1 Introduction

The continuous-variable quantum key distribution protocols allow for the parties to establish an unconditionally secure communication over the traditional telecommunication networks [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In comparison to discrete-variable (DV) QKD, the CVQKD schemes do not require single-photon devices, a fact that allows us to implement in practice by standard devices [21,22,23,24,25,26,27,28,29,30,31]. Despite the several attractive benefits of CVQKD [1,2,3, 9,10,11,12,13,14], these protocols still require significant performance improvements to be comparable with that of the traditional telecommunications [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49]. For this purpose, the multicarrier CVQKD has been defined through the adaptive multicarrier quadrature division (AMQD) modulation [5]. In a multicarrier CVQKD system, the Gaussian input CVs are granulated into subcarrier Gaussian CVs via the inverse Fourier transform, which are decoded by the receiver by the unitary CVQFT (continuous-variable quantum Fourier transform) operation [5]. Precisely, the multicarrier transmission divides the physical Gaussian link into several Gaussian sub-channels, where each sub-channel is dedicated for the conveying of a Gaussian subcarrier CV. In particular, the multicarrier transmission injects several benefits to CVQKD, such as improved secret key rates, higher tolerable excess noise, and enhanced transmission distances [5, 6, 8, 9, 29,30,31, 50,51,52,53,54,55,56,57]. Specifically, the benefits of the multicarrier CVQKD modulation has been extended to a multiple-access CVQKD through the AMQD-MQA (multiuser quadrature allocation) scheme [8], in which the simultaneous reliable transmission of the legal users is handled through the sophisticated allocation mechanism of the Gaussian subcarriers. The SVD-assisted (singular value decomposition) AMQD injects further extra degrees of freedom into the transmission [29, 52], which can also be exploited in a multiple-access multicarrier CVQKD.

The achievable secret key rates in a multicarrier CVQKD setting have been rigorously derived in [6]. These secret key rates also confirm the multimode bounds determined in [43] (see the results on fundamental rate-loss scaling in quantum optical communications in [43]). For further information on the bounds of private quantum communications, we suggest [44].

Here, we provide a natural representation of the subcarrier CV transmission and show that it also allows us to utilize the physical attributes into the sub-channel modeling. The angular domain representation is a useful tool in traditional telecommunications to model the physical signal propagation through a communication channel. The angular domain representation is aimed at revealing and verifying the connections of the physical layer and the mathematical channel model in different scenarios, avoiding the use of an inaccurate channel representation [58,59,60]. Here, we show that similar benefits can be brought up to multicarrier CVQKD. We define the subcarrier domain representation for multicarrier CVQKD. The subcarrier domain utilizes the phase-space angles of the Gaussian subcarrier CV to construct the model of a Gaussian sub-channel and to build an appropriate statistical model of subcarrier transmission. The subcarrier domain is an adequate application for multicarrier CVQKD since it is a natural representation of the CVQFT operation. The key behind the subcarrier domain representation is the Fourier operation, which has a central role in a multicarrier CVQKD setting since this operation makes possible the construction of Gaussian subcarrier CVs from the single-carriers and Gaussian sub-channels from the physical Gaussian link.

Particularly, the CVQFT transformation not just opens the door for the characterization of the subcarrier domain of a Gaussian sub-channel but also provides us a framework to study the effects of psychical layer transmission in an experimental multicarrier CVQKD scenario. The subcarrier domain utilizes physical attributes such as the phase space angle into the description of 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.

The novel contributions of our manuscript are as follows:

  1. 1.

    We define the subcarrier domain of multicarrier continuous-variable quantum key distribution.

  2. 2.

    The subcarrier domain injects physical attributes to the description of the subcarrier transmission.

  3. 3.

    We prove that the subcarrier domain is a natural representation of the subcarrier-level transmission in a multicarrier CVQKD scheme.

  4. 4.

    We 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.

  5. 5.

    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.

This paper is organized as follows. In Sect. 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, Sect. 5 concludes the results. Supplemental information is included in the Appendix.

2 Preliminaries

The basic terms and notations of multicarrier CVQKD are summarized in Sect. A.1 of the Appendix. The detailed description of AMQD and AMQD-MQA can be found in [5] and [8]. For the secret key rate formulas, see [6].

3 Subcarrier Domain of Multicarrier CVQKD

Proposition 1

(Subcarrier domain representation of multicarrier transmission.) For the i-th Gaussian sub-channel \({ {{\mathcal {N}}}}_{i} \), the subcarrier domain representation is \({{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( { {{\mathcal {N}}}}_{i} \right) \right) =UT_{i} \left( { {{\mathcal {N}}}}_{i} \right) U\), where U is the CVQFT operator.

Proof

The proofs throughout assume l Gaussian sub-channels for the multicarrier transmission. The angles of the \({\left| \phi _{i} \right\rangle } \) transmitted and the \({\left| \phi '_{i} \right\rangle } \) received subcarrier CVs in the phase space \({{{\mathcal {S}}}}\) are denoted by \(\theta _{i}^{\mathrm{*}} \in \left[ 0,2\pi \right] \), and \(\theta _{i} \in \left[ 0,2\pi \right] \), respectively.

Specifically, first, we express \(F\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) \) as

$$\begin{aligned} \begin{aligned} F\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right)&=\sum \limits _{l}{F\left( {{T}_{i}}\left( {{{\mathcal {N}}}_{i}} \right) \right) } \\&=\sum \limits _{i=0}^{l-1}{\sum \limits _{k=0}^{l-1}{{{T}_{k}}{{e}^{\tfrac{-i2\pi ik}{l}}}}}, \end{aligned} \end{aligned}$$
(1)

from which \(F\left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \) is yielded as

$$\begin{aligned} F\left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) =\sum _{k=0}^{l-1}T_{k} e^{{\textstyle \frac{-\mathrm{i}2\pi ik}{l}} } . \end{aligned}$$
(2)

