Equation 12 · The Bit Comes Back Before the Bearing
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The single-axis U(1) choice above is deliberate rather than a simplification smuggled in for convenience. A compass, in the ordinary sense of the word, reports one angle against one fixed axis; a full SU(2) orientation, tracking three Euler angles against a Cartesian triad, is the harder and more general problem Peres and Scudo actually solved for a single quantum carrier; a bounded-size probe transmits a frame with a fidelity that improves with the carrier’s own total angular momentum but never reaches unity at finite size [ 10 ] . Restricting to one axis keeps the generator a single Hermitian operator with a single real eigenvalue spectrum, which is exactly the structure the quantum…
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The single-axis U(1) choice above is deliberate rather than a simplification smuggled in for convenience. A compass, in the ordinary sense of the word, reports one angle against one fixed axis; a full SU(2) orientation, tracking three Euler angles against a Cartesian triad, is the harder and more general problem Peres and Scudo actually solved for a single quantum carrier; a bounded-size probe transmits a frame with a fidelity that improves with the carrier’s own total angular momentum but never reaches unity at finite size [ 10 ] . Restricting to one axis keeps the generator a single Hermitian operator with a single real eigenvalue spectrum, which is exactly the structure the quantum Cramér-Rao bound below is built for; nothing in the argument that follows depends on stopping at one axis, and a full three-axis version would simply replace a single Fisher information with a Fisher information matrix, without changing which mechanism is doing the work.
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