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Theoretical Computer Science ••• (••••) •••–•••

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Theoretical Computer Science www.elsevier.com/locate/tcs

Multicarrier continuous-variable quantum key distribution Laszlo Gyongyosi a,b,c,∗,1 a b c

School of Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, UK Department of Networked Systems and Services, Budapest University of Technology and Economics, Budapest, H-1117, Hungary MTA-BME Information Systems Research Group, Hungarian Academy of Sciences, Budapest, H-1051, Hungary

a r t i c l e

i n f o

Article history: Received 6 November 2018 Received in revised form 8 August 2019 Accepted 22 November 2019 Available online xxxx Communicated by M. Hirvensalo Keywords: Quantum key distribution Quantum cryptography Quantum communications Continuous-variables Quantum Shannon theory

a b s t r a c t The multicarrier continuous-variable quantum key distribution (CVQKD) protocol is deﬁned. In a CVQKD protocol, the information is conveyed by coherent quantum states. The quantum continuous variables are sent through a noisy quantum channel. For a quantum channel with additive-multiplicative noise both additive and multiplicative disturbances are present in the transmission. The multiplicative disturbance is an inherent attribute of diverse physical environments. Physical links with additive and multiplicative disturbances can represent a more general approach than purely additive noise links in several practical scenarios. In a standard CVQKD setting, the noise is modeled as an additive white Gaussian noise caused by an eavesdropper (Gaussian quantum link). As a corollary, standard CVQKD protocols are not optimal for arbitrary Gaussian quantum channels if multiplicative disturbances are also present in the physical link. Here, we deﬁne the adaptive multicarrier quadrature division (AMQD) modulation technique for CVQKD. The AMQD method is optimal for arbitrary Gaussian quantum channels with arbitrary multiplicative disturbances. The protocol granulates the Gaussian random input into Gaussian subcarrier continuous variables in the encoding phase, which are then decoded by a continuous unitary transformation. The subcarrier coherent variables formulate subchannels from the physical link which leads to improved transmission eﬃciency, higher tolerable loss, and excess noise in comparison to standard CVQKD protocols. We also derive the security proof of multicarrier CVQKD at optimal Gaussian attacks in the ﬁnite-size and asymptotic regimes. © 2019 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction Continuous-variable (CV) quantum key distribution (QKD) systems [1] allow for the establishment of an unconditionally secure quantum communication over the current standard telecommunication networks [2,4,8–10,24–31,45–75,78–86]. CVQKD systems possess several beneﬁts and advantages over the DV (discrete variable) protocols, since they do not require specialized devices or special requirements in an experimental scenario [24–31,48–55]. CVQKD systems are based on continuous variables such as Gaussian random position and momentum quadratures [3,24–31,48–55] in the phase space. The Gaussian modulated coherent states are transmitted over a noisy quantum channel, where the presence of an eavesdropper adds a white Gaussian noise to the transmission [19–21,59,60]. In our framework we are focusing on Gaussian quantum

* 1

Correspondence to: Laszlo Gyongyosi. E-mail address: [email protected]. Parts of this work were presented in a conference proceedings [41].

https://doi.org/10.1016/j.tcs.2019.11.026 0304-3975/© 2019 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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channels with arbitrary random multiplicative noise. It practically means, that an input coherent quantum system is modiﬁed by not just the additive white Gaussian noise of an eavesdropper, but also by a multiplicative disturbance via the random channel transmittance of the link (Note in our CVQKD-based description the Gaussian quantum link with a random channel transmittance is referred to as a Gaussian link throughout). The random channel transmittance coeﬃcient is drawn from an arbitrary distribution [56–58,63–67] that is determined by the actual physical-layer attributes of the quantum link. Since these assumptions are particularly convenient for wireless quantum communications [9,10,42–44] the adaptive multicarrier quadrature division (AMQD) [41] provides a useful framework for the implementation and optimization of CVQKD over wireless quantum channels, and free-space optical (FSO) [15,42–44] quantum channels. In CVQKD protocols, a Gaussian modulation is a robust and easily applicable ﬁnding in a practical scenario, and allows for the implementation of the protocol in the experiment; however, CVQKD is still sensitive to the imperfections of the transmission and the practical devices [24–31,35–40,48–55,87,89]. The performance of the protocol is strongly determined by the excess noise of the quantum channel, and the transmittance parameter of the physical link [24–31,73,74,76,77] (speciﬁcally, the Gaussian noise of the quantum channel models the eavesdropper’s optimal entangling cloner attack [4, 8,24–31], and the channel is referred to as a Gaussian quantum channel). Since the amount of tolerable loss and the excess noise are central parameters from the viewpoint of the running of CVQKD, it would be desirable to make some optimization steps in the encoding and decoding process to override the current limitations and to improve the quality of the quantum-level transmission. Our aim is to provide a solution to this problem by introducing the AMQD modulation technique for CVQKD, which can be applied both in one-way and two-way CVQKD to increase the tolerable loss and excess noise. In traditional telecommunications, OFDM (orthogonal frequency-division multiplexing) is a well-known and widely applied technique for improving the bandwidth eﬃciency over noisy communication networks [19–23]. In an OFDM scheme, the information is encoded in multiple carrier frequencies, and its main advantage over single-carrier transmission is that the subcarrier-based transmission can attenuate and overwhelm the problems of diverse and unfavorable channel conditions. OFDM systems have been admitted to be a useful encoding method in traditional networking; however, no similar method exists for CVQKD. If a similar solution were available for continuous variables, one could enjoy similar beneﬁts in a quantum-communication scenario; however, up to this point no analogous answer exists for quantum-level transmission. With this in mind, we deﬁne the multicarrier continuous-variable quantum key distribution protocol through the AMQD, which works on continuous variables and for which similar beneﬁts can be reached in the process of quantum-level information transmission. In the standard CVQKD coding scenario, Alice, the sender, modulates and separately transmits each coherent state in the phase space. This standard modulation scheme is referred as single-carrier modulation throughout, consistent to its traditional meaning. The key idea behind AMQD modulation is as follows. Alice, draws a zero-mean, circular symmetric complex Gaussian random vector, which is then transformed by the inverse Fourier operation. At a given modulation variance, Alice prepares her Gaussian subcarrier CVs, which are then fed into the channel. Bob, the receiver, applies the inverse unitary of Alice’s operation, which makes it possible for him to recover the noisy version of Alice’s input coherent states. This kind of communication will be referred as multicarrier CVQKD modulation [10–18,41]. The Gaussian subcarrier CV states sent through the channel, which overall allows higher tolerable loss and excess noise at a given modulation variance. The Gaussian quantum channel can be viewed as several parallel Gaussian quantum channels, called sub-channels, each dedicated for the transmission of a given subcarrier with an independent, and signiﬁcantly lower noise variance. The information transmission capability of the sub-channels is diverse, depending on the variance of the subcarrier CV, which allows for the development of smart adaptive modulation techniques for the proposed multicarrier quadrature division-encoding technique. The idea behind this is to use only Gaussian sub-channels with low noise level for the transmission, and to not send any valuable information over the so noisy sub-channels. The result of the adaptive allocation is a better performance of the protocol at low SNRs (signal-to-noise ratio) and higher tolerable loss, which are crucial cornerstones in an experimental CVQKD that operates in practice at very low SNRs. The AMQD modulation granulates Alice’s initial Gaussian states into several subcarrier Gaussian CVs, which divide the physical channel into several Gaussian sub-channels. Bob applies an inverse continuous unitary operation, which allows him to obtain Alice’s initial (noisy) coherent states. The proposed AMQD modulation offers several important features, but the main improvement is in the quality of the quantum-level transmission, since the subcarriers allow a more eﬃcient communication over the same quantum channel at a given modulation variance. The novel contributions of our manuscript are as follows: 1. We deﬁne the multicarrier CVQKD framework for Gaussian quantum channels with a random multiplicative noise. 2. We characterize the operations and study the performance of multicarrier CVQKD. 3. We conceive a modulation-variance adaption technique which provides optimal capacity-achieving communication over a Gaussian quantum link with an arbitrary distribution of the channel transmittance coeﬃcient. 4. We provide a security proof of the multicarrier CVQKD protocol in the ﬁnite-size and asymptotic regimes. The AMQD framework allows us to deﬁne multiple-access multicarrier CVQKD [10], to prove the improved secret key rates and security thresholds of multicarrier CVQKD [11], and due to the fact that the multicarrier modulation injects several extra degrees of freedom into the transmission, it is possible to introduce advanced phenomena that are not available in

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a single-carrier CVQKD setting such as singular value decomposition assisted multicarrier CVQKD [14], multidimensional manifold extraction [12], and advanced quadrature detection techniques [13]. This paper is organized as follows. In Section 2 the system model is summarized. In Section 3, the adaptive modulation variance allocation mechanism is discussed. Section 4 provides the security proof. Section 5 studies the performance of the protocol. Finally, Section 6 concludes the results. Supplemental information is included in the Appendix. 2. System model For some basic deﬁnitions of multicarrier CVQKD, see also [41]. 2.1. Preliminaries In the standard single-carrier modulation scheme, the input coherent state |ϕi = |xi + ip i is a Gaussian state in the phase space S , with i.i.d. Gaussian random position and momentum quadratures xi ∈ N 0, σω20 , p i ∈ N 0, σω20 , where

σω20 is the modulation variance. The coherent state |ϕi in the phase space S can be modeled as a zero-mean, circular sym metric complex Gaussian random variable z ∈ CN 0, σω2z , with variance σω2z = E | z|2 , and with i.i.d. real and imaginary zero-mean Gaussian random components, Re ( zi ) ∈ N 0, σω20 , Im ( zi ) ∈ N 0, σω20 .

In the single-carrier scenario, the transmission of this complex variable over the Gaussian quantum channel N can be characterized by the T (N ) normalized complex transmittance variable

T (N ) = ReT (N ) + iImT (N ) ∈ C, (1) √ √ where 0 ≤ ReT (N ) ≤ 1/ 2 stands for the transmission of the position quadrature, 0 ≤ ImT (N ) ≤ 1/ 2 is the transmission of the momentum quadrature, with relation

ReT (N ) = ImT (N )

(2)

by our convention. The 0 ≤ | T (N )| ≤ 1 magnitude of the T (N ) complex variable is

| T (N )| =

ReT (N )2 + ImT (N )2 =

√

2ReT (N ) ∈ R

(3)

and the squared magnitude of T (N ) is

| T (N )|2 = ReT (N )2 + ImT (N )2 = 2ReT (N )2 ∈ R.

(4)

Assuming a 0 dB loss, the quadrature transmittance is parameterized with ReT (N ) = ImT (N ) =

| T (N )| = | T (N )|2 = 1.

√1 2

and

(5)

In CVQKD, at a given input x, the output y of a Gaussian quantum link with random multiplicative disturbance [88] T (N ) ∈ D drawn from an arbitrary distribution D , can be expressed as

y = T (N ) x + , (6) 2 2 2 where ∈ CN 0, σ , σ = E || is the noise variable that models the Gaussian noise (i.e., the noise of eavesdropper in a CVQKD setting) of the physical quantum link. The additive noise circular complex Gaussian is a2 zero-mean, symmetric 2 random variable, with independent quadrature components x ∈ N 0, σN , p ∈ N 0, σN .

