Iterative Secret Key Rate Adapting with Error Minimization for

Iterative Secret Key Rate Adapting with Error Minimization for Continuous-Variable Quantum Key Distribution Laszlo Gyongyosi 1

Quantum Technologies Laboratory Department of Networked Systems and Services Budapest University of Technology and Economics 2 Magyar tudosok krt., Budapest, H-1117 Hungary 2 MTA-BME Information Systems Research Group Hungarian Academy of Sciences 7 Nador st., Budapest, H-1051 Hungary [email protected]

Abstract We define an iterative error-minimizing secret key adapting method for multicarrier CVQKD. A multicarrier CVQKD protocol uses Gaussian subcarrier quantum continuous variables (CVs) for the transmission. The proposed method allows for the parties to reach a given target secret key rate with minimized error rate through the Gaussian subchannels by a sub-channel adaption procedure. The adaption algorithm iteratively determines the optimal transmit conditions to achieve the target secret key rate and the minimal error rate over the sub-channels. The solution requires no complex calculations or computational tools, allowing for easy implementation for experimental CVQKD scenarios. Keywords: quantum key distribution; continuous variables; CVQKD; AMQD; AMQDMQA; quantum Shannon theory.

1

1 Introduction Continuous-variable quantum key distribution (CVQKD) provides an easily implementable solution to realize unconditional secure communication over the current telecommunication networks [10–22]. CVQKD does not require single-photon sources and detectors and can be implemented in an experimental scenario by standard devices [1], [9–26], [30–37]. In a CVQKD setting, the information is carried by a continuous-variable quantum state that is defined in the phase space via the position and momentum quadratures. In an experimental CVQKD scenario, the CV quantum states have a Gaussian random distribution, and the quantum channel between the sender (Alice) and receiver (Bob) is also Gaussian, because the presence of an eavesdropper (Eve) adds a white Gaussian noise into the transmission [19-41]. The CVQKD protocols have several smart properties. However, the relevant performance attributes of experimental CVQKD (i.e., secret key rates, transmission distances, tolerable excess noise, etc.) still require significant improvements. For this purpose, the multicarrier CVQKD has been recently introduced through the adaptive quadrature division modulation (AMQD) [2]. The multicarrier CVQKD scheme injects several additional degrees of freedom into the transmission, which are not available for a standard (single-carrier) CVQKD setting. In particular, the extra benefits and resources allow the realization of improved secret key rates and a higher amount of tolerable losses with unconditional security. These results also made possible to utilize several significant phenomena for CVQKD that are unavailable in a standard CVQKD protocol (such as single layer transmission [4], enhanced security thresholds [5], multidimensional manifold extraction [6], characterization of the subcarrier domain [7], adaptive quadrature detection and subchannel estimation techniques [8], extensive utilization of distribution statistics and random matrix formalism [9], and advanced statistical approaches for performance improvements [10], [4142]). In this work, we define an iterative error-minimizing secret key adaption method for multicarrier CVQKD. The proposed secret key adaption algorithm iteratively determines the optimal transmit conditions at a given target secret key rate to realize minimal error transmission over the sub-channels. At a given transmission rate of private classical information (private rate), the method determines and selects that sub-channel from the set of available sub-channels for the transmission of the quadratures, which sub-channel provides a minimal error rate. The secret key adaption successively utilizes private rate curves for the sub-channels. The curves also define an adaption region for each sub-channel. Particularly, the adaption region provides a base for the iterative, private rate increment method utilized by our secret key rate adaption algorithm. The iterative sub-channel selection procedure depends on the actual target private rate and the noise levels of the sub-channels. The scheme provably yields a minimized error rate transmission for all sub-channels while achieving the selected target secret key rate. We demonstrate the results through the framework of AMQD and also extend the results to the multiple-access multicarrier CVQKD. This paper is organized as follows. In Section 2, preliminary findings are summarized. Section 3 discusses the iterative secret key adaption scheme. Section 4 extends the results to a mul2

tiuser setting. Finally, Section 5 concludes the results. A numerical evidence is included in the Supplemental Information.

2 Preliminaries In Section 2, the notations and basic terms are summarized. For further information, see the detailed descriptions of [2–10].

2.1

Multicarrier CVQKD

The following description assumes a single user, and the use of n Gaussian sub-channels  i for the transmission of the subcarriers, from which only l sub-channels will carry valuable information. In the single-carrier modulation scheme, the j-th input single-carrier state j j = x j +ip j is a Gaussian state in the phase space  , with i.i.d. Gaussian random position and momentum quadratures

(

x j Î  0, sw2

0

), p

j

(

Î  0, sw2

0

),

(1)

where sw2 is the modulation variance of the quadratures. In the multicarrier scenario, the in0

formation is carried by Gaussian subcarrier CVs, fi = x i +ipi , via quadratures

x i Î  ( 0, sw2 ) , pi Î  ( 0, sw2 ) ,

(2)

where sw2 is the modulation variance of the subcarrier quadratures, which are transmitted through a noisy Gaussian sub-channel  i . Precisely, each  i Gaussian sub-channel is dedicated for the transmission of one Gaussian subcarrier CV from the n subcarrier CVs. (Note: index i refers to a subcarrier CV, index j to a single-carrier CV, respectively.) The single-carrier CV state j j in  can be modeled as a zero-mean, circular symmetric com-

æ ö plex Gaussian random variable z j Î  ççç 0, sw2 ÷÷÷ , with a variance zj ø è 2ù é sw2 =  ê z j ú = 2sw2 , zj 0 ë û and with i.i.d. real and imaginary zero-mean Gaussian random components

(

Re ( z j ) Î  0, sw2

0

) , Im ( z

j

) Î  ( 0, sw2

0

).

(3)

(4)

In the multicarrier CVQKD scenario, let n be the number of Alice’s input single-carrier Gaussian states. The n input coherent states are modeled by an n-dimensional, zero-mean, circular symmetric complex random Gaussian vector T

z = x + ip = ( z 0 , , z n -1 ) Î  ( 0, Kz ) ,

3

(5)

where each z j is a zero-mean, circular symmetric complex Gaussian random variable

æ z j Î  ççç 0, sw2 zj è

÷÷ö , z = x + ip . j j ÷ø j

(6)

In the first step of AMQD, Alice applies the inverse FFT (fast Fourier transform) operation to vector z (see (5)), which results in an n-dimensional zero-mean, circular symmetric complex T

Gaussian random vector d , d Î  ( 0, Kd ) , d = (d0 , , dn -1 ) , precisely as d=F

-1

(z) = e

dT AAT d 2

(

=e

where

s2 d02 ++dn2 -1 w0

(

) ,

2

(7)

)

di = xd + ipd , di Î  0, sd2 , i

i

i

(8)

é 2ù where sw2 =  ê di ú = 2sw2 , thus the position and momentum quadratures of fi di ë û

are i.i.d.

Gaussian random variables with a constant variance sw2 for all  i , i = 0, , l - 1 sub-channels:

Re ( di ) = xd Î  ( 0, sw2 ) , Im ( di ) = pd Î  ( 0, sw2 ) , i

(9)

i

where Kd =  éê dd† ùú ,  éë d ùû =  éê e ig d ùú = e ig éë d ùû ,  éê ddT ë ë û ë û

ù =  éê e ig d (e ig d )T úû êë

ù ú = e i2 g úû

é ddT ù for êë úû

any g Î éë 0, 2p ùû . The T (  ) transmittance vector of  in the multicarrier transmission is T T (  ) = éëT0 (  0 ) , ,Tn -1 (  n -1 ) ûù Î  n ,

(10)

Ti (  i ) = Re (Ti (  i ) ) + i Im (Ti (  i ) ) Î  ,

(11)

where

is a complex variable, which quantifies the position and momentum quadrature transmission (i.e., gain) of the i-th Gaussian sub-channel  i , in the phase space  , with real and imaginary parts

0 £ Re Ti (  i ) £ 1

2 , and 0 £ Im Ti (  i ) £ 1

2.

(12)

Particularly, the Ti (  i ) variable has the squared magnitude of

Ti (  i )

2

2

2

= ReTi (  i ) + Im Ti (  i ) Î  ,

(13)

Re Ti (  i ) = Im Ti (  i ) .

