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Optics Express

Tighter Heisenberg limits for phase estimation in noisy environments based on the quantum Ziv-Zakai bound

Wei Ye, Peng Xiao, Yunbin Yan, Kai Han, Xiang Zhu, Xiaofan Xu, Jie Ren, Ying Xia, WenWen Hu, Xuan Rao, Shoukang Chang

Published Jan 15, 2025• 2 citations
Publisher

Abstract

In exploring the precision limits of quantum metrology, the quest for a tighter Heisenberg limit in real-world environments becomes a key challenge. For this reason, in this paper, we propose a tighter Heisenberg limit for phase estimation, called the photon loss (PL)-type bound and the photon diffusion (PD)-type bound in realistic scenarios under the framework of quantum Ziv-Zakai bound (QZZB). In order to demonstrate the superiority of the proposed Heisenberg limits for phase estimation, as a comparison, we also introduce the Margolus-Levitin (ML)-type and Mandelstam-Tamm (MT)-type bounds [Phys. Rev. A90, 043818 (2014)10.1103/PhysRevA.90.043818] based on the QZZB framework when considering the same initial states, i.e., a coherent state, a superposition state of the vacuum and general Fock states, and an even coherent state. The simulation results show that for photon-loss scenario, the PL-type bounds for all given initial states are closer to the QZZB than the ML-type and MT-type bounds, thereby exhibiting a tighter Heisenberg limit. In contrast, for the phase-diffusion scenario, when the phase diffusion strength exceeds a certain threshold, the tightness of PD-type bounds for the coherent state and superposition state can present better than that of the ML-type and MT-type bounds. Furthermore, the tighter QZZB over the QCRB can be achieved using the superposition state in photon loss or the even coherent state in phase diffusion.