In an experimental telecommunication scenario, a multiplicative disturbance can be present in several dimensions (time, frequency and space). In the proposed CVQKD setting, the multiplicative disturbance is a direct consequence of the transmission medium, and it is not a result of an eavesdropping activity. Contrary to an eavesdropping which always results in an additive white Gaussian noise in any CVQKD setting (assuming an optimal attack), the consequence of the multiplicative disturbance is the so-called fading effect. In practical communications, the multiplicative disturbance is an inherent attribute of many physical environments; a well-known example is the wireless setting. However, the multiplicative disturbance T (N ) of (6) can occur in an arbitrary physical quantum link N , because T (N ) is a random variable drawn from a particular distribution D . For example, in a Rayleigh fading channel [88], T (N ) is a zero-mean circular complex Gaussian random variable T (N ) ∈ CN 0, σ T2(N ) , with a particular variance

σT2(N ) .

In the multicarrier CVQKD case, the Gaussian quantum channel is divided into n Gaussian sub-channels Ni , i = 2 {0 . . . n − 1}, each with an independent noise variance σN , each for the transmission of a continuous variable subcarrier i |φi , which leads to the output for the i-th sub-channel of N :

y i = T ( N i ) x i + i , i = 0 . . . n − 1, 2 where T (Ni ) ∈ C , i ∈ CN 0, σ . i

(7)

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2.1.1. Terms of multicarrier quadrature division modulation state A Gaussian modulated i-th coherent |ϕi = |xi + ip i can be rewritten as a zero-mean, circular symmetric complex Gaussian random variable zi ∈ CN 0, σω2z

i

,

σω2z = E|zi |2 , as i

zi = xi + ip i ,

(8)

and then the CV state is as

|ϕ i = | z i . i ϕi

(9) iϕ 2 i ϕ 2 zi has the same distribution of zi for any ϕi , i.e., E [zi ] = E e i zi = Ee i [zi ] and σzi = E | zi | . The

The variable e probability density function (PDF) of zi is

1

f ( zi ) = where | zi | =

− zi 2

e

2πσω20

2 2σω 0

= f ( xi , p i ) =

1 2πσω20

− xi 2 + p i 2

e

2 2σω 0

(10)

,

x2i + p 2i is the magnitude, which is a Rayleigh random variable with density

f (| zi |) =

| zi | 2

σω z

− z i 2

e

2 2σω

zi

, | z i | ≥ 0,

(11)

i

while | zi |2 = x2i + p 2i the squared magnitude is exponentially distributed with density

− zi 2

1

f | z i |2 =

σω2z

e

2 σω z

i

, | z i |2 ≥ 0 .

(12)

i

The i-th subcarrier CV is deﬁned as

|φi = |IFFT ( zi ) = F −1 ( zi ) = |di ,

(13)

where IFFT stands for the Inverse Fast Fourier Transform, and subcarriercontinuous variable |φ i in (13) is also a zero-

σd2i = E |di |2 , di = xdi + ipdi , where are i.i.d. zero-mean Gaussian random variables, and σω2 F is the variance of the Fourier

mean, circular symmetric complex Gaussian random variable di ∈ CN 0, σd2 ,

i

xdi ∈ N 0, σω2 F , pdi ∈ N 0, σω2 F transformed Gaussian signal. The inverse of (13) yields the single-carrier CV from the subcarrier CV as follows:

|ϕi = CVQFT (|φi ) = F (|di ) = F F −1 ( zi ) = | zi ,

(14)

where CVQFT is the continuous-variable QFT operation, see Section 2.2 and Section 2.3. 2.2. Continuous-variable quantum Fourier transform In terms of the CV scenario, by a convention the |x position quadrature could be used as a computational basis. (The continuous-variable quantum Fourier transformation will be abbreviated as CVQFT.) Let the Gaussian variable be [41]

1

g (x) =

√

σ 2π

∞

−x2

e 2σ 2 , x ∈ N 0, σ 2 ,

(15)

where −∞ g (x) dx = 1. The Fourier transform of (15) is expressed as

F ( g (x)) = G (ω) = e

−ω 2 σ 2 2

, ω ∈ N 0, σ 2 .

(16)

Between the Fourier transform F (x) and the inverse Fourier transform function F −1 (ω), the connection is as follows [20, 21]:

∞ F (x) = −∞

where

F −1 (ω) e −ixω dω,

(17)

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F

−1

5

∞ F (x) e ixω dx.

1 2π

(ω ) =

(18)

−∞

Gaussian random vector g ∈ 0, Kg , g = ( g 0 , . . . , gn−1 ) T , where Kg = E gg T is the covariance matrix, the Fourier transform and its inverse are related by

∞ ∞ F ( g) =

∞ ...

−∞ −∞

F −1 (w) e −ig·w dw,

−∞

(19)

where w ∈ (0, Kw ) is a n-dimensional Gaussian random vector with covariance matrix Kw = E ww T , and

F

−1

∞ ∞ 1

( w) =

∞ ...

(2π )n −∞ −∞

F (g) e ig·w dg.

(20)

−∞

Assuming |x position computational basis, function F (·) acts on the coherent input |ϕi as follows [9]:

F (|ϕi ) = F ( x|ϕi ) → p |ϕi ,

(21)

where

∞ x|ϕi =

√1

2π

p |ϕi dpe ipx ,

(22)

x|ϕi dxe −ipx

(23)

−∞

and

∞ p |ϕ i =

√1

2π

−∞

are wavefunctions in the position space x, and momentum space p for the state |ϕi , respectively. The notation x|ψ stands for the inner product [9], which can be rewritten as

∞ x|ψ = −∞

f x − x ψ x dx ,

(24)

where f x − x = x|x and ψ (x) = x|ψ. Assuming a complete set of orthonormal wavefunctions {ϕn } [9], the arbitrary wavefunction ψ is as ψ = n c i ϕi , from which

∞ ψ (q) =

c ( p )u ( p ; q) dp ,

(25)

−∞

where

c ( p ) = u ( p ; q) , ψ (q)

∞

= u ( p ; q) ,

c ( p )u ( p ; q) dp

−∞

∞ =

(26)

f p − p c p dp .

−∞

The inverse function of (21) is deﬁned as

F −1 ( p |ϕi ) = x|ϕi .

(27)

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Fig. 1. (a) The input Gaussian state in the position basis (a) and (b) the CVQFT transformed signal in the momentum basis, where

σ F2i = 1/σω2i [41].

2.3. CVQFT in a multicarrier CVQKD Let us derive the Fourier transform of the Gaussian input [20,21,41]. First, we rewrite (10) in the position basis x as g (x) =

1 2π σω20

e

−x2 a2

, where a2 = 2σω20 . Then, the Fourier transformed signal G ( p ) is precisely evaluated as

F ( g (x))

= G ( p) 1

∞

=√

2π

1

=√

2π

1

=√

2π

1

=√

2π

1

=√

2π

g (x) e −ipx dx

−∞

e

−x2 a2

e −ipx dx

2πσω20 −∞

∞

1

e

−x2 a2

−ipx

(28) dx

2πσω20 −∞ 1

e−

2πσω20

1

∞

1

2π

e

2 − p2 2σ F

a2 p 2 4

∞

e

−

x + iap a 2

2

dx

−∞

.

σ F2

The Fourier transform G ( p ) of the Gaussian signal g (x) is also Gaussian in the conjugate-variable space, with variance σ F2 = 1/σω20 . The F (|ϕi ) Fourier transform of the coherent Gaussian state |ϕi is also a Gaussian state. Since the position

and momentum quadratures are Fourier-transform pairs, it follows that as the modulation variance σω2 of the input Gaussian signal increases, the variance σ F2 of the Fourier transformed signal decreases. From the uncertainty principle [41], it can be concluded that if x = σω0 (i.e., the uncertainty of the Gaussian is proportional to the standard deviation) and p = 1/σω0 , then x p = 1. The normalized Gaussian functions g (x) =

1 2π σω2

i

√

2σωi

e

−x2 2 2σω i

√

and G ( p ) =

−

2σωi

2π σω2

e

p2

2σ 2 Fi

are shown in Fig. 1 [41]. In terms

i

of the quadratures, the momentum p is the Fourier transform of the position x, and the position x is the inverse-Fourier transform of the momentum p. 2.4. Gaussian sub-channels in a multicarrier CVQKD The characterization of the Gaussian sub-channels is proposed in Proposition 1 [41].

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Proposition 1. The AMQD divides channel N into n Gaussian sub-channels Ni , i = {0, . . . , n − 1}, with independent n−1the2Gaussian 2 2 noise variances σN , where n1 i =0 σN = σN . i

i

Proof. Let n be the number of Alice’s input Gaussian states. The n input coherent states are modeled by an n-dimensional, zero-mean, circular symmetric complex random Gaussian vector

z = x + ip = ( z0 , . . . , zn−1 ) T ∈ CN (0, Kz ) ,

(29)

where each zi can be modeled as a zero-mean, circular symmetric complex Gaussian random variable zi ∈ CN 0, σω2z

i

,

zi = xi + ip i . The real and imaginary variables (i.e., the position and momentum quadratures) formulate n-dimensional real Gaussian random vectors, x = (x0 , . . . , xn−1 ) T and p = ( p 0 , . . . , pn−1 ) T , with zero-mean Gaussian random variables

1

f ( xi ) =

√

σω0 2π

− xi 2 2 2σω 0

e

, f ( pi ) =

1

√

σω0 2π

e

− pi 2 2 2σω 0

(30)

,

where σω20 is the stands for single-carrier modulation variance (precisely, the variance of the real and imaginary components of zi ), while Kz is the n × n Hermitian covariance matrix of z:

Kz = E zz† ,

(31)

where z† is the adjoint of z. For vector z,

E [z] = E e i γ z = Ee i γ [z]

holds, and

E zz for any

T

iγ

(32)

T

iγ

=E e z e z

= Ee i2γ zzT ,

(33)

γ ∈ [0, 2π ]. The density of z is as follows (if Kz is invertible):

1 f (z) = π n det e −z Kz

†

Kz −1 z

(34)

.

A n-dimensional Gaussian random vector is expressed as x = As, where A is an (invertible) linear transform from Rn to Rn , and s is an standard Gaussian random vector N (0, 1)n . This vector is characterized by its covariance matrix n-dimensional C (x) = E xx T = AA T , as has density

f (x) = √

2π

1 n

det AA T

e

xT x − T 2 AA

(35)

.

The Fourier transformation F (·) of the n-dimensional Gaussian random vector v = ( v 0 , . . . , v n−1 ) T results in the n-dimensional Gaussian random vector m = (m0 , . . . , mn−1 ) T :

m = F (v) = e

−mT AA T m 2

=e

−σω2 m20 +...+mn2−1 0

(36)

.