(14)

where The Fourier-transformed transmittance of the i-th sub-channel  i (resulted from CVQFT operation at Bob) is denoted by 2

F (Ti (  i ) ) . 4

(15)

The

n-dimensional

zero-mean,

circular

symmetric

complex

Gaussian

noise

vector

2 D Î  ( 0, sD ) , of the quantum channel  , is evaluated as n

T

D = ( D0 , , Dn -1 ) Î  ( 0, KD ) ,

(16)

where

KD =  éê DD† ùú , û ë with independent, zero-mean Gaussian random components

(

)

(17)

(

Dx Î  0, s 2 , and Dp Î  0, s 2 i

i

i

i

),

(18)

with variance s 2 , for each Di of a Gaussian sub-channel  i , which identifies the Gaussian i

noise of the i-th sub-channel  i on the quadrature components x i , pi in the phase space  .

(

2 Thus F ( D ) Î  0, sD

i

) , where

2 sD = 2s 2 . i

(19)

i

The CVQFT-transformed noise vector can be rewritten as T

F ( D ) = ( F ( D0 ) , , F ( Dn -1 ) ) , with independent components F ( Dx

i

) Î  ( 0, s2 ) i

and F ( Dp

i

(20)

) Î  ( 0, s2 )

on the quadra-

i

tures, for each F ( Di ) .

2.1.1

Multiuser Quadrature Allocation (MQA)

In a MQA multiple access multicarrier CVQKD, a given user U k , k = 0, , K - 1 , where K is the number of total users, is characterized via m subcarriers, formulating an U logical chank

nel of U k ,

where U

k ,i

T (21) U = éê U ,0 , , U ,m-1 ùú , k ë k k û is the i-th sub-channel of U . For a detailed description of MQA for multicarrier k

CVQKD see [3]. The general model of AMQD-MQA is depicted in Fig. 1 [3], [5].

5

AMQD

z1 Alice 1

 zK Alice K

CV singlecarriers

CV subcarriers

j1

f1

 jK

U -1

 Rate selection

( CVQFT† )

j1¢

f1¢





(AWGN)

fn



U



fn¢

( CVQFT )

jK¢

Bob 1

 Bob K

Figure 1. The AMQD-MQA multiple access scheme with multiple independent transmitters and multiple receivers [3]. The modulated Gaussian CV single-carriers are transformed by a unitary operation (inverse CVQFT) at the  encoder, which outputs the n Gaussian subcarrier CVs. The parties send the jk single-carrier Gaussian CVs with variance sw20,k to Alice. In the rate-selection phase, the encoder determines the transmit users. The data states of the transmit users are then fed into the CVQFT† operation. The fi

Gaussian subcarrier CVs have a vari-

ance sw2 per quadrature components. The Gaussian CVs are decoded by the CVQFT unitary operation. Each jk¢

2.2

is received by Bob k (AWGN – additive white Gaussian noise).

Private Classical Rate Curves

The secret key adaption method utilizes r + 2 rate curves for the sub-channels, defined via set (22)  = { Rmin (  i ) , R (q ) , Rmax (  i )} , q = 0  r - 1 . Specifically, for all sub-channels  i , i = 0, , l - 1 a given rate curve is selected from  (22) according to the sub-channel conditions. In particular, a R ( q ) rate curve refers to the transmission rate of private classical information (private rate) over  i , with the relation Rmin (  i ) < R ( 0 )  < R ( r - 1 ) < Rmax (  i ) ,

(23)

where R ( q ) is referred to as the target private rate at an R (q - 1 ) actual private rate in an iteration procedure. Without loss of generality, assuming a reverse reconciliation, a given target private rate R ( q ) is defined as

R (q ) £ P (  i ) 1 = lim max ( cAB (  i ) - cBE (  i ) ) , n ¥ n "pi , ri

(24)

where P (  i ) is the private classical capacity of  i , cAB (  i ) and cBE (  i ) are the Holevo information of Alice (transmitter) and Bob (receiver), and Bob and Eve (eavesdropper), respectively.

6

3 Secret Key Rate Adapting with Minimized Error Rate Theorem 1. A given target secret key rate S * (  ) =

l -1

å i =0 Ri (  i ) ,

where Ri (  i ) is the

private rate of sub-channel  i , can be achieved over the l sub-channels such that the error rate of all sub-channels is minimized. Proof. The proof focuses on a single sub-channel  i for the private transmission of a single quadrature component x i (or pi ), which refers to a x i Î  ( 0, sw2 ) position or a pi Î  ( 0, sw2 ) momentum quadrature of the i-th subcarrier, respectively. Let

ni (  i ) = s 2

F (Ti (  i ) )

i

2

(25)

of the i -th, i = 0, , l - 1 sub-channel  i . Let Rmin (  i ) and Rmax (  i ) be the minimal and maximal private classical information transmission rates selected for  i . The private rates are referring to the transmission of a given quadrature x i . These rate curves, R ( q ) > R ( q - 1 ) , allow us to reach a target secret key rate S * ( 

)

over

the sub-channels with a minimized error rate in a multicarrier CVQKD setting. Specifically, it requires a rigorously defined iterative condition on the selection of the sub-channels for each target rate R ( q ) . From Rmin (  i ) and Rmax (  i ) , an adaption region A can be characterized with r rate curves inside the region. Define r rate curves for the transmission of x i in the region of A

A = éë Rmin (  i ) , Rmax (  i ) ùû ,

(26)

as R (q ) , q = 0 r - 1 ,

(27)

such that (23) holds. Precisely, at a given private rate R (  i ) , (25) is referred to as ni ( R (  i ) ) ,

ni ( R (  i ) ) = sR2 where sR2

( i )

2

( i )

F (T ( R (  i ) ) ) ,

(28)

is the noise variance of  i at R (  i ) , while F (T ( R (  i ) ) ) is the transmittance

coefficient of  i at R (  i ) . Note that in function of (28), after a scaling the rate curves of (27) are almost parallel to Rmin (  i ) . By theory, at R (  i ) = 0 (28) is directly defined from s 2 is i

the noise variance and Ti (  i ) is the transmittance coefficient of  i , as given in (25). In function of ni ( R (  i ) ) , the SNR of  i at a given R (  i ) is expressed as [17] 7

SNR ( R (  i ) ) = 10 log10

1 . ni ( R (  i ) )

(29)

To step forward, we have to focus on the behavior of parameter ni ( ⋅ ) at an increased transmission rate. Particularly, let R  (  i ) be the current private rate and R (  i ) be the target private rate for a  sub-channel  i , such that R  (  i ) < R (  i ) < R (  i ) . Then let dn ( R (  i ) ) identify the i

cumulative ni ( R (  i ) ) parameter of  i at an increased (target) private rate R (  i ) , evaluated via the following iteration:  dn ( R (  i ) ) = dn ( R  i ) ) + Dn ( R (  i ) , R (  i ) ) , (  i i i  where Dn identifies the difference of ni at R (  i ) and R (  i ) as i  Dn = ni ( R (  i ) ) - ni ( R (  i ) ) , i

(30)

(31)

the iteration (30) at no transmission, R (  i ) = 0 , identifies ni (  i ) as

dn ( R (  i ) = 0 ) = ni (  i ) , i

(32)

while for any R (  i ) > R  ( i ) > 0 ,

ni ( R (  i ) ) < ni ( R  ( i )) .