Next, we recall the attributes of a multicarrier CVQKD transmission from [5]. In particular, assuming l Gaussian sub-channels, the output \({\mathbf {y}}\) is precisely as follows:

$$\begin{aligned} \begin{aligned} {\mathbf {y}}&=F\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) F\left( {\mathbf {d}} \right) +F\left( \Delta \right) \\&=F\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) F\left( {{F}^{-1}}\left( {\mathbf {z}} \right) \right) +F\left( \Delta \right) \\&=\left( F\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) F \right) {\mathbf {d}}+F\left( \Delta \right) \\&=\sum \limits _{l}{F\left( {{T}_{i}}\left( {{{\mathcal {N}}}_{i}} \right) \right) {{d}_{i}}+F\left( {{\Delta }_{i}} \right) }, \end{aligned} \end{aligned}$$
(3)

where

$$\begin{aligned} F\left( {\mathbf {d}} \right) =F\left( {{F}^{-1}}\left( {\mathbf {z}} \right) \right) ={\mathbf {z}}. \end{aligned}$$
(4)

The l columns of the \(l\times l\) unitary matrix F formulate basis vectors, which are referred to as the domain \({{\mathcal R}}_{\phi _{i} } \), from which the subcarrier domain representation \({{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{\mathcal {N}}}_{i} \right) \right) \) of \(T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \) is defined as

$$\begin{aligned} {{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{\mathcal {N}}}_{i} \right) \right) =F\left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) F. \end{aligned}$$
(5)

Thus, (3) can be rewritten as

$$\begin{aligned} \begin{aligned} {{{\mathbf {y}}}^{{{{\mathcal {R}}}_{\phi }}}}&={{{\mathcal {R}}}_{\phi }}\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) {\mathbf {d}}+F\left( \Delta \right) \\&=\sum \limits _{l}{{{{\mathcal {R}}}_{{{\phi }_{i}}}}\left( {{T}_{i}}\left( {{{\mathcal {N}}}_{i}} \right) \right) {{d}_{i}}+F\left( {{\Delta }_{i}} \right) }, \end{aligned} \end{aligned}$$
(6)

where \({{{\mathbf {y}}}^{{{{\mathcal {R}}}_{\phi }}}}\) is referred to as the subcarrier domain representation of \({\mathbf {y}}\).

Particularly, from (5) follows that \({{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \) can be expressed as

$$\begin{aligned} {{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{\mathcal {N}}}_{i} \right) \right) =UT_{i} \left( {{{\mathcal {N}}}}_{i} \right) U, \end{aligned}$$
(7)

where U is an \(l\times l\) unitary matrix as

$$\begin{aligned} U=F, \end{aligned}$$
(8)

where F refers to the CVQFT operator which for l subcarriers can be expressed by an \(l\times l\) matrix, as

$$\begin{aligned} U={\textstyle \frac{1}{\sqrt{l} }} e^{{\textstyle \frac{-\mathrm{i}2\pi ik}{l}} } , i,k=0,\ldots ,l-1, \end{aligned}$$
(9)

thus,

$$\begin{aligned} {{{\mathcal {R}}}_{{{\phi }_{i}}}}\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) =\sum \limits _{i=1}^{l}{U\left( {{T}_{i}}\left( {{{\mathcal {N}}}_{i}} \right) \right) }. \end{aligned}$$
(10)

To conclude, the results in (1) and (2) can be rewritten as

$$\begin{aligned} U\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) =\sum \limits _{i=1}^{l}{U\left( {{T}_{i}}\left( {{{\mathcal {N}}}_{i}} \right) \right) }, \end{aligned}$$
(11)

thus

$$\begin{aligned} F\left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) =U\left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) . \end{aligned}$$
(12)

Specifically, an arbitrary distributed \({{{\mathcal {R}}}_{\phi }}\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) \) can be approximated via an averaging over the following statistics:

$$\begin{aligned} {{{\mathcal {S}}}}\left( {{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \right) \in {{{\mathcal {C}}}{{\mathcal {N}}}}\left( 0,\sigma _{{\mathbf {T}}\left( {{{\mathcal {N}}}}\right) }^{2} \right) , \end{aligned}$$
(13)

by theory.

Since the unitary U operation does not change the distribution of \({{{\mathcal {S}}}}\left( {\mathbf {T}}\left( {{\mathcal {N}}}\right) \right) \), an arbitrarily distributed \({\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \) can be approximated via an averaging over the statistics of

$$\begin{aligned} {{{\mathcal {S}}}}\left( {\mathbf {T}}\left( {{\mathcal {N}}}\right) \right) \in {{{\mathcal {C}}}{{\mathcal {N}}}}\left( 0,\sigma _{{\mathbf {T}}\left( {{{\mathcal {N}}}}\right) }^{2} \right) . \end{aligned}$$
(14)

\(\square \)

Theorem 1

(Subcarrier domain of a Gaussian sub-channel). The \({{\mathcal R}}_{\phi _{i} } \) subcarrier domain representation of \({{{\mathcal {N}}}}_{i} \), \(i=0\ldots l-1\), is \({{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) =\sum _{k}\mathrm{A} \left( {{{\mathcal {N}}}}_{i} \right) b\left( {k/ l} \right) ^{\dag } b\left( \cos \theta _{i} \right) b\left( \cos \theta _{i}^{\mathrm{*}} \right) ^{\dag } b\left( {i /l} \right) \), \(k=0\ldots l-1\), where \(b\left( \cdot \right) \) is an orthonormal basis vector of \({{{\mathcal {R}}}}_{\phi _{i} } \), \(\theta _{i}^{\mathrm{*}} \) and \(\theta _{i} \) are the phase-space angles of \({\left| \phi _{i} \right\rangle } \) and \({\left| \phi '_{i} \right\rangle } \), \(\mathrm{A} \left( {{{\mathcal {N}}}}_{i} \right) =x_{i} \), where \(x_{i} \) is a real variable, \(x_{i} \ge 0\).

Proof

The b basis vectors of \({{{\mathcal {R}}}}_{\phi _{i} } \) are evaluated as follows. Let \(\theta _{i}^{\mathrm{*}} \in \left[ 0,2\pi \right] \) refer to the angle of the i-th noise-free input Gaussian subcarrier CV \({\left| \phi _{i} \right\rangle } \) in \({{{\mathcal {S}}}}\). The angle of the i-th noisy subcarrier CV \(\left| {{\phi _{i}^{\prime }}} \right\rangle \) is referred to as \(\theta _{i} \in \left[ 0,2\pi \right] \), \(\theta _{i} \ne \theta _{i}^{\mathrm{*}} \).

