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Physical review. D/Physical review. D.

Effect of the gravitational redshift on the precision of phase estimation

Zhen-Dong Wei, Wei Han, Yingjie Zhang, Zhong‐Xiao Man, Yun‐Jie Xia, Heng Fan

Published Jan 9, 2025• 9 citations
Publisher

Abstract

Gravitational effects are inevitable as photons propagate in curved spacetime, and especially gravitational redshift can cause the distortion of photon wave packets, so it is very meaningful and necessary to study the controllable parameter estimation under the gravitational-redshift effect. Here, in correlated noisy channels under the effect of gravitational redshift, we analyze the precision of phase estimation of two-qubit system states which is introduced by a symplectic matrix for two-mode entangled states (two types of Werner-like states). We find that the gravitational redshift could decrease the precision of phase estimation of these two-qubit system states. Interestingly, by adjusting the correlation parameter of the channels and the initial entanglement, the precision of phase estimation in noisy channels under the influence of gravitational redshift could be enhanced. The larger the initial entanglement and the stronger the correlation of the channels, the higher the precision of phase estimation obtained. Particularly, for two types of Werner-like initial states, the phase estimation of the system is more robust for the Werner-like initial state $\mathrm{cos}(\frac{\ensuremath{\alpha}}{2})|01⟩+{e}^{i\ensuremath{\phi}}\mathrm{sin}(\frac{\ensuremath{\alpha}}{2})|10⟩$ as the strong enough gravitational-redshift effect.

