Optimal Detection of Rotations about Unknown Axes by Coherent and Anticoherent States
John Martin, Stefan Weigert, Olivier Giraud
Abstract
Coherent and anticoherent states of spin systems up to spin <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>j</mml:mi></mml:math>. We identify optimal rotosensors (i) for arbitrary rotation angles in the case of spin quantum numbers up to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>7</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math> and (ii) for small rotation angles in the case of spin quantum numbers up to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:math>. 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>j</mml:mi></mml:math>.