#  Optimal Detection of Rotations about Unknown Axes by Coherent and
Anticoherent States

John Martin  Institut de Physique Nucléaire, Atomique et de Spectroscopie,
CESAM, University of Liège, B-4000 Liège, Belgium<EMAIL_ADDRESS>Stefan
Weigert  Department of Mathematics, University of York, UK-York YO10 5DD,
United Kingdom<EMAIL_ADDRESS>Olivier Giraud  Université Paris-
Saclay, CNRS, LPTMS, 91405 Orsay, France  olivier.giraud@universite-paris-
saclay.fr

###### Abstract

Coherent and anticoherent states of spin systems up to spin $j=2$ are known to
be optimal in order to detect rotations by a known angle but unknown rotation
axis. These optimal quantum rotosensors are characterized by minimal fidelity,
given by the overlap of a state before and after a rotation, averaged over all
directions in space. We calculate a closed-form expression for the average
fidelity in terms of anticoherent measures, valid for arbitrary values of the
quantum number $j$. We identify optimal rotosensors (i) for arbitrary rotation
angles in the case of spin quantum numbers up to $j=7/2$ and (ii) for small
rotation angles in the case of spin quantum numbers up to $j=5$. The closed-
form expression we derive allows us to explain the central role of
anticoherence measures in the problem of optimal detection of rotation angles
for arbitrary values of $j$.

##  1 Introduction and main result

Historically, advances in measurement techniques often are the reason for
physics to progress. Over time,_metrology_ has developed as a subject of its
own, especially in the context of defining standard units of measurement for
physical quantities.

Quantum theory provides new perspectives on measurements, ranging from
fundamental limitations on measurements [1], new opportunities [2] as well as
technical challenges and even philosophical quagmires [3]. From a practical
point of view, quantum information science requires ever better control of
microscopic systems and, hence, measurements which are as accurate as
possible. More specifically, quantum metrology [4] aims at finding bounds on
the achievable measurement precision and at identifying states which would be
optimal for quantum measurements or other specific tasks. The optimal
transmission of a Cartesian frame [5] or the efficient detection of
inhomogeneous magnetic fields [6] are typical examples. While the classical
Cramér-Rao theorem [7, 8] provides a lower bound on the variance of random
estimators by means of the Fisher information, its quantum-mechanical
counterpart provides bounds for quantum parameter estimation theory [9]. The
quantum Cramér-Rao bound is expressed as the inverse of the quantum Fisher
information, which can be geometrically interpreted as the (Bures) distance
between two quantum states differing by an infinitesimal amount in their
parameter [10, 11]. It provides lower bounds on the variance of any quantum
operator whose measurement aims at estimating the parameter. Optimal
measurement is achieved by maximizing the quantum Fisher information over
parameter-dependent states.

The quantum Cramér-Rao bound was calculated for instance in the reference
frame alignment problem [12]. This problem involves estimating rotations about
unknown axes. It has been shown in [13] that spin states with vanishing spin
expectation value and isotropic variances of the spin components are valuable
for estimating such rotations, as they saturate the quantum Cramér-Rao bound
for _any_ axis. Also, recently, the problem of characterizing a rotation about
an unknown direction encoded into a spin-$j$ state has been considered in
[14].

In this paper, we are interested to determine whether a quantum system has
undergone a rotation $R_{\mathbf{n}}(\eta)$ by a _known_ angle $\eta$ about an
_unknown_ axis $\mathbf{n}$. Suppose first that we apply the rotation by
$\eta$ to an initial state $|\psi\rangle$ about a _known_ axis and perform a
measurement of the projector $|\psi\rangle\langle\psi|$ in the rotated state
$R_{\mathbf{n}}(\eta)|\psi\rangle$. The expectation value of the observable
$|\psi\rangle\langle\psi|$ is given by

$F_{|\psi\rangle}(\eta,\mathbf{n})=|\langle\psi|R_{\mathbf{n}}(\eta)|\psi\rangle|^{2}\,,$
(1)

i.e. by the fidelity between the initial state and the final state. The
fidelity $F_{|\psi\rangle}(\eta,\mathbf{n})$ equals the probability to find
the quantum system in the initial state after the rotation. Thus, the
probability to detect that the rotation has occurred is given by the quantity
$1-F_{|\psi\rangle}(\eta,\mathbf{n})$. Therefore, the measurement will be most
sensitive if the rotation is applied to states $|\psi\rangle$ which _minimize_
the expression (1) for given angle and rotation axis.

Next, suppose that only the rotation angle $\eta$ is well-defined while the
rotation axis is not known, as described in [15]. This situation occurs, for
example, when spins prepared in the state $|\psi\rangle$ are—during the
measurement sequence—subjected to a magnetic field whose direction randomly
fluctuates on a time scale much larger than the Larmor period. Measuring the
observable $|\psi\rangle\langle\psi|$ on an ensemble of identically prepared
systems will now produce a value of the fidelity (1) averaged over all
possible spatial directions $\mathbf{n}$. Then, the most suitable quantum
states $|\psi\rangle$—called _optimal_ _quantum rotosensors_ in [15]—are
determined by the requirement that the _average fidelity_

${\cal
F}_{|\psi\rangle}(\eta)=\frac{1}{4\pi}\int_{\mathcal{S}^{2}}F_{|\psi\rangle}(\eta,\mathbf{n})\,d\mathbf{n}\,,$
(2)

achieve its minimum, for a given value of the parameter $\eta$.

The fidelity (1) and its average (2) also play a role when setting up
experiments which aim to determine an unknown rotation angle as accurately as
possible. This is explained in more detail in Appendix A.

For the spin values $j=1/2,1,3/2,2$, optimal quantum rotosensors have been
identified [15], using an approach which combines analytical and numerical
methods. For rotation angles $\eta$ close to $\pi$, the average fidelity is
minimized systematically by _coherent_ spin states. Coherent spin states are
strongly localized in phase space and entirely specified by a spatial
direction into which they point on the Bloch sphere [16]. For small rotation
angles $\eta$, the average fidelity is minimized by _anticoherent_ states,
which are characterized by the fact that they do not manifest any privileged
direction; in this respect, they are as distinct as possible from coherent
states [17]. The role of anticoherent states for optimal detection of
rotations has also been observed and was subsequently quantified in terms of
quantum Fisher information in [13]. Between these two extreme cases of
$\eta\sim 0$ and $\eta\sim\pi$, optimal states are neither coherent nor
anticoherent in general. From an experimental point of view, anticoherent and
other non-classical spin states have been created using a variety of physical
systems. For instance, anticoherent states of quantum light fields have been
generated using orbital angular momentum states of single photons with their
usefulness for quantum metrology being established in [18]. Non-classical spin
states—including Schrödinger cat states (c.f. Sec. 4)—of highly magnetic
dysprosium atoms with spin quantum number $j=8$ have been created in order to
enhance the precision of a magnetometer [19].

The main result of the present paper is a closed-form expression of the
average fidelity ${\cal F}_{|\psi\rangle}(\eta)$, valid for arbitrary values
of $j$. A rather general argument, based solely on the symmetries of the
average fidelity ${\cal F}_{|\psi\rangle}(\eta)$, shows that it must be a
linear combination of the form

${\cal F}_{|\psi\rangle}(\eta)=\varphi_{0}^{(j)}(\eta)+\sum_{t=1}^{\lfloor
j\rfloor}\varphi_{t}^{(j)}(\eta)\,\mathcal{A}_{t}(|\psi\rangle),$ (3)

as explained in detail in Sec. 2. In this expression, the ${\cal
A}_{t}(|\psi\rangle)$ are the anticoherence measures of a state
$|\psi\rangle$, introduced in [20] and given explicitly in Eq. (10), while the
real-valued functions $\varphi_{t}^{(j)}(\eta)$ are trigonometric polynomials
independent of $|\psi\rangle$, and $\lfloor j\rfloor$ is the largest integer
smaller than or equal to $j$. The main challenge is to calculate the
$\eta$-dependent coefficients $\varphi_{t}^{(j)}(\eta)$, which we do in Sec.
3.

In earlier works, the average fidelity ${\cal F}_{|\psi\rangle}(\eta)$ had
been expressed as a sum of functions of $\eta$ weighted by _state-dependent_
coefficients, upon representing the state in the polarization-tensor basis
[15]. The advantage of relation (3) is that the average fidelity depends on
the state under consideration only through its measures of anticoherence, and
thus it directly relates to the degree of coherence or anticoherence of the
state. Expression (3) allows us to identify optimal quantum rotosensors for
spin quantum numbers up to $j=5,$ thereby confirming the role played by
coherent and anticoherent states beyond $j=2$. Readers mainly interested in
the optimal quantum rotosensors may want to directly consult Sec. 4.

Let us outline the overall argument leading to the expression of the average
fidelity ${\cal F}_{|\psi\rangle}(\eta)$ in (3). In Sec. 2, we introduce a
number of tools and concepts feeding into the derivation of (3): first, we
discuss the symmetries built into the average fidelity ${\cal
F}_{|\psi\rangle}(\eta)$, followed by a brief summary of the Majorana
representation which enables us to interpret spin-$j$ states as completely
symmetric states of $N=2j$ qubits. This perspective allows us to introduce,
for $1\leqslant t\leqslant\lfloor j\rfloor$, the anticoherence measure
$\mathcal{A}_{t}(|\psi\rangle)$, defined as the linear entropy of the
$t$-qubit reduced density matrix of $|\psi\rangle\langle\psi|$. To actually
carry out the integration in Eq. (2), we will use a tensor representation (see
Sec. 2.5) of mixed spin-$j$ states generalizing the Bloch representation. In
addition, this representation also enables us to exploit the symmetries of the
average fidelity which can only depend on expressions invariant under
$\mathrm{SU}(2)$ rotations. As shown in Sec. 2.6, it is then possible to
establish a linear relation between these invariants and the anticoherence
measures $\mathcal{A}_{t}(|\psi\rangle)$, which finally leads to (3).

Section 3 is dedicated to deriving explicit expressions for the functions
$\varphi_{t}^{(j)}(\eta)$. This will be done in two ways: the first one is
based on the fact that anticoherence measures are explicitly known for certain
states, so that the functions $\varphi_{t}^{(j)}(\eta)$ appear as solutions of
a linear system of equations. The second approach makes use of representations
of the Lorentz group and allows us to obtain a general closed expression. In
Sec. 4 we make use of this closed-form expression to identify the optimal
quantum rotosensors. We conclude with a brief summary given in Sec. 5.

##  2 Concepts and tools

In this section, we introduce the tools that will be needed to address the
optimality problem described in the Introduction.

###  2.1 Notation

Quantum systems with integer or half-integer spin $j$ are described by states
$|\psi\rangle$ of the Hilbert space $\mathbb{C}^{N+1}$ with $N=2j$, which
carries a $(N+1)$-dimensional representation of the group SU$(2)$. The
components of the angular momentum operator ${\bf J}$ satisfy
$[J_{k},J_{\ell}]=i\varepsilon_{k\ell m}J_{m}$, $k,\ell,m\in\\{x,y,z\\}$,
where $\varepsilon_{k\ell m}$ is the Levi-Civita symbol. Denoting unit vectors
in $\mathbb{R}^{3}$ by

$\mathbf{n}=\begin{pmatrix}\sin\theta\cos\phi\\\ \sin\theta\sin\phi\\\
\cos\theta\end{pmatrix}\,,\quad\theta\in[0,\pi]\,,\quad\phi\in[0,2\pi[\,,$ (4)

the operator

$R_{\mathbf{n}}(\eta)=e^{-i\eta\mathbf{J}\boldsymbol{\cdot}\mathbf{n}}$ (5)

describes a rotation by an angle $\eta\in[0,4\pi[$ about the direction
$\mathbf{n}$.

###  2.2 Symmetries

By definition, the average fidelity in (2) is a positive function of the angle
$\eta$ and of the state $|\psi\rangle$ and possesses three symmetries: it is
$2\pi$-periodic in $\eta$, symmetric about $\eta=\pi$, and invariant under
rotation of $|\psi\rangle$.

Periodicity with period $2\pi$ comes from the fact that
$R_{\mathbf{n}}(2\pi)=(-1)^{N}$. Symmetry about $\eta=\pi$ is equivalent to

${\cal F}_{|\psi\rangle}(\eta)={\cal F}_{|\psi\rangle}(2\pi-\eta)\,,$ (6)

which can be shown using
$R_{\mathbf{n}}(2\pi-\eta)=(-1)^{N}R_{-\mathbf{n}}(\eta)$ and the fact that
the set of directions averaged over in (2) is the same irrespective of the
sign of the unit vector ${\bf n}$ since the fidelity (1) is given by the the
_squared_ modulus of the overlap between the states $|\psi\rangle$ and
$R_{\mathbf{n}}(\eta)|\psi\rangle$.

Invariance under rotation of $|\psi\rangle$ can be understood in the following
way. Let
$R_{\mathbf{m}}(\chi)=e^{-i\chi\mathbf{J}\boldsymbol{\cdot}\mathbf{m}}$ be a
unitary operator representing a rotation in $\mathbb{R}^{3}$ by an angle
$\chi\in[0,4\pi[$ about the direction $\mathbf{m}$, acting on a state
$|\psi\rangle\in\mathbb{C}^{N+1}$. Then the average fidelities ${\cal F}$
associated with the states $|\psi\rangle$ and $|\psi^{R}\rangle\equiv
R_{\mathbf{m}}(\chi)|\psi\rangle$ are equal. Indeed, we have

$F_{|\psi^{R}\rangle}(\eta,\mathbf{n})=\langle\psi|R_{\mathbf{m}}(\chi)^{\dagger}R_{\mathbf{n}}(\eta)R_{\mathbf{m}}(\chi)|\psi\rangle$
(7)

and

$\displaystyle
R_{\mathbf{m}}(\chi)^{\dagger}R_{\mathbf{n}}(\eta)R_{\mathbf{m}}(\chi)$
$\displaystyle=e^{-i\eta(R_{\mathbf{m}}(\chi)^{\dagger}\mathbf{J}R_{\mathbf{m}}(\chi))\boldsymbol{\cdot}\mathbf{n}}$
$\displaystyle=e^{-i\eta(R\mathbf{J})\boldsymbol{\cdot}\mathbf{n}}=e^{-i\eta\mathbf{J}\boldsymbol{\cdot}\mathbf{n}^{R}}\,,$
(8)

with ${\bf n}^{R}\equiv R^{T}\mathbf{n}$ the vector obtained by the rotation
$R\in$ SO$(3)$ associated with $R_{\mathbf{m}}(\chi)$. Due to the invariance
under rotations of the unit-ball region $\mathcal{S}^{2}$ appearing in (2)
(invariance of the Haar measure used), the result of the integration will be
the same, leading to

$\displaystyle{\cal F}_{|\psi^{R}\rangle}(\eta)$
$\displaystyle=\frac{1}{4\pi}\int_{\mathcal{S}^{2}}F_{|\psi^{R}\rangle}(\eta,\mathbf{n})\,d\mathbf{n}$
$\displaystyle=\frac{1}{4\pi}\int_{\mathcal{S}^{2}}F_{|\psi\rangle}(\eta,\mathbf{n})\,d\mathbf{n}={\cal
F}_{|\psi\rangle}(\eta)\,.$ (9)

This invariance of the fidelity can be seen in a geometrically appealing way
by use of the Majorana representation, which we consider now.

