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Quantum communications and quantum metrology in the spacetime of a rotating planet
- Jan Kohlrus^{1}Email authorView ORCID ID profile,
- David Edward Bruschi^{2, 3},
- Jorma Louko^{1} and
- Ivette Fuentes^{1, 4}
Received: 31 January 2017
Accepted: 11 April 2017
Published: 20 April 2017
Abstract
We study how quantum systems that propagate in the spacetime of a rotating planet are affected by the curved background. Spacetime curvature affects wavepackets of photons propagating from Earth to a satellite, and the changes in the wavepacket encode the parameters of the spacetime. This allows us to evaluate quantitatively how quantum communications are affected by the curved spacetime background of the Earth and to achieve precise measurements of Earth’s Schwarzschild radius and equatorial angular velocity. We then provide a comparison with the state of the art in parameter estimation obtained through classical means. Satellite to satellite communications and future directions are also discussed.
Keywords
- satellite communications
- quantum metrology
- Kerr spacetime
1 Introduction
Quantum communications is a rapidly growing field which promises several technical improvements to current classical communications. One example is the use of quantum cryptography which would make communications more secure, thanks to more robust protocols than the classical ones [1]. More fundamental aspects can also be studied using quantum communications. For example, the interplay between quantum physics and relativity can be probed through quantum communications between moving observers and within schemes in a curved spacetime background [2]. The results of the measurements can then be compared with the predictions obtained by theories that were developed in the overlap of quantum physics and relativity, the most well known and understood of these being Quantum Field Theory (QFT) in curved spacetime [3].
Knowing quantitatively how quantum communications are affected by the curved spacetime background would enable to compensate undesirable relativistic effects in future quantum technologies. Precise values of the necessary corrections in such quantum communication setups can only be obtained with an accurate knowledge of the spacetime parameters. We thus need to employ techniques from the field of quantum metrology, which aims at exploiting quantum resources, such as entanglement, to estimate physical parameters [4]. Within a standard estimation protocol, an input quantum state undergoes a transformation that encodes the parameter to be estimated. The resulting state of this transformation is then compared, by means of the fidelity, to a neighbouring state which is infinitesimally close in terms of the parameter. One can define a distance between these two states that is directly related to the Quantum Fisher Information (QFI), which in turn is directly related to the maximum precision one can obtain in an estimation scheme. A final measurement provides an estimation of the value of the parameter in a single-shot run [4].
Typical applications of quantum metrology range from phase estimation in quantum optics to estimating the gravitational potential with Bose Einstein Condensates (BECs) [5, 6]. However, when estimating relativistic parameters, gravity usually appears as an external potential, or a phase modification, which does not overcome the inherent inconsistency between quantum physics and relativity [7]. Recently, this gap has been bridged and quantum field theory in curved spacetime has been employed as the core framework to compute the ultimate bounds on ultra-precise measurements of relativistic parameters. In particular, it was shown that it is possible to use the shifting induced on the frequency distribution of single photons ascending the gravitational potential of a static planet to estimate with great precision the distance between a user based on Earth and one on a satellite [2, 8]. In this case, gravity isn’t affecting the quantum state as the simple addition of a phase. The effects due to curved spacetime can therefore not be explained by a simple ad hoc implementation of proper time in a classical quantum mechanics scheme. Furthermore, it was shown that these effects can have potentially high impact on specific types of quantum key distribution (QKD) protocols [2]. This direction has the potential of leading towards the development of new relativistic and quantum technologies aimed at testing the predictions of quantum field theory in curved spacetime in space-based experiments with satellites.
