Optimal Detection of Rotations about Unknown
Axes by Coherent and Anticoherent States
John Martin
1
, Stefan Weigert
2
, and Olivier Giraud
3
1
Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, University of Liège, B-4000 Liège, Belgium
2
Department of Mathematics, University of York, UK-York YO10 5DD, United Kingdom
3
Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
Coherent and anticoherent states of spin sys-
tems up to spin j = 2 are known to be opti-
mal in order to detect rotations by a known
angle but unknown rotation axis. These opti-
mal 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 val-
ues 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 of-
ten are the reason for physics to progress. Over time,
metrology has developed as a subject of its own, es-
pecially in the context of defining standard units of
measurement for physical quantities.
Quantum theory provides new perspectives on mea-
surements, 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 infor-
mation science requires ever better control of micro-
scopic systems and, hence, measurements which are
as accurate as possible. More specifically, quantum
metrology [4] aims at finding bounds on the achiev-
able 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 inho-
mogeneous magnetic fields [6] are typical examples.
John Martin: jmartin@uliege.b e
Stefan Weigert: stefan.weigert@york.ac.uk
Olivier Giraud: olivier.giraud@universite-paris-saclay.fr
While the classical Cramér-Rao theorem [7, 8] pro-
vides a lower bound on the variance of random estima-
tors by means of the Fisher information, its quantum-
mechanical counterpart provides bounds for quan-
tum parameter estimation theory [9]. The quantum
Cramér-Rao bound is expressed as the inverse of the
quantum Fisher information, which can be geometri-
cally 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 mea-
surement 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 prob-
lem [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 satu-
rate the quantum Cramér-Rao bound for any axis.
Also, recently, the problem of characterizing a rota-
tion 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
n
(η) by a known angle η about an unknown axis
n. Suppose first that we apply the rotation by η to
an initial state |ψi about a known axis and perform
a measurement of the projector |ψihψ| in the rotated
state R
n
(η)|ψi. The expectation value of the observ-
able |ψihψ| is given by
F
|ψi
(η, n) = |hψ|R
n
(η)|ψi|
2
, (1)
i.e. by the fidelity between the initial state and the
final state. The fidelity F
|ψi
(η, n) equals the proba-
bility 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 quan-
tity 1 − F
|ψi
(η, n). Therefore, the measurement will
be most sensitive if the rotation is applied to states
|ψi which minimize the expression (1) for given angle
and rotation axis.
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arXiv:1909.08355v2 [quant-ph] 18 Jun 2020
Next, suppose that only the rotation angle η is well-
defined while the rotation axis is not known, as de-
scribed in [15]. This situation occurs, for example,
when spins prepared in the state |ψi 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 |ψihψ| on an ensemble of identically pre-
pared systems will now produce a value of the fidelity
(1) averaged over all possible spatial directions n.
Then, the most suitable quantum states |ψi—called
optimal quantum rotosensors in [15]—are determined
by the requirement that the average fidelity
F
|ψi
(η) =
1
4π
Z
S
2
F
|ψi
(η, n) dn , (2)
achieve its minimum, for a given value of the param-
eter η.
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 quan-
tum rotosensors have been identified [15], using an
approach which combines analytical and numerical
methods. For rotation angles η close to π, the av-
erage fidelity is minimized systematically by coherent
spin states. Coherent spin states are strongly local-
ized in phase space and entirely specified by a spatial
direction into which they point on the Bloch sphere
[16]. For small rotation angles η, the average fidelity
is minimized by anticoherent states, which are char-
acterized by the fact that they do not manifest any
privileged direction; in this respect, they are as dis-
tinct as possible from coherent states [17]. The role
of anticoherent states for optimal detection of rota-
tions has also been observed and was subsequently
quantified in terms of quantum Fisher information
in [13]. Between these two extreme cases of η ∼ 0 and
η ∼ π, optimal states are neither coherent nor anti-
coherent in general. From an experimental point of
view, anticoherent and other non-classical spin states
have been created using a variety of physical sys-
tems. For instance, anticoherent states of quantum
light fields have been generated using orbital angular
momentum states of single photons with their useful-
ness 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 magne-
tometer [19].