2

In the ﬁrst step of AMQD, Alice applies the inverse FFT operation to vector z (see (29)), which results in an n-dimensional zero-mean, circular symmetric complex Gaussian random vector d, d ∈ CN (0, Kd ), d = (d0 , . . . , dn−1 ) T , precisely as

d = F −1 (z)

d T AA T d =e 2

=e

σω20 d20 +...+dn2−1 2

where di = xdi + i pdi , di ∈ CN 0, σ

2 di

(37)

,

, and the position and momentum quadratures of |φi , and Kd = E dd , E [d] = †

T = Ee i2γ ddT for any γ ∈ [0, 2π ]. E e i γ d = Ee i γ [d], and E ddT = E e i γ d e i γ d In the next step, Alice modulates the coherent Gaussian subcarriers as follows:

|φi = |di = F −1 (z) .

(38)

The result of (37) deﬁnes n independent Ni Gaussian sub-channels, each with noise variance σN , one for each subcarrier i coherent state |φi . After subcarriers are transmitted through the noisy channel, Bob applies the CVQFT, which results the

CV him the noisy version ϕi = zi of Alice’s input zi . 2

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On Bob’s side, the received system y is an n-dimensional zero-mean, circular symmetric complex Gaussian random vector y ∈ CN 0, E yy† . The m-th element of y is ym , expressed as follows:

ym = F (T (N )) zm + F ()

= F (T (N )) F F −1 ( zm ) + F () = F ( T i (Ni )) F (di ) + F (i ) ,

(39)

n

where

T (N ) = [T 0 (N0 ) , . . . , T n−1 (Nn−1 )] T ∈ Cn ,

(40)

T i (Ni ) = Re ( T i (Ni )) + iIm ( T i (Ni )) ∈ C,

(41)

where

is a complex variable, which quantiﬁes the position and momentum quadrature transmission √(i.e., gain) of the i-th Gaussian √ sub-channel Ni , in the phase space S , with real and imaginary parts 0 ≤ ReT N 1 / 2, 0 ≤ ImT i (Ni ) ≤ 1/ 2. The ≤ ( ) i i

T i (Ni ) variable has a magnitude of | T i (Ni )| = ReT i (Ni )2 + ImT i (Ni )2 ∈ R, where ReT i (Ni ) = ImT i (Ni ), by our convention. √ The CVQFT-transformed channel transmission parameters are (upscaled by n) expressed by the complex vector:

F (T (N )) =

n −1

F ( T i (Ni )) =

i =0

n −1 n −1

Tke

−i2π ik n

∈ Cn ,

(42)

i =0 k =0

where F (T (N )) is the Fourier transform of (40). The n-dimensional F () complex vector is evaluated as

F () = e

− F () T C( F ()) F () 2

− F 0 2 σ 2

=e

N0

+···+ F n−1 2 σ 2

Nn−1

2

(43)

,

which is the Fourier transform of the n-dimensional zero-mean, circular symmetric complex Gaussian noise vector ∈ CN 0, σ2 n ,

= (0 , . . . , n−1 ) T ∈ CN (0, C ()) ,

(44) 2 2 where C () = E † , with independent, zero-mean Gaussian random components xi ∈ N 0, σN , p i ∈ N 0, σN i i

2 with variance σN , for each i , which identiﬁes the Gaussian noise of the i-th sub-channel Ni on the quadrature compoi nents in the phase space S . The CVQFT-transformed noise vector in (43) can be rewritten as

F () = ( F (0 ) , . . . , F (n−1 )) T , (45) with independent components F xi ∈ N 0, σ F2(N ) and F p i ∈ N 0, σ F2(N ) on the quadratures, for each F (i ). It i

i

also deﬁnes an n-dimensional zero-mean, circular symmetric complex Gaussian random vector F () ∈ CN (0, C ( F ())) with a covariance matrix

C ( F ()) = E F () F ()† , and the noise variance

(46)

σ F2(N ) of the independent Fourier-transformed quadratures is evaluated as

n −1

σ F2(N ) In×n =

1 n

i =0

σ F2(Ni )

= σ F2(N ) =

(47) 1 2 σN

,

2 σN is the noise variance of the Gaussian quantum channel N , In×n is the n × n identity matrix, hence F (i ) ∈ 2 CN 0, σ F (i ) , and σ F2(i ) = E | F (i )|2 , with independent noise variance σ F2(N ) on the quadrature components (For

where

simplicity, the notation of In×n will be omitted from the description.).

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An AMQD block is formulated from n Gaussian subcarrier continuous variables, as follows:

y [ j] = F (T (N )) F (d) [ j] + F () [ j] , j = 1, . . . , n,

(48) where j is the index of the AMQD block, F (T (N )) is deﬁned in (42), F (d) = F F −1 (z) , where F −1 (z) is shown in (37),

while

y [ j] = ( y 1 [ j] , . . . , yn [ j]) T , F (d) [ j] = ( F (d1 ) [ j] , . . . , F (dn ) [ j]) T ,

(49)

T

F () [ j] = ( F (1 ) [ j] , . . . , F (n ) [ j]) . The squared magnitude

τ = F (d) [ j] 2

(50)

identiﬁes an exponentially distributed variable, with density f (τ ) = 1/2σω2n e −τ /2σω , and from the Parseval theorem [19] it follows that 2

E [τ ] ≤ n2σω2 ,

(51)

while the average quadrature modulation variance is n −1

σω2 =

1 n

σω2i = σω20 ,

(52)

i =0

where σω2i is the modulation variance of the quadratures of the subcarrier |φi transmitted by sub-channel Ni . The transformed vector y in (39) and (49) clearly demonstrates that the physical Gaussian channel is, in fact, divided into n Gaussian quantum channels with independent noise variances. Each Ni Gaussian sub-channel is dedicated for the transmission of one Gaussian subcarrier CV from the n subcarrier CVs. 2 2.5. Eavesdropping Eve’s optimal entangling cloner attack [4] in the multicarrier modulation setting is described as follows. Let the quadratures of the i-th subcarrier |φi transmitted by Ni be

xin,i , p in,i , xin,i ∈ N 0, σω2i , p in,i ∈ N 0, σω2i ,

(53)

where σω2i is the modulation variance of the CVQFT transformed subcarrier CVs. In the general attacker model, Eve is equipped with n EPR ancilla pairs | E B ⊗n each with variance W i , where E i is the i-th ancilla with quadrature components

2 2 x E ,i , p E ,i , x E ,i ∈ N 0, σω2i + σN , p E ,i ∈ N 0, σω2i + σN . i i

(54)

The part B i is sent back to Ni ; system B i is characterized via the following quadratures:

2 2 x B ,i , p B ,i , x B ,i ∈ N 0, σω2i + σN , p B ,i ∈ N 0, σω2i + σN . i i

(55)

Eve’s Gaussian attack in the multicarrier scenario is summarized as follows. Eve attacks each sub-channel with a BS (beam

2

splitter) with transmittance T E ve,i ∈ C , 0 < T E ve,i < 1, and an entangled ancilla | E B with variance W . The quadratures of the i-th sub-channel are xin,i , p in,i , Eve’s quadratures are x E ,i , p E ,i , Bob’s received noisy quadratures are x B ,i , p B ,i .

2 Each sub-channel is characterized with a Gaussian noise N 0, σN 2

i

on the quadrature components, with independent noise

variance σN . As shown in (44), Eve’s optimal Gaussian attacks deﬁne an n-dimensional zero-mean, circular symmetric i complex Gaussian random noise vector. In the AMQD scenario, the appropriate noise vector is given by (43) and (46), as F () ∈ CN (0, C ( F ())), and F (i ) ∈

CN 0, σ F2(i ) , respectively.

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2.6. Crosstalk noise The crosstalk is an additive noise in multicarrier transmission that can occur between adjacent sub-channels. Crosstalk noise can result from a nonperfect sub-channel separation and sub-channel ﬁltering procedure. As a result of interchannel crosstalk, some information from neighboring channels can leak, which acts as noise to the receiver. In practice, crosstalk arises from the imperfections of optical ﬁlters, optical switches or other optical components, e.g., from the imperfect isolation of different wavelength ports [32]. Crosstalk noise has no effect on the security of AMQD modulation, since it allows no more leaking of information to an eavesdropper than single-carrier CVQKD protocols (see Section 4). Let us assume that the correlation between Alice and Bob is quantiﬁed via the χ ( A : B ) = H ( B ) − H ( B | A ) Holevo information. The crosstalk information γ (Ni ) on the i-th sub-channel Ni , i = {0 . . . n − 1}, is evaluated from the interference noise of the neighboring sub-channels as follows:

γ (Ni ) =

x jχ A j : B j ,

j =i

(56)

where 0 ≤ x j ≤ 1, χ A j : B j = H B j − H B j A j is the Holevo information of Alice and Bob conveyed by the neighboring sub-channel N j , for all j = i. The average crosstalk on the n sub-channels is expressed as

γ (N ) =

1 n

n

x jχ A j : B j .

(57)

j =i

Since the crosstalk noise

γi (Ni ) on sub-channel Ni acts for Bob as Gaussian the Central Limit noise (which is justiﬁed by 2 γi ∈ CN 0, σγ2i with variance σγ2i = E γi . This additional

Theorem, more precisely for i → ∞), it can be modeled by

noise does not change the rate formulas of AMQD (see (69) and (75)), because this additional noise is already included in 2 the sub-channel’s noise variance σN . i In other words, the mutual information between Alice and Bob is completely characterized for a given Ni by the noise 2 variance σN , which can be decomposed as i

2

σNi = σ E2ve,i + σγ2i ,

(58)

where σ E2ve,i is the noise variance of Eve’s optimal Gaussian attack, see also Section 4. Assuming that n sub-channels have been allocated for the multicarrier transmission, the I ( B : E ) mutual information of Eve and Bob, and the χ ( B : E ) Holevo information of Eve and Bob, in the presence of a nonzero crosstalk between the sub-channels changes as follows:

I ( B : E )γi = I ( B : E ) +

T E ve,i 2 γ (Ni ) n

= I (B : E) +

T E ve,i 2 x j I A j : B j , j =i

n

with

γ (Ni ) =

j =i

(59)

x j I A j : B j , and

χ ( B : E )γi = χ ( B : E ) +

T E ve,i 2 γ (Ni ) n

T E ve,i 2 x j χ A j : B j , = χ (B : E) + n

(60)

j =i

2 where T E ve,i is the squared magnitude of Eve’s normalized complex transmittance T E ve,i = ReT E ve,i + iImT E ve,i ∈ C , √ √ where 0 ≤ ReT E ve,i ≤ 1/ 2, 0 ≤ ImT E ve,i ≤ 1/ 2 that characterizes the attack of sub-channel Ni , and the quantity γ (Ni ) is calculated with respect to the n sub-channels (a nonzero crosstalk can also occur on an unused sub-channel, hence all sub-channels have to be taken into consideration in the RHS of (60)). For detailed security proof of AMQD see Section 4. 2.7. Multicarrier continuous-variable quantum key distribution protocol The run of the multicarrier quadrature division is sketched as follows. In the initial phase, Alice draws an n-dimensional, zero-mean circular Gaussian random vector z = x + ip = ( z0 , . . . , zn−1 ) T ∈ CN (0, Kz ), zi = xi + ip i , symmetric complex 2 2 with xi ∈ N 0, σω0 , p i ∈ N 0, σω0 are i.i.d. Gaussian random variables that identiﬁes the x position and p momentum quadratures in the phase space S , while

