Perspectives of microwave quantum key distribution in open-air
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Perspectives of microwave quantum key distribution in open-air
F. Fesquet,1, 2, ∗ F. Kronowetter,1, 2, 3 M. Renger,1, 2 Q. Chen,1, 2 K. Honasoge,1, 2
O. Gargiulo,1, 2 Y. Nojiri,1, 2 A. Marx,1 F. Deppe,1, 2, 4 R. Gross,1, 2, 4 and K. G. Fedorov1, 2, †
1
Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany
2
Physik-Department, Technische Universität München, 85748 Garching, Germany
3
Rohde & Schwarz GmbH & Co. KG, Mühldorfstraße 15, 81671 Munich, Germany
4
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 Munich, Germany
One of the cornerstones of quantum communication is the unconditionally secure distribution of classical
keys between remote parties. This key feature of quantum technology is based on the quantum properties of
propagating electromagnetic waves, such as entanglement, or the no-cloning theorem. However, these quantum
resources are known to be susceptible to noise and losses, which are omnipresent in open-air communication
scenarios. In this work, we theoretically investigate the perspectives of continuous-variable open-air quantum
arXiv:2203.05530v1 [quant-ph] 10 Mar 2022
key distribution at microwave frequencies. In particular, we present a model describing the coupling of propagat-
ing microwaves with a noisy environment. Using a protocol based on displaced squeezed states, we demonstrate
that continuous-variable quantum key distribution with propagating microwaves can be unconditionally secure
at room temperature up to distances of around 200 meters. Moreover, we show that microwaves can potentially
outperform conventional quantum key distribution at telecom wavelength at imperfect weather conditions.
I. INTRODUCTION microwave-to-optical frequency transducers, a natural choice
is to consider quantum communication and QKD in the as-
Quantum key distribution (QKD) can be defined as a sociated microwave regime. Here, superconducting Joseph-
method to securely exchange a common key between two son parametric amplifiers (JPAs) represent a robust source of
parties, conventionally referred to as Alice and Bob. QKD quantum states in the form of squeezed microwaves. Flux-
has attracted increasing interest over the last decades due to driven JPAs routinely generate microwave squeezed states
the promise of unconditionally secure communication while with squeezing levels up to 10 dB below the vacuum limit
maintaining high secret key rates [1, 2]. The security of [17–19].
commonly used classical encryption schemes, or key agree- In this work, we focus on a theoretical analysis of a partic-
ment protocols, is based on asymmetric mathematical prob- ular one-way communication prepare-and-measure CV-QKD
lems which cannot be easily inverted by classical methods. protocol with Gaussian modulation [20] for the microwave
Prominent examples are the Diffie-Hellman algorithm or the wavelength range of 30-300 mm, corresponding to the fre-
RSA code [3, 4]. In contrast, QKD relies on the fact that un- quency range of 1-10 GHz. We analyze the potential of
conditional security is provided by the fundamental laws of this protocol for open-air microwave quantum communica-
quantum mechanics, taking advantage of unique quantum re- tion (MQC), considering realistic atmospheric conditions. We
sources, such as entanglement [5], or notably, the no-cloning compare its performance to a traditional implementation at the
theorem [6]. In QKD based on continuous-variable (CV) telecom wavelength of 1550 nm (193.55 THz). We model the
quantum states (CV-QKD), one encodes information in con- signal readout with a homodyne detection, since the informa-
jugate electromagnetic field quadratures, according to various tion is encoded only in one of the two electromagnetic field
specific protocols [7]. In particular, CV-QKD protocols pro- quadratures. In the last step, signal detection is followed by a
vide a powerful alternative to QKD based on discrete-variable one-way classical reconciliation, or error correction, protocol.
(DV) quantum states (DV-QKD), due to potentially higher se- There, an important distinction must be made between either
cret key rates and being compatible with the existing commu- direct reconciliation (DR) or reverse reconciliation (RR) [21].
nication infrastructure [8, 9]. In DR, the one-way communication is performed from Alice
First successful realizations of QKD protocols have been to Bob. As a result, Alice’s key is used as a reference which
implemented at the near-infrared regime at so-called telecom Bob tries to estimate from the data he obtained after the com-
wavelengths (780-850 nm and 1520-1600 nm wavelength) munication. On contrary in RR, the one-way communication
[10], where a significant progress has been achieved with the goes from Bob to Alice, where Bob’s measured key is esti-
recent realization of ground-to-satellite QKD networks [11]. mated by Alice.
A particular reason for this choice of frequencies is the low In Sections II and III, we introduce all relevant elements
atmospheric absorption on the order of 1.0 × 10−2 dB/km for a practical implementation of the CV-QKD protocol in the
[12]. However, in the light of recent impressive achieve- microwave regime. We perform a quantitative analysis of the
ments with superconducting quantum circuits operating in the secret key rate and secure distance, assuming either DR or RR
microwave regime [13–16] and due to the lack of efficient with an ideal reconciliation efficiency in the asymptotic case,
i.e., in the case of Alice and Bob exchanging an infinitely long
key. We additionally present a model based on beam splitter
transformations to describe the coupling of propagating mi-
∗
florian.fesquet@wmi.badw.de crowaves with a bright microwave thermal background. Our
†
kirill.fedorov@wmi.badw.de results show that microwave CV-QKD can be well suited for2
Alice p and encoded symbols. This imposes an additional constraint
σs2 + σA 2
= σas2
, where we denote the squeezed and anti-
DA(ai) ai q
ki squeezed quadrature variance as σs2 and σas 2
, respectively. The
resulting displaced squeezed state propagates through a lossy
SA(xq,p) and noisy quantum channel N before being received and mea-
sured by Bob with a local homodyne measurement. After re-
DA(ai) ai peating this process for each of Alice’s symbols, Bob obtains
p
a measured key K0 = {k10 , ..., ki0 , ..., kn0 }, representing an es-
timation of Alice’s key K.