In this work we extend the analysis carried out in previous works which investigated quantum estimation techniques in scenarios where photons are exchanged between Earth and a satellite [8]. There, the Earth was assumed as static and the effects on the propagation of the photons depend only on the Schwarzschild radius of the Earth. Here we consider a rotating planet, and we model the metric outside the mass distribution by the well known Kerr metric [9]. The transformation induced by the curvature on the traveling photon reduces to a beam-splitter, a well known linear transformation in quantum optics [10]. We can therefore restrict ourselves to Gaussian states and employ the powerful covariance matrix formalism that allows to achieve analytical insight in scenarios that involve Gaussian states and linear unitary transformations [11, 12]. In particular, we seek out the effects of rotation on previously employed entanglement-swapping protocols [2, 8].
We find the error bound on the equatorial angular velocity of the Earth and compare it with that achieved with cutting edge technology. The rate of improvement of quantum optical technologies and the rapid increase of the control over quantum systems suggest that in the near future our scheme might provide a reliable way to outperform current technologies based on classical means.
The paper is organised as follows. In Section 2, we present the process of exchanging photons between Earth and a satellite, we characterise and model the system, and we give the mathematical formalism that is going to be relevant for the general relativistic calculations that will follow. In Section 3, we derive the expression of the frequency shift for the photon travelling through the Kerr spacetime. Section 4 consists of the relativistic quantum metrology calculations. It introduces the relevant perturbative quantities that are affecting the states, and derives the Quantum Fisher Information (QFI) for the system studied, and hence the estimated error bounds for the spacetime parameters. Section 5 introduces the satellite to satellite scheme and the related precision estimations are computed in the same fashion as in the Earth to satellite case. Finally, Section 6 briefly discusses how the effects computed in this work can affect a simple QKD protocol, specifically comparing the magnitude of the effect with what has been found in [2].
Throughout the whole paper we employ geometrical units \(G=1=c\). Relevant constants are restored when needed for the sake of clarity. Vectors and matrices are denoted in bold characters. Vectors are written using the usual differential geometry notation [13], namely \(\boldsymbol{X}=(X^{t}, X^{r}, X^{\theta}, X^{\phi}) = X^{t} \partial_{t} + X^{r} \partial_{r} + X^{\theta} \partial_{\theta} + X^{\phi} \partial_{\phi}\). Einstein’s summation convention is assumed on repeated Greek indices. A and B indices denote evaluations at Alice’s and Bob’s events respectively.
2 Introduction to the formalism
2.1 Description of the experiment
2.2 Wave packet characterisation
3 Frequency shift in Kerr spacetime
3.1 Preliminaries
3.2 Frequency shift formula
A last relevant check is to verify the absence of frequency shift in flat spacetime. Unsurprisingly, we get from the relevant limit of (21) that in Minkowski spacetime \(f_{M}=1\).
4 Quantum estimation of rotation parameters of the Earth
In this section, we apply quantum estimation techniques to find the ultimate bounds on the precision of measurements of parameters of the Earth.
4.1 Summary of spacetime parameters
Dimensionless perturbative parameters in the frequency shift formula
Quantity (N. Units) | Quantity (S.I.) | Value | Orbit |
---|---|---|---|
\(M/r_{A}\) | \(GM/(r_{A}c^{2})\) | 6.95 × 10^{−10} | / |
\(M/r_{B}\) | \(GM/(r_{B}c^{2})\) | 1.05 × 10^{−10} | GEO |
5.29 × 10^{−10} | LEO | ||
\(a/r_{A}\) | \(a/r_{A}\) | 5.11 × 10^{−7} | / |
\(a/r_{B}\) | \(a/r_{B}\) | 7.74 × 10^{−8} | GEO |
3.89 × 10^{−7} | LEO | ||
\(r_{A}\omega_{A}\) | \(r_{A}\omega_{A}/c\) | 1.55 × 10^{−6} | / |
We have used the following values: \(a = 3.26\mbox{ m}\), \(\omega_{A}=7.29 \times 10^{-5}\mbox{ rad/s}\), \(r_{A}=6\mbox{,}378\mbox{ km}\), \(M=5.97 \times10^{24}\mbox{ kg}\). Furthermore we consider two orbits for satellites, low Earth orbits \(r_{B}(\mathrm{LEO}) = r_{A} + 2\mbox{,}000\mbox{ km}\) and geostationary ones \(r_{B}(\mathrm{GEO}) = r_{A} + 35\mbox{,}784\mbox{ km}\).