DOI
10.1364/oe.541384

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In exploring the precision limits of quantum metrology, the quest for a tighter Heisenberg limit in real-world environments becomes a key challenge. For this reason, in this paper, we propose a tighter Heisenberg limit for phase estimation, called the photon loss (PL)-type bound and the photon diffusion (PD)-type bound in realistic scenarios under the framework of quantum Ziv-Zakai bound (QZZB). In order to demonstrate the superiority of the proposed Heisenberg limits for phase estimation, as a comparison, we also introduce the Margolus-Levitin (ML)-type and Mandelstam-Tamm (MT)-type bounds [Phys. Rev. A90, 043818 (2014)10.1103/PhysRevA.90.043818] based on the QZZB framework when considering the same initial states, i.e., a coherent state, a superposition state of the vacuum and general Fock states, and an even coherent state. The simulation results show that for photon-loss scenario, the PL-type bounds for all given initial states are closer to the QZZB than the ML-type and MT-type bounds, thereby exhibiting a tighter Heisenberg limit. In contrast, for the phase-diffusion scenario, when the phase diffusion strength exceeds a certain threshold, the tightness of PD-type bounds for the coherent state and superposition state can present better than that of the ML-type and MT-type bounds. Furthermore, the tighter QZZB over the QCRB can be achieved using the superposition state in photon loss or the even coherent state in phase diffusion.
In this work, we address the central problem about how to effectively find the available precision limit of unknown parameters. In the framework of the quantum Ziv-Zakai bound (QZZB), we employ noiseless linear amplification (NLA)techniques to an initial coherent state (CS) as the probe state, and focus on whether the phase estimation performance is improved significantly in noisy scenarios, involving the photon-loss and phase-diffusion cases. More importantly, we also obtain two kinds of Heisenberg error limits of the QZZB with the NLA-based CS in these noisy scenarios, making comparisons with both the Margolus-Levitin (ML) type bound and the Mandelstam-Tamm (MT) type bound. Our analytical results show that in cases of photon loss and phase diffusion, the phase estimation performance of the QZZB can be improved remarkably by increasing the NLA gain factor. Particularly, the improvement is more pronounced with severe photon losses. Furthermore in minimal photon losses, our Heisenberg error limit shows better compactness than the cases of the ML-type and MT-type bounds. Our findings will provide an useful guidance for accomplishing more complex quantum information processing tasks.
In the highly non-Gaussian regime, the quantum Ziv-Zakai bound (QZZB) provides a lower bound on the available precision, demonstrating the better performance compared with the quantum Cramér-Rao bound. However, evaluating the impact of a noisy environment on the QZZB without applying certain approximations proposed by Tsang [Phys. Rev. Lett.108, 230401 (2012)10.1103/PhysRevLett.108.230401] remains a difficult challenge. In this paper, we not only derive the asymptotically tight QZZB for phase estimation with the photon loss and the phase diffusion by invoking the variational method and the technique of integration within an ordered product of operators, but also show its estimation performance for several different Gaussian resources, such as a coherent state (CS), a single-mode squeezed vacuum state (SMSVS) and a two-mode squeezed vacuum state (TMSVS). In this asymptotically tight situation, our results indicate that compared with the SMSVS and the TMSVS, the QZZB for the CS always shows the better estimation performance under the photon-loss environment. More interestingly, for the phase-diffusion environment, the estimation performance of the QZZB for the TMSVS can be better than that for the CS throughout a wide range of phase-diffusion strength. Our findings will provide an useful guidance for investigating the noisy quantum parameter estimation.
I propose quantum versions of the Ziv-Zakai bounds as alternatives to the widely used quantum Cramér-Rao bounds for quantum parameter estimation. From a simple form of the proposed bounds, I derive both a Heisenberg error limit that scales with the average energy and a limit similar to the quantum Cramér-Rao bound that scales with the energy variance. These results are further illustrated by applying the bound to a few examples of optical phase estimation, which show that a quantum Ziv-Zakai bound can be much higher and thus tighter than a quantum Cramér-Rao bound for states with highly non-gaussian photon-number statistics in certain regimes and also stay close to the latter where the latter is expected to be tight.
Critical quantum metrology with equilibrium states predicts quantum-enhanced sensitivity only in the vicinity of criticality, where large prior information about the parameter is required. By employing quantum Ziv-Zakai bounds, we derive a limit on the mean-square error in critical quantum metrology. For second-order quantum phase transitions, we show that the precision predicted by the Cramér-Rao bound offers no substantial improvement over the prior standard deviation. Thus, the critical quantum sensor's precision can only achieve a constant gain compared to the prior standard deviation, even without performing any measurement. We elucidate the fundamental limitation on the achievable precision in critical quantum metrology in the context of local sensing, even without considering state-preparation costs or noise. Thus, the super-Heisenberg-limited sensitivity at criticality arises from precise prior knowledge rather than a genuine gain due to criticality. Our work provides a practical framework for assessing critical quantum metrology and a routine for studying quantum sensing with many-body systems.
We propose quantum versions of the Bell-Ziv-Zakai lower bounds for the error in multiparameter estimation.As an application, we consider measurement of a time-varying optical phase signal with stationary Gaussian prior statistics and a power-law spectrum ∼1=jωj p , with p > 1.With no other assumptions, we show that the mean-square error has a lower bound scaling as 1=N 2ðp-1Þ=ðpþ1Þ , where N is the time-averaged mean photon flux.Moreover, we show that this scaling is achievable by sampling and interpolation, for any p > 1.This bound is thus a rigorous generalization of the Heisenberg limit, for measurement of a single unknown optical phase, to a stochastically varying optical phase.
Abstract Sensing and imaging are among the most important applications of quantum information science. To investigate their fundamental limits and the possibility of quantum enhancements, for decades researchers have relied on the quantum Cramér-Rao lower error bounds pioneered by Helstrom. Recent work, however, has called into question the tightness of those bounds for highly nonclassical states in the non-asymptotic regime, and better methods are now needed to assess the attainable quantum limits in reality. Here we propose a new class of quantum bounds called quantum Weiss-Weinstein bounds, which include Cramér-Rao-type inequalities as special cases but can also be significantly tighter to the attainable error. We demonstrate the superiority of our bounds through the derivation of a Heisenberg limit and phase estimation examples.