t$-design exists. The
configurations that maximize $t$ for a given $N$ are called optimal designs,
and we use $t$ to denote the degree of an optimal $N$-point design in the
following. No analytical relationship is known between $N$ and $t$: it is
known that for a $t$-design, the number of points $N$ is at least proportional
to $t^{2}$; whereas, for some orders $t$ the only known constructions have $N$
scaling proportionally with $t^{3}$. As a function of $N$, the order $t$ is
non-monotonic. The results for $1\leq N\leq 100$ are summarized in [371].

From the numerical data thus far one can conclude that the maximum value of
$M$and $t$ coincide. We therefore conjecture that if an optimal spherical
design of order $t$ exists for some $N$, then one can find an $M$th-order
unpolarized $N$-photon state with $M=t$.

The next thing one can note is that an optimal $t$-design does not necessarily
give a $t$th-order unpolarized state. Quite often the configurations are
similar, e.g. regular polygons with their surface normals along the polar
axis, but displaced from each other along the axis by certain distances.
However, these distances often need to be fine-tuned for an optimal $t$-design
to become a maximally-unpolarized state. The Platonic solids are exceptions to
this observation. That the optimal configurations for $t$-designs and
maximally unpolarized states do not always coincide underscores the mystery
that the optimal $t$ and maximal $M$ always seem to be equal for any $N$.

The Thomson problem consists of arranging $N$ identical point charges on the
surface of a sphere so that the electrostatic potential energy of the entire
configuration is minimized. The problem can be generalized to potential
energies that are proportional to $r^{-d}$, where $r$ is the Euclidean
distance between the charges. The choice $d=1$ is the Thomson problem,
corresponding to the usual Coulomb potential; the case $d\rightarrow\infty$ is
called Tammes problem [372]. For small $S$, up to 3, the Thomson
configurations are identical to the optimal spherical $t$-design and to the
maximally unpolarized states. For larger $S$, they differ in general and the
degree of unpolarization of the Thomson states is lower than the maximum.
Different from the two previous cases, the solution to the Thomson problem
appears to be unique for every $S$ [373].

The Queens of quantumness are the states that maximize the Hilbert-Schmidt
distance to the closest point of the convex hull of the mixed SU(2) coherent
states [370]. This convex hull defines the subspace of classical states.
Therefore, the states maximizing the distance to the nearest point on this
hull can be thought of as having maximally quantum characteristics. In Ref.
[370] it is claimed that the Queens can be seen as the least classical (or
most quantum) of all states given this metric. Although we have used another
figure of merit, our approach and that in [370] share the view that the states
_most different_ from SU(2) coherent states are the most quantum.

When we interpret our subspace $\mathcal{H}_{S}$ as the symmetric subspace of
a system of $S$ qubits, the Kings seem to also be closely linked to other
intriguing problems, such as maximally entangled symmetric states [374, 375]
and $k$-maximally mixed states [376, 377].

###  8.4 Metrological applications

The main goal of quantum metrology is to measure a physical magnitude with
surprising precision by exploiting quantum resources [378]. In particular,
tailoring polarization states to better detect SU(2) rotations is quite a
relevant problem with direct applications to magnetometry [379, 380, 381],
polarimetry [382, 383], and metrology in general [384].

