Overcoming sloppiness for enhanced metrology in a Mach–Zehnder interferometer

Multi-parameter statistical models may depend only on some functions of the parameters that are fewer than the number of initial parameters themselves. Such sloppy statistical models are characterized by a degenerate Fisher Information matrix, indicating that it is impossible to simultaneously estimate all the parameters. In a quantum setting, once an encoding is fixed, the same can happen for the Quantum Fisher Information matrix computed from a sloppy quantum statistical model. In addition to sloppiness, however, further issues of quantum incompatibility can arise. We take a fully Gaussian case-study to investigate the topic, showing that by appropriately scrambling the quantum states in between the encoding of two phase-shift parameters a Mach–Zehnder interferometer, not only sloppiness can be lifted, but also the quantum incompatibility can be put identically to zero, maintaining an enhanced scaling of precision and the covariance of the model with respect to exact values of the parameters.