###  2.3 Majorana representation of pure spin states

The Majorana representation establishes a one-to-one correspondence between
spin-$j$ states and $N=2j$-qubit states that are invariant under permutation
of their constituent qubits (see e.g. [21, 22, 23]). It allows to
geometrically visualise a pure spin-$j$ state as $N$ points on the unit sphere
associated with the Bloch vectors of the $N$ qubits. The Majorana points are
often referred to as stars, and the whole set of Majorana points of a given
state as its Majorana constellation. Considering a spin-$j$ state
$|\psi\rangle$ as an $N$-qubit state, any local unitary (LU) operation
$U=u^{\otimes N}$ with $u\in\mathrm{SU}(2)$ transforms $|\psi\rangle$ into a
state whose Majorana constellation is obtained by the constellation of
$|\psi\rangle$ rotated by the SO$(3)$ rotation associated with $u$. Spin-
coherent states take a very simple form in the Majorana representation, as
they can be seen as the tensor product $|\phi\rangle^{\otimes N}$ of some
spin-$1/2$ state $|\phi\rangle$. Their constellation thus reduces to an
$N$-fold degenerate point.

The fidelity (1) is given by the squared modulus of the overlap between
$|\psi\rangle$ and $R_{\mathbf{n}}(\eta)|\psi\rangle$. Since the Majorana
constellation of $R_{\mathbf{n}}(\eta)|\psi\rangle$ is obtained by rigidly
rotating that of $|\psi\rangle$, the fidelity (1) only depends on the relative
positions of these two sets of points. The _average_ transition probability
${\cal F}_{|\psi\rangle}(\eta)$ is obtained by integrating over all possible
constellations obtained by rigid rotations of the Majorana constellation of
$|\psi\rangle$, and therefore it must be invariant under LU. In other words,
the equality (9) takes the form ${\cal F}_{|\psi\rangle}(\eta)={\cal
F}_{u^{\otimes N}|\psi\rangle}(\eta)$.

###  2.4 Anticoherence measures

An order-$t$ _anticoherent_ state $|\chi\rangle$ is defined by the property
that $\langle\chi|(\mathbf{J}\boldsymbol{\cdot}\mathbf{n})^{k}|\chi\rangle$ is
independent of the vector $\mathbf{n}$ for all $k=1,\ldots,t$. In the Majorana
representation, it is characterized by the fact that its $t$-qubit reduced
density matrix is the maximally mixed state in the symmetric sector [24].

The degree of coherence or $t$-anticoherence of a spin-$j$ pure state
$|\psi\rangle$ can be measured by the quantities
$\mathcal{A}_{t}(|\psi\rangle)$, which are positive-valued functions of
$|\psi\rangle$ [20]. Let $\rho_{t}=\mathrm{tr}_{\neg
t}\left[|\psi\rangle\langle\psi|\right]$ be the $t$-qubit reduced density
matrix of the state $|\psi\rangle$ interpreted as a $N$-qubit symmetric state
with $N=2j$; it is obtained by taking the partial trace over all but $t$
qubits (it does not matter which qubits are traced over since $|\psi\rangle$
is a symmetric state). The measures $\mathcal{A}_{t}(|\psi\rangle)$ are
defined as the rescaled linear entropies

$\mathcal{A}_{t}(|\psi\rangle)=\frac{t+1}{t}\left(1-\mathrm{tr}\left[\rho_{t}^{2}\right]\right)\,,$
(10)

where $\mathrm{tr}\left[\rho_{t}^{2}\right]$ is the purity of $\rho_{t}$.
Thus, anticoherence measures are quartic in the state $|\psi\rangle$ and range
from $0$ to $1$, and are invariant under SU$(2)$ rotations. Spin-coherent
states are characterized by pure reduced states and thus are the only states
such that $\mathcal{A}_{t}=0$. Anticoherent states to order $t$ are
characterized by $\rho_{t}=\mathbb{1}/(t+1)$ and thus are the only states such
that $\mathcal{A}_{t}=1$. In particular, if a state $|\psi\rangle$ is
anticoherent to some order $t$, then it is necessarily anticoherent to all
lower orders $t^{\prime}=1,\ldots,t$ since reductions of the maximally mixed
state are maximally mixed.

While for any state we have $0\leqslant\mathcal{A}_{t}\leqslant 1$, not all
possible tuples $(\mathcal{A}_{1},\mathcal{A}_{2},\ldots)$ are realised by a
physical state $|\psi\rangle$. For instance, since $\mathcal{A}_{t}=1$ implies
that $\mathcal{A}_{t^{\prime}}=1$ for all $t^{\prime}\leqslant t$, the choice
$\mathcal{A}_{2}=1$ and $\mathcal{A}_{1}<1$ cannot correspond to any state. We
denote the domain of admissible values of the measures $\mathcal{A}_{t}$ by
$\Omega$.

###  2.5 Tensor representation of mixed states

We now introduce a tensor representation of an arbitrary (possibly mixed)
spin-$j$ state $\rho$ acting on a $(N+1)$-dimensional Hilbert space with
$N=2j$, following [24]. Any state can be expanded as

$\rho=\frac{1}{2^{N}}\,x_{\mu_{1}\mu_{2}\ldots\mu_{N}}S_{\mu_{1}\mu_{2}\ldots\mu_{N}}.$
(11)

Here and in what follows, we use Einstein summation convention for repeated
indices, with Greek indices running from $0$ to $3$ and Latin indices running
from $1$ to $3$. Here, the $S_{\mu_{1}\mu_{2}\ldots\mu_{N}}$ are
$(N+1)\times(N+1)$ Hermitian matrices invariant under permutation of the
indices.

The $x_{\mu_{1}\mu_{2}\ldots\mu_{N}}$ are real coefficients also invariant
under permutation of their indices, which enjoy what we call the tracelessness
property

$\sum_{a=1}^{3}x_{aa\mu_{3}\ldots\mu_{N}}=x_{00\mu_{3}\ldots\mu_{N}}\,,\quad\forall\;\mu_{3},\ldots,\mu_{N}.$
(12)

Whenever $x_{\mu_{1}\mu_{2}\ldots\mu_{N}}$ has some indices equal to $0$, we
take the liberty to omit them, so that e.g. for a spin-$3$ state $x_{110200}$
may be written $x_{112}$ (recall that the order of the indices does not
matter). In the case of a spin-coherent state given by its unit Bloch vector
$\mathbf{n}=(n_{1},n_{2},n_{3})$, the coefficients in (11) are simply given by
$x_{\mu_{1}\mu_{2}\ldots\mu_{N}}=n_{\mu_{1}}n_{\mu_{2}}\ldots n_{\mu_{N}}$,
with $n_{0}=1$.

In the following, we will make use of two essential properties of the tensor
representation. Namely, let us consider a state $\rho$ with coordinates
$x_{\mu_{1}\mu_{2}\ldots\mu_{N}}$ in the expansion (11). Then, the tensor
coordinates of the $t$-qubit reduced state $\rho_{t}$ in the expansion (11)
are simply given by
$x_{\mu_{1}\mu_{2}\ldots\mu_{t}}=x_{\mu_{1}\mu_{2}\ldots\mu_{t}0...0}$. Thus,
since we omit the zeros in the string $\mu_{1}\mu_{2}\ldots\mu_{N}$, the
tensor coordinates of $\rho_{t}$ and $\rho$ coincide for any string of
$k\leqslant t$ nonzero indices.

The second property we use is that for states $\rho$ and $\rho^{\prime}$ in
the form (11) with tensor coordinates respectively
$x_{\mu_{1}\mu_{2}\ldots\mu_{N}}$ and
$x^{\prime}_{\mu_{1}\mu_{2}\ldots\mu_{N}}$ we have

$\mathrm{tr}\left[\rho\rho^{\prime}\right]=\frac{1}{2^{N}}\sum_{\mu_{1},\mu_{2},...,\mu_{N}}x_{\mu_{1}\mu_{2}\ldots\mu_{N}}x^{\prime}_{\mu_{1}\mu_{2}\ldots\mu_{N}}.$
(13)

Note that this equality holds despite the fact that the
$S_{\mu_{1}\mu_{2}\ldots\mu_{N}}$ are not orthogonal; this property follows
from the fact that these matrices form a $2^{N}$-tight frame, see [24]. In
particular, for a pure state $\rho=|\psi\rangle\langle\psi|$, the equality
$\mathrm{tr}\rho^{2}=1$ translates into

$\sum_{\mu_{1},\mu_{2},...,\mu_{N}}x_{\mu_{1}\mu_{2}\ldots\mu_{N}}^{2}=2^{N},$
(14)

while the purity of the reduced density matrix $\rho_{t}$ reads

$\mathrm{tr}\left[\rho_{t}^{2}\right]=\frac{1}{2^{t}}\sum_{\mu_{1},\mu_{2},...,\mu_{t}}x_{\mu_{1}\mu_{2}\ldots\mu_{t}}^{2}\,.$
(15)

The normalization condition $\mathrm{tr}\left[\rho\right]=1$ imposes
$x_{00\ldots 0}=1$. A consequence of (12) is then that
$\sum_{a=1}^{3}x_{aa}=1$.

###  2.6 SU$(2)$-Invariants

If $u\in$ SU(2) and $R\in$ SO(3) is the corresponding rotation matrix, then
the tensor coordinates of $U\rho U^{\dagger}$ with $U=u^{\otimes N}$ are the
$\mathsf{R}_{\mu_{1}\nu_{1}}\ldots\mathsf{R}_{\mu_{N}\nu_{N}}x_{\nu_{1}\ldots\nu_{N}}$
where $\mathsf{R}$ is the $4\times 4$ orthogonal matrix

$\mathsf{R}=\left(\begin{array}[]{c|c}\begin{array}[]{c}1\end{array}&\begin{array}[]{c}0\end{array}\\\
\hline\cr\begin{array}[]{c}0\end{array}&\begin{array}[]{c}R\end{array}\end{array}\right).$
(16)

That is, $x_{\mu_{1}\mu_{2}\ldots\mu_{N}}$ transforms as a tensor. Under such
transformations, $x_{\mu}x_{\mu}$ goes into
$\mathsf{R}_{\mu\nu}\mathsf{R}_{\mu\nu^{\prime}}x_{\nu}x_{\nu^{\prime}}=(\mathsf{R}^{T}\mathsf{R})_{\nu^{\prime}\nu}x_{\nu}x_{\nu^{\prime}}=x_{\nu}x_{\nu}$,
where the last equality comes from orthogonality of $\mathsf{R}$. Thus
$x_{\mu}x_{\mu}$ is an SU(2) invariant. Similarly, $x_{\mu}x_{\mu\nu}x_{\nu}$
and, more generally, any product of the $x_{\mu_{1}\mu_{2}\ldots\mu_{N}}$ such
that all indices are contracted (i.e. summed from $0$ to $3$), are invariant
under SU(2) action on $\rho$. One can then show by induction that products of
terms $x_{a_{1}a_{2}\ldots a_{k}}$ with $k\leqslant N$ where all indices
appear in pairs and are summed from $1$ to $3$ are also SU(2) invariant. For
instance, $x_{a}x_{a}$, $x_{ab}x_{ab}$, $x_{ab}x_{bc}x_{ca}$,
$x_{a}x_{ab}x_{b}$ are such invariants.

Invariants of degree 1 in $x$ are of the form $x_{a_{1}a_{2}\ldots a_{2k}}$,
where the $a_{i}$ appear in pairs. Since the order of indices is not relevant,
these invariants are in fact of the form $x_{a_{1}a_{1}a_{2}a_{2}\ldots
a_{k}a_{k}}$. Because of Eq. (12), each pair can be replaced by zeros in the
string, so that $x_{a_{1}a_{1}a_{2}a_{2}\ldots a_{k}a_{k}}=x_{00\ldots 0}=1$.
Therefore, there is no invariant of degree 1. The invariants of degree 2 are
products of the form $x_{a_{1}a_{2}\ldots a_{k}}x_{b_{1}b_{2}\ldots
b_{k^{\prime}}}$ where indices appear in pairs and are summed from $1$ to $3$.
If the two indices of a pair appear in the same index string
($a_{1}a_{2}\ldots a_{k}$ or $b_{1}b_{2}\ldots b_{k^{\prime}}$), then from Eq.
(12), they can again be replaced by zeros and discarded. Thus the invariants
of degree 2 are $\kappa_{1}=x_{a}x_{a}$, $\kappa_{2}=x_{ab}x_{ab}$, and more
generally, for $1\leqslant r\leqslant N$,

$\kappa_{r}=x_{a_{1}a_{2}...a_{r}}x_{a_{1}a_{2}...a_{r}}.$ (17)

Using (10) and (15) one can express the invariants $\kappa_{r}$ in terms of a
linear combination of the $\mathcal{A}_{t}$. Indeed, grouping together terms
with the same number of nonzero indices in (15) yields

$\mathrm{tr}\left[\rho_{t}^{2}\right]=\frac{1}{2^{t}}\sum_{\mu_{1},\mu_{2},...,\mu_{t}}x_{\mu_{1}\mu_{2}\ldots\mu_{t}}^{2}=\frac{1}{2^{t}}\sum_{r=0}^{t}\binom{t}{r}\kappa_{r}\,.$
(18)

Inverting that relation via the binomial inversion formula, we obtain

$\kappa_{r}=\sum_{t=0}^{r}(-1)^{t+r}\,2^{t}\binom{r}{t}\mathrm{tr}\left[\rho_{t}^{2}\right]\,,$
(19)

and by use of (10) we finally can express the $\mathrm{SU}(2)$-invariants in
terms of anticoherence measures,

$\kappa_{r}=\sum_{t=0}^{r}(-1)^{t+r}\,2^{t}\binom{r}{t}\left(1-\frac{t}{t+1}\mathcal{A}_{t}\right)$
(20)

for $r=1,\ldots,N$.