The main result of the present paper is a closed-
form expression of the average fidelity F
|ψi
(η), valid
for arbitrary values of j. A rather general argument,
based solely on the symmetries of the average fidelity
F
|ψi
(η), shows that it must be a linear combination
of the form
F
|ψi
(η) = ϕ
(j)
0
(η) +
bjc
X
t=1
ϕ
(j)
t
(η) A
t
(|ψi), (3)
as explained in detail in Sec. 2. In this expression, the
A
t
(|ψi) are the anticoherence measures of a state |ψi,
introduced in [20] and given explicitly in Eq. (10),
while the real-valued functions ϕ
(j)
t
(η) are trigono-
metric polynomials independent of |ψi, and bjc is the
largest integer smaller than or equal to j. The main
challenge is to calculate the η-dependent coefficients
ϕ
(j)
t
(η), which we do in Sec. 3.
In earlier works, the average fidelity F
|ψi
(η) had
been expressed as a sum of functions of η weighted
by state-dependent coefficients, upon representing the
state in the polarization-tensor basis [15]. The advan-
tage of relation (3) is that the average fidelity depends
on the state under consideration only through its mea-
sures 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 F
|ψi
(η) 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 F
|ψi
(η),
followed by a brief summary of the Majorana repre-
sentation which enables us to interpret spin-j states as
completely symmetric states of N = 2j qubits. This
perspective allows us to introduce, for 1 6 t 6 bjc,
the anticoherence measure A
t
(|ψi), defined as the
linear entropy of the t-qubit reduced density ma-
trix of |ψihψ|. 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 fi-
delity which can only depend on expressions invariant
under 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 A
t
(|ψi),
which finally leads to (3).
Section 3 is dedicated to deriving explicit expres-
sions for the functions ϕ
(j)
t
(η). This will be done in
two ways: the first one is based on the fact that an-
ticoherence measures are explicitly known for certain
states, so that the functions ϕ
(j)
t
(η) appear as solu-
tions of a linear system of equations. The second ap-
proach makes use of representations of the Lorentz
group and allows us to obtain a general closed ex-
pression. In Sec. 4 we make use of this closed-form
expression to identify the optimal quantum rotosen-
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sors. 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 |ψi of the Hilbert space 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 J satisfy [J
k
, J
`
] =
iε
k`m
J
m
, k, `, m ∈ {x, y, z}, where ε
k`m
is the Levi-
Civita symbol. Denoting unit vectors in R
3
by
n =
sin θ cos φ
sin θ sin φ
cos θ
, θ ∈ [0, π] , φ ∈ [0, 2π[ , (4)
the operator
R
n
(η) = e
−iηJ· n
(5)
describes a rotation by an angle η ∈ [0, 4π[ about the
direction n.
2.2 Symmetries
By definition, the average fidelity in (2) is a positive
function of the angle η and of the state |ψi and pos-
sesses three symmetries: it is 2π-periodic in η, sym-
metric about η = π, and invariant under rotation of
|ψi.
Periodicity with period 2π comes from the fact that
R
n
(2π) = (−1)
N
. Symmetry about η = π is equiva-
lent to
F
|ψi
(η) = F
|ψi
(2π − η) , (6)
which can be shown using R
n
(2π−η) = (−1)
N
R
−n
(η)
and the fact that the set of directions averaged over in
(2) is the same irrespective of the sign of the unit vec-
tor n since the fidelity (1) is given by the the squared
modulus of the overlap between the states |ψi and
R
n
(η)|ψi.
Invariance under rotation of |ψi can be understood
in the following way. Let R
m
(χ) = e
−iχJ·m
be a
unitary operator representing a rotation in R
3
by an
angle χ ∈ [0, 4π[ about the direction m, acting on
a state |ψi ∈ C
N+1
. Then the average fidelities F
associated with the states |ψi and |ψ
R
i ≡ R
m
(χ)|ψi
are equal. Indeed, we have
F
|ψ
R
i
(η, n) = hψ|R
m
(χ)
†
R
n
(η)R
m
(χ)|ψi (7)
and
R
m
(χ)
†
R
n
(η)R
m
(χ) = e
−iη(R
m
(χ)
†
JR
m
(χ))·n
= e
−iη(RJ)·n
= e
−iηJ·n
R
, (8)
with n
R
≡ R
T
n the vector obtained by the rotation
R ∈ SO(3) associated with R
m
(χ). Due to the invari-
ance under rotations of the unit-ball region S
2
ap-
pearing in (2) (invariance of the Haar measure used),
the result of the integration will be the same, leading
to
F
|ψ
R
i
(η) =
1
4π
Z
S
2
F
|ψ
R
i
(η, n) dn
=
1
4π
Z
S
2
F
|ψi
(η, n) dn = F
|ψi
(η) . (9)
This invariance of the fidelity can be seen in a geo-
metrically appealing way by use of the Majorana rep-
resentation, 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 al-
lows 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 of-
ten referred to as stars, and the whole set of Majo-
rana points of a given state as its Majorana constella-
tion. Considering a spin-j state |ψi as an N-qubit
state, any local unitary (LU) operation U = u
⊗N
with u ∈ SU(2) transforms |ψi into a state whose
Majorana constellation is obtained by the constella-
tion of |ψi 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 |φi
⊗N
of some spin-1/2 state
|φi. Their constellation thus reduces to an N-fold de-
generate point.
