Statistical quadrature evolution by inference for multicarrier continuous-variable quantum key distribution

We define the statistical quadrature evolution (QE) method for multicarrier continuous-variable quantum key distribution (CVQKD). A multicarrier CVQKD protocol uses Gaussian subcarrier quantum continuous variables for information transmission. The QE scheme utilizes the theory of mathematical statistics and statistical information processing. The QE model is based on the Gaussian quadrature inference (GQI) framework to provide a minimal error estimate of the CV state quadratures. The QE method evaluates a unique and stable estimation of the non-observable continuous input from the measurement results and through the statistical inference method yielded from the GQI framework. We prove that the QE method minimizes the overall expected error by an estimator function and provides a viable, easily implementable, and computationally efficient way to maximize the extractable information from the observed data. The QE framework can be established in an arbitrary CVQKD protocol and measurement setting and is implementable by standard low-complexity functions, which is particularly convenient for experimental CVQKD.

Here, we propose the statistical quadrature evolution (QE) method for multicarrier CVQKD. The QE extends the results of the Gaussian Quadrature Inference (GQI) [10] to evolve a minimal error estimate of the continuous input regime from noisy discrete variables and estimates. The QE method utilizes the theory of mathematical statistics and the fundamentals of statistical information processing. While the GQI operates on the measured noisy subcarrier CVs to estimate the continuous spectrum of the non-observable input subcarriers, the QE scheme functionally builds on the GQI output and on the measured discrete noisy subcarrier variables to evolve the continuous regime of the non-observable single-carrier inputs. The QE method achieves a theoretically minimized magnitude error to evolve the non-observable single-carrier continuous variable Gaussian quadratures from the observed discrete variables. The theoretical minimum error of QE is functionally provided by the GQI method [10] and by an optimal estimator function applied to the measurement results.
The QE block evaluates a unique estimation of the non-observable continuous input regime via the noisy subcarrier variables yielded from the measurements and by the subcarrier estimations yielded from the GQI. Precisely, the QE model utilizes a corresponding estimator function that provides an optimal least-square formulation 1 (optimal leastsquares estimator). In fact, the estimator function is a linear operator whose coefficients depend on the GQI output. The QE framework also solves the problem of the optimal least-square estimate being a non-linear function of the observed data (e.g., discrete subcarrier variables and GQI output). Particularly, it is explained by the mathematical fact that the optimal least-square estimator is a linear function of the observed data if both the non-observable input (Gaussian CV states) and the observed data (measured CV states) are jointly Gaussian [30,32], which is exactly the case in a CVQKD setting. The QE framework provides a viable, easily implementable, and computationally efficient way to maximize the extractable information from the observed data. Using the fundamental theory that stands behind the GQI and QE blocks, the proposed QE output provided by the least-square formulation is always a unique and stable solution, which minimizes the overall expected estimation error for each individual component of the input vector. Specifically, by exploiting the statistical framework of multicarrier CVQKD, we prove that QE in a multicarrier CVQKD setting achieves a vanishing error as the number of subcarriers dedicated to a given user increases. We derive the corresponding expected error and expected variance using the statistical model of multicarrier CVQKD and demonstrate the results through numerical evidence. The error and error variances and the corresponding covariances are derived via computationally efficient, easily implementable functions.
The QE framework can be established in an arbitrary CVQKD protocol, and can be applied with homodyne or heterodyne measurement settings. The methods of the framework are implementable by standard low-complexity functions, which are particularly convenient for experimental CVQKD.
The novel contributions of our manuscript are as follows.
4. The QE method minimizes the overall expected error by an estimator function and provides a viable, easily implementable, and computationally efficient way to maximize the extractable information from the observed data. 5. The QE framework can be established in an arbitrary CVQKD protocol and measurement setting and is implementable by standard low-complexity functions, which is particularly convenient for experimental CVQKD.
This paper is organized as follows. In Sect. 2, preliminary findings are summarized. Section 3 discusses the QE method for multiple access multicarrier CVQKD and derives the achievable statistical secret key rates. Numerical evidence is included in Sect. 4. Finally, Sect. 5 concludes the results. Supplemental information is included in the Appendix.