###  2.7 General form of the average fidelity

Let us now explain why the average fidelity ${\cal F}_{|\psi\rangle}(\eta)$
given in Eq. (3) is a linear combination of the lowest $\left\lfloor
j\right\rfloor$ anticoherent measures $\mathcal{A}_{t}$. Due to its rotational
symmetry, the average fidelity ${\cal F}_{|\psi\rangle}(\eta)$—when considered
as a function of the tensor coordinates $x_{\mu_{1}\mu_{2}\ldots\mu_{N}}$—can
only involve invariants constructed from these coordinates. With ${\cal
F}_{|\psi\rangle}(\eta)$ being quadratic in $\rho=|\psi\rangle\langle\psi|$,
it must also be quadratic in $x$. As there is no invariant of degree 1, the
only invariants that can appear in the expression of ${\cal
F}_{|\psi\rangle}(\eta)$ are the invariants $\kappa_{r}$ defined in (17).
Since the quantity ${\cal F}_{|\psi\rangle}(\eta)$ is quadratic it must be a
linear combination of the coefficients $\kappa_{r}$ which, according to Eq.
(20), implies that ${\cal F}_{|\psi\rangle}(\eta)$ is also a linear
combination of the $\mathcal{A}_{t}$. Furthermore, the identity

$\mathrm{tr}\left[\rho_{t}^{2}\right]=\mathrm{tr}\left[\rho_{N-t}^{2}\right]\,,$
(21)

which holds for any pure state, means that the anticoherence measures
$\mathcal{A}_{t}$ for $t>N/2$ can be expressed in terms of the measures
$\mathcal{A}_{t}$ for $t<N/2$. Therefore, (3) is the most general form the
fidelity ${\cal F}_{|\psi\rangle}(\eta)$ can take, with the dependence in
$\eta$ being only in the coefficients of the measures $\mathcal{A}_{t}$.

###  2.8 Generalizations

It is worth stressing that the form (3) for the average fidelity also holds
for more general types of average fidelity

$\frac{1}{4\pi}\int_{S^{2}}|\langle\psi|U_{\mathbf{n}}(\eta)|\psi\rangle|^{2}\,d\mathbf{n}$
(22)

between a state $|\psi\rangle$ and its image under the unitary

$U_{\mathbf{n}}(\eta)=e^{-i\eta\,f(\mathbf{J}\boldsymbol{\cdot}\mathbf{n})},$
(23)

where $f$ is an arbitrary real analytic function, ensuring that
$f(\mathbf{J}\boldsymbol{\cdot}\mathbf{n})$ is an Hermitian operator. Indeed,
from an argument similar to that of Sec. 2.6, the generalized fidelity (22)
can be expressed as a function of the $\kappa_{r}$ and hence of the
$\mathcal{A}_{t}$. An interesting case is when $U_{\mathbf{n}}(\eta)$ is a
spin-squeezing operator, which corresponds to choosing
$f(\mathbf{J}\boldsymbol{\cdot}\mathbf{n})=(\mathbf{J}\boldsymbol{\cdot}\mathbf{n})^{2}$.
Moreover, if we now consider the quantities

$\frac{1}{4\pi}\int_{S^{2}}|\langle\psi|U_{\mathbf{n}}(\eta)|\psi\rangle|^{2k}\,d\mathbf{n}$
(24)

with integer $k\geqslant 2$, the same arguments show that they are linear
combinations of higher-order invariants, leading to generalizations of the
relation (20).

##  3 Closed form of the average fidelity

In this section we derive the angular functions $\varphi_{t}^{(j)}(\eta)$,
which characterize the fidelity through (3), in two different ways. The first
method (subsection 3.1) is based on the fact that anticoherence measures can
be evaluated explicitly for Dicke states. The second method (subsection 3.2)
exploits a tensor representation of spin states [24] which uses Feynman rules
from relativistic spin theory. These approaches are independent and we
checked, for all integers and half-integers $j$ up to $26$, that as expected
they yield the same angular functions. Technical detail is delegated to
appendices in both cases.

###  3.1 Derivation based on anticoherence measures for Dicke states

In the following, we will work in the standard angular momentum basis of
$\mathbb{C}^{N+1}$, for positive integer or half-integer value of $j=N/2$. It
consists of the Dicke states $\left\\{|j,m\rangle,\left|m\right|\leqslant
j\right\\}$ given by the common eigenstates of $\mathbf{J}^{2}$, the square of
the angular momentum operator $\mathbf{J}$, and of its $z$-component $J_{z}$.
In this basis, any spin-$j$ state $|\psi\rangle$ can be expanded as

$|\psi\rangle=\sum_{m=-j}^{j}c_{m}\,|j,m\rangle\,,$ (25)

with $c_{m}\in\mathbb{C}$ and $\sum_{m=-j}^{j}|c_{m}|^{2}=1$.

The first derivation is based on the fact that both the measures of
$t$-anticoherence $\mathcal{A}_{t}(|j,m\rangle)$ and the average fidelities
${\cal F}_{|j,m\rangle}(\eta)$ can be determined explicitly for Dicke states.
Their measures of $t$-anticoherence are given by

$\mathcal{A}_{t}(|j,m\rangle)=\frac{t+1}{t}\left[1-\frac{\sum_{\ell=0}^{t}\binom{j+m}{t-\ell}^{2}\binom{j-m}{j-m-\ell}^{2}}{\binom{2j}{t}^{2}}\right].$
(26)

They can readily be obtained from the purities
$\mathrm{tr}\left[\rho_{t}^{2}\right]$ for a state of the form (25), which
were calculated in [20] in terms of the coefficients $c_{m}$ and read

$\mathrm{tr}\left[\rho_{t}^{2}\right]=\sum_{q,\ell=0}^{t}\left|\sum_{k=0}^{2j-t}c_{j-k-\ell}^{*}\,c_{j-k-q}\,\Gamma_{k}^{\ell
q}\right|^{2}$ (27)

with

$\Gamma_{k}^{\ell
q}=\frac{\sqrt{\binom{2j-k-q}{t-q}\binom{2j-k-\ell}{t-\ell}\binom{k+q}{k}\binom{k+\ell}{k}}}{\binom{2j}{t}}.$
(28)

As for the fidelity, the calculation is done in Appendix B and yields

${\cal
F}_{|j,m\rangle}(\eta)=\frac{1}{(2j+1)^{2}}\sum_{\ell=0}^{2j}(2\ell+1)(C_{jm\ell
0}^{jm}\,\chi_{\ell}^{j}(\eta))^{2}\,,$ (29)

with Clebsch-Gordan coefficients $C_{jm\ell 0}^{jm}$ and the functions
$\chi_{\ell}^{j}(\eta)$ defined in Eqs. (77)–(78). The angular functions
$\varphi_{t}^{(j)}(\eta)$ are then solutions of the system of linear equations

$\left\\{\begin{array}[]{l}{\cal
F}_{|j,m\rangle}(\eta)=\varphi_{0}^{(j)}(\eta)+\sum_{t=1}^{\lfloor
j\rfloor}\varphi_{t}^{(j)}(\eta)\,\mathcal{A}_{t}(|j,m\rangle)\\\\[8.0pt]
\mathrm{for}\;\,m=j,j-1,\ldots,j-\lfloor j\rfloor.\end{array}\right.$ (30)

This system can easily be solved for the lowest values of $j$. A general (but
formal) solution can then be obtained by inverting the system (30).

###  3.2 Derivation based on relativistic Feynman rules and tensor
representation of spin states

The second approach allows us to derive a closed-form expression for the
functions $\varphi_{t}^{(j)}(\eta)$. It is based on an expansion of the
operator

$\Pi^{(j)}(q)\equiv(q_{0}^{2}-|{\mathbf{q}}|^{2})^{j}\,e^{-2\theta_{q}\,\hat{{\bf
q}}\boldsymbol{\cdot}{\bf J}},$ (31)

with $\tanh\theta_{q}=-|{\mathbf{q}}|/q_{0}$ and $\hat{{\bf
q}}={\mathbf{q}}/|{\mathbf{q}}|$, as a multivariate polynomial in the
variables $q_{0},q_{1},q_{2},q_{3}$. This operator is a $(N+1)$-dimensional
representation (with $N=2j$) of a Lorentz boost in the direction of the
4-vector $q=(q_{0},{\mathbf{q}})=(q_{0},q_{1},q_{2},q_{3})$. As shown in [25],
it can be written as

$\Pi^{(j)}(q)=(-1)^{2j}q_{\mu_{1}}q_{\mu_{2}}\ldots
q_{\mu_{2j}}S_{\mu_{1}\mu_{2}\ldots\mu_{2j}}.$ (32)

The identification of Eqs. (31) and (32) defines the $(N+1)\times(N+1)$
matrices $S_{\mu_{1}\ldots\mu_{N}}$ appearing in (11) (see [24] for detail).
Taking

$q_{0}=i\cot(\eta/2)\quad\mbox{and}\quad q_{i}=n_{i}\,,\quad i=1,2,3\,,$ (33)

in (31), we see that $\Pi^{(j)}(q)$ reduces to a rotation operator,

$R_{\mathbf{n}}(\eta)=e^{-i\eta\mathbf{J}\boldsymbol{\cdot}\mathbf{n}}=\frac{\Pi^{(j)}(q)}{m^{N}}$
(34)

with

$m^{2}=q_{0}^{2}-|{\mathbf{q}}|^{2}=-\frac{1}{\sin^{2}(\eta/2)}.$ (35)

Moreover, for a state $\rho$ given by (11) we have

$\mathrm{tr}\left[\rho\,\Pi^{(j)}(q)\right]=(-1)^{N}x_{\mu_{1}\mu_{2}\ldots\mu_{N}}q_{\mu_{1}}\ldots
q_{\mu_{N}},$ (36)

according to Eq. (24) of [24], which holds for any 4-vector $q$. Thus, with
$\rho=|\psi\rangle\langle\psi|$, using the identity (34) and the expansion
(32) for the rotation operator in (1) allows us to explicitly perform the
integral in Eq. (2), resulting in

$\displaystyle{\cal F}_{|\psi\rangle}(\eta)={}$
$\displaystyle\frac{1}{4\pi}\int_{\mathcal{S}^{2}}|\langle\psi|R_{\mathbf{n}}(\eta)|\psi\rangle|^{2}\,d\mathbf{n}$
(37) $\displaystyle={}$
$\displaystyle\frac{1}{4\pi}\int_{\mathcal{S}^{2}}\left|\textrm{tr}\left[\rho\frac{\Pi^{(j)}(q)}{m^{N}}\right]\right|^{2}d\mathbf{n}$
$\displaystyle={}$
$\displaystyle(-1)^{N}\frac{x_{\mu_{1}\ldots\mu_{N}}x_{\nu_{1}\ldots\nu_{N}}}{4\pi}$
$\displaystyle\times\int_{\mathcal{S}^{2}}\frac{q_{\mu_{1}}\ldots
q_{\mu_{N}}q_{\nu_{1}}^{*}\ldots q_{\nu_{N}}^{*}}{m^{2N}}\,d\mathbf{n},$

where $*$ denotes complex conjugation (which acts on $q_{0}$ only because of
the choice (33) and using $|m|^{2}=-m^{2}$). Each term $q_{\mu_{1}}\ldots
q_{\nu_{N}}^{*}$ with $2(N-k)$ indices equal to 0 is proportional to

$\frac{q_{0}^{2(N-k)}}{m^{2N}}=(-1)^{k}\sin^{2k}\left(\frac{\eta}{2}\right)\cos^{2(N-k)}\left(\frac{\eta}{2}\right).$
(38)

For the remaining $2k$ nonzero indices, we have from (33) that $q_{i}=n_{i}$,
so that (37) involves an integral of the form

$\frac{1}{4\pi}\int_{\mathcal{S}^{2}}n_{a_{1}}n_{a_{2}}\ldots
n_{a_{2k}}\,d\mathbf{n}\,,\qquad 1\leqslant a_{i}\leqslant 3\,.$ (39)

These integrals are performed in Appendix C. The integrals (39) are in fact
precisely given by the tensor coordinates $x_{a_{1}a_{2}\ldots a_{2k}}^{(0)}$
of the maximally mixed state, whose expression is explicitly known. One can
therefore rewrite (37) as

$\displaystyle{\cal
F}_{|\psi\rangle}(\eta)=\sum_{k=0}^{N}(-1)^{N}\frac{q_{0}^{2(N-k)}}{m^{2N}}$
(40)
$\displaystyle\times\sum_{\genfrac{}{}{0.0pt}{1}{\boldsymbol{\mu}{,}\boldsymbol{\nu}}{2(N-k)\textrm{zeros}}{}}(-1)^{\textrm{nr
of 0 in
}\boldsymbol{\nu}}x_{\mu_{1}\ldots\mu_{N}\nu_{1}\ldots\nu_{N}}^{(0)}x_{\mu_{1}\ldots\mu_{N}}x_{\nu_{1}\ldots\nu_{N}}\,,$

where the sum over $\boldsymbol{\mu}{,}\boldsymbol{\nu}$ runs over all strings
of indices (between 0 and 3) containing $2(N-k)$ zeros. An explicit expression
for this sum is derived in Appendix C, leading to the compact expression

${\cal
F}_{|\psi\rangle}(\eta)=\sum_{k=0}^{N}\sin^{2k}\left(\frac{\eta}{2}\right)\cos^{2(N-k)}\left(\frac{\eta}{2}\right)\sum_{t=0}^{N}a_{t,k}^{(j)}\;\mathrm{tr}\left[\rho_{t}^{2}\right]\,,$
(41)

with numbers

$a_{t,k}^{(j)}=\frac{4^{t}(-1)^{k+t}\binom{2N}{2k}\binom{k}{t}\binom{2N-2t}{N-t}}{(2k+1)\binom{2N}{N}}.$
(42)

Note that the sum over $k$ in (41) can start at $k=t$ because the factor
$\binom{k}{t}$ in $a_{t,k}^{(j)}$ implies that $a_{t,k}^{(j)}=0$ for $t>k$.
Using the symmetry
$\mathrm{tr}\left[\rho_{t}^{2}\right]=\mathrm{tr}\left[\rho_{N-t}^{2}\right]$
we may rewrite (41) as