σω20 is the modulation variance (at a single-carrier transmission). In the next

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Fig. 2. The AMQD modulation scheme. Alice draws an n-dimensional, zero-mean, circular symmetric complex Gaussian random vector z, which are then inverse Fourier-transformed by F −1 . The resulting vector d encodes the subcarrier quadratures for the Gaussian modulation. In the decoding, Bob applies the U unitary CVQFT on the n subcarriers to recover the noisy version of Alice’s original variable as a continuous variable in the phase space. The quantum link N (abbreviated as AWGN - Additive White Gaussian Noise - link) is characterized via an additive Gaussian noise with arbitrary random transmittance coeﬃcients (multiplicative disturbance).

step, Alice applies the inverse FFT on z, that gives her the results of an n-dimensional, zero-mean circular symmetric complex Gaussian random vector d = x + ip = (d0 , . . . , dn−1 ) T ∈ CN (0, Kd ). According to d, she prepares the |φ1...n Gaussian subcarrier CVs, by modulating with σω2 = σω20 the position and momentum quadratures, where |φi is the i-th subcarrier continuous variable. The n subcarrier coherent states |φi divide the physical Gaussian quantum channel into n Gaussian 2 quantum channels, each equipped with an independent noise variance σN .

i

In the decoding phase, Bob applies the CVQFT unitary operation U on the received noisy Gaussian subcarrier CVs, φi ,

which results him the noisy coherent state versions of Alice’s Gaussian variables, ϕ0 ...n−1 = z0 ...n−1 = z , and the Fourier

transformed sub-channel noise variance σ F2(N ) . The CVQFT-transformed | F ( T i (Ni ))|2 transmission parameters of the Gausi sian sub-channels are strongly diverse (see Section 3), which makes available the use of an adaptive variance modulation to improve the tolerable noise and excess noise. The steps of multicarrier quadrature division modulation are summarized in Fig. 2. (Note: In the two-way protocol, Bob sends the subcarrier CVs to Alice, who generates coupled Gaussian CVs with her BS. The resulting Gaussian subcarrier CV is then sent back to Bob, who applies the CVQFT operation. Alice also applies a CVQFT operation on her system.) The reason behind the improvement is that the dB limits of the transmission over the Gaussian quantum channel can signiﬁcantly be extended by the Fourier-transformed multicarrier continuous variables. The adaptive modulation sends information through only the l high-quality sub-channels from the total n, while the n − l noisy sub-channels transmit no information, as it is proposed in the next section. 3. Adaptive modulation Theorem 1. The ratio

n−1

2 ν i = σN /| F ( T i (Ni ))|2 , where | F ( T i (Ni ))|2 =

k=0

Tke

, i = 0 . . . n − 1, provides the σω2 = ν E ve −

−i2π ik 2 n

min (νi ) optimal constant modulation variance for the Gaussian sub-channels, where ν E ve is a security bound of an optimal Gaussian attack. Proof. Note the proof throughout the section focuses on the transmission of classical information. The rigorous derivation of the private classical capacity and the achievable secret key rates of AMQD are detailed in [11]. The proof consists of two parts. First, we show that there exists an optimal constant modulation variance for the sub channels with average σω2 = n1 n σω2i = ν E ve − (ν ), ν E ve is the security bound of Eve’s optimal Gaussian attack, and ν is

ν=

1 n

νi =

n

1

n

2 σN

σ2

i N , = n−1 −i2π ik 2 n n−1 n−1 | F T i (Ni ))|2 ( n = 0 i i =0 k =0 T k e

n−1

−i2π ik 2

(61)

n−1

−i2π ik 2

2 n is the noise variance of Ni , while k=0 T k e n is for the i-th sub-channel Ni , σN k=0 T k e i the squared magnitude of the sum of the Fourier coeﬃcients of the n Gaussian sub-channels. Then, we show that for low SNRs, the optimal solution is ν = min (νi ). The modulation variances of each of the Ni sub-channels (zero or nonzero) are dependent on the value of ν E ve . This parameter is deﬁned as

where

2 ν i = σN / i

1

ν E ve = , λ

(62)

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where λ is the Lagrange multiplier as

−1 2 −1 n ∗ 2 1 n ∗ −i2nπ ik λ = F TN = Tk e , n

(63)

i =0 k =0

∗ is the expected transmittance of the n sub-channels under an optimal Gaussian attack. where T N From λ, and the σω2i modulation variances of the Ni sub-channels, a Lagrangian can be constructed as

2

2

L λ, σω0 . . . σωn−1 =

n −1

log2

i =0

n −1 σω2i | F ( T i (Ni ))|2 1+ σω2i . − λ 2 σN i =0

(64)

By the Kuhn-Tucker condition [19,22,23], follows that ∂ L2 = 0 if only the i-th sub-channel gets a non-zero modulation variance, σωi > 0, while ∂ L2 ∂ σωi 2

∂ σωi

≤ 0, if the sub-channel gets zero modulation variance, σω2i = 0.

After some calculations, one gets the following average modulation variance: n −1

2

σω =

1 n

ν E ve −

i =0

2 σN

| F ( T i (Ni ))|2

n −1

=

1 n

(ν E ve − νi ),

(65)

i =0

where ν is shown in (61). One can readily see that in (65), each sub-channel is allocated by a different modulation variance, depending on the actual value of | F ( T i (Ni ))|2 . The reason for this is as follows. Only those l < n sub-channels can transmit information for which νi < ν E ve ; otherwise, the channel gets zero modulation variance. (In general, this kind of strategy is called water-ﬁlling [19,22,23].) Since it is not a reasonable assumption in a practical CVQKD that the transmitter has an exact knowledge about the state of each Gaussian sub-channels, at this point we have to introduce a more ﬂexible technique. In fact, it is not a required condition to calculate with the exact νi parameters and modulation variances σω2i for the sub-channels. A simpliﬁed solution exists: allocate a constant modulation variance which νi < ν E ve is satisﬁed: l −1

2

σω =

1 l

ν E ve −

i =0

σω2 for those l Ni sub-channels, for

2 σN

maxi | F ( T i (Ni ))|2

= ν E ve − min (νi ) .

(66)

2

At a given ν E ve bound, it is enough to ﬁnd a given maxi F T 0...l−1 N0...l−1 for those Ni sub-channels, for which νi < ν E ve hold (Note: It can be determined in a pre-calibration phase prior to the main run of the protocol. In practice, this phase can be established in one simple step by sending empty “pilot” continuous variables over the sub-channels that contain no valuable information.), and then allocating the same modulation variance σω2 for all of these sub-channels. Particularly, the signiﬁcance of the adaptive-variance modulation proposed in (66) is crucial for low SNRs, which is precisely the case in a long-distance scenario, since the information transmission capability of the Gaussian sub-channels become very sensitive in the low SNR regimes [9,19,22,23]. At low SNRs the constant allocation provides an optimal solution [22,23], because its performance is very close to the exact allocation and can be performed with no exact knowledge about the state of the sub-channels. It is particularly convenient, since the AMQD modulation scheme allocates a constant modulation variance for the good Gaussian sub-channels. The algorithm of the optimal constant [22,23] modulation variance adaption is summarized in Algorithm 1. Algorithm 1 Optimal modulation variance.

2 1. Let the squared magnitudes of the Fourier-transformed sub-channel transmittance coeﬃcients given in an ordered list L = F T 0...l N0...l−1 so

that | F ( T i (Ni ))|2 ≥ | F ( T i +1 (Ni +1 ))|2 and νi ≤ νi +1 , and let λ be the Lagrange coeﬃcient and 2 2 denoted by L (χ ), for which F T χ Nχ > λ. For L (i > χ ), F T χ +1...n Nχ +1...n ≤ λ.

2. Determine the constant modulation variance

σω2 as σω2 =

3. If νχ +1 ≥ σω + ν1 then L (χ ) = L (χ − 1), and determine 2

4. If νχ +1 < σω2 + ν1 , then L (χ ) = L (χ + 1), compute

The rate over the

1

i =1

2 χ σωi . σω as σω2 =

χ

χ +1

2

log2 1 +

χ + 1 Gaussian sub-channels is R =

ν E ve = 1/λ. Let χ be the largest index in the list,

σ 2 F T i N i 2 − log2 1+ ω 2

1

χ −1

2

χ −1 σωi .

2|

2

σω F ( T i (Ni ))| 2 σN

n−1

, where | F ( T i (Ni ))|2 =

χ +1 log2 1 +

k=0

Tke

.

−i2π ik 2 n

σω2 | F ( T i (Ni ))|2 , which approximates the complex 2 σN

σN domain capacity by precisely /ln 2. χ +1 2 The modulation variance adaption scheme over a Gaussian quantum channel (with random channel transmittance) is summarized in Fig. 3. The parameter νi of the Gaussian sub-channels is depicted in yellow. If νi is under a critical limit

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Fig. 3. The constant modulation variance allocation mechanism over a Gaussian quantum channel with random multiplicative disturbances. If the i-th sub-channel Ni is noisy, i.e., νi ≥ ν E ve , Alice will not use that sub-channel, i.e., the modulation variance σω2i of |φi is 0. Only those sub-channels will be used for the transmission for which νi is under the critical bound ν E ve (red dashed line). Assuming l sub-channels with νi < ν E ve , the modulation variance 2 2 2 2 for these sub-channels is chosen to be a constant σω2i = ν E ve − min (ν0 , . . . , νn−1 ), where l σωi = l σω < nσω0 , and σω0 is the single-carrier modulation variance. (For interpretation of the colors in the ﬁgure(s), the reader is referred to the web version of this article.)

Fig. 4. The constant modulation variance allocation in function of the normalized squared magnitude. As the number of subcarrier goes to inﬁnity, the rate goes arbitrary close to the real capacity of the Gaussian quantum channel. Only those sub-channels get nonzero modulation variance σω2 for which νi < ν E ve holds.

ν E ve , the channel is assumed to be useful and can be used for information transmission. If νi < ν E ve , Alice allocates a constant modulation variance σω2i = σω2 according to (66), for the Gaussian sub-channel Ni . If νi ≥ ν E ve , Alice allocates zero modulation variance for Ni , i.e., σω2i = 0.

The AMQD is equipped with all of those properties that allow it to meet the requirements of an experimental protocol, speciﬁcally in the low SNR regimes. As a corollary, the transmission eﬃciency signiﬁcantly can be boosted in an experimental long-distance CVQKD scenario. The adaptive modulation variance can be rephrased in terms of the normalized n1 | T (N )|2 squared magnitude of the channel transmittance T (N ), where n is the number of subcarrier CVs. Let

ν E ve −

σ2

1

2 ν = σN /F

n

2 | T (N )|2 , and σω2 n1 | T (N )|2 =

N 2 . In this approach, the optimal modulation variance is max F n1 | T (N )|2 1 | T (N )|2 2

σω2 (x) dx = σω2 ,

(67)

0

where x = n1 | T (N )|2 , see (66).