The salient feature of maximally unpolarized states is their ability to detect
small, but arbitrary, SU(2) transformations with optimal resolution. This was
already anticipated in [385], where the authors specifically found that, for
photon numbers 4, 6, 8, 12 and 20, the states corresponding to regular
polyhedron Majorana constellations best signaled misalignments between two
Cartesian reference frames. To understand this, it is instructive to look at
related states, namely the NOON states. Such NOON states are known to have the
highest sensitivity to small rotations about the $\hat{S}_{3}$-axis among all
states with a fixed excitation $S$ [386]. This can be easily understood by
looking at their Majorana constellation, which is apparent from Fig. 3 and
consists of $2S$ equidistantly placed points around the equator. A rotation
around the $\hat{S}_{3}$-axis is described by the unitary operator
$\hat{U}(\phi)=\exp(-i\phi\hat{S}_{3})$; therefore, for $\phi=\pi/(2S)$ the
states $|\mathrm{NOON}\rangle$ and $\hat{U}(\phi)|\mathrm{NOON}\rangle$ are
orthogonal, while for $\vartheta=q\pi/S$ they are parallel, where $q$ is an
integer. Thus, it should not come as a surprise that NOON states are optimal
for detecting small rotations around the $\hat{S}_{3}$-axis, in the interval
$0\leq\vartheta\leq\pi/(2S)$. If the rotation axis lies in the equatorial
plane, then a rotation of $\pi$ is needed to obtain a parallel state,
irrespective of $S$. When $S$ is a half integer this only happens when the
axis intersects one of the Majorana points, and when $S$ is an integer this
happens if the axis either intersects either a point or is the intersector
between two points. Thus, the rotation resolution is highly directional for a
NOON state.

This is precisely the advantage of maximally unpolarized states: having a high
degree of spherical symmetry, they resolve rotations around any axis
approximately equally well. It may not be obvious from their appearance that
they have high sensitivity to small rotations about an arbitrary axis. To
substantiate this claim, recall that the action $\tau$ needed to make a state
$|\Psi\rangle$ evolve so that
$|\langle\Psi|\exp(i\hat{A}\tau)|\Psi\rangle|^{2}=1-\epsilon$, where
$\epsilon$ is a small, positive, real number, and $\hat{A}$ is Hermitian, is
inversely proportional to the variance $\mathrm{Var}(\hat{A})$ [387]. The
relation connecting the evolution speed $d\epsilon/d\tau$ and the variance is
sometimes called the quantum speed limit [388, 389]. A NOON state in the
$\hat{S}_{3}$ basis has maximal variance
$\langle\mathrm{Var}(\hat{S}_{3})\rangle=S^{2}$ for a fixed $S$ and is thus
the state with maximal sensitivity to a rotation around the $\hat{S}_{3}$
axis. However, the variances $\mathrm{Var}(\hat{S}_{1})$ and
$\mathrm{Var}(\hat{S}_{2})$ are only $S/2$ and thus the state is rather
insensitive to rotations around those axes (or around any rotation axis in the
$\hat{S}_{1}$-$\hat{S}_{2}$ plane). In contrast, all the Kings of Quantumness
have isotropic variances equal to $S(S+1)/3$; namely, close to the maximum
possible variance. The quantum speed limit theorem thus asserts that, having a
large and isotropic variance of the Stokes operator, these states are rather
sensitive to rotations around any axis $\hat{S}_{{\bf n}}$ [390, 391].

Another way of explaIning the sensitivity to a rotation around an arbitrary
axis is to observe that, since these states have maximal spherical symmetry,
they become parallel, or almost parallel, for relatively small rotations
around several axes. To quantify this statement one could use the Fisher
information and the Cramér-Rao bound to assess the uncertainty in estimating
the rotation direction and the rotation angle [388, 389].

##  9 Conclusions

Quantum polarization provides an excellent landscape to flaunt nonclassical
features. This is because the polarization degree of freedom is easily
accessible both theoretically and experimentally. Moreover, we have shown that
this variable finds parallels with many different topics of quantum physics.

As polarization is the manifestation of photon spin, it in turn contributes,
together with the orbital angular momentum (OAM), to the total angular
momentum of light. It should come as no surprise that the methods developed
for polarization characterization lend themselves, virtually without
modification, to the analysis of OAM. In fact, since it is at present
experimentally easier to generate specific OAM states than the corresponding
polarization states (which are, in general, highly entangled), some of the
theory developed for polarization has actually been tested with OAM as the
model system.

Light has been an excellent laboratory for research in quantum theory. We hope
that the results summarized in this contribution show how this is particularly
true for light polarization.

