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arXiv:2310.13045 · Quantum

Perfect quantum protractors

Michał Piotrak, Marek Kopciuch, Arash Dezhang Fard, Magdalena Smolis, Szymon Pustelny, Kamil Korzekwa

Published Sep 3, 2024• 3 citations
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Abstract

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.

DOI
10.22331/q-2024-09-03-1459

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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.
In this work we study various notions of uncertainty for angular momentum in the spin- s representation of SU(2). We characterize the ‘uncertainty regions’ given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s . Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s , since they are saturated by states taken from the Holstein–Primakoff approximation. We show that, for all s , all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius s ( s + 1 ) . Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s . For large s the Maassen–Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length r ( s ) , where r ( s ) / s → 1 as s → ∞ .
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.
In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the 'uncertainty regions' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty.
The quantum rotor represents, after the harmonic oscillator, the next obvious quantum system to study the complementary pair of variables: the angular momentum and the unitary shift operator in angular momentum. Proper quantification of uncertainties and the incompatibility of these two operators are thus essential for applications of rotorlike quantum systems. While angular momentum uncertainty is characterized by variance, several uncertainty measures have been proposed for the shift operator, with dispersion the simplest example. We establish a hierarchy of those measures and corresponding uncertainty relations which are all perfectly or almost perfectly saturated by a tomographically complete set of von Mises states. Building on the interpretation of dispersion as the moment of inertia of the unit ring we then show that the other measures also possess the same mechanical interpretation. This unifying perspective allows us to express all measures as a particular instance of a single generic angular uncertainty measure. The importance of these measures is then highlighted by applying the simplest two of them to derive optimal simultaneous measurements of the angular momentum and the shift operator. Finally, we argue that the model of quantum rotor extends beyond its mechanical meaning with promising applications in the fields of singular optics, superconductive circuits with a Josephson junction, or optimal pulse shaping in the time-frequency domain. Our findings lay the groundwork for quantum-information and metrological applications of the quantum rotor and point to its interdisciplinary nature.
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.
Uncertainty relations are key components in the understanding of the nature of quantum mechanics. In particular, entropic relations are preferred in the study of angular position and angular momentum states. We propose a new form of angle–angular momentum state that provides, for all practical purposes, a lower bound on the entropic uncertainty relation, , for any given angular uncertainty, thus improving upon previous bounds. We establish this by comparing this sum with the absolute minimum value determined by a global numerical search. These states are convenient to work with both analytically and experimentally, which suggests that they may be of use for quantum information purposes.