Gaussian quadrature inference
In this section, we summarize the basic terms of GQI; for the further details, see [10].
According to the GQI framework, the m where x U k , j ∈ N 0, σ 2 ω 0 , p U k , j ∈ N 0, σ 2 ω 0 are Gaussian random quadratures, σ 2 ω 0 is the single-carrier modulation variance [8], and let the m subcarrier CVs of U k be referred via where where σ 2 N U k ,i is the noise variance of N U k ,i , and Let n be the number of single carriers, n → ∞, and let where and where σ 2 ω 0 , σ 2 ω are the single-carrier and multicarrier modulation variances, respectively. Statistically, in a multicarrier CVQKD setting, the following relation can be written between a single-carrier x U k , j and subcarrier x U k ,i Gaussian quadrature component and Specifically, for any σ 2 ω < σ 2 ω 0 , it follows that = 1 (11) and therefore, x U k ,i in (9) can be rewritten as Note that in (9) it is assumed that the integral of (10) exists and is invertible; thus, x U k , j is either square integrable or absolutely integrable [30]. The x U k ,i noisy version of (9) is available for Bob via a corresponding M measurement operator (e.g., homodyne or heterodyne measurement) performed on the noisy φ i CV state, as In particular, the S e −iθ ϕ U k , j spectral density of x U k , j can be defined via the which is a statistical measure of the strength of the fluctuations of the subcarrier components [10,30]. It can be verified that (15) is analogous to the power spectrum e where e −iθ ϕ U k , j ≥ 0 is a real function of θ ϕ U k , j , such that where A x U k , j (·) is the autocorrelation function (autocorrelation sequence [30,32]) of x U k , j , expressed as Without loss of generality, (15) and (17), allow us to write Using (20), the estimation of U −1 x U k , j , where U −1 (·) is the inverse CVQFT unitary operation, is expressed as which, by using (17), can be further evaluated as In particular, E U −1 x U k , j allows us to uniquely specify A x U k ,i (g) of a noisy subcarrier quadrature x U k ,i as follows.
For a noisy subcarrier quadrature where is defined in (8), while A x U k , j (g) of x U k , j is as where G i e iθ ϕ U k , j is defined as where Note that T i N U k ,i can be determined via a pilot CV state-based channel estimation procedure (pilot: contains no valuable information); for details, see [20]. The Gaussian quadrature inference (GQI) for multicarrier CVQKD [10]. User U k is equipped with a logical channel The H x U k , j entropy rate of the Gaussian quadrature component x U k , j of ϕ U k , j is evaluated as The GQI scheme [10] for multicarrier CVQKD is briefly summarized in Fig. 1.
Note that for a Gaussian WSS x (n), since H (·) is a functional of P x (·) [30,32], and 3 Statistical quadrature evolution by Gaussian quadrature inference Theorem 1 (Quadrature evolution by Gaussian quadrature inference). Let ϕ U k , j = x U k , j + i p U k , j be the jth input CV of user U k , k = 0, . . . , K − 1, let E U −1 ϕ U k , j be the output of GQI, and let κ U k , j = F M φ U k ,i , i = 0, . . . , m − 1 be the F-transformed m noisy subcarrier CVs of U k at a measurement M. The E ϕ U k , j estimate of ϕ U k , j is E ϕ U k , j = ξ j κ U k , j , where ξ j is an optimal least-squares estimator yielded from E U −1 ϕ U k , j .
Proof Let For simplicity, in later parts we refer only to the quadrature component x U k , j of ϕ U k , j , which formulates the d-dimensional vector x U k as Let x U k ,i be the ith subcarrier quadrature component, resulting from a measurement M, where φ U k ,i is the ith noisy Gaussian subcarrier CV of U k . Let refer to the F-transformed (FFT) m noisy subcarrier CVs of U k associated with ϕ U k , j . Applying the F-operation for the The QE block uses the GQI output E U −1 x U k , j , the κ U k , j elements, and an optimal least-squares estimator ξ j , evaluated via the covariances cov Let E x U k , j refer to the estimate of x U k , j , and let error component e j be defined as with an expected error variance E σ 2 ξ j , In particular, to determine the d-dimensional estimator ξ , we introduce the covariance matrices cov x U k x U k , cov κ κ (39) and the cross-correlation [30] matrix Specifically, based on a d-dimensional x U k , the covariance matrix cov x U k x U k is a d ×d dimensional matrix, evaluated as where E U −1 x U k , j is the GQI output. Based on the d-dimensional κ, and a GQI output E U −1 x U k , j , the cov κ κ covariance matrix is expressed as where where G j e iθ ϕ U k , j is as and where such that for a noisy observation κ U k , j , Without loss of generality, the cross-correlation matrix cov x U k κ is evaluated as (23) as where * is the linear convolution [27,30], and whileÂ x U k ,i (·) refers to the autocorrelation coefficient associated with the x U k ,i subcarrier of user U k . As follows, (48) can be rewritten as by fundamental theory [30,32]. Particularly, from (43) and (48), the ξ optimal least-squares estimator for d-dimensional vectors is defined via cov x U k κ and cov κ κ as from which E x U k is defined precisely as Specifically, the estimation E x U k in (53) provides a d-dimensional error vector which has a covariance cov ee as by some fundamental theory [30,32]. The A i (g) autocorrelation coefficients are determined via the inference method of the GQE framework [10], as where λ 0 is a Lagrangian coefficient.
The QE quadrature evolution scheme is depicted in Fig. 2.