Acknowledgments. Over the years, the ideas in this paper have been further
developed and expanded with questions, suggestions, criticism, and advice from
many colleagues. Particular thanks for help in various ways goes to G. S.
Agarwal, F. Bouchard, R. W. Boyd, A. S. Chirkin, J. H. Eberly, A. T. Friberg,
R. J. Glauber, D. F. V. James, V. P. Karassiov, Y. H. Kim, E. Karimi, N.
Korolkova, Ch. Marquardt, C. Müller, A. Normann, Ł. Rudnicki, Ch. Silberhorn,
W. P. Schleich, and J. Söderholm.

Funding. Ministerio de Ciencia e Innovación (PGC2018-099183-B-I00); Consejo
Nacional de Ciencia y Tecnologia de Mexico (254127); Fundacja na rzecz Nauki
Polskiej (2018/MAB/5); Canada National Sciences and Engineering Research
Council; Walter C. Sumner Foundation; Lachlan Gilchrist Fellowship Fund;
Michael Smith Foreign Study Supplement; Mitacs Globalink; Ministry of
Education and Science of the Russian Federation (Mega Grant 14.W03.31.0032).

Disclosures. The authors declare no conflicts of interest.

##  Appendix A Coherent states

The coherent-state approach is not just a convenient mathematical tool, but it
also helps one to understand how physical properties of a system are reflected
by the geometrical structure of the related phase space. The reader interested
can check more details in the pertinent monographs [117, 116]

Let $G$ be a Lie group (connected and simply connected, with finite dimension
$n$), which is the dynamical group of a given quantum system. Let $T$ be a
unitary irreducible representation (irrep) of $G$ acting on the Hilbert space
$\mathcal{H}$:

$|\Psi_{g}\rangle=\hat{T}(g)|\Psi_{0}\rangle\,,\qquad\qquad g\in G\,,$ (174)

where $|\Psi_{0}\rangle$ is a fixed vector in $\mathcal{H}$ of the
representation $\hat{T}(g)$. The subgroup $H\in G$ such that

$\hat{T}(h)|\Psi_{0}\rangle=e^{i\alpha(h)}|\Psi_{0}\rangle\,,\qquad\qquad\forall
h\in H\,,$ (175)

is called the isotropy group (or little group) of the state
$|\Psi_{0}\rangle$. Since the states of a quantum system are defined up to
multiplication by a global phase, any element of the orbit $|\Psi_{g}\rangle$
of the state $|\Psi_{0}\rangle$ can be put in correspondence with an element
of the coset space $X=G/H$. We thus define the set of coherent states for the
group $G$ with respect to the fiducial vector $|\Psi_{0}\rangle$ by

$|\Omega\rangle=\hat{T}(\Omega)|\Psi_{0}\rangle\,,$ (176)

where $\Omega\in X$; that is, $g=\Omega h$.

An essential property is that the coherent states form an (overcomplete) basis
in the state space of the system

$\int_{X}d\mu(\Omega)\,|\Omega\rangle\langle\Omega|=\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}\,,$ (177)

where $d\mu(\Omega)$ is the invariant integration measure on $X$.

In principle, coherent states can be generated from any state
$|\Psi_{0}\rangle$. However, the states $|\Psi_{0}\rangle$ having the largest
isotropy groups generate coherent states closest to the classical states in
the sense that they minimize the uncertainty relations.

An important class of coherent states corresponds to the coset spaces $X$ that
are homogeneous Kähler manifolds. Then, a natural symplectic structure can be
introduced on $X$, so that it can be considered the phase space of a classical
dynamical system [392].

