Singular Layer Transmission for Continuous-Variable Quantum

Singular Layer Transmission for Continuous-Variable Quantum Key Distribution Laszlo Gyongyosi 1

Quantum Technologies Laboratory, Department of Telecommunications Budapest University of Technology and Economics 2 Magyar tudosok krt, Budapest, H-1117, Hungary 2 Information Systems Research Group, Mathematics and Natural Sciences Hungarian Academy of Sciences Budapest, H-1518, Hungary [email protected]

Abstract We develop a singular layer transmission model for continuous-variable quantum key distribution (CVQKD). In CVQKD, the transmit information is carried by continuousvariable (CV) quantum states, particularly by Gaussian random distributed position and momentum quadratures. The reliable transmission of the quadrature components over a noisy link is a cornerstone of CVQKD protocols. The proposed singular layer uses the singular value decomposition of the Gaussian quantum channel, which yields an additional degree of freedom for the phase space transmission. This additional degree of freedom can further be exploited in a multiple-access scenario. The singular layer defines the eigenchannels of the Gaussian physical link, which can be used for the simultaneous reliable transmission of multiple user data streams. Our transmission model also includes the singular interference avoider (SIA) precoding scheme. The proposed SIA precoding scheme prevents the eigenchannel interference to reach an optimal transmission over a Gaussian link. We demonstrate the results through the adaptive multicarrier quadrature division– multiuser quadrature allocation (AMQD-MQA) CVQKD multiple-access scheme. We define the singular model of AMQD-MQA and characterize the properties of the eigenchannel interference. We propose the SIA precoding of Gaussian random quadratures and the optimal decoding at the receiver. We show a random phase space constellation scheme for the Gaussian sub-channels. The singular layer transmission provides improved simultaneous transmission rates for the users with unconditional security in a multiple-access scenario, particularly in crucial low signal-to-noise ratio regimes. Keywords: quantum cryptography, continuous variables, quantum Shannon theory

1

1 Introduction The continuous-variable quantum key distribution (CVQKD) protocols allow for the legal parties to transmit information with unconditional security over the current, well-established standard telecommunication networks. The CVQKD protocols use continuous quantum systems for the information transmission, practically Gaussian random distributed position and momentum quadratures in the phase space [1-15]. These quadratures identify a continuous-variable (CV) quantum state in the phase space, which are then transmitted over a noisy quantum link (e.g., via an optical fiber or a wireless optical channel [16-17]), which is attacked by an eavesdropper [13-14]. Because the optimal attack against CVQKD protocols is a Gaussian attack, the noise of the physical quantum link can be provably modeled as an additive white Gaussian noise and the link as a Gaussian channel. The CVQKD protocols are equipped with several benefits in comparison with the discrete-variable (DV) QKD; however, in comparison with the traditional telecommunication protocols, the efficiency of CVQKD still requires significant improvements. An enhancement is particularly crucial and essential for the low signal-to-noise ratio (SNR) regimes, at which the experimental long-distance CVQKD protocols are operating. For this purpose, the adaptive multicarrier quadrature division (AMQD) modulation scheme has been recently introduced for CVQKD [4], which injects several convenient abilities into CVQKD regarding the transmission rates, distances, and tolerable noise level, similar to the orthogonal frequency-division multiplexing (OFDM) [10] of traditional networking. In particular, the AMQD modulation granulates the transmit information into Gaussian subcarrier CVs and formulates Gaussian sub-channels from the physical Gaussian link, leading to improved transmission distances, higher secret key rates, and enhanced tolerable excess noise. The benefits of AMQD have also been extended to a multiuser scenario within the framework of the AMQD–multiuser quadrature allocation (MQA) multiple-access scheme. The AMQD-MQA [3] uses the Gaussian subcarriers and the AMQD modulation for the reliable simultaneous transmission of the users’ input CVs and provides capacityachieving multiple-access communication for the parties over a shared physical Gaussian quantum channel. The singular value decomposition (SVD) is a well-known tool in linear algebra, with a widespread application from mathematical statistic to signal processing. In this work, we show that the benefits of SVD can also be exploited in a CVQKD scenario. We reveal that the information transmission of CVQKD over the Gaussian quantum channel can be rephrased in terms of SVD, which is called the singular layer transmission model throughout. The singular layer injects an additional degree of freedom into the transmission, which can be exploited in the protocol. We define the singular layer of AMQD and AMQD-MQA and prove that the singular layer allows for the parties to reach improved transmission distances and higher secret key rates. Specifically, the singular layer defines a higher-level layer above the physical layer and transforms the quantum channel onto a set of eigenchannels. From the layer structure, it follows that the singular layer transmission does not require any change in the lower layers, that is, the physical layer transmission remains untouched, because the SVD defines a purely logical layer over the physical layer transmission. In Section 3, we reveal how the singular layer transmission can be exploited in the

2

multicarrier-level transmission of AMQD. We also define the singular layer model of AMQDMQA multiple access, and by exploiting the tools of the quantum Shannon theory, we study the achievable reliable transmission rates. We characterize the optimal phase space decoding process and an information-theoretical optimal decoder for the Gaussian random phase space quadratures. At a partial channel side information (e.g., an exact state information about the Gaussian sub-channels is not available for the parties), the eigenchannels can interfere with one another; thus, the handling of an interference avoider is a crucial point in the SVD-level approach of a CVQKD protocol. Because the interference cancellation, in theory, could require an unlimited modulation variance because the interference vector could occur in the arbitrary domains of the phase space, a carefully designed interference avoider is a critical requirement on the SVD-level. For this purpose, in Section 4, we define the singular interference avoider (SIA) scheme. In Section 5, we show a random phase space coding scheme for the transmission of Gaussian subcarrier CVs over the Gaussian sub-channels, which provides an unconditional security and an optimal partition of the phase space. The benefits of the proposed singular layer model can be exploited most significantly in the crucial low-SNR regimes, which is in fact particularly important in the implementation of long-distance CVQKD. This paper is organized as follows. Section 2 summarizes the preliminary findings. Section 3 discusses the singular layer transmission model. Section 4 studies the SIA scheme and the optimal phase space decoding process. Section 5 defines a phase space coding scheme for the transmission of Gaussian subcarrier CVs. Finally, Section 6 concludes the results. Supplemental information is included in the Appendix.

2 Preliminaries 2.1

Basic terms and notations

Before we would start to discuss the singular layer model, we summarize the basic notations of the AMQD-MQA multiple access CVQKD scheme from Section 2.1 of [3]. Further related background material can be found in [4.] The input of k-th user U k is a Gaussian CV state y k Î  , where  stands for the phase space. The noise of the Gaussian quantum channel  is denoted by D , which acts independently on the x position, and p momentum quadratures. These Gaussian CV states, along with the noise of the quantum channel, can be modeled as Gaussian random continuous-variables. A single-carrier Gaussian modulated CV state y in the phase space  is modeled as a zeromean, circular symmetric complex Gaussian random variable z Î  ( 0, sz2 ), with i.i.d. zero

(

mean, Gaussian random quadrature components x , p Î  0, sw2

0

) , where

sw2 is the modulation 0

variance. The variance of z is

é 2ù sz2 =  ê z ú = 2sw2 . 0 ë û

3

(1)

2 The D Î  ( 0, sD ) noise variable of the Gaussian channel  with i.i.d. zero-mean, Gaussian

random components on the position and momentum quadratures Dx , Dp Î  ( 0, s 2

)

has the

variance of

é 2ù 2 sD =  ê D ú = 2s 2 . ë û

(2)

A f Î  Gaussian subcarrier CV state is also modeled by a zero-mean, circular symmetric Gaussian random variable d Î  ( 0, sd2 ), with i.i.d. zero mean, Gaussian random quadrature components xd , pd Î  ( 0, sw2 ) , where sw2 is the modulation variance of the Gaussian subcarrier CV state. The Gaussian subcarrier random variable d has variance

é 2ù sd2 =  ê d ú = 2sw2 . ë û

(3)

Let assume that K independent users are in the CVQKD system. The subset of allocated users is denoted by  Í K . Only the allocated users can transmit information in a given (particularly j-th) AMQD block. An AMQD block consist of l Gaussian subcarrier CVs (assuming an optimal Gaussian collective attack [13-13], only l sub-channels have high quality from the n, these are referred as good sub-channels throughout, for details see [3] and [4]). Following the formalism of AMQD [4], the variables of an allocated user U k , k = 1, ,  , where  is the cardinality of the subset  , are as follows. The i-th Gaussian modulated input coher-

(

ent state of U k is referred as jk ,i = x k ,i + ipk ,i , where x k ,i Î  0, sw2

0,k

),

(

pk ,i Î  0, sw2

0,k

)

are the position and momentum quadratures with variance sw2 , respectively. This CV state can 0,k

be rewritten as a zero-mean, æ ö é z k ,i Î  ççç 0, sw2 ÷÷÷ , sw2 = ê  z k ,i z k ,i ø zk ,i è ë

circular symmetric complex Gaussian random variable 2ù ú , as û

z k ,i = x k ,i + ipk ,i ,

(4)

jk ,i = z k ,i .

(5)

thus

The

variable

e iji z k ,i

has

the

same

distribution

of

zk ,i

for

any

ji ,

i.e.,

2ù é  ëé z k ,i ûù =  êé e iji zk ,i úù = e iji ëé z k ,i ûù and sz2 =  ê zk ,i ú . The density of zk ,i is ë û k ,i ë û

f ( z k ,i ) = where zk ,i =

1 2 psw2

e

æ 2 ÷ö -ççç zk ,i ÷÷÷ çè ø

- x 2 + p2 k ,i k ,i

2 s2 w0,k

2 s2 w0,k

(

= f ( x k ,i , pk ,i ) =

0,k

1 2 psw2

e

) ,

0,k

x k2,i + pk2,i is the magnitude, which is a Rayleigh random variable with density

4

(6)

- z k ,i

f ( z k ,i while the z k ,i

2

zk ,i

)= s

2 wz k ,i

e

2

2 s2 wz k ,i

, z k ,i ³ 0 ,

(7)

= x k2,i + pk2,i squared magnitude is exponentially distributed with density - zk ,i

f

(z )= 2

k ,i

1 sw2

e

2

s2 wz k ,i

, z k ,i

2

³ 0.

(8)

zk ,i

The i-th Gaussian subcarrier CV of user U k is defined as

fi = IFFT ( zk ,i ) = F -1 ( z k ,i ) = di ,

(9)

where IFFT stands for the Inverse Fast Fourier Transform, and subcarrier continuous-variable fi in Equation (9) is also a zero-mean, circular symmetric complex Gaussian random variable

(

)

é 2ù di Î  0, sd2 , sd2 =  ê di ú . The quadrature components of the modulated Gaussian subcari i ë û rier CVs are referred by xd , pd Î  ( 0, sw2 ) , where sw2 is the constant modulation variance of AMQD that is used in the transmission phase (Note: the constant modulation variance is provably the optimal solution in the low-SNR regimes, because the performance is very close to the exact allocation [3-4], [18-19], [21-23]. This property will not change in the singular layer of AMQD and AMQD-MQA.). Assuming K independent users in the AMQD-MQA, who transmit the single-carrier Gaussian CVs to an encoder  , (9) modifies as follows

fi = CVQFT† ( z k ,i ) = F -1 ( z k ,i ) = di ,

(10)

where CVQFT† refers to the continuous-variable inverse quantum Fourier transform (QFT). The inverse of (9) results the single-carrier CV from the subcarrier CV as follows:

jk ,i = CVQFT ( fi

) = F ( di ) =

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

(11)

where CVQFT is the continuous-variable QFT operation. Let zk be a d-dimensional, zero-mean, circular symmetric complex random Gaussian vector of

Uk , zk = xk + ipk = ( zk ,1, , zk ,d ) Î  ( 0, Kz T

k

),

(12)

where Kz is the d ´ d Hermitian covariance matrix of zk , Kz =  éê zk zk† ùú , and zk† stands for k k ë û the adjoint of zk . Each zk ,i variable is a zero-mean, circular symmetric complex Gaussian ran-

æ dom variable z k ,i Î  ççç 0, sw2 z k ,i è

ö÷ ÷ , z = x k ,i + ipk ,i . The real and imaginary variables (i.e., the ø÷ k ,i 5

position and momentum quadratures) formulate d-dimensional real Gaussian random vectors, T

T

xk = ( x k ,1, , x k ,d ) and pk = ( pk ,1, , pk ,d ) , with zero-mean Gaussian random variables -x k ,i 2

1

x k ,i =

sw

0,k

where sw2

2p

e

2 s2 w0,k

- pk ,i 2

1

, pk ,i =

sw

0,k

2p

e

2 s2 w0,k

,

(13)

is the stands for single-carrier modulation variance (precisely, the variance of the real 0,k

and imaginary components of z k ,i ). For vector zk ,

 éë zk ùû =  éê e ig zk ùú = e ig éë zk ùû ë û

(14)

T ù é  éê zk zTk ùú =  ê e ig zk (e ig zk ) ú = e i2 g éê zk zTk ùú , ë û ë û êë ûú

(15)

holds, and

for any g Î éë 0, 2p ùû . The density of zk is as follows (if Kz is invertible): k

f ( zk ) =

1 pd det Kz

e

1 -zk† Kz z k k

.