quantum channel In order to obtain a common secret key, Alice and Bob
’
tE perform a one-way classical postprocessing. The first step,
ki know as sifting of the keys, consists of producing compat-
n ible data, discarding any measurement where encoding and
p BAE(tE) measurement basis disagree. The second step, commonly re-
ferred to as parameter estimation, allows to obtain an upper
Bob Eve bound on the amount of information lost in the quantum chan-
nel. Then, a classical reconciliation algorithm, depending on
Figure 1. General scheme of a QKD protocol based on dis- whether DR or RR has been chosen, is used to generate a com-
placed squeezed states. Here, Alice starts by generating a random mon key. This performance of this algorithm is characterized
continuous-variable number ki corresponding to Alice’s symbol and
with a reconciliation efficiency β. Alice and Bob further pro-
randomly choosing among one of the two possible encoding bases, q
or p. This procedure results in propagating states which are squeezed
ceed to a confirmation step to validate a recovered common
along one or the other quadrature with the complex squeezing ampli- key. Finally, a classical privacy amplification algorithm pro-
tude ξ. Every symbol ki is encoded via a displacement amplitude αi , duces a secret key, discarding any eavesdropped bits common
such that |αi | = |ki |. The resulting displaced squeezed state is sent key.
through a quantum channel N with transmissivity τE . This channel To describe the quantum channel N , we quantify losses us-
is assumed to be under full control of a potential eavesdropper, Eve, ing a quantum channel transmissivity τE and quantum channel
who also induces extra noise photons n̄. At the end, Bob receives a noise, n̄ which represents an average noise photon number re-
state which he measures in a random basis q or p in order to obtain ferred to the output of the channel. Generally, these losses and
an estimate, ki0 , of the original symbol.
noise represent interactions between the propagating states
and environment. In the worst case, from the standpoint of
security, the quantum channel is under the full control of a
short-distance open-air communication scenarios. In particu- potential eavesdropper, Eve. This implies that our estimated
lar, in Section IV we find that the microwave CV-QKD pro- secure bit rates is a lower bound of realistically achievable se-
tocol may produce higher secure key rates than its telecom cure bit rates. Here, we analyze an asymptotic case, where
counterpart. We conclude our discussion in Section V with a Alice communicates an infinitely long key, n → ∞. Using
study of weather effects for the particular common cases of this assumption, we can avoid finite-size effects [22] and re-
rain and haze. We find that MQC secure communication dis- strict our analysis to collective Gaussian attacks [23], while
tances are almost unchanged as compared to ideal dry weather considering them as the most general attacks.
conditions. This is in striking contrast to the telecom proto- In collective Gaussian attacks, all physical Gaussian states
cols. remain Gaussian states throughout the quantum communica-
tion. Additionally, Eve is assumed to interact individually
with each state sent by Alice and to store all her extracted
II. QUANTUM KEY DISTRIBUTION PROTOCOL WITH states in a quantum memory before applying a joint measure-
CONTINUOUS VARIABLES ment on her state ensemble at the end of the classical postpro-
cessing. Then, Eve’s eavesdropping attack can be extended,
First, we consider a prepare-and-measure CV-QKD proto- i.e., dilated, into an entangling cloner attack [24]. In this at-
col, independent of a particular hardware platform and fre- tack, Eve starts with a two-mode squeezed (TMS) vacuum
quency range, as described in Ref. 20. A corresponding state [25] for each incoming state from Alice. One of these
scheme is shown in Fig. 1. Here, Alice transmits to Bob modes is coupled to the quantum channel via a beam split-
a Gaussian-modulated random key K = {k1 , ..., ki , ..., kn }. ter with transmissivity τE , while Eve preserves the other un-
This key is a string of real numbers ki , randomly chosen from coupled mode (for details, see appendix A). After interaction
2 with Alice’s signal, this coupled mode contains partial infor-
a Gaussian distribution with variance σA . To this end, Al-
ice prepares a q-squeezed or p-squeezed state [15], with both mation on the communicated key. A general TMS state fea-
states having an equal chance of being selected. Each sym- tures quantum entanglement, implying that both Eve’s modes
bol, ki , is then encoded as a displacement amplitude, αi , of are strongly correlated. Then, Eve can use these correlations
each squeezed state such that |αi | = |ki |. Averaging over to extract as much information as possible on the sent key.
various states of Alice results in a thermal state, preventing We model our CV-QKD protocol using an input state ρ̂in for
Eve from extracting any information on the encoding basis each symbol ki . This input state contains three modes. The3
first mode is used by Alice to generate the displaced squeezed or phase-sensitive amplification of incident signals [28]. The
states. The remaining two modes describe Eve’s TMS state. latter regime is directly related to the generation of squeezed
The first mode of Eve’s TMS state locally looks like a thermal microwave states. According to Cave’s theory of noise in li-
state coinciding with the environmental noise state, but pos- near amplifiers [29], phase-insensitive bosonic amplifiers are
sessing quantum entanglement with the third, retained, Eve’s quantum-limited in the sense that they add at least half a noise
mode. Following this formalism, we denote the final state photon to an input signal. In contrast, a phase-sensitive ampli-
ρ̂out , which is also a three-mode state, where its first mode fier can, in principle, achieve noiseless amplification. In prac-
corresponds to the local state which Bob receives after Eve’s tice, JPAs have proven to approach both these limits, which
attack. Similarly, the remaining two modes describe Eve’s makes them well qualified for MQC applications. In experi-
modes after her attack. The final state can be written as ments, JPAs operating in the GHz regime were shown to reach
† noise levels on the order of 0.1 added noise photons in the
ρ̂out = T̂AE ρ̂in T̂AE , phase-sensitive regime [17]. Presently, the noise performance
(1)
T̂AE = B̂AE (τE ) D̂A (αi ) ŜA (ξ) , of JPAs is limited by fabrication imperfections, pump-induced
noise [30, 31], or higher-order nonlinearities [32]. The dis-
where B̂AE (τE ) is a beam splitter operator (see appendix A) placement operation required by our CV-QKD protocol can
with transmissivity τE , which describes coupling Alice’s be experimentally realized by applying strong coherent drive
mode to environment (Eve’s mode). Similarly, D̂A (αi ) repre- tones to cryogenic directional couplers [26]. Ultimately, the
sents the displacement operator [26] applied to Alice’s mode, combination of JPAs with subsequent directional couplers, al-
with the displacement amplitude encoding a specific symbol, lows one to generate displaced squeezed states with a desired
|αi | = |ki |. Additionally, ŜA (ξ) corresponds to the squeeze displacement amplitude α.
operator [15] acting on Alice’s mode, and ξ is the complex
squeezing amplitude.