Let us first consider the canonical example of the Heisenberg-Weyl group
$W_{1}$, which is the dynamical symmetry group for a mode of the quantized
radiation field. The associated Lie algebra $\mathfrak{w}_{1}$ is defined by
the canonical commutation relations

$[\hat{a},\hat{a}^{\dagger}]=\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}\,.$ (178)

A general element of this algebra has the form $t\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}+i(\alpha^{\ast}\hat{a}-\alpha\hat{a}^{\dagger})$,
where $t\in\mathbb{R}$ and $\alpha\in\mathbb{C}$. Therefore, the elements of
the group are obtained via the exponential map

$\hat{T}(t,\alpha)=e^{it}\hat{D}(\alpha)\,,\qquad\qquad\hat{D}(\alpha)=\exp(\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a})\,.$
(179)

An element of this group is specified by $g=(t,\alpha)$ and
$\alpha\in\mathbb{C}$ and the operators $\hat{T}(g)=\hat{T}(t,\alpha)$ are an
irrep of the Hilbert space of Fock states $\\{|n\rangle\\}$. The phase space
is the complex plane $\mathbb{C}=W_{1}/U(1)$, and the (quadrature) coherent
states are:

$|\alpha\rangle=\hat{D}(\alpha)|0\rangle,$ (180)

with the vacuum $|0\rangle$ being the fiducial state.

The expansion in the Fock basis can be easily obtained by disentangling the
displacement operator $\hat{D}(\alpha)$; the result is

$|\alpha\rangle=e^{-|\alpha|^{2}/2}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle\,,$
(181)

which exhibits the well-known Poissonian statistics of a radiation field with
average value $\bar{N}=\lvert\alpha\rvert^{2}$.

If we consider the transformation of the operators of the algebra
$\mathfrak{w}_{1}$ by the elements of the group $W_{1}$, it is not difficult
to show that:

$\hat{D}(\alpha)\,\hat{a}\,\hat{D}^{\dagger}(\alpha)=\hat{a}-\alpha\;\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}\,,$ (182)

and by applying (182) to the coherent states $|\alpha\rangle$ we see

$\hat{a}|\alpha\rangle=\alpha|\alpha\rangle,$ (183)

which is another equivalent definition of the coherent states.

The resolution of unity in terms of $|\alpha\rangle$ is

$\int_{\mathbb{C}}d\mu(\alpha)\;|\alpha\rangle\langle\alpha|=\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}},\qquad\qquad
d\mu(\alpha)=\frac{1}{\pi}\,d\alpha\,,$ (184)

and, as a consequence, we can expand an arbitrary state using (184):

$|\Psi\rangle=\int_{\mathbb{C}}d\mu(\alpha)\;\Psi(\alpha^{\ast})|\alpha\rangle,\qquad\qquad\Psi(\alpha^{\ast})=\langle\alpha|\Psi\rangle\,.$
(185)

The function $\Psi(\alpha^{\ast})$ is the coherent-state wave function of the
state $|\Psi\rangle$.

The coherent states saturate the Heisenberg uncertainty relation

$\mathrm{Var}(_{)}{\alpha}{\hat{x}}\;\mathrm{Var}(_{)}{\alpha}{\hat{p}}\geq
1/2$ (186)

where the canonical position-momentum operators are implicitly defined by
$\hat{a}=(\hat{x}+i\hat{p})/\sqrt{2}$. For this reason, coherent states are
often considered the _most classical_ states of the harmonic oscillator.

Next, we consider the SU(2) symmetry, whose Lie algebra $\mathfrak{su}(2)$ is
spanned by the operators $\\{\hat{S}_{1},\hat{S}_{2},\hat{S}_{3}\\}$, with
commutation relations

$[\hat{S}_{k},\hat{S}_{\ell}]=i\epsilon_{klm}\hat{S}_{m}\,.$ (187)

The irreps are labeled by the index $S$ $(S=0,\tfrac{1}{2},1,\dots)$ and the
carrier space $\mathcal{H}_{S}$ is spanned by the standard basis
$\\{|S,m\rangle\mid m=S,\dots,-S)\\}$. The highest weight state is
$|S,S\rangle$ and is annihilated by the ladder operator $\hat{S}_{+}$. The
isotropy subgroup for this state consists of all the elements of the form
$\exp(i\psi\hat{S}_{3})$, so it is isomorphic to U(1). The coset space is then
SU(2)/U(1), which is the unit sphere $\mathcal{S}_{2}$, and it is exactly the
same as the classical phase space, the natural arena to describe the dynamics.