(16)

k

A d-dimensional Gaussian random vector is expressed as xk = As , where A is an (invertible) linear transform from d to d , and s is a d-dimensional standard Gaussian random vector  ( 0,1 )d . This vector is characterized by its covariance matrix Kx =  éê xk xTk ùú = AAT , as k ë û -

xk =

1

(

2p

d

)

(

T

det AA

)

xT xk k 2 AAT

e (

).

(17) T

The Fourier transformation F (⋅ ) of an l-dimensional Gaussian random vector v = ( v1, , vl ) T

results in the d-dimensional Gaussian random vector m = ( m1, , md ) : m = F (v) = e

- mT AAT m 2

(

=e

-s2 m12 ++m 2 d w0 2

) .

(18)

In the first step of AMQD, Alice applies the inverse FFT operation to vector zk (see Equation (12)), which outputs an l-dimensional zero-mean, circular symmetric complex Gaussian random T

vector d , d Î  ( 0, Kd ) , d = (d1, , dl ) , as d=F

(

-1

( zk ) = e

)

dT AAT d 2

(

=e

s2 d12 ++d 2 l w0 2

) ,

where di = xd + ipd , di Î  0, sd2 , and the position and momentum quadratures of fi i

i

i

i.i.d. Gaussian random variables 6

(19) are

xd Î  ( 0, sF2 ) , pd Î  ( 0, sF2 ) , i

(20)

i

T ù é where Kd =  éê dd† ùú ,  éë d ùû =  éê e ig d ùú = e ig éë d ùû , and  éê ddT ùú =  ê e ig d (e ig d ) ú = e i2 g éê ddT ùú ë û ë û ë û ë û êë ûú for any g Î éë 0, 2p ùû . The coherent Gaussian subcarrier CV is

fi = di = F -1 ( z k ) .

(21)

The result of Equation (19) defines l independent  i Gaussian sub-channels, each with noise variance s 2 , one for each subcarrier coherent state fi . After the CV subcarriers are transmiti

ted through the noisy quantum channel, Bob applies the CVQFT, which results him the noisy version jk¢ = zk¢ of the input z k of U k . The m-th element of d-dimensional zero-mean, circular symmetric complex Gaussian random output vector yk Î  0,  éê yk yk† ùú of U k , is as follows: ë û

(

)

yk , m = F ( T (  ) ) z k , m + F ( D )

= F ( T (  ) ) F ( F -1 ( z k , m ) ) + F ( D ) =

where F ( T ( 

))

(22)

ål F (Ti (  i ))F (di ) + F ( Di ),

is the Fourier transform of the l-dimensional complex channel transmission

vector T

T (  ) = (T1 (  1 ) ,Tl ( l ) ) Î  l ,

(23)

where

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

(24)

is a complex variable, called transmittance coefficient, which quantifies the position and momentum quadrature transmission (i.e., gain) of the i-th Gaussian sub-channel  i , in the phase space

 , with (normalized) real and imaginary parts 0 £ Re Ti (  i ) £ 1 The Ti (  i ) variable has a magnitude of Ti (  i ) =

2 , 0 £ Im Ti (  i ) £ 1

2.

Re Ti (  i ) + Im Ti (  i ) Î  , where 2

2

Re Ti (  i ) = Im Ti (  i ) , by our convention.

For the l sub-channels, the F ( D ) complex vector is evaluated as

F (D) = e

T -F ( D ) KF ( D )F ( D ) 2

=e

æç 2 2 -ççç F ( D1 ) s2 ++F ( Dl ) s2 çç 1 l è 2

ö÷ ÷÷ ÷÷ ø÷÷

,

(25)

which is the Fourier transform of the l-dimensional zero-mean, circular symmetric complex GausT 2 sian noise vector D Î  ( 0, sD )d , D = ( D1, , Dl ) Î  ( 0, KD ) , where KD =  éëê DD† ùúû , with

7

(

)

(

independent, zero-mean Gaussian random components Dx Î  0, s 2 , Dp Î  0, s 2 i

i

i

i

)

with

variance s 2 for each Di . These identify the Gaussian noise of sub-channel  i on the quadrai

ture components in the phase space  . The CVQFT-transformed noise vector can be rewritten as

F ( D ) = ( F ( D1 ) , , F ( Dl ) ) , T

F ( Dx

i

with

Fourier-transformed

) Î  ( 0, sF2 (  ) ) =  ( 0, s2 ) , i

F ( Dp

i

i

quadrature

) Î  ( 0, sF2 (  ) ) =  ( 0, s2 ) i

i

noise

components

for each F ( Di ) ,

which defines a d-dimensional zero-mean, circular symmetric complex Gaussian random vector

(

)

F ( D ) Î  0, KF ( D ) =  ( 0, KD ) with a covariance matrix †ù é KF ( D ) =  ê F ( D ) F ( D ) ú . ë û

(26)

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

y éë j ùû = F ( T ( 

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

(27)

where j is the index of the AMQD block, F ( d ) = F ( F -1 ( z ) ) , for F -1 ( z ) see (19), while T y éë j ùû = ( y1 éë j ùû , , yd éë j ùû ) , T F ( d ) éë j ùû = ( F ( d1 ) éë j ùû , , F (dl ) éë j ùû ) ,

(28)

T F ( D ) éë j ùû = ( F ( D1 ) éë j ùû , , F ( Dl ) éë j ùû ) .

The squared magnitude t = F ( d ) éë j ùû f ( t ) = ( 1 2sw2n )e

2

is an exponentially distributed variable, with density

-t 2sw2

, and from the Parseval theorem [18-20] follows, that  éë t ùû £ n 2sw2 , while the average quadrature modulation variance of the Gaussian subcarriers is n

sw2 =

1 n

å sw2 i =1

i

= sw2 .

(29)

0

Eve’s attack on sub-channel  i is modeled by the TEve,i normalized complex transmittance

TEve,i = ReTEve,i + i Im TEve,i Î  , where 0 £ Re TEve,i £ 1

2 , 0 £ Im TEve,i £ 1

2.

Assuming that s subcarriers dedicated for user U k , a logical channel U can be defined for the k

given AMQD block as T U éë j ùû = éë  1, , s ùû . k

2.2

(30)

SVD on a Hilbert space

The SVD of an M ´ N real or complex matrix M is a factorization of M , as follows:

M = U å V -1 ,

8

(31)

where U is an M ´ M unitary matrix, V -1 is an N ´ N unitary matrix, and G is an M ´ N diagonal matrix [18-20]. The li nonnegative real diagonal entries of å are the singular values of

M , and the columns of U and V -1 are referred as the left-singular vectors (eigenvectors of

MM† ) and right-singular vectors of M (eigenvectors of MM† ). The li singular values are the square roots of the nonzero eigenvalues of MM† or MM† . In particular, the SVD of Equation (31) can be extended to a (separable) Hilbert space  via a bounded operator  (⋅ ) , using a partial isometry U , a unitary V , a measure space Z , and a nonnegative measureable  , such that

 ( M ) = U GV -1 ,

(32)

where G is a multiplication by  on Z2 , which can be decomposed into

 ( M ) = UV -1 ⋅ V GV -1 ,

(33)

where UV -1 identifies a partial isometry and V GV -1 is positive.

3 Singular Layer for CVQKD Theorem 1 (Additional degree of freedom for AMQD via singular layer transmission). The additional degree of freedom of the singular layer provides an improved R transmission rate over the l 2 æ ö sw2 ¢¢ F (Ti (  i ) ) ÷ ÷÷ , Gaussian sub-channels  i , upper bounded by C AMQD (  ) = max å l log2 ççç 1 + i 2 2 s  + sg "i çè ÷÷ø æ ö where sw2 ¢¢ = nEve - ççç s 2 max li2 ÷÷÷ , li is the i-th eigenchannel of F ( T ) , max li2 is the largest è ø n min n min †

eigenvalue of F ( T ) F ( T ) , and sg2 is the interference noise. Proof. In terms of AMQD [4], the output yk ,m refers to the m-th message of user U k , and it is expressed as yk ,m = F ( T ( 

) ) z k ,m

+ F (D) .

(34)

The singular layer consists of a pre-unitary F1 (U 1 ) (scaled FFT operation (scaled CVQFT), independent from the IFFT operation F (U 1 )) (see Figs. 1 and 2) and a post-unitary U 2-1 (CVQFT operation, independent from the U CVQFT† operation) that perform the pre- and post-transform. The pre-unitary F1 (U 1 ) transforms such that the input will be sent through the

li eigenchannels of the Gaussian link, whereas U 2-1 performs its inverse. Note that the pre- F1 (U 1 ) and post-U 2-1 unitaries are the not inverse of F and U but F1-1 (U 1-1 ) and U 2 , respectively. In particular, these unitaries define the set S1 of singular operators, as follows:

9

S1 = { F1,U 2-1 } .

(35)

Specifically, if each transmit user sends a single-carrier Gaussian CV signal to an encoder  , then the pre-operator is the unitary U 1 , the CVQFT operation, whereas the unitary post-operator is achieved by the inverse CVQFT operation U 2-1 , defining the set S2 of singular operators as

S2 = {U 1,U 2-1 } . The subindices of the operators

{F1,U 2-1 }

(36)

and {U 1,U 2-1 } are different in each Si , i = 1, 2 be-

cause these operators are not the inverse of each other. These operators are determined by the SVD of F ( T ) , which is evaluated as

F ( T ) = U 2 GF1-1 ,

(37)

where F1-1, F1 Î Kin ´Kin and U 2 ,U 2-1 Î Kout ´Kout , Kin , and Kout refer to the number of sender and receiver users such that Kin £ Kout , F1-1F1 = F1F1-1 = I , and U 2U 2-1 = U 2-1U 2 = I . The term G Î  is a diagonal matrix with nonnegative real diagonal elements l1 ³ l2 ³  ln ,

(38)

min

which are called the eigenchannels of F ( T ) = U 2 GF1-1 , where

n min = min ( K in , Kout ) ,

(39)

which equals to the rank of F ( T ) , where

Kin £ Kout ,

(40)

by an initial assumption. (Note: the eigenchannels are also called the ordered singular values of F ( T ) .) In terms of the li eigenchannels, F ( T ) can be precisely rewritten as

ån

F (T) =

min

liU 2,i F1,-i 1 ,

(41)

where liU 2,i F1,-i 1 are rank-one matrices. In fact, the n min squared eigenchannels li2 are the eigenvalues of the matrix †

F ( T ) F ( T ) = U 2 GGTU 2-1 ,

(42)

where GT is the transpose of G . The unitary F1 (U 1 ) applied on the input data z Î  ( 0, Kz ) defines an s stream matrix s = ( s1, , sn

T

min

)

= F1 ( z ) Î  ( 0, Ks ) ,

with n min data streams si . A stream variable si identifies the CV state si

(43) in the phase space

 as

si ¢ = liU 2,i F1,-i 1 si .

(44)

The noisy s ¢ =  ( s ) stream vector is

s ¢ = ( s1¢, , sn¢

T

min

)

= F1 ( z ¢ ) Î  ( 0, Ks + Kg + KD ) , 10

(45)

where Kg is the covariance of the g Î  ( 0, Kg ) eigenchannel interference and

s ¢ = F ( T ) s = U 2 GF1-1 s =

ån

min

liU 2,i F1,-i 1 si .