B. Microwave antennas
III. EXPERIMENTAL SETUP CONSIDERATIONS In order to couple propagating microwave states, generated
at millikelvin temperatures, to the open-air quantum channel
An open-air implementation of the above introduced CV- one requires a microwave interface between the correspond-
QKD protocol requires various different aspects to be taken ing cryogenic environment and the open-air medium. A mi-
into account. In this work, we focus on the central compo- crowave antenna serves as such kind of interface. The antenna
nents to realize such an open-air MQC and present an associ- may be modelled by a transmission line of spatially varying
ated generic scheme in Fig. 2. We analyze the generation and impedance connecting the 50 Ω-matched cryogenic circuits to
detection of propagating squeezed states at millikelvin tem- open-air channels with characteristic impedance of 377 Ω. A
peratures. Detection is modelled by a homodyne quadrature central figure of merit of the transmitter and receiver antennas
measurement with quantum efficiency, η. Coupling of the is their passive antenna gain, G. In general, for microwave
squeezed states to the open-air environment (atmosphere) is antennas, the gain reads [33]
modelled with two antennas with corresponding gain coeffi-
cients. The environment is assumed to be at ambient temper- G = ηrad D, (2)
ature, T = 300 K, and is described by frequency-dependent
absorption losses. The latter may also be subject to imperfect where 0 ≤ ηrad ≤ 1 is the radiation efficiency and accounts
weather conditions, as it will be discussed later. for the antenna losses, while D represents the antenna direc-
tivity. The latter expresses the ability of the antenna to fo-
cus the emitted power into a specific direction and strongly
A. Generation of quantum microwave states depends on the antenna geometry. An antenna with a well-
defined physical aperture area, A, has the directivity
4πA
The experimental realization of the CV-QKD protocol D= eA , (3)
λ2
requires the generation of propagating displaced squeezed
states. In the microwave regime, flux-driven Josephson para- where A is determined by the size and shape of the antenna,
metric amplifiers (JPAs) provide an opportunity to gener- λ the signal wavelength, and eA the aperture efficiency, de-
ate squeezed states with tunable squeezing level and angle fined as ratio between the effective aperture and physical
[17, 27]. Typically, JPAs consist of a coplanar waveguide λ/4 aperture areas. Cryogenic to open-air transmission of mi-
resonator which is short-circuited to ground by a direct current crowave signals is a current technological challenge. First
superconducting quantum interference device (dc-SQUID). proposals already exists [34]. For communication distances
This dc-SQUID acts can be described as nonlinear inductance of around 50 m, an open-air geometric attenuation of signals,
which can be modulated by applying an external magnetic also known as the path loss, is around 80 dB (see Sec. III D) at
flux resulting in a flux-tunability of the resonance frequency of the frequency of 5 GHz. In general, the path loss can be com-
the JPA. Applying a microwave pump signal inductively cou- pensated by using transmitter and receiver antennas with suf-
pled to the λ/4 resonator, JPAs can provide phase-insensitive ficient gain. For instance, a parabolic transmitter and receiver4
Alice Bob
d readout
JPA
source transmitter losses & noise receiver detector
S Gt τE nth Gr h
Figure 2. Schematics of main components for an open-air MQC. Source denotes a squeezing generator, typically implemented with a JPA in
a cryogenic environment. Transmitter and receiver represent corresponding microwave antennas with gains Gt and Gr , respectively. These
antennas belong to different communication parties, Alice and Bob, and are separated by a distance d. Atmospheric absorption losses are
quantified using transmissivity τE which couples the quantum communication channel to the open-air environment with the thermal noise
photon number n̄th . Readout is modelled as a homodyne detector with an overall quantum efficiency η.
antennas with a diameter of around Dant = 2 m could com- this case, we can use the Friis formula to estimate the total
pensate the aforementioned path loss. A more detailed analy- amplification noise namp of the detection chain. For instance,
sis on antennas designs, gains, and related path losses goes be- for the case of two chained amplifiers and in the limit large
yond the scope of this paper and is discussed elsewhere [35]. amplification, G1,2 1, the total amplification noise reads
Here, we assume that antenna gains fully compensate for the n2
path loss and focus on the effects of atmospheric absorption namp = n1 + , (5)
losses as the main source of communication imperfections. G1
where n1 and G1 are the noise photon number and gain of the
first amplifier in the chain, while n2 describes the input noise
C. Quantum efficiency of the detection chain photon number of the second amplifier. Thus, the total noise
namp depends mainly on the noise properties of the first am-
In order to finalize the prepare-and-measure CV-QKD pro- plifier. For homodyne detectors at telecom wavelengths, the
tocol, one has to perform a homodyne quadrature measure- quantum efficiency is usually modeled by additional losses,
ment. In the microwave regime, this task requires usage of lin- introduced by a non-unity transmissivity within a beam split-
ear amplifiers with a certain quantum efficiency, ηmw , to quan- ter model. Both approaches are known to be equivalent as
tify the amplification chain noise performance. The quantum described in Ref. 32.