DOI
10.1103/physrevd.111.026007

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Gravitational effects are inevitable as photons propagate in curved spacetime, and especially gravitational redshift can cause the distortion of photon wave packets, so it is very meaningful and necessary to study the controllable parameter estimation under the gravitational-redshift effect. Here, in correlated noisy channels under the effect of gravitational redshift, we analyze the precision of phase estimation of two-qubit system states which is introduced by a symplectic matrix for two-mode entangled states (two types of Werner-like states). We find that the gravitational redshift could decrease the precision of phase estimation of these two-qubit system states. Interestingly, by adjusting the correlation parameter of the channels and the initial entanglement, the precision of phase estimation in noisy channels under the influence of gravitational redshift could be enhanced. The larger the initial entanglement and the stronger the correlation of the channels, the higher the precision of phase estimation obtained. Particularly, for two types of Werner-like initial states, the phase estimation of the system is more robust for the Werner-like initial state cos(\ensuremathα2)∣01⟩+ei\ensuremathϕsin(\ensuremathα2)∣10⟩\mathrm{cos}(\frac{\ensuremath{\alpha}}{2})|01⟩+{e}^{i\ensuremath{\phi}}\mathrm{sin}(\frac{\ensuremath{\alpha}}{2})|10⟩cos(2\ensuremathα​)∣01⟩+ei\ensuremathϕsin(2\ensuremathα​)∣10⟩ as the strong enough gravitational-redshift effect.
As photons propagate through curved spacetime, gravitational effects become unavoidable. In particular, gravitational redshift can induce significant distortion in photon wave packets, making it es?sential to investigate parameter estimation within this context. While previous research has focused on single-parameter estimation using the quantum Cramer-Rao bound, the multiparameter scenario remains largely unexplored. In this work, we investigate multiparameter quantum estimation for a photon system subject to gravitational redshift under both amplitude-damping and Ohmic-like dephasing channels. Our analysis reveals that the quantum Cramer-Rao bound fails to provide a tight error bound for the two-parameter estimation involving the initial phase and weight parameters inboth types of noisy channels. To overcome this limitation, we numerically compute two tighter error bounds, i.e., the Holevo Cramer-Rao bound and the Nagaoka bound, when utilizing a semidefinite program. We demonstrate that the Nagaoka bound yields the tightest error bound among all considered bounds, consistent with the general hierarchy of multiparameter quantum estimation. Furthermore, for the three-parameter estimation, including the initial weight parameter, the phase parameter, and the strength of gravitational redshift, we observe significantly enhanced estimation precision in the strong-coupling regime compared to the weak-coupling regime under the amplitude-damping channel. Similarly, in the Ohmic-like dephasing channel, the sub-Ohmic regime consistently affords higher precision than the Ohmic and super-Ohmic regimes.
The gravitational frequency shift of light is well understood in the theory of classical electromagnetism. Nevertheless, its description in quantum theory is not yet fully developed. Recent work pointed out inconsistencies in previously developed models aimed at describing the gravitational redshift as an effective multi-mode mixer (MMM) acting on modes of light, but so far a complete solution of these issues was not obtained. Here, we identify the root cause of the MMM model's inconsistency and provide two complementary approaches to correct it: a "natural" one from a field-theoretic perspective, and another adapted to the language of quantum mechanics of finite-dimensional systems. We show that the second approach allows for modeling of the redshift in a multi-mode transmission setup as a quantum channel that can be characterized using standard quantum information-theoretic techniques when restricting the input states to Gaussian states of light.
Abstract Gravitational redshift is discussed in the context of quantum photons propagating in curved spacetime. A brief introduction to modelling realistic photons is first presented and the effect of gravity on the spectrum computed for photons largely confined along the direction of propagation. It is then shown that redshift-induced transformations on photon operators with sharp momenta are not unitary, while a unitary transformation can be constructed for realistic photons with finite bandwidth. The unitary transformation obtained is then characterized as a multimode mixing operation, which is a generalized rotation of the Hilbert-space basis. Finally, applications of these results are discussed with focus on performance of quantum communication protocols, exploitation of the effects for quantum metrology and sensing, as well as potential for tests of fundamental science.
Gravitational redshift can cause the distortion of photon wave packets; thus, the non-Markovian speedup evolution of a photon system must be controlled by considering gravitational redshift. We analyze the non-Markovian speedup dynamics of a photon system in the amplitude-damping channel or pure-dephasing channel under the effect of gravitational redshift, which is modelled as a beam-splitter operation. We show that the gravitational redshift weakens the non-Markovian dynamical behavior in both channels. For the amplitude-damping channel, the gravitational redshift promotes further speedup evolution of a photon system when the initial coherence of the system is nonzero. A stronger gravitational redshift and larger initial coherence make improving the evolution speed of the photon system easier. For the pure-dephasing channel, the non-Markovian dynamical behavior decreases owing to the gravitational redshift; however, the evolution speed of the photon is not influenced by the gravitational redshift.
We analyze the domain of validity of a quantum optical model that describes the effects of gravitational redshift on the quantum state of photons that propagate in curved spacetime. This model assumes that the modes defining the initial state of the photon are mixed with an auxiliary environment mode via an effective multimode mixer. We find that the model, as proposed, is consistent only to first order for small redshift, where the range of validity is conditional not only to the gravitational parameters, but also to those that define the photonic modes.We identify the problem and provide a partial solution in terms of a necessary condition on the transformation matrix representing the process, which requires the use of a number of auxiliary modes that is at least equal to the number of modes that define the photonic state. We conclude by discussing implications for theoretical quantum optics and photonics in curved spacetime, as well as for the development of quantum technologies.
Abstract We analyze the domain of validity of a quantum optical model that describes the effects of gravitational redshift on the quantum state of photons that propagate in curved spacetime. This model assumes that the modes defining the initial state of the photon are mixed with an auxiliary environment mode via an effective multimode mixer. We find that the model, as proposed, is consistent only to first order for small redshift, where the range of validity is conditional not only to the gravitational parameters, but also to those that define the photonic modes. We identify the problem and provide a partial solution in terms of a necessary condition on the transformation matrix representing the process, which requires the use of a number of auxiliary modes that is at least equal to the number of modes that define the photonic state. We conclude by discussing implications for theoretical quantum optics and photonics in curved spacetime, as well as for the development of quantum technologies.