The fidelity (1) is given by the squared modulus
of the overlap between |ψi and R
n
(η)|ψi. Since the
Majorana constellation of R
n
(η)|ψi is obtained by
rigidly rotating that of |ψi, the fidelity (1) only de-
pends on the relative positions of these two sets of
points. The average transition probability F
|ψi
(η) is
obtained by integrating over all possible constellations
obtained by rigid rotations of the Majorana constel-
lation of |ψi, and therefore it must be invariant under
LU. In other words, the equality (9) takes the form
F
|ψi
(η) = F
u
⊗N
|ψi
(η).
2.4 Anticoherence measures
An order-t anticoherent state |χi is defined by the
property that hχ|(J · n)
k
|χi is independent of the vec-
tor n for all k = 1, . . . , t. In the Majorana represen-
tation, it is characterized by the fact that its t-qubit
reduced density matrix is the maximally mixed state
in the symmetric sector [24].
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The degree of coherence or t-anticoherence of a
spin-j pure state |ψi can be measured by the quan-
tities A
t
(|ψi), which are positive-valued functions of
|ψi [20]. Let ρ
t
= tr
¬t
[|ψihψ|] be the t-qubit reduced
density matrix of the state |ψi 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 |ψi is a
symmetric state). The measures A
t
(|ψi) are defined
as the rescaled linear entropies
A
t
(|ψi) =
t + 1
t
1 − tr
ρ
2
t
, (10)
where tr
ρ
2
t
is the purity of ρ
t
. Thus, anticoher-
ence measures are quartic in the state |ψi and range
from 0 to 1, and are invariant under SU(2) rota-
tions. Spin-coherent states are characterized by pure
reduced states and thus are the only states such that
A
t
= 0. Anticoherent states to order t are character-
ized by ρ
t
= 1/(t+1) and thus are the only states such
that A
t
= 1. In particular, if a state |ψi is anticoher-
ent to some order t, then it is necessarily anticoherent
to all lower orders t
0
= 1, . . . , t since reductions of the
maximally mixed state are maximally mixed.
While for any state we have 0 6 A
t
6 1, not all
possible tuples (A
1
, A
2
, . . .) are realised by a physi-
cal state |ψi. For instance, since A
t
= 1 implies that
A
t
0
= 1 for all t
0
6 t, the choice A
2
= 1 and A
1
< 1
cannot correspond to any state. We denote the do-
main of admissible values of the measures A
t
by Ω.
2.5 Tensor representation of mixed states
We now introduce a tensor representation of an ar-
bitrary (possibly mixed) spin-j state ρ acting on a
(N + 1)-dimensional Hilbert space with N = 2j, fol-
lowing [24]. Any state can be expanded as
ρ =
1
2
N
x
µ
1
µ
2
...µ
N
S
µ
1
µ
2
...µ
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
µ
1
µ
2
...µ
N
are (N + 1) × (N + 1) Her-
mitian matrices invariant under permutation of the
indices.
The x
µ
1
µ
2
...µ
N
are real coefficients also invariant un-
der permutation of their indices, which enjoy what we
call the tracelessness property
3
X
a=1
x
aaµ
3
...µ
N
= x
00µ
3
...µ
N
, ∀ µ
3
, . . . , µ
N
. (12)
Whenever x
µ
1
µ
2
...µ
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
n = (n
1
, n
2
, n
3
), the coefficients in (11) are simply
given by x
µ
1
µ
2
...µ
N
= n
µ
1
n
µ
2
. . . n
µ
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 ρ with coordinates x
µ
1
µ
2
...µ
N
in
the expansion (11). Then, the tensor coordinates of
the t-qubit reduced state ρ
t
in the expansion (11) are
simply given by x
µ
1
µ
2
...µ
t
= x
µ
1
µ
2
...µ
t
0...0
. Thus, since
we omit the zeros in the string µ
1
µ
2
. . . µ
N
, the tensor
coordinates of ρ
t
and ρ coincide for any string of k 6 t
nonzero indices.