efficiency is defined as the ratio between vacuum fluctuations
and fluctuations in output signals resulting from additional
noise photons namp due to amplification, where namp is re- D. Losses and noise budget
ferred to the input of the detection chain. Therefore, we can
express the quantum efficiency as [30] We conclude this section with a brief analysis of losses and
1 noise in open-air communication channels, where losses scale
ηmw = . (4) with the communication distance. We distinguish between
1 + 2namp
two categories of losses: (i) the path loss which represent ge-
State-of-the-art travelling wave parametric amplifiers (TW- ometric attenuation of propagating signals and (ii) absorption
PAs) allow for phase-insensitive amplification with high gain losses due to coupling to the environment, such as the atmo-
values (∼ 20 dB) and broad bandwidths (∼ 3 GHz) at cryo- spheric absorption losses or weather-induced losses. For sig-
genic temperatures. These TWPAs are also potentially able nals transmitted and received via the antennas, the path loss
to approach the quantum-limited regime characterized by Lp , describing the fraction of the initial signal power lost dur-
namp = 0.5 for phase-insensitive mode of operation [29]. ing the communication, is commonly described using the Friis
Conversely, as mentioned in Ref. 36, a phase-sensitive lin- transmission formula [33]
ear amplifier could achieve noiseless amplification of a single 2 !
quadrature, at the cost of deamplifying the conjugate quadra- λ
Lp = 10 log10 Gt Gr . (6)
ture. Such a detection scheme can be used to implement a 4πd
microwave homodyne detection similar to its optical counter-
part [37]. In cryogenic microwave experiments, one typically Here, Gt (Gr ) is the transmitter (receiver) antenna gain, λ the
uses chained quantum-limited amplifiers followed by cryo- wavelength of the communication signals, and d the propaga-
genic high-electron-mobility transistor (HEMT) amplifiers. In tion distance. The absorption and scattering power losses can5
be modeled via a single effective beam splitter with transmis- a
0 1 2 3 4 5
sivity τeff given by K (bits/symbol) environmental photons nth
0 345 690 1035 1380
squeezing level S (dB)
−γd/10
τE = 10 , (7) 15
where γ is the specific attenuation (dB/km) for a respective 12
loss mechanism. In our case, we attribute these losses to at- 9
mospheric absorption and weather conditions, such as rain or
6
haze. Empirical models show that for microwave frequencies
around fmw ' 5 GHz, propagation losses mainly arise due to 3
molecular oxygen absorption [38]. For the ideal case of dry 0
0.4 0.6 0.8 1.0 0 0.05 0.10 0.15 0.20
weather, we estimate the corresponding specific attenuation transmissivity tE noise photons n
of γmw = 6.3 × 10−3 dB/km [38]. To describe the coupling
of the propagating quantum bosonic signal â to the noisy en-
b
vironmental modes, we use the input-output formalism. The 0 1 2 3 4 5
output signal mode â0 can be expressed as K (bits/symbol) environmental photons nth
√ 0 345 690 1035 1380
√
squeezing level S (dB)
â0 = τE â + 1 − τE ĥenv , (8) 15
12
where ĥenv corresponds to the environmental thermal mode.
The latter may be a vacuum or thermal state, depending on the 9
carrier frequency and environment temperature. For a thermal 6
background, the average thermal noise photon number n̄th per 3
mode is given by the Planck distribution as
0
0.0 0.2 0.4 0.6 0.8 1.0 0 0.05 0.10 0.15 0.20
1
n̄th = , (9) transmissivity tE noise photons n
hf
exp kB T −1
Figure 3. Secret key K of the CV-QKD protocol plotted as a func-
where h is the Planck constant, kB is the Boltzmann constant, tion of the squeezing level S (measured in dB below the vacuum
f the signal frequency, and T the background temperature. limit), transmissivity τE , and noise n̄, for ideal detection efficiency
At last, it is instructive to mention open-air losses at tele- ηmw = 1. Panels a and b show the cases of DR and RR, respec-
com wavelengths. We emphasize that Eq. 6 is also applica- tively. Number of environmental noise photons, n̄, is shown for a
ble in the optical frequency range. Then, Gt and Gr cor- fixed communication distance, d = 200 m, and microwave specific
respond to the effective passive gain of optical lenses used attenuation, γmw = 6.3 × 10−3 dB/km. Grey areas represent the
regions of negative keys, i.e., insecure communication.
to focus and collect optical beams. Typical telecom wave-
lengths (780 −850 nm, and 1520 −1600 nm) are chosen to
suffer from the lowest possible atmospheric absorption losses.
At the telecom wavelength of 1550 nm, absorption losses of Bob’s measured key. Additionally, χE is the Holevo quan-
less than 1.0 × 10−2 dB/km can be reached[12]. In this case, tity [39] of Eve and gives an upper bound on the information
open-air attenuation is mainly caused by scattering losses, that Eve obtained during the quantum communication (see ap-
such as Rayleigh or Mie scattering [12]. The corresponding pendix A). For the sake of simplicity, we assume perfect a rec-
open-air specific attenuation is γtel = 2.02 × 10−1 dB/km. onciliation efficiency β = 1. It should be noted that experi-
We discuss the additional attenuation due to rain and haze in mental values β > 0.9 have been obtained [40]. A positive
more detail in Sec. V. value of K indicates a secure communication, as Alice and
Bob share more information than Eve can in principle obtain.