1 | T (N )|2 and parameter n 1 2 1 2 2 2 ν = σN / F n | T (N )| is shown. The maximal value of | F ( T )| is obtained at n | T (N )| → 0, because as n increases, the normalized quantity represents a ﬁner sampling of T (N ) of the physical quantum link.

In Fig. 4, the adaptive modulation variance technique in terms of the normalized quantity [19] 2

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For the l Gaussian sub-channels Ni , νi < ν E ve , the mutual information is evaluated as follows. Assuming a single-carrier 2 modulation over a Gaussian channel N with noise variance σN , and modulation variance σω20 , let the R single (N ) maximal rate between Alice and Bob is as

R single (N ) = log2

σω20 | T (N )|2 1+ . 2 σN

(68)

For the l Gaussian sub-channels R AM Q D (N ) is evaluated as

R AM Q D (N ) = max ∀i

l −1

log2

i =0

σω2i | F ( T i (Ni ))|2 1+ , 2 σN

(69)

l−1

2 2 2 where σN = σN Il×l = 1l i =0 σNi follows from (46) for the noise variance of the l Gaussian sub-channels. For the averaged modulation variance of the l sub-channels, the relation

σω2

l −1

σω2 =

1 l

σω2i < σω20

(70)

i =0

follows, see (52). The multicarrier modulation transmits the same amount of information, hence at (70), 1 l

l

σωi | F ( T (Ni ))| holds, i.e., 2

2

R AM Q D (N ) = R single (N ) .

σω20 | T (N )|2 = (71)

σω2 = σω20 . In this case, the 2 2 | σ F T N , hence: ( ( ))| i l ωi

Furthermore, (71) can be increased by improving the average modulation variance in (70) up to relation between (68) at

σω20 , and (69) at σω2 = σω20 , is σω20 | T (N )|2 <

1 l

R AM Q D (N ) > R single (N ) ,

(72)

which shows that better rates can be reached by the AMQD modulation at a given noise variance variance σω20 , in comparison to the single-carrier modulation.

2

σN and modulation

2 sum of the squared magnitude of the Fourier transLet 0 ≤ ≤ 1 be the probability that the l | F ( T i (Ni ))| 2 formed channel transmission coeﬃcients of the l Gaussian sub-channels pick up the maximum l | F ( T i (Ni ))| = 2 max∀i l | F ( T i (Ni ))| :

= Pr

2

| F ( T i (Ni ))| = max ∀i

l

R AM Q D (N ) = log2 1 +

maxi

| F ( T i (Ni ))|

(73)

.

l

From this, the achievable rate is

2

l

| F ( T i (Ni ))|2 · SNR .

(74)

It can be concluded that the adaptive modulation variance increases the performance, especially in low SNR regimes, in which situation a constant modulation variance for all sub-channels is optimal. It also optimizes the received modulation variance at the decoder. Finally, in terms of (67) the rate is evaluated as 1 | T (N )|2 2

R AM Q D (N ) =

log2 1 + 0

1

σω2 (x) F

n

2 | T (N )|2

2 σN

dx,

(75)

where x = (N )| and n → ∞. The proof is concluded here. 2 1 |T n

2

4. Security proof The results on the ﬁnite-size and asymptotic security of AMQD are summarized in Theorems 2 and 3. The derivations conﬁrm that under an optimal coherent Gaussian (general) or collective Gaussian attack, any crosstalk among the subchannels cannot allow more information to be leaked to an eavesdropper than the single-carrier CVQKD.

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The ﬁnite-size and asymptotic analyses assume a ﬁnite number l of sub-channels for the information transmission and for the statistical averaging (i.e., a ﬁnite number of Gaussian subcarrier CVs available for the transmission). The ﬁnite-size regime analysis considers an optimal coherent Gaussian attack, while the asymptotic regime analysis utilizes an optimal collective Gaussian attack. In the coherent Gaussian attack, Eve interacts via an arbitrary unitary operation collectively with the CV quantum states and applies an arbitrary (assumed to be collective) measurement before the reconciliation process [5]. In the collective attack, Eve performs the same unitary individually on the CV quantum states and applies a collective measurement using her local quantum memory [33]. The security proof also considers the presence of the cross-talk effect between the Gaussian sub-channels, since it represents a valid issue in a multicarrier setting, while it has no relevance in a single-carrier setting. The secret key rates of AMQD conﬁrm the multimode bounds determined in [73] (see the results on fundamental rate-loss scaling in quantum optical communications in [73]). For further information on the bounds of private quantum communications, we suggest [74]. For both the ﬁnite-size and the asymptotic regime settings, as the quantum-level transmission is closed and the CVs are measured, post-processing operations are applied to the measurement results (similar to a single-carrier CVQKD protocol [63]) to minimize Eve’s knowledge. A detailed security analysis of the proposed protocol with an extension to realistic scenarios can be found in [11]. 4.1. Finite-size regime Theorem 2 (Finite-size security of AMQD). AMQD is unconditionally secure against arbitrary ﬁnite-size coherent attacks. Proof. The proof assumes the use of l sub-channels for the transmission from the total n. By utilizing the framework of [5], in the ﬁnite-size regime, any coherent attack against a CVQKD protocol can be reduced to an individual Gaussian attack. In the individual attack, Eve interacts individually with CV quantum states and measures them individually before the reconciliation. Thus, it is enough to show that an individual Gaussian attack is optimal against the multicarrier protocol and also that the multicarrier CVQKD does not increase Eve’s knowledge in the ﬁnite-size regime. The proof also assumes a reverse reconciliation (Bob starts the reconciliation process); hence, Eve’s correlation will be quantiﬁed by the preserved correlation with Bob’s system B. Let N be the block-size used by Eve for the attack, set as

N |K | ,

(76)

where | K | is the size of the ﬁnal key K . Using the formalism of [5], we assume that Eve applies an arbitrary ﬁnite-size coherent attack against the multicarrier CVQKD protocol. Let be H ( ·| ·) is the conditional entropy function, and let H ( B | A )Ni be the conditional entropy of Alice and Bob over sub-channel Ni , which is bounded by subadditivity of entropy function [5] as

H ( B | A )Ni ≤

H ( B z | A )Ni =

z

H ( B z | A z )Ni ,

(77)

z

where A and B are the systems of Alice and Bob formulated via N components, z = 1, . . . , N, B z is a z-th component of B, and from the strong subadditivity property, H ( B z | A )Ni is evaluated as

H ( B z | A )Ni = H ( B z | A 1 , . . . , A N )Ni ≤ H ( B z | A z )Ni .

(78)

Assuming that A and B have distributions of some mixture of A z and B z components such that z is drawn from a uniform distribution [5], the term H ( B z | A z )Ni in (78) can be rewritten as

H ( B z | A z )Ni = H ( B | A , z)Ni ≤ H ( B | A )Ni ,

(79)

therefore (77) can be rewritten via (79) for a given Ni as

H ( B | A )Ni ≤ N H ( B | A )Ni . The I ( A : B ) averaged mutual information between Alice and Bob averaged over the l sub-channels is yielded as

I ( A : B) =

=

1 l

I ( A : B )Ni

l

1 l

(80)

H ( B )Ni − H ( B | A )Ni ,

l

where I ( A : B )Ni is the mutual information between Alice and Bob evaluated over an i-th the sub-channel Ni . The I ( B : E ) averaged mutual information of Bob and Eve is deﬁned as

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16

I (B : E) =

=

1 l

I ( B : E )Ni

l

1 l

(81)

H ( B )Ni − H ( B | E )Ni ,

l

where I ( B : E )Ni is as

I ( B : E )Ni = H ( B )Ni − H ( B | E )Ni .

(82)

γ (Ni ) > 0, (81) can be rewritten via (59) as 1 T E ve,i 2 γ (Ni ) I ( B : E )N +

For any nonzero crosstalk noise

I ( B : E )γ˜ =

=

1

i

l

n

l

1 l

I ( B : E )Ni +

l

n

(83)

1 n

T E ve,i 2 x j I A j : B j .

n

j =i

For a given Ni , let A ∗ and B ∗ refer to the systems of Alice and Bob formulated via unmeasured subcarrier quadratures with a Gaussian distribution with a particular H ( B ∗ | A ∗ )Ni . Then, from (82), we have

H B ∗ A ∗ N − I ( B : E )Ni − H ( B )Ni = H B ∗ A ∗ N + H ( B | E )Ni , i i

(84)

thus a statistical averaging over the l sub-channels yields

1 l

=

l

1 H B ∗ A∗ N − I ( B : E )Ni − H ( B )Ni i l

1 l

H B

∗

l

l

∗

1

A N + i l

(85) H ( B | E )Ni ≥ 2N H 0 ,

l

where H 0 is the entropy of the single-carrier quadrature of the vacuum state for an harmonic oscillator, by some fundamental theory [6,7]. Thus, the amount of information leaked to Eve is upper bounded as

I ( B : E )γ˜ ≤

1 l

l

1 H B ∗ A∗ N + H ( B )Ni − 2N H 0 , i l

(86)

l

which lower bounds to secret key rate S AM Q D at a reverse reconciliation as

S AM Q D = I ( A : B ) − I ( B : E )γ˜

(87)

such that

S AM Q D ≥ S AM Q D ,

(88)

where S AM Q D is a lower bound, as

S AM Q D = −2N H 0 −

1 l

H ( B | A )Ni −

l

1 l

H B ∗ A∗ N . i

(89)

l

From a symmetry relation [5],

1 l

l

1 H B ∗ A∗ N = H ( B | A )N i , i l

(90)

l

thus (89) can be rewritten as

S AM Q D ≥ −2N H 0 − 2

1 l

H ( B | A )N i .

l

(91)

Since the protocol uses a Gaussian modulation, 1l l H ( B | A )Ni is upper bounded via the entropy Gaussian distribution G which has a maximized entropy for a given variance, as

1 l

l

H ( B | A )Ni ≤

1 l

l

H G ( B | A )Ni ,

1 l

l

H G ( B | A )Ni of the

(92)

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17

thus (89) can be rewritten as

S AM Q D ≥ −2N H 0 − 2

1 l

H G ( B | A )Ni .

(93)

l

For a given Ni , let B˜ i be the estimate of Bob’s system on Alice’s side from her system A i , evaluated as A B B˜ i = i 2 i A i ,

(94)

Ai

with estimation error ε B˜ i ,

ε B˜ i = B i − B˜ i .

(95)

Then,

1 l

l

where H G

H ( B | A )N i =

1 l

H

1 ε B˜ i A i ≤ H G ε B˜ i A i , l

l

ε B˜ i A i has a variance of

σ 2 ˜

ε B i Ai

2 = ε B˜ i = B 2i −

A i B i 2 A 2i

(96)

l

(97)

.

