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Physical Review A

Optimal quantum rotosensors

Chryssomalis Chryssomalakos, H. Hernández-Coronado

Published May 25, 2017• 34 citations
Publisher

Abstract

We look for optimal quantum rotosensors, i.e., quantum spin states that are optimal in detecting rotations by a given angle. The exact quantity to be minimized is the probability that the rotated state projects onto the original one, averaged uniformly over all rotation axes. We show analytically that, for small rotation angles, the solution is given by anticoherent states. Numerical analysis shows that, for spin $s=1$ and $3/2$, there exists, in each case, a critical rotation angle beyond which the minimum is instead achieved by the coherent states, the transition between the two being discontinuous. For $s=2$, a third degenerate minimum appears in between the above two, consisting of pairs of antipodal points, regardless of the angle between their axes. We also determine the worst rotosensors, i.e., those states that maximize the above quantity; interestingly, the same set of states shows up but for different ranges of the rotation angle.

DOI
10.1103/physreva.95.052125

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We look for optimal quantum rotosensors, i.e., quantum spin states that are optimal in detecting rotations by a given angle. The exact quantity to be minimized is the probability that the rotated state projects onto the original one, averaged uniformly over all rotation axes. We show analytically that, for small rotation angles, the solution is given by anticoherent states. Numerical analysis shows that, for spin s=1s=1s=1 and 3/23/23/2, there exists, in each case, a critical rotation angle beyond which the minimum is instead achieved by the coherent states, the transition between the two being discontinuous. For s=2s=2s=2, a third degenerate minimum appears in between the above two, consisting of pairs of antipodal points, regardless of the angle between their axes. We also determine the worst rotosensors, i.e., those states that maximize the above quantity; interestingly, the same set of states shows up but for different ranges of the rotation angle.
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 .
The efficiency of a quantum metrology protocol can be significantly diminished by the interaction of the system with its environment, leading to a loss of purity and, as a result, a mixed state for the probing system. An example is the measurement of a magnetic field through the rotation of a spin that is subject to decoherence due to its coupling to a surrounding spin or bosonic bath. In this work we define mixed optimal quantum rotosensors (OQRs) as mixed spin-jjj states that achieve maximum sensitivity to estimate infinitesimal rotations, when the rotation axis is unknown. We study two scenarios, where the probe states saturate the averaged fidelity or the averaged quantum Cram\'er-Rao bound, the latter giving the ultimate sensitivity. We find that mixed OQRs can achieve sensitivity equal to that of pure states and are obtained by mixing states from linear subspaces of anticoherent states. We present several examples of mixed OQRs and their associated anticoherent subspaces. We also show that OQRs maximize entanglement in a specific sense, preserving the known relation between entanglement and optimal rotation sensitivity for pure states, even in the context of mixed states. Our results highlight the interconnection between quantum metrology of rotations, anticoherence, and entanglement in mixed spin states.
Abstract A multitude of engineering and scientific applications require the estimation of an unknown axis’ rotation, known as the reference frame alignment problem. These problems can be categorized using the quantum Cramer–Rao bound. Detecting rotations is typically focused on smaller angles, as errors are more significant in this range. Therefore, selecting the optimal combination of measurement techniques based on a quantum entangled state for a particular setup is a critical step. This paper proposes a linear combination of anticoherence and coherence measures based on the generalized GHZ entanglement states. The anticoherence measures, optimizes the spin system’s average fidelity in detecting rotated states, while the overlap between states in any spin number is minimal. The optimal quantum rotosensors are characterized by the minimal average fidelity given by the overlap before and after averaging over all directions. The individual subsystems’ states have substantial overlap with GHZ states, making the generalized GHZ state the entanglement quantum spin-j state with coherence decay rate ϵ beneficial in finding the optimal rotation detection on noncoherent entanglement state.
Spin anticoherent states acquired recently a lot of attention as the most "quantum" states. Some coherent and anticoherent spin states are known as optimal quantum rotosensors. In this work, we introduce a measure of quantumness for orthonormal bases of spin states, determined by the average anticoherence of individual vectors and the Wehrl entropy. In this way, we identify the most coherent and most quantum states, which lead to orthogonal measurements of extreme quantumness. Their symmetries can be revealed using the Majorana stellar representation, which provides an intuitive geometrical representation of a pure state by points on a sphere. Results obtained lead to maximally (minimally) entangled bases in the 2j+1 dimensional symmetric subspace of the 22j dimensional space of states of multipartite systems composed of 2j qubits. Some bases found are iso-coherent as they consist of all states of the same degree of spin-coherence.
In this paper we introduce and investigate the concept of a perfect quantum protractor , a pure quantum state |ψ⟩∈H that generates three different orthogonal bases of H under rotations around each of the three perpendicular axes. Such states can be understood as pure states of maximal uncertainty with regards to the three components of the angular momentum operator, as we prove that they maximise various entropic and variance-based measures of such uncertainty. We argue that perfect quantum protractors can only exist for systems with a well-defined total angular momentum j , and we prove that they do not exist for j∈{1/2,2,5/2} , but they do exist for j∈{1,3/2,3} (with numerical evidence for their existence when j=7/2 ). We also explain that perfect quantum protractors form an optimal resource for a metrological task of estimating the angle of rotation around (or the strength of magnetic field along) one of the three perpendicular axes, when the axis is not a priori known. Finally, we demonstrate this metrological utility by performing an experiment with warm atomic vapours of rubidium-87, where we prepare a perfect quantum protractor for a spin-1 system, let it precess around x , y or z axis, and then employ it to optimally estimate the rotation angle.
Many protocols require precise rotation measurement. Here we present a general class of states that surpass the shot-noise limit for measuring rotation around arbitrary axes. We then derive a quantum Cram\'er-Rao bound for simultaneously estimating all three parameters of a rotation (e.g., the Euler angles) and discuss states that achieve Heisenberg-limited sensitivities for all parameters; the bound is saturated by ``anticoherent'' states [Zimba, Electron. J. Theor. Phys. 3, 143 (2006)] (we are reluctant to use ``anticoherent'' to describe the states, but the name has become commonplace over the last decade). Anticoherent states have garnered much attention in recent years, and we elucidate a geometrical technique for finding new examples of such states. Finally, we discuss the potential for divergences in multiparameter estimation due to singularities in spherical coordinate systems. Our results are useful for a variety of quantum metrology and quantum communication applications.