(33)

To conclude, from (30) follows that a rate increment from R  (  i ) to R (  i ) also increases  dn ( R  (  i ) ) by Dn ( R (  i ) , R (  i ) ) , thus for any R (  i ) > R  (  i ) , the following relation i

i

holds [17]:

dn ( R (  i ) ) > dn ( R  ( i )) . i i

(34)

Let us then define r private transmission curves in the adaption region A of  i . The aim of the iterative secret key adaption scheme is to provide a rate increment in each step by selecting that sub-channel  i , for which (30) is minimal. Specifically, it is a convenient approach because this sub-channel provides the best condition for the transmission.  As we show, by using this sub-channel, the increased rate R can be achieved with a minimized error rate, but at the same time, it keeps the target secret key rate. Therefore, applying the subchannel selection procedure with respect to the iterative condition of (30), a desired target secret key rate can be achieved such that the transmission is adapted to not just the sub-channel conditions, but also to yield a minimized error rate for all sub-channels. Applying (30) for a Rmin (  i ) private rate is as follows. An Rmin (  i ) target private rate over

 i , dn ( Rmin (  i ) ) is yielded via ni (  i ) and Dn ( Rmin (  i ) , R ( 0 ) ) derived from R ( 0 ) Î A i

i

as

dn ( Rmin (  i ) ) = ni (  i ) + Dn ( Rmin (  i ) , R ( 0 ) ) . i

i

(35)

Applying (30) to the r rate curves R ( q ) Î A , q = 0 r - 2 at a target rate R ( q ) results in

dn ( R ( q ) ) as i

8

dn ( R (q ) ) = dn ( R (q - 1 ) ) + Dn ( R ( q ) , R ( q + 1 ) ) , i

i

(36)

i

Specifically, in each step for a given target R ( q ) , the method selects that  i , for which (36) is minimal, because that sub-channel provides the best conditions. Thus, the secret key adaption is an iterative process and depends on the dn ( R ( q - 1 ) ) parameter obtained at R ( q - 1 ) and on i

Dn ( R ( q ) , R ( q + 1 ) ) . i

In particular, for q = 0 , (36) yields dn ( R ( 0 ) ) = dn ( Rmin (  i ) ) + Dn ( R ( 0 ) , R ( 1 ) ) i

i

i

(37)

= ni (  i ) + Dn ( Rmin (  i ) , R ( 0 ) ) + Dn ( R ( 0 ) , R ( 1 ) ) , i

i

where R ( 1 ) Î A . While, for q = r - 1 , (30) results in

dn ( R ( r - 1 ) ) = dn ( R ( r - 2 ) ) + Dn ( R ( r - 1 ) , Rmax (  i ) ) i i i r -2 æ ö÷ çç = ç ni (  i ) + Dn ( Rmin (  i ) , R ( 0 ) ) + å Dn ( R ( k ) , R ( k + 1 ) ) ÷÷÷ (38) i i çè ø÷ k =0 +Dn ( R ( r - 1 ) , Rmax (  i ) ) , i

where R ( k ) Î A , k = 0  r - 2 . For Rmax (  i ) , by definition dn ( Rmax (  i ) ) = +¥ [17]. i

The distribution of a sample set dn ( R ( q ) ) for m sub-channels, i = 0, , m - 1 , in a low-SNR i

CVQKD scenario is illustrated in Fig. 2(a). The SNR in Fig. 2(b) is derived from the dn ( R (q ) ) i

set of the m sub-channels as SNR ( ni ( R (q ) ) ) = 10 log10 ( 1 ( ni ( R (q ) ) ) ) . dn ( R ( q ) ) 1

10

0,8

8

0,6

6

SNR

Value

i

0,4

4

0,2

2

0

0

0

200

400

600

800

1000

0

Data unit index (a)

200

400

600

800

1000

Data unit index (b)

Figure 2. The distribution of a low-SNR set dn ( R (q ) ) , i = 0, , m - 1 , for m = 1000 subi

channels (a). The SNR is derived from dn ( R ( q ) ) for all sub-channels (b). i

Next, we show that for an arbitrary target private rate R (  i ) > 0 , the iterative condition on

dn ( R (  i ) ) (see (36)) provides a minimized error rate over the selected  i . i

9

æ ö Let  be the bit error rate, and let  çç R ( q )d R q -1 ÷÷÷ refer to the bit error rate of  i at tar( ( ) ) è ø ni get private rate R ( q ) , at an actual rate R ( q - 1 ) , and dn ( R ( q - 1 ) ) . Then, at a given R ( q ) , i

selecting that  i , i = 0, , l - 1 from the total l, for which dn ( R ( q - 1 ) ) is minimal, as i

x ( R ( q - 1 ) ) = min dn ( R (q - 1 ) ) "i

(39)

i

yields a minimized bit error rate at a given R ( q ) over the selected  i as

(

 R ( q )x

( R(q -1 ) )

) =  çèæç R

min

ö÷ ,

(  i )d

÷÷ ni ( R ( q -1 ) ) ø

(40)

æ ö where  çç Rmin (  i )d R q -1 ÷÷÷ is defined as )) ø è ni ( ( æ ö÷ ç  çç Rmin (  i )n  +æçç D R  ,R 0 + q -1 D R k ,R k +1 ö÷÷ ÷÷÷ ) )÷÷÷ ÷ ç ni ( min ( i ) ( ) ) å ni ( ( ) ( i( i ) ç (41) çèç çè øø k =0 = 21 erfc  ( ni (  i ) + ( Dn ( Rmin (  i ) , R ( 0 ) ) +  + Dn ( R ( q - 1 ) , R ( q ) ) ) ), i

i

where function  ( ⋅ ) is evaluated as

 ( ni (  i ) + ( Dn ( Rmin (  i ) , R ( 0 ) ) +  + Dn ( R (q - 1 ) , R (q ) ) ) ) i i æ SNR ( n (  ) ) - é SNR ( n ( R ( 0 ) ) ) - SNR ( n ( R (  ) ) ) ù ö÷ çç i i i i min i ë û ÷÷ ç ÷÷ , æ q -1 ö÷ = çç ÷÷ çç - ççç å é SNR ( ni ( R ( k + 1 ) ) ) - SNR ( ni ( R ( k ) ) ) ù ÷÷ ÷÷ û ÷÷ çç çè k = 0 ë ÷ø ø è

(42)

where SNR ( ni ( R ( x ) ) ) = 10 log10 ( 1 ( ni ( R ( x ) ) ) ) , while erfc ( ⋅ ) is the complementary error function

erfc ( x ) =

2 p

¥

òx

2

e -t dt .

(43)

The aim of the error minimization procedure is to achieve (40) for all R (⋅ ) via the selection of that  i for which dn ( R ( q - 1 ) ) is minimal. i

Let R ( k + 1 )d

ni

( R( k ) )

refer to a private rate R ( k + 1 ) over at dn ( R ( k ) ) with respect to the i

transmission of a single quadrature component x i (or pi ). First, we apply (39) to the minimal rate Rmin (  i ) by the selection of that  i , for which

ni (  i ) is minimal, thus x ( 0 ) = min dn ( 0 ) = min ni , "i

which yields

(

 Rmin (  i )x

(0)

(44)

"i

i

) =  (R

min

(  i )n (  ) ). i

(45)

i

Similarly, at R ( 0 ) , that sub-channel is selected for the R ( 0 ) rate transmission, for which

dn ( Rmin (  i ) ) is minimal. i

10

In particular, due to the iterative determination of dn ( Rmin (  i ) ) , the corresponding

(

 R ( 0 )x

( Rmin (  i ) )

i

) is yielded as æ  (R ( 0) =  çç R ) ( ( )) è

(  i )n (  )+D

min

x Rmin  i

i

i

ni

÷ö .

( Rmin (  i ),R ( 0 ) ) ÷÷ø

(46)

Precisely, for R ( q ) , q = 1  r - 1 , therefore, in each step, that sub-channel selected for the

(

transmission, for which x ( R ( q - 1 ) ) is minimal, ensures that the resulting  R ( q )x

( R(q -1 ) )

) is

evaluated as

(

 R ( q )x

( R(q -1 ) )

) =  æççè R

min

ö÷ .

(  i )d

ni

( R (q -2 ) )+Dni ( R (q -1 ),R(q ) ) ø÷÷

(47)

Putting the pieces together, the utilization of (47) for q = 1 is as æ ö  R ( 1 )x R 0 =  çç Rmin (  i )d R  +D R 0 ,R 1 ÷÷÷ , ( ( )) è ni ( min ( i ) ) ni ( ( ) ( ) ) ø

(

)

(48)

while for q = r - 1 ,

(

 R ( r - 1 )x and finally, for Rmax (  i )x

(

( R( r -2

( R ( r -1 ) )

 Rmax (  i )x

æ =  çç R ) )) è

min

÷÷ö , ÷ ni ( R ( r - 3 ) )+Dni ( R ( r -2 ),R ( r -1 ) ) ø

(  i )d

(49)

, the corresponding error rate is

( R( r -1 ))

) =  æççè R

min

(  i )d

ni

ö÷ .

( R( r -2 ) )+Dni ( R ( r -1 ),Rmax (  i ) ) ø÷÷

(50)

The  bit error rates at private rates, Rmin (  i ) , R ( 0 ) , , Rmin ( r - 1 ) , Rmax (  i ) , for a given sub-channel  i in function of ni (low-SNR scenario) are summarized in Fig. 3 for the range

ni = éë 0.1, 0.3 ùû (a), and ni = éë 0.3, 0.9 ùû (b).