Statistical secret key rates
statistical secret key rate converges to the P M U k statistical private classical information of M U k , as m → ∞.
Proof To derive the statistical secret key rate from the statistical quadrature evolution, we can directly apply the results to the achievable secret key rates of the GQI block from [10].
Let sub-index ( j, Z , m) refer to the jth optimal single carrier at Z autocorrelation coefficients and m sub-channels in M U k dedicated to U k . Recalling the results of Theorem 3 and Lemma 1 from [10], the following relation holds at m and m + 1 sub-channels in M U k : where D AB (·) is the relative entropy between Alice and Bob, whilex refer to the spectral densities of Bob and Eve. Therefore, without loss of generality, refer to the statistical secret key rate of U k at m sub-channels in M (m) U k . For the GQI method, it is proven [10] where M (m) refer to the logical channel of U k at m, and m +1 sub-channels, andx ( j,Z ,m),U k ,x ( j,Z ,m),U k are the optimal quadratures of Alice and Bob, respectively. Therefore, at a reverse reconciliation, for Z → ∞ and m → ∞ where D BE (·) is the relative entropy between Bob and Eve.

Numerical evidence
This section proposes numerical evidence to demonstrate the results of the GQI and QE phases through a multiuser multicarrier CVQKD environment (AMQD-MQA [12]). The numerical evidence serves demonstration purposes.

System parameters
The numerical evidence focuses on the probability distributions of the CV states and subcarriers and studies the statistical properties and effects of modulation variance. It also revises the noise characteristic of the sub-channels and the expected errors of the quadrature estimation procedure. The single-carrier inputs of user U k have a modulation variance of σ 2 ω 0 and formulate a d-dimensional input vector x U k (31).
The jth single carrier is dedicated to a single-carrier channel The single carriers are granulated into m subcarriers, where the ith subcarrier is and has a modulation variance of σ 2 ω . The m sub-channels, The T N U k ,i sub-channel transmittance coefficients are estimated in a pre-communication phase via the subcarrier spreading technique [20]. The subcarrier spreading is an iterative method that uses p x pilot CV states to statistically determine the sub-channel transmittance coefficients with minimal theoretical error.
The outputs of the N U k ,i sub-channels (noisy Gaussian subcarriers) are referred to as where σ 2 The output of the GQI block is as which at a direct-GQI (DGQI [10]) can be rewritten as where * is the linear convolution, while function β i,ε provides an ε minimal magnitude error, where and x U k , j , x U k , j are the input-output single-carrier quadratures, F −1 (·) is the inverse FFT operation, while where C 0 is arbitrarily set to unity [30] and P is the number of C y coefficients, while As follows, using (67),σ 2 where The F-transformed subcarriers formulate a d-dimensional vector κ (34), with jth element as and with variance The output of the QE block is a d-dimensional vector, with jth element with variancẽ where β i,ε is given in (70).
The jth single carrier is estimated from E U −1 x U k , j and κ U k , j as given in (36).
The e estimation error is a d-dimensional vector with jth element e j , see (37). The E σ 2 ξ j expected error variance is evaluated via (38).