The second property we use is that for states ρ and
ρ
0
in the form (11) with tensor coordinates respec-
tively x
µ
1
µ
2
...µ
N
and x
0
µ
1
µ
2
...µ
N
we have
tr [ρρ
0
] =
1
2
N
X
µ
1
,µ
2
,...,µ
N
x
µ
1
µ
2
...µ
N
x
0
µ
1
µ
2
...µ
N
. (13)
Note that this equality holds despite the fact that
the S
µ
1
µ
2
...µ
N
are not orthogonal; this property fol-
lows from the fact that these matrices form a 2
N
-
tight frame, see [24]. In particular, for a pure state
ρ = |ψihψ|, the equality trρ
2
= 1 translates into
X
µ
1
,µ
2
,...,µ
N
x
2
µ
1
µ
2
...µ
N
= 2
N
, (14)
while the purity of the reduced density matrix ρ
t
reads
tr
ρ
2
t
=
1
2
t
X
µ
1
,µ
2
,...,µ
t
x
2
µ
1
µ
2
...µ
t
. (15)
The normalization condition tr [ρ] = 1 imposes
x
00...0
= 1. A consequence of (12) is then that
P
3
a=1
x
aa
= 1.
2.6 SU(2)-Invariants
If u ∈ SU(2) and R ∈ SO(3) is the corresponding
rotation matrix, then the tensor coordinates of UρU
†
with U = u
⊗N
are the R
µ
1
ν
1
. . . R
µ
N
ν
N
x
ν
1
...ν
N
where
R is the 4 × 4 orthogonal matrix
R =
1 0
0 R
!
. (16)
That is, x
µ
1
µ
2
...µ
N
transforms as a tensor. Under
such transformations, x
µ
x
µ
goes into R
µν
R
µν
0
x
ν
x
ν
0
=
(R
T
R)
ν
0
ν
x
ν
x
ν
0
= x
ν
x
ν
, where the last equality comes
from orthogonality of R. Thus x
µ
x
µ
is an SU(2) in-
variant. Similarly, x
µ
x
µν
x
ν
and, more generally, any
product of the x
µ
1
µ
2
...µ
N
such that all indices are con-
tracted (i.e. summed from 0 to 3), are invariant under
SU(2) action on ρ. One can then show by induction
that products of terms x
a
1
a
2
...a
k
with k 6 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
...a
2k
,
where the a
i
appear in pairs. Since the order of in-
dices is not relevant, these invariants are in fact of
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the form x
a
1
a
1
a
2
a
2
...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
...a
k
a
k
= x
00...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
...a
k
x
b
1
b
2
...b
k
0
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
. . . a
k
or b
1
b
2
. . . b
k
0
), then from Eq. (12), they
can again be replaced by zeros and discarded. Thus
the invariants of degree 2 are κ
1
= x
a
x
a
, κ
2
= x
ab
x
ab
,
and more generally, for 1 6 r 6 N,
κ
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
κ
r
in terms of a linear combination of the A
t
. In-
deed, grouping together terms with the same number
of nonzero indices in (15) yields
tr
ρ
2
t
=
1
2
t
X
µ
1
,µ
2
,...,µ
t
x
2
µ
1
µ
2
...µ
t
=
1
2
t
t
X
r=0
t
r
κ
r
.
(18)
Inverting that relation via the binomial inversion for-
mula, we obtain
κ
r
=
r
X
t=0
(−1)
t+r
2
t
r
t
tr
ρ
2
t
, (19)
and by use of (10) we finally can express the SU(2)-
invariants in terms of anticoherence measures,
κ
r
=
r
X
t=0
(−1)
t+r
2
t
r
t
1 −
t
t + 1
A
t
(20)
for r = 1, . . . , N.
2.7 General form of the average fidelity
Let us now explain why the average fidelity F
|ψi
(η)
given in Eq. (3) is a linear combination of the low-
est bjc anticoherent measures A
t
. Due to its rota-
tional symmetry, the average fidelity F
|ψi
(η)—when
considered as a function of the tensor coordinates
x
µ
1
µ
2
...µ
N
—can only involve invariants constructed
from these coordinates. With F
|ψi
(η) being quadratic
in ρ = |ψihψ|, 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 F
|ψi
(η) are the invari-
ants κ
r
defined in (17). Since the quantity F
|ψi
(η)
is quadratic it must be a linear combination of the
coefficients κ
r
which, according to Eq. (20), implies
that F
|ψi
(η) is also a linear combination of the A
t
.
