Subcarrier Domain of Multicarrier Continuous-Variable Quantum Key Distribution

We propose the subcarrier domain of multicarrier continuous-variable (CV) quantum key distribution (QKD). In a multicarrier CVQKD scheme, the information is granulated into Gaussian subcarrier CVs and the physical Gaussian link is divided into Gaussian sub-channels. The sub-channels are dedicated for the conveying of the subcarrier CVs. The angular domain utilizes the phase-space angles of the Gaussian subcarrier CVs to construct the physical model of a Gaussian sub-channel. The subcarrier domain injects physical attributes to the description of the subcarrier transmission. We prove that the subcarrier domain is a natural representation of the subcarrier-level transmission in a multicarrier CVQKD scheme. We also extend the subcarrier domain to a multiple-access multicarrier CVQKD setting. We demonstrate the results through the adaptive multicarrier quadrature-division (AMQD) CVQKD scheme and the AMQD-MQA (multiuser quadrature allocation) multiple-access multicarrier scheme. The subcarrier domain representation provides a general apparatus that can be utilized for an arbitrary multicarrier CVQKD scenario. The framework is particularly convenient for experimental multicarrier CVQKD scenarios.

1 Introduction the transmission. Thus, the subcarrier domain representation takes into account not just the theoretical model but also the physical level of the subcarrier transmission. Since the subcarrier domain is a natural representation of a multicarrier CVQKD transmission, it allows us to extend it to a multiple-access multicarrier CVQKD setting. Furthermore, the subcarrier domain model provides a general framework for any experiential multicarrier CVQKD.
This paper is organized as follows. In Section 2, some preliminaries are briefly summarized. Section 3 discusses the subcarrier domain representation for multicarrier CVQKD. Section 4 extends the subcarrier domain for multiple-access multiuser CVQKD. Finally, Section 5 concludes the results. Supplemental information is included in the Appendix.

Multicarrier CVQKD
In this section we very briefly summarize the basic notations of AMQD from [2]. The following description assumes a single user, and the use of n Gaussian sub-channels for the transmission of the subcarriers, from which only l sub-channels will carry valuable information.
.The transmittance vector of in the multicarrier transmission is where is a complex variable, which quantifies the position and momentum quadrature transmission (i.e., gain) of the i-th Gaussian sub-channel , in the phase space , with real and imaginary parts Particularly, the variable has the squared magnitude of Re Im where ( ) (

Re Im
The Fourier-transformed transmittance of the i-th sub-channel (resulted from CVQFT operation at Bob) is denoted by 4 The n-dimensional zero-mean, circular symmetric complex Gaussian noise vector of the quantum channel , is evaluated as with independent, zero-mean Gaussian random components , and , of a Gaussian sub-channel , which identifies the Gaussian noise of the i-th sub-channel on the quadrature components in the phase space .
The CVQFT-transformed noise vector can be rewritten as with independent components and on the quadratures, for each . Precisely, it also defines an n-dimensional zero-mean, circular symmetric complex Gaussian random vector

Subcarrier Domain of Multicarrier CVQKD
The proofs throughout assume l Gaussian sub-channels for the multicarrier transmission. The angles of the i f transmitted and the i f¢ received subcarrier CVs in the phase space are denoted by , and , respectively.
from which is yielded as Next, we recall the attributes of a multicarrier CVQKD transmission from [2]. In particular, assuming l Gaussian sub-channels, the output y is precisely as follows: The l columns of the l unitary matrix formulate basis vectors, which are referred to as the domain , from which the subcarrier domain representation of is defined as Thus, (19) can be rewritten as where is referred to as the subcarrier domain representation of . Particularly, from (21) follows that can be expressed as where U is an l unitary matrix as where F refers to the CVQFT operator which for l subcarriers can be expressed by an l matrix, as To conclude, the results in (17) and (18) can be rewritten as thus, Specifically, an arbitrary distributed can be approximated via an averaging over the following statistics: ( ) ( by theory.
Since the unitary U operation does not change the distribution of , an arbitrarily distributed can be approximated via an averaging over the statistics of ■ Theorem 1 (Subcarrier domain of a Gaussian sub-channel). The subcarrier domain repre- , where is an orthonormal basis vector of , and are the phase-space The b basis vectors of are evaluated as follows: Let In particular, for the subcarrier domain representation, the scaled CVQFT operation defines the basis at l Gaussian sub-channels as an matrix: while for the input angle also defines an matrix as Precisely, the difference of the cos functions of the i-th transmitted and the received angles is defined as Let and be the basis vectors of [21][22][23], then for the where [21] ( ) cos cos 1 cos cos 1 sin cos cos In particular, using (36), after some calculations, the result in (34) can be rewritten as one can find that for l , can be rewritten as [21] The function ( ) and ( ) k l b is an matrix as 1 l´( Precisely, using the orthonormal basis of (42), the result in (36) can be rewritten as Specifically, the expression of (45) allows us to redefine the plot of (40) to express ( ) (49) In particular, the parameter is called the virtual gain of the sub-channel transmittance coefficient, and without loss of generality, it is defined as where is a real variable, .
By exploiting the properties of the Fourier transform [21][22][23], for a given and , can be rewritten as Specifically, in (52)

Statistics of Subcarrier Domain Sub-channel Transmission
, where the cardinality of sets identifies the number of non-zero rows and columns of .
First, we recall from For a given , the values of angle has the following statistical impacts on Without loss of generality, let parameters i and k be fixed as where is a real variable. 0 C > Particularly, for the sub-channels, the and let s be the number of sub-channels for which Specifically, for , let determine the value of k as ¶ Let us define a initial subset with In this setting, as cos , the cardinality of  increases, 1 i W  0 : cos 1 : , while as , the cardinality of  decreases, thus cos 1 In particular, as (63) holds, the range of k expands from C to the full domain of , and 0, 2 where a is an average which around the ( ) ( [21][22][23].
The impact of cos Fig. 4 for . The maximum transmittance is normalized to unit for . Statistically, the convergence of improves the range of k and decreases the sub-channel transmittance (see ).

For
, the transmittance picks up the maximum at k C (red) in a narrow range of k . Statistically, as cos , the transmittance significantly decreases, moving stochastically around an average a (dashed grey line) within the full range . )) for an arbitrary distribution, by theory [21].
The rank in (66) basically changes in function of the number l of Gaussian sub-channels utilized for the multicarrier transmission since the increasing l results in more non-zero elements in [21][22][23]. On the other hand, the rank in Let be the number of transmitter and receiver users in a multiple access multicarrier CVQKD [3], and let Z be the dimensional input of the users. The Gaussian CV subcar- where stands for the inverse CVQFT unitary operation.  [2] and [3][4][5][6].) Specifically, the output Y in a setting is then yielded as , where the basis vectors are precisely as The maximum values of ( )