Figure 3. The  bit error rates at Rmin (  i ) , R ( 0 ) , , R ( r - 1 ) , Rmax (  i ) in a low-SNR scenario for a given sub-channel  i at ni = éë 0.1, 0.3 ùû (a) and ni = éë 0.3, 0.9 ùû (b). The iterative secret key adapting method with the error minimization is summarized as follows.

11

Algorithm (Iterative Secret Key Adapting) 1. Let Rmin (  i ), Rmax (  i ) be the minimal and maximal private transmission rates selected for all sub-channels  i , i = 0, , l - 1 . 2. For a given  i , let ni (  i ) = s 2

F (T (  i ) )

i

2

at noise variance

s 2 and transmission coefficient F (Ti (  i ) ) . Define r curves R ( q ) , i

q = 0 r - 1 in the adaption region A = éë Rmin (  i ), Rmax (  i ) ùû , with relation Rmin (  i ) < R ( 0 ) < R ( r - 1 ) < Rmax (  i ) .

3. For target private rate Rmin (  i ) , determine dn ( Rmin (  i ) ) = ni (  i ) + Dn ( Rmin (  i ), R ( 0 ) ) . i

i

dn ( Rmin (  i ) ) as i

For

R (q )

the

q = 0 r - 2 curves of the adaption region A , compute dn ( R (q ) ) as i

dn ( R (q ) ) = dn ( R ( q - 1 ) ) + Dn ( R ( q ), R ( q + 1 ) ) . At q = r - 1 , use i

i

i

dn ( R ( r - 1 ) ) = dn ( R ( r - 2 ) ) + Dn ( R ( r - 1 ), Rmax (  i ) ) . i

i

At

i

Rmax (  i ) , use dn ( Rmax (  i ) ) = +¥ . i

4. Utilize the adaption method: At a target rate R ( q ) , q = 0 r - 1 , select that sub-channel  i for which dn ( R ( q - 1 ) ) is minimal. At i

Rmin (  i ) , select that  i for which ni (  i ) is minimal. At R ( 0 ) select  i for which dn ( Rmin (  i ) ) is minimal, while at Rmax (  i ) , sei

lect  i for which dn ( R ( r - 1 ) ) is minimal. i

5. Repeat the steps until S * (  ) £

l -1

å i = 0 Ri (  i ) ,

where S * ( 

)

is the

desired secret key rate over the l sub-channels, and Ri (  i ) is the private transmission rate of  i at a minimized  for all  i .

■ The resulting  bit error rates of the secret key rate adapting method for a given sub-channel  i in function of ni are summarized in Fig. 4. Parameter ni is scaled for the SNR with 5 dB -SNR ( ni ( R ( x ) ) ) 10 steps in the range of éë 15, -5 ùû as ni ( Ri ( x ) ) = 10

12

Figure 4. The  bit error rate of secret key adapting for a sub-channel  i in function of -SNR ( ni ( R ( x ) ) ) 10

ni ( Ri ( x ) ) = 10

. The adaption is made via r rate curves R ( q ) , q = 0  r - 1 ,

Rmin (  i ) < R ( 0 )  < R ( r - 1 ) < Rmax (  i ) in the A = éë Rmin (  i ) , Rmax (  i ) ùû adaption region (shaded area).

4 Multiuser Multicarrier CVQKD Scenario This section extends the results for a multiuser multicarrier CVQKD scenario. Lemma 1. The secret key adaption can be extended to a U k , k = 0, , K - 1 multiuser setting, where K is the number of users, to achieve target secret key rate S * ( U

k

)

with minimized er-

ror rate over the m sub-channels of U of U k , for "k . k

Proof. T The proof focuses on a given logical channel U = éê U ,0 , , U ,m-1 ùú of a user U k , where k ë k k û U ,i , i = 0, , m - 1 is the i-th sub-channel. k

Let K be the number of transmit users, and select a given U k , k = 0, , K - 1 . Let S * ( U be the target secret key rate of users over U , and let R ( U k

The steps of the extension are summarized as follows.

13

k ,i

) be the private rate of

U

k

k ,i

) .

Let

ni ( U where s 2

U k ,i

U

k ,i

k ,i

ni ( R ( U

(

R U

) = s2

is the noise variance of U

. At a given rate curve R ( U

where s 2

k ,i

k ,i

k ,i

U k ,i

k ,i

, while Ti ( U

coefficient of U

k ,i

at R ( U

the set U of m U k

k ,i

k ,i

k ,i

k ,i

)

))

2

,

(51)

is the transmittance coefficient of

) > 0, )) = sR2 ( 

U k ,i

is the noise variance at R ( U

)

F (T ( U

k ,i

)

),

F (T ( R ( U

k ,i

)))

while Ti ( R ( U

k ,i

2

,

(52)

))

is the transmittance

) , respectively. Apply the secret key rate adaption method over

, sub-channels of U , until k

m -1

S

*

( U ) £ å R ( U ,i ) . k

(53)

k

i =0

Apply the steps for all transmit users U k for their U sets of m sub-channels. Therefore, one k

can utilize (47) at a given R ( U and from (32) follows dn

i

) = Ri (q ) , where Ri (q ) refers to the R (q ) curve of U ,i , ( R ( U ,i ) = 0 ) = ni ( U ,i ) , which yields the error rate for U ,i of k ,i

k

k

k

k

U as k

æ ö÷ ç æ ÷ö÷ =  çç R (  q -1 æç ö÷ ÷ ÷  çç R ( U ,i ) . (54) ) U k ,i n  ç Dn ( Rmin ( U ,i ),Ri ( 0 ) )+ å Dn ( Ri ( k ),Ri ( k +1 ) )÷÷ ÷ ç min k x ( R ( q -1 ) ) ÷ è ø i ( U k ,i )+ç ÷÷ ÷ ÷ k i çè ççè i øø k =0 ■

4.1

Variance Adaption for an Equalized Error Rate for Users

In this section, we propose a modulation variance adaption method to achieve an equally minimized error rate for the sub-channels of a given user. The results can be extended to an arbitrary number of users. Theorem 2. For all U k , k = 0, , K - 1 , the error rate of the U channels of U

k

k ,i

, i = 0, , m - 1 sub-

of user U k can be equally minimized via the a sw2 = sw2 + Ds2 modulation wi

variance correction, where Ds 2 > 0 . wi

Proof. Let

R  ( U k ,i ) = Ri (q - 1 ) , q = 0, , r - 1 ,

14

(55)

and R ( U

k ,i

) be the target private rate, R ( U ,i ) = Ri (q ) ,

(56)

k

where Ri (q ) refers to the R ( q ) curve of U identify Ri ( -1 ) = Rmin ( U

k ,i

, with relation to Ri (q - 1 ) < Ri (q ) , and let us

) and Ri ( r ) = Rmax ( U ,i ) . R ( U ,i ) , the minimal dn ( R  ( U ,i ) )

k ,i

Precisely, for a given U k at

k

k

i

parameter for the U

k

k ,i

,

i = 0, , m - 1 sub-channels of the set U is evaluated as k

xU = min dn ( R  ( U k ,i ) ) . k i "i Î

(57)

Uk

Let sw2 refer to the input modulation variance of the i-th subcarrier of U k . Specifically, using the expression of xU in (57), the sw2 modulation variance of the i-th subcarrier is corrected by k

Ds2 as wi

Ds2 = dn ( R  ( U k ,i ) ) - xU k , i w

(58)

i

yielding a modulation variance increment

sw2 = sw2 + Ds2

(59)

wi

for the input of U

k ,i

.

In particular, using (59), the jd

ni

R ( U

k ,i

( R ( U ,i ) ) k

resulting dn parameter for U i

at a target rate

) is therefore jd

ni

( R ( U ,i ) ) = dn ( R ( U ,i )) - xU k

i

k

( ( R (  ))) SNR increment of

from which the DSNR jd

U k ,i

ni

is as

k ,i

U

k

k ,i

,

(60)

at a given R ( U

k ,i

) = R (q )

( ( R (  )))

DSNR jd

ni

= 10 log10

U k ,i

1

xU

(61)

k

- ( ni ( U

k ,i

) + ( Dn ( Rmin ( U ,i ), R ( 0 ) ) +  + Dn ( R (q ), R (q + 1)) ) ), i

k

i

where  (⋅ ) is specified in (42).

