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arXiv:2210.10926 · Physical Review A

Optimal time for sensing in open quantum systems

Zain H. Saleem, Anil Shaji, Stephen K. Gray

Published Aug 16, 2023• 14 citations
arXivHTMLPDF

Abstract

We study the time-dependent quantum Fisher information (QFI) in an open quantum system satisfying the Gorini-Kossakowski-Sudarshan-Lindblad master equation. Most of our work concerns a simple three-level model for a single probe coupled to a sink, although some results for multiple probes are also given. We also study the dynamics of the system from an effective non-Hermitian dynamics standpoint and use it to understand the scaling of the QFI when multiple probes are used. A focus of our work is how the QFI is maximized at certain times, suggesting that the best precision in parameter estimation can be achieved by focusing on these times. The propagation of errors analysis allows us to confirm and better understand this idea. We also propose an optimum-time-based parameter estimation procedure involving relatively low-resource-consuming measurements followed by higher-resource-consuming measurements and demonstrate it in simulation.

DOI
10.1103/physreva.108.022413

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We study the time-dependent quantum Fisher information (QFI) in an open quantum system satisfying the Gorini-Kossakowski-Sudarshan-Lindblad master equation. Most of our work concerns a simple three-level model for a single probe coupled to a sink, although some results for multiple probes are also given. We also study the dynamics of the system from an effective non-Hermitian dynamics standpoint and use it to understand the scaling of the QFI when multiple probes are used. A focus of our work is how the QFI is maximized at certain times, suggesting that the best precision in parameter estimation can be achieved by focusing on these times. The propagation of errors analysis allows us to confirm and better understand this idea. We also propose an optimum-time-based parameter estimation procedure involving relatively low-resource-consuming measurements followed by higher-resource-consuming measurements and demonstrate it in simulation.
Quantum metrology studies the ultimate limit of precision in estimating a physical quantity if quantum strategies are exploited. Here we investigate the evolution of a two-level atom as a detector which interacts with a massless scalar field using the master equation approach for open quantum system. We employ local quantum estimation theory to estimate the Unruh temperature when probed by a uniformly accelerated detector in the Minkowski vacuum. In particular, we evaluate the Fisher information (FI) for population measurement, maximize its value over all possible detector preparations and evolution times, and compare its behavior with that of the quantum Fisher information (QFI). We find that the optimal precision of estimation is achieved when the detector evolves for a long enough time. Furthermore, we find that in this case the FI for population measurement is independent of initial preparations of the detector and is exactly equal to the QFI, which means that population measurement is optimal. This result demonstrates that the achievement of the ultimate bound of precision imposed by quantum mechanics is possible. Finally, we note that the same configuration is also available to the maximum of the QFI itself.
We consider the general problem of estimating an unknown control parameter of an open quantum system. We establish a direct relation between the evolution of both system and environment and the precision with which the parameter can be estimated. We show that when the open quantum system undergoes a first-order dynamical phase transition the quantum Fisher information (QFI), which gives the upper bound on the achievable precision of any measurement of the system and environment, becomes quadratic in observation time (cf. ``Heisenberg scaling''). In fact, the QFI is identical to the variance of the dynamical observable that characterizes the phases that coexist at the transition, and enhanced scaling is a consequence of the divergence of the variance of this observable at the transition point. This identification makes it possible to establish the finite time scaling of the QFI. Near the transition the QFI is quadratic in time for times shorter than the correlation time of the dynamics. In the regime of enhanced scaling the optimal measurement whose precision is given by the QFI involves measuring both system and output. As a particular realization of these ideas, we describe a theoretical scheme for quantum enhanced estimation of an optical phase shift using the photons being emitted from a quantum system near the coexistence of dynamical phases with distinct photon emission rates.
The fundamental metrological limits of temperature sensing in open quantum systems remain largely unresolved, particularly regarding the role of non-Gaussian quantum resources. In this letter, we establish analytic bounds on the quantum Fisher information (QFI) for temperature estimation using non-Gaussian states undergoing dissipative bosonic evolution. By focusing on the short-time regime governed by a time-local master equation, we derive precise scaling laws that elucidate when and how non-Gaussian probes decisively outperform Gaussian states under identical energy constraints. Our analysis uncovers a distinct linear-in-time QFI enhancement unique to Fock states, in contrast to the inherently weaker, quadratic scaling of Gaussian probes. These theoretical insights are substantiated through exact numerical simulations and mapped onto experimentally accessible platforms such as circuit QED. Our results not only clarify the quantum thermometric advantage of non-Gaussianity but also chart a realistic pathway toward harnessing it in noisy quantum technologies.
Temperature as an environmental parameter influences the evolution of an open quantum system. In detail, temperature lies in Lindblad operator of quantum master equation that the evolution of an open quantum system follows. Hence, one can implement a temperature estimation of thermal baths through a measurement of quantum Fisher information about temperature brought from quantum states. Such a method by calculating quantum Fisher information about a parameter to estimate its value avoids measuring the parameter directly and it does not change the value of the parameter due to making measurements. In this paper, we consider a model consisting of a XXZ spin-[Formula: see text] chain coupled locally to independent thermal baths with different temperature. Based on the model, we investigate optimal temperature estimation for thermal baths with respect to an open quantum system subjected to non-steady states. We first study optimal probe time for temperature estimation in the case of non-steady states and find that the optimal time shows different features for different types of system variables. It proves that in a certain duration there exists a tradeoff between the trial times and the attaining amount of Fisher information in each trial. In addition, we pay attention to an issue on optimal probe states. We demonstrate that in many cases the optimal states are not always the maximally entangled states and even maybe the separable states, which is related with the measuring time, system couplings.
Quantum parameter estimation with Hermitian systems has been applied in various fields, but there are relatively few results concerning non-Hermitian systems. Here, we study the quantum parameter estimation for general non-Hermitian Hamiltonians and derive an intuitive expression of quantum Fisher information (QFI) for pure states. Furthermore, we propose the condition for optimal measurements, which is applicable to both Hermitian and non-Hermitian Hamiltonians. To illustrate these results, we calculate and study the QFI of a specific parity-time (PT)(\mathcal{PT})(PT) -symmetric non-Hermitian Hamiltonian and give the optimal measurement. Surprisingly, we find some interesting properties of this PT\mathcal{PT}PT-symmetric Hamiltonian QFI, such as the mutations in QFI at exceptional point. Moreover, we also compare the variance of estimation generated by the optimal measurement with the theoretical precision bound to verify the condition for optimal measurements we proposed.
We investigate the sensing capacity of non-equilibrium dynamics in quantum systems exhibiting Bloch oscillations. By focusing on the resource efficiency of the probe, quantified by quantum Fisher information, we find different scaling behaviors in two different phases, namely localized and extended. Our results provide a quantitative ansatz for quantum Fisher information in terms of time, probe size, and the number of excitations. In the long-time regime, the quantum Fisher information is a quadratic function of time, touching the Heisenberg limit. The system size scaling drastically depends on the phase changing from quantum-enhanced scaling in the extended phase to size-independent behavior in the localized phase. Furthermore, increasing the number of excitations always enhances the precision of the probe, although, in the interacting systems the enhancement becomes less eminent than the non-interacting probes. This is due to the induced localization by increasing the interaction between the excitations. We show that a simple particle configuration measurement together with a maximum likelihood estimation can closely reach the ultimate precision limit in both single- and multi-particle probes.