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arXiv:2512.13640 · Advanced Quantum Technologies

Quadratic and cubic scrambling in the estimation of two successive phase-shifts

Manju, Stefano Olivares, Matteo G. A. Paris

quant-phSubmitted Dec 15, 2025• Announced Dec 16, 2025• 0 citations
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Abstract

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.

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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.
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.
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.
We address the precise characterization of coupling strength of nonlinear media in continuous-variable (CV) quantum systems using coherent and squeezed vacuum states as Gaussian probes, together with their photon-added and photon-subtracted counterparts as non-Gaussian probes. We consider three main classes of nonlinear Hamiltonians, namely quadrature nonlinearities, generalized squeezing and Kerr-type interactions. By analytically evaluating the quantum Fisher information (QFI), we compare the performance of Gaussian and non-Gaussian probes and assess the optimal probe based on the probe parameters, energy resource and non-Gaussianity. Our results are twofold as follows: first, for coherent-state family, the improvement provided by photon addition at fixed coherent amplitude originates mainly from the extra energy carried by the probe and does not provide a genuine metrological resource, since the same precision can be achieved by a Gaussian coherent-state signal of a larger energy, which can be more easily produced. Second, in contrast, photon addition and subtraction become effective resources when applied to already nonclassical states such as squeezed vacuum states. In this case, they lead to a significant enhancement of the QFI, particularly for higher-order interactions. Although Gaussian squeezed states remain optimal at equal energy constraint, photon-added and photon-subtracted squeezed states achieve comparable sensitives with significantly lower squeezing requirements. Since large squeezing level remains experimentally challenging, these non-Gaussian probes offer a practical route towards enhanced estimation of the nonlinear coupling strength within currently accessible squeezing regimes.
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cram\'er-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optimally tuned squeezing and displacement fulfill the measurement compatibility condition and enable the simultaneous estimation of all three parameters, with an advantage over individual estimation schemes that quickly rises with increasing mean energy of the probes.
Active optical media leading to interaction Hamiltonians of the form H=λ˜(a+a†)ζ represent a crucial resource for quantum optical technology. In this paper, we address the characterization of those nonlinear media using quantum probes, as opposed to semiclassical ones. In particular, we investigate how squeezed probes may improve individual and joint estimation of the nonlinear coupling λ˜ and of the nonlinearity order ζ. Upon using tools from quantum estimation, we show that: (i) the two parameters are compatible, i.e., the may be jointly estimated without additional quantum noise; (ii) the use of squeezed probes improves precision at fixed overall energy of the probe; (iii) for low energy probes, squeezed vacuum represent the most convenient choice, whereas for increasing energy an optimal squeezing fraction may be determined; (iv) using optimized quantum probes, the scaling of the corresponding precision with energy improves, both for individual and joint estimation of the two parameters, compared to semiclassical coherent probes. We conclude that quantum probes represent a resource to enhance precision in the characterization of nonlinear media, and foresee potential applications with current technology.