Modulation and noise
This analysis focuses on the probability distributions, the statistical properties of the input CV states and subcarriers, and the noise characteristics.
In Fig. 3a, the Gaussian single-carrier inputs x U k , j ∈ N 0, σ 2 ω 0 , σ 2 ω 0 = 225, of user U k for a d = 1000 dimensional input vector x U k are depicted. In Fig. 3b, the quadrature component noise The data unit index refers to single-carrier and subcarrier CV units, respectively.
In Fig. 4, the x U k ,i ∈ N 0, σ 2 In the next phase, the GQI block operates on the x U k ,i , 0 ≤ i ≤ m − 1, m noisy subcarriers to determine E U −1 x U k , j with minimal theoretical error.

GQI method
This analysis studies the distribution statistics of the GQI output and the FFT-transformed subcarrier elements.
, output of the GQI block is shown in Fig. 5a.
The ith subcarrier is estimated via In Fig. 5b, the κ U k , j elements are depicted, The GQI output elements provide vanishing magnitude error [10], which will be further utilized in the QE block, since the QE block operates on E U −1 x U k , j and κ U k , j , 0 ≤ j ≤ d − 1, to determine E x U k , j ∈ N 0,σ 2 ω 0 .

QE method
This analysis reveals the distribution statistics of the QE output and the expected estimation error and expected error variance. In Fig. 6a, the output of the QE block, E x U k , j ∈ N 0,σ 2 ω 0 ,σ 2 In Fig. 6b, the elements of the estimation error vector are depicted for d = 1000, The cov x U k, j κ U k , j cov −1 κ U k , j κ U k , j κ U k, j estimate approximates the jth input element, x U k , j , with vanishing error, and the e j components converge to zero.
In Fig. 7, the E σ 2 ξ j expected error variances of the noise vector elements e j derived from cov ee are depicted, d = 1000.  In Fig. 8a, a given e j in function of m is depicted. In Fig. 8b, the E σ 2 ξ j expected error variance is depicted, at 20 ≤ m ≤ 45, σ 2 N U k ,i = 16, σ 2 ω 0 = 225 and d = 40. The QE block provides a theoretical error minimum for the quadrature estimation, based on the GQE method. As m increases, E σ 2 ξ j converges to zero.

Conclusions
We defined the QE method for multicarrier CVQKD. The QE scheme extends the results of the GQI to evolve a minimal error estimate of the continuous input regime from noisy discrete variables and estimates. The QE model achieves a theoretically minimized magnitude error to evolve the non-observable continuous variable quadratures from the measured values with a vanishing error. The QE output is always a unique and stable solution, which minimizes the overall expected estimation error for each individual component of the input. We proved that in a multicarrier CVQKD setting the QE method achieves a vanishing error as the number of subcarriers increases. We derived the corresponding expected error and expected variance via the statistical model. The QE scheme provides a viable, easily implementable, and computationally efficient way to maximize the extractable information from the observed data.

Compliance with ethical standards
Author contributions L.GY. designed the protocol and wrote the manuscript. L.GY. and S.I. analyzed the results. All authors reviewed the manuscript.

Conflict of interest We have no competing interests.
Ethical statement This work did not involve any active collection of human data.