Since the irreps can be written in terms of the Euler angles as

$\hat{T}^{(S)}(\phi,\theta,\psi)=\exp(-i\phi\hat{S}_{3})\exp(-i\theta\hat{S}_{2})\exp(-i\psi\hat{S}_{3})\,,$
(188)

it is clear that the elements of SU(2)/U(1) can be represented as in (113);
i.e.,

$\hat{D}(\mathbf{n})=e^{-i\phi\hat{S}_{3}}e^{-i\theta\hat{S}_{2}}=\exp\left[\tfrac{1}{2}\theta(\hat{S}_{+}e^{-i\phi}-\hat{S}_{-}e^{i\phi})\right]\,,$
(189)

so they depend on the spherical angles $(\theta,\phi)$ determining the unit
vector $\mathbf{n}$. The system of coherent states is thus

$|S,\mathbf{n}\rangle=\hat{D}(\mathbf{n})|S,S\rangle\,.$ (190)

One can write the operator $\hat{D}(\mathbf{n})$ in a disentangled form to
obtain another useful parametrization of the SU(2) coherent states:

$|S,\zeta\rangle=(1+|\zeta|^{2})^{-S}\exp(\zeta\hat{S}_{+})|S,-S\rangle\,,$
(191)

with $\zeta=\tan(\theta/2)e^{i\phi}$ being the stereographic projection from
the north pole of the sphere $\mathcal{S}^{2}$ onto the complex plane
$\mathbb{C}$, tangent to the south pole.

Expanding the exponential, we obtain the expression for the SU(2) coherent
states in terms of the angular momentum basis $|S,m\rangle$

$|S,\mathbf{n}\rangle=\sum_{m=-S}^{S}\binom{2S}{S+m}^{1/2}[\sin(\theta/2)]^{S-m}\,[\cos(\theta/2)]^{S+m}\,\exp[-i(S+m)\phi]|S,m\rangle\,.$
(192)

SU(2) coherent states are not orthogonal; their overlap is

$\lvert\langle\mathbf{n}_{1}|\mathbf{n}_{2}\rangle\rvert^{2}=\left[\tfrac{1}{2}(1+\mathbf{n}_{1}\cdot\mathbf{n}_{2})\right]^{2S}\,,$
(193)

which has been used in (84).

The resolution of the identity now reads

$\int_{\mathcal{S}_{2}}d\mu(\mathbf{n})\,|S,\mathbf{n}\rangle\langle
S,\mathbf{n}|=\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}},\qquad\qquad
d\mu(\mathbf{n})=\frac{2S+1}{4\pi}\sin\theta d\theta d\phi\,,$ (194)

which allows for an expansion

$|\Psi\rangle=\int d\mu(\mathbf{n})\;\Psi(\mathbf{n})|S,\mathbf{n}\rangle\,,$
(195)

where $\Psi(\mathbf{n})$ is the coherent-state wave function.

Finally, as discussed in Section 3.2, the SU(2) coherent states minimize the
Heisenberg uncertainty relation.

##  Appendix B Quasiprobability distributions on the sphere

Presenting quantum mechanics as a statistical theory on a classical phase
space has attracted a great deal of attention since the very early days of
this discipline.

The main ingredient for any successful phase-space method is a _bona fide_
mapping that relates operators with functions defined on a smooth manifold
$\mathcal{M}$, the phase space of the system, endowed with a very precise
mathematical structure [393]. This mapping, first suggested by Weyl [394], is
not unique. In fact, a whole family of $s$-parametrized functions can be
assigned to each operator and the choice of a particular element of the family
depends on its convenience for each problem. In particular, the time-honored
quasiprobability distributions are the functions connected with the density
operator. The most common choices of $s$ are $+1$, 0, and $-1$, which
correspond to the $P$ (Glauber-Sudarshan) [395, 396], $W$ (Wigner) [397], and
$Q$ (Husimi) [180] functions, respectively.

