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arXiv:1609.02092

Unruh acceleration effect on the precision of parameter estimation

NASSER METWALLY

Published Sep 7, 2016• 2 citations
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Abstract

The dynamics of Fisher information for an accelerated system initially prepared in the $X$-state is discussed. An analytical solution, which consists of three parts: classical, the average over all pure states and a mixture of pure states is derived for the general state and for Werner state. It is shown that, the Unruh acceleration has a depleting effect on the Fisher information. This depletion depends on the degree of entanglement of the initial state settings. For the $X$-state, for some intervals of Unruh acceleration, the Fisher information remains constant, irrespective to the Unruh acceleration. In general, the possibility of estimating the state's parameters decreases as the acceleration increases. However, the precision of estimation can be maximized for certain values of the Unruh acceleration. We also investigate the contribution of the different parts of the Fisher information on the dynamics of the total Fisher information.

DOI
10.48550/arxiv.1609.02092

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The dynamics of Fisher information for an accelerated system initially prepared in the XXX-state is discussed. An analytical solution, which consists of three parts: classical, the average over all pure states and a mixture of pure states is derived for the general state and for Werner state. It is shown that, the Unruh acceleration has a depleting effect on the Fisher information. This depletion depends on the degree of entanglement of the initial state settings. For the XXX-state, for some intervals of Unruh acceleration, the Fisher information remains constant, irrespective to the Unruh acceleration. In general, the possibility of estimating the state's parameters decreases as the acceleration increases. However, the precision of estimation can be maximized for certain values of the Unruh acceleration. We also investigate the contribution of the different parts of the Fisher information on the dynamics of the total Fisher information.
In this paper, Fisher information for an accelerated system initially prepared in the X-state is discussed. An analytical solution, which consists of three parts: classical, the average over all pure states and a mixture of pure states, is derived for the general state and for Werner state. It is shown that the Unruh acceleration has a depleting effect on the Fisher information. This depletion depends on the degree of entanglement of the initial state settings. For the X-state, for some intervals of Unruh acceleration, the Fisher information remains constant, irrespective to the Unruh acceleration. In general, the possibility of estimating the state’s parameters decreases as the acceleration increases. However, the precision of estimation can be maximized for certain values of the Unruh acceleration. We also investigate the contribution of the different parts of the Fisher information on the dynamics of the total Fisher information.
The goal of quantum metrology is the exploitation of quantum resources, like entanglement or quantum coherence, in the fundamental task of parameter estimation. Here we consider the question of the estimation of the Unruh temperature in the scenario of relativistic quantum metrology. Specifically, we study two distinct cases. First, a single Unruh-DeWitt detector interacting with a scalar quantum field undergoes an uniform acceleration for a finite amount of proper time, and the role of coherence in the estimation process is analyzed. After this, we consider two initially entangled detectors, one of which is inertial while the other one undergoes acceleration. Our results show that the maximum of the Fisher information, thus characterizing the maximum possible precision according to Crammér-Rao bound, occurs only for small accelerations, while it decreases fast when acceleration increases. Moreover, the role of initial coherence ---in the single detector case---, or entanglement ---in the two detectors case---, is to decrease Fisher information. Therefore, under the considered protocol, internal coherence (or entanglement) is not a resource for estimating Unruh temperature. These unexpected results show that a detection of the Unruh effect can be even more challenge than previously thought. Finally, by considering the connection between Unruh effect and Hawking radiation, we discuss how our results can be understood in the context of the estimation of Hawking temperature.
We study the quantum Fisher information (QFI) of acceleration, in the open quantum systems, for a two-level atom with the circular motion coupled to a massless scalar field in the Minkowski vacuum without and with a reflecting boundary in the ultra-relativistic limit. As we amplify a, the saturation time decreases for θ≠π, but first increases and then decreases for θ=π. Without a boundary, there exists a peak value of QFI with a certain time. The QFI varies with the initial state parameter θ, and firstly takes peak value in the ground state of the atom. The variation of QFI with respect to θ gradually fades away with the evolution of time. With a boundary, the detection range of acceleration has been expanded. The QFI firstly takes the maximum in the excited state of the atom. In addition, we study the QFI of temperature for a static atom immersed in a thermal bath without and with a boundary. The relation between the saturation time and T is similar to a. Without a boundary, the QFI of temperature is similar to that of acceleration. With a boundary, the QFI firstly takes peak value in the ground state of the atom, which is different from the behavior of acceleration. The results provide references for the detection of Unruh-like effect.
We study the quantum metrology for a pair of entangled Unruh-Dewitt detectors when one of them is accelerated and coupled to a massless scalar field. Comparing with previous schemes, our model requires only local interaction and avoids the use of cavities in the probe state preparation process. We show that the probe state preparation and the interaction between the accelerated detector and the external field have significant effects on the value of quantum Fisher information, correspondingly pose variable ultimate limit of precision in the estimation of Unruh effect. We find that the precision of the estimation can be improved by a larger effective coupling strength and a longer interaction time. Alternatively, the energy gap of the detector has a range that can provide us a better precision. Thus we may adjust those parameters and attain a higher precision in the estimation. We also find that an extremely high acceleration is not required in the quantum metrology process.
An appropriate weak value can be used to amplify weak physical parameters and improve the precision of parameter estimation in the process of quantum weak measurement. The increase of the precision of such a parameter estimation may originate from the entanglement in the system. This paper uses Fisher information to study the influence of the entanglement in the system and the entanglement between the system and the pointer on the estimation precision of the coupling parameters between the system and the pointer in the process of weak measurement. The entanglement of the entangled pure state of the GHZ-like state and the entangled mixed state affected by the depolarization noise and the influence of the entanglement between the system and the pointer on the parameter estimation are analyzed. The results show that the Fisher information quantity increases with the increase of the initial state entanglement degree of the system when the initial state of the system is an entangled pure state or an entangled mixed state affected by depolarization noise, and both the Fisher information quantity and the post-selection probability reach their maximum values when the initial and final state of the system are both the maximum entangled states; but the weaker the entanglement between the system and the pointer, the more the Fisher information obtained in the measurement will be and the higher the accuracy of parameter estimation. These research results show that the entanglement in the system will improve the precision of parameter estimation, while the entanglement between the system and the pointer will reduce the precision of parameter estimation.
We analyze quantum parameter estimation by studying the dynamics of the quantum Fisher information (QFI) for two classes of parameters, acceleration and initial-state weight, in an Unruh-DeWitt detector undergoing four distinct noninertial motions: linear, cusped, catenary, and circular trajectories, respectively. We assume that the detector is initialized in a pure superposition state with a weight parameter $\ensuremath{\theta}$ characterizing the probability of the detector occupying each state. Our results reveal that, over long evolution times, the QFI for the acceleration parameter converges to a nonnegative asymptotic value that depends sensitively on the trajectory, whereas the QFI for the weight parameter decays to zero as the system thermalizes. Importantly, for sufficiently large accelerations, one can attain the optimal precision in estimating the acceleration parameter within a finite interaction time, eliminating the need for infinitely long measurements. Comparing trajectories, we find that for small accelerations (relative to the detector's energy gap), linear motion yields the highest QFI for $\ensuremath{\theta}$, while for large accelerations, circular motion becomes optimal for estimating $\ensuremath{\theta}$. By contrast, circular motion offers the best precision for estimating acceleration itself in both the small- and large-acceleration regimes (the latter only at very long times). These contrasting behaviors of QFI across trajectories suggest a novel metrological protocol for inferring the underlying noninertial motion of a quantum probe.