$\displaystyle{\cal F}_{|\psi\rangle}(\eta)={}$
$\displaystyle\sum_{k=t}^{N}\sin^{2k}\left(\frac{\eta}{2}\right)\cos^{2(N-k)}\left(\frac{\eta}{2}\right)$
(43) $\displaystyle\times\sum_{t=0}^{\lfloor
j\rfloor}\left(a_{t,k}^{(j)}+a_{N-t,k}^{(j)}\right)\left(1-\frac{\delta_{jt}}{2}\right)\mathrm{tr}\left[\rho_{t}^{2}\right].$

From (10) we obtain a relation between $\mathcal{A}_{t}$ and
$\mathrm{tr}\left[\rho_{t}^{2}\right]$, namely
$\mathrm{tr}\left[\rho_{t}^{2}\right]=1-\frac{t}{t+1}\mathcal{A}_{t}$, which
yields the explicit expression of the polynomials $\varphi_{t}^{(j)}(\eta)$ in
Eq. (3) as

$\varphi_{t}^{(j)}(\eta)=\sum_{k=t}^{N}b_{t,k}^{(j)}\,\sin^{2k}\left(\frac{\eta}{2}\right)\cos^{2(N-k)}\left(\frac{\eta}{2}\right),$
(44)

with coefficients

$b_{t,k}^{(j)}=\left\\{\begin{array}[]{ll}{\displaystyle-\frac{t}{t+1}\left(a_{t,k}^{(j)}+a_{N-t,k}^{(j)}\right)\left(1-\frac{\delta_{jt}}{2}\right)}&t\neq
0\\\ {\displaystyle\frac{\binom{N}{k}}{2k+1}}&t=0\,.\end{array}\right.$ (45)

Note that although $q_{0}$ and $m$ are not well-defined for $\eta=0$, the
ratio in (38) always is, so that the expression above is valid over the whole
range of values of $\eta$. For spin-coherent states, all $\mathcal{A}_{t}$
vanish and thus ${\cal F}_{|\psi\rangle}(\eta)=\varphi_{0}^{(j)}(\eta)$ from
Eq. (3), which coincides with the expression obtained in [15]. For the
smallest values of $j$, we recover the functions obtained in Section 3.1. In
the following section, we will use the functions $\varphi_{t}^{(j)}(\eta)$
given in (44) to identify optimal quantum rotosensors.

##  4 Optimal quantum rotosensors

###  4.1 Preliminary remarks

We now address the question of finding the states $|\psi\rangle$ which
minimize the average fidelity ${\cal F}_{|\psi\rangle}(\eta)$ for fixed
rotation angles $\eta$. According to Eq. (3), the fidelity is a _linear_
function of the anticoherence measures $\mathcal{A}_{t}$ with $1\leqslant
t\leqslant\lfloor j\rfloor$. Linearity, when combined with the fact that the
domain $\Omega$, over which the measures $\mathcal{A}_{t}$ vary, is _bounded_
implies that the fidelity must attain its minimum on the boundary. The
minimization problem thus amounts to characterizing this domain $\Omega$.
Unfortunately, even for the smallest values of $j$, no simple descriptions of
this domain are known.

We will first determine the states minimizing the $2\pi$-periodic average
fidelity for values of $j$ up to $j=7/2$, with the rotation angle taking
values in the interval $\eta\in[0,\pi]$ (which is sufficient due to the
symmetry (6)). Then we will examine the limiting case of angles $\eta$ close
to $0$ for arbitrary values of the quantum number $j$. Throughout this
section, we will expand arbitrary states with spin $j$ in terms of the Dicke
states, as shown in Eq. (25).

For spins up to $j=2$ the states minimizing the average fidelity ${\cal
F}_{|\psi\rangle}(\eta)$ are known [15]. In Sec. 4.2, we show that our
approach based on the expression (3) correctly reproduces these results. Then,
in Sec. 4.3, we consider the minimization problem for spin quantum numbers up
to $j=7/2$, mainly identifying the optimal rotosensors within various ranges
of the rotation angle $\eta$ by numerical techniques. More specifically, for a
fixed angle $\eta$, ${\cal F}_{|\psi\rangle}(\eta)$ is a function of the
$\mathcal{A}_{t}$ which can be parametrized by the complex coefficients
$c_{m}$ entering the expansion (25) of the state $|\psi\rangle$ in the Dicke
basis (see Eq. (27)). We search numerically for the minimum value of ${\cal
F}_{|\psi\rangle}(\eta)$ with respect to the $c_{m}$, taking into account the
normalization condition $\sum_{m}|c_{m}|^{2}=1$. In most cases this numerical
search converges towards states which have simple analytic expressions which
are the ones that we give. For each value of $j$, we performed this search at
about 1000 evenly spaced values of $\eta$ in order to explore the whole range
of rotation angles. Whenever we find a region of values of $\eta$ in which
$|\psi_{1}\rangle$ is the optimal state adjacent to a region where
$|\psi_{2}\rangle$ is optimal, at the critical angle separating these two
regions, one should have ${\cal F}_{|\psi_{1}\rangle}(\eta)={\cal
F}_{|\psi_{2}\rangle}(\eta)$ because the average fidelity ${\cal
F}_{|\psi\rangle}(\eta)$ is a continuous function of $|\psi\rangle$.
Therefore, the critical angle is a solution of the equation

$\sum_{t=1}^{\lfloor
j\rfloor}\varphi_{t}^{(j)}(\eta)\,\mathcal{A}_{t}(|\psi_{1}\rangle)=\sum_{t=1}^{\lfloor
j\rfloor}\varphi_{t}^{(j)}(\eta)\,\mathcal{A}_{t}(|\psi_{2}\rangle).$ (46)

###  4.2 Rotosensors for arbitrary rotation angles $\eta$ and $j\leqslant 2$

####  4.2.1 $j=1/2$

For a spin $1/2$, all pure states are coherent: each state $|\psi\rangle$ can
be obtained by a suitable rotation of the state
$|\tfrac{1}{2},\tfrac{1}{2}\rangle$. Since the fidelity is invariant under
rotation, all states are equally sensitive to detect rotations for any angle
$\eta$.

####  4.2.2 $j=1$

For $j=1$, the expansion (3) takes the form

${\cal
F}_{|\psi\rangle}(\eta)=\varphi_{0}^{(j)}(\eta)+\varphi_{1}^{(1)}(\eta)\,\mathcal{A}_{1}\,,$
(47)

with

$\displaystyle\varphi_{0}^{(1)}(\eta)={}$
$\displaystyle\frac{1}{15}\big{(}6\cos(\eta)+\cos(2\eta)+8\big{)},$ (48)
$\displaystyle\varphi_{1}^{(1)}(\eta)={}$
$\displaystyle-\frac{1}{15}\big{(}2\cos(\eta)-3\cos(2\eta)+1\big{)}.$

The first strictly positive zero of $\varphi_{1}^{(1)}(\eta)$ is given by
$\eta_{0}=\arccos(-2/3)$. In the interval $\eta\in[0,\eta_{0}[$, where
$\varphi_{1}^{(1)}(\eta)$ is negative, the fidelity ${\cal
F}_{|\psi\rangle}(\eta)$ is minimized by states with $\mathcal{A}_{1}=1$, i.e.
by $1$-anticoherent states. For $\eta=\eta_{0}$, the fidelity takes the same
value for all states $|\psi\rangle$, namely ${\cal
F}_{|\psi\rangle}(\eta_{0})=\varphi_{0}^{(1)}(\eta_{0})=7/27$. For rotation
angles in the the remaining interval, $\eta\in]\eta_{0},\pi]$, where
$\varphi_{1}^{(1)}(\eta)$ is positive, ${\cal F}_{|\psi\rangle}(\eta)$ is
minimized for states with $\mathcal{A}_{1}=0$, i.e. coherent states. Thus, we
indeed recover the results obtained in [15].

####  4.2.3 $j=3/2$

In this case, the average fidelity (3) reads

${\cal
F}_{|\psi\rangle}(\eta)=\varphi_{0}^{(3/2)}(\eta)+\varphi_{1}^{(3/2)}(\eta)\,\mathcal{A}_{1}\,,$
(49)

with

$\displaystyle\varphi_{0}^{(3/2)}(\eta)={}$
$\displaystyle\frac{1}{70}\big{(}\cos(3\eta)+8\cos(2\eta)+29\cos(\eta)+32\big{)},$
(50) $\displaystyle\varphi_{1}^{(3/2)}(\eta)={}$
$\displaystyle\frac{3}{70}\big{(}3\cos(3\eta)+3\cos(2\eta)-4\cos(\eta)-2\big{)}.$

The situation is basically the same as for $j=1$. The first strictly positive
zero of the coefficient $\varphi_{1}^{(3/2)}(\eta)$ is found to be
$\eta_{0}=\arccos(\frac{-9+\sqrt{21}}{12})$. Hence, in the interval
$\eta\in[0,\eta_{0}[$ where $\varphi_{1}^{(3/2)}(\eta)$ is negative, the
fidelity ${\cal F}_{|\psi\rangle}(\eta)$ is minimal for $1$-anticoherent
states. At the value $\eta=\eta_{0}$, the fidelity takes the same value for
all states $|\psi\rangle$, namely, ${\cal
F}_{|\psi\rangle}(\eta_{0})=\varphi_{0}^{(3/2)}(\eta_{0})=(33+2\sqrt{21})/80$.
Otherwise, ${\cal F}_{|\psi\rangle}(\eta)$ is minimized for coherent states,
thereby reproducing earlier results [15].

####  4.2.4 $j=2$

For $j=2$, the fidelity (3) is a linear combination of three terms,

${\cal
F}_{|\psi\rangle}(\eta)=\varphi_{0}^{(2)}(\eta)+\varphi_{1}^{(2)}(\eta)\,\mathcal{A}_{1}+\varphi_{2}^{(2)}(\eta)\,\mathcal{A}_{2}\,,$
(51)

with the angular functions $\varphi_{k}^{(2)},k=0,1,2$, displayed in Appendix
D. They all take negative values in the interval $\eta\in[0,\eta_{0}]$, with
$\eta_{0}\approx 1.2122$ the first strictly positive zero of
$\varphi_{1}^{(2)}(\eta)$. The tetrahedron state

$|\psi^{\mathrm{tet}}\rangle=\frac{1}{2}\left(|2,-2\rangle+i\sqrt{2}\,|2,0\rangle+|2,2\rangle\right),$
(52)

whose Majorana points lie at the vertices of a regular tetrahedron, is
$2$-anticoherent, and for $j=2$ it is the only state (up to LU) with
$\mathcal{A}_{1}=\mathcal{A}_{2}=1$ [26]; hence it provides the optimal
rotosensor for angles in the interval $\eta\in[0,\eta_{0}]$. For larger angles
of rotation comprised between $1.68374$ and $2.44264$, we find numerically
that an optimal state is the Schrödinger cat state

$|\psi^{\mathrm{cat}}\rangle=\frac{1}{\sqrt{2}}\left(|2,-2\rangle+|2,2\rangle\right)\,,$
(53)

which is only 1-anticoherent, with $\mathcal{A}_{1}=1$ and
$\mathcal{A}_{2}=3/4$. For values $\eta\gtrsim 2.44264$, the optimal state is
a coherent state.

We thus obtain numerically three intervals with three distinct optimal states
corresponding to $(\mathcal{A}_{1},\mathcal{A}_{2})=(1,1),(1,3/4)$, and
$(0,0)$, respectively. In order to find the critical angles, we solve Eq.
(46). The angle $\eta_{1}$ separating the first two regions is a solution of
$\varphi_{2}^{(2)}(\eta)=0$. The first positive zero of
$\varphi_{2}^{(2)}(\eta)$ is $\eta_{1}=2\arctan(\sqrt{9-2\sqrt{15}})\approx
1.68374$, which coincides with the numerically obtained value. The angle
$\eta_{2}$ at which the second and third region touch, is a zero of
$\varphi_{1}^{(2)}(\eta)+\tfrac{3}{4}\,\varphi_{2}^{(2)}(\eta)$. Its first
strictly positive zero is given by

$\eta_{2}=2\arctan\left(\sqrt{-\frac{a+102b}{a-38b}}\right)\,,$ (54)

with $a=19\ 6^{2/3}+\sqrt[3]{6}\left(223-35\sqrt{7}\right)^{2/3}$ and
$b=\sqrt[3]{223-35\sqrt{7}}$, and we have indeed $\eta_{2}\approx 2.44264$.
The results we obtained are summarized in Fig. 1; they agree with the findings
of [15].

It is noteworthy that the state (53) is not the only state with anticoherence
measures $\mathcal{A}_{1}=1$ and $\mathcal{A}_{2}=3/4$. For instance, any
state of the form

$|\psi\rangle=\frac{c_{1}|2,-1\rangle+c_{2}|2,0\rangle-
c_{1}^{*}|2,1\rangle}{\sqrt{2|c_{1}|^{2}+|c_{2}|^{2}}}\,$ (55)

with $c_{1}\in\mathbb{C}$ and $c_{2}\in\mathbb{R}$ come with the same measures
of anticoherence, as readily follows from Eq. (27). These states are thus also
optimal in the interval $\eta\in[\eta_{1},\eta_{2}]$, thereby removing the
uniqueness of optimal rotosensors observed for $j=1$ and $j=3/2$.

Figure 1: Average fidelity ${\cal F}_{|\psi\rangle}(\eta)$ (top, red solid
curve) and measures of anticoherence ${\cal A}_{t}$ (bottom) for optimal
states with $j=2$, as functions of the rotation angle $\eta$; the values of
the measures $\mathcal{A}_{t}$ for the optimal states are discontinuous at the
values $\eta_{1}\approx 1.68374$ and $\eta_{2}\approx 2.44264$ (see text for
details). The dashed curve on top shows the average fidelity
$\varphi_{0}^{(2)}(\eta)$ for coherent states. The blue (red) shaded area
shows the range of rotation angles for which anticoherent states to order
$\lfloor j\rfloor$ (coherent states) are optimal.