4.2 Quantification of the frequency shift
4.3 Quantum Fisher information (QFI) and single parameter estimation
4.4 Optimal bounds for the error on spacetime parameters
For orbits at altitude around \(L\sim\frac{r_{A}}{2}\) however, the Schwarzschild term \(\delta_{S}\) in (34) vanishes, we then need to add the lowest order terms from \(\delta_{c}\). These are several orders of magnitude smaller than \(\delta_{S}\), therefore satellites orbiting at these altitudes are not recommended for the experiments proposed here since the precision they would provide for the measurement of the Schwarzschild radius is significantly lower. This result is new compared to the study carried in [8], it comes from taking into account special relativistic effects due to our observers’ motions.
5 Satellite to satellite communication
6 Quantum bit error rate (QBER) in a simple QKD protocol
In the Earth to satellite setup, the contribution of the rotation to δ in (24) is negligible for most orbits. We obtain a QBER of order 10^{−6} for communications to LEO orbits and 10^{−4} to GEO orbits. For orbits at radii \(r_{B} \sim\frac{3}{2} r_{A}\) however, the rotation term becomes dominant and the QBER shrinks to ∼10^{−10}. Hence, these orbits are recommended to reduce the QBER in Earth to satellite quantum communications.
In the satellite to satellite case, the Schwarzschild part of the shift is always dominant. The value of the shift between a LEO and a GEO satellite is similar to the GEO orbits case in the Earth to satellite scheme, hence the value for the QBER for quantum communications between a LEO and a GEO satellite is of order 10^{−4} too. However, taking into account atmospheric effects in the ground to satellite case would make the satellite to satellite scheme more accurate.
7 Conclusion
In this paper we have derived an expression for the general relativistic frequency shift of a photon travelling through Earth’s rotating surrounding spacetime. We have specialised to photons travelling with vanishing angular velocities from an equatorial laboratory on Earth towards a satellite revolving in the equatorial plane of the Kerr spacetime. This study provides analytical insight and successfully extends previous results obtained for Schwarzschild spacetime [2, 8]. We have found that including the rotation of the Earth does not change previous estimates obtained for the Schwarzschild radius in a quantum metrology scheme. However, we were able to estimate the precision for the quantum measurement of the equatorial angular velocity of the Earth. We find that the error bound predicted for the equatorial angular velocity of the Earth can exceed the precision obtained with the state of the art when high values of squeezing and a large number of probe systems (or measurements) are employed. Suitably chosen signals, such as frequency comb, instead of Gaussian-shaped frequency distributions, could also improve precision [24, 25]. Taking into account special relativistic effects, we have also found a specific class of circular orbits where the frequency of the received photons remains almost unchanged. For quantum metrology purposes these orbits have to be avoided since the quantum state of the photons is less perturbed, yet they are very useful for minimal curved spacetime disturbance channels for quantum communication. To complete our analysis, we have added a study of the error bounds for the same parameters when communication occurs between two satellites, which has relevance for practical implementations of many quantum information schemes, such as proposed implementations of QKD through satellite nodes [26]. We conclude that recent advances in quantum technologies, which include the ability to create larger values of squeezing, show the promising opportunities of improving the state of the art for measurements of physical parameters of the Earth.
Declarations
Acknowledgements
J. Louko was supported in part by STFC (Theory Consolidated Grant ST/J000388/1). D. E. Bruschi was partially supported by the I-CORE Program of the Planning and Budgeting Committee and the Israel Science Foundation (grant No. 1937/12), as well as by the Israel Science Foundation personal grant No. 24/12. D. E. Bruschi would also like to thank the University of Vienna for hospitality.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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