In particular, for the subcarrier domain representation, the scaled CVQFT operation defines the \(b\left( \cdot \right) \) basis at l Gaussian sub-channels as an \(l\times 1\) matrix:

$$\begin{aligned} b\left( \cos {{\theta }_{i}} \right) =\tfrac{1}{\sqrt{l}}\left( \begin{matrix} 1 \\ {{e}^{-i2\pi \cos {{\theta }_{i}}}} \\ {{e}^{-i2\pi 2\cos {{\theta }_{i}}}} \\ \vdots \\ {{e}^{-i2\pi \left( l-1 \right) \cos {{\theta }_{i}}}} \\ \end{matrix} \right) , \end{aligned}$$
(15)

while for the input angle \(\theta _{i}^{{*}} \) also defines an \(l\times 1\) matrix as

$$\begin{aligned} b\left( \cos \theta _{i}^{\text {*}} \right) =\tfrac{1}{\sqrt{l}}\left( \begin{matrix} 1 \\ {{e}^{-i2\pi \cos \theta _{i}^{{*}}}} \\ {{e}^{-i2\pi 2\cos \theta _{i}^{{*}}}} \\ \vdots \\ {{e}^{-i2\pi \left( l-1 \right) \cos \theta _{i}^{\text {*}}}} \\ \end{matrix} \right) . \end{aligned}$$
(16)

Precisely, the difference of the cos functions of the i-th \(\theta _{i}^{\mathrm{*}} \) transmitted and the \(\theta _{i} \) received angles is defined as

$$\begin{aligned} \tau _{i} =\cos \theta _{i} -\cos \theta _{i}^{\mathrm{*}} . \end{aligned}$$
(17)

Let \(b\left( \cos \theta _{i} \right) \) and \(b\left( \cos \theta _{i}^{\mathrm{*}} \right) \) be the basis vectors of \(\cos \theta _{i} ,\cos \theta _{i}^{\mathrm{*}} \) [58,59,60], then for the \(\Omega _{i} =\theta _{i} -\theta _{i}^{\mathrm{*}} \) angle

$$\begin{aligned} \left| \cos \Omega _{i} \right| =\left| b\left( \cos \theta _{i}^{\mathrm{*}} \right) ^{\dag } b\left( \cos \theta _{i} \right) \right| , \end{aligned}$$
(18)

and

$$\begin{aligned} \left| \cos \Omega _{i} \right| =\left| f\left( \tau _{i} \right) \right| , \end{aligned}$$
(19)

where [58]

$$\begin{aligned} f\left( \tau _{i} \right) ={\textstyle \frac{1}{l}} e^{\mathrm{i}\pi \left( l-1\right) \left( \cos \theta _{i} -\cos \theta _{i}^{\mathrm{*}} \right) } {\textstyle \frac{\sin \left( \pi l\left( \cos \theta _{i} -\cos \theta _{i}^{\mathrm{*}} \right) \right) }{\sin \left( \pi \left( \cos \theta _{i} -\cos \theta _{i}^{\mathrm{*}} \right) \right) }} . \end{aligned}$$
(20)

In particular, using (20), after some calculations, the result in (18) can be rewritten as

$$\begin{aligned} \left| \cos \Omega _{i} \right| =\left| {\textstyle \frac{\sin \left( \pi l\left( \cos \theta _{i} -\cos \theta _{i}^{\mathrm{*}} \right) \right) }{l\sin \left( \pi \left( \cos \theta _{i} -\cos \theta _{i}^{\mathrm{*}} \right) \right) }} \right| . \end{aligned}$$
(21)

Specifically, by expressing (20) via the formula of

$$\begin{aligned} \mathrm{sinc}\left( x\right) =\sin \left( \pi x\right) {\textstyle \frac{1}{\pi x}} , \end{aligned}$$
(22)

one can find that for \(l\rightarrow \infty \), \(f\left( \tau _{i} \right) \) can be rewritten as [58]

$$\begin{aligned} \mathop {\lim }\limits _{l\rightarrow \infty } f\left( \tau _{i} \right) =e^{\mathrm{i}\pi l\tau _{i} } \mathrm{sinc}\left( l\tau _{i} \right) . \end{aligned}$$
(23)

The function \(\left| f\left( \tau _{i} \right) \right| \) for different values of \(\tau _{i} \) is depicted in Fig. 1. The function picks up the \(\left| f\left( \tau _{i} \right) \right| =1\) maximum at \(\tau _{i} =0\), with a period \(r=1\). For l sub-channels, a period yields l values.

Fig. 1
figure 1

The function \(\left| f\left( \tau _{i} \right) \right| \) (blue) for different values of \(\tau _{i} \). The period of the function is \(r=1\). The sinc function (green) is approximated with an arbitrary precision in the asymptotic limit of \(l\rightarrow \infty \)

Full size image
Fig. 2
figure 2

The \({{{\mathcal {R}}}}_{\phi _{i} } \) subcarrier domain representation of \({{{\mathcal {N}}}}_{i} \), at \(\theta _{i}^{\text {*}}={\pi }/{2}\;\), \(l=2\). The function \(\left| f\left( \tau _{i} \right) \right| \) picks up the maximum at \(\tau _{i} =0\)

Full size image

Next, we utilize function \(f\left( \cdot \right) \) to derive the \({{{\mathcal {R}}}}_{\phi _{i} } \) subcarrier domain representation of \({{{\mathcal {N}}}}_{i} \). Function \(f\left( \tau _{i} \right) \) at a given \(\theta _{i} \) formulates a plot

$$\begin{aligned} \mathrm{p:}\left( \cos \theta _{i} ,\left| f\left( \tau _{i} \right) \right| \right) , \end{aligned}$$
(24)

where from the \(r=1\) periodicity of \(f\left( \cdot \right) \) follows that the main loops are obtained at

$$\begin{aligned} \cos \theta _{i} =\cos \theta _{i}^{\mathrm{*}} . \end{aligned}$$
(25)

The plot \(\left| f\left( \tau _{i} \right) \right| \) as a function of \(\cos \theta _{i} \) is depicted in Fig. 2, for \(\theta _{i}^{\text {*}}={\pi }/{2}\;\), \(l=2\).

From the bases \(b\left( \cdot \right) \) of (15) and (16), the Fourier bases \(b\left( {\textstyle \frac{k}{l}} \right) \) and \(b\left( {\textstyle \frac{i}{l}} \right) \), \(i,k=0,\ldots , l-1\), is defined as follows.