###  4.3 Rotosensors for $5/2\leqslant j\leqslant 7/2$

####  4.3.1 $j=5/2$

For $j=5/2$, there is no anticoherent state of order $2$ but only of order $1$
[12]. Numerical optimization shows that the optimal state for small angles of
rotation is the $1$-anticoherent state with the largest measure of
$2$-anticoherence, that is given by

$|\psi\rangle=\frac{1}{\sqrt{2}}\left(|\tfrac{5}{2},-\tfrac{3}{2}\rangle+|\tfrac{5}{2},\tfrac{3}{2}\rangle\right),$
(56)

and has $\mathcal{A}_{1}=1$ and $\mathcal{A}_{2}=99/100$. This state is found
to be optimal up to a critical angle $\eta_{1}\approx 1.49697$, which is
obtained from Eq. (46) and coincides with the first strictly positive zero of
$\varphi_{2}^{(5/2)}(\eta)$. It is worth noting that the optimal state (56)
was also found to be the most non-classical spin state for $j=5/2$, both in
the sense that it maximizes the quantumness [27] and that it minimizes the
cumulative multipole distribution [28, 29]. The Majorana constellation of this
state defines a triangular bipyramid, which is a spherical $1$-design [30,
31], thus corresponding to the arrangement of point charges on the surface of
a sphere which minimize the Coulomb electrostatic potential energy (solution
to Thomson’s problem for 5 point charges, see [32]).

For larger angles of rotation ranging between $\eta_{1}$ and $\eta_{2}\approx
2.2521$, we find that an optimal state is

$|\psi^{\mathrm{cat}}\rangle=\frac{1}{\sqrt{2}}\left(|\tfrac{5}{2},-\tfrac{5}{2}\rangle+|\tfrac{5}{2},\tfrac{5}{2}\rangle\right)\,;$
(57)

unlike in the case $j=2$, we found this state for $j=5/2$ to be the only state
(up to LU) with $\mathcal{A}_{1}=1$ and $\mathcal{A}_{2}=3/4$. For
$\eta\in[\eta_{2},\pi]$, we find that coherent states are optimal. The
transition occurs at the first strictly positive zero $\eta_{2}$ of
$\varphi_{1}^{(5/2)}(\eta)+\tfrac{3}{4}\,\varphi_{2}^{(5/2)}(\eta)$. Our
results are summarized in Fig. 2.

Figure 2: Average fidelity ${\cal F}_{|\psi\rangle}(\eta)$ (top, red solid
curve) and measures of anticoherence ${\cal A}_{t}$ (bottom) for optimal
states with $j=5/2$, as functions of the rotation angle $\eta$; the values of
the measures $\mathcal{A}_{t}$ for the optimal states are discontinuous at the
values $\eta_{1}\approx 1.49697$ and $\eta_{2}\approx 2.2521$ (see text for
details). The dashed curve on top shows the average fidelity
$\varphi_{0}^{(5/2)}(\eta)$ for coherent states. Shaded areas are defined as
in Fig. 1.

####  4.3.2 $j=3$

Anticoherent states of order $3$ do exist for $j=3$. They are all connected by
rotation to the octahedron state

$|\psi^{\mathrm{oct}}\rangle=\frac{1}{\sqrt{2}}\left(|3,-2\rangle+|3,2\rangle\right),$
(58)

whose Majorana points lie at the vertices of a regular octahedron. Therefore,
the state (58) is, at small $\eta$, the unique optimal quantum rotosensor (up
to LU) for $j=3$. Numerical optimization shows that the octahedron state is
optimal up to a critical angle $\eta_{1}\approx 1.3635$ coinciding with the
first strictly positive zero of
$\tfrac{1}{4}\,\varphi_{2}^{(3)}(\eta)+\tfrac{1}{3}\,\varphi_{3}^{(3)}(\eta)$,
and that, for larger angles, the state

$|\psi^{\mathrm{cat}}\rangle=\frac{1}{\sqrt{2}}\left(|3,-3\rangle+|3,3\rangle\right)$
(59)

with $\mathcal{A}_{1}=1$, $\mathcal{A}_{2}=3/4$ and $\mathcal{A}_{3}=2/3$ is
optimal up to a critical angle $\eta_{2}\approx 2.04367$ coinciding with the
first strictly positive zero of
$\varphi_{1}^{(3)}(\eta)+\tfrac{3}{4}\,\varphi_{2}^{(3)}(\eta)+\tfrac{2}{3}\,\varphi_{3}^{(3)}(\eta)$.
We found that this is the only spin-$3$ state (up to LU) with
$\mathcal{A}_{1}=1$, $\mathcal{A}_{2}=3/4$ and $\mathcal{A}_{3}=2/3$. Coherent
states are found to be optimal for angles of rotation in the ranges
$[\eta_{2},\eta_{3}]$ and $[\eta_{4},\pi]$ with $\eta_{3}\approx 2.35881$ and
$\eta_{4}\approx 2.65576$ coinciding with the second and third strictly
positive zeros of
$\varphi_{1}^{(3)}(\eta)+\varphi_{2}^{(3)}(\eta)+\varphi_{3}^{(3)}(\eta)$. In
the range $[\eta_{3},\eta_{4}]$, the octahedron state (58) becomes again
optimal (although the three functions $\varphi_{k}^{(3)}$ for $k=1,2,3$ are
not simultaneously negative in that range). Our results are displayed in Fig.
3.

Figure 3: Average fidelity ${\cal F}_{|\psi\rangle}(\eta)$ (top, red solid
curve) and measures of anticoherence ${\cal A}_{t}$ (bottom) for optimal
states with $j=3$, as functions of the rotation angle $\eta$; the values of
the measures $\mathcal{A}_{t}$ for the optimal states are discontinuous at the
values $\eta_{1}\approx 1.3635$, $\eta_{2}\approx 2.04367$, $\eta_{3}\approx
2.35881$ and $\eta_{4}\approx 2.65576$ (see text for details). The dashed
curve on top shows the average fidelity $\varphi_{0}^{(3)}(\eta)$ for coherent
states. Shaded areas are defined as in Fig. 1.

####  4.3.3 $j=7/2$

This is the smallest spin quantum number for which a smooth variation of the
optimal state with $\eta$ is observed, resulting in the complex behaviour
displayed in Figs. 4 and 5. There are no anticoherent states to order $3$ for
$j=7/2$, but there exist anticoherent states to order $2$. The optimal state
for small angles of rotation (by which we mean here $\eta\to 0$) turns out to
be one of those. Numerical optimization yields the state

$|\psi\rangle=\sqrt{\tfrac{2}{9}}\,|\tfrac{7}{2},-\tfrac{7}{2}\rangle-\sqrt{\tfrac{7}{18}}\,|\tfrac{7}{2},-\tfrac{1}{2}\rangle-\sqrt{\tfrac{7}{18}}\,|\tfrac{7}{2},\tfrac{5}{2}\rangle$
(60)

with measures of anticoherence $\mathcal{A}_{1}=\mathcal{A}_{2}=1$ and
$\mathcal{A}_{3}=1198/1215$. This is not the state with the highest measure of
$3$-anticoherence, as the state

$|\psi\rangle=\frac{1}{\sqrt{2}}\left(|\tfrac{7}{2},-\tfrac{5}{2}\rangle+|\tfrac{7}{2},\tfrac{5}{2}\rangle\right),$
(61)

has measures of anticoherence $\mathcal{A}_{1}=1$, $\mathcal{A}_{2}=195/196$
and $\mathcal{A}_{3}=146/147>1198/1215$. The latter state is found to be
optimal for $\eta\in[\eta_{1},\eta_{2}]$ with $\eta_{1}\approx 0.71718$ (not
identified) and $\eta_{2}\approx 1.24169$ coinciding with the first strictly
positive zero of
$\tfrac{12}{49}\,\varphi_{2}^{(7/2)}(\eta)+\tfrac{16}{49}\,\varphi_{3}^{(7/2)}(\eta)$.
The state

$|\psi^{\mathrm{cat}}\rangle=\frac{1}{\sqrt{2}}\left(|\tfrac{7}{2},-\tfrac{7}{2}\rangle+|\tfrac{7}{2},\tfrac{7}{2}\rangle\right)$
(62)

with $\mathcal{A}_{1}=1$, $\mathcal{A}_{2}=3/4$ and $\mathcal{A}_{3}=2/3$ is
found to be optimal for $\eta\in[\eta_{2},\eta_{3}]$ and
$\eta\in[\eta_{4},\eta_{5}]$ with $\eta_{3}\approx 1.60141$ and
$\eta_{4}\approx 1.88334$ coinciding with the third and fourth strictly
positive zeros of $\varphi_{1}^{(7/2)}(\eta)$ and $\eta_{5}\approx 2.41684$
with the first strictly positive zero of
$\varphi_{1}^{(7/2)}(\eta)+\tfrac{3}{4}\,\varphi_{2}^{(7/2)}(\eta)+\tfrac{2}{3}\,\varphi_{3}^{(7/2)}(\eta)$.
In the interval $[\eta_{5},\pi]$, coherent states are found to be optimal.

Figure 4: Average fidelity ${\cal F}_{|\psi\rangle}(\eta)$ (top, red solid
curve) and measures of anticoherence ${\cal A}_{t}$ (bottom) for optimal
states with $j=7/2$, as functions of the rotation angle $\eta$. The dashed
curve on top shows the average fidelity $\varphi_{0}^{(7/2)}(\eta)$ for
coherent states. Shaded areas are defined as in Fig. 1.  Figure 5: Measures of
anticoherence ${\cal A}_{t}$ for optimal states with $j=7/2$, as functions of
the rotation angle $\eta\in[0,0.8]$.

###  4.4 Rotosensors for small rotation angles $\eta$ and arbitrary values of
$j$

####  4.4.1 Angular functions at small angles

According to Secs. 4.2 and 4.3 optimal rotosensors for integer values of spin
($j=1,2,3$) are given by $j$-anticoherent states while for half-integer spin
($j=3/2,5/2,7/2$) the fidelity is optimized by states which are anticoherent
of order $t=1,1,2$, respectively, and possess large anticoherence measures
${\cal A}_{t}$ for values of $t$ up to $t=\left\lfloor j\right\rfloor$. This
fact can be understood quite generally through the behaviour of the functions
$\varphi_{t}^{(j)}(\eta)$ at small $\eta$ for arbitrary values of $j$. In the
vicinity of $\eta=0$, the functions $\varphi_{t}^{(j)}(\eta)$ given in Eq.
(44) take the form

$\varphi_{t}^{(j)}(\eta)=\frac{b_{t,t}^{(j)}}{2^{2t}}\,\eta^{2t}+\mathcal{O}(\eta^{2t+2}),$
(63)

with coefficients $b_{t,t}^{(j)}$ given by Eq. (45). These coefficients are
strictly negative for all $t\geqslant 1$ and all $j=N/2$, since
$a_{t,t}^{(j)}>0$ and $a_{N-t,t}^{(j)}$ is either $0$ for $t<N/2$ or positive
for $t=N/2$. This implies that all functions $\varphi_{t}^{(j)}(\eta)$ are
negative in some interval around $\eta=0$. Thus, the fidelity ${\cal
F}_{|\psi\rangle}(\eta)$ is a linear combination of the $\mathcal{A}_{t}$ with
negative coefficients in that interval. Since
$0\leqslant\mathcal{A}_{t}\leqslant 1$, it follows that if there exists a
state with $\mathcal{A}_{t}=1$ for all $t\leqslant\lfloor j\rfloor$—that is,
an anticoherent state to order $\lfloor j\rfloor$—then this state provides an
optimal quantum rotosensor for $\eta$ in that interval.

This interval can be made more specific, at least for the lowest values of
$j$. Let $\eta_{0}$ denote the first zero of $\varphi_{1}^{(j)}(\eta)$.
Numerical results up to $j=85$ indicate that all functions
$\varphi_{t}^{(j)}(\eta)$ for $t=1,\ldots,\lfloor j\rfloor$ are negative for
$\eta\in[0,\eta_{0}]$, so that an anticoherent state to order $\lfloor
j\rfloor$ (if it exists) is optimal in the whole interval $[0,\eta_{0}]$. As
shown in Fig. 6, $\eta_{0}$ is found to scale as $3\pi/(4j)$ for large $j$. A
simple explanation for this is that the expansion of the function
$\varphi_{1}^{(j)}(\eta)$ as $\sum_{k}a_{k}\cos(k\eta)$ is dominated by the
term $a_{2j}\cos(2j\eta)$ (note however that $\eta_{0}$ is even better
approximated by $9/(4j)$). Conversely, the states maximizing ${\cal
F}_{|\psi\rangle}(\eta)$ for small angles of rotation are the states with
$\mathcal{A}_{t}=0$ for all $t$, i.e. coherent states.

Figure 6: First zero $\eta_{0}$ of the functions $\varphi_{1}^{(j)}(\eta)$
(blue dots) as a function of $j$: for $j=1$ and for $j\geqslant 5/2$, the
values are well approximated by $\eta_{0}\approx 3\pi/(4j)$ (pink dashes).

To see whether any general pattern emerges, we now identify optimal small-
angle rotosensors for the next few values of the spin quantum numbers.

####  4.4.2 $j=4$

For $j=4$, there is no anticoherent state to order $t=4$. We find that the
optimal state for small angles of rotation is the $3$-anticoherent state

$|\psi\rangle=\sqrt{\tfrac{5}{24}}\,|4,-4\rangle-\sqrt{\tfrac{7}{12}}\,|4,0\rangle-\sqrt{\tfrac{5}{24}}\,|4,4\rangle,$
(64)

with $\mathcal{A}_{1}=\mathcal{A}_{2}=\mathcal{A}_{3}=1$ and
$\mathcal{A}_{4}=281/288$.

####  4.4.3 $j=9/2$

For $j=9/2$, there is no anticoherent state to order $t\geqslant 3$. The
anticoherent states of order $t=2$ with the largest $\mathcal{A}_{3}$ are
found to be of the form

$|\psi\rangle=\tfrac{\sqrt{13}}{8}\,|\tfrac{9}{2},-\tfrac{9}{2}\rangle+e^{i\chi}\sqrt{\tfrac{15}{32}}\,|\tfrac{9}{2},-\tfrac{1}{2}\rangle-\tfrac{\sqrt{21}}{8}\,|\tfrac{9}{2},\tfrac{7}{2}\rangle,$
(65)

with $\chi\in[0,\pi/2]$. Their measures of anticoherence are
$\mathcal{A}_{1}=\mathcal{A}_{2}=1$, $\mathcal{A}_{3}=2347/2352$ and
$\mathcal{A}_{4}=5\left(355609+175\sqrt{273}\cos(2\chi)\right)/1806336$. Among
these states, the one with $\chi=0$ has the largest value of $\mathcal{A}_{4}$
and numerical results suggest that this is the optimal state for small angles
of rotation.