Furthermore, the identity
tr
ρ
2
t
= tr
ρ
2
N−t
, (21)
which holds for any pure state, means that the antico-
herence measures A
t
for t > N/2 can be expressed in
terms of the measures A
t
for t < N/2. Therefore, (3)
is the most general form the fidelity F
|ψi
(η) can take,
with the dependence in η being only in the coefficients
of the measures 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
1
4π
Z
S
2
|hψ|U
n
(η)|ψi|
2
dn (22)
between a state |ψi and its image under the unitary
U
n
(η) = e
−iη f(J·n)
, (23)
where f is an arbitrary real analytic function, ensur-
ing that f(J · n) is an Hermitian operator. Indeed,
from an argument similar to that of Sec. 2.6, the gen-
eralized fidelity (22) can be expressed as a function
of the κ
r
and hence of the A
t
. An interesting case is
when U
n
(η) is a spin-squeezing operator, which cor-
responds to choosing f(J · n) = (J · n)
2
. Moreover, if
we now consider the quantities
1
4π
Z
S
2
|hψ|U
n
(η)|ψi|
2k
dn (24)
with integer k > 2, the same arguments show that
they are linear combinations of higher-order invari-
ants, leading to generalizations of the relation (20).
3 Closed form of the average fidelity
In this section we derive the angular functions ϕ
(j)
t
(η),
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 represen-
tation 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 mea-
sures for Dicke states
In the following, we will work in the standard angu-
lar momentum basis of C
N+1
, for positive integer or
half-integer value of j = N/2. It consists of the Dicke
states {|j, mi, |m| 6 j} given by the common eigen-
states of J
2
, the square of the angular momentum
operator J, and of its z-component J
z
. In this basis,
any spin-j state |ψi can be expanded as
|ψi =
j
X
m=−j
c
m
|j, mi, (25)
with c
m
∈ C and
P
j
m=−j
|c
m
|
2
= 1.
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The first derivation is based on the fact that both
the measures of t-anticoherence A
t
(|j, mi) and the av-
erage fidelities F
|j,mi
(η) can be determined explicitly
for Dicke states. Their measures of t-anticoherence
are given by
A
t
(|j, mi) =
t + 1
t
1 −
P
t
`=0
j+m
t−`
2
j−m
j−m−`
2
2j
t
2
.
(26)
They can readily be obtained from the purities tr
ρ
2
t
for a state of the form (25), which were calculated in
[20] in terms of the coefficients c
m
and read
tr
ρ
2
t
=
t
X
q,`=0
2j−t
X
k=0
c
∗
j−k−`
c
j−k−q
Γ
`q
k
2
(27)
with
Γ
`q
k
=
q
2j−k−q
t−q
2j−k−`
t−`
k+q
k
k+`
k
2j
t
. (28)
As for the fidelity, the calculation is done in Appendix
B and yields
F
|j,mi
(η) =
1
(2j + 1)
2
2j
X
`=0
(2` + 1)(C
jm
jm`0
χ
j
`
(η))
2
,
(29)
with Clebsch-Gordan coefficients C
jm
jm`0
and the func-
tions χ
j
`
(η) defined in Eqs. (77)–(78). The angular
functions ϕ
(j)
t
(η) are then solutions of the system of
linear equations
F
|j,mi
(η) = ϕ
(j)
0
(η) +
P
bjc
t=1
ϕ
(j)
t
(η) A
t
(|j, mi)
for m = j, j − 1, . . . , j − bjc.