The set \(\mathrm{S}_{b} \) of the \({{{\mathcal {R}}}}_{\phi } \) orthonormal basis over the \({{{\mathcal {C}}}}^{l} \) complex space of the \({{{\mathcal {R}}}}_{\phi } \) subcarrier domain representation can be defined as

$$\begin{aligned} \mathrm{S}_{b} =\left\{ b\left( 0\right) ,b\left( {\textstyle \frac{1}{l}} \right) ,\ldots ,b\left( {\textstyle \frac{l-1}{l}} \right) \right\} \in {{{\mathcal {C}}}}^{l} , \end{aligned}$$
(26)

and \(b\left( {\textstyle \frac{k}{l}} \right) \) is an \(l\times 1\) matrix as

$$\begin{aligned} b\left( 0 \right) =\tfrac{1}{\sqrt{l}}\left( \begin{matrix} 1 \\ 1 \\ 1 \\ \vdots \\ 1 \\ \end{matrix} \right) , \end{aligned}$$
(27)

and

$$\begin{aligned} b\left( \tfrac{k}{l} \right) =\tfrac{1}{\sqrt{l}}\left( \begin{matrix} 1 \\ {{e}^{\tfrac{-i2\pi k}{l}}} \\ {{e}^{\tfrac{-i2\pi 2k}{l}}} \\ \vdots \\ {{e}^{\tfrac{-i2\pi \left( l-1 \right) k}{l}}} \\ \end{matrix} \right) , \end{aligned}$$
(28)

while the \(b\left( {\textstyle \frac{i}{l}} \right) \)\(l\times 1\) matrix is precisely as

$$\begin{aligned} b\left( \tfrac{i}{l} \right) =\tfrac{1}{\sqrt{l}}\left( \begin{matrix} 1 \\ {{e}^{\tfrac{-i2\pi i}{l}}} \\ {{e}^{\tfrac{-i2\pi i2}{l}}} \\ \vdots \\ {{e}^{\tfrac{-i2\pi i\left( l-1 \right) }{l}}} \\ \end{matrix} \right) . \end{aligned}$$
(29)

Precisely, using the orthonormal basis of (26), the result in (20) can be rewritten as

$$\begin{aligned} f\left( \tau _{i} \right) =b\left( 0\right) ^{\dag } b\left( \tau _{i} \right) . \end{aligned}$$
(30)

Specifically, the expression of (30) allows us to redefine the plot of (24) to express \(b\left( {\textstyle \frac{k}{l}} \right) \) as follows:

$$\begin{aligned} \mathrm{p}_{b\left( {\textstyle \frac{k}{l}} \right) } \mathrm{:}\left( \cos \theta _{i} ,\left| f\left( \cos \theta _{i} -{\textstyle \frac{k}{l}} \right) \right| \right) , \end{aligned}$$
(31)

and thus the maximum values are obtained at

$$\begin{aligned} \cos \theta _{i} ={\textstyle \frac{k}{l}} . \end{aligned}$$
(32)

Particularly, at a given l, evaluating f at \(k=1,\ldots ,l-1\) yields the following values [58]:

$$\begin{aligned} f\left( {\textstyle \frac{k}{l}} \right) =0, \end{aligned}$$
(33)

and

$$\begin{aligned} f\left( {\textstyle \frac{-k}{l}} \right) =f\left( {\textstyle \frac{l-k}{l}} \right) . \end{aligned}$$
(34)

In particular, the \(\mathrm{A} \left( {{{\mathcal {N}}}}_{i} \right) \) parameter is called the virtual gain of the \({{{\mathcal {N}}}}_{i} \) sub-channel transmittance coefficient, and without loss of generality, it is defined as

$$\begin{aligned} \mathrm{A} \left( {{{\mathcal {N}}}}_{i} \right) =x_{i} , \end{aligned}$$
(35)

where \(x_{i} \) is a real variable, \(x_{i} \ge 0\).

From (15), (16), and (35), \({\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \) can be expressed as

$$\begin{aligned} {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) =\sum _{i=0}^{l-1}\mathrm{A} \left( {{{\mathcal {N}}}}_{i} \right) b\left( \cos \theta _{i} \right) b\left( \cos \theta _{i}^{\mathrm{*}} \right) ^{\dag } . \end{aligned}$$
(36)

By exploiting the properties of the Fourier transform [58,59,60], for a given \(\cos \theta _{i}^{\mathrm{*}} \) and \(\cos \theta _{i} \), \({{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \) can be rewritten as

$$\begin{aligned} \begin{aligned} {{{\mathcal {R}}}_{\phi }}\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right)&=\sum \limits _{i=0}^{l-1}{\sum \limits _{k=0}^{l-1}{b{{\left( \tfrac{k}{l} \right) }^{\dagger }}{{T}_{i}}\left( {{{\mathcal {N}}}_{i}} \right) b\left( \tfrac{i}{l} \right) }} \\&=\sum \limits _{i=0}^{l-1}{\sum \limits _{k=0}^{l-1}{b{{\left( \tfrac{k}{l} \right) }^{\dagger }}\mathrm{A} \left( {{{\mathcal {N}}}_{i}} \right) b\left( \cos {{\theta }_{i}} \right) b{{\left( \cos \theta _{i}^{\text {*}} \right) }^{\dagger }}b\left( \tfrac{i}{l} \right) }} \\&=\sum \limits _{i=0}^{l-1}{\sum \limits _{k=0}^{l-1}{{{x}_{i}}b{{\left( \tfrac{k}{l} \right) }^{\dagger }}b\left( \cos {{\theta }_{i}} \right) b{{\left( \cos \theta _{i}^{\text {*}} \right) }^{\dagger }}b\left( \tfrac{i}{l} \right) }}. \end{aligned} \end{aligned}$$
(37)

Specifically, in (37), for the representation of term \(b{{\left( \cos \theta _{i}^{\text {*}} \right) }^{\dagger }}b\left( {i}/{l}\; \right) \) [58], set \(S^{{{\mathcal R}}_{\phi _{i} } } \) can be defined for the \({{\mathcal R}}_{\phi _{i} } \) domain of \({{{\mathcal {N}}}}_{i} \) as

$$\begin{aligned} S^{{{{\mathcal {R}}}}_{\phi } } :\left| \cos \theta _{i} -\left( {\textstyle \frac{k}{l}} \right) \right| <{\textstyle \frac{1}{l}} . \end{aligned}$$
(38)

The set \(S^{{{{\mathcal {R}}}}_{\phi } } \) in the subcarrier domain representation for \(\theta _{i}^{\text {*}}={\pi }/{2}\;\), \(l=2\) and \(k=0,\ldots , l-1\), is illustrated by the dashed area in Fig. 3. The \(b\left( 0\right) ,b\left( {\textstyle \frac{1}{l}} \right) ,\ldots ,b\left( {\textstyle \frac{l-k}{l}} \right) \) basis vectors of \({{{\mathcal {R}}}}_{\phi } \) for \(l=2\) are also depicted, evaluated via (31).

