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arXiv:2610.32724

Optimal probes for estimating the dephasing of a quantum spin system

Leonardo Ratini, Stefano Carretta, Francesco Albarelli

quant-phSubmitted Oct 23, 2026• Announced Oct 26, 2026• 0 citations
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Abstract

We investigate the optimal estimation of dephasing in finite-dimensional quantum spin systems. We first discuss some properties of the optimal probe states and present an efficient numerical algorithm to compute the optimal precision in dephasing estimation and identify the optimal probe states. For the dephasing channel arising from Gaussian phase diffusion, a spin cat state is known to be optimal for small dephasing. We analytically determine the range of dephasing strengths over which the cat state remains globally optimal for spins $J=1$ and $J=3/2$, and derive upper bounds on its optimality region for larger spins. We then compare numerically several probe states proposed in the literature, showing in particular that the simple class of probes obtained from uniform superpositions of all levels performs near-optimally for stronger dephasing. Finally, we apply our framework to a central-spin model coupled to a nuclear-spin bath, whose dynamics we simulate using the cluster-correlation expansion. Our results show that, while the cat state is optimal for weak Gaussian dephasing, pulse-controlled non-Markovian dynamics can favor states with amplitudes distributed over multiple spin levels, and in some cases the uniform state outperforms the other fixed probes considered. This shows how the choice of probe state depends both on the dephasing strength and on the dynamics.