(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 ϕ
(j)
t
(η). It is based on an
expansion of the operator
Π
(j)
(q) ≡ (q
2
0
− |q|
2
)
j
e
−2θ
q
ˆ
q·J
, (31)
with tanh θ
q
= −|q|/q
0
and
ˆ
q = q/|q|, as a multi-
variate 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
, q) = (q
0
, q
1
, q
2
, q
3
). As shown in [25],
it can be written as
Π
(j)
(q) = (−1)
2j
q
µ
1
q
µ
2
. . . q
µ
2j
S
µ
1
µ
2
...µ
2j
. (32)
The identification of Eqs. (31) and (32) defines the
(N + 1) ×(N + 1) matrices S
µ
1
...µ
N
appearing in (11)
(see [24] for detail). Taking
q
0
= i cot(η/2) and q
i
= n
i
, i = 1, 2, 3 , (33)
in (31), we see that Π
(j)
(q) reduces to a rotation op-
erator,
R
n
(η) = e
−iηJ· n
=
Π
(j)
(q)
m
N
(34)
with
m
2
= q
2
0
− |q|
2
= −
1
sin
2
(η/2)
. (35)
Moreover, for a state ρ given by (11) we have
tr
h
ρ Π
(j)
(q)
i
= (−1)
N
x
µ
1
µ
2
...µ
N
q
µ
1
. . . q
µ
N
, (36)
according to Eq. (24) of [24], which holds for any 4-
vector q. Thus, with ρ = |ψihψ|, 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
F
|ψi
(η) =
1
4π
Z
S
2
|hψ|R
n
(η)|ψi|
2
dn
=
1
4π
Z
S
2
tr
ρ
Π
(j)
(q)
m
N
2
dn
= (−1)
N
x
µ
1
...µ
N
x
ν
1
...ν
N
4π
×
Z
S
2
q
µ
1
. . . q
µ
N
q
∗
ν
1
. . . q
∗
ν
N
m
2N
dn,
(37)
where ∗ denotes complex conjugation (which acts on
q
0
only because of the choice (33) and using |m|
2
=
−m
2
). Each term q
µ
1
. . . q
∗
ν
N
with 2(N − k) indices
equal to 0 is proportional to
q
2(N−k)
0
m
2N
= (−1)
k
sin
2k
η
2
cos
2(N−k)
η
2
. (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
1
4π
Z
S
2
n
a
1
n
a
2
. . . n
a
2k
dn , 1 6 a
i
6 3 . (39)
These integrals are performed in Appendix C. The in-
tegrals (39) are in fact precisely given by the tensor
coordinates x
(0)
a
1
a
2
...a
2k
of the maximally mixed state,
whose expression is explicitly known. One can there-
fore rewrite (37) as
F
|ψi
(η) =
N
X
k=0
(−1)
N
q
2(N−k)
0
m
2N
×
X
µ,ν
2(N−k)zeros
(−1)
nr of 0 in ν
x
(0)
µ
1
...µ
N
ν
1
...ν
N
x
µ
1
...µ
N
x
ν
1
...ν
N
,
(40)
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where the sum over µ,ν runs over all strings of in-
dices (between 0 and 3) containing 2(N − k) zeros.
An explicit expression for this sum is derived in Ap-
pendix C, leading to the compact expression
F
|ψi
(η) =
N
X
k=0
sin
2k
η
2
cos
2(N−k)
η
2
N
X
t=0
a
(j)
t,k
tr
ρ
2
t
,
(41)
with numbers
a
(j)
t,k
=
4
t
(−1)
k+t
2N
2k
k
t
2N−2t
N−t
(2k + 1)
2N
N
. (42)
Note that the sum over k in (41) can start at k = t
because the factor
k
t
in a
(j)
t,k
implies that a
(j)
t,k
= 0
for t > k. Using the symmetry tr
ρ
2
t
= tr
ρ
2
N−t
we
may rewrite (41) as
F
|ψi
(η) =
N
X
k=t
sin
2k
η
2
cos
2(N−k)
η
2
×
bjc
X
t=0
a
(j)
t,k
+ a
(j)
N−t,k
1 −
δ
jt
2
tr
ρ
2
t
.
(43)
From (10) we obtain a relation between A
t
and tr
ρ
2
t
,
namely tr
ρ
2
t
= 1 −
t
t+1
A
t
, which yields the explicit
expression of the polynomials ϕ
(j)
t
(η) in Eq. (3) as
ϕ
(j)
t
(η) =
N
X
k=t
b
(j)
t,k
sin
2k
η
2
cos
2(N−k)
η
2
, (44)
with coefficients
b
(j)
t,k
=
−
t
t + 1
a
(j)
t,k
+ a
(j)
N−t,k
1 −
δ
jt
2
t 6= 0
N
k
2k + 1
t = 0 .
(45)
Note that although q
0
and m are not well-defined for
η = 0, the ratio in (38) always is, so that the expres-
sion above is valid over the whole range of values of
η. For spin-coherent states, all A
t
vanish and thus
F
|ψi
(η) = ϕ
(j)
0
(η) from Eq. (3), which coincides with
the expression obtained in [15]. For the smallest val-
ues of j, we recover the functions obtained in Section
3.1. In the following section, we will use the functions
ϕ
(j)
t
(η) given in (44) to identify optimal quantum ro-
tosensors.
4 Optimal quantum rotosensors
4.1 Preliminary remarks
We now address the question of finding the states |ψi
which minimize the average fidelity F
|ψi
(η) for fixed
rotation angles η. According to Eq. (3), the fidelity
is a linear function of the anticoherence measures A
t
with 1 6 t 6 bjc. Linearity, when combined with
the fact that the domain Ω, over which the measures
A
t
vary, is bounded implies that the fidelity must at-
tain its minimum on the boundary. The minimization
problem thus amounts to characterizing this domain
Ω. Unfortunately, even for the smallest values of j,
no simple descriptions of this domain are known.