Data accessibility statement This work does not have any experimental data.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

A.1 Multicarrier CVQKD
First, we summarize the basic notations of AMQD [8]. The following description assumes a single user, and the use of n Gaussian sub-channels N i for the transmission of the subcarriers, from which only l sub-channels will carry valuable information.
In the single-carrier modulation scheme, the jth input single-carrier state ϕ j = x j + i p j is a Gaussian state in the phase space S, with i.i.d. Gaussian random position and momentum quadratures x j ∈ N 0, σ 2 ω 0 , p j ∈ N 0, σ 2 ω 0 , where σ 2 ω 0 is the modulation variance of the quadratures (For simplicity, σ 2 ω 0 is referred to as the single-carrier modulation variance, throughout). Particularly, this Gaussian single carrier is transmitted through a Gaussian quantum channel N . In the multicarrier scenario, the information is carried by Gaussian subcarrier CVs, where σ 2 ω is the modulation variance of the subcarrier quadratures, which are transmitted through a noisy Gaussian sub-channel N i . Each N i Gaussian sub-channel is dedicated for the transmission of one Gaussian subcarrier CV from the n subcarrier CVs. (Note: index i refers to the subcarriers, while index j to the single carriers throughout the manuscript.) The single-carrier state ϕ j in the phase space S can be modeled as a zero-mean, circular symmetric complex Gaussian random variable z j ∈ CN 0, σ 2 ω z j , with variance σ 2 ω z j = E z j 2 , and with i.i.d. real and imaginary zero-mean Gaussian random components Re z j ∈ N 0, σ 2 ω 0 , Im z j ∈ N 0, σ 2 ω 0 . In the multicarrier CVQKD scenario, let n be the number of Alice's input single-carrier Gaussian states. The n input coherent states are modeled by an n-dimensional, zero-mean, circular symmetric complex random Gaussian vector where each z j can be modeled as a zero-mean, circular symmetric complex Gaussian random variable Specifically, the real and imaginary variables (i.e., the position and momentum quadratures) formulate n-dimensional real Gaussian random vectors, x = (x 1 , . . . , x n ) T and p = ( p 1 , . . . , p n ) T , with zero-mean Gaussian random variables with densities f (x j ) and f ( p j ) as where K z is the n × n Hermitian covariance matrix of z: while z † is the adjoint of z.
For vector z, holds, and for any γ ∈ [0, 2π ]. The density of z is as follows (if K z is invertible): A n-dimensional Gaussian random vector is expressed as x = As, where A is an (invertible) linear transform from R n to R n , and s is an n-dimensional standard Gaussian random vector N (0, 1) n . This vector is characterized by its covariance matrix K x = E xx T = AA T , and has density The Fourier transformation F (·) of the n-dimensional Gaussian random vector v = (v 1 , . . . , v n ) T results in the n-dimensional Gaussian random vector m = (m 1 , . . . , m n ) T , as: In the first step of AMQD, Alice applies the inverse FFT (fast Fourier transform) operation to vector z (see (A.1)), which results in an n-dimensional zero-mean, circular symmetric complex Gaussian random vector d, where σ 2 The T (N ) transmittance vector of N in the multicarrier transmission is is a complex variable, which quantifies the position and momentum quadrature transmission (i.e., gain) of the ith Gaussian sub-channel N i , in the phase space S, with real and imaginary parts Particularly, the T i (N i ) variable has the squared magnitude of The Fourier-transformed transmittance of the ith sub-channel N i (resulted from CVQFT operation at Bob) is denoted by The n-dimensional zero-mean, circular symmetric complex Gaussian noise vector ∈ CN 0, σ 2 n of the quantum channel N is evaluated as with independent, zero-mean Gaussian random components and with variance σ 2 N i , for each i of a Gaussian sub-channel N i , which identifies the Gaussian noise of the ith sub-channel N i on the quadrature components in the phase space S.
The CVQFT-transformed noise vector can be rewritten as It also defines an n-dimensional zero-mean, circular symmetric complex Gaussian random vector F ( ) ∈ CN 0, K F( ) with a covariance matrix where K F( ) = K , by theory. At a constant subcarrier modulation variance σ 2 ω i for the n Gaussian subcarrier CVs, the corresponding relation is where σ 2 ω i is the modulation variance of the quadratures of the subcarrier |φ i transmitted by sub-channel N i . Assuming l good Gaussian sub-channels from the n with constant quadrature modulation variance σ 2 ω i , where σ 2 ω i = 0 for the ith unused sub-channel, In particular, from the relation of (A.27), for the transmittance parameters the following relation follows at a given modulation variance σ 2 ω 0 , precisely, where |T (N )| 2 is the transmittance of N in a single-carrier scenario, and For the method of the determination of these l Gaussian sub-channels, see [8]. Alice's ith Gaussian subcarrier is expressed as (A.30) Fig. 9 The AMQD-MQA multiple access scheme with multiple independent transmitters and multiple receivers [12]. The modulated Gaussian CV single carriers are transformed by a unitary operation (inverse CVQFT) at the E encoder, which outputs the n Gaussian subcarrier CVs for the transmission. The parties send the |ϕ k single-carrier Gaussian CVs with variance σ 2 ω 0,k to Alice. In the rateselection phase, the encoder determines the transmit users. The data states of the transmit users are then fed into the CVQFT † operation. The |φ i Gaussian subcarrier CVs have a variance σ 2 ω per quadrature components. The Gaussian CVs are decoded by the CVQFT unitary operation. Each ϕ k is received by Bob k