Putting the pieces together, from (37), \({{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \) of a given Gaussian sub-channel \({{\mathcal {N}}}_{i} \) is yielded as

$$\begin{aligned} \begin{aligned} {{{\mathcal {R}}}_{{{\phi }_{i}}}}\left( {{T}_{i}}\left( {{{\mathcal {N}}}_{i}} \right) \right)&=\sum \limits _{k=0}^{l-1}{b{{\left( \tfrac{k}{l} \right) }^{\dagger }}{{T}_{i}}\left( {{{\mathcal {N}}}_{i}} \right) b\left( \tfrac{i}{l} \right) } \\&=\sum \limits _{k=0}^{l-1}{\mathrm{A} \left( {{{\mathcal {N}}}_{i}} \right) b{{\left( \tfrac{k}{l} \right) }^{\dagger }}b\left( \cos {{\theta }_{i}} \right) b{{\left( \cos \theta _{i}^{\text {*}} \right) }^{\dagger }}b\left( \tfrac{i}{l} \right) }. \end{aligned} \end{aligned}$$
(39)

\(\square \)

Fig. 3
figure 3

The set \(S^{{{{\mathcal {R}}}}_{\phi } } \) (dashed areas) for \(\theta _{i}^{\text {*}}={\pi }/{2}\;\), \(l=2\) Gaussian sub-channels and \(k=0,\ldots l-1\). The curves (red, green) depict the basis vectors \(b\left( 0\right) ,b\left( {\textstyle \frac{1}{l}} \right) \) of \({{{\mathcal {R}}}}_{\phi } \)

Full size image

3.1 Statistics of Subcarrier Domain Sub-channel Transmission

Theorem 2

(Transmittance of the Gaussian sub-channels). For arbitrary \(\theta _{i}^{\mathrm{*}} \), the magnitude \(\left| {{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \right| \) of \({{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{\mathcal {N}}}_{i} \right) \right) \) of a given \({{{\mathcal {N}}}}_{i} \) is maximized in the asymptotic limit of \(\cos \Omega _{i} \rightarrow 1\), where \(\Omega _{i} =\theta _{i} -\theta _{i}^{\mathrm{*}} \). Averaging over the statistics \({{{\mathcal {S}}}}\left( {{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \right) \in {{{\mathcal {C}}}{{\mathcal {N}}}}\left( 0,\sigma _{{\mathbf {T}}\left( {{{\mathcal {N}}}}\right) }^{2} \right) \), \(rank\left( {{\mathcal S}}\left( {{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \right) \right) =\min \left( \left| S_{i} \right| ,\left| S_{k} \right| \right) \), where the cardinality of sets \(S_{i} ,S_{k} \) identifies the number of non-zero rows and columns of \({{{\mathcal {R}}}}_{\phi _{i} } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \).

Proof

First, we recall \({{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \) from (39) and express \(\left| {{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \right| \) as

$$\begin{aligned} \left| {{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \right| =\left| \sum _{k=0}^{l-1}x_{i} b\left( {\textstyle \frac{k}{l}} \right) ^{\dag } b\left( \cos \theta _{i} \right) b\left( \cos \theta _{i}^{\mathrm{*}} \right) ^{\dag } b\left( {\textstyle \frac{i}{l}} \right) \right| . \end{aligned}$$
(40)

For a given \(\theta _{i}^{\mathrm{*}} \), the values of angle \(\theta _{i} \) has the following statistical impacts on (40).

Without loss of generality, let parameters i and k be fixed as

$$\begin{aligned} i,k=\left\{ C,C\right\} , \end{aligned}$$
(41)

where \(C>0\) is a real variable.

Particularly, for the \({{{\mathcal {N}}}}_{i} \) sub-channels, the \(\left| {{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \right| \) magnitudes formulate a set

$$\begin{aligned} \partial =\left\{ \left| {{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \right| , i=0,\ldots ,l-1\right\} . \end{aligned}$$
(42)

Let

$$\begin{aligned} \Omega _{i} =\theta _{i} -\theta _{i}^{\mathrm{*}} , \end{aligned}$$
(43)

and let s be the number of sub-channels for which \(\left| {{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \right| \approx 0\).

Specifically, for \(\partial \), let determine \(\Omega _{i} \) the value of k as

$$\begin{aligned} k=\left\{ \begin{array}{l} {k\in \left[ 0,2C\right] ,\; \text {if}\mathrm{\; }\left| \Omega _{i} \right| \rightarrow \pi } \\ {k=i=C,\; \text {if}\mathrm{\; }\left| \Omega _{i} \right| =0\mathrm{\; \; }} \end{array}\right\} . \end{aligned}$$
(44)

Let us define a \({{{\mathcal {G}}}}_{0} \) initial subset with \(\left| {{{\mathcal {G}}}}_{0} \right| =s_{0} \), as

$$\begin{aligned} {{{\mathcal {G}}}}_{0} =\left\{ \left| {{{\mathcal {R}}}}_{\phi _{j} } \left( T_{j} \left( {{{\mathcal {N}}}}_{j} \right) \right) \right| ,j=0,\ldots ,s_{0} -1\right\} \subseteq \partial , \end{aligned}$$
(45)

where

$$\begin{aligned} \left| {{{\mathcal {R}}}}_{\phi _{j} } \left( T_{j} \left( {{{\mathcal {N}}}}_{j} \right) \right) \right| \approx 0. \end{aligned}$$
(46)