Using (97), the lower bound (93) is evaluated for an arbitrary coherent attack, as

S AM Q D ≥ N log2

1 l

l

N0

σ 2

(98)

,

ε B˜ i A i

where N 0 is the vacuum variance, which bound is equal to the bound if Eve apply a ﬁnite-size Gaussian individual attack against the protocol. From the σω2 < σω20 modulation variance relation (see also Theorem 1), it follows the amount of information leaked to Eve in a single-carrier modulation at modulation variance (81)) are related as

σω20 and a multicarrier CVQKD at modulation variance σω2 (see

I ( B : E ) ≤ I ( B : E )single ,

(99)

where I ( B : E )single is the mutual information of Eve in a single-carrier CVQKD at a reverse reconciliation and modulation variance σω20 ,

I ( B : E )single = H ( B )single − H ( B | E )single .

(100)

On the other hand, for any nonzero γ˜ (see (57)), the mutual information in (81) and (83) are related as

I ( B : E )γ˜ ≥ I ( B : E ) ,

(101)

therefore from (99) and (101), one ﬁnds

I ( B : E )γ˜ ≤ I ( B : E )single ,

(102)

which conﬁrms that a multicarrier CVQKD with modulation variance σω2 and γ˜ > 0, cannot leak more information to an eavesdropper in the ﬁnite-size regime than a single-carrier CVQKD with modulation variance σω20 at an optimal ﬁnite-size Gaussian individual attack. The proof is concluded here. 2 4.2. Asymptotic regime Theorem 3 (Asymptotic security of AMQD). AMQD is unconditionally secure against an optimal collective Gaussian attack in the asymptotic regime. Proof. The proof also assumes reverse reconciliation scenario. First, we express the eavesdropper information under an optimal Gaussian attack in an AMQD modulation. Then, we show that this attack – besides the fact that it maximizes Eve’s Holevo information – cannot leak more information to an eavesdropper than the single-carrier case. (The optimality of collective Gaussian attacks for single-carrier CVQKD has already

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18

been proven in [33,34] and the proof follows for the multicarrier case of AMQD. Therefore, we focus mainly on the analysis of the amount of leakable information in a multicarrier CVQKD scenario, assuming an optimal collective Gaussian attack.) The notations used in our proof are summarized as follows. Let ρ ∈ H be the density matrix of a Gaussian state, and let ρ ∈ H be the density matrix of a non-Gaussian state with the same covariance matrix and displacement vector. Let p (x) refer to the Gaussian probability distribution, and p (x) to an arbitrary probability distribution with the same ﬁrst and second momenta as p (x). Let ρ A B be a Gaussian state between Alice and Bob, andlet ρA B refer to an arbitrary shared state. For these density matrices, the relation between the joint entropy is H (ρ A B ) ≥ H ρ A B . Let’s assume that a Gaussian density matrix ρi is transmitted through sub-channel Ni , and the shared state between Alice, Bob and Eve that is related to the i-th sub-channel is a pure system ρiA B E . The measurement process is characterized as follows. Let X i be Alice’s classical variable that is encoded into the input system A i , which is a density matrix ρi , and let B i be the classical variable of Bob that results from a measurement M i = M i , B i B applied on ρi , where M i , B i are positive

i

¯ ¯ operators, i M i , B i M i , B i = I . The conditional entropy between A i and B i is H ( A i | B i ) = H A i − H ( B i ), where A i refers to the measured density matrix ρ¯i [33]. Let’s denote Eve’s optimal Gaussian attack on the i-th sub-channel Ni by the CPTP map Ti . Taking arbitrary density matrices ρ and σ , the relation D ( ρ σ) ≥ D ( Ti (ρ ) Ti (σ )) follows, where D ( · ·) is the relative entropy function, along with the entropic relation H ( A ) − H A ≥ H T ( A ) − H T A [33]. We express the Holevo information of the eavesdropper χ ( B i : E i )γ ,Ni on a sub-channel Ni , assuming reverse reconciliation, and optimal Gaussian attack, where B i is Bob’s variable from the i-th sub-channel, while E i identiﬁes Eve’s variable. During the optimal Gaussian attack against AMQD, Eve attacks each sub-channel separately with a beam splitter that has transmittance T E ve,i . 2 2 Assuming an average modulation variance σω2 = 1l l σωi < σω0 in the AMQD setting with l sub-channels for the trans†

mittance (σω2i -s are constant values due to the optimality consumption, see Theorem 1), the following relation holds for the Holevo information of the eavesdropper:

(B : E) = H (E) − H ( E| B) χ =

1 l

( H ( E i ) − H ( E i | B i ))

(103)

l

≤ χ ( B : E )single = H ( E ) − H ( E | B )single . This relation follows from that in an AMQD modulation H ( E | X ) ≥ H ( E | X )single , because the same amount of information is transmitted via modulation variance σω2 < σω20 over the l sub-channels. At this point, we add the γ > 0 crosstalk into the evaluation. Assuming the use of sub-channel Ni , the optimal Gaussian attack leads to

2

χ ( B : E )γ ,Ni = χ ( B : E )Ni + T E ve,i γ (Ni ) = χ ( B : E )Ni +

T E ve,i 2 x j χ A j : B j .

(104)

j =i

The average modulation

(B : E) = χ

1 l

ω on the l sub-channels leads to average Holevo information

χ ( B : E )Ni ,

(105)

l

while the average amount of crosstalk over all the n sub-channels is evaluated as

γ (N ) = =

1 n

γ (N i )

n

1 n

n

x jχ A j : B j ,

j =i

( B : E )γ over all the n sub-channels, as which together leads to the averaged Holevo information χ

(106)

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L. Gyongyosi / Theoretical Computer Science ••• (••••) •••–•••

( B : E )γ = χ = = =

1 l

χ ( B : E )Ni +

l

1 l

χ ( B : E )Ni +

l

1 l

χ ( B : E )Ni +

l

1 l

χ ( B : E )Ni +

l

19

1

T E ve,i 2 γ (Ni )

n

n

1

T E ve,i 2 x j χ A j : B j

n

j =i

n

1 n

| T E ve |2 x j χ A j : B j

(107)

j =i

1 n

| T E ve |2 γ (Ni )

n

(N ) , ( B : E ) + | T E ve | γ =χ 2

2

where | T E ve |2 = n1 n T E ve,i . For this quantity, the relation

( B : E )γ ≤ χ ( B : E )single χ

(108)

(N ) > 0 follows, since for γ

(B : E) = H (E) − H ( E| B) χ ( B : E )γ = H ( E ) − H ( E | B )γ <χ

(109)

≤ χ ( B : E )single = H ( E ) − H ( E | B )single , and

H ( E | B ) > H ( E | B )γ ≥ H ( E | B )single .

(110)

(N ) > 0. The RHS of (110) is veriﬁed by the fact that a nonzero crosstalk The relation H ( E | B ) > H ( E | B )γ holds for any γ

has no relevance on the mutual information of Alice and Bob, i.e., communicating at a constant modulation variance σω2 < σω20 on the l sub-channels, the relation H ( B | A ) = H ( B | A )γ = H ( B | A )single follows for an AMQD modulation. The ﬁrst equality follows from the fact that the presence of any nonzero crosstalk does not change the mutual information of the legal parties, while the second comes from that in an AMQD modulation the same amount of information is transmitted as the single-carrier CVQKD via lower modulation variance (see (70)), from which (110) is immediately concluded. Finally, we show that the collective Gaussian attack is optimal in the AMQD modulation. Fixing the ﬁrst and second (N ) > 0, by moments, under an optimal Gaussian attack for all sub-channels Ni , i = 1 . . . n, and with a nonzero crosstalk γ the optimality consumption one obtains nonnegative ≥ 0 for the difference of Eve’s Holevo information that results from an optimal Gaussian attack and an arbitrary attack:

B : E γ = χ ( B : E )γ − χ 1 = H (Ei ) − H Ei − H ( Ei| Bi) − H Ei Bi l

+

l

1 n

n

l

⎞ 2 2 T E ve,i x j χ A j : B j − T E ve,i x j χ A j : B j ⎠ ⎝ ⎛

j =i

j =i

= H (E) − H E − H ( E| B) − H E B (N ) − | T E ve |2 γ (N ) + | T E ve |2 γ = H ( A B ) − H A B − H ( A B | X ) − H A B X (N ) − | T E ve |2 γ (N ) + | T E ve |2 γ = H ( A B ) − H A B − H A¯ B¯ − H A¯ B¯ + H ( X ) − H X (N ) − | T E ve |2 γ (N ) , + | T E ve |2 γ

(111)

B : E γ is Eve’s average Holevo information in an AMQD modulation for an arbitrary attack, χ ( B : E )γ is evaluated where χ (N ) and | T E ve |2 γ (N ) refer to the leaked valuable crosstalk information under an optimal Gaussian by (107), while | T E ve |2 γ attack and an arbitrary attack, respectively.

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20

(N ) > 0 acts as a Gaussian noise for the receiver that is already included in the subSince any nonzero crosstalk γ 2 , see (58), the next equation is straightforward for the minimized mutual information under an channel noise variance σN i optimal collective Gaussian attack:

I ( A : B) − I A : B

= = −

1

I Aj : Bj −

l

H A j − H A j

l

H A j B j − H A j B j

1 l

l

l

I Aj : Bj

l

1

l

=

H B j − H B j

+

−

H B j − H B j

l

(112)

l

H A j B j − H A j B j

l

≤ 0. Thus, Eve’s collective Gaussian attack remains optimal in an AMQD modulation (i.e., Eve’s Holevo information is maximized), and the presence of a nonzero crosstalk has no effect on the mutual information between Alice and Bob. On the other hand, Eve’s Holevo information could be slightly improved by a nonzero crosstalk, to be precise by (N ), however an AMQD modulation does not allow leaking of more information to Eve than the single-carrier | T E ve |2 γ CVQKD. The argumentation behind this is as follows. The inequality

( B : E )γ ≥ 0 χ ( B : E )single − χ

(N ) ≥ 0 ( B : E ) + | T E ve |2 γ χ ( B : E )single − χ

(113)

directly comes from (107), which combined with (110) leads to

(N ) , ( B : E ) ≥ | T E ve |2 γ χ ( B : E )single − χ

(114)

which reveals that in an AMQD modulation, the degradation of Eve’s Holevo information in comparison to the single-carrier CVQKD is always equal to or greater than the valuable information that is leaked from the crosstalk to Eve, thus

(B : E) < χ ( B : E )γ < χ ( B : E )single − | T E ve |2 γ (N ) , χ

(115)

(N ), the inequality hence for any | T E ve |2 γ

χ ( B : E )γ ≤ χ ( B : E )single

(116)

immediately follows. These results lead to the S AM Q D secret key rates for reverse reconciliation at an AMQD modulation, under an optimal collective Gaussian attack:

( B : E )γ, S AM Q D = I ( A : B ) − χ

(117)

and, if Bob is allowed to perform a collective measurement, then the secret key rate is

( B : E )γ. S AM Q D = χ ( A : B ) − χ

(118)

These results prove the unconditional security of AMQD against optimal collective Gaussian attacks, which concludes the proof. 2 5. Numerical evaluation 5.1. Tolerable excess noise Theorem 4 (Tolerable excess noise of the AMQD modulation). At a given |T (N )|2 , N tol, AM Q D ≥ N tol,single , where N tol,single and N tol, AM Q D are the tolerable excess noise of single-carrier CVQKD and AMQD.