æ Therefore, the  çç R ( U ,i ) k çè xU k

÷÷ö error rate for all  is U k ,i at an arbitrary R ( U k ,i )d R  ÷ø ni (  ( U k ,i ) )

equally minimized by xU via (41) as k

15

æ ö  çç R ( U ,i ) ÷÷÷ k çè xU ø k æ ÷÷ö = min  çç R ( U ,i ) k dn ( R "i ÎU ( Uk ,i )) ÷ø èç i  k æ ö =  çç Rmin ( U ,i ) ÷÷÷ , i = 0, , m - 1. k çè  dn ( R ( U k ,i ) ) ø i  The formula of (62) proves the minimal error rate at arbitrary S * ( U

(62)

k

) over

U for all U k

k ,i

sub-channels of U k . Without loss of generality, the results can be extended for all K users to achieve minimized equalized error rate over all U , k = 0, , K - 1 logical channels. k

In a single user setting, e.g., K = 1 , the method provides an equal, minimized error rate over the l sub-channels. ■ A numerical evidence is included in the Supplemental Information.

5 Conclusions We defined an iterative secret key adaption method for multicarrier CVQKD. The scheme provides a minimized error rate using the utilization of an adaptive private classical information transmission through the sub-channels. The private classical transmission is realized through pre-defined private rate curves, which characterize an adaption region for each sub-channel to find the best conditions for the transmission at a given private classical rate. The method allows us to reach a given target secret key rate with optimal transmit conditions and minimized error rate for all sub-channels. The scheme requires no complex calculations or sophisticated computational tools, allowing for easy implementation for experimental CVQKD scenarios.

Acknowledgements This work was partially supported by the GOP-1.1.1-11-2012-0092 (Secure quantum key distribution between two units on optical fiber network) project sponsored by the EU and European Structural Fund, by the Hungarian Scientific Research Fund - OTKA K-112125, and by the COST Action MP1006.

References [1] [2]

S. Imre and L. Gyongyosi. Advanced Quantum Communications - An Engineering Approach. Wiley-IEEE Press (New Jersey, USA), (2012). L. Gyongyosi, Adaptive Multicarrier Quadrature Division Modulation for Continuousvariable Quantum Key Distribution, arXiv:1310.1608 (2013). 16

[3] [4] [5] [6] [7] [8] [9] [10] [11]

[12] [13] [14] [15] [16] [17] [18] [19]

[20] [21]

[22]

L. Gyongyosi, Multiuser Quadrature Allocation for Continuous-Variable Quantum Key Distribution, arXiv:1312.3614 (2013). L. Gyongyosi, Singular Layer Transmission for Continuous-Variable Quantum Key Distribution, arXiv:1402.5110 (2014). L. Gyongyosi, Security Thresholds of Multicarrier Continuous-Variable Quantum Key Distribution, arXiv:1404.7109 (2014). L. Gyongyosi, Multidimensional Manifold Extraction for Multicarrier ContinuousVariable Quantum Key Distribution, arXiv:1405.6948 (2014). L. Gyongyosi, Subcarrier Domain of Multicarrier Continuous-Variable Quantum Key Distribution, arXiv:1406.6949 (2014). L. Gyongyosi, Adaptive Quadrature Detection for Multicarrier Continuous-Variable Quantum Key Distribution, arXiv:1408.6493 (2014). L. Gyongyosi, Distribution Statistics and Random Matrix Formalism for Multicarrier Continuous-Variable Quantum Key Distribution, arXiv:1410.8273 (2014). L. Gyongyosi, Gaussian Quadrature Inference for Multicarrier Continuous-Variable Quantum Key Distribution, arXiv:1504.05574 (2015). L. Gyongyosi, S. Imre, Geometrical Analysis of Physically Allowed Quantum Cloning Transformations for Quantum Cryptography, Information Sciences, Elsevier, pp. 123, DOI: 10.1016/j.ins.2014.07.010 (2014). S. Pirandola, R. Garcia-Patron, S. L. Braunstein and S. Lloyd. Phys. Rev. Lett. 102 050503. (2009). S. Pirandola, A. Serafini and S. Lloyd. Phys. Rev. A 79 052327. (2009). S. Pirandola, S. L. Braunstein and S. Lloyd. Phys. Rev. Lett. 101 200504 (2008). C. Weedbrook, S. Pirandola, S. Lloyd and T. Ralph. Phys. Rev. Lett. 105 110501 (2010). C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. Ralph, J. Shapiro, and S. Lloyd. Rev. Mod. Phys. 84, 621 (2012). H. Rohling, C. Fellenberg, Successive Bit Loading Concept, OFDM Concepts for Future Communication Systems, Springer (2011). L. Gyongyosi, Scalar Reconciliation for Gaussian Modulation of Two-Way Continuous-variable Quantum Key Distribution, arXiv:1308.1391 (2013). P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, E. Diamanti, Experimental demonstration of long-distance continuous-variable quantum key distribution, arXiv:1210.6216v1 (2012). M. Navascues, F. Grosshans, and A. Acin. Optimality of Gaussian Attacks in Continuous-variable Quantum Cryptography, Phys. Rev. Lett. 97, 190502 (2006). R. Garcia-Patron and N. J. Cerf. Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution. Phys. Rev. Lett. 97, 190503 (2006). F. Grosshans, Collective attacks and unconditional security in continuous variable quantum key distribution. Phys. Rev. Lett. 94, 020504 (2005). 17

[23] M R A Adcock, P Høyer, and B C Sanders, Limitations on continuous-variable quantum algorithms with Fourier transforms, New Journal of Physics 11 103035 (2009) [24] S. Pirandola, S. Mancini, S. Lloyd, and S. L. Braunstein, Continuous-variable Quantum Cryptography using Two-Way Quantum Communication, arXiv:quantph/0611167v3 (2008). [25] L. Hanzo, H. Haas, S. Imre, D. O'Brien, M. Rupp, L. Gyongyosi. Wireless Myths, Realities, and Futures: From 3G/4G to Optical and Quantum Wireless, Proceedings of the IEEE, Volume: 100, Issue: Special Centennial Issue, pp. 1853-1888. (2012). [26] D. Tse and P. Viswanath. Fundamentals of Wireless Communication, Cambridge University Press, (2005). [27] D. Middlet, An Introduction to Statistical Communication Theory: An IEEE Press Classic Reissue, Hardcover, IEEE, ISBN-10: 0780311787, ISBN-13: 978-0780311787 (1960) [28] S. Kay, Fundamentals of Statistical Signal Processing, Volumes I-III, Prentice Hall, (2013) [29] O. S. Jahromi, Multirate Statistical Signal Processing, ISBN-10 1-4020-5316-9, Springer (2007). [30] G. Heinzel, A. Rudiger, R. Schilling, Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new at-top windows, http://hdl.handle.net/11858/00-001M-0000-0013557A-5 (2002). [31] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, ISBN : 0-521-43108-5, Cambridge University Press (1993). [32] S. Imre, F. Balazs: Quantum Computing and Communications – An Engineering Approach, John Wiley and Sons Ltd, ISBN 0-470-86902-X, 283 pages (2005). [33] D. Petz, Quantum Information Theory and Quantum Statistics, Springer-Verlag, Heidelberg, Hiv: 6. (2008). [34] R. V. Meter, Quantum Networking, John Wiley and Sons Ltd, ISBN 1118648927, 9781118648926 (2014). [35] L. Gyongyosi, S. Imre: Properties of the Quantum Channel, arXiv:1208.1270 (2012). [36] K Wang, XT Yu, SL Lu, YX Gong, Quantum wireless multihop communication based on arbitrary Bell pairs and teleportation, Phys. Rev A, (2014). [37] Babar, Zunaira, Ng, Soon Xin and Hanzo, Lajos, EXIT-Chart Aided Near-Capacity Quantum Turbo Code Design. IEEE Transactions on Vehicular Technology (submitted) (2014). [38] Botsinis, Panagiotis, Alanis, Dimitrios, Ng, Soon Xin and Hanzo, Lajos LowComplexity Soft-Output Quantum-Assisted Multi-User Detection for Direct-Sequence Spreading and Slow Subcarrier-Hopping Aided SDMA-OFDM Systems. IEEE Access, PP, (99), doi:10.1109/ACCESS.2014.2322013 (2014).