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We investigate the optimal estimation of dephasing in finite-dimensional quantum spin systems. We first discuss some properties of the optimal probe states and present an efficient numerical algorithm to compute the optimal precision in dephasing estimation and identify the optimal probe states. For the dephasing channel arising from Gaussian phase diffusion, a spin cat state is known to be optimal for small dephasing. We analytically determine the range of dephasing strengths over which the cat state remains globally optimal for spins J=1J=1J=1 and J=3/2J=3/2J=3/2, and derive upper bounds on its optimality region for larger spins. We then compare numerically several probe states proposed in the literature, showing in particular that the simple class of probes obtained from uniform superpositions of all levels performs near-optimally for stronger dephasing. Finally, we apply our framework to a central-spin model coupled to a nuclear-spin bath, whose dynamics we simulate using the cluster-correlation expansion. Our results show that, while the cat state is optimal for weak Gaussian dephasing, pulse-controlled non-Markovian dynamics can favor states with amplitudes distributed over multiple spin levels, and in some cases the uniform state outperforms the other fixed probes considered. This shows how the choice of probe state depends both on the dephasing strength and on the dynamics.
We investigate the estimation of dephasing-induced decoherence in continuous-variable quantum systems using non-Gaussian probe states. By purifying the open system, we identify optimal probes, specifically squeezed cat and symmetric squeezed compass states, via quantum Fisher information. These results are in agreement with numerical simulation. In settings where the intra-cavity field is inaccessible and standard measurements are impractical, utilizing an ancilla approach where a qubit traverses or interacts with the cavity field, leading to measurement of the qubit, hence allowing estimation of the dephasing rate via Wigner function reconstruction or less costly marginal distribution.
The spin-boson model, describing a two-level system coupled to a bosonic bath, is extensively studied as a paradigmatic dissipative quantum system, exhibiting rich dynamical behavior and even a localization transition in the strong coupling regime. Here, we additionally consider dephasing as a source of Markovian dissipation on top of the non-Markovian dynamics due to an Ohmic bath, and investigate the dynamics of the spin. We show that the characteristic frequency of the spin dynamics, while strongly renormalized by the bosonic bath, changes in a simple fashion (or does not change at all) with dephasing. To obtain these results, we develop an exact nonperturbative method known as the stochastic Schr\"odinger equation, mimicking the Ohmic bath via a stochastic magnetic field combined with the Lindblad quantum master equation due to dephasing, which allows us to numerically compute the dynamics. Furthermore, we derive weak-coupling analytic results utilizing the well-known noninteracting blip approximation. Our findings are relevant to quantum simulation of the spin-boson model in the regime of strong coupling in trapped ions and circuit QED architectures among others.
Quantum metrology faces major challenges in noisy environments, where decoherence rapidly degrades useful quantum resources. We investigate the dynamics of the precision limits given by the quantum Fisher information (QFI) for phase estimation under spatially correlated dephasing. We characterize the dynamics of the QFI by the sensitivity and degradation indicators that can be obtained as analytical expressions derived using perturbative theory treatment. These short-time and weak-noise formulas yield analytic insight into how collective-spin moments govern both (i) the noiseless sensitivity and (ii) the leading noise-induced degradation of metrological usefulness. We identify a trade-off that is intrinsic to our commuting encoding-noise structure. We analyze the QFI dynamics for the Gaussian spin state (GSS) superpositions, encompassing spin coherent state (SCS), Dicke state superpositions, spin-squeezed states, and GHZ-like states. Predictions from indicators of the QFI dynamics are compared to both the quantum Cramér--Rao bound and the measurement-specific sensitivity bounds for an optimal parameter and interrogation time under a finite total time resource. When possible, we analytically derive the measurement-specific sensitivity bounds for spin-projection and parity-based measurements.
Abstract Quantum probing is the art of exploiting simple quantum systems interacting with a complex environment to extract precise information about some environmental parameters, e.g. the temperature of the environment or its spectral density. Here we analyze the performance of a single-qubit probe in characterizing Ohmic bosonic environments at thermal equilibrium. In particular, we analyze the effects of tuning the interaction Hamiltonian between the probe and the environment, going beyond the traditional paradigm of pure dephasing. In the weak-coupling and short-time regime, we address the dynamics of the probe analytically, whereas numerical simulations are employed in the strong coupling and long-time regime. We then evaluate the quantum Fisher information for the estimation of the cutoff frequency and the temperature of the environment. Our results provide clear evidence that pure dephasing is not optimal, unless we focus attention to short times. In particular, we found several working regimes where the presence of a transverse interaction improves the maximum attainable precision, i.e. it increases the quantum Fisher information. We also explore the role of the initial state of the probe and of the probe characteristic frequency in determining the estimation precision, thus providing quantitative guidelines to design optimized detection to characterize bosonic environments at the quantum level.
Molecular nanomagnets are quantum spin systems potentially serving as qudits for future quantum technologies thanks to their many accessible low-energy states. At low temperatures, the primary source of error in these systems is pure dephasing, caused by their interactions with the bath of surrounding nuclear spins degrees of freedom. Most importantly, as the system’s dimensionality grows going from qubits to qudits, the control and mitigation of decoherence become more challenging. Here, we analyze the characteristics of pure dephasing in molecular qudits under spin-echo sequences. We use a realistic description of their interaction with the bath, whose non-Markovian dynamics is accurately computed by the cluster correlation expansion technique. First, we show that the differences in the expectation values of the local spin operators on the eigenstates of the qudit are the source of decoherence in these systems. Indeed, we demonstrate that this is a necessary and sufficient condition to prevent the decay of coherence with time, also introducing a parameter to quantify the deviation from such ideal condition. We illustrate this with two paradigmatic systems: a single giant spin and a composite antiferromagnetic spin system. We then advance a proposal for optimized nanomagnets, identifying key ingredients for engineering robust qudits for quantum technologies.
Abstract The geometry of quantum states provides a unifying framework for estimation processes based on quantum probes, and it establishes the ultimate bounds of the achievable precision. We show a relation between the statistical distance between infinitesimally close quantum states and the second order variation of the coherence of the optimal measurement basis with respect to the state of the probe. In quantum phase estimation protocols, this leads to propose coherence as the relevant resource that one has to engineer and control to optimize the estimation precision. Furthermore, the main object of the theory i.e. the symmetric logarithmic derivative, in many cases allows one to identify a proper factorization of the whole Hilbert space in two subsystems. The factorization allows one to discuss the role of coherence versus correlations in estimation protocols; to show how certain estimation processes can be completely or effectively described within a single-qubit subsystem; and to derive lower bounds for the scaling of the estimation precision with the number of probes used. We illustrate how the framework works for both noiseless and noisy estimation procedures, in particular those based on multi-qubit GHZ-states. Finally we succinctly analyze estimation protocols based on zero-temperature critical behavior. We identify the coherence that is at the heart of their efficiency, and we show how it exhibits the non-analyticities and scaling behavior proper of a large class of quantum phase transitions.