IV. SECURITY ANALYSIS In Fig. 3 we show results of numerical evaluation of the se-
cret key as a function of the transmissivity τE and noise pho-
A. Secret key
tons n̄ in the quantum channel. Remarkably, in the DR case a
secure communication cannot exist when τE exceeds a thresh-
old value of 0.5, which illustrates the well-known result that
In order to assess the experimental feasibility of our CV- secure CV-QKD communication in DR schemes are limited
QKD protocol, it is mandatory to analyze its security. The to 3 dB of losses [41, 42]. The reason for this fact is that
latter is quantified by the secret key K, which represents the communication with DR cannot be secure when Eve receives
amount of secure information per communicated symbol and more than 50% of Alice’s information. In this scenario, Eve
reads as effectively replaces Bob as the communication partner. As il-
K = β · I (A:B) − χE . (10) lustrated in Fig. 3, this limit can be entirely circumvented by
using the RR scheme. Then, Bob’s measured key is treated as
Here, I (A:B) is the mutual information between Alice and a reference and Alice needs to correct her own key according
Bob and measures correlations between Alice’s sent key and to it. For RR, if we imagine that Eve only induces losses dur-6
0 1 2 3 4 5 0 50 100 150
secret key K (bits/symbol) dc (m)
a
squeezing level S (dB)
15 15 1.0 3
telecom
secret key rate R (Gbits/s)
100 m DR 2
12 12
0.8 microwave
9 80 m 1
9
0.6 dc
0
mw
6 6 50 m
3
0.4 telecom
3 3 16 m 2 microwave
0.2
0 0 1
50 100 150 200 50 100 150 200 dc
0 0
distance d (m) distance d (m) 2 4 6 8 10 50 100 150 200
b 3
1.0
telecom
secret key rate R (Gbits/s)
Figure 4. Secret key K of the CV-QKD protocol as a function of RR 2
communication distance d and squeezing level S. Left (right) plot 0.8 microwave
corresponds to the DR (RR) cases, respectively. Squeezing is given 80 m 1
0.6 dc
in dB below the vacuum level. Detection efficiency is assumed to 0
mw
be ideal, ηmw = 1. We assume the average environmental noise 3
0.4 telecom
photon number n̄th = 1250 and transmission losses γE = γmw ' 50 m 2
6.3 × 10−3 dB/km. Grey areas represent the regions of negative microwave
0.2 16 m
keys, i.e., insecure communication. 1
dc
0 0
2 4 6 8 10 50 100 150 200
∆ f mw (GHz) distance d (m)
ing the quantum communication, Alice has always more in-
formation than Eve on Bob’s measured key. This is because, Figure 5. Crossover distance dc between microwave and telecom
in her attack, Eve is assumed to induce losses by using a beam CV-QKD. Panels a and b illustrate the DR and RR cases, re-
splitter to get part of the signals sent by Alice. As a result, spectively. For the telecom and microwave wavelengths, we as-
Eve can only obtain a fraction of Alice’s information. If Eve sume transmission losses γtel ' 2.02 × 10−1 dB/km, and γmw '
couples noise photons in addition to the previous losses, the 6.3 × 10−3 dB/km, respectively. For both DR and RR, the secret
correlations between Alice’s sent key and Bob’s measured key key rates R of both detection cases are shown on the right column as
decrease. At the same time, Eve gains more information on a function of communication distance d.
Bob’s measured key. In particular, the communication is se-
cure up to a noise photon threshold value n̄ of 0.183 for both
reconciliation cases. This result is consistent with the well- sequently, the effects of coupled noise largely outweigh the
known Pirandola-Laurenza-Ottaviani-Banchi (PLOB) upper- effects of losses and make the RR and DR cases more similar.
bounds for Gaussian channels [43]. This noise threshold cor-
For a more practical evaluation of the QKD performance,
responds to the crossover of the quantum channel capacity
one typically uses a secret key rate R0 . The latter evaluates the
from finite values to zero. It is also important to note that
amount of secure bits per second that can be obtained from the
these noise numbers do not account for noise photons which
communication protocol. Under the asymptotic case assump-
can be added by Bob during his measurements. Finally, we
tion, one can express the secret key rate R0 in bits per second
observe that an increase in squeezing level results in an in-
using the secret key as
crease of the secret key. This increase can be understood as
a decrease of the displacement uncertainty encoding the sym-
bols, while also allowing for higher displacement amplitudes R0 = fr · K, (11)
[20].
Next, we extend our analysis of the microwave CV-QKD where fr represents the repetition rate (in symbols per sec-
protocol to variable communication distances d, for both DR ond). This rate which encompasses all information post-
and RR. To this end, we use Eq. 8 in combination with the spe- processing steps, such as sifting, parameter estimations [5, 7],
cific attenuation given in Sec.III D to convert communication and experimental bandwidths of the involved devices. We use
distances d into corresponding transmissivities τE . The corre- an upper bound on the secret key rate, R, derived from the
sponding secret keys are shown in Fig. 4. Remarkably, we ob- Shannon-Hartley theorem and the Nyquist rate [44]
serve positive secret key values over communication distances
of up to 200 m, in both DR and RR. These results suggest the
experimental feasibility of microwave QKD in open-air con- R0 ≤ R = 2 ∆f · K, (12)
ditions. No major distinction in communication distances is
observed between the reconciliation cases, although one could where ∆f denotes the experimental detection bandwidth.
intuitively expect RR to yield larger distances according to our This upper bound becomes especially useful when comparing
previous discussion. This behavior originates from the pres- different physical QKD platforms, as it is going to be dis-
ence of the bright microwave thermal background which cou- cussed in the next section.
ples to propagating states during the communication. Con-7
B. Comparison between telecom and microwave frequency tations reach secure key rates up to a few Mbits per sec-
ond [51–53]. Aside from finite quantum detection efficien-
Here, we now compare the microwave CV-QKD perfor- cies and bandwidths, practical secret key rates are also lim-
mance to that of QKD at telecom frequencies. For this pur- ited by various factors such as actual experimental repetition
pose, we define and numerically compute a communication rates [7], device-induced noise [52], finite size effects [22], or
crossover distance dc post-processing [54]. Nevertheless, the demonstrated results
make MQC relevant for short-distance classical communica-
dc := max (d) , (13) tion protocols such as Wifi 802.11 standard (maximal com-
Rmw ≥Rtel munication distance ' 70 m), Bluetooth 5.0 ( ' 240 m), or
more recent technologies such as 5G (' 305 m) because of
where d corresponds to communication distance, while Rmw
matching frequency ranges, distances, and technological in-
and Rtel are the secret key rates for microwave and telecom
frastructure.