18

[39] Botsinis, Panagiotis, Ng, Soon Xin and Hanzo, Lajos Fixed-complexity quantumassisted multi-user detection for CDMA and SDMA. IEEE Transactions on Communications, vol. 62, (no. 3), pp. 990-1000, doi:10.1109/TCOMM.2014.012514.130615 (2014). [40] L. Gyongyosi, S. Imre, Adaptive multicarrier quadrature division modulation for longdistance continuous-variable quantum key distribution, Proc. SPIE 9123, Quantum Information and Computation XII, 912307; doi:10.1117/12.2050095, From Conference Volume 9123, Quantum Information and Computation XII, Baltimore, Maryland, USA (2014). [41] L. Gyongyosi: Statistical Quadrature Evolution by Inference for Continuous-Variable Quantum Key Distribution, arXiv:1603.09247 (2016). [42] L. Gyongyosi, S. Imre, Gaussian Quadrature Inference for Multicarrier ContinuousVariable Quantum Key Distribution, SPIE Quantum Information and Computation XIV, 17 - 21 Apr 2016, Baltimore, Maryland, USA (2016).

19

Supplemental Information S.1 Numerical Evidence This section proposes numerical evidence to demonstrate the results through a multiuser multicarrier CVQKD environment (AMQD-MQA [3]). The numerical evidence serves demonstration purposes.

S.1.1

Parameters

To demonstrate the results of Section 4.1, let U k be a given user with m sub-channels. The parameters of the numerical evidence are summarized as follows. The single-carrier inputs of user U k ,

xU

k,j

(

Î  0, sw2

0

),

(S.1)

 have a modulation variance of sw2 and formulate a d -dimensional input vector xU . k

0

The j-th single-carrier is dedicated to a single-carrier channel U transmittance coefficient is depicted by T ( U

k ,j

),

k,j

. The single-carrier channel

j = 0, , d - 1 , where d is the dimension of

the input vector. The single-carriers are granulated into m subcarriers, where the i-th subcarrier is

xU

Î  ( 0, sw2 ) ,

(S.2)

, i = 0, , m - 1 ,

(S.3)

k ,i

and has a modulation variance of sw2 . The m sub-channels,

U

k ,i

formulate the U logical channel of user U k , k

(

The Dx Î  0, s 2 i

variance of U

U k ,i

k ,i

) noise of

T U = éê U ,0 , , U ,m -1 ùú . k ë k k û

U

k ,i

(S.4)

is added to the subcarriers, where s 2

is the noise

U k ,i

.

For a given sub-channel U

k ,i

of U k , parameters ni ( U

k ,i

)

and ni ( R ( U

k ,i

))

k ,i

))

are evaluated

as

ni ( U

k ,i

) = s2

U k ,i

and

20

F (T ( U

2

,

(S.5)

ni ( R ( U where the T ( U

k ,i

)

k ,i

) ) = sR2 ( 

U k ,i

)

F (T ( R ( U

k ,i

)))

2

,

sub-channel transmittance coefficients are estimated in a pre-

communication phase via the subcarrier spreading technique [8]. The dn ( R ( U i

k ,i

))

(S.6)

of U

k ,i

parameters are determined from ni ( U

k ,i

)

for all sub-channels

via the iterative method of Theorem 2.

S.1.2

Modulation Variance Adaption

The analysis focuses on a low-SNR CVQKD scenario. An initial low-SNR set of dn ( R  ( U k ,i ) ) , i

i = 0, , m - 1 of user U k from a low-SNR scenario is illustrated in Fig. S.1(a). The minimum of the set is xU = min dn ( R  ( U k ,i ) ) , from which the modulation variance correction for k i "i Î Uk

U

k ,i

is Ds2 = dn ( R  ( U k ,i ) ) - xU k . The corresponding Dsw2 (see (58)), i = 0, , m - 1 vali w i

i

ues determined from the dn ( R  ( U k ,i ) ) elements are depicted in Fig. S.1(b). i dn ( R  ( U k ,i ) ) i

1

Ds 2

0,8

Value

0,8

Value

wi

1

0,6 0,4

0,6 0,4

0,2

0,2

xU 0

k

0 0

200

400

600

800

1000

0

200

400

600

Data unit index

Data unit index

(a)

(b)

800

1000

Figure S.1. An initial low-SNR set of dn ( R  ( U k ,i ) ) of user U k , i = 0, , m - 1 , m = 1000 . i The minimum of set dn ( R  ( U k ,i ) ) is xU k (red solid line) (a). The resulting Dsw2 variance i i

correction, i = 0, , m - 1 , m = 1000 determined from the dn ( R  ( U k ,i ) ) elements (b). i In a low-SNR setting due to the value range of dn ( R  ( U k ,i ) ) , i = 0, , m - 1 , the required i

Ds2 variance correction for the subcarriers is therefore is negligible. wi

21

The xU

, i = 0, , m - 1 input quadratures of user U k with a constant variance sw2 are illus-

k ,i

trated in Fig. S.2(a). Applying the result of Ds2 , the xU wi

k ,i

, i = 0, , m - 1 input quadratures

of user U k at the sw2 = sw2 + Ds 2 increased variance are depicted in Fig. S.2(b). i

wi

xU ,i , sw2 k

50

50

30

30

10

10

Value

Value

xU ,i , sw2 k

-10 -30

i

-10 -30 -50

-50

0

200

400

600

800

1000

0

200

400

600

Data unit index

Data unit index

(a)

(b)

Figure S.2. The xU

k ,i

800

1000

, i = 0, , m - 1 , m = 1000 input quadratures of user U k at a constant

variance sw2 = 64 (a). The xU

k ,i

, i = 0, , m - 1 , m = 1000 input quadratures of user U k at

increased variance sw2 = sw2 + Ds 2 (b). i

wi

The variance adaption of xU

k ,i

, i = 0, , m - 1 requires only a moderate Ds2 for all subcarriwi

ers to achieve an equalized, significantly lower error rate through the sub-channels.

S.1.3

SNR Differences

The effects of the Ds2 variance correction and the resulting jd

ni

wi

( R ( U ,i ) ) k

can be expressed

in terms of the resulting SNR change.

( ) = SNR ( s ) - SNR ( s ) , i = 0, , m - 1 SNR difference for the input quad-

The DSNR sw2

2 wi

i

{xU ,i , xU ,i }

ratures

k

k

( ( R (  )))

DSNR jd

ni

U k ,i

at

2 w

sw2

and

sw2 = sw2 + Ds2 i

is depicted in Fig. S.3(a). The

wi

SNR differences (see (61)) achieved at a given jd

ni

i = 0, , m - 1 are illustrated in Fig. S.3(b).

22

( R ( U ,i ) ) , k

( ( R (  )))

( )

DSNR sw2

DSNR jd

8

8

6

6

SNR

10

SNR

10

4

4

2

2

0

0

0

200

400

600

U k ,i

ni

i

800

1000

0

200

Data unit index

400

600

800

1000

Data unit index

(a)

(b)

( )

Figure S.3. The DSNR sw2 , i = 0, , m - 1 , m = 1000 parameter for i

{xU ,i , xU ,i } k

k

at sw2

( ( R (  ))) parameter, i = 0, , m - 1 , m = 1000 (b).

and sw2 (a). The DSNR jd

ni

i

U k ,i

( ) input SNR difference results in an improved D ( j ( R (  ))) parameter

The DSNR sw2

SNR

i

via jd

ni

S.1.4

dn

U k ,i

i

( R ( U ,i ) ) for all i. k

Error Rate Minimization

( R ( U ,i ) ) are compared in ( R ( U ,i )) , i = 0, , m - 1 are illus-

The BER values for the initial low-SNR set and for the set of jd

ni

Fig 4. The BER of U

k ,i

at the initial low-SNR set of dn

i

k

trated in Fig. S.4(a). The BER of the m sub-channels

jd

ni

( R ( U ,i ) ) , is depicted in Fig. S.4(b). k

23

k

U

k ,i

,

i = 0, , m - 1

at

ni

100

100

10-1

10-1

10-2

10-2

10-3

10-4

10-5

10-5 200

400

600

800

k

10-3

10-4

0

( R ( U ,i ) )

jd

BER

BER

dn ( R  ( U k ,i ) ) i

1000

0

200

Data unit index

400

600

800

1000

Data unit index

(a)

(b)

Figure S.4. The BER for the initial low-SNR set dn ( R  ( U k ,i ) ) , i = 0, , m - 1 , m = 1000 i (the minimum of the set is depicted by the green solid line) (a). The BER values for

jd

ni

( R ( U ,i ) ) , i = 0, , m - 1 , m = 1000 k

As follows, as xU

k

(b).

is determined and applied in the iteration procedure, the resulting BER is

equally minimized for all U

k ,i

sub-channels of user U k .