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21

Proof. The eﬃciency of the adaptive multicarrier division technique can be approached by how eﬃciently the allocated modulation variances are utilized at a given secret key rate. Assuming a single-carrier quadrature encoding-scheme with eﬃciency

ηsingle =

bR

σω20

(119)

,

where b = 2 is the number of quadrature bases (position and momentum bases), R is the secret key rate, and σω20 is the modulation variance used for the modulation of n coherent states. In the adaptive multicarrier division technique, the rate R given in (119) can be reached with eﬃciency:

η AM Q D =

bR

σω2

(120)

,

where σω2 is the constant modulation variance used for the l subcarriers. If σω2 < σω20 holds at a given R, the multicarrier modulation scheme transmits the same amount of information such as the single-carrier modulation scheme, with increased eﬃciency, η AM Q D > ηsingle . The improvement in the tolerable loss is as follows. In the single-carrier modulation scheme, at the 3.1 dB of signal attenuation the quantum channel makes it impossible to transmit any information. In the multicarrier modulation, the tolerable loss is more than 3.1 dB at the same modulation variance σω20 (justiﬁed by the convention of full width at half maximum (FWHM)). For σω20 and AMQD modulation, the −3.1 dB attenuation brings up diverse amount of loss, which is equivalent to the use of an improved virtual modulation variance A σω20 , where A > 1. At σω20 , a higher modulation variance A σω20 can be virtually simulated by the multicarrier transmission, which results in a higher tolerable loss (dBs) and a higher tolerable excess noise in overall. Let the input Gaussian signal be

g 1 (x) = The

1

√

e

σω 0 2 π

−x2 2 2σω 0

(121)

.

σω2 < σω20 modulation variance of the l sub-channels leads to another input Gaussian signal of g 2 (x) =

−x2

1

2

√

σω 2 π

e 2σω .

(122)

From (122) the virtual Gaussian signal at a modulation variance

g 3 (x) = √

1 A σω0

√

2π

e

−x2 2 2 A σω 0

σω20 is as follows (123)

,

where A σω20 > σω20 , and A > 1, which is evaluated as

A=

1 l

2 l | F ( T i (Ni ))|

=

| T (N )|2

1 l

l−1 l−1 −i2π ik 2 l i =0 k =0 T k e | T (N )|2

.

(124)

The FWHMs of these Gaussian signals are as follows:

√

F W H M ( g 1 (x)) = 2 2 ln 2σω0 ,

(125)

F W H M ( g 2 (x)) = 2 2 ln 2σω

(126)

√

and

F W H M ( g 3 (x)) =

√

√

A2 2 ln 2σω0 .

(127)

These results are summarized in Fig. 5. The improvement in the excess noise is as follows [8]. In the single-carrier case, at a given T E ve and W , the excess noise is as

N single =

( W − 1) | T E ve |2 1 − | T E ve |2

,

where | T E ve |2 < | T |2 ∈ [0, 1] is the squared magnitude of the complex variable T E ve .

(128)

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22

Fig. 5. (a) The FWHM of the single-carrier Gaussian√ signal √ (blue) and the Gaussian signal of the subcarrier transmission (red). (b) The FWHM of the virtual Gaussian signal at a modulation variance of σω20 is A2 2 ln 2σω0 , A > 1. The AMQD allows more tolerable loss at a given modulation variance σω20 .

In the AMQD modulation, at a given W , the excess noise is reduced to

N AM Q D =

2 1 n F T E ve ,i n , 2 1 − n1 n F T E ve,i

( W − 1)

(129)

where

F T E ve,i =

n −1

T E ve,k e

−i2π ik

(130)

n

k =0

and

n −1

2

−i2π ik 1 1 F T E ve,i 2 = 1 | F ( T i )|2 , T E ve,k e n ≤ | T E ve |2 < | T |2 ≤ n n n n n n

(131)

k =0

which relation is justiﬁed by the CVQFT transformation. Since, at a given W , the tolerable excess noise depends only on T E ve , from (131) it follows, that in the AMQD modulation scheme higher amount of excess noise can be tolerated. The improvement in the tolerable excess noise can be approached by the ratio κ ≥ 1 between the excess noise N single and N AM Q D as:

κ= =

N single N AM Q D

( W − 1) | T E ve |2 1 − | T E ve |2

1−

2 n F T E ve ,i 2 1

1 n

( W − 1) n n F T E ve,i 2 | T E ve |2 − | T E ve |2 n1 n F T E ve,i = 2 1 − | T E ve |2 1 F T E ve,i 2 n F T E ve ,i n n n 2 | T E ve |2 − | T E ve |2 n1 n F T E ve,i = . 2 | T |2 n1 n F T E ve,i

(132)

The improvement in the tolerable excess noise is N tol, AM Q D = α N tol,single , where N tol,single is the tolerable excess noise of a single-carrier CVQKD, α = xκ = xN single / N AM Q D ≥ 1, and N AM Q D / N single ≤ x ≤ 1, yielding N tol, AM Q D ≥ N tol,single . 2 5.2. Secret key rates To express the S secret key rates of AMQD, let l Gaussian sub-channels with νi < ν E ve , and let the variance of Eve’s EPR-ancilla be W [4,8]. For a detailed proof on the achievable secret key rates and security thresholds of AMQD, see [11]. 5.2.1. One-way CVQKD Homodyne measurement, reverse reconciliation Assuming that the protocol is a one-way multicarrier CVQKD system and Bob applies a homodyne measurement setting for the measurement of the quadratures, at a reverse reconciliation the resulting secret key rate is as

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1

R R ,hom

S one− way = where

b=

1−

2

log2

1 l

1 l

1−

W

l

−

W +1

| F ( T i (Ni ))| b 2

2

| F ( T i (Ni ))|

W+

l

1 l

2

log2

W +1 2

−

W −1 2

23

log2

W −1

(133)

2

| F ( T i (Ni ))|2 .

(134)

l

Homodyne measurement, direct reconciliation Considering that the protocol is a one-way multicarrier CVQKD system and Bob applies a homodyne measurement setting for the measurement of the quadratures, at a direct reconciliation the resulting secret key rate is as D R ,hom

S one− way =

1

| F ( T i (Ni ))|2 e log2 2 2 1 − 1l b l | F ( T i (Ni ))| # # # # W b/e + 1 W b/e + 1 W b/e − 1 W b/e − 1 + − log2 log2 1

− where

e=

1−

2

W +1 2

1 l

l

l

2

log2

W +1

−

2

2

| F ( T i (Ni ))|

+

l

W −1 2

1 l

2

log2

W −1

2

2

(135)

,

2

| F ( T i (Ni ))|

W.

(136)

l

5.2.2. Two-way CVQKD Homodyne measurement, reverse reconciliation As the protocol is a two-way multicarrier CVQKD system and Bob applies a homodyne measurement setting for the measurement of the quadratures, at a reverse reconciliation the resulting secret key rate is as

R R ,hom

S t wo− way =

1 2

1 2 2 2 | | + F T N F T N ( ( ))| ( ( ))| i i i i l l l l 1 2 2 1− l l | F ( T i (Ni ))| W +1 W +1 W −1 W −1 . − − log2 log2

1−

log2

1

2

2

2

(137)

2

Homodyne measurement, direct reconciliation Assuming that the protocol is a two-way multicarrier CVQKD system and Bob applies a homodyne measurement setting for the measurement of the quadratures, at a direct reconciliation the resulting secret key rate is as D R ,hom

S t wo− way =

1 2

log2

| F ( T i (Ni ))|2 − 2 2 1 − 1l l | F ( T i (Ni ))| 1 l

l

W +1 2

log2

W +1 2

−

W −1 2

log2

W −1 2

.

(138)

For the achievable secret key rates of AMQD at the measurement apparatuses and protocol settings, see [11]. 5.3. Wireless quantum links In this section we derive the P (N ) private classical capacity (an upper bound on the maximal achievable secret key rate S (N ), S (N ) ≤ P (N )) over an FSO quantum link [15] N in an AMQD setting, at a reverse reconciliation. Lemma 1. A multicarrier CVQKD over an FSO channel yields achievable private classical capacity P (N ) = A B and χ B E are avB E ) where l is the number of sub-channels, f is the reconciliation eﬃciency, χ limn→∞ n1 max∀i ( f ( χAB ) − χ eraged Holevo information between Alice and Bob, and Bob and Eve. Proof. In the proof we assume that parties (Bob and Eve) are equipped with quantum memory and are allowed to perform joint measurement (that justiﬁes the use of Holevo information throughout the proof). In an FSO setting, the T (Ni ) is the complex channel transmittance coeﬃcient for a sub-channel Ni is as [15]

T (Ni ) = ηi I i + i ηi I i ,

(139)

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24

where ηi is the effective photocurrent conversion ratio of the receiver, while I i is the normalized irradiance [42–44] for Ni , with the gamma-gamma (GG) probability density

f (Ii ) =

2(ab)

a+b 2

a+b 2 −1

(a) (b)

Ii

#

K a−b 2 abI i ,

(140)

where K v (·) is the modiﬁed Bessel function of the second kind and of order v [42–44], (·) is the Gamma function, while a ≥ 0, and b ≥ 0 are the distribution-shaping parameters expressed as

$

a = exp

0.492 1 + 0.18d2 + 0.5612/5

7/6

% −1 −1

(141)

,

and

$

b = exp

0.512 1 + 0.9d2 + 0.6212/5

5/6

%−1 −1

(142)

,

where

7/6 2 = 1.23C 2 x j L 11/6

(143)

is the Rytov variance, C is the altitude-dependent turbulence strength, L is the length of the link, and |k| is the optical wave number, while d is deﬁned as 2

d=

|k| D 2 /4L ,

(144)

where D is the receiver’s aperture diameter [15,42–44]. Coeﬃcient i is deﬁned as

i = (ηi E [I i ])2 = ηi2 .

(145)

Let assume that a and b In function of optimal coeﬃcients λ∗SNR and

ϑi = F ( γ ) = F (ηi )2 F ( I i )2

(146)

is the optimal γ coeﬃcient, averaged for the l sub-channels in the FSO channel, the optimal modulation variance where γ σω2 of a Gaussian sub-carrier quadrature component for all sub-channels is evaluated as

σω2 =

1

λ∗SNR

−

1

ϑ

(147)

,

where λ∗SNR is a Lagrange multiplier λ that satisﬁes the average power constraint c ∗

c∗ : σω2 = SNR∗ = E

x|γ

σω2 (γ ) ,

(148)

where σω2 refers to the average input power associated to an i-th input quadrature component x = xi in a multicarrier setting, evaluated as

1

λ∗SNR = λ τ

a+b 5 4 −8

e

# −2 ab λ/

1

(149)

SNR∗

λ, , where is averaged over the l sub-channels in an FSO channel, with optimal values of τ, a+b

τ=

5

(a) (b) 4 − 8 √ a+b 5 2 π (ab) 2 − 4

(150)

and

= F ( = F η)2 ,

(151)

=η , where η is the optimal effective photocurrent conversion ratio of the receiver, and where SNR is an averaged SNR evaluated as 2

∗

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SNR∗ ≈

1

λ τ

a+b − 13 4 8

e

# −2 ab λ/

25

(152)

,

λ is an optimal Lagrange multiplier. where After some calculations, the ν E ve security bound of Eve’s optimal collective Gaussian attack in a multicarrier CVQKD setting over an FSO channel can be rewritten as

ν E ve =

1

=

λ∗

SNR

τ SNR∗ a+b 5 λ 4 − 8 e −2

# . ab λ/

(153)

Therefore, utilizing the modulation adaption scheme from Section 3 (see (61)-(66)), the optimal channels in an FSO setting is as

ν = min (νi ) =

1

ϑ

ν coeﬃcient for all sub(154)

.