frequencies, respectively. According to Eq. 12 , it is relevant
to optimize the detection bandwidth to achieve high secret key
rates. To this end, we assume an experimental state-of-the- V. WEATHER INDUCED LOSSES EFFECTS
art broadband squeezing generation and detection at 1550 nm
wavelength over a bandwidth of ∆ftel = 1.2 GHz with a A. Non-optimal weather conditions
quantum efficiency of ηtel = 0.53, as shown in Ref. 45. In
this experiment, the authors also report a squeezing level of
3 dB, which we will use as a common level of vacuum squeez- So far, we have investigated open-air CV-QKD under ideal
ing for both the microwave and telecom regimes. We com- weather conditions. However, it is well-known that realistic,
pute the corresponding crossover distance as a function of non-optimal weather conditions may drastically affect absorp-
the microwave detection bandwidth ∆fmw and quantum ef- tion losses for propagating signals. Such effects are especially
ficiency ηmw . The corresponding results are shown in Fig. 5 prominent in the telecom frequency range. Therefore, it is na-
for both DR and RR. Interestingly, we observe that the mi- tural to investigate effects of these non-optimal conditions on
crowave CV-QKD protocol can outperform the telecom coun- the MQC as well. Specifically, we focus on two non-ideal
terpart for realistic values of ∆fmw and ηmw . A clear distinc- weather scenarios: rain and haze. In the context of microwave
tion can be seen between the two reconciliation cases. For communication, the ITU-P. 838 and ITU-P. 840 recommenda-
the DR case, it is beneficial to aim at the quantum efficiency tions provide empirical prediction models for the induced at-
close to unity and large detection bandwidths. The situation tenuation on propagating microwave signals due to rainfall
is noticeably different in RR. For the latter, we observe that and haze, respectively. More precisely, the specific attenua-
above a certain detection bandwidth, the optimal quantum ef- tion γmw,r due to rain along a horizontal path can be expressed
ficiency is no longer unity. Instead, there exists an optimal as [55]
detection noise added by Bob, which maximizes the secret γmw,r = k (f ) Rrα(f ) , (14)
key rate depending on the detection bandwidth. The exis-
tence of an optimal quantum efficiency is a remarkable fea- where k and α are coefficients which depend on the commu-
ture of RR, which arises when Bob couples additional noise nication microwave frequency f , while Rr (mm/h) is the rain
during his measurements [46]. To illustrate the influence of rate. The haze specific attenuation γh can be obtained from
the quantum efficiency and the detection bandwidth, we envi- the liquid water concentration M (g/cm3 ) from a linear rela-
sion two different microwave homodyne detection cases im- tionship as [56]
plemented by a phase-sensitive amplifier. First, we choose a
γmw,h = Kl (f, T ) M , (15)
high detection bandwidth ∆fmw = 3 GHz with a quantum ef-
ficiency of ηmw = 0.345. This case is motivated by the exist- where Kl ((dB/km) / (g/cm3 )) is the linear attenuation that
ing state-of-the-art superconducting TWPA devices operated depends on the considered microwave frequency f and water
in the phase-insensitive regime [47, 48]. The second case con- temperature T in the atmosphere. The liquid water concentra-
siders a detection bandwidth of ∆fmw = 1.2 GHz = ∆ftel , tion can be related to a physically more intuitive quantity, the
and we choose a quantum efficiency of ηmw = 0.695, such so-called visibility V (km). The latter represents the distance
that both cases yield the same DR crossover distance. This at which the light intensity from an object drops to 2% of its
case originates from recent results on broadband squeezing in initial value. For a non-polluted environment, one can link
the microwave regime [49, 50]. By using this set of already two aforementioned quantities as [57]
experimentally feasible parameters, we can reach a crossover a b
distance of dc = 16 m for both cases. For RR, we observe M= , (16)
that the crossover distance can be increased to dc = 25 m. V
The reason is that RR benefits from a quantum efficiency where a = − log (0.02) /99 and b = 0.92−1 . For the tele-
below unity. Remarkably, high secret key rates R of a few com frequencies, rain causes a wavelength-independent atten-
Gbits per second can be reached for all of the previously men- uation. The specific attenuation γtel,r can be expressed for a
tioned set of parameters. However, we stress that the com- horizontal path as [12]
puted secret key rates R are merely upper bounds for realis-
tically achievable rates. Existing telecom QKD implemen- γtel,r = k Rrα , (17)8
a ideal weather b 3
ideal weather
103
1000 10
103 3
1000
10
100 telecom, 100 100 telecom,
100
λ = 1550 nm λ = 1550 nm
10 10
10 microwave, 10 microwave,
secret key rate R (Mbits/s)
λ = 6 cm 1 λ = 6 cm 1
5x103 6x103 7x103 105
1 1
heavy rain heavy rain
103
1000 3 33
10
1000
10
10
100 100
10 10
1 1
light haze light haze
1000
103 1033
1000
10
100 100
10 10
1 1
1 10 100 1000
103 1 10 100 103
1000 104
10000 105
100000
distance d (m) distance d (m)
Figure 6. Secret key rates of the CV-QKD protocol for various weather conditions. Telecom, and microwave secret key rates R are computed in
DR (panel a) and in RR (panel b) as a function of the communication distance d for the squeezing levels of Stel = Smw = 3 dB. Three different
weather conditions are considered: ideal weather conditions (visibility of 23 km), heavy rain with a rain rate of 7 mm/h, and light haze with a
visibility of 4 km. The choice of quantum efficiency and detection bandwidth is the same as for the ideal weather conditions. For the telecom
case, we consider the total transmission losses γtel ' 2.02 × 10−1 dB/km (optimal), γtel ' 4.17 dB/km (rain), and γtel ' 1.55 dB/km
(haze). For the microwave case, we assume the transmission losses γmw ' 6.3 × 10−3 dB/km (optimal), γmw ' 1.22 × 10−2 dB/km
(rain), and γmw ' 6.4 × 10−3 dB/km (haze).
where Rr is the rain rate, k = 1.076, and α = 0.67. The haze- short-distance microwave QKD could potentially yield higher
specific attenuation is empirically derived similarly to the mi- secret key rates as compared to the telecom case. The rea-
crowave case. Once again, visibility determines the specific son is that microwave QKD benefits from higher experimen-
attenuation γtel,r . Empirical models for Mie scattering show tal bandwidths and lower losses due to weather imperfections.
that [12, 58] We note that telecom QKD allows for secure communication
−p(V ) over much larger distances, up to d ' 140 km using RR.