(46)

Without loss of generality, using the operations F1 , F1-1 , and U 2-1 , Equation (34) can be rewritten as

U 2-1 ( yk ,m ) = F1 ( (U 2 GF1-1 )UF ( z k ,m ) )U 2-1 + U 2-1 (U ( D ) + gi ) = U 2 GF1-1z k ,m + U 2-1 (U ( D ) ) + U 2-1 ( gi ) ,

(47)

where U 2 and U 2-1 unitaries stand for the CVQFT and inverse CVQFT operations, U refers to the CVQFT operation, and U 2-1 (U ( D ) ) has the same distribution as D ,

U 2-1 (U ( D ) ) Î  ( 0, KD ) ,

(48)

U 2-1 ( gi ) Î  ( 0, Kg ) .

(49)

whereas i

Thus, for simplicity, the term of U 2-1 (U ( D ) ) will be called U ( D ) , and

F1 ( z )

2

2

= z ,

(50)

by theory (see also the definitions of AMQD [4] and AMQD-MQA [3]). In case of AMQD, the subcarrier transmission is realized through the l good  i Gaussian sub-channels with a constant modulation variance sw2 . In the singular layer, the transmission is interpreted through the n min eigenchannels li , with a modulation variance sw2 ¢ ¹ sw2 . The unitary F1 makes the transmission of data stream si through the li eigenchannel possible. In particular, if the encoder  knows the SVD decomposition

F ( T ) = U 2 GF1-1 ,

(51)

then it is possible for  to send the pre-transformed data streams si through the li eigenchannels without interference gi . At a partial channel side information, the li eigenchannels of G interfere with one another; thus, the streams will arrive non-orthogonally to the decoder which requires an interference cancellation at  . é The Fourier-transformed interference U 2-1 ( gi ) , U 2-1 ( gi ) Î  ( 0, Kg ) , Kg = sg2 = ê  gi i i i ë 2 2 1 å n sg = sg , is defined in the  phase space precisely as n min

min

, 2

ù ú, û

i

æ nmin ö÷ ç U 2-1 ( gi ) = U 2-1 çç å ljU 2, j F1,-j1 ÷÷÷ s j , çè j ¹i ø÷

(52)

where U 2-1 refers to the post-unitary inverse CVQFT operation. (Note: The gi ( li ) eigenchannel interference is not related to the crosstalk noise [4] gi (  i ) , which could occur between the  i Gaussian sub-channels during the transmission. In particular, the crosstalk noise is included in the noise variance s 2 of the Gaussian sub-channels; see [4].) 11

The unitary post-operation is performed by U 2-1 at  on the noisy stream vector s ¢ as

U 2-1 s ¢ = z ¢ = j .

(53)

In Fig. 1, the elements of the additional layer injected by the SVD are depicted above the functional components of AMQD-MQA. AMQD

F1-1

s

Modulated CV subcarriers

d1 z 1,  , z K

F1

F

 Rate selection

( FFT )



 (s)

Gaussian modulation

 fn

U2 j1¢

f1¢

f1

dn

( IFFT )

G 



(AWGN)

fn¢

U ( CVQFT )

U 2-1 ( CVQFT ) †

Bob 1

 jK¢

 Bob K

F ( T ) = U 2 GF1-1 Figure 1. The singular layer transmission model of the K  K AMQD-MQA multiple-access scheme. The model consists of a single encoder and K-independent receivers. The SVD of AMQD is expressed by the matrix F ( T ) = U 2 GF1-1 . The pre- and post-unitaries of the singular layer are the F1 scaled FFT operation and its unitary inverse CV operation U 2-1 . In the second setting of AMQD-MQA, F ( T ) is precisely evaluated as

F ( T ) = U 2 GU 1-1 .

(54)

Thus, without loss of generality,

U 2-1 ( yk ,m ) = U 1 ( (U 2 GF1-1 )UF ( z k ,m ) )U 2-1 + U 2-1 (U ( D ) + gi )

(55)

= U 2 GF1-1z k ,m + U 2-1 (U ( D ) ) + U 2-1 ( gi ) . The SVD decomposition for the second setting of AMQD-MQA is depicted in Fig. 2. AMQD

U 1-1

CV singlecarriers

z1 Alice 1

 zK Alice K

j1

 jK

s

G

( CVQFT )

 Rate selection

U -1 ( CVQFT† )

 fn

U2

j1¢

f1¢

f1

U1

 (s)

 (AWGN)



U

U 2-1

fn¢

( CVQFT )

( CVQFT† )

 jK¢

Bob 1

 Bob K

F ( T ) = U 2 GU 1-1 Figure 2. The singular layer transmission model of the K  K AMQD-MQA, with Kindependent transmitters and K-independent receivers. The SVD of AMQD is expressed by the matrix F ( T ) = U 2 GU 1-1 . The pre- and post-unitaries of the singular layer are the U 1 CVQFT operation and the inverse CVQFT operation U 2-1 .

12

In the AMQD framework, the optimal solution is to allocate the constant modulation variance

sw2 ¢ for the n min eigenchannel li because in the low-SNR regimes, it is provably the optimal solution [4]. Assuming the constant modulation variance sw2 for the l sub-channels

sw2 ¢ =

ån

min

sw2 ¢ =

(

i

1 l

ål sw2

i

= sw2

= nEve - s 2 max F (Ti (  i ) ) where max F (Ti (  i ) ) "i

2

"i

2

)

(56)

,

identifies the strongest Gaussian sub-channel  i in AMQD, see [4],

and

sw2 =

1 1+c

sw2 ¢¢ =

1 æ ç nEve çè 1+c ç

æ öö - ççç s 2 max li2 ÷÷ , ÷÷ ÷÷ è øø n min

(57)

where c > 0 is a constant, sw2 ¢¢ is the constant sub-channel modulation variance at a constant modulation variance sw2 ¢ for the i-th eigenchannel li in the singular layer, which is yielded as i

æ ö sw2 ¢ = m - ççç s 2 max li2 ÷÷÷ i è ø n min 2 1 = n sw ¢ min

= =

1 n min

(58)

sw2

1 1 æ çç nEve n min 1+c ç è

æ öö - ççç s 2 max li2 ÷÷ , ÷÷ ÷÷ è øø n min †

where max li2 is the is the largest eigenvalue of F ( T ) F ( T ) , thus m is chosen as follows: n min

m=

æ 1 ç nEve n min ( 1+c ) ç èç

æ öö + ( n min ( 1 + c ) - 1 ) ççç s 2 max li2 ÷÷ ÷÷ , ÷÷ è øø n min

(59)

where nEve = 1 l , and l is the Lagrange coefficient of Eve’s optimal collective Gaussian attack, see [4] From (58), the achievable C ( 

)

capacity of  over the n min eigenchannels li is precisely:

æ log2 çç 1 + çè min æ = å n log2 ççç 1 + çè min

C ( ) =

ån

ö÷ ÷÷ s 2 + sg2 ø ÷ sw2 ¢ li2 i

1 nmin

sw2 ¢li2

s 2 + sg2

ö÷ ÷÷ = ÷ø

æ å nmin log2 ççèç1 +

ö÷ 2 2 ÷ ÷, n min ( s  + sg ) ø sw2 ¢li2

(60)

where sg2 is the eigenchannel interference, which is vanishing sg2  0 , in an ideal scenario when the SVD of F ( T ) is available for a precoding or postcoding at the transmitter or at the receiver, respectively. In particular, sg2 is also vanishing at the partial channel side information when only a statistical approximation S ( F ( T ) ) of F ( T ) is available for the parties (see Section 4), in which the phenomenon is much more significantly present (and also significant by practical purposes) in the low-SNR regimes, as it will be also proven in Section 4.

13

In particular, the singular layer transmission is allocated with the modulation variance sw2 ¢ of i

Equation (58) such that the constraint of Equation (56) is satisfied. At n min eigenchannels, the covariance matrix Kk , s of user U k is evaluated as follows:

{

k ,1

{

k ,1

}F

-1 1

Kk , s = F1diag sw2 ¢ , , sw2 ¢

or

k ,nmin

Kk , s = U 1diag sw2 ¢ , , sw2 ¢

k ,nmin

}U

(61)

-1 1 ,

(62)

where the modulation variance sw2 ¢ is allocated to the i-th eigenchannel li of U k , which leads to k ,i

achievable capacity for U k as

(

C  Uk

) = log2 det ( IK =

out

+

1 s 2

†

F ( Tk ) Kk , sF ( Tk )

å log2 ( 1 + SNIRk,ilk2,i ),

)

(63)

n min

where SNIR is the signal-to-noise-plus-interference ratio. The SNIR is precisely evaluated as SNIRi =

sw2 ¢ i

s 2 + sg2

=

1 nmin

sw2

,

s 2 + sg2

(64)

where sg2 is the variance of eigenchannel interference U 2-1 ( gi ) Î  ( 0, Kg ) . For Kin transmit users, the resulting capacity is

(

C (  ) = log2 det IK

out

+

1 s 2

†

F ( T ) KsF ( T )

)

æ ö †÷ ç = log2 det çç IK + å 12 F ( Tk ) Kk , sF ( Tk ) ÷÷÷ s çè out ÷ø Kin  =

(65)

Kin

å å log2 (1 + SNIRk,ilk2,i ),

k =1 n min

according to the capacity region of AMQD-MQA defined in [3]. In fact, assuming that the strongest eigenchannel is max li and all li eigenchannels pick up this n min

maximum value in the low-SNR regimes

( sg2

 0 ), the achievable capacity is calculated as fol-

lows:

C ( ) »

å max li2 ⋅ log2 e ⋅

n min

n min

sw2 ¢ i

s 2

,

(66)

where

max li2 n min

å nmin sw2i¢ s 2

Thus,

14

= max li2 n min

sw2 s 2

.

(67)

2 li2 > max å l F (Ti (  i ) ) . å max n "i

n min

(68)

min

To see it, we exploit a property of AMQD. Let 0 £ p £ 1 be the probability that the 2 ål F (Ti (  i ))

sion

sum of the squared magnitude of the Fourier-transformed channel transmis-

coefficients

ål F (Ti (  i ))

2

of

the

l

Gaussian

sub-channels

pick

up

the

maximum

2

= max å l F (Ti (  i ) ) : "i

é p =  ê Pr ë

(å

l

F (Ti (  i ) )

2

= max å l F (Ti (  i ) )

2

"i

)ùúû .

(69)

From this, the achievable rate over  is precisely 2 æ max å l F (Ti (  i ) ) ö ÷÷ ç i ç RAMQD (  ) = p log2 ç 1 + SNR ÷÷ , p èç ø÷

where SNR =

sw2 s 2

(70)

, which can be further evaluated into 2

RAMQD (  ) = max å l F (Ti (  i ) ) ⋅ log2 e ⋅ SNR . i

(71)

Assuming that li eigenchannels are allocated with the modulation variance sw2 ¢ , and assuming the probability that each squared eigenchannel li2 equals max li2 , n min

é æ ÷öù ç p =  êê Pr çç å li2 = å max li2 ÷÷÷ úú , n ÷ø úû êë çè n min n min min one obtains rate, calculated as follows: æ å nmax li2 ÷÷ö çç n min min Rmax l2 (  ) = p log2 çç 1 + ⋅ SNR ÷÷ p ÷÷ i ççè n min ø 2 = å max li ⋅ log2 e ⋅ SNR n min

=

(73)

n min

å max li2 ⋅ log2 e ⋅

n min

(72)

n min

sw2 ¢ i

s 2

,

and from which Equation (66) is concluded. As a conclusion, by performing the modulation variance allocation mechanism for the l Gaussian sub-channels from [3-4] and by redefining n min = s 2 max F (Ti (  i ) ) "i

¢ = n min

s 2 max li2

,

2

from [4] as (74)

n min

the constant modulation variance

sw2 ¢¢

is derived as follows:

¢ sw2 ¢¢ = nEve - n min = nEve -

s 2 max li2

n min

15

,

(75)

where nEve = 1 l , and l is the Lagrange coefficient, with

æ 1 ¢ = s 2 ççç sw2 ¢¢ - sw2 = n min - n min 2 F (Ti (  i ) ) çè max "i The C AMQD ( 

)

1 max li2

n min

÷÷ö ÷÷ . ÷ø

(76)

achievable capacity over the l Gaussian sub-channels is increased in comparison

with that of [4] to 2 l æ sw2 ¢¢ F (Ti (  i ) ) ö ÷÷ C AMQD (  ) = max å log2 ççç 1 + i ÷ s 2 "i ç è ø÷÷ i =1 æç æ ö ÷÷ö s2 çç 2 çç  ÷÷÷ F (Ti (  i ) ) ÷ ÷ n ç Eve ÷ ç 2 ÷ l çç max li ÷ ç ÷÷ ÷ø n min èç ÷÷, = max å log2 ççç 1 + ÷÷ s 2 "i ç i =1 ÷÷ çç ÷÷ çè ø

(77)

with a total constraint 1 l

ål sw2 ¢¢ = sw2 ¢¢ i

= ( 1 + c ) sw2 > sw2 .

