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 .
Comparison of quantum objects is the task of determining relational properties, such as whether two unknown objects are the same or different, without identifying the objects themselves. Multiple copies of unknown states or multiple queries to unknown channels are natural resources for improving comparison, and the optimal strategy for pure-state comparison with multiple copies is known. For unitary-channel comparison with multiple queries, however, the optimal strategy has been unclear because different queries can be arranged in parallel, sequentially, adaptively, or in more general causal structures. We study comparison of two unknown
d-dimensional unitary channels using
N1 and
N2 queries to the two channels, respectively, under the promise that they are either identical Haar-random unitaries or independent Haar-random unitaries. We optimize over valid general testers, a class containing ordinary quantum testers, classically controlled causal orders, and indefinite-causal-order processes. We characterize the optimal minimum-error and one-sided unambiguous comparison probabilities for arbitrary finite query numbers. The upper bound in fact holds for a larger positive relaxation, while the optimum is attained by a valid parallel tester. The optimum is determined by a finite representation-theoretic parameter built from Young diagrams and Littlewood--Richardson coefficients. For fixed
N1, the performance is saturated for every
N2≥(d−1)N1. By contrast, the corresponding pure-state comparison parameter has no finite-
N2 saturation and reaches its known-reference limit only asymptotically, yielding continued success-probability improvement in the nontrivial decision regime. This highlights a sharp difference between comparison tasks for states and channels, analogous to distinctions known in quantum discrimination.
For a system of N spins there are quantum states that can encode a direction in an intrinsic way. Information on this direction can later be decoded by means of a quantum measurement. We present here the optimal encoding and decoding procedure using the fidelity as a figure of merit. We compute the maximal fidelity and prove that it is directly related to the largest zeros of the Legendre and Jacobi polynomials. We show that this maximal fidelity approaches unity quadratically in
1/N. We also discuss this result in terms of the dimension of the encoding Hilbert space.