S.2 Notations The notations of the manuscript are summarized in Table S.1. Table S.1. Summary of notations. Notation i

Description Index for the i-th subcarrier Gaussian CV, fi = x i + ipi . Index

j

for

the

j-th

Gaussian

single-carrier

CV,

j j = x j + ip j .

Number of Gaussian sub-channels  i for the transmission l

of the Gaussian subcarriers. The overall number of the subchannels is n. The remaining n - l sub-channels do not transmit valuable information.

24

x i , pi

x i¢, pi¢ x j , pj x j¢ , p j¢ x A,i , pA,i

fi , fi¢

z Î  ( 0, Kz )

Position and momentum quadratures of the i-th Gaussian subcarrier, fi = x i + ipi . Noisy position and momentum quadratures of Bob’s i-th noisy subcarrier Gaussian CV, fi¢ = x i¢ + ipi¢ . Position and momentum quadratures of the j-th Gaussian single-carrier jj = x j + ip j . Noisy position and momentum quadratures of Bob’s j-th recovered single-carrier Gaussian CV jj¢ = x j¢ + ip j¢ . Alice’s quadratures in the transmission of the i-th subcarrier. Transmitted and received Gaussian subcarriers. A d-dimensional input CV vector to transmit valuable information. A

d-dimensional

noisy

output

vector,

T

z ¢T

M

z ¢T = A† z + ( F d ( D ) ) = ( z 0¢ , , zd¢ -1 ) , z j¢ =

(

1 l

å i =0 F (Tj,i (  j,i ) )) z j l -1

where

( (

+ F ( D ) Î  0, 2 sw2 + s 2 0

)).

Measurement operator, homodyne or heterodyne measurement. Set of private rates, identifies r + 2 rate curves for a given sub-channel  i , as



 = { Rmin (  i ) , R (q ) , Rmax (  i )} , q = 0  r - 1 , where Rmin (  i ) < R ( 0 )  < R ( r - 1 ) < Rmax (  i ) . A target private rate, refers to a target transmission rate of

R (q )

private classical information over  i . Also identified by

R ( i ) . A current private rate, refers to a current transmission rate

R (q - 1 )

of private classical information over  i . Also identified by R  ( i ) .

P ( i ) cAB (  i )

Private classical capacity of a sub-channel  i . Holevo

information

25

of

Alice

(transmitter)

and

Bob

(receiver), with respect to sub-channel  i .

cBE (  i )

S

*

( )

Holevo information of Bob and Eve (eavesdropper), with respect to sub-channel  i . Target secret key rate S * (  ) =

l -1

å i =0 Ri (  i ) ,

where

Ri (  i ) is the private rate of sub-channel  i . Adaption region, A = éë Rmin (  i ) , Rmax (  i ) ùû , contains r

A

private rate curves for the iteration. SNR (signal to noise ratio) of  i at private rate R (  i ) ,

SNR ( R (  i ) ) = 10 log10 SNR ( R (  i ) )

where ni ( R (  i ) ) = sR2

1 , ni ( R (  i ) ) 2

F (T ( R (  i ) ) ) , sR2

( i )

( i )

is

the noise variance of  i at R (  i ) , while F (T ( R (  i ) ) ) is the transmittance coefficient of  i at R (  i ) . Cumulative ni ( R (  i ) ) of  i at an increased (target) private rate R (  i ) , evaluated as

dn ( R (  i ) ) i

Dn

i

 dn ( R (  i ) ) = dn ( R  i ) ) + Dn ( R (  i ) , R (  i ) ) , (  i i i

where R  (  i ) is the current private rate, R (  i ) is the  target private rate for  i , R  ( i ) < R ( i ) < R ( i ) .  Difference of ni at R ( i ) and R ( i ) ,  R ( i ) > R ( i ) > 0 ,  Dn = ni ( R (  i ) ) - ni ( R (  i ) ) . i

Bit error rate of  i . At target private rate R ( q ) at a



current rate R ( q - 1 ) , and dn ( R ( q - 1 ) ) , is expressed as i

æ ö  çç R (q )d R q -1 ÷÷÷ . )) ø è ni ( ( Identifies that  i , i = 0, , l - 1 from the total l, for x ( R (q - 1) )

which dn ( R ( q - 1 ) ) is minimal at a given R ( q ) , i

x ( R ( q - 1 ) ) = min dn ( R ( q - 1 ) ) , "i

i

where x ( 0 ) = min dn ( 0 ) = min ni . "i

26

i

"i

A target private rate R ( q ) over

R ( q )d

ni ( R ( q -1 ) )

i

at a current

dn ( R ( q - 1 ) ) . i

æ ö  çç Rmin (  i )d R q -1 ÷÷÷ )) ø è ni ( (

Minimal bit error rate, achievable at Rmin (  i ) , at a given

dn ( R ( q - 1 ) ) . i

Function, evaluates SNR quantities from the input ni (⋅ ) parameters, defined as  ( ni (  i ) + ( Dn ( Rmin (  i ) , R ( 0 ) ) +  + Dn ( R (q - 1 ) , R ( q ) ) ) ) i i æ ö çç SNR ( ni (  i ) ) - éë SNR ( ni ( R ( 0 ) ) ) - SNR ( ni ( Rmin (  i ) ) ) ùû ÷÷ ÷ çç ÷ q 1 æ ö ÷÷ , =ç çç - ççç å é SNR ( ni ( R ( k + 1 ) ) ) - SNR ( ni ( R ( k ) ) ) ù ÷÷÷ ÷÷ ë û ÷ ÷ ÷ø çè k = 0 ççè ø÷

 (⋅)

where SNR ( ni ( R ( x ) ) ) = 10 log10 ( 1 ( ni ( R ( x ) ) ) ) . erfc ( ⋅ )

Complementary error function, erfc ( x ) =

2 p

¥

òx

2

e -t dt .

A given user, k = 0, , K - 1 .

Uk

Logical channel of user U k , k = 0, , K - 1, where K is the number of total users,

U

T U = éê U ,0 , , U ,m -1 ùú , k ë k k û

k

and U

k ,i

is the i-th sub-channel of U , m is the number k

of subcarriers dedicated to U k . Target secret key rate of U , k

S

*

m -1

( U )

S

k

where R ( U

R  ( U k ,i )

*

( U ) = å R ( U ,i ) , k

i =0

) is the private rate of U ,i . A current private rate of U ,i , R  ( U ,i ) = Ri (q - 1 ) . A target private rate of U ,i , R ( U ,i ) = Ri (q ) where k ,i

k

k

k

k

R ( U

k

,i )

i

k

Ri (q ) refers to the R ( q ) rate of U

k ,i

Parameter dn of U i

27

k ,i

, with relation

Ri ( -1 ) = Rmin ( U

Ri (q - 1 ) < Ri (q ) ,

Ri ( r ) = Rmax ( U dn ( Ri ( q ) )

k

k ,i

),

).

k ,i

at private rate R ( U

k ,i

).

and

Identifies the minimal dn ( R  ( U k ,i ) ) parameter for the i

xU

U

k

k ,i

, i = 0, , m - 1 sub-channels of the set U , as k

xU = min dn ( R  ( U k ,i ) ) . k i "i Î Uk

Modulation variance adaption at rate R  ( U k ,i ) ,

Ds2

Ds2 = dn ( R  ( U k ,i ) ) - xU k . i w

wi

i

A corrected modulation variance, sw2 = sw2 + Ds2 , where wi

sw2

sw2 is the initial input subcarrier modulation variance, and

Ds2 = dn ( R  ( U k ,i ) ) - xU k . i w i

jd

ni

( R ( U ,i ) ) k

( ( R (  )))

DSNR jd

U k ,i

ni

æ  çç R ( U ,i ) k xU èç k F

ö÷ ÷ ø÷

The resulting dn

R ( U

k ,i

parameter for U

i

) , jd ( R ( U ,i ) ) = dn ( R ( U ,i )) - xU ni

k

i

SNR increment of U

R ( U

k ,i

at a target rate

k ,i

k ,i

k

, achieved by Ds2

.

k

at a given

wi

) = R (q ) .