(78)

The higher-layer manipulations of AMQD can be exploited in a multicarrier scenario, using the l Gaussian sub-channels for the transmission. In particular, the singular layer leads to improved capacity and improved user rates in the multiuser scheme of AMQD-MQA. However, the modulation variance allocation mechanism requires some refinements in comparison with that of [4]. The algorithm for the enhanced modulation variance allocation is precisely as follows. Algorithm 1. Perform the singular layer transmission via the unitary pre- and post operators F1, U 2-1 , or U 1,U 2-1 , where F1 is the scaled FFT operation, U 1 is the CVQFT operation, and U 2-1 is the inverse CVQFT,

respectively.

F ( T ) = U 2 GF1-1

Compute

( F ( T ) = U 2 GU 1-1 ), where G Î  is a diagonal matrix with nonnegative

real

diagonal

l1 ³ l2 ³  ln

elements

min

,

where

n min = min ( K in , Kout ) , at Kin transmit and Kout receiver users and l good Gaussian sub-channels. 2. Determine the n min squared eigenchannels li2 from G or from the †

eigenvalues of the matrix F ( T ) F ( T ) = U 2 GGTU 2-1 , where GT is the transpose of G . 3. Determine max li2 , where n min

ån

2

min

max li2 > max å l F (Ti (  i ) ) , n min

"i

and compute the constant modulation variance sw2 ¢ for the li ei-

16

æ ö genchannels as sw2 ¢ = m - ççç s 2 max li2 ÷÷÷ , where m is chosen as i è ø n min æ æ öö m = n 11+c ççç nEve + ( n min ( 1 + c ) - 1 ) ççç s 2 max li2 ÷÷ ÷÷ , with a total ÷÷ )è è øø n min min ( constraint sw2 ¢ =

ån

4. Determine SNIR as

min

sw2 ¢ =

å nmin sw2i¢ s 2

1 l

i

+ s g2

=

low-SNR regimes is R (  ) £

ål sw2

i

sw2 s 2

ån

+ sg2

min

= sw2 . . The achievable rate in the

max li2 ⋅ log2 e ⋅ n min

sw2 ¢ i

s 2

, sg2  0 ,

if all li eigenchannels pick up the theoretical maximum max li2 . n min

5. Perform the allocation of the l Gaussian sub-channels with the æ ö ¢ = nEve - çç s 2 max li2 ÷÷ , modulation variance sw2 ¢¢ = nEve - n min ÷ çè ø n min

¢ = s 2 max li2 , nEve = 1 l , and l is the Lagrange cowhere n min n min

efficient.

Then

æ l RAMQD (  ) = max å i =1 log2 ççç 1 + "i çè

sw2 ¢¢ F (Ti (  i ) ) i

s 2

+ sg2

2

ö÷ ÷÷ ø÷÷

over the Gaussian sub-channels, where sg2  0 in the low-SNR regimes. 6. If P < nk , perform the compensation of a nonideal Gaussian modulation by nk¢ = nk - P for the l Gaussian sub-channels, where 2

¢ = s 2 max F (Ti (  i ) ) - s 2 max li2 and nk P = n min - n min "i

n min

is the constant correction term of the noise level of the subchannels. Note that in step 5, the allocation mechanism of the Gaussian sub-channels can be found in [4], and in step 6, the method for the compensation of the imperfections of a nonideal Gaussian input modulation is handled by the proposed algorithm of [3]. The results, in fact, conclude that from the SVD of F ( T ) = U 2 GF1-1 and F ( T ) = U 2 GU 1-1 , the

n min additional degree of freedom can be exploited to reach an improved performance in the multicarrier transmission of AMQD. ■ The effect of the singular layer on the modulation variance allocation mechanism of the Gaussian sub-channels is illustrated in Fig. 3.

17

Figure 3. The effect of singular layer on the sub-channel modulation variance. In an AMQD modulation, the Gaussian subcarrier CVs are transmitted over l good (i.e., ni < nEve ) Gaussian sub-channels  i . The sub-channel noise n min is decreased from n min = s 2 max F (Ti (  i ) )

2

"i

¢ = to n min

s 2

max li2 n min

, which results in an increased modulation variance

sw2 ¢¢

¢ and = nEve - n min

a decreased constant correction nk¢ = nk - P in the noise level. The total constraint is increased to

1 l

ål sw2 ¢¢ = sw2 ¢¢ > sw2 . i

4 SIA at a Partial Channel Side Information 4.1

Transmission at a partial channel side information

Lemma 1. At a partial channel side information, there exists an F ( T ) -independent pre-unitary operation Q such that F ( T ) = xK S ( F ( T ) ) xK-1 , where S ( F ( T ) ) = xK-1 GxK out

model of F ( T ) , xK-1 and xK out

in

in

out

in

is the statistical

are unitary operators that formulate the input covariance matrix

Ks = xK ÃxK-1 , and à is a diagonal matrix. At S ( F ( T ) ) , sg2  0 in the low-SNR regimes. in

in

Proof. At a partial channel side information, the exact SVD of F ( T ) is not available for the encoder; †

thus, the eigenvectors of F ( T ) F ( T ) are not available for the pre-unitary operation F1 and U 1 . In this case, the encoder  can use an F ( T ) -independent pre-unitary Q . In particular, the transmission is allocated with the modulation variance of Equation (58) such that the constraint of Equation (56) is satisfied, which at Kin transmit users and input vector s = ( s1, , sK leading to the covariance matrix 18

T

in

)

,

{

Ks = Qdiag sw2 ¢ , , sw2 ¢ 1

}Q , †

Kin

(79)

and the R reliable transmission rate is † ù é R (  ) £  ê log2 det IK + 12 F ( T ) KsF ( T ) ú , (80) out s  ë û which can be simply verified by a sphere packing argument [18-20]. Using Equation (79) with the assumption Kout = 1 and Q = IK , the maximal rate leads to

(

)

in

Equation (60). Thus, the capacity of the Gaussian quantum channel can be achieved at an F ( T ) -independent unitary Q because the covariance matrix Ks in Equation (79) can be chosen as a function f (⋅ ) of the distribution of F ( T ) . At a partial channel side information, Ks requires the transmitter to build a statistical model

S ( F ( T ) ) = f ( F ( T ) ) about the distribution of F ( T ) , which allows to reach sg2  0 in the low-SNR regimes. It then leads to a reliable transmission rate of é max R (  ) = max  ê log2 det IK + 2 out Ks :Tr ( Ks )£sw ë = å  éê log2 ( 1 + SNRi li2 ) ùú. ë û

(

1 s 2

)

† ù F ( T ) KsF ( T ) ú û

(81)

n min

The Ks of the input has to be chosen such that it matches S ( F ( T ) ) rather than the exact realization of the distribution of F ( T ) [18] because at a partial channel side information, only

S ( F ( T ) ) is available for Alice. In particular, by exploiting the fact that Eve attacks all of the

 i Gaussian sub-channels, and the distribution of F ( T ) formulates a time-independent deterministic process,  can choose an F ( T ) -independent Q such that

Q = S ( F1 ) or Q = S (U 1 ) .

(82)

Thus, S ( F ( T ) ) allows for Alice to use

S ( F ( T ) ) =  éê F2 GF1-1 ùú ë û

(83)

or

S ( F ( T ) ) =  éêU 2 GU 1-1 ùú . (84) ë û The results in Equations (83) and (84) allow for the transmitter to determine the eigenvalues of † S ( F ( T ) ) S ( F ( T ) ) =  éê F2 GGT F2-1 ùú ë û

(85)

or † S ( F ( T ) ) S ( F ( T ) ) =  éêU 2 GGTU 2-1 ùú , (86) ë û In fact, operation Q transforms the input to transmit them over the eigendirections of

†

F ( T ) F ( T ) ; thus, the allocation algorithm of Theorem 1 can be directly applied in the input, which precisely leads to the maximal rate of Equation (77).

19

In particular, if the distribution of F ( T ) cannot be modeled as a deterministic process, the Alice still can handle the situation via S ( F ( T ) ) as follows. In this case, the entries of matrix

S ( F ( T ) ) are statistically independent with a zero mean. Thus, it formulates another (i.e., different from the SVD of F ( T ) ) matrix as

S ( F ( T ) ) = xK-1 GxK , out

where xK-1 and xK out

in

(87)

in

are unitary operators. Thus,

F ( T ) = xK S ( F ( T ) ) xK-1 ,

(88)

Q = xK ,

(89)

out

in

and the pre-unitary Q is in

which leads to the Ks covariance matrix

Ks = xK ÃxK-1 , in

(90)

in

where à is a diagonal matrix that identifies the sw2 ¢ modulation variance components. Without loss of generality, the S ( F ( T ) ) = xK-1 GxK , and Ks leads to achievable capacity

( (

in

out

é  ê log2 det IK + out Ks :Tr ( ) ë é = max  ê log2 det IK + 2 out Ks :Tr ( Ks )£sw ë

C ( ) =

max

Ks £sw2

1 s 2 1 s 2

)

† ù S ( F ( T ) ) xK-1 Ks xK S ( F ( T ) ) ú in in û

)

† ù S ( F ( T ) ) xK-1 xK ÃxK-1 xK S ( F ( T ) ) ú , in in in in û

(91)

where the diagonal entries of à are chosen such that the optimality condition

à = sw2 ¢ is met, where sw2 ¢ =

ån

min

1 Kin

IK

(92)

in

sw2 ¢ with the constraint of Equation (56). i

Putting the pieces together, (91) leads to † é ù C (  ) = max  ê log2 det IK + 12 xK S ( F ( T ) ) ÃS ( F ( T ) ) xK-1 ú 2 out out s out  Ã :Tr ( Ã )£sw ë û † ù é 1 = max  ê log2 det IK + 2 S ( F ( T ) ) ÃS ( F ( T ) ) ú out s Ã :Tr ( Ã )£sw2 ë û 2 1 sw ¢ é æ öù † K n =  ê log2 det ççç IK + in 2min F ( T ) F ( T ) ÷÷÷ ú (93) ê out s  Kin ÷ø úû ç è ë 1 é sw2 ¢ æ öù K n 2÷ ú ÷ =  êê å log2 ççç 1 + in min l 2 i ÷ ú s ÷ ç  è ø úû êë nmin 1 s2 é æ Kin nmin w ¢ ÷öùú, 2÷ ç ê l = å  log2 çç 1 + ÷ i ÷ ê s 2 çè ø ûú n min ë

( (

)

)

where li2 are the ordered squared, random singular values of F ( T ) . For the random distribution of li2 , from the Jensen inequality [18], it precisely follows that

20

1 é æ ööù s2 æ ÷÷ ú Kin n min w ¢ ç 1 ç 2 ÷÷ ê ç ç l (94) £ n min  ê log2 ç 1 + ÷÷ å ç 2 2 i ú, ÷÷ n s s ç ç min  ÷÷ øû êë è è øø úû n min which demonstrates that Equation (93) is maximized if S ( F ( T ) ) identifies a random process, in

é æ å  êê log2 çèççç1 + n min ë

1 Kin n min

öù

sw2 ¢

li2 ÷÷÷ ú ÷ú

fact, if the distribution of li2 converges into a  uniform distribution. In particular, in long-distance CVQKD scenarios, the transmission is realized in the low-SNR regimes, and we focus specifically on this segment. In the low-SNR regime, the term of 1 Kin nmin

sw2 ¢

s 2

li2 allows us to rewrite Equation (93) as

é æ C (  ) = å  ê log2 ççç 1 + ê çè n min ë =å

1 Kin nmin

s 2

n min

1 Kin nmin

=

1 nmin

sw2 ¢

s 2

ö li2 ÷÷÷ » ÷ø

1 Kin n min

s 2

sw2 ¢

s 2

öù li2 ÷÷÷ ú ÷ø úû

(

)

† ù é  êTr F ( T ) F ( T ) ú log2 e ë û

Kin 1 n min

sw2 ¢

 éê li2 ùú log2 e ë û

( Kin ⋅Kout )

= Kout 1 Kin nmin

sw2 ¢

s 2 2 sw ¢

s 2

æ where log2 ççç 1 + çè

sw2 ¢

1 Kin nmin

sw2 ¢

s 2

(95)

log2 e

log2 e,

li2 log2 e , by theory. The result in Equation (95) reveals

that in the low-SNR regimes, the distribution of li2 has no relevance [18] because  éê li2 ùú does not ë û 2 deal with the correlation among li ; and, the inequality of Equation (94) vanishes in the lowSNR CVQKD scenarios. Note, that assuming that Kin = Kout = K , from the law of large numbers the capacity at S ( F ( T ) ) can be approximated as

C (  ) = Kc ⋅  éë SNR ùû ,

(96)

where c is a constant. ■

4.2

Postcoding interference cancellation

Lemma 2 (Interference cancellation at the decoder). If S ( F ( T ) ) =  éêU 2 GF1-1 ùú is available at ë û -1 -1 the decoder, then the interference U 2 ( gi ) = U 2 gi can be cancelled by operator i as

i ( si¢ s j¢

) = ^i ,

where ^i =

n min

{ s j¢ }j =1 ,

j ¹ i consists of n min CV states s j¢ , such that each

is orthogonal to the subspace  i

{ljU 2, j F1,-j1 },

spanned by the n min - 1 eigenchannel vectors

j = 1, , n min , j ¹ i .