####  4.4.4 $j=5$

For $j=5$, there is no anticoherent state to order $t\geqslant 4$. We find
that the optimal state for small angles is the $3$-anticoherent state

$|\psi\rangle=\sqrt{\tfrac{5}{16}}\,|5,-4\rangle+\sqrt{\tfrac{3}{8}}\,|5,0\rangle-\sqrt{\tfrac{5}{16}}\,|5,4\rangle,$
(66)

with $\mathcal{A}_{1}=\mathcal{A}_{2}=\mathcal{A}_{3}=1$,
$\mathcal{A}_{4}=895/896$ and $\mathcal{A}_{5}=1097/1120$.

$j$ | $|\psi^{\mathrm{optimal}}\rangle$ | $\mathcal{A}_{t}$ | Interval
---|---|---|---
$1$ | $\begin{array}[]{c}|\psi^{\mathrm{cat}}\rangle\\\ \mathrm{any\leavevmode\nobreak\ state}\\\ |j,j\rangle\end{array}$ | $\begin{array}[]{c}\mathcal{A}_{1}=1\\\ 0\leqslant\mathcal{A}_{1}\leqslant 1\\\ \mathcal{A}_{1}=0\end{array}$ | $\begin{array}[]{c}\eta\in[0,\eta_{0}[\\\ \eta=\eta_{0}\\\ \eta\in[\eta_{0},\pi]\end{array}$
$3/2$ | $\begin{array}[]{c}|\psi^{\mathrm{cat}}\rangle\\\ \mathrm{any\leavevmode\nobreak\ state}\\\ |j,j\rangle\end{array}$ | $\begin{array}[]{c}\mathcal{A}_{1}=1\\\ 0\leqslant\mathcal{A}_{1}\leqslant 1\\\ \mathcal{A}_{1}=0\end{array}$ | $\begin{array}[]{c}\eta\in[0,\eta_{0}[\\\ \eta=\eta_{0}\\\ \eta\in[\eta_{0},\pi]\end{array}$
$2$ | $\begin{array}[]{c}|\psi^{\mathrm{tet}}\rangle\\\ |\psi^{\mathrm{cat}}\rangle\\\ |j,j\rangle\end{array}$ | $\begin{array}[]{c}\mathcal{A}_{1}=\mathcal{A}_{2}=1\\\ \mathcal{A}_{1}=1,\mathcal{A}_{2}=3/4\\\ \mathcal{A}_{1}=\mathcal{A}_{2}=0\end{array}$ | $\begin{array}[]{c}\eta\in[0,\eta_{1}],\eta_{1}\approx 1.68374\\\ \eta\in[\eta_{1},\eta_{2}]\\\ \eta\in[\eta_{2},\pi],\eta_{2}\approx 2.44264\end{array}$
$5/2$ | $\begin{array}[]{c}\mathrm{Eq}.\leavevmode\nobreak\ \eqref{s52}\\\ |\psi^{\mathrm{cat}}\rangle\\\ |j,j\rangle\end{array}$ | $\begin{array}[]{c}\mathcal{A}_{1}=1,\mathcal{A}_{2}=99/100\\\ \mathcal{A}_{1}=1,\mathcal{A}_{2}=3/4\\\ \mathcal{A}_{1}=\mathcal{A}_{2}=0\end{array}$ | $\begin{array}[]{c}\eta\in[0,\eta_{1}],\eta_{1}\approx 1.49697\\\ \eta\in[\eta_{1},\eta_{2}]\\\ \ \eta\in[\eta_{2},\pi],\eta_{2}\approx 2.2521\end{array}$
$3$ | $\begin{array}[]{c}|\psi^{\mathrm{oct}}\rangle\\\ |\psi^{\mathrm{cat}}\rangle\\\ |j,j\rangle\end{array}$ | $\begin{array}[]{c}\mathcal{A}_{1}=\mathcal{A}_{2}=\mathcal{A}_{3}=1\\\ \mathcal{A}_{1}=1,\mathcal{A}_{2}=3/4,\mathcal{A}_{3}=2/3\\\ \mathcal{A}_{1}=\mathcal{A}_{2}=\mathcal{A}_{3}=0\end{array}$ | $\begin{array}[]{c}\eta\in[0,\eta_{1}]\cup[\eta_{3},\eta_{4}],\eta_{3}\approx 2.35881\\\ \eta\in[\eta_{1},\eta_{2}],\eta_{1}\approx 1.3635,\eta_{2}\approx 2.04367\\\ \eta\in[\eta_{2},\eta_{3}]\cup[\eta_{4},\pi],\eta_{4}\approx 2.65576\end{array}$
$7/2$ | $\begin{array}[]{c}\mathrm{Eq}.\leavevmode\nobreak\ \eqref{s72AC}\\\ -\\\ |\psi^{\mathrm{cat}}\rangle\\\ -\\\ |j,j\rangle\end{array}$ | $\begin{array}[]{c}\mathcal{A}_{1}=\mathcal{A}_{2}=1,\mathcal{A}_{3}=1198/1215\\\ \tfrac{195}{196}\leqslant\mathcal{A}_{2}\leqslant 1,\tfrac{1198}{1215}\leqslant\mathcal{A}_{3}\leqslant\tfrac{146}{147},\,\mathrm{see\leavevmode\nobreak\ Fig.}\leavevmode\nobreak\ \ref{figj72bis}\\\ \mathcal{A}_{1}=1,\mathcal{A}_{2}=3/4,\mathcal{A}_{3}=2/3\\\ \mathrm{see\leavevmode\nobreak\ Fig.}\leavevmode\nobreak\ \ref{figj72}\\\ \mathcal{A}_{1}=\mathcal{A}_{2}=\mathcal{A}_{3}=0\end{array}$ | $\begin{array}[]{c}\eta\to 0\\\ \eta\in[0,\eta_{1}],\eta_{1}\approx 0.71718\\\ \eta\in[\eta_{2},\eta_{3}]\cup[\eta_{4},\eta_{5}],\eta_{2}\approx 1.24169\\\ \eta\in[\eta_{3},\eta_{4}],\eta_{3}\approx 1.60141,\eta_{4}\approx 1.88334\\\ \eta\in[\eta_{5},\pi],\eta_{5}\approx 2.41684\end{array}$
Table 1: Summary of the results of Secs. 4.2 and 4.3 on optimal states for
$1\leqslant j\leqslant 7/2$. Here, $\eta_{0}$ denotes the first strictly
positive zero of $\varphi_{1}^{(j)}(\eta)$, $|\psi^{\mathrm{tet}}\rangle$
defined for $j=2$ is given by Eq. (52), $|\psi^{\mathrm{oct}}\rangle$ defined
for $j=3$ is given by Eq. (58), and
$|\psi^{\mathrm{cat}}\rangle=\frac{1}{\sqrt{2}}\left(|j,-j\rangle+|j,j\rangle\right)$
for any $j$. The state $|j,j\rangle$ has been taken as an example of coherent
state. Note that optimal states given here are not necessarily unique (states
not related by a rotation can have the same $\mathcal{A}_{t}$).

####  4.4.5 Arbitrary values of $j$

As was mentioned earlier, if an anticoherent state to order $\lfloor j\rfloor$
exists for a given $j$, then this state gives rise to an optimal quantum
rotosensor for $\eta\in[0,\eta_{0}]$. This applies to values $j=1,3/2,2$ and
$j=3$, which are the only cases where existence of anticoherent states to
order $t=\lfloor j\rfloor$ has been established (see e.g. [33, 20]).

The situation is less straightforward if such a state is not known to exist
from the outset. The only general conclusion one can draw is that minimizing
the average fidelity ${\cal F}_{|\psi\rangle}(\eta)$ for a fixed angle
$\eta\in[0,\eta_{0}]$ corresponds to maximizing the measures $\mathcal{A}_{t}$
within the domain $\Omega$ (by definition, $\Omega$ is the set of all
reachable $\mathcal{A}_{t}$ so that by changing $|\psi\rangle$, we will remain
within $\Omega$). In this sense, the more anticoherent a state is, the more
sensitive it will be as a quantum rotosensor. In general, varying
$|\psi\rangle$ will change all anticoherence measures simultaneously. The
challenge is to determine whether a state with given values of the measures
$\mathcal{A}_{t}$ exists and, if it does, to identify it.

The maximal order of anticoherence that a spin-$j$ state can display is
generally much smaller than $\lfloor j\rfloor$, typically $t\sim 2\sqrt{j}$
for large spins $j$ [33]. Numerical results for $j\lesssim 100$ seem to
suggest that the pairs $(t,j)$ for which a $t$-anticoherent spin-$j$ state
exists coincide with those for which a $2j$-points spherical $t$-design exists
in three dimensions [34]. The latter have been tabulated up to $j=50$ [31].
For example, the first pairs $(t,j)$ for $j\leqslant 4$ are given by $(1,1)$,
$(1,3/2)$, $(2,2)$, $(1,5/2)$, $(3,3)$, $(2,7/2)$, $(3,4)$.

##  5 Summary and conclusions

The main result of this work is a closed-form expression (3) for the fidelity
${\cal F}_{|\psi\rangle}(\eta)$ between a state and its image under a rotation
by an angle $\eta$ about an axis ${\bf n}$, averaged over all rotation axes.
The expression takes the form of a linear combination of anticoherence
measures $\mathcal{A}_{t}$, with explicit $\eta$-dependent coefficients. It
follows that not only spin-$j$ states which are related by a global rotation
of the axes come with the same average fidelity, but more generally all states
with identical purities of their reduced density matrices (calculated for any
subset of their $2j$ constituent spin-$1/2$ in the Majorana representation).
This gives an explanation for the observation of [15] that optimal states are
not necessarily unique. Moreover, since the fidelity is linear in the
anticoherence measures, optimal states correspond to values of
$\mathcal{A}_{t}$ on the boundary of the domain $\Omega$ of admissible values.
This shows the relevance of characterizing the domain $\Omega$.

The expression (3) allows us to characterize states which optimally detect
rotations by their degree of coherence or anticoherence. At small angles
$\eta\leqslant\eta_{0}$, where the coefficients of the measures
$\mathcal{A}_{t}$ are all negative, optimality of detection of rotations goes
hand in hand with high degrees of anticoherence. For angles close to
$\eta=\pi$, however, numerical results support the claim that optimality is
achieved throughout by spin coherent states.

We also performed a systematic investigation of states minimizing the average
fidelity for small values of $j$, for all integers and half-integers from
$j=1/2$ to $j=5$. Table 1 summarizes our findings for the lowest values of
$j$. At small rotation angle, all optimal states were found to have a maximal
lowest anticoherence measure: $\mathcal{A}_{1}=1$. These states, which are
anticoherent to order $1$, exist for any value of $j$, and one may conjecture
that they should, in fact, be optimal for arbitrary values of $j$. More
generally, for all values of $j$ investigated and for $\eta\leqslant\eta_{0}$,
the optimal states turned out to have, for each $t>1$, the largest admissible
anticoherence measure $\mathcal{A}_{t}$ compatible with fixed values of the
lower measures $\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{t-1}$.
Whether this property holds in general remains an open question.

Note that natural generalizations of this problem, such as maximization of the
average fidelity, can also be addressed by our approach. For instance, for
small rotation angles $\eta\in[0,\eta_{0}]$, where all
$\varphi_{t}^{(j)}(\eta)$ with $t\geqslant 1$ are negative, the average
fidelity is maximal for coherent states. For rotation angles close to
$\eta=\pi$, numerical results indicate that the $1$-anticoherent state
$|\psi^{\mathrm{cat}}\rangle=\frac{1}{\sqrt{2}}\left(|j,-j\rangle+|j,j\rangle\right)$
is optimal for all $j$ up to $17/2$.

OG and SW thank the hospitality of the University of Liège, where this work
has been initiated.

##  Appendix A Fidelity in parameter estimation theory of rotations

It was shown in [13] that minimizing the uncertainty in the measurement of an
_unknown_ angle about a _known_ rotation axis is equivalent to identifying the
states which minimize the fidelity $F_{|\psi\rangle}(\eta,\mathbf{n})$,
assuming small rotation angles and using parameter estimation theory. To see
this, first expand the fidelity as

$|\langle\psi|R_{\mathbf{n}}(\eta)|\psi\rangle|^{2}=1-\eta^{2}(\Delta
J_{\mathbf{n}})^{2}+{\cal O}(\eta^{4})\,,$ (67)

where $(\Delta J_{\mathbf{n}})^{2}$ is the variance of
$J_{\mathbf{n}}\equiv\mathbf{J}\boldsymbol{\cdot}\mathbf{n}$ in the state
$|\psi\rangle$. Solving (67) for the angle $\eta$ will, upon measuring the
fidelity in any state $|\psi\rangle$, result in an estimate of $\eta$. Second,
the accuracy of this value depends on the initial state $|\psi\rangle$: using
error propagation, one finds that the variance of the estimator is
approximately given by

$\left(\Delta\eta\right){}^{2}\approx\frac{1}{\left(2\Delta J_{{\bf
n}}\right)^{2}}\,.$ (68)

Thus, states $|\psi\rangle$ with _large_ values of the variance _$\left(\Delta
J_{{\bf n}}\right)^{2}$_ are seen to minimize the uncertainty of the angle
$\eta$. According to Eq. (67), these states also _minimize_ the fidelity
$F_{|\psi\rangle}(\eta,\mathbf{n})$.