We will first determine the states minimizing the
2π-periodic average fidelity for values of j up to
j = 7/2, with the rotation angle taking values in the
interval η ∈ [0, π] (which is sufficient due to the sym-
metry (6)). Then we will examine the limiting case
of angles η close to 0 for arbitrary values of the quan-
tum number j. Throughout this section, we will ex-
pand 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 av-
erage fidelity F
|ψi
(η) 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 quan-
tum numbers up to j = 7/2, mainly identifying the
optimal rotosensors within various ranges of the ro-
tation angle η by numerical techniques. More specifi-
cally, for a fixed angle η, F
|ψi
(η) is a function of the
A
t
which can be parametrized by the complex coef-
ficients c
m
entering the expansion (25) of the state
|ψi in the Dicke basis (see Eq. (27)). We search nu-
merically for the minimum value of F
|ψi
(η) with re-
spect to the c
m
, taking into account the normalization
condition
P
m
|c
m
|
2
= 1. In most cases this numeri-
cal 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 η in order to explore the
whole range of rotation angles. Whenever we find a
region of values of η in which |ψ
1
i is the optimal state
adjacent to a region where |ψ
2
i is optimal, at the crit-
ical angle separating these two regions, one should
have F
|ψ
1
i
(η) = F
|ψ
2
i
(η) because the average fidelity
F
|ψi
(η) is a continuous function of |ψi. Therefore, the
critical angle is a solution of the equation
bjc
X
t=1
ϕ
(j)
t
(η) A
t
(|ψ
1
i) =
bjc
X
t=1
ϕ
(j)
t
(η) A
t
(|ψ
2
i). (46)
4.2 Rotosensors for arbitrary rotation angles η
and j 6 2
4.2.1 j = 1/2
For a spin 1/2, all pure states are coherent: each state
|ψi can be obtained by a suitable rotation of the state
|
1
2
,
1
2
i. Since the fidelity is invariant under rotation,
all states are equally sensitive to detect rotations for
any angle η.
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4.2.2 j = 1
For j = 1, the expansion (3) takes the form
F
|ψi
(η) = ϕ
(j)
0
(η) + ϕ
(1)
1
(η) A
1
, (47)
with
ϕ
(1)
0
(η) =
1
15
6 cos(η) + cos(2η) + 8
,
ϕ
(1)
1
(η) = −
1
15
2 cos(η) − 3 cos(2η) + 1
.
(48)
The first strictly positive zero of ϕ
(1)
1
(η) is given by
η
0
= arccos(−2/3). In the interval η ∈ [0, η
0
[, where
ϕ
(1)
1
(η) is negative, the fidelity F
|ψi
(η) is minimized
by states with A
1
= 1, i.e. by 1-anticoherent states.
For η = η
0
, the fidelity takes the same value for
all states |ψi, namely F
|ψi
(η
0
) = ϕ
(1)
0
(η
0
) = 7/27.
For rotation angles in the the remaining interval,
η ∈]η
0
, π], where ϕ
(1)
1
(η) is positive, F
|ψi
(η) is min-
imized for states with 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
F
|ψi
(η) = ϕ
(3/2)
0
(η) + ϕ
(3/2)
1
(η) A
1
, (49)
with
ϕ
(3/2)
0
(η) =
1
70
cos(3η) + 8 cos(2η) + 29 cos(η) + 32
,
ϕ
(3/2)
1
(η) =
3
70
3 cos(3η) + 3 cos(2η) − 4 cos(η) − 2
.
(50)
The situation is basically the same as for j = 1. The
first strictly positive zero of the coefficient ϕ
(3/2)
1
(η)
is found to be η
0
= arccos(
−9+
√
21
12
). Hence, in the
interval η ∈ [0, η
0
[ where ϕ
(3/2)
1
(η) is negative, the
fidelity F
|ψi
(η) is minimal for 1-anticoherent states.
