Limits and security of free-space quantum communications

Open Access

Limits and security of free-space quantum communications

Stefano Pirandola

Phys. Rev. Research 3, 013279 – Published 25 March 2021

Abstract

The study of free-space quantum communications requires tools from quantum information theory, optics, and turbulence theory. Here we combine these tools to bound the ultimate rates for key and entanglement distribution through a free-space link, where the propagation of quantum systems is generally affected by diffraction, atmospheric extinction, turbulence, pointing errors, and background noise. Besides establishing ultimate limits, we also show that the composable secret-key rate achievable by a suitable (pilot-guided and postselected) coherent-state protocol is sufficiently close to these limits, therefore showing the suitability of free-space channels for high-rate quantum key distribution. Our paper provides analytical tools for assessing the composable finite-size security of coherent-state protocols in general conditions from the standard assumption of a stable communication channel (as is typical in fiber-based connections) to the more challenging scenario of a fading channel (as is typical in free-space links).

Received 9 September 2020
Accepted 5 March 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.013279

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum cryptography Quantum entanglement

Quantum Information, Science & Technology

Authors & Affiliations

Stefano Pirandola

Department of Computer Science, University of York, York YO10 5GH, United Kingdom

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 3, Iss. 1 — March - May 2021

Subject Areas

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Free-space communication from a transmitter (Tx) to a receiver (Rx) separated by distance $z$ . The transmitter generates a Gaussian beam with spot size $w_{0}$ and mean number of photons ${\bar{n}}_{T}$ . The propagation of the beam is affected by diffraction, atmospheric extinction $η_{atm}$ , and turbulence/pointing errors, so its short-term spot-size $w_{st}$ is randomly deflected by $r$ from the aperture center of the receiver, with an associated transmissivity $η_{st} (r)$ . The beam is also affected by an additional attenuation, given by the efficiency $η_{eff}$ of the receiver. In total, transmitter and receiver are connected by an instantaneous lossy channel with transmissivity $τ (r) = η_{st} (r) η_{atm} η_{eff}$ as in Eq. (17). Besides loss, we also consider noise. In particular, thermal noise ${\bar{n}}_{B}$ is collected by the field of view of the Rx and further noise ${\bar{n}}_{ex}$ may be locally generated by setup imperfections. As a result, the detector is hit by ${\bar{n}}_{R} = τ (r) {\bar{n}}_{T} + \bar{n}$ mean photons whose $\bar{n} = η_{eff} {\bar{n}}_{B} + {\bar{n}}_{ex}$ are due to thermal noise.
Reuse & Permissions
Figure 2
Performance of free-space quantum communications in terms of bits per channel use versus distance. (a) We consider the general worst-case scenario with untrusted loss and noise at the receiver ( $η_{eff} = 0.5$ , ${\bar{n}}_{ex} = 0.05$ ). We plot the ultimate loss-based upper bound of Eq. (27) for nighttime (top red line) and daytime (red dashed line). This is compared with the bounds explicitly accounting for thermal noise $\bar{n} = η_{eff} {\bar{n}}_{B} + {\bar{n}}_{ex}$ . In particular, we plot the thermal upper bound of Eq. (33) for nighttime (black solid line) and daytime (black dashed line), as well as the thermal lower bound of Eq. (36) for nighttime (blue solid line) and daytime (blue dashed line). (b) Same comparison as in (a) but considering a noiseless receiver ( $η_{eff} = 0.5$ , ${\bar{n}}_{ex} = 0$ ). For nighttime, the upper and lower thermal bounds coincide with loss-based upper bound (solid red line). For daytime, the performances are instead separate (dashed lines). (c) Same comparison as in (a) but considering an ideal lossless and noiseless receiver ( $η_{eff} = 1$ , ${\bar{n}}_{ex} = 0$ ). As in (b), the two thermal bounds collapse in the loss-based upper bound during night time (solid red line). Performances are different during daytime (dashed lines). Other parameters are $R_{0} = \infty$ (collimated Gaussian beam), $λ = 800 nm$ , $w_{0} = a_{R} = 5 cm$ , $Ω_{fov} = 10^{- 10} sr$ , $Δ t = 10 ns$ , and $Δ λ = 1 nm$ . We consider $h = 30 m$ , so $C_{n}^{2} ≃ 1.28 (2.06) \times 10^{- 14} m^{- 2 / 3}$ for night (day), and we have ${\bar{n}}_{B} ≃ 4.75 \times 10^{- 8}$ ( $\times 10^{- 3}$ ) at night (cloudy day).
Reuse & Permissions
Figure 3
For daytime and fixed distance $z = 1 km$ , we plot the thermal bounds in Eqs. (33) and (36) as a function of the receiver's aperture $a_{R}$ . We assume an ideal receiver ( ${\bar{n}}_{ex} = 0$ and $η_{eff} = 1$ ). Other parameters are as in Fig. 2.
Reuse & Permissions
Figure 4
General description of the protocol and worst-case eavesdropping scenario. Alice's modulated coherent state $|α〉$ , with ${\bar{n}}_{T}$ mean photons, is subject to channel loss $η_{ch}$ and background noise ${\bar{n}}_{B}$ , before entering the receiver with quantum efficiency $η_{eff}$ and setup noise ${\bar{n}}_{ex}$ . Bob's detects ${\bar{n}}_{R} = τ {\bar{n}}_{T} + \bar{n}$ mean photons, where $τ = η_{ch} η_{eff}$ is the total transmissivity of the link and $\bar{n} = η_{eff} {\bar{n}}_{B} + {\bar{n}}_{ex}$ is the total number of thermal photons. The input-output relation for the quadratures is given by Eq. (48). In the worst-case scenario, Eve collects all the leakage and controls all thermal noise. This is equivalent to assume that she replaces the channel with a beam splitter with transmissivity $τ$ and thermal input ${\bar{n}}_{e} = \bar{n} / (1 - τ)$ . The latter is part of a TMSV state in her hands, whose output is stored in a quantum memory.
Reuse & Permissions
Figure 5
Defading of data. See text for details.
Reuse & Permissions
Figure 6
Composable secret-key rates (bits/use) versus distance (m) for free-space QKD in the regime of weak turbulence and for cloudy daytime operation. We consider a coherent-state protocol with heterodyne detection, pilot-guided and operated in postselection as described in the main text. Physical and protocol parameters are listed in Tables 1 and 2. (a) We plot the secret key rate of Eq. (154) assuming a TLO (black curves) and an LLO (blue curves). In particular, we plot the performances at fixed postselection threshold $f_{th} = 0.84$ and fixed input modulation, $μ = 20$ for TLO and $μ = 8.4$ for LLO (solid curves). These are chosen to optimize the rates at the maximum distance ( $z = 1066 m$ ). We compare these performances with those achievable by optimizing the rates over $μ$ and $f_{th}$ at each distance (dashed curves). (b) We plot the rate of Eq. (156) against general attacks for TLO (black line) and LLO (blue line). These performances are not optimized and refer to fixed threshold $f_{th} = 0.84$ and input modulation ( $μ = 20$ for TLO and $μ = 8.4$ for LLO).
Reuse & Permissions

Physical Review Research