Let us generalize the argument to the case in which the rotation axis is
_unknown_. We will see that the states producing the most reliable
results—i.e. the smallest variance in the angle estimator—are those which
minimize the average fidelity ${\cal F}_{|\psi\rangle}(\eta)$. It is
convenient to describe the randomness in the rotation axis in terms of a
quantum channel (see for instance [35] for the use of channels in quantum
estimation theory). Suppose we prepare the pure initial state
$\rho_{0}=|\psi\rangle\langle\psi|$ and send it through the $\eta$-dependent
channel $\Lambda_{\eta}(\cdot)$,

$\rho_{\eta}=\Lambda_{\eta}(\rho_{0})=\frac{1}{4\pi}\int_{{{\cal
S}^{2}}}R_{\mathbf{n}}(\eta)\rho_{0}R_{\mathbf{n}}^{\dagger}(\eta)\,d{\bf
n}\,,$ (69)

which describes rotations by $\eta$ about all possible rotation axes. Next, we
measure the projector $\rho_{0}=|\psi\rangle\langle\psi|$. Assuming the
rotation angle to be small, $\eta\ll 1$, the probability to still find the
propagated state $\rho_{\eta}$ in the initial state $\rho_{0}$ is given by

$\langle\rho_{0}\rangle_{\eta}=\mathrm{tr}\left[\rho_{0}\rho_{\eta}\right]=1-\eta^{2}V+{\cal
O}(\eta^{4})\,,$ (70)

where $V$ is the variance of
$J_{\mathbf{n}}\equiv\mathbf{J}\boldsymbol{\cdot}\mathbf{n}$, averaged over
all directions,

$V=\frac{1}{4\pi}\int_{{\cal
S}^{2}}\left(\langle\psi|J_{\mathbf{n}}^{2}|\psi\rangle-\langle\psi|J_{\mathbf{n}}|\psi\rangle^{2}\right)\,d{\bf
n}\,.$ (71)

Using $\rho_{0}^{2}=\rho_{0}$ and the relation (70), the variance of the
measurement outcomes is found to be

$\left(\Delta\rho_{0}\right)_{\eta}^{2}=\mathrm{tr}\left[\rho_{0}^{2}\rho_{\eta}\right]-\left(\mathrm{tr}\left[\rho_{0}\rho_{\eta}\right]\right)^{2}=\eta^{2}V+{\cal
O}(\eta^{4})\,.$ (72)

Now using again the error propagation formula instrumental in the derivation
of Eq. (68) about _known_ rotation axes _,_ its generalization to _unknown_
axes is given by

$\left(\Delta\eta\right){}^{2}=\frac{\left(\Delta\rho_{0}\right)_{\eta}^{2}}{|\partial\langle\rho_{0}\rangle_{\eta}/\partial\eta|^{2}}+{\cal
O}(\eta^{2})\approx\frac{1}{4V}\,.$ (73)

This result concludes the argument we wish to provide: it is of physical
interest to minimize the average fidelity

${\cal
F}_{|\psi\rangle}(\eta)\equiv\mathrm{tr}\left[\rho_{0}\rho_{\eta}\right]\approx
1-\eta^{2}V\,,$ (74)

since the states which do so are those states which allow one to most
accurately estimate a (small) rotation angle about unknown axes.

##  Appendix B Average fidelity for Dicke states

For Dicke states $|j,m\rangle$ (common eigenstates of $\mathbf{J}^{2}$ and
$J_{z}$), the average fidelity (2) reads

$\displaystyle{\cal F}_{|j,m\rangle}(\eta)$
$\displaystyle{}=\frac{1}{4\pi}\int_{\mathcal{S}^{2}}|\langle
j,m|R_{\mathbf{n}}(\eta)|j,m\rangle|^{2}\,d\mathbf{n}$ (75)
$\displaystyle{}=\frac{1}{4\pi}\int_{\mathcal{S}^{2}}|U_{mm}^{j}(\eta,\mathbf{n})|^{2}\,d\mathbf{n}$

with $U_{mm}^{j}(\eta,\mathbf{n})\equiv U_{mm}^{j}$ a matrix element of the
rotation operator in the angle-axis parametrization given by

$U_{mm}^{j}=\frac{\sqrt{4\pi}}{2j+1}\sum_{\lambda,\mu}(-i)^{\lambda}\sqrt{2\lambda+1}\chi_{\lambda}^{j}(\eta)C_{jm\lambda\mu}^{jm}Y_{\lambda}^{m}(\mathbf{n})$
(76)

where $C_{jm\lambda\mu}^{jm}$ are Clebsch-Gordan coefficients,
$Y_{\lambda}^{m}(\mathbf{n})$ are spherical harmonics and
$\chi_{\lambda}^{j}(\eta)$ are the generalized characters of order $\lambda$
of the irreducible representations of rank $j$ of the rotation group [36].
These are defined by

$\chi_{\lambda}^{j}(\eta)=\sqrt{\tfrac{(2j+1)(2j-\lambda)!}{(2j+\lambda+1)!}}\sin^{\lambda}\left(\tfrac{\eta}{2}\right)\left(\tfrac{d}{d\cos\left(\tfrac{\eta}{2}\right)}\right)^{\lambda}\chi^{j}(\eta)$
(77)

with the characters

$\chi^{j}(\eta)=\frac{(4j+2)!!}{2(4j+1)!!}\,P_{2j}^{\big{(}\tfrac{1}{2},\tfrac{1}{2}\big{)}}\left(\cos\left(\tfrac{\eta}{2}\right)\right)$
(78)

where $P_{n}^{(\alpha,\beta)}$ are Jacobi polynomials. Taking the modulus
squared of (76) and integrating over all directions by using orthonormality of
the spherical harmonics, we readily get Eq. (29).

##  Appendix C Explicit calculation of the angular functions
$\varphi_{t}^{(j)}(\eta)$

###  C.1 Matrices $S_{\mu_{1}\mu_{2}\ldots\mu_{N}}$

The matrices $S_{\mu_{1}\mu_{2}\ldots\mu_{N}}$ with $N=2j$ appearing in the
expansion (11) can be obtained by expanding the $(j,0)$ representation of a
Lorentz boost,

$\Pi^{(j)}(q)\equiv(q_{0}^{2}-|{\mathbf{q}}|^{2})^{j}\,e^{-2\theta_{q}\,\hat{{\bf
q}}\boldsymbol{\cdot}{\bf J}},$ (79)

with $\theta_{q}=\mathrm{arctanh}(-|{\mathbf{q}}|/q_{0})$ and $\hat{{\bf
q}}={\mathbf{q}}/|{\mathbf{q}}|$. This expansion takes the form of a
multivariate polynomial in the variables $q_{0},q_{1},q_{2},q_{3}$,

$\Pi^{(j)}(q)=(-1)^{2j}q_{\mu_{1}}q_{\mu_{2}}\ldots
q_{\mu_{2j}}S_{\mu_{1}\mu_{2}\ldots\mu_{2j}},$ (80)

where the coefficients are the $(N+1)\times(N+1)$ matrices
$S_{\mu_{1}\mu_{2}\ldots\mu_{N}}$ [24].

###  C.2 Tensor coordinates of the maximally mixed state

The maximally mixed state $\rho_{0}=\mathbb{1}/(N+1)$ can be expanded along
(11) with coefficients $x_{\mu_{1}\mu_{2}\ldots\mu_{N}}^{(0)}$. The coherent
state decomposition of the maximally mixed state,
$\rho_{0}=\frac{1}{4\pi}\int_{\mathcal{S}^{2}}|\mathbf{n}\rangle\langle\mathbf{n}|\,d\mathbf{n}$,
yields the identity

$x_{\mu_{1}\mu_{2}\ldots\mu_{N}}^{(0)}=\frac{1}{4\pi}\int_{\mathcal{S}^{2}}n_{\mu_{1}}n_{\mu_{2}}\ldots
n_{\mu_{N}}\,d\mathbf{n}.$ (81)

Using our convention not to write indices when they are equal to 0, we have,
irrespective of spin size, $x_{0}^{(0)}=1$, $x_{aa}^{(0)}=1/3$,
$x_{aaaa}^{(0)}=1/5$ and $x_{aabb}^{(0)}=1/15$ for $a\neq b$. More generally,
the coefficients of the maximally mixed state are given by the polynomial
identity (cf. Eq. (27) of [24])

$x_{\mu_{1}\mu_{2}\ldots\mu_{N}}^{(0)}q_{\mu_{1}}\ldots
q_{\mu_{N}}=\sum_{k=0}^{j}\frac{\binom{N}{2k}}{2k+1}q_{0}^{N-2k}|{\mathbf{q}}|^{2k}\,,$
(82)

which leads to

$x_{a_{1}a_{2}\ldots
a_{N}}^{(0)}=\frac{1}{N+1}\frac{\binom{j}{p_{1}/2,p_{2}/2,p_{3}/2}}{\binom{N}{p_{1},p_{2},p_{3}}}\,,$
(83)

where $p_{i}$ denotes the number of $i$ in $\\{a_{1},a_{2},\ldots,a_{N}\\}$
and the terms in the fraction are multinomial coefficients (by convention the
right-hand side evaluates to zero if some $p_{i}$ is not even).

###  C.3 Average fidelity in terms of tensor coordinates

According to Eq. (40), the average fidelity can be written as a double sum,

$\displaystyle{\cal
F}_{|\psi\rangle}(\eta)=\sum_{k=0}^{N}(-1)^{N}\frac{q_{0}^{2(N-k)}}{m^{2N}}$
(84)
$\displaystyle\times\sum_{\genfrac{}{}{0.0pt}{1}{\boldsymbol{\mu}\bm{,}\boldsymbol{\nu}}{2(N-k)\textrm{zeros}}{}}(-1)^{\textrm{nr
of 0 in
}\boldsymbol{\nu}}x_{\mu_{1}\ldots\mu_{N}\nu_{1}\ldots\nu_{N}}^{(0)}x_{\mu_{1}\ldots\mu_{N}}x_{\nu_{1}\ldots\nu_{N}}\,.$

We now wish to show that the second sum which runs over all strings of indices
(between 0 and 3) containing $2(N-k)$ zeros can evaluated explicitly leading
to the simpler form for ${\cal F}_{|\psi\rangle}(\eta)$ given in Eq. (99) at
the end of this section.

The sum runs over terms containing $2(N-k)$ zeros, that is, $2k$ non-zero
indices. We split it into terms containing $r$ nonzero indices in
$\boldsymbol{\mu}$ and $2k-r$ in $\boldsymbol{\nu}$. At fixed $k$ we have

$\displaystyle\sum_{\genfrac{}{}{0.0pt}{1}{\boldsymbol{\mu}\bm{,}\boldsymbol{\nu}}{2(N-k)\textrm{zeros}}{}}(-1)^{\textrm{nr
of 0 in
}\boldsymbol{\nu}}x_{\mu_{1}\ldots\mu_{N}}x_{\nu_{1}\ldots\nu_{N}}x_{\mu_{1}\ldots\mu_{N}\nu_{1}\ldots\nu_{N}}^{(0)}$
$\displaystyle=\sum_{r=2k-N}^{N}(-1)^{N-2k+r}\binom{N}{r}\binom{N}{2k-r}\times$
$\displaystyle\qquad\times\sum_{a_{i},b_{i}}x_{a_{1}...a_{r}}x_{b_{1}...b_{2k-r}}x_{a_{1}...a_{r}b_{1}...b_{2k-r}}^{(0)}.$
(85)

We now evaluate the sums
$\sum_{a_{i},b_{i}}x_{a_{1}...a_{r}}x_{b_{1}...b_{2k-r}}x_{a_{1}...a_{r}b_{1}...b_{2k-r}}^{(0)}$.
We may suppose that $r\leqslant 2k-r$. Using (83), we see that the nonzero
indices $a_{i}$ and $b_{i}$ must occur in pairs. Indices $a_{i}$ are either
paired with indices $a_{k}$ or indices $b_{k}$. We can then split the sum
according to the number of pairings of the form $(a_{i},b_{i})$ (all other
pairings are then within the $a_{i}$ or within the $b_{i}$). Let us first
consider the case $k=r$. We are going to show that

$\displaystyle\sum_{a_{i},b_{i}}$
$\displaystyle{}x_{a_{1}...a_{r}}x_{b_{1}...b_{r}}x_{a_{1}...a_{r}b_{1}...b_{r}}^{(0)}=$
(86) $\displaystyle{}\lambda_{0}\sum_{a_{i}}x_{a_{1}...a_{r}}^{2}$
$\displaystyle{}+\lambda_{1}\sum_{a_{i}}\left(\sum_{b}x_{a_{1}...a_{r-2}bb}\right)^{2}$
$\displaystyle{}+\lambda_{2}\sum_{a_{i}}\left(\sum_{b_{1},b_{2}}x_{a_{1}...a_{r-4}b_{1}b_{1}b_{2}b_{2}}\right)^{2}+\ldots$

with

$\lambda_{q}=\frac{2^{r-2q}r!^{2}}{(2r+1)!}\binom{r}{r-2q,q,q}.$ (87)

We first use the explicit expression (83) of
$x_{a_{1}...a_{r}b_{1}...b_{r}}^{(0)}$ to get an equation equivalent to (86),
namely

$\displaystyle\sum_{c_{i}}x_{c_{1}...c_{r}}x_{c_{r+1}...c_{2r}}\frac{\binom{2r}{r}\binom{r}{p_{1}/2\,p_{2}/2\,p_{3}/2}}{\binom{2r}{p_{1}\,p_{2}\,p_{3}}}=2^{r}\sum_{a_{i}}x_{a_{1}...a_{r}}^{2}$
$\displaystyle+2^{r-2}\binom{r}{r-2,1,1}\sum_{a_{i}}\left(\sum_{b}x_{a_{1}...a_{r-2}bb}\right)^{2}+\cdots$
$\displaystyle+2^{r-2q}\binom{r}{r-2q,q,q}\sum_{a_{i}}\left(\sum_{b}x_{a_{1}...a_{r-2q}b_{1}b_{1}...b_{q}b_{q}}\right)^{2}$
$\displaystyle+\cdots,$ (88)