C λ These distances are significantly reduced when the effects of
γλ,h = , (18) rain and haze are considered. For these weather conditions,
V 550
the maximum secure telecom communication distances are
where C = 39.1 log (e), λ (nm) corresponds to a certain strongly reduced to 300 m (7 km) and 800 m (2 km), in the
telecom wavelength, and p is a scattering coefficient that DR (RR) respectively. Conversely, for microwave frequen-
depends on the considered visibility range and varies from 0 cies, the maximum secure communication distance is almost
to 1.6. [12, 58]. unchanged in both reconciliation cases compared to that ob-
tained for optimal weather conditions, highlighting the robust-
ness of microwave CV-QKD to weather effects. The most sig-
nificant difference arises when considering the effect of light
B. Effects of weather conditions haze. Remarkably, haze induces little-to-no microwave losses.
Even strong haze and fog only weakly disturbs microwave sig-
In order to study the effect of non-optimal weather condi- nals by creating a small additional attenuation of around 10−3
tions on the CV-QKD secure key rates, we consider two spe- dB/km. The latter holds even when visibility is reduced to less
cific situations: (i) heavy rain with the rate Rr = 7 mm/h than 500 m. In contrast, reducing the visibility below 4 km
and (ii) light haze with a visibility V = 4 km. We com- would generate large losses (more than 10 dB/km) for the
pare the telecom and microwave secret key rates in Fig. 6. For telecom frequency, preventing the possibility of any relevant
the detection bandwidth and quantum efficiency, we stick to secure quantum communication. These results indicate that
the previously analyzed set of parameters (∆ftel = 1.2 GHz, an ideal quantum open-air communication network could con-
ηtel = 0.53 and ∆fmw = 3 GHz, ηmw = 0.345). We find that sist of a combination of microwave-based channels for short9
distances (d ≤ 200 m) and telecom-based channels for long first need to consider the matrix representing the beam splitter
distances (d > 200 m). operator
√ √
√ τ I2 , 1√− τ I2
VI. DISCUSSION B (τ ) = . (A1)
− 1 − τ I2 , τ I2
In conclusion, we have performed a comprehensive anal- Here, τ is the transmissivity associated with the beam splitter
ysis of microwave CV-QKD and demonstrate its potential and I2 denotes the 2×2 identity matrix. Further, we introduce
for applications in open-air conditions. We have shown the direct sum for matrices A and B as
that quantum microwaves can yield positive secret key rates
A, 0
for short-distance communication for both DR and RR. A⊕B= . (A2)
0, B
Our calculations rely on empirical models for microwave
and telecom atmospheric absorption losses. We have esti- Since we assume all states in the protocol to be Gaussian,
mated the related microwave and telecom specific attenua- these states are characterized by their displacement vector x̂
tion for optimal weather conditions to 6.3 × 10−3 dB/km and and covariance matrix V [7]. In this formalism, the displace-
2.02 × 10−1 dB/km, respectively. In our analysis, we have ment vector of an N -mode Gaussian state reads as
assumed microwave homodyne detection based on state-of-
the-art TWPAs. Our model for the CV-QKD protocol predicts x̂ = (q̂1 , p̂1 , . . . , q̂N , p̂N ) , (A3)
positive secret key rates for the microwave regime over dis-
tances of around 200 m. We have employed this model to where q̂i , p̂i are the conjugate quadrature operators of the ith
compare the microwave and telecom cases for different detec- mode. The displacement vector fulfills the commutation rela-
tion quantum efficiencies and bandwidths. Our results show tion
that, based on parameters of state-of-the-art technology, the
i
microwave CV-QKD can potentially outperform the telecom Ωij ,
[x̂i , x̂j ] =
implementations for short distances of around 30 m in terms 2
of the secret key rates. From our analysis, it appears that both N (A4)
M 0, 1
reconciliation scenarios are relevant. In particular, DR is fa- Ω= .
−1 , 0
i=1
vored for high quantum efficiencies, while RR allows for ap-
plications with rather lower detection quantum efficiencies η. We use the expression x̄ to refer to the expectation value of
The RR case also exhibits a nontrivial dependence of the se- the displacement vector, i.e.,
cret key rate R on η, which can be explained by the positive
impact of detection noise on the protocol security. x̄ = hx̂i. (A5)
Finally, we have considered the open-air CV-QKD proto-
col under non-ideal weather conditions of rain and haze. We The elements of the covariance matrix of a mode are com-
have found that these non-idealities strongly reduce the secure puted as
communication distance for the telecom regime, from 140 km
to several hundred meters. Remarkably, the microwave open- Vij = hx̂i x̂j + x̂i x̂j i/2 − hx̂i i hx̂j i. (A6)
air CV-QKD protocol appears to be largely immune to these
Using the previously introduced matrices in combination with
weather imperfections with its secure communication dis-
Eq. 1, we can express the mean displacement vector of Bob’s
tances staying mostly unchanged. We envision that our re-
mode (Eve’s mode) x̄B (x̄E ) as well as the covariance matrix
sults to serve as a motivation for building first prototypes of
of Bob’s mode (Eve’s mode) VB (VE ) as
secure microwave quantum local area networks. Our analysis
also establishes the foundations for hybrid networks, where T T T
(x̄B , x̄E ) = Σ · (x̄0 , x̄E,in ) + ΣE · x̄A , 0̄E,in ,
short-distance secure communication is carried out by mi-
crowave signals. Such hybrid network offer the advantage
VB , CBE
(A7)
= Σ · (V0 ⊕ VE,in ) · ΣT ,
of providing potential high secret key rates and robustness to CT BE , VE
weather imperfections, while switching to telecom setups for
long-distance communication. Short-distance MQC secure with
platforms could also complement current classical microwave ΣE = B (τE ) ⊕ I2 ,
communication technologies such as Wifi, Bluetooth, and 5G
due to the intrinsic frequency and range compatibilities. ΣA = R (ϕ/2) · S (r) ⊕ I4 , (A8)
Σ = ΣE · ΣA .