Utilizing the modulation variance adaption (66), the P (Ni ) private classical capacity of a sub-channel Ni at a reverse reconciliation with reconciliation eﬃciency f is as

1

P (Ni ) = lim

n→∞

where

max f χ A B ,i − χ B E ,i ,

(155)

n ∀ p i ,ρi

χ A B ,i is the Holevo information between Alice and Bob over Ni , evaluated as i 1 4 , χ A B ,i ≥ SNR log i SNRi 16(ai b i )2

2 where i = F (i ) = F (ηi )2 , ai and b i characterizes Ni , SNRi = σω2i /σN , while i Bob and Eve with respect to an i-th sub-channel Ni , as

χ B E ,i =

2 λi − 1 G

2

i =1

−

4 λi − 1 G

i =3

2

(156)

χ B E ,i is the Holevo information between

(157)

,

where function G (·) is as [15,61,62]

G (x) = (x + 1) log2 (x + 1) − x log2 x, √ √ with λ21,2 = 12 A ± A 2 − 4B , λ23,4 = 12 C ± C 2 − 4D , λ5 = 1, where

(158)

A = V 2 (1 − 2T i ) + 2T i + T i2 ( V + κline )2 , B=

T i2 ( V

(159)

2

κline + 1) ,

1

C=

√

2 A κM + B + 1 + 2κ M V

( T i ( V + κtot ))2 2 √ V + B κM , D= T i ( V + κtot )

B + T i ( V + κline ) + 2T i V 2 − 1 ,

(160) (161)

(162)

where T i is the channel transmittance of Ni , V = σω20 + 1,

κM is the detector-added error, κline is the total channel-added

noise, while κtot is the overall noise κtot = κline + κ M T1 [15,62]. i The SNR B E ,i is the SNR of the logical sub-channel N B E ,i , as

SNR B E ,i = where

σω2 , 2 σN B E ,i

(163)

2 σω2 is the modulation variance of Bob’s noisy sub-carrier component, while σN is variance of Eve’s channel. B E ,i

(N ) private classical capacity over the l sub-channels is as Thus an averaged P

P˜ (N ) = lim

1

n→∞

A B ≥ where χ

16(ab)2

max

n ∀i

SNR∗ log4

l −1

1 l

i =0 1 SNR∗

P (Ni ) ≈ lim

n→∞

1

max f

n ∀i

χ˜ A B − χ˜ B E ,

(164)

is an averaged Holevo information between Alice and Bob, while the averaged Holevo

B E ≈ quantity between Bob and Eve is as χ

1 l

l−1

i =0

χ B E ,i , where χ B E ,i is shown in (157). The proof is concluded here. 2

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6. Conclusions Here, we deﬁned the framework of multicarrier CVQKD. The AMQD modulation is designed for arbitrary quantum links with additive and multiplicative disturbances. We also investigated an adaptive modulation variance allocation mechanism for the scheme and proved the security of the protocol against optimal Gaussian attacks in the ﬁnite-size and asymptotic regimes. The method provides a useful framework for the implementation of CVQKD over practical quantum channels in diverse environments. Statements Ethics statement This work did not involve any active collection of human data. Data accessibility statement This work does not have any experimental data. Competing ﬁnancial interests statement We have no competing ﬁnancial interests. Competing interests statement We have no competing interests. Authors’ contributions LGY conceived the mathematical models, interpreted the computational results, and wrote the paper. LGY implemented and performed the calculations. Declaration of competing interest There is no any competing interests. Acknowledgements The research reported in this paper has been supported by the National Research, Development and Innovation Fund (TUDFO/51757/2019-ITM, Thematic Excellence Program). This work was partially supported by the National Research Development and Innovation Oﬃce of Hungary (Project No. 2017-1.2.1-NKP-2017-00001), by the Hungarian Scientiﬁc Research Fund – OTKA K-112125 and in part by the BME Artiﬁcial Intelligence FIKP grant of EMMI (BME FIKP-MI/SC). Appendix A A.1. Abbreviations The abbreviations of the manuscript are summarized as follows. AMQD AWGN BS CV CVQFT CVQKD DR DV FFT FWHM ICVQFT IFFT OFDM

Adaptive Multicarrier Quadrature Division Additive White Gaussian Noise Beam Splitter Continuous-Variable Continuous-Variable Quantum Fourier Transform Continuous-Variable Quantum Key Distribution Direct-Reconciliation Discrete Variable Fast Fourier Transform Full Width at Half Maximum Inverse CVQFT Inverse Fast Fourier Transform Orthogonal Frequency-Division Multiplexing

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ll Table A.1 Summary of notations. Notation

Description

F (·) F −1 (·)

z = x + ip

The CVQFT transformation, applied on Bob’s side, continuous variable U unitary operation, F (·) = CVQFT (·). Inverse FFT transform, applied on Alice’s side, F −1 (·) = IFFT (·). Single-carrier modulation variance. Multicarrier modulation variance. Average modulation variance of the l Gaussian sub-channels Ni . An n-dimensional, zero-mean, circular symmetric complex random Gaussian vector, Kz = E zz† , where zi = xi + ip i ,

d = F −1 ( z)

x = (x0 , . . . , xn−1 ) T and p = ( p 0 , . . . , pn−1 ) T , and xi ∈ N 0, σω20 , p i ∈ N 0, σω20 are i.i.d. zero-mean Gaussian random variables. An n-dimensional, zero-mean, circular symmetric complex random Gaussian vector, Kd = E dd† , d = (d0 , . . . , dn−1 ) T ,

σω20 σω2 = 1l l σω2i

are i.i.d. zero-mean Gaussian random variables, σω2 F = 1/σω20 . The i-th component is di ∈ CN 0, σ , with variance σ = E |di |2 . −1

The i-th subcarrier Gaussian CV, |φi = F ( zi ) , where F −1 stands for the inverse FFT.

Bob’s decoded Gaussian state, the CVQFT-transformed |φi subcarrier CV, |ϕi = F (|φi ) = F F −1 ( zi ) .

di = xi + ip i , xi ∈ N 0, σω2 F , p i ∈ N 0, σω2 F

|φi |ϕi N N i , i = 0, . . . , n − 1 T (N ) ∈ C T (Ni ) ∈ C T E ve T E ve,i W ∈ CN (0, C ())

ϕ i

F (T (N )) F ()

y [ j], j = 1, . . . , n.

τ = F (d) [ j] 2 2 νi = σN /| F ( T i )|2 ν E ve = 1/λ

L = | F ( T 0...n−1 )|2 N single

2 di

2 di

Gaussian quantum channel in the single-carrier transmission. Gaussian sub-channels in the multicarrier transmission. Channel transmittance, normalized complex variable. The real part identiﬁes the position quadrature transmission, the imaginary part stand for the transmittance of the position quadrature. Transmittance of the i-th sub-channel. Eve’s transmittance, T E ve = 1 − T (N ). Eve’s transmittance for the i-th subcarrier CV. Variance of Eve’s EPR ancilla used in the entangling cloner attack. Gaussian noise of the quantum channel N , zero-mean, circular complex random vector with variance symmetric Gaussian

2 2 C () = E † , and with quadrature components xi ∈ N 0, σN , p i ∈ N 0, σN . i

i Bob’s coherent Gaussian state in the phase space, ϕi = xi + i p i . Modeled as a zero-mean, circular symmetric complex Gaussian random variable zi ∈ CN 0, σz2i , σz2i = E | zi |2 , zi = xi + ip i , with quadrature components xi ∈ N 0, σω20 , p i ∈ N 0, σω20 , which are i.i.d. zero-mean Gaussian random variables. The CVQFT transform of transmittance matrix T (N ) ∈ Cn , n-dimensional complex vector. The CVQFT transform of vector , F () ∈ CN(0, C ( F ())) . An n-dimensional zero-mean, circular symmetric complex Gaussian random vector, where C ( F ()) = E F () F ()† , and the quadrature components are F xi ∈ N 0, σ F2(N ) , i 2 F p i ∈ N 0, σ F2(N ) , where σ F2(N ) < σN . i An AMQD block. Formulated by n Gaussian subcarrier continuous variables, where j is the index of the AMQD block, y [ j] = F (T (N )) F (d) [ j] + F () [ j] and y [ j] = ( y 1 [ j] , . . . , yn [ j]) T F (d) [ j] = ( F (d1 ) [ j] , . . . , F (dn ) [ j]) T , F () [ j] = ( F (1 ) [ j] , . . . , F (n ) [ j]) T . Exponentially distributed variable, E [τ ] ≤ n2σω2 .

n−1

Ratio of noise variance and Fourier transformed channel transmittance, where | F ( T i )|2 =

( W −1) | T E ve |2 1−| T E ve |2

( W −1)

2 1 n F T E ve ,i n . F T E ve,i 2

N AM Q D

The excess noise in the AMQD modulation scheme, N A M Q D =

κ

Ratio of single-carrier and AMQD excess noise, κ = N single / N A M Q D ≥ 1. Tolerable excess noise in AMQD modulation, where α = xκ ≥ 1, N tol, A M Q D = α N tol,single Single-carrier CVQKD and AMQD eﬃciency. Transmission rate of classical information. Secret key rate.

N tol, A M Q D R S

QFT QKD RR SNR

Tke

, i = 0 . . . n − 1.

−i2π ik 2 n

Security bound of the optimal Gaussian attack, where λ is the Lagrange coeﬃcient. The probability that the sum of the squared magnitudes of the Fourier transformed coeﬃcients of the l Gaussian sub-channels 2 2 picks up a maximum value, Pr l | F ( T i )| = l max∀i | F ( T i )| . Ordered list of the squared magnitudes so that | F ( T i )|2 ≥ | F ( T i +1 )|2 and νi ≤ νi +1 . Excess noise at a single-carrier transmission, where T E ve is Eve’s transmittance, W is the variance of the EPR-ancilla, N single =

ηsingle , η A M Q D

k=0

Quantum Fourier Transform Quantum Key Distribution Reverse Reconciliation Signal to Noise Ratio

A.2. Notations The notations of the manuscript are summarized in Table A.1. References [1] S. Pirandola, et al., Advances in quantum cryptography, arXiv:1906.01645, 2019.

1− n1

n

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