where $p_{i}$ is the number of $i$ in $\\{c_{1},c_{2},\ldots,c_{2r}\\}$ and
terms with $p_{i}$ odd are zero. In order to prove Eq. (88), we just observe
that it represents two different ways of counting the same quantity. Indeed,
let $\eta_{i}=\\{a_{i},\epsilon_{i},\epsilon_{i}^{\prime}\\}$ for $1\leqslant
i\leqslant r$ be triplets with $1\leqslant a_{i}\leqslant 3$ and
$0\leqslant\epsilon_{i},\epsilon_{i}^{\prime}\leqslant 1$. To a given set
$\mathbf{\eta}=\\{\eta_{1},\ldots,\eta_{r}\\}$ we associate a term of the form
$x_{c_{1}...c_{r}}y_{c_{r+1}...c_{2r}}$ where the $c_{i}$ occur in pairs
$(a_{1},a_{1}),(a_{2},a_{2}),\ldots,(a_{r},a_{r})$. In a pair $(a_{i},a_{i})$,
the first $a_{i}$ is assigned to be an index of $x$ if $\epsilon_{i}=0$, of
$y$ if $\epsilon_{i}=1$ (and similarly the second $a_{i}$ in the pair is an
index of $x$ if $\epsilon_{i}^{\prime}=0$, of $y$ otherwise). For instance,
$\eta=(a,0,0)$ corresponds to a term $x_{aa\ldots}y_{\ldots}$ and
$\eta=(a,0,1)$ corresponds to a term $x_{a\ldots}y_{a\ldots}$. In order that
$x$ and $y$ have the same number $r$ of indices we need to have
$\sum_{i}(\epsilon_{i}+\epsilon_{i}^{\prime})=r$, so that among the
$\epsilon_{i},\epsilon_{i}^{\prime}$ there are $r$ 0’s and $r$ 1’s. Each
$\mathbf{\eta}=\\{\eta_{1},\ldots,\eta_{r}\\}$ such that
$\sum_{i}(\epsilon_{i}+\epsilon_{i}^{\prime})=r$ then corresponds to a unique
term of the form $x_{c_{1}...c_{r}}y_{c_{r+1}...c_{2r}}$. Consider now, for
some $q\leqslant r$, all $\mathbf{\eta}$ with
$\sum_{i}(\epsilon_{i}+\epsilon_{i}^{\prime})=r$ for which
$\epsilon_{i}=\epsilon_{i}^{\prime}=0$ for exactly $q$ values of $i$. These
correspond to terms $x_{c_{1}...c_{r}}y_{c_{r+1}...c_{2r}}$ such that exactly
$q$ pairs $(a_{i},a_{i})$ appear as indices of $x$, $q$ pairs appear as
indices of $y$, and $r-2q$ are distributed over $x$ and $y$, i.e. terms of the
form $x_{a_{1}a_{2}...a_{r-2q}b_{1}b_{1}b_{2}b_{2}\ldots
b_{q}b_{q}}y_{a_{1}a_{2}...a_{r-2q}c_{1}c_{1}c_{2}c_{2}\ldots c_{q}c_{q}}$.
Replacing $y$ by $x$, all these terms are those appearing in the right-hand
side of (88). In fact, each sum on the right-hand side of (88) can be
interpreted as the sum over all $\eta_{i}$ such that
$\sum_{i}(\epsilon_{i}+\epsilon_{i}^{\prime})=r$ and
$\epsilon_{i}=\epsilon_{i}^{\prime}=0$ for exactly $q$ values of $i$. For
instance the first term on the right-hand side of (88) corresponds to terms
$q=0$, where all pairs $(a_{i},a_{i})$ are distributed over the two different
strings of indices (and then of course replacing $y$ by $x$). The prefactors
correspond to the ways of choosing the positions of a given set of pairs: the
multinomial coefficient corresponds to the choice of positions of the indices
among the $r$ indices of $x_{a_{1}a_{2}...a_{r-2q}b_{1}b_{1}b_{2}b_{2}\ldots
b_{q}b_{q}}$. The factor $2^{r-2q}$ corresponds to choosing between $x$ and
$y$ for the $r-2q$ indices $a_{i}$ which are distributed over $x$ and $y$.

The same sum can be expressed as the left-hand-side of (88) if we now first
sum over all strings $c_{1}\leqslant c_{2}\leqslant\cdots\leqslant c_{2r}$,
which implies dividing by the number of permutations
$\binom{2r}{p_{1},p_{2},p_{3}}$, then consider all possible positions of the
$a_{i}$ over the $r$ pairs, which implies multiplying by the number of
permutations of the pairs $\binom{r}{p_{1}/2,p_{2}/2,p_{3}/2}$, and finally
choose the $r$ entries among the $\epsilon_{i}$ and $\epsilon_{i}^{\prime}$
that will take the value 0, hence the factor $\binom{2r}{r}$. Thus (88) holds,
which proves (86). The tracelessness condition (12) then allows us to reduce
the sums over $b$ in (86) to invariants $\kappa_{r}$. We simply get

$\displaystyle\sum_{a_{i},b_{i}}x_{a_{1}...a_{r}}x_{b_{1}...b_{r}}x_{a_{1}...a_{r}b_{1}...b_{r}}^{(0)}$
(89)
$\displaystyle\qquad=\lambda_{0}\kappa_{r}+\lambda_{1}\kappa_{r-2}+\lambda_{2}\kappa_{r-4}+\ldots$

Following exactly the same procedure from (86) to (89) in the case where the
strings of indices of $x$ and $y$ have different lengths, we obtain the more
general expression

$\displaystyle\sum_{a_{i},b_{i}}x_{a_{1}...a_{r}}x_{b_{1}...b_{2k-r}}x_{a_{1}...a_{r}b_{1}...b_{2k-r}}^{(0)}$
$\displaystyle=\frac{r!(2k-r)!}{(2k+1)!}\sum_{q=0}^{\lfloor\frac{r}{2}\rfloor}2^{r-2q}\binom{k}{r-2q,q,q+k-r}\kappa_{r-2q}.$
(90)

From (84) we finally get

$\displaystyle{\cal F}_{|\psi\rangle}(\eta)={}$
$\displaystyle\sum_{k=0}^{N}(-1)^{k}\sin^{2k}\left(\frac{\eta}{2}\right)\cos^{2(N-k)}\left(\frac{\eta}{2}\right)\sum_{r=0}^{2k}(-1)^{r}\frac{N!^{2}}{(N-r)!(N-2k+r)!(2k+1)!}$
(91)
$\displaystyle\times\sum_{q=0}^{\lfloor\frac{r}{2}\rfloor}2^{r-2q}\binom{k}{r-2q,q,q+k-r}\kappa_{r-2q}.$

Changing the summation over $r$ to a summation over $s=r-2q$, we get

$\displaystyle{\cal F}_{|\psi\rangle}(\eta)={}$
$\displaystyle\sum_{k=0}^{N}(-1)^{k}\sin^{2k}\left(\frac{\eta}{2}\right)\cos^{2(N-k)}\left(\frac{\eta}{2}\right)\sum_{s=0}^{2k}(-2)^{s}\kappa_{s}$
(92)
$\displaystyle\times\sum_{q=0}^{k-\lfloor\frac{s+1}{2}\rfloor}\binom{N}{s+2q}\binom{N}{2k-2q-s}\frac{(s+2q)!(2k-2q-s)!}{(2k+1)!}\binom{k}{s,q,k-q-s}.$

Because of the multinomial coefficient at the end of (92), the sum over $s$
can be restricted to $s\leqslant k$ and the sum over $q$ to $q\leqslant k-s$,
yielding

$\displaystyle{\cal F}_{|\psi\rangle}(\eta)={}$
$\displaystyle\sum_{k=0}^{N}(-1)^{k}\sin^{2k}\left(\frac{\eta}{2}\right)\cos^{2(N-k)}\left(\frac{\eta}{2}\right)\frac{N!^{2}k!}{(2k+1)!}$
(93)
$\displaystyle\times\sum_{s=0}^{k}\frac{(-2)^{s}}{s!(2N-2k)!(k-s)!}\sum_{q=0}^{k-s}\binom{2N-2k}{N-s-2q}\binom{k-s}{q}\kappa_{s}.$

Grouping the $\kappa_{s}$ together by changing the order of the sum we get

$\displaystyle{\cal F}_{|\psi\rangle}(\eta)={}$ $\displaystyle
N!^{2}\sum_{s=0}^{N}\frac{(-2)^{s}\kappa_{s}}{s!}\sum_{k=s}^{N}(-1)^{k}\sin^{2k}\left(\frac{\eta}{2}\right)\cos^{2(N-k)}\left(\frac{\eta}{2}\right)\frac{k!}{(2k+1)!}$
(94)
$\displaystyle\times\sum_{q=0}^{k-s}\frac{1}{(N-s-2q)!(N-2k+s+2q)!(k-s-q)!q!}.$

Because of the sum over $q$ from 0 to $k-s$, we can make the sum over $k$
start at 0. We then use (19) to express the $\kappa_{s}$ in terms of
$\mathrm{tr}\left[\rho_{t}^{2}\right]$. This gives

$\displaystyle{\cal F}_{|\psi\rangle}(\eta)={}$ $\displaystyle
N!^{2}\sum_{t=0}^{N}\frac{(-2)^{t}}{t!}\mathrm{tr}\left[\rho_{t}^{2}\right]\sum_{k=0}^{N}(-1)^{k}\sin^{2k}\left(\frac{\eta}{2}\right)\cos^{2(N-k)}\left(\frac{\eta}{2}\right)\frac{k!}{(2k+1)!}$
(95)
$\displaystyle\times\sum_{s=t}^{N}\frac{2^{s}}{(s-t)!}\sum_{q=0}^{k-s}\frac{1}{(N-s-2q)!(N-2k+s+2q)!(k-s-q)!q!}.$

It turns out that the sums in the second line of this expression can be
performed. Indeed, the identity

$\sum_{s=t}^{N}\frac{2^{s}}{(s-t)!}\sum_{q=0}^{k-s}\frac{1}{(N-s-2q)!(N-2k+s+2q)!(k-s-q)!q!}=\frac{2^{t}(2N-2t)!}{(N-t)!^{2}(k-t)!(2N-2k)!}$
(96)

holds for arbitrary $N,t,k$. This can be proved as follows. First change
variables $N\to N-t$, $k\to k-t$ and $s\to s-t$, so that showing (96) amounts
to showing

$\sum_{s=0}^{k}\frac{2^{s}}{s!}\sum_{q=0}^{k-s}\frac{1}{(N-s-2q)!(N-2k+s+2q)!(k-s-q)!q!}=\frac{(2N)!}{N!^{2}k!(2N-2k)!}$
(97)

(the upper bound of the sum over $s$ can be changed from $N$ to $k$ since
terms $s>k$ do not contribute). Equation (97) can be rewritten

$\sum_{s=0}^{k}\sum_{q=0}^{k-s}2^{s}\binom{k}{s}\binom{k-s}{q}\binom{2N-2k}{N-s-2q}=\binom{2N}{N}.$
(98)

Such an identity can be proven by writing
$(1+x)^{2N}=(1+2x+x^{2})^{k}(1+x)^{2N-2k}$ for any $k$ and any $x$, and
expanding the first factor using multinomial coefficients and the second one
using binomial coefficients:

$\displaystyle(1+x)^{2N}=$ $\displaystyle(1+2x+x^{2})^{k}(1+x)^{2N-2k}$
$\displaystyle=\sum_{s,q}\binom{k}{s,q,k-s-q}(2x)^{s}(x^{2})^{q}\sum_{u}\binom{2N-2k}{u}x^{u}$
$\displaystyle=\sum_{s,q,u}2^{s}\binom{k}{s}\binom{k-s}{q}\binom{2N-2k}{u}x^{u+s+2q}$

(the boundaries of the sums are taken care of by the binomial coefficients
which vanish outside a certain range of parameters). Identifying the
coefficients of the term in $x^{N}$ readily gives (98).

Using (96), Eq. (95) finally reduces to

${\cal
F}_{|\psi\rangle}(\eta)=\frac{1}{2N+1}\frac{1}{\binom{2N}{N}}\sum_{t=0}^{N}(-4)^{t}\binom{2N-2t}{N-t}\mathrm{tr}\left[\rho_{t}^{2}\right]\sum_{k=0}^{N}(-1)^{k}\sin^{2k}\left(\frac{\eta}{2}\right)\cos^{2(N-k)}\left(\frac{\eta}{2}\right)\binom{2N+1}{2k+1}\binom{k}{t}.$
(99)

##  Appendix D Angular functions for $j=2$

Evaluating the expression (44) for $j=2$ leads to these three angular
functions:

$\displaystyle\varphi_{0}^{(2)}(\eta)={}$
$\displaystyle\frac{1}{315}\left(130\cos(\eta)+46\cos(2\eta)+10\cos(3\eta)+\cos(4\eta)+128\right),$
(100) $\displaystyle\varphi_{1}^{(2)}(\eta)={}$
$\displaystyle-\frac{4}{315}\left(10\cos(\eta)-11\cos(2\eta)+16\cos(3\eta)-20\cos(4\eta)+5\right),$
$\displaystyle\varphi_{2}^{(2)}(\eta)={}$
$\displaystyle-\frac{64}{105}\sin^{4}\left(\frac{\eta}{2}\right)(10\cos(\eta)+5\cos(2\eta)+6)\,.$

##  Appendix E Sample code

We give here a short sample code written in Mathematica™ to find an optimal
state for $j=5/2$ and $\eta=0.5$.

⬇

1(* Angular functions, see Eqs. (42), (45) and (44) *)

2a[n_,t_,k_]:=((-1)^(k+t)*4^t)/(2*k+1)*(Binomial[2*n,2*k]*Binomial[2*n-2*t,n-t]
Binomial[k,t])/Binomial[2*n,n];

3b[n_,t_,k_]:=If[t==0,Binomial[n,k]/(1+2*k),-(t/(t+1))*If[t==n/2,1/2,1]*(a[n,t,k]+a[n,n-t,k])];

4phi[n_,t_,eta_]:=Sum[Sin[eta/2]^(2*k)*Cos[eta/2]^(2*(n-k))*b[n,t,k],{k,0,n}];

5

6(* Measures of anticoherence of a pure state of the form (25),

7 see Eqs. (10) and (27) *)

8A[cm__,n_,t_]:=(t+1)/t*(1-Sum[Abs[Sum[Conjugate[cm[[k+l+1]]]*cm[[k+q+1]]*Sqrt[Binomial[k+l,k]*Binomial[n-k-l,t-l]*Binomial[k+q,k]*Binomial[n-k-q,t-q]]/Binomial[n,t],{k,0,n-t}]^2],{q,0,t},{l,0,t}])

9

10(* Average fidelity, see Eq. (3) *)

11F[cm__,n_,eta_]:=phi[n,0,eta]+Sum[phi[n,t,eta]*A[cm,n,t],{t,1,Floor[n/2]}];

12

13(* Normalized state expressed in the Dicke basis for j=5/2 *)

14j=5/2; n=2*j;

15cm=Normalize@(Array[r,n+1,0]+I*Array[i,n+1,0]);

16

17(* Search for an optimal state for eta=0.5 *)

18eta=0.5;

19f=Simplify[ComplexExpand[F[cm,n,eta]]];

20sol=NMinimize[f,Array[r,n+1,0]~Join~Array[i,n+1,0],AccuracyGoal->25,PrecisionGoal->25];

21

22(* Minimal average fidelity *)

23Re@sol[[1]]

24

25(* Optimal state in the Dicke basis *)

26cmsol=cm/.sol[[2]]

27

28(* Measures of anticoherence of order 1 and 2 *)

29A[cmsol,n,1]

30A[cmsol,n,2]

The evaluation of the code with Mathematica 12.0 yields the output

⬇

10.453337

2

3{0.189461+0.48194 I, -0.155904-0.0488534 I, 0.0828666+0.00440845 I,
0.374917+0.583967 I, 0.257018-0.0504367 I, -0.165557+0.347373 I}

4

51.

6

70.99

The state that is found, with measures of anticoherence $\mathcal{A}_{1}=1$
and $\mathcal{A}_{2}=99/100$, can be shown to be related by a rotation to the
state (56).

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