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.
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.
Abstract Multiparameter quantum estimation theory plays a crucial role in advancing quantum metrology. Recent studies focused on fundamental challenges such as enhancing precision in the presence of incompatibility or sloppiness, yet the relationship between these features remains poorly understood. In this work, we explore the connection between sloppiness and incompatibility by introducing an adjustable scrambling operation for parameter encoding. Using a minimal yet versatile two-parameter qubit model, we examine the trade-off between sloppiness and incompatibility and discuss: (1) how information scrambling can improve estimation, and (2) how the correlations between the parameters and the incompatibility between the symmetric logarithmic derivatives impose constraints on the ultimate quantum limits to precision. Through analytical optimization, we identify strategies to mitigate these constraints and enhance estimation efficiency. We also compare the performance of joint parameter estimation to strategies involving successive separate estimation steps, demonstrating that the ultimate precision can be achieved when sloppiness is minimized. Our results provide a unified perspective on the trade-offs inherent to multiparameter qubit statistical models, offering practical insights for optimizing experimental designs.
Multiparameter quantum estimation becomes challenging when the parameters are incompatible, i.e., when their respective symmetric logarithmic derivatives do not commute, or when the model is sloppy, meaning that the quantum probe depends only on combinations of parameters leading to a degenerate or ill-conditioned Fisher information matrix. In this work, we explore the use of scrambling operations between parameter encoding to overcome sloppiness. We consider a bosonic model with two phase-shift parameters and analyze the performance of second- and third-order nonlinear scrambling using two classes of probe states: squeezed vacuum states and coherent states. Our results demonstrate that nonlinear scrambling mitigates sloppiness, increases compatibility, and improves overall estimation precision. We find third-order nonlinearity to be more effective than second-order under both fixed-probe and fixed-energy constraints. Furthermore, by comparing joint estimation to a stepwise estimation strategy, we show that a threshold for nonlinear coupling exists. For coherent probes, joint estimation outperforms the stepwise strategy if the nonlinearity is sufficiently large, while for squeezed probes, this advantage is observed specifically with third-order nonlinearity.
Abstract When two successive squeezing operations with the same phase are applied to a field mode, reliably estimating the amplitude of each is impossible because the output state depends solely on their sum. In this case, the quantum statistical model becomes sloppy, and the quantum Fisher information matrix turns singular. However, estimation of both parameters becomes feasible if the quantum state is subjected to an appropriate scrambling operation between the two squeezing operations. In this work, we analyze in detail the effects of a phase-shift scrambling transformation, optimized to reduce sloppiness and maximize the overall estimation precision using only classical probes. We also compare the optimized precision bounds of joint estimation with those of stepwise estimation methods, finding that joint estimation retains an advantage despite the quantum noise induced by the residual parameter incompatibility. Finally, we analyze the precision achievable by general-dyne detection and find that it approaches the optimal precision when one of the two squeezing amplitudes is large enough.
Critical quantum metrology exploits the hypersensitivity of quantum systems near phase transitions to achieve enhanced precision in parameter estimation. While single-parameter estimation near critical points is well established, the simultaneous estimation of multiple parameters, which is essential for practical sensing applications, remains challenging. This difficulty arises from sloppiness, a phenomenon that typically renders the quantum Fisher information matrix (QFIM) singular or nearly singular. In this work, we demonstrate that multiparameter critical metrology is not only feasible but can also retain divergent precision scaling, provided one accepts a trade-off in the scaling exponent. Using the ground state of the single-cavity Dicke model (DM), we show that two Hamiltonian parameters can be simultaneously estimated with a scalar variance bound scaling as the square root of the critical parameter. This overcomes the inherent sloppiness by leveraging higher-order contributions to the QFIM. To recover the optimal quadratic scaling, we introduce the Dicke dimer (DD) with photon hopping. In this extended model, a triple point in the phase diagram enables the simultaneous closure of two excitation gaps, which effectively increases the rank of the QFIM and restores the ideal single-parameter scaling for specific parameter pairs. Furthermore, we extend our analysis to dissipative settings subject to photon loss. Finally, we establish a connection between the derived critical scalings and the fundamental state preparation time, providing a unified framework to operationally compare different sensing strategies. Our results demonstrate that critical quantum metrology can be made robust against dissipation and scalable to multiparameter scenarios, paving the way for practical quantum sensors operating near phase transitions.
Abstract The Mach–Zehnder interferometer is a fundamental tool for measuring phase shifts between two light paths, serving as a crucial prototype for achieving high-precision measurements in various scientific and technological applications. In this study, we analyze different models for estimating the relative phase shift in a general two-arm Mach–Zehnder interferometer. We demonstrated that single-parameter estimation models can be reduced from the two-parameter estimation model by imposing appropriate constraints on the parameter space. To make quantum Fisher information of the single-parameter estimation models meaningful, the corresponding constraints must be guaranteed in the experiment implementation. Furthermore, we apply the quantum Fisher information approach to analyze the Mach–Zehnder interferometer with the an input state composed of a displaced squeezed vacuum state and a coherent state, providing insights into the precision limits of such configurations.
Multiparameter quantum estimation faces two fundamental obstacles: sloppiness, i.e., anisotropy of the quantum Fisher information matrix (QFIM) that renders some parameter directions insensitive, and incompatibility, the non-commutativity of optimal measurements for different parameters. The trade-off bound CT captures their joint impact on precision, but it has remained unclear how the distribution of incompatibility across parameter planes affects its overall cost. Here we separate the total amount of incompatibility from its location. We introduce a dimensionless quantity Gn(F) that measures the alignment between the incompatibility distribution and the eigenvalues of the QFIM, and show how the Frobenius scale of the incompatibility contribution factorizes. We obtain a bound and prove the incompatibility cost lies between this bound and a rank-dependent multiple thereof. We also prove that at fixed sloppiness, or equivalently fixed Fisher volume, concentrating incompatibility into a single parameter plane reduces the optimized trade-off cost because the Fisher geometry can then be reshaped to allocate more Fisher area to that plane. A qutrit SU(2) encoding numerically confirms that states with larger incompatibility strength can nevertheless incur a smaller cost if the matching factor G is sufficiently small. Our results establish that the distribution of incompatibility relative to the Fisher eigenbasis is a central diagnostic for multiparameter estimation, beyond the total incompatibility strength.