Optimize information over quantum measurements

For a differentiable state family $\rho_\lambda$, the symmetric logarithmic derivative $L_\lambda$ is Hermitian and satisfies a symmetrized operator equation. This is the quantum counterpart of relating a probability derivative to its logarithmic derivative. The symmetrization matters because density matrices and their derivatives need not commute. Quantum Fisher information is the expectation of $L_\lambda^2$. It is nonnegative and carries the inverse square of the parameter unit, just as classical information does.

For full-rank states the SLD is uniquely determined. If the state has a kernel, some operator components can be undetermined without changing information on the supported state; rank-changing points need care. An eigenbasis measurement of the SLD attains the quantum information locally for the regular one-parameter problem. Its basis can depend on the unknown parameter, so this statement describes a local optimum rather than an automatically implementable detector covering every parameter value. No actual instrument is specified merely by calculating the optimum.

For a normalized pure state, differentiating the state vector gives a tangent. The component parallel to the state changes only its overall phase and cannot distinguish physical states. Subtracting that component leaves the physically meaningful tangent, whose squared norm gives quantum Fisher information up to a factor of four. Under encoding $\lvert\psi_\lambda\rangle=\exp(-i\lambda G)\lvert\psi_0\rangle$ with a parameter-independent Hermitian generator, the result is four times its variance in the input state. An eigenstate of the generator therefore has zero information.

For a qubit phase rotation generated by $G=\sigma_z/2$, the state at polar angle $\theta$ has generator variance $\sin^2\theta/4$ and phase information $\sin^2\theta$. The equator maximizes it, while either pole gives zero. The azimuth chooses the initial direction but does not change this phase information. This result describes a pure state with unitary encoding. Replacing a noisy mixed sensor by a point on the surface of the Bloch sphere would erase the physical decoherence that the information calculation is supposed to include.

Measuring the z populations of this rotating pure qubit gives no phase dependence at any polar angle. Its classical phase information is zero even when the quantum information is positive. A rotated readout can access the transverse motion. Thus a failed detector does not prove the preparation lacks metrological value. Conversely, a large calculated quantum information is only an upper limit on what a restricted detector can extract. Include detection noise and accessible operations before claiming an experimental advantage.

Information also depends on which quantity is being estimated. If phase is an angular-frequency offset multiplied by a known interrogation time, frequency information gains the square of that time. Repetitions multiply information for independent identical copies. Neither conversion creates information for free: longer coherent interaction costs time and may increase decoherence, while more copies cost preparation and measurement resources. State the generator normalization before comparing papers, since doubling the generator doubles the phase response and quadruples the numerical information at a fixed parameter convention.