21

Proof. Let the noisy output CV stream si ¢ Î  to be given as

si ¢ = si + U 2-1 gi + U Di ,

(97)

where U Di = U ( Di ) is the CVQFT-transformed noise of the Gaussian channel  (for further information see [4]), and U 2-1 gi

is the inverse CVQFT-transformed interference of the

li , i = 1, , nmin eigenchannels, evaluated as é æ nmin öù ÷ -1 ç ê gi =  êU 2 çç å ljU 2, j F1,-j1 ÷÷÷ úú s j . (98). èç j ¹i ø÷ ûú ëê Assuming that the decoder knows S ( F ( T ) ) =  éêU 2 GF1-1 ùú , it is possible to construct a i opë û erator such that i ( si¢ ) = ^i , (99) U 2-1

where the set

^i =

n min

{ s j¢ }j =1 ,

consists of n min CV states s j¢ , such that each s j¢

j ¹i

(100)

is orthogonal to the subspace  i spanned

by the n min - 1 eigenchannel vectors, calculated as follows:

{ljU 2, j F1,-j1 }, j = 1, , nmin , j ¹ i .

(101)

In particular, operator i defines a dS ^ ´ Kout matrix, and the i-th row of i form an orthonori

mal basis bi of

^i .

Without loss of generality, the dS ^ rows of ^i , i

ì b = ïíb1, , bd ïîï Si^

are all orthogonal to the subspace  i

{ljU 2, j F1,-j1 } ,

üï ý, ïþï

(102)

spanned by the n min - 1 eigenchannel vectors

j ¹ i.

Specifically, if the SVD of S ( F ( T ) ) =  éêU 2 GF1-1 ùú is not the linear combination of any other ë û -1 SVDs of F (Tj ) , j ¹ i , then U 2 gi can be cancelled by applying i on s j¢ , resulting in

i si ¢ = i liU 2,i F1,-i 1 si + iU 2-1 gi + iU Di , where iU 2-1 gi = Æ , and iU Di

identifies a Gaussian noise vector iU Di

(103) in the phase

space with a distribution

iU Di Î  ( 0, KD ) . i

In particular, the noise given in Equation (104) then leads to SNR, calculated as follows: é s 2  ( F (T (  ) ) ) 2 ù ú. SNR =  ê w ¢¢ i 2 i i s êë úû The decoder-side interference cancellation can be rewritten by an operator Wi , for all i, as

22

(104)

(105).

Wi = i † i ( liU 2,i F1,-i 1 ) .

(106)

Let -I be the pseudoinverse operation [18-20]. Because for the SVD of F (Ti ) , the relation -I

( liU 2,i F1,-i 1 )

† † æ ö-1 = çç ( liU 2,i F1,-i 1 ) ( liU 2,i F1,-i 1 ) ÷÷÷ ( liU 2,i F1,-i 1 ) è ø

=(

liU 2,i F1,-i 1

(107)

-1

)

holds because F ( T ) is a square matrix and invertible. †

It leads to the conclusion that ( liU 2,i F1,-i 1 ) liU 2,i F1,-i 1 = I is invertible; thus, liU 2,i F1,-i 1 is linearly independent (i.e., it cannot be expressed as the linear combination of any other SVDs

( ljU 2, j F1,-j1 ) ,

j ¹ i ), for "i .

Thus, the overall stream-level rate at the utilization of projector i is

Rstream ( l ) £ 

(å

n min

(

log2 1 +

sw2 ¢li2 s 2

)) .

(108)

These results conclude the proof on the decoder-side eigenchannel interference cancellation. ■ Note that the application of operator Wi requires the knowledge of S ( F ( T ) ) =  éêU 2 GF1-1 ùú at ë û the decoder. Thus, this type of interference cancellation is suboptimal. This problem will be resolved in Theorem 2.

4.3

SIA precoding

Theorem 2 (SIA precoding of Gaussian random quadratures). The U 2-1 gi

eigenchannel inter-

ference can be cancelled by a precoding operator V at encoder  , which defines a phase space constellation  such that the transmitted difference of the equivalence class EC k ji

ki = min { EC k ji - aU 2-1 gi of the Gaussian CV ji

}

is the minimal

and the interference term

aU 2-1 gi , where a = sw2 sw2 + s 2 . Operator V is optimal at S ( F ( T ) ) for all settings of AMQD-MQA. Proof. At a S ( F ( T ) ) partial channel side information, the streams that sent over the eigenchannels cannot be perfectly separated orthogonally by F1 (scaled FFT operation) and U 1 . Assuming an interference-free transmission, the decoder receives the noisy CV state ji¢ = ji + U Di . In particular, in the AMQD setting, the inference is Fourier-transformed by the receiver, denoted by U gi . Thus, the noisy Gaussian CV ji¢ is ji¢ = ji + U 2-1 gi + U Di .

23

As illustrated in Fig. 4, the eigenchannel interference U 2-1 ( gi ) is an additional noise in the phase space  , added to the transmit user Gaussian CV state ji .

Figure 4. At eigenchannel interference, the user signal ji which is then distorted into receives ji¢ = ji + U Di

is transformed into ji + U 2-1 gi ,

ji¢ = ji + U 2-1 gi + U Di . With no interference, the decoder (x, position quadrature; p, momentum quadrature).

Assume that a constellation  in  exists and user U i transmits a Gaussian CV ji

of  . In

the naive approach, the  encoder compensates U 2-1 gi as follows. It encodes ji as

ki = ji + ( -U 2-1 gi The CV state ki

)=

ji - U 2-1 gi .

(109)

is then sent to the receiver, which results in the noisy ki¢ as follows:

ki¢ = ki + U 2-1 gi + U Di = ji + U Di .

(110)

In particular, the naive precoding approach is strictly suboptimal because the energy increase sg2

i

required in Equation (109) is unbounded by theory [18-20], because the transmit user Gaussian CV ji

and the inference U 2-1 gi

can be arbitrary distant from each other in  . Thus, the

compensation of U 2-1 gi could require an unbounded additional modulation variance sg2 . i

To avoid this problem, the precoding operator V defines the W equivalence class set of the ji Gaussian CV states in the phase space with dEC equivalence class EC k ji

CV states. Denoting

the CV states of a set W by EC k ji , W is defined as

W º { EC k ji

dEC

}k =1 .

(111)

The phase space constellation  constructed by V is the superset of W equivalence class sets, as follows:

 º

 W =  { EC k D

D

where D refers to the phase space domain where ki

. The naive approach is illustrated in Fig. 5.

24

ji

dEC

}k =1,

(112)

(more precisely, U 2-1 gi ) could occur in

Figure 5. The naive precoding approach. (a) The transmitted ki Gaussian CV ji

is the difference of the user

and the interference U 2-1 gi . The naive approach is strictly suboptimal be-

cause the energy increase sg2 required for ki = ji - U 2-1 gi

could be unbounded, depending

i

on the difference of ji

and U 2-1 gi . (b) The CV state ki

under interference U 2-1 gi

is

transformed into ji . The origin at the decoder is shifted onto U 2-1 gi . Operator V chooses an EC k ji

from W , for which EC k ji - U 2-1 gi

pares a transmit Gaussian CV state ki

from EC k ji

is minimal, and pre-

such that

ki = min { EC k ji - U 2-1 gi

}.

(113)

In particular, in this approach, the required modulation variance sg2 is bounded because ali

though the difference ji

- U 2-1

gi

can be arbitrary, by choosing the closest EC k ji

U 2-1 gi , the difference of EC k ji - U 2-1 gi

to

is restricted and minimized, which also results in

a minimized additional modulation variance sg2 . i

We define an optimal phase space constellation  for V, which leads to capacity-achieving communication over the over the Gaussian quantum channel  . We formulate the problem directly in terms of ji

single-carrier Gaussian CVs. The proof assumes a modulation variance sw2 for

the ji transmit Gaussian CVs, with total constraint

1 l

ål sw2

i

= sw2 .

By a simple sphere packing argument, for a d-dimensional random phase space code

V =

d

{ ji }i =1 , at a modulation variance

radius

d sw2 . From the law of large numbers, the received noisy phase space code V ¢ =

will lie within a sphere Q of radius radius

sw2 , all of these Gaussian CV states lie in a sphere Q of d

{ ji¢ }i =1

d ( sw2 + s 2 ) , and a noisy ji¢ will lie in a sphere Q of

s 2 around the transmitted CV state ji .

Assuming a maximum likelihood nearest neighbor decision [18-20] rule decoder  (referred to as a throughout), ji¢ will be decoded to the transmit Gaussian CV state ji closest to a ji¢ , where 25

sw2 2 sw + s 2

a=

,

(114)

which is then, for a d-dimensional case precisely leads to

aV ¢ - V

2

= aF ( D ) + ( a - 1 ) V

2

= a2d s 2 + ( a - 1 ) d sw2 d sw2 s 2

=

(115)

.

sw2 + s 2

In particular, for decoder a , the transmit Gaussian ji = EC k ji

lies inside an Q around

a ji¢ of the radius, sw2 s 2

r =

sw2 + s 2

.

(116)

Specifically, the probability p that another random Gaussian CV jj , j ¹ i , lies inside the Q of a ji¢ is as follows:

p=

sw2 s 2

1 sw2

sw2 + s 2

s 2 2 sw + s 2

=

.

(117)

Thus, for aV ¢ , this probability is

p=

(

sw2 + s 2

1 p

.

(118)

KV

{ Vk }k =1 ,

By the union bound, for K V d-dimensional Vk s,

KV
sw .

(128)

In particular, by a random coding argumentation, the hmin minimal distance between ji jj , j ¹ i , is hmin = 2sw because at sw2 , a random CV ji

and

lies in an uncertainty sphere of

radius sw . For a , the radius of Q around a ji¢ is r , see Equation (116), thus Q lives inside the sw -radius sphere. The precoding operator V includes the construction of an optimal  and the encoding of

ki = min { EC k ji - aU 2-1 gi

} , where a  0

in the low-SNR regime; thus,

ki = EC k ji .

(129)

From the structure of W , the optimality of the precoding operator V straightforwardly follows because the EC k ji phase space symbols in  are restricted to domain D . In the first setting of AMQD-MQA, operation V is performed on the di variables that identify the fi CV states in the phase space. It is illustrated in Fig. 6.