A.2 Multiuser multicarrier CVQKD
In a MQA multiple access multicarrier CVQKD, a given user U k , k = 0, . . . , K − 1, where K is the number of total users, is characterized via m subcarriers, formulating an M U k logical channel of U k , For a detailed description of MQA for multicarrier CVQKD, see [12]. The general model of AMQD-MQA is depicted in Fig. 9  The notations of the manuscript are summarized in Table 1.

Number of autocorrelation coefficients
C xx An n × n covariance matrix associated with x (n) N n ( f E (x)) Number of n-tuples (x 1 , . . . , x n ) ∈ X n with a given empirical density for the estimator function, U −1 is the inverse CVQFT operation M Measurement operator, homodyne or heterodyne measurement θ ϕ U k , j θ ϕ U k , j = π/ , where = σ 2 ω0 /σ 2 ω , and σ 2 ω0 , σ 2 ω are the single-carrier and multicarrier modulation variances, σ 2 ω < σ 2 ω0 , ≥ 1, θ ϕ U k , j ≤ π Table 1 continued where T * N is the expected transmittance of the l sub-channels under an optimal Gaussian attack λ i Optimal Lagrange multipliers , evaluated at a direct-GQI as where * is the linear convolution, while function β i,ε provides an ε minimal magnitude error ε = arg min ε max , where , and x U k , j , x U k , j are the input, output single-carrier quadratures, F −1 (·) is the inverse FFT operation [1], while β i,ε = 1 + P y=1 C y cos (y Q i ), where C 0 is arbitrarily set to unity, P is the number of C y coefficients, while Variance of QE output, E x U k , j ∈ N 0,σ 2 ω0 , evaluated asσ 2 Function, provides an ε minimal magnitude error, for x U k , j are the input, output single-carrier quadratures, F −1 (·) is the inverse FFT operation  The variable of a single-carrier Gaussian CV state, z ∈ CN 0, σ 2 z , |ϕ i ∈ S. Zero-mean, circular symmetric complex Gaussian random variable, σ 2 z = E |z| 2 = 2σ 2 ω0 , with i.i.d. zero mean, Gaussian random quadrature components x, p ∈ N 0, σ 2 ω0 , where σ 2 ω0 is the variance The noise variable of the Gaussian channel N , ∈ CN 0, σ 2 , with i.i.d. zero-mean, Gaussian random noise components on the position and momentum quadratures x , p ∈ N 0, σ 2 N , The variable of a Gaussian subcarrier CV state, d ∈ CN 0, σ 2 d , |φ i ∈ S. Zero-mean, circular symmetric Gaussian random variable, σ 2 d = E |d| 2 = 2σ 2 ω , with i.i.d. zero mean, Gaussian random quadrature components x d , p d ∈ N 0, σ 2 ω , where σ 2 ω is the (constant) modulation variance of the Gaussian subcarrier CV state The inverse CVQFT transformation, F −1 (·) = CVQFT † (·), applied by the encoder, continuous-variable unitary operation The ith Gaussian subcarrier CV of user U k , where IFFT stands for the Inverse Fast Fourier Transform,