In this setting, as \(\cos \Omega _{i} \rightarrow 1\), the cardinality of \({{{\mathcal {G}}}}\) increases,

$$\begin{aligned} {{{\mathcal {G}}}}:\cos \Omega _{i} \rightarrow 1:\left| {{\mathcal G}}\right| =s>\left| {{{\mathcal {G}}}}_{0} \right| =s_{0} , \end{aligned}$$
(47)

while as \(\cos \Omega _{i} \rightarrow -1\), the cardinality of \({{{\mathcal {G}}}}\) decreases, thus

$$\begin{aligned} {{{\mathcal {G}}}}:\cos \Omega _{i} \rightarrow -1:\left| {{\mathcal G}}\right| =s<\left| {{{\mathcal {G}}}}_{0} \right| =s_{0} . \end{aligned}$$
(48)

In particular, as (48) holds, the range of k expands from C to the full domain of \(k=\left[ 0,2C\right] \), and \(\left| {{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \right| \) decreases, thus

$$\begin{aligned} \left| {{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \right| \approx a, \end{aligned}$$
(49)

where a is an average which around the \(\left| {{\mathcal R}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \right| \) coefficients stochastically moves [58,59,60].

The impact of \(\cos \Omega _{i} \rightarrow -1\) on \(\left| {{\mathcal R}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \right| \) is depicted in Fig. 4 for \(i,k=\left\{ C,C\right\} \), \(i=C=0.5l\). The maximum transmittance is normalized to unit for \(k=0.5C\). Statistically, the convergence of \(\cos \Omega _{i} \rightarrow -1\) improves the range of k and decreases the sub-channel transmittance (see (44)).

Fig. 4
figure 4

The impact of \(\cos \Omega _{i} \rightarrow -1\) on \(\left| {{{\mathcal {R}}}}_{\phi _{i} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \right| \) for a fixed \(i=C=0.5l\) on a sub-channel \({{{\mathcal {N}}}}_{i} \). The parameter range changes to \(i,k=\left\{ C,\left[ 0,2C\right] \right\} ,C=0.5l\), from \(i,k=\left\{ C,C\right\} \). For \(\cos \Omega _{i} \rightarrow 1\), the transmittance picks up the maximum at \(k=C=0.5l\) (red) in a narrow range of \(k\approx C\). Statistically, as \(\cos \Omega _{i} \rightarrow -1\), the transmittance significantly decreases, moving stochastically around an average a (dashed grey line) within the full range \(k=\left[ 0,2C\right] \)

Full size image

Particularly, the degrees of freedom in \({{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \) can be evaluated through the rank of \({{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \).

Let us identify the number of non-zero rows and columns of \({{{\mathcal {R}}}}_{\phi _{i} } \left( {\mathbf {T}}\left( {{\mathcal {N}}}\right) \right) \) via \(\left| S_{i} \right| \) and \(\left| S_{k} \right| \), of sets \(S_{i} ,S_{k} \), respectively. By averaging [58,59,60] over the statistics of

$$\begin{aligned} {{{\mathcal {S}}}}\left( {{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \right) \in {{{\mathcal {C}}}{{\mathcal {N}}}}\left( 0,\sigma _{T\left( {{\mathcal {N}}}\right) }^{2} \right) , \end{aligned}$$
(50)

thus the rank of \({{{\mathcal {S}}}}\left( {{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \right) \) without loss of generality is expressed as

$$\begin{aligned} \begin{aligned} \text {rank}\left( {\mathcal {S}}\left( {{{\mathcal {R}}}_{\phi }}\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) \right) \right)&=\min \left( \left| {{S}_{i}} \right| ,\left| {{S}_{k}} \right| \right) \\&\approx \min \left( \sum \nolimits _{l}{\cos {{\theta }_{i}},\sum \nolimits _{l}{\cos \theta _{i}^{\text {*}}}} \right) , \end{aligned} \end{aligned}$$
(51)

for an arbitrary distribution, by theory [58].

The rank in (51) basically changes in function of the number l of \({{{\mathcal {N}}}}_{i} \) Gaussian sub-channels utilized for the multicarrier transmission since the increasing l results in more non-zero elements in \({{{\mathcal {S}}}}\left( {{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{\mathcal {N}}}\right) \right) \right) \) [58,59,60]. On the other hand, the rank in (51) also changes in function of \(\theta _{i} \). Specifically, as \(\cos \Omega _{i} \rightarrow 1\), the matrix \({{{\mathcal {S}}}}\left( {{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \right) \) will have significantly decreased number of non-zero entries (see (47)), while for \(\cos \Omega _{i} \rightarrow -1\), the rank increases because the number of non-zero entries in (51) increases [58] (see (48)).

These statements can be directly extended to the diversity, since the \(\text {div}\left( \cdot \right) \) diversity function of \({{{\mathcal {S}}}}\left( {{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \right) \) is evaluated via the number of non-zero entries in \({{\mathcal S}}\left( {{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \right) \),

$$\begin{aligned} \text {div}\left( {\mathcal {S}}\left( {{{\mathcal {R}}}_{\phi }}\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) \right) \right) =\bigcup \limits _{\forall i,k}{{{E}_{i,k}}\left( {\mathcal {S}}\left( {{{\mathcal {R}}}_{\phi }}\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) \right) \right) \ne 0}, \end{aligned}$$
(52)

where \(E_{i,k} \) identifies an \(\left( i,k\right) \) entry of \({{{\mathcal {S}}}}\left( {{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \right) \right) \). Precisely, from (52) follows that \(\text {div}\left( {{{\mathcal {S}}}}\left( {{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{\mathcal {N}}}\right) \right) \right) \right) \) increases with the number l of Gaussian sub-channels. \(\square \)

4 Subcarrier Domain of Multiuser Multicarrier CVQKD

Lemma 1

(Subcarrier domain of multiple-access multicarrier CVQKD). The \({{{\mathcal {R}}}}_{\phi } \left( {\mathbf {T}}\left( {{\mathcal {N}}}\right) \right) \) of \({\mathbf {T}}\left( {{{\mathcal {N}}}}\right) \) in a \(K_{in} ,K_{out} \) multiuser setting is \({{{\mathcal {R}}}}_{\phi }^{K_{in} ,K_{out} } \left( {\mathbf {T}}\left( {{\mathcal {N}}}\right) \right) =U_{K_{out} } {\mathbf {T}}\left( {{\mathcal {N}}}\right) U_{K_{out} } \), where \(U_{K_{out} } \) is a \(K_{out} \times K_{out} \) unitary \(U_{K_{out} } ={\textstyle \frac{1}{\sqrt{K_{out} } }} e^{{\textstyle \frac{-\mathrm{i}2\pi ik}{K_{out} }} } \), \(i,k=0,\ldots ,K_{out} -1\).