27

AMQD

F1-1

s

Modulated CV subcarriers

d1

F ( IFFT )

 (s)

G

V

dn

Precoding

Gaussian modulation

 fn

j1¢

f1¢

f1



U2

 (AWGN)

U 2-1

U

 fn¢

( CVQFT )

( CVQFT† )

 jK¢

Bob 1

 Bob K

F ( T ) = U 2 GF1-1 Figure 6. The first setting of AMQD-MQA with a precoding operator V. The SVD of AMQD is expressed by the matrix F ( T ) = U 2 GF1-1 . As depicted in Fig. 7, in the second setting of AMQD-MQA, the users transmit single-carrier Gaussian CV states to the encoder  . Hence, operation V acts on the EC k ji = ji Gaussian CV states. After the precoding process, the CV subcarriers are sent through the  i Gaussian sub-channels, see Section 3. AMQD

U 1-1 U -1 ( CVQFT† )

s CV subcarriers

G

Precoding

 fn

U2

f1¢

f1

V

 (s)

 (AWGN)

j1¢

U 2-1

U

 fn¢

( CVQFT )

( CVQFT† )

 jK¢

Bob 1

 Bob K

F ( T ) = U 2 GU 1-1 Figure 7. The second setting of AMQD-MQA with precoding operator V . The SVD of AMQD is expressed by the matrix F ( T ) = U 2 GU 1-1 . For a , the number of uncertainty spheres that can be packed into the sphere of radius

sw2 ,

by a simple sphere packing argument, yields

C AMQD (  ) = log2

( (

Vol Q æç æç Vol çç Qçç çç çç è è

sw2

))

2 s2 sw  2 + s2 sw 

öö ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷÷ øø

,

(130)

where Vol (⋅ ) stands for the volume of the sphere Q . As it can be concluded, within the AMQD framework, the formula of Equation (130) exactly coincidences with the capacity of the l Gaussian sub-channels  i [4]. The phase space constellation  is depicted in Fig. 8. For the a decoder, the r -radius uncertainty sphere Q around a ji¢ will lie in the sw -radius sphere around EC k ji . 28

Figure 8. The phase space constellation  =

 W =  { EC k D

D

ji

dEC

}k =1

at the use of the precod-

ing operator V. The CV states of W are labeled as EC k ji . The hmin minimal distance between the constellation points is 2sw . In the encoding phase, the precoding is performed via the Fourier-transformed interference aU 2-1 gi , where a = sw2 sw2 + s 2 , and a random ji sphere of radius sw . The eigenchannel interference shifts the origin into aU 2-1 gi decoding, the transmit Gaussian CV

r =

ji

lies in a at a . In

lies in the uncertainty sphere of radius

sw2 s 2 sw2 + s 2 around a ji¢ . The noisy a ji¢ will be decoded to the transmit CV state

ji closest to a ji¢ (x, position quadrature; p, momentum quadrature). Note that for constellation  , there is no another EC k jj , j ¹ i constellation point within the uncertainty sphere of a ji¢ . Using a decoder  (with no a -constraint), the received noisy ji¢ would lie outside the sw sphere of ji . Let us to focus specifically on the crucial low-SNR regime. By taking the limit lim a  0 , s 2 ¥

(131)

Equation (123) picks up the formula of ki = min { EC k ji

Thus, 29

}.

(132)

aU 2-1 gi = Æ

(133)

ki = EC k ji .

(134)

and The a reliable transmission over an Gaussian channel strictly lower bounds the SNR, particularly SNR ³ -1.59 dB [18-20]. In fact, Equations (131)–(134) are also lower bounded by this constraint. To summarize, at V , the achievable overall capacity in the singular layer at a partial channel side information S ( F ( T ) ) , with a covariance matrix

{

}Q

Ks = Qdiag sw2 ¢ , , sw2 ¢ is as follows:

Kin

1

†

(135)

(

)ùúû ,

(

) ùúû .

† é C (  ) =  ê log2 det IK + 12 F ( T ) KsF ( T ) out s ë thus, the available transmission rate is † é R (  ) £  ê log2 det IK + 12 F ( T ) KsF ( T ) out s  ë

(136)

(137) ■

4.4

Phase space decoding of Gaussian random quadratures

Lemma 3. The decoder a is optimal for the phase space decoding of Gaussian random quadratures since it maximizes the SNIR. Proof. Let assume

that

user

Uk

utilizes

a

an

decoder

for

his

d-dimensional

output

T yk = éë yk ,1, , yk ,d ùû . Let  U k the d-dimensional channel vector of user U k , and ck to be the sum of noise U ( Dk ) and the U 2-1 ( gk ) interference terms

ck = U ( gk ) + U 2-1 ( Dk ) ,

(138)

with invertible covariance matrix

Kc = ( s 2 + sg2 ) IK k

out

n min

å sw2 ¢ F ( Tj ( 

+

j ¹k

j

Uj

)) F ( T (  )) , j

Uj

†

(139)

for which

( Kc

k

-1

+ xx† )

1 = Kc k

1 1 Kxx† Kc c

k

k

1 1+ x† Kx c

,

(140)

k

which further can be decomposed as

Kc = U å U † , k

where U is a unitary, and å is a diagonal matrix. User U k ’s a decoder first utilizes 30

(141)

K0.5 = U å 0.5 U † , c

(142)

k

( (

0.5 F Tk  U k where å 0.5 is a diagonal matrix with the square rooted elements of å , and Kc

( (

) ) zk + U 2-1 ( gk ) + U ( Dk ) ,

on his system yk = F Tk  U k

F ( Dk ) Î  ( 0, KD

k

( (

=F Tk  U k

(

0, 2sw2 0

†

Uk

k

-0.5 ck

where z k Î 

where U 2-1 ( gk ) Î  ( 0, Kg

k

),

))

and

) , which precisely leads to

( K F ( T (  ) )) K = ( K F ( T (  ) )) K -0.5 ck

k

-0.5 yk ck

( Tk (  U ) ) zk + ( K-c 0.5F ( Tk (  U ) )) K-c 0.5ck † † ) ) K-c 1F ( Tk (  U ) ) zk + F ( Tk (  U ) ) K-c 1 (U ( Dk ) + U 2-1 ( gk ) ),

k

Uk

†

†

-0.5 F ck

k

k

k

k

k

k

) , thus 

(143)

k

k

a

can be rewritten as

( (

1 a = KF Tk  U k c k

)) .

(144)

In particular, the result in Equation (144) satisfies that

I ( z k : yk ) = I ( z k : a† yk ) ,

(145)

where z k and yk are independent conditioned on a† yk . In fact, in Equation (145), a† yk provides a sufficient statistic S ( a† yk ) to decode z k from yk , since projecting the vector yk onto the unit vector

( (

vk = F Tk  U k yields

))

( (

))

(146)

) ) zk + ck ,

(147)

F Tk  U k

( (

vk† yk = F Ti  U k where ci Î  ( 0, 2s 2 + sg2 ) .

The optimality also requires from a that the estimate zk of z k from vector yk minimizes the 2ù é mean square error  ê z k - zk ú . ë û The estimation zk can be rewritten as

( (

) ) zk + Ac†ck ,

zk = Ac† yk = Ac†F Tk  U k

(148)

where c = 1 , where A is a constant [18] expressed as

A=

2 2ù é  ê zk ú c†F Tk  U k ë û 2 2ù é  ê z k ú c†F Tk  U k + c† Kc c k ë û

( ( )) ( ( ))

†

( ( )) c + , † F ( Tk (  Uk ) ) c F Tk  U k

(149)

which leads to 2ù é  ê zk ú ë û E

))

2

=1+

2ù é  ê z k ú c†F Tk  U k ë û c† Kc c

))

2

=1+

2ù é  ê z k ú c†F Tk  U k ë û c† KD + Kg c

( ( k

( (

(

31

i

)

(150)

.

As follows, for

( (

1 c = KF Tk  U k c k

) ),

(151)

,

(152)

the error E is minimized as

E=

2ù é  ê zk ú ë û

( (

1+ SNIR F Tk  U k

))

2

thus the SNIR is maximized in 1 SNIR = sw2 ¢¢ K. c

(153)

k

Particularly, in the low-SNR regimes n min

( (

å sw2 ¢ F Tj  j ¹k

j

Uj

)) ( (

F Tj 

Uj

))

†

 0,

(154)

thus Equation (139) can be re-evaluated as

Kc = s 2 IK . k

(155)

out

Specifically, from the covariance matrix Kc , the signal-to-noise-plus-interference ratio in the k

multicarrier transmission is

( (

Precisely, if F T  U k

é SNIR = SNR =  ê sw2 ¢¢ ( s 2 IK out ëê

-1 ù

)

ú. ûú

(156)

) ) is time-invariant then the total rate that can be achieved by

Kout in-

stances of a equals to the overall mutual information. We apply the chain rule [18] of mutual information, from which

I ( z : y ) = I ( z 1,  , z K : y ) in

= I ( z 1 : y ) + I ( z 2 : y z 1 ) +  + I ( z K : y z 1 , , z K in

in -1

)

(157)

where

I ( z k : y z1, , z k -1 ) = I ( zk : y ¢ ) ,

(158)

such that k -1

( (

y ¢ = y - å F Tj  j =1

( (

= F Tk 

Uk

Uj

)) z

j

) ) zk + å F ( Tj (  )) + U ( D ) . Uj

(159)

j >k

Thus, Equation (158) can be rewritten as I ( z k : y z1, , zk -1 ) = I ( zk : a y ¢ ) ,

(160)

where the resulting function I ( z k : y z1, , zk -1 ) precisely picks up the information theoretic maximum. ■

32

5 Phase Space Coding for the Gaussian Subcarrier CVs Theorem 3. The transmission of fi channels

can

be

optimized

by

a

Gaussian subcarrier CVs over the  i Gaussian subrandom

phase

æ P (  ) = çç f1d , P2 f1d , , Pl f1d çè  P ( 1 ) P ( 2 )  P ( l )   

space

constellation

ö ÷÷÷ , where Pi , i = 2, , l ø

 (  )

as

is a random

permutation operator, and d (  ) = d  (  ) is the cardinality of  (  i ) . The optimality func i  j tion of  ( 

)

is o (  (  ) ) =

å ( nEve -

di

2

) , where d

is the normalized codeword differ-

i

l

ence. Proof. Assume that Alice and Bob agreed on the l Gaussian sub-channels,  i . We rewrite again the output of  i is as follows [3-4]:

yi = F (T (  i ) ) F (di ) + U ( Di ) ,

(161)

where variable di is the Gaussian subcarrier input of  i (for further details, see Section 2.1 and [4]). T

T

For two l-dimensional input code words dA = ( dA,1, dA,l )

and dB = (dB,1, dB,l ) , the prob-

ability that dA is distorted onto dB [18] on the output of the l Gaussian sub-channels conditioned on F ( T ( 

))

is as follows: Pr ( dA  dB F ( T ( 

æ

) ) ) = Q ççç è

sw2 ¢¢ 2s 2

å F (Ti (  i ))

2

l

di

2

÷÷ö , ÷÷ø

(162)

where Q is the Gaussian tail function, sw2 ¢¢ is the modulation variance derived in Equation (75), and di is the normalized difference [18] of dA,i and dB,i , calculated as follows: di =

1 s2 w ¢¢ s2 

(dA,i - dB,i ) .

(163)

Assuming the case that in Equation (162), the condition sw2 ¢¢

2s 2

å F (Ti (  i ) )

2

di

2

c

1 l 2R

.

(165)

Thus,

> cl

1 l l 2Rl

(166)

for any constant c > 0 and for an arbitrary pair of dA and dB . Thus, Equation (165) is a sufficient condition on the optimality on the structure of the phase space constellation  (  i ) . 33

In particular, at the achievable rate R per  i , the cardinality of  (  i ) is as follows:

 (  i ) = 2R .

(167)

Thus, each  (  i ) is precisely defined with 2R CV states fi

for each  i Gaussian sub-

channels, and  (  i ) contains 2lR constellation points fi in overall, that is,

 (  ) = 2lR .

(168)

Specifically, evaluating the Q ( ⋅ ) Gaussian tail function at

min

"F (Ti (  i ) )

F (Ti (  i ) ) leads to the

worst-case scenario as follows: æ Q çç çè

sw2 ¢¢

min

å F (Ti (  i ))

2s 2

"F (Ti (  i ) )

2

di

ö÷ ÷÷ , ø÷

2

l

(169)

such that for the l  i Gaussian sub-channels, the following constraint is satisfied: æ å log2 çççè1 + l

F ( T( 

))

2

sw2 ¢¢

s 2

ö÷ ÷= ø÷

æ å log2 çççè1 + l

2

F (Ti (  i ) ) sw2 ¢¢ s 2

ö÷ ÷ ³ lR , ø÷

(170)

where R is the average rate supported by each  i Gaussian sub-channels. Using Equation (163), the constraint of Equation (170) can be rewritten as follows [18]: æ Q (  ) ö å log2 ççèç1 + i di 2 i ø÷÷÷ ³ lR ,

(171)

l

where 2

2

F (Ti (  i ) ) di sw2 ¢¢

Qi (  i ) =

(172)

s 2

and 1 "Qi ³0 2

min

å Qi

=

l

sw2 ¢¢

min

"F (Ti (  i ) )

2s 2

å F (Ti (  i ))

2

di

2

.

