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-
Accepted in Quantum 2020-06-16, click title to verify. Published under CC-BY 4.0. 2