For the relevant case of the SU(2) dynamical symmetry, the $s$-parametrized
Weyl-Stratanovich map

$\hat{A}\mapsto
W_{A}^{(s)}(\mathbf{n})=\mathop{\mathrm{Tr}}\nolimits[\hat{A}\,\hat{w}^{(s)}(\mathbf{n})],$
(196)

puts in one-to-one correspondence each operator $\hat{A}$ invariantly acting
on $\mathcal{H}_{S}$ with a function on the sphere $\mathcal{S}_{2}$. The
corresponding kernels $\hat{w}^{(s)}$ are defined as [169, 398, 290, 179]

$\hat{w}^{(s)}(\mathbf{n})=\sqrt{\frac{4\pi}{2S+1}}\sum_{K=0}^{2S}\sum_{q=-K}^{K}(C_{SS,K0}^{SS})^{-s}\;Y_{Kq}^{\ast}(\mathbf{n})\,\hat{T}_{Kq}^{S}\,,$
(197)

where $Y_{Kq}(\mathbf{n})$ are the spherical harmonics,
$C_{S_{1}m_{1},S_{2}m_{2}}^{Sm}$ the Clebsch-Gordan coefficients and
$\hat{T}_{Kq}^{S}$ the irreducible tensor operators

$\hat{T}_{Kq}^{S}=\sqrt{\frac{2K+1}{2S+1}}\sum_{m,m^{\prime}=-S}^{S}C_{Sm,Kq}^{Sm^{\prime}}\,|S,m^{\prime}\rangle\langle
S,m|\,.$ (198)

As expected, the kernels are properly normalized

$\mathop{\mathrm{Tr}}\nolimits[\hat{w}^{(s)}(\mathbf{n})]=1\,,\qquad\frac{2S+1}{4\pi}\int_{\mathcal{S}_{2}}d\mathbf{n}\;\hat{w}^{(s)}(\mathbf{n})=\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}\,,$ (199)

with $d\mathbf{n}=\sin\theta\ d\theta\ d\phi$ the invariant measure on the
sphere.

Consequently, the symbol of $\hat{A}$ can be concisely expressed as

$W_{A}^{(s)}(\mathbf{n})=\sqrt{\frac{4\pi}{2S+1}}\sum_{K=0}^{2S}\sum_{q=-K}^{K}(C_{SS,K0}^{SS})^{-s}A_{Kq}\;Y_{LM}^{\ast}(\mathbf{n})\,,$
(200)

where $A_{Kq}=\mathop{\mathrm{Tr}}\nolimits(\hat{A}\hat{T}_{Kq}^{S\dagger})$.

The traditional SU(2) quasiprobability distributions are simply the
$s$-symbols of the density operator $\hat{\varrho}$. The value $s=0$
corresponds to the standard Wigner function, while $s=\pm 1$ leads to the $P$
and $Q$-functions respectively, defined as dual coefficients in the basis of
spin coherent states $|S,\mathbf{n}\rangle$ [116], according to

$Q(\mathbf{n})=\langle\mathbf{n}|\hat{\varrho}|\mathbf{n}\rangle\,,\qquad\qquad\hat{\varrho}=\frac{2S+1}{4\pi}\int_{\mathcal{S}_{2}}d\mathbf{n}\;P(\mathbf{n})\;|\mathbf{n}\rangle\langle\mathbf{n}|\,.$
(201)

The symbols $W_{A}^{(s)}(\mathbf{n})$ are covariant under SU(2)
transformations and provide the overlap relation

$\mathop{\mathrm{Tr}}\nolimits(\hat{\varrho}\hat{A})=\frac{2S+1}{4\pi}\int_{\mathcal{S}_{2}}d\mathbf{n}\,W_{\varrho}^{(s)}(\mathbf{n})\,W_{A}^{(-s)}(\mathbf{n})\,.$
(202)

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