At the value η = η
0
, the fidelity takes the same value
for all states |ψi, namely, F
|ψi
(η
0
) = ϕ
(3/2)
0
(η
0
) =
(33 + 2
√
21)/80. Otherwise, F
|ψi
(η) 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,
F
|ψi
(η) = ϕ
(2)
0
(η) + ϕ
(2)
1
(η) A
1
+ ϕ
(2)
2
(η) A
2
, (51)
with the angular functions ϕ
(2)
k
, k = 0, 1, 2, displayed
in Appendix D. They all take negative values in the
interval η ∈ [0, η
0
], with η
0
≈ 1.2122 the first strictly
positive zero of ϕ
(2)
1
(η). The tetrahedron state
|ψ
tet
i =
1
2
|2, −2i + i
√
2 |2, 0i + |2, 2i
, (52)
whose Majorana points lie at the vertices of a regu-
lar tetrahedron, is 2-anticoherent, and for j = 2 it is
the only state (up to LU) with A
1
= A
2
= 1 [26];
hence it provides the optimal rotosensor for angles in
the interval η ∈ [0, η
0
]. For larger angles of rotation
comprised between 1.68374 and 2.44264, we find nu-
merically that an optimal state is the Schrödinger cat
state
|ψ
cat
i =
1
√
2
(|2, −2i + |2, 2i) , (53)
which is only 1-anticoherent, with A
1
= 1 and A
2
=
3/4. For values η & 2.44264, the optimal state is a
coherent state.
We thus obtain numerically three intervals with
three distinct optimal states corresponding to
(A
1
, A
2
) = (1, 1), (1, 3/4), and (0, 0), respectively. In
order to find the critical angles, we solve Eq. (46).
The angle η
1
separating the first two regions is a so-
lution of ϕ
(2)
2
(η) = 0. The first positive zero of ϕ
(2)
2
(η)
is η
1
= 2 arctan(
p
9 − 2
√
15) ≈ 1.68374, which coin-
cides with the numerically obtained value. The angle
η
2
at which the second and third region touch, is a
zero of ϕ
(2)
1
(η) +
3
4
ϕ
(2)
2
(η). Its first strictly positive
zero is given by
η
2
= 2 arctan
r
−
a + 102b
a − 38b
!
, (54)
with a = 19 6
2/3
+
3
√
6
223 − 35
√
7
2/3
and b =
3
p
223 − 35
√
7, and we have indeed η
2
≈ 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 A
1
= 1 and A
2
=
3/4. For instance, any state of the form
|ψi =
c
1
|2, −1i + c
2
|2, 0i − c
∗
1
|2, 1i
p
2|c
1
|
2
+ |c
2
|
2
(55)
with c
1
∈ C and c
2
∈ R come with the same measures
of anticoherence, as readily follows from Eq. (27).
These states are thus also optimal in the interval
η ∈ [η
1
, η
2
], thereby removing the uniqueness of opti-
mal rotosensors observed for j = 1 and j = 3/2.
4.3 Rotosensors for 5/2 6 j 6 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 rota-
tion is the 1-anticoherent state with the largest mea-
sure of 2-anticoherence, that is given by
|ψi =
1
√
2
|
5
2
, −
3
2
i + |
5
2
,
3
2
i
, (56)
and has A
1
= 1 and A
2
= 99/100. This state is found
to be optimal up to a critical angle η
1
≈ 1.49697,
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πη
2
η
1
π
4
0
1
0.8
0.6
0.4
0.2
0
t = 2
t = 1
η
A
t
1
0.8
0.6
0.4
0.2
0
F
|ψi
(η)
Figure 1: Average fidelity F
|ψi
(η) (top, red solid curve) and
measures of anticoherence A
t
(bottom) for optimal states
with j = 2, as functions of the rotation angle η; the values
of the measures A
t
for the optimal states are discontinuous
at the values η
1
≈ 1.68374 and η
2
≈ 2.44264 (see text for
details). The dashed curve on top shows the average fidelity
ϕ
(2)
0
(η) for coherent states. The blue (red) shaded area shows
the range of rotation angles for which anticoherent states to
order bjc (coherent states) are optimal.
which is obtained from Eq. (46) and coincides with
the first strictly positive zero of ϕ
(5/2)
2
(η). 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 distribu-
tion [28, 29]. The Majorana constellation of this state
defines a triangular bipyramid, which is a spherical
1-design [30, 31], thus corresponding to the arrange-
ment 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 η
1
and
η
2
≈ 2.2521, we find that an optimal state is
|ψ
cat
i =
1
√
2
|
5
2
, −
5
2
i + |
5
2
,
5
2
i
; (57)
unlike in the case j = 2, we found this state for j =
5/2 to be the only state (up to LU) with A
1
= 1 and
A
2
= 3/4. For η ∈ [η
2
, π], we find that coherent states
are optimal. The transition occurs at the first strictly
p