(173)

l

In particular, from the modulation variance allocation algorithm of Section 3, some calculations reveal that

å F (Ti (  i )) "F (T (  ) ) min i

i

2

can be reevaluated precisely as

l

2

å F (Ti (  i ) ) "F (T (  ) ) min i

i

l

=

å l

1 s2 w ¢¢ s2 

æ çç u çè Eve

1 di

2

ö - 1 ÷÷. ÷ø

(174)

From Equations (173) and (174), the Gaussian tail function in Equation (169) can be rewritten as [18] æ 2 ö Q çç 21 å nEve - di ÷÷÷ (175) ÷ø çè l

(

)

and æ

å log2 ççèç uEve l

1 di

2

ö÷ ÷= ø÷

æ

å log2 ççèç 1 + l

Qi (  i ) ö ÷ di

2

÷ = lR . ø÷

(176)

This constraint on uEve allows us to design phase space constellation for each  i Gaussian subchannels, such that the achievable rate R over each sub-channel is maximized. With these results 34

in mind, we design a phase space constellation for reliable simultaneous transmission over the Gaussian sub-channels, assuming the presence of an optimal Gaussian collective attack [4]. As one can readily conclude, the optimality of sub-channel phase space constellation  (  ) can be straightforwardly derived from Equations (175) and (176). The optimality of  (  ) can explicitly be quantified via the optimality function o ( ⋅ ) , calculated as follows:

o (  (  ) ) =

å ( nEve -

di

2

l

).

Assuming that  (  ) contains l phase space constellations  (  i ) with 2R

(177)

fi

Gaussian

subcarrier CVs in each  (  i ) , but with independent  (  i ) sub-channel constellations for each  i , leads to [18] 2

å di

2

=

l -1

å

1 s2 w ¢¢ s2 

l -1

(dA,i - dB,i )

=0

(178)

and

æ

å log2 ççèç uEve l

Thus, the optimality function converges to zero, o (  ( 

1 di

2

ö ÷÷÷ = 0 . ø

(179)

)) = 0 .

(180)

From Equation (180), it straightforwardly follows that independent phase space constellations  (  i ) for the Gaussian sub-channels  i cannot lead to an optimal solution. In particular, a more sophisticated phase space constellation is required for a nonzero optimality function. Specifically, it can be achieved by the so-called random permutation coding [18-20], in which each  (  i ) of  i maximizes o (  (  i ) ) for "i . In a phase space constellation P , each  i Gaussian sub-channel has a different random phase space constellation P (  i ) such that if the two Gaussian subcarrier CVs fi

and fj , j ¹ i

are close to each other in P (  i ) of  i , then in the P (  j ) of  j , the distance  between fi

and fj

in P (  j ) is at least double the minimum distance of these Gaussian CVs in

P (  i ) . Without loss of generality, for two sub-channels  1 and  2 , with two reference constellation symbols fi as follows:

and fj , j ¹ i in P (  1 ) of  1 , the ¶ minimal distance function [18] is defined

¶ ( P (  1 ) ) = min ( dA,i - dA, j ) , "dA,i

whereas for sub-channel  2 , the distance between fi and fj

35

(181) is precisely yielded as follows:

¶ ( P (  2 ) ) = min (dA,i - dA, j ) "dA,i

(182)

=  ⋅ ¶ ( P (  1 ) ), where  ³ 2 .

Smoothing the nonnegativity condition in Equation (173), a lower bound on o ( P (  i ) ) is yielded as follows [18]:

o ( P (  i ) ) ³ l 2R d1l

2l

- å di

2

,

(183)

l

¢ = s 2 max li2 ³ 0 , where for n min n min

å di

2

 0.

(184)

l

Thus, the lower bound is

o ( P (  i ) ) ³ l 2R d1l

2l

.

(185)

The constellation P (  j ) of  j is a result of a random one-to-one permutation map P on

P (  i ) of  i , i ¹ j , as follows: P (  j ) = P ( P (  i ) ) . For an l-dimensional input d of  1l , a random constellation point fi

(186) in P (  1 ) is selected

for  1 . Then for the remaining l - 1 Gaussian sub-channel  2l , a random permutation map is applied for Pi for  i . Thus, the d (  ) constellation points f1d  i  ( i )

of P ( 

)

formulate P ( 

P (  ) = P (  1 ) , , P ( l ) æ , P2 f1d , , Pl f1d = çç f1d çè  P ( 1 ) P ( 2 )  P ( l )   

)

as follows:

÷ö÷ . ÷ø

(187)

In particular, an important property of Equation (187) is that the Pi permutation operators are drawn from a  uniform distribution, thus each Pi operator occurs with probability

p=

1 2lR

.

(188)

Some calculations yield that for the average inverse of Equation (166), the following upper bound holds for the l - 1 random permutation operators Pi taken on a pair of Gaussian subcarriers

{d1, d2 } : é ù ê 1 1 ⋅ å d1 -d2 2 P2 (d1 )-P2 (d2 ) 2  Pl (d1 )-Pl (d2 ) 2 úú £ l l Rl , 2l ê 2lR ( 2lR -1 ) êë úû d1 ¹d2 where for the uniform at random Pi , i = 2 l , permutation operators, the relation æ ö÷ çç 1 1 å çç å d1 -d2 2 P2 (d1 )-P2 (d2 ) 2  Pl (d1 )-Pl (d2 ) 2 ÷÷÷÷ £ l l Rl 2lR 2lR ø d1 è d1 ¹d2 P

36

(189)

(190)

is straightforwardly yielded. The structure of the phase space constellation P (  i ) for two sub-channels 1 and  2 with

d P (  ) = 16 [18] and  = 2 , is illustrated in Fig. 9. i 

Figure 9. The phase space constellations P (  1 ) and P (  2 ) for Gaussian subcarrier CVs, at two Gaussian sub-channels  1 and  2 , at  = 2 . In each constellation, d P (  ) = 16 , i 

fi , i = 1, , d (  ) , where P (  2 ) = P ( P (  1 ) ) and P (  1 ) = P -1P (  2 ) , and P is a  i random permutation operator drawn from a  uniform distribution (x, position quadrature; p, momentum quadrature). Assuming that d1 =  = dl is satisfied, the Q function can be evaluated as follows:

æ Q çç çè

1 2

å ( nEve -

2j

£ d1 j

2

£ dj +1

2j

2

l

)

÷÷ö = l 2R - 1 d 2 , ( ) i ÷÷ø

(191)

d1 £  £ dl , with a largest index j such that

whereas for an ordered structure

dj

di

, where dl +1 = ¥ , Equation (173) is reduced to a maximization prob-

lem, as

æ ç max çç j 2Rl d1  dj çè

(

2

)

1 j

j ö 2÷ - å di ÷÷÷ . ÷ø i =1

(192)

Note that j = l results in d1 =  = dl . Thus, the condition coincidences with Equation (191). ■

37

6 Conclusions We proposed a singular layer transmission model for CVQKD. The CVQKD protocols have been introduced for practical reasons, most importantly for their easy implementation over standard telecommunication networks and flexible encoding and decoding capabilities. The singular layer of CVQKD injects an extra degree of freedom into the transmission to extend the achievable distances and to improve the secret key rates. The benefits of the singular layer have been demonstrated through the AMQD and AMQD-MQA schemes, which use Gaussian subcarrier CVs for the transmission. We derived the rate formulas and studied the properties of an optimal phase space constellation. The SIA scheme provides a solution for the eigenchannel interference cancellation, which phenomenon occurs at a partial channel side information. We also analyzed the properties of an information-theoretical optimal phase space decoding and defined a phase space constellation for the Gaussian sub-channels, which is provably optimal and offers unconditional security for the transmission. The results confirm that the additional degree of freedom brought in by the singular layer can be significantly exploited in the crucial low-SNR regimes, which is particularly convenient for long-distance CVQKD scenarios.

Acknowledgements The author would like to thank Professor Sandor Imre for useful discussions. The results are supported by the grant COST Action MP1006.

References [1]

[2] [3] [4] [5] [6] [7] [8] [9]

S. Pirandola, S. Mancini, S. Lloyd, and S. L. Braunstein, Continuous-variable Quantum Cryptography using Two-Way Quantum Communication, arXiv:quantph/0611167v3 (2008). F. Grosshans, P. Grangier, Reverse reconciliation protocols for quantum cryptography with continuous variables, arXiv:quant-ph/0204127v1 (2002). L. Gyongyosi, Multiuser Quadrature Allocation for Continuous-Variable Quantum Key Distribution, arXiv:1312.3614 (2013). L. Gyongyosi, Adaptive Multicarrier Quadrature Division Modulation for Continuousvariable Quantum Key Distribution, arXiv:1310.1608 (2013). 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). 38

[10] William Shieh and Ivan Djordjevic. OFDM for Optical Communications. Elsevier (2010). [11] L. Gyongyosi, Scalar Reconciliation for Gaussian Modulation of Two-Way Continuousvariable Quantum Key Distribution, arXiv:1308.1391 (2013). [12] 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). [13] M. Navascues, F. Grosshans, and A. Acin. Optimality of Gaussian Attacks in Continuous-variable Quantum Cryptography, Phys. Rev. Lett. 97, 190502 (2006). [14] R. Garcia-Patron and N. J. Cerf. Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution. Phys. Rev. Lett. 97, 190503 (2006). [15] 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) [16] 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). [17] S. Imre and L. Gyongyosi. Advanced Quantum Communications - An Engineering Approach. Wiley-IEEE Press (New Jersey, USA), (2012). [18] D. Tse and P. Viswanath. Fundamentals of Wireless Communication, Cambridge University Press, (2005). [19] David Middlet, An Introduction to Statistical Communication Theory: An IEEE Press Classic Reissue, Hardcover, IEEE, ISBN-10: 0780311787, ISBN-13: 978-0780311787 (1960) [20] Steven Kay, Fundamentals of Statistical Signal Processing, Volumes I-III, Prentice Hall, (2013) [21] P. S. Chow, Bandwidth optimized digital transmission techniques for spectrally shaped channels with impulse noise, Ph.D. thesis, Stanford University (1993). [22] W. Yu, J. M. Cioffi, On Constant Power Water-filling (2001) [23] A. Lozano, A. M. Tulino, S. Verdu, Optimum Power Allocation for Parallel Gaussian Channels With Arbitrary Input Distributions, IEEE Transactions on Information Theory, Vol. 52, No. 7 (2006).

39

Supplemental Information S.1 Notations The notations of the manuscript are summarized in Table S.1. Table S.1. Summary of notations. Notation

Description Bound operator for the extension of SVD onto a Hilbert

 ( M ) = U GV -1

space  ,  ( M ) = UV -1 ⋅ V GV -1 , where U is a partial isometry, V is unitary, and a measure space Z . Sets of singular operators S1 = { F1,U 2-1 } , S2 = {U 1,U 2-1 } .

S1 , S2

The

SVD

of

F (T) ,

where

F1-1, F1 Î Kin ´Kin

and

U 2 ,U 2-1 Î Kout ´Kout are unitary matrices, Kin and Kout

F ( T ) = U 2 GF1-1 ,

refer to the number of sender and receiver users such that

F ( T ) = U 2 GU 1-1

Kin £ Kout ,

F1-1F1 = F1F1-1 = I , U 2U 2-1 = U 2-1U 2 = I ,

and G Î  is a diagonal matrix with non-negative real diagonal elements li , F ( T ) =

ån

min

liU 2,i F1,-i 1 .