Proof

Let \(K_{in} ,K_{out} \) be the number of transmitter and receiver users in a multiple access multicarrier CVQKD [8], and let \({\mathbf {Z}}\) be the \(K_{in} \) dimensional input of the \(K_{in} \) users. The Gaussian CV subcarriers formulate the \(K_{in} \) dimensional vector

$$\begin{aligned} {\mathbf {D}}=U_{K_{in} } {\mathbf {Z}}, \end{aligned}$$
(53)

where \(U_{K_{in} } \) stands for the inverse CVQFT unitary operation.

The \(U_{K_{in} } \) and \(U_{K_{out} } \), \(K_{in} \times K_{in} \), \(K_{out} \times K_{out} \) unitary matrices at l Gaussian sub-channels are as

$$\begin{aligned} U_{K_{in} } ={\textstyle \frac{1}{\sqrt{K_{in} } }} e^{{\textstyle \frac{\mathrm{i}2\pi ik}{K_{in} }} } , i,k=0,\ldots ,K_{in} -1, \end{aligned}$$
(54)

and

$$\begin{aligned} U_{K_{out} } ={\textstyle \frac{1}{\sqrt{K_{out} } }} e^{{\textstyle \frac{-\mathrm{i}2\pi ik}{K_{out} }} } , i,k=0,\ldots ,K_{out} -1, \end{aligned}$$
(55)

which unitary is the CVQFT operation (For further details, see the properties of the multicarrier CVQKD modulation [5, 6, 8, 9].).

Specifically, the output \({\mathbf {Y}}\) in a \(K_{in} ,K_{out} \) setting is then yielded as

$$\begin{aligned} \begin{aligned} {\mathbf {Y}}&={{U}_{{{K}_{out}}}}{\mathbf {T}}\left( {\mathcal {N}} \right) \left( {{U}_{{{K}_{out}}}}{\mathbf {D}} \right) +{{U}_{{{K}_{out}}}}\Delta \\&=\left( {{U}_{{{K}_{out}}}}{\mathbf {T}}\left( {\mathcal {N}} \right) {{U}_{{{K}_{out}}}} \right) {\mathbf {D}}+{{U}_{{{K}_{out}}}}\Delta \\&={\mathcal {R}}_{\phi }^{{{K}_{in}},{{K}_{out}}}\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) {\mathbf {D}}+{{U}_{{{K}_{out}}}}\Delta , \end{aligned} \end{aligned}$$
(56)

thus without loss of generality

$$\begin{aligned} {\mathcal {R}}_{\phi }^{{{K}_{in}},{{K}_{out}}}\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) ={{U}_{{{K}_{out}}}}{\mathbf {T}}\left( {\mathcal {N}} \right) {{U}_{{{K}_{out}}}}. \end{aligned}$$
(57)

Particularly, by rewriting (37), \({{\mathcal R}}_{\phi }^{K_{in} ,K_{out} } \left( {\mathbf {T}}\left( {{\mathcal {N}}}\right) \right) \) can be expressed as

$$\begin{aligned} \begin{aligned}&{\mathcal {R}}_{\phi }^{{{K}_{in}},{{K}_{out}}}\left( {\mathbf {T}}\left( {\mathcal {N}} \right) \right) \\&\quad =\sum \limits _{i=0}^{{{K}_{in}}-1}{\sum \limits _{k=0}^{{{K}_{out}}-1} {{{b}_{{{K}_{out}}}}{{\left( \tfrac{k}{l} \right) }^{\dagger }}{{T}_{i}}\left( {{{\mathcal {N}}}_{i}} \right) {{b}_{{{K}_{in}}}}\left( \tfrac{i}{l} \right) }} \\&\quad =\sum \limits _{i=0}^{{{K}_{in}}-1}{\sum \limits _{k=0}^{{{K}_{out}}-1} {{{b}_{{{K}_{out}}}}{{\left( \tfrac{k}{l} \right) }^{\dagger }}\mathrm{A}\left( {{{\mathcal {N}}}_{i}} \right) {{b}_{{{K}_{out}}}}\left( \cos {{\theta }_{i}} \right) {{b}_{{{K}_{in}}}}{{\left( \cos \theta _{i}^{\text {*}} \right) }^{\dagger }}{{b}_{{{K}_{in}}}}\left( \tfrac{i}{l} \right) }} \\&\quad =\sum \limits _{i=0}^{{{K}_{in}}-1} {\sum \limits _{k=0}^{{{K}_{out}}-1}{{{x}_{i}}{{b}_{{{K}_{out}}}}{{\left( \tfrac{k}{l} \right) }^{\dagger }}{{b}_{{{K}_{out}}}}\left( \cos {{\theta }_{i}} \right) {{b}_{{{K}_{in}}}}{{\left( \cos \theta _{i}^{\text {*}} \right) }^{\dagger }}{{b}_{{{K}_{in}}}}\left( \tfrac{i}{l} \right) }}, \end{aligned} \end{aligned}$$
(58)

where the basis vectors are precisely as

$$\begin{aligned} {{b}_{{{K}_{out}}}}\left( \cos {{\theta }_{i}} \right) =\tfrac{1}{\sqrt{{{K}_{out}}}}\left( \begin{matrix} 1 \\ {{e}^{\tfrac{-i2\pi l\cos {{\theta }_{i}}}{{{K}_{out}}}}} \\ {{e}^{\tfrac{-i2\pi 2l\cos {{\theta }_{i}}}{{{K}_{out}}}}} \\ \vdots \\ {{e}^{\tfrac{-i2\pi \left( {{K}_{out}}-1 \right) l\cos {{\theta }_{i}}}{{{K}_{out}}}}} \\ \end{matrix} \right) \end{aligned}$$
(59)