Here, the dot · represents a matrix multiplication and x̄0 (V0 )
Appendix A: Description of Bob’s and Eve’s quantum states the mean displacement vector (covariance matrix) of the ini-
tial vacuum state. Furthermore, x̄A represents Alice’s mean
In this section, we provide details about the states of Bob displacement vector, and x̄E,in represents Eve’s initial mean
and Eve. To describe Bob’s and Eve’s quantum states, we displacement vector of her TMS state. Additionally, r and ϕ10
correspond to the squeezing factor and squeezing angle of the with the same probability for both outcomes (P (C = 0) =
generated squeezed states by Alice, respectively. Correlations P (C = 1) = 1/2). Since c is given by a discrete variable and
between Bob’s and Eve’s individual states are described by ki by a continuous variable, we use a mixed joint probability
the submatrix CBE . Finally, R corresponds to a 2D rotation density function which gives
matrix while Ssq is a 2×2 matrix, which we calculate as
fA,C (ki , c) = fA|C (ki |c) p (C = c)
R (ϕ/2) · Ssq (r) = = fA (ki ) p (C = c)
(A15)
ki2
cos (ϕ/2) , sin (ϕ/2) exp (−r) , 0 1 1
· . =p exp − ,
−sin (ϕ/2) , cos (ϕ/2) 0, exp (r) 2πσA 2 2σA 2 2
(A9)
Using Eq. A7, the variance of Bob’s states reads where ρ̂kEi is the density matrix of an individual state obtained
by Eve from the entangling cloner attack. Moreover, fA is
1 the probability density function of the random variable A rep-
VB = τE VA + (1 − τE ) (1 + 2nEve ) I2
4 resenting Alice’s random choice for ki . Additionally, fA|C
(A10)
1 is the probability density function of a random variable A|C
= τE VA + (1 − τE ) + n̄ I2 ,
4 representing Alice’s random choice for ki conditioned on the
value of C. Note that we use fA|C = fA , since Alice uses
where VA represents Alice’s state variance while the covari- the same random variable to get ki independently of the value
ance matrix of Eve’s mode coupled to Alice’s mode is given taken by C. From this description, we write Eve’s Holevo
by 0.25 (1 + 2nEve ) I2 . Lastly, we incorporate the noise of quantity for the DR case as
the amplification chain by using the following input-output
formalism for a bosonic signal mode â [29] X 1Z ∞
χE,DR = S (ρ̂avg,E ) − fA (ki ) S ρ̂kEi dki ,
0 √ √ c=0,1
2 −∞
â = G â + G − 1 ĥ†amp . (A11) (A16)
where S is the von Neumann entropy. In order to compute
Here, G is the gain of the amplification chain and ĥamp is an χE,RR , we need to compute the covariance matrix of Eve’s
environmental mode, modelled as a thermal state. For G 1, mode after Bob has performed his measurement on either the
this results in the final covariance matrix for Bob: q or p quadrature. Following Ref. 42, the covariance matrix
of each individual mode of Eve after Bob’s measurement is
1
VB = τE VA + (1 − τE ) + n̄ + n̄g I2 . (A12) derived as
4
1
Here, n̄g = n̄amp /2 is the added quadrature noise from the VkE,B
i
= Vavg,E − 2 CEB · Π · CTEB , (A17)
σB
amplification chain expressed in an average photon number.
2
In the previous equation, the covariance matrix has been di- where σB = τE e2r /4 + n̄ + n̄g + (1 − τE ) /4. Additionally,
vided by the gain G as this gain can always be determined Π ∈ {Πq , Πp } is a projective measurement operator in phase
from calibration measurements. In the third step, we compute space, meaning that
the mutual information I (A:B) using the expression
1 0
Πq = (q-quadrature measured) ,
I (A:B) = h (B) − h (B|A) , (A13) 0 0
(A18)
0 0
where h denotes the differential entropy. Local measurements Πp = (p-quadrature measured) .
0 1
of Bob on individual states he receives during the communi-
cation are represented by a classical random variable B|A. Finally, CEB represents the correlations between Eve’s mode,
Then, B is a classical random variable representing Bob’s which she used during her entangling cloner attack, and Bob’s
overall measurements over all received states (i.e., represen- mode. One can derive that
ting Bob’s final key estimation K0 = {k10 , . . . , kN
0
} ). Further- T
CEB = C1 I2 , C2 σz , (A19)
more, to compute Holevo quantity in DR χE,DR and in RR
χE,RR , we first start by finding Eve’s average state. To this where σz is the Z Pauli matrix, and
end, we introduce an integer c ∈ {0 , 1} describing the choice √ √
C1 = − τE 1 − τE e2r /4 − n̄tot ,
of Bob’s measurement basis, q or p. Note that due to the sift-
√
q (A20)
ing step, Bob’s measurement basis matches Alice’s encoding 2
C2 = 1 − τE (n̄tot ) − 1.
basis. Eve’s average state reads as
X Z ∞ In the previous expression, we used the notation n̄tot =
ρ̂avg,E = fA,C (ki , c) ρ̂kEi dki , (A14) (n̄ + n̄g ) / (1 − τE ) + 1/4. Finally, for the RR case one can
c=0,1 −∞ express Eve’s Holevo quantity as
X 1Z ∞
where fA,C (ki , c) is the probability of Alice encoding a sym- χE,RR = S (ρ̂avg,E ) − fA (ki ) S ρ̂kE,B
i
dki .
bol ki in a measurement basis according to c. Addition- c=0,1
2 −∞
ally, C represents a binary random variable used to obtain c (A21)11
Here, ρ̂kE,B
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via Germany’s Excellence Strategy (EXC-2111-390814868), [7] F. Laudenbach, C. Pacher, C.-H. F. Fung, A. Poppe, M. Peev,
the Elite Network of Bavaria through the program ExQM, the B. Schrenk, M. Hentschel, P. Walther, and H. Hübel,
EU Flagship project QMiCS (Grant No. 820505), and the Continuous-Variable Quantum Key Distribution with Gaussian
German Federal Ministry of Education and Research via the Modulation-The Theory of Practical Implementations, Adv.
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Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y.
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imental implementations. K.G.F., A.M., and R.G. supervised
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this work. F.F., and K.G.F. wrote the manuscript. All au- Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins,
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Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang,
T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri,
COMPETING INTERESTS K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov,
F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh,
S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi,
The authors declare no competing interests.
K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Nee-
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A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh,
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