The non-negative real diagonal elements of the diagonal

l1 ³ l2 ³  ln

li2

n min

s

si

min

matrix

GÎ ,

called

the

eigenchannels

of

F ( T ) = U 2 GF1-1 . The n min squared eigenchannels li2 are the eigenvalues of †

F ( T ) F ( T ) = U 2 GGTU 2-1 . n min = min ( K in , Kout ) , equals to the rank of F ( T ) , where Kin £ Kout . Stream matrix, s = ( s1, , sn

T

min

)

Î  ( 0, Ks ) , defined by

the unitary F1 (U 1 ) applied on z Î  ( 0, Kz ) . A stream variable si that identifies the CV state si phase space  . Expressed as

40

in the

si ¢ = liU 2,i F1,-i 1 si , and

s¢ = F ( T ) s = The

ån

min

liU 2,i F1,-i 1 si .

Fourier-transformed

(

eigenchannel

)

2ù é KU -1 ( g ) = sg2 =  ê gi ú , 2 i i ë û

U 2-1 ( gi ) Î  0, KU -1 ( g ) , 2 i

U 2-1 ( gi )

U 2-1 ( gi ) = U 2-1

(å

n min j ¹i

interference,

)

ljU 2, j F1,-j1 s j .

The

variance

sg2  0 , in the low-SNR regimes. 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 position and momentum quadratures

Dx , Dp Î  ( 0, s 2 ) ,

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 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 ( ⋅ )

Inverse FFT transform, applied by the encoder.

sw2 sw2 =

1 l

coder, continuous-variable unitary operation. tinuous-variable unitary operation.

Single-carrier modulation variance. 0

ål sw2

i

Multicarrier modulation variance. Average modulation variance of the l Gaussian sub-channels  i .

41

The i-th Gaussian subcarrier CV of user U k , where IFFT stands for the Inverse Fast Fourier Transform, fi Î  ,

= F

-1

( z k ,i )

( ) Î  ( 0, s ) ,

é 2ù sd2 =  ê di ú , i ë û

di Î  0, sd2 ,

fi = IFFT ( zk ,i )

i

= di .

xd

i

2 wF

(

pd Î  0, sw2 i

F

)

di = xd + ipd , i

are

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 subcarrier CV, expressed as F ( di

)=

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

Gaussian quantum channel.

 i , i = 1, , n

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

quantifies

the position and momentum quadrature transmission, with

Ti (  i )

(normalized)

real

0 £ ReTi (  i ) £ 1

2,

and

imaginary

0 £ Im Ti (  i ) £ 1

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. The subset of allocated users,  Í K . Only the allocated

 ÍK

users can transmit information in a given (particularly the j-th) AMQD block. The cardinality of subset  is  . An allocated user from subset  Í K .

U k , k = 1, , 

A d-dimensional, zero-mean, circular symmetric complex T

z = x + ip = ( z 1,  , z d )

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

42

T

(

T

x = ( x1, , xd ) , p = ( p1, , pd ) , with x i Î  0, sw2

(

pi Î  0, sw2

0

0

),

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

An l-dimensional, zero-mean, circular symmetric complex random Gaussian vector of the l Gaussian subcarrier CVs, T  ( 0, Kd ) , Kd =  éê dd† ùú , d = (d1, , dl ) , di = x i + ipi , ë û

(

d = F -1 ( z )

x i , pi Î  0, sw2 variables,

F

)

are i.i.d. zero-mean Gaussian random

sw2 = 1 sw2 . F

(

The

i-th

component

is

0

)

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

(

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

yk , m

yk , m =

ål F (Ti (  i ))F (di ) + F ( Di ) .

T Fourier transform of T (  ) = ëéT1 (  1 ) ,Tl ( l ) ûù Î  l ,

F ( T (  ))

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 )

2

An

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

TEve,i = ReTEve,i + i Im TEve,i Î  ,

0 £ Im TEve,i £ 1

Rk

2 , 0 £ TEve,i

2

0 £ ReTEve,i £ 1

2,

< 1.

Transmission rate of user U k . A di subcarrier in an AMQD block. For subset  Í K

di

with  users and l Gaussian sub-channels for the transmission, di =

43

1 n



å k =1 zke

-i2 pik n ,i

= 1, , l .

The min { n1, , nl } minimum of the ni sub-channel coeffi-

n min

F (Ti (  i ) )

Modulation

sw2 = nEve - n min ( d )p

nEve =

sw2

2

cients, where ni = s 2

1 l

variance,

, l = F (T*

)

2

=

1 n

and ni < nEve .

n

n

å i =1 å k =1Tk*e

(x )

,

-i 2 pik n

where 2

and

T* is the expected transmittance of the Gaussian sub-

channels under an optimal Gaussian collective attack. Additional sub-channel coefficient for the correction of modulation imperfections. For an ideal Gaussian modulation, nk = 0 , while for an arbitrary p ( x )

nk

(

nk = n min 1 -  ( d )p

(x )

) , where k =

distribution

(

1

nEve -n min  ( d )p

(x )

-1

)

.

The set of  i Gaussian sub-channels from the set of l good T

U éë j ùû = éë  1, ,  s ùû k

sub-channels that transmit the s subcarriers of user U k in the j-th AMQD block. The constant modulation variance sw2 ¢ for eigenchannel li , i

sw2 ¢ i

æ ö evaluated as sw2 ¢ = m - ççç s 2 max li2 ÷÷÷ = i è ø n min total constraint sw2 ¢ =

ån

min

sw2 ¢ = i

1 l

1 n min

ål sw2

sw2 ¢ , with a

= sw2 .

i

The modulation variance of the AMQD multicarrier transmission

in

the

SVD

environment.

Expressed

as

æ ö sw2 ¢¢ = nEve - ççç s 2 max li2 ÷÷÷ , where li is the i-th eigenè ø n min sw2 ¢¢

channel of F ( T ) , max li2 is the largest eigenvalue of n min

†

F (T)F (T) , 1 l

with

a

total

constraint

ål sw2 ¢¢ = sw2 ¢¢ > sw2 . i

The covariance matrix Kk , s of user U k . In the first setting of Kk , s

AMQD-MQA

{

it

it is evaluated as Kk , s 44

expressed

as

} F . In the second setting, = U diag { s , , s }U .

Kk , s = F1diag sw2 ¢ , , sw2 ¢ k ,1

is

k ,nmin

1

-1 1

2 wk¢ ,1

2 wk¢ ,n

min

-1 1

The minimum of the ni¢ -s of the l Gaussian sub-channels

¢ n min

¢ =  i , obtained at the SVD of F ( T ) , n min

s 2 max li2

.

n min

Difference

P

n min

of

¢ , n min

and 2

¢ = s 2 max F (Ti (  i ) ) - s 2 max li2 . P = n min - n min "i

n min

The statistical model of F ( T ) at a partial channel side information, S ( F ( T ) ) = xK-1 GxK , where xK-1 in

out

S ( F ( T ))

out

and xK

in

are unitaries that formulate the input covariance matrix

Ks = xK ÃxK-1 , in

while

Ã

is

a

diagonal

matrix,

in

{

Ks = Qdiag sw2 ¢ , , sw2 ¢ 1

Kin

}Q . †

Diagonal matrix, the entries of à are chosen such that

à = sw2 ¢

1 Kin

IK

in

à = sw2 ¢

1 Kin

IK , where sw2 ¢ = in

ån

min

sw2 ¢ is the total coni

straint. Projector,

i

i ( si¢

) = ^i ,

where set

^i =

n min

{ s j¢ }j =1 ,

j ¹ i consists of n min CV states s j¢ , such that each s j¢ is orthogonal to the sub-space  i spanned by the n min - 1 eigenchannels { ljU 2, j F1,-j1 } , j = 1, , n min , j ¹ i .

Wi

V ki ki

The operator of the decoder-side interference cancellation,

Wi = i † i ( liU 2,i F1,-i 1 ) . Encoder-side interference cancellation operator. Defines a phase space constellation  and performs the precoding. Naive precoding with unbounded additional modulation variance, ki = ji + ( -U 2-1 gi

)=

Precoding

boundary

with

a

ki = min { EC k ji - U 2-1 gi

ji - U 2-1 gi . condition,

}.

Optimal precoding with a boundary condition and maxiki

mum likelihood nearest neighbor decision rule at a ,

ki = min { EC k ji - aU 2-1 gi

45

}.

Equivalence class of ji . Identifies a replication of ji

EC k ji

the phase space by EC k ji

in

of W .

Parameter of a maximum likelihood nearest neighbor deci-

a

sion decoder a , a = sw2 sw2 + s 2 .

W

Equivalence class set, W º { EC k ji

dEC

}k =1 .

Phase space constellation  constructed by V . Superset



 W =  { EC k

of W equivalence classes  =

D

V =

d

{ ji }i =1

dEC

}k =1.

ji

D

A d-dimensional random phase space code. Radius of an uncertainty sphere around a ji¢ , in which

r

sw2 s 2

the transmit Gaussian ji lies, r =

sw2 + s 2

.

Maximum likelihood nearest neighbor decision rule decoder, with a

a

perr =

constraint, a = sw2 s 2

sw2 + s 2

sw2 2 sw + s 2

, and error probability

.

Minimal distance between arbitrary constellation points,

hmin

hmin = 2sw . The

sum

of

U ( Di ) Î  ( 0, KD

terms

(

i

)

U 2-1 ( gi ) Î  0, KU -1 ( g ) , 2 i ci Î  ( 0, Kc ) ,

ci

= s 2 IK

Kin

out

( (

+ å sw2 ¢¢F Tj U j ¹i

j

1 a = KF ( Ti ( U c i

i

Kc = KD + KU -1 ( g ) i i i 2 †

j

)) F ( T (  )) , j

and

Uj

)) .

æ The Gaussian tail function, Q çç çè

Q

and

ci = U 2-1 ( gi ) + U ( Di ) ,

where

i

)

1 2

å ( nEve -

di

2

l

ö

) ø÷÷÷÷ .

Random phase space permutation constellation for the tran-

P ( 

)

smission of the Gaussian subcarriers, expressed as æ P (  ) = P (  1 ) , , P ( l ) = çç f1d , P2 f1d , , Pl f1d çè  P ( 1 )  P ( 2 )  P ( l )   

where fi

ö÷ ÷, ø÷

are the Gaussian subcarrier CVs, Pi , i = 2, , l

46

is a random permutation operator, d (  ) = d (  ) is the  i j  cardinality

 (  i ) .

of

o (  (  ) ) =

å ( nEve -

The

di

2

l

optimality

function

is

).

Cardinality of  (  i ) .

d P (  ) i 

The normalized difference of two Gaussian subcarriers dA,i

di

1

and dB,i , di =

s2 w ¢¢ s2 

(dA,i - dB,i ) .

The optimality function, quantifies the optimality of the o (  ( 

))

phase

space

o (  (  ) ) =

 (  ) ,

constellation

å ( nEve -

di

l

2

as

).

Difference function of Gaussian subcarriers (phase space symbols)

¶

dA,i

and

dA, j

in

constellations

P (  k ) ,

k = 1, , l . For two Gaussian sub-channels 1 and  2 , ¶ ( P (  1 ) ) = min (dA,i - dA, j ) , "dA,i

¶ ( P (  2 ) ) =  ⋅ ¶ ( P (  1 ) ) , where  ³ 2 , j ¹ i .

zi

Estimation of input

zi = Ac† yi = Ac†F ( Ti ( U

a , such that

at decoder i

) ) zi + Ac†ci ,

minimizes the

2ù é mean square error  ê z i - zi ú , where ë û

zi

A=

2 é 2ù  ê zi ú c†F Ti U i ë û 2 2ù z i ú c†F Ti U + c† Kc c i i û

( (

é ê ë

( (

1 c = KF ( Ti ( U c i

i

))

†

))

( ( i )) c + , † F ( Ti ( U ) ) c i

) ) , and

Kc = s 2 IK

F Ti U

i

out

in the low-SNR

regimes.

S.2 Abbreviations AMQD AWGN CV CVQFT

Adaptive Multicarrier Quadrature Division Additive White Gaussian Noise Continuous-Variable Continuous-Variable Quantum Fourier Transform

47

DV EC FFT IFFT MQA OFDM OFDMA QKD SIA SNR SNIR SVD

Discrete-Variable Equivalence Class Fast Fourier Transform Inverse Fast Fourier Transform Multiuser Quadrature Allocation Orthogonal Frequency-Division Multiplexing OFDM Multiple Access Quantum Key Distribution Singular Interference Avoider Signal-to-Noise Ratio Signal-to-Noise-plus-Interference Ratio Singular Value Decomposition

48