An equally minimized bit error rate for all U arbitrary R ( U

k ,i

)d

ni

( R ( Uk ,i ))

k ,i

at an

.

Fourier transform (FFT). A d-dimensional input vector of user U k , k = 0, , K - 1 ,

 jU

 jU = ( jU ,0 , , jU k

k

k

T

k ,d -1

single-carrier CV, jU

k,j

)

, where jU

= xU

k,j

+ ipU

k ,j

k,j

refers to the j-th , and

{xU , j , pU , j } k

k

are Gaussian random quadratures.

 xU

 A d-dimensional vector of U k , xU = ( xU ,0 , , xU k

k

where xU

k,j

is the quadrature component of jU

Noisy subcarrier, discrete variable, xU¢

xU¢

k ,i

k

k ,i

M is a measurement operator, and fU¢

k,j

= M ( fU¢ k ,i

k ,i

Input subcarrier vector, xU

28

k ,i

,

k ,i

) , where

is the i-th noisy

= ( xU ,0 , , xU k

)

.

Gaussian subcarrier CV of U k .

xU

T

k ,d -1

T

k ,m -1

)

.

xU¢

Output subcarrier vector, xU¢

k ,i

k ,i

= ( xU¢ ,0 , , xU¢ k

T

k ,m -1

)

.

The variable of a single-carrier Gaussian CV state, ji Î  . Zero-mean, circular symmetric complex Gaussian

z Î  ( 0, sz2 )

é 2ù random variable, sz2 =  ê z ú = 2sw2 , with i.i.d. zero 0 ë û mean,

Gaussian

(

x , p Î  0, sw2

0

random

) , where s

2 w0

quadrature

components

is the variance.

The noise variable of the Gaussian channel  , with i.i.d.

D Î  (

)

2 0, sD

zero-mean, Gaussian random noise components on the posi-

Dx , Dp Î  ( 0, s 2 ) ,

tion and momentum quadratures

é 2ù 2 sD =  ê D ú = 2s 2 . ë û The variable of a Gaussian subcarrier CV state, fi Î  . Zero-mean, circular symmetric Gaussian random variable,

d Î  (

0, sd2

)

é 2ù sd2 =  ê d ú = 2sw2 , with i.i.d. zero mean, Gaussian ranë û dom quadrature components xd , pd Î  ( 0, sw2 ) , where sw2 is the (constant) modulation variance of the Gaussian subcarrier CV state.

F -1 ( ⋅ ) = CVQFT† ( ⋅ )

The inverse CVQFT transformation, applied by the en-

F ( ⋅ ) = CVQFT ( ⋅ )

The CVQFT transformation, applied by the decoder, con-

F

-1

(⋅ ) = IFFT (⋅ ) sw2

sw2 =

1 l

coder, continuous-variable unitary operation. tinuous-variable unitary operation. Inverse FFT transform, applied by the encoder. Single-carrier modulation variance.

0

ål sw2

i

Multicarrier modulation variance. Average modulation variance of the l Gaussian sub-channels  i . The i-th Gaussian subcarrier CV of user U k , where IFFT

fi = IFFT ( z k ,i )

= F -1 ( z k ,i ) = di .

stands for the Inverse Fast Fourier Transform, fi Î  ,

( ) Î  ( 0, s ) ,

é 2ù sd2 =  ê di ú , i ë û

di Î  0, sd2 , i

xd

i

2 wF

29

(

pd Î  0, sw2 i

F

)

are

di = xd + ipd , i

i.i.d.

i

zero-mean

Gaussian random quadrature components, and sw2

is the

F

variance of the Fourier transformed Gaussian state.

jk ,i = CVQFT ( fi

The decoded single-carrier CV of user U k from the subcar-

)

rier CV, expressed as F ( di



)=

F ( F -1 ( z k , i ) ) = z k ,i .

Gaussian quantum channel.

 i , i = 0, , n - 1

Gaussian sub-channels. Channel transmittance, normalized complex random variable, T (  ) = Re T (  ) + i Im T (  ) Î  . The real part

T ( )

identifies the position quadrature transmission, the imaginary part identifies the transmittance of the position quadrature. Transmittance coefficient of Gaussian sub-channel

Ti (  i ) = Re (Ti (  i ) ) + i Im (Ti (  i ) ) Î  ,

i ,

quantifies

the position and momentum quadrature transmission, with

Ti (  i )

(normalized)

real

0 £ Re Ti (  i ) £ 1

and

imaginary

0 £ Im Ti (  i ) £ 1

2,

parts

2,

where

ReTi (  i ) = Im Ti (  i ) . TEve

Eve’s transmittance, TEve = 1 - T (  ) .

TEve,i

Eve’s transmittance for the i-th subcarrier CV. A d-dimensional, zero-mean, circular symmetric complex

T

z = x + ip = ( z 0 , , zd -1 )

random Gaussian vector that models d Gaussian CV input states,  ( 0, Kz ) , Kz =  éê zz† ùú , where z j = x j + ip j , ë û T

x = ( x 0 , , xd -1 ) ,

(

p j Î  0, sw2

0

T

p = ( p0 , , pd -1 ) ,

(

x j Î  0, sw2

0

),

) i.i.d. zero-mean Gaussian random variables.

An l-dimensional, zero-mean, circular symmetric complex random Gaussian vector,  ( 0, Kd ) , Kd =  éê dd† ùú , ë û

d = F -1 ( z )

T

d = (d0 , , dl -1 ) , di = x i + ipi ,

(

x i , pi Î  0, sw2

F

)

are

i.i.d. zero-mean Gaussian random variables. The i-th com-

(

)

é 2ù ponent is di Î  0, sd2 , sd2 =  ê di ú . i i ë û

30

(

yk Î  0,  éê yk yk† ùú ë û

)

A d-dimensional zero-mean, circular symmetric complex Gaussian random vector. The m-th element of the k-th user’s vector yk , expressed as

y k ,m

yk , m =

ål F (Ti (  i ))F (di ) + F ( Di ) .

Fourier F (T( 

))

transform

of

T T (  ) = éëT0 (  0 ) ,Tl -1 ( l -1 ) ùû Î  l ,

the

complex

transmittance vector. Complex vector, expressed as F ( D ) = e

F (D)

2

, with

†ù é covariance matrix KF ( D ) =  ê F ( D ) F ( D ) ú . ë û

AMQD block, y éë j ùû = F ( T ( 

y éë j ùû

t = F ( d ) éë j ùû

T -F ( D ) KF ( D )F ( D )

An

2

exponentially

f ( t ) = ( 1 2sw2n )e

) ) F ( d ) éë j ùû + F ( D ) éë j ùû .

distributed

-t

2sw2

variable,

with

density

,  éë t ùû £ n 2sw2 .

Eve’s transmittance on the Gaussian sub-channel  i ,

TEve,i = Re TEve,i + i Im TEve,i Î  ,

TEve,i

0 £ Im TEve,i £ 1

di

2 , 0 £ TEve,i

2

0 £ Re TEve,i £ 1

2,

< 1.

A di subcarrier in an AMQD block. The min { n 0 , , nl -1 } minimum of the ni sub-channel co-

n min

F (Ti (  i ) )

efficients, where ni = s 2

2

and ni < nEve .

Constant modulation variance, sw2 = nEve - n min ( d )p

nEve =

sw2

1 l

, l = F (T*

)

2

=

1 n

n -1

n -1

å i =0 å k =0Tk*e

-i 2 pik n

(x )

,

2

and

T* is the expected transmittance of the Gaussian subchannels under an optimal Gaussian collective attack.

S.3 Abbreviations AMQD AWGN CV

Adaptive Multicarrier Quadrature Division Additive White Gaussian Noise Continuous-Variable

31

CVQFT CVQKD DV FFT ICVQFT IFFT MQA QFT QKD SNR

Continuous-Variable Quantum Fourier Transform Continuous-Variable Quantum Key Distribution Discrete Variable Fast Fourier Transform Inverse CVQFT Inverse Fast Fourier Transform Multiuser Quadrature Allocation Quantum Fourier Transform Quantum Key Distribution Signal to Noise Ratio

32