and

$$\begin{aligned} {{b}_{{{K}_{in}}}}\left( \cos \theta _{i}^{\text {*}} \right) =\tfrac{1}{\sqrt{{{K}_{in}}}}\left( \begin{matrix} 1 \\ {{e}^{\tfrac{-i2\pi l\cos \theta _{i}^{\text {*}}}{{{K}_{in}}}}} \\ {{e}^{\tfrac{-i2\pi 2l\cos \theta _{i}^{\text {*}}}{{{K}_{in}}}}} \\ \vdots \\ {{e}^{\tfrac{-i2\pi \left( {{K}_{in}}-1 \right) l\cos \theta _{i}^{\text {*}}}{{{K}_{in}}}}} \\ \end{matrix} \right) , \end{aligned}$$
(60)

thus

$$\begin{aligned} {{b}_{{{K}_{out}}}}\left( \tfrac{k}{l} \right) =\tfrac{1}{\sqrt{{{K}_{out}}}}\left( \begin{matrix} 1 \\ {{e}^{\tfrac{-i2\pi k}{{{K}_{out}}}}} \\ {{e}^{\tfrac{-i2\pi 2k}{{{K}_{out}}}}} \\ \vdots \\ {{e}^{\tfrac{-i2\pi \left( {{K}_{out}}-1 \right) k}{{{K}_{out}}}}} \\ \end{matrix} \right) , \end{aligned}$$
(61)

and

$$\begin{aligned} {{b}_{{{K}_{in}}}}\left( \tfrac{i}{l} \right) =\tfrac{1}{\sqrt{{{K}_{in}}}}\left( \begin{matrix} 1 \\ {{e}^{\tfrac{-i2\pi i}{{{K}_{in}}}}} \\ {{e}^{\tfrac{-i2\pi i2}{{{K}_{in}}}}} \\ \vdots \\ {{e}^{\tfrac{-i2\pi i\left( {{K}_{in}}-1 \right) }{{{K}_{in}}}}} \\ \end{matrix} \right) . \end{aligned}$$
(62)

Without loss of generality, the function f from (20) can be rewritten as

$$\begin{aligned} f^{K_{out} } \left( \tau _{i} \right) ={\textstyle \frac{1}{K_{out} }} e^{{\textstyle \frac{\mathrm{i}\pi l\left( K_{out} -1\right) \tau _{i} }{K_{out} }} } {\textstyle \frac{\sin \left( \pi l\tau _{i} \right) }{\sin \left( \pi {\textstyle \frac{l}{K_{out} }} \tau _{i} \right) }} , \end{aligned}$$
(63)

with

$$\begin{aligned} \left| \cos \Omega _{i} \right| =\left| {\textstyle \frac{\sin \left( \pi l\left( \cos \theta _{i} -\cos \theta _{i}^{\mathrm{*}} \right) \right) }{K_{out} \sin \left( \pi {\textstyle \frac{l}{K_{out} }} \left( \cos \theta _{i} -\cos \theta _{i}^{\mathrm{*}} \right) \right) }} \right| , \end{aligned}$$
(64)

such that

$$\begin{aligned} f^{K_{out} } \left( {\textstyle \frac{k}{l}} \right) =0, \end{aligned}$$
(65)

and

$$\begin{aligned} f^{K_{out} } \left( {\textstyle \frac{-k}{l}} \right) =f^{K_{out} } \left( {\textstyle \frac{K_{out} -k}{l}} \right) , k=1,\ldots ,K_{out} -1. \end{aligned}$$
(66)

The maximum values of \(f^{K_{out} } \left( {\textstyle \frac{k}{l}} \right) \) are obtained at

$$\begin{aligned} \cos {{\theta }_{i}}={k}/{l}\;\text { }\bmod {{{K}_{out}}}/{l}\;. \end{aligned}$$
(67)

The subcarrier domain representation \({{{\mathcal {R}}}}_{\phi }^{K_{in} ,K_{out} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \) via \(\left| f^{K_{out} } \left( \tau _{i} \right) \right| \) for \(K_{out} >l\), at \(\theta _{i}^{\text {*}}={\pi }/{2}\;\), \(l=2\) is shown in Fig. 5.

Fig. 5
figure 5

The function \(\left| f^{K_{out} } \left( \tau _{i} \right) \right| \) of \({{{\mathcal {R}}}}_{\phi }^{K_{in} ,K_{out} } \left( T_{i} \left( {{{\mathcal {N}}}}_{i} \right) \right) \) for \(K_{out} >l\), at \(\theta _{i}^{\text {*}}={\pi }/{2}\;\), \(l=2\)

Full size image

The sets \(\mathrm{S}_{b_{K_{in} } } \), \(\mathrm{S}_{b_{K_{out} } } \) of the \(b_{K_{in} } \), \(b_{K_{out} } \) orthonormal bases of a \(K_{in} ,K_{out} \) setting are as

$$\begin{aligned} \mathrm{S}_{b_{K_{in} } } =\left\{ b_{K_{in} } \left( 0\right) ,b_{K_{in} } \left( {\textstyle \frac{1}{l}} \right) ,\ldots ,b_{K_{in} } \left( {\textstyle \frac{K_{in} -1}{l}} \right) \right\} \in {{{\mathcal {C}}}}^{K_{in} } , \end{aligned}$$
(68)

and

$$\begin{aligned} \mathrm{S}_{b_{K_{out} } } =\left\{ b_{K_{out} } \left( 0\right) ,b_{K_{out} } \left( {\textstyle \frac{1}{l}} \right) ,\ldots ,b_{K_{out} } \left( {\textstyle \frac{K_{out} -1}{l}} \right) \right\} \in {{{\mathcal {C}}}}^{K_{out} } . \end{aligned}$$
(69)

\(\square \)

5 Conclusions

We defined the subcarrier domain for multicarrier CVQKD. In a multicarrier CVQKD protocol, the characterization of the subcarrier domain of a Gaussian sub-channel is provided by the unitary CVQFT transformation, which has a central role in multicarrier CVQKD. The subcarrier domain injects physical attributes to the mathematical model of the Gaussian sub-channels. It provides a natural representation of multicarrier CVQKD and allows us to extend it to a multiple-access multicarrier CVQKD setting. The subcarrier domain representation is a general framework that can be utilized for an arbitrary multicarrier CVQKD scenario. The subcarrier domain also offers an apparatus to formulate the psychical model of the sub-channel transmission, which is particularly convenient for an experimental multicarrier CVQKD scenario.