Decentralized continuous-variable quantum secret sharing

Abstract

Quantum secret sharing (QSS) is a method that allows multiple users to obtain pieces of a secret key from a dealer, but the complete key can only be recovered if all users collaborate. However, the status of the dealer in the system brings potential security risks to the communication network. To solve this problem, we propose a decentralized quantum secret sharing (DQSS) scheme that allows any honest user in the system to become a dealer and send a partial key to other users through an insecure quantum channel. The position of the dealer in the user chain in this scheme is no longer limited to the end of the chain, but can be located at any position. This scheme can be realized with local local oscillator (LLO) using discretely modulated coherent states (DMCSs). With this approach, the multi-user quantum secret sharing scheme can be implemented with any honest user acting as a dealer in a quantum communication network.

Appendices

Appendix: Phase noise analyses

In this part, we will firstly introduce the LLO protocol and consider the phase noise. Then, we will analyze the change from partially untrusted noise to trusted noise caused by Bob (Dealer) end local calibration.

From the [16], the phase noise $\xi _{\text {phase}}$ in LLO CV QKD can be expressed as

$$\begin{aligned} \xi _{\text {phase}}=2 V_{A}\left( 1-e^{-\frac{V_{\text {est}}}{2}}\right) , \end{aligned}$$

(A1)

where $V_{A}$ denotes the modulation variance, $V_{\textrm{est}}$ is the variance of the phase noise, which can be calculated by the real phase rotation value $\theta _{S}$ and the estimated value ${\hat{\theta }}_{S}$

$$\begin{aligned} V_{\textrm{est}}={\text {var}}\left( \theta _{S}-{\hat{\theta }}_{S}\right) . \end{aligned}$$

(A2)

We denote $E_{\textrm{R}}$ as the amplitude of the reference pulse, $\chi$ as the total noise variance from Alice’s input, T as the channel transmittance, $\varepsilon _{0}$ as the channel excess noise of the phase reference, $\mu$ as the detection efficiency, $v_{\textrm{el}}$ as the electronics noise of the imperfect detector, and it consists of the following parts:

$$\begin{aligned} V_{\textrm{est}} = V_{\textrm{error}} + V_{\textrm{drift}} + V_{\textrm{channel}}; \end{aligned}$$

(A3)

then, we analyze the three part of Eq. (A3) independently.

$V_{\textrm{error}}$ denotes the variance generated by the error between the real phase rotation value $\theta _{{R}}$ and estimated value ${\hat{\theta }}_{R}$

$$\begin{aligned} V_{\text {error}}={\text {var}}\left( \theta _{R}-{\hat{\theta }}_{R}\right) =\frac{\chi +1}{E_{R}^{2}} \end{aligned}$$

(A4)

where $\chi$ is the total noise of phase reference can be calculated by

$$\begin{aligned} \chi =\frac{1}{T}-1+\varepsilon _{0}+\frac{2-\mu +2 v_{\textrm{el}}}{T \mu } \end{aligned}$$

(A5)

where first three items are the components of channel-added noise in CV QKD, and the last item denotes the heterodyne detector noise.

$V_{\textrm{drift}}$ represents the variance of the relative phase drift between the two lasers between $\varDelta T$

$$\begin{aligned} V_{\textrm{drift}}=2 \pi \left( \varDelta v_{A}+\varDelta v_{B}\right) \varDelta T \end{aligned}$$

(A6)

where $\varDelta v_A$ and $\varDelta v_B$ correspond to the linewidths of the two free-running lasers.

$V_{\textrm{channel}}$ represent the variance of the noise which is caused by the drift of phase accumulation between the signal pulse and the phase reference.

Therefore, when $V_{\textrm{est}} \rightarrow 0$, the phase noise can be approximated as

$$\begin{aligned} \xi _{\text {phase}}&=V_{A} V_{\text {est}}=V_{A}\left( V_{\textrm{drift}}+V_{\text {channel}}+V_{\text {error}}\right) \nonumber \\&=\xi _{\textrm{drift}}+\xi _{\text {channel}}+\xi _{\text {error}}. \end{aligned}$$

(A7)

Furthermore, we will analyze the trusted noise after calibration. Since the excess noise $\xi$ is a component element of the channel-added noise which is part of the total added noise, it can be calculated by

$$\begin{aligned} \xi =\xi _{\text {phase}}+\xi _{\text {rest}}, \end{aligned}$$

(A8)

Then, the total channel-added noise $\chi _{\text {line}}$ can be calculated as

$$\begin{aligned} \chi _{\text {line}}=1 / T-1+\xi , \end{aligned}$$

(A9)

and the total detection-added noise $\chi _{\text {het}}$ can be calculated as

$$\begin{aligned} \chi _{\text {het}}=\left( 2-\mu +2 v_{\textrm{el}}\right) / \mu . \end{aligned}$$

(A10)

Combined with the CV QKD noise calculation method, the total noise is

$$\begin{aligned} \chi _{\text {tot}}=\chi _{\text {line}}+\frac{\chi _{\text {het}}}{T} \end{aligned}$$

(A11)

Then, we try to distinguish the trusted part and the untrusted part among them. We regard the $\chi _{\text {het}}$ is the trusted noise based on a well calibration of $\mu$ and $v_{\textrm{el}}$. Meanwhile, a part of the phase reference measurement noise $\xi _{\textrm{error}}$ can be well-calibrated locally on Bob’s side so it can be regarded as trusted noise [17, 24].

According to (A11), the total noise of phase reference can be written as

$$\begin{aligned} \chi =\chi ^{\textrm{u}}+\frac{\chi ^{T}}{T} \end{aligned}$$

(A12)

where $\chi ^{u}=1 / T-1+\varepsilon _{0}$, $\chi ^{T}=\left( 2-\mu +2 v_{\textrm{el}}\right) / \mu$.

Combined (A4), (A5), (A7) and (A12), the measurement noise of phase reference can be written with the same form as

$$\begin{aligned} \xi _{\text {error}}=V_{A} V_{\text {error}}=\xi _{\text {error}}^{u}+\frac{\xi _{\text {error}}^{T}}{T}, \end{aligned}$$

(A13)

where

$$\begin{aligned} \xi _{\text {error}}^{u}&=V_{A}\left( \frac{\chi ^{u}+1}{E_{R}^{2}}\right) =V_{A}\left( \frac{1+T \varepsilon _{0}}{T E_{R}^{2}}\right) ,\end{aligned}$$

(A14)

$$\begin{aligned} \xi _{\text {error}}^{T}&=V_{A}\left( \frac{\chi ^{T}}{E_{R}^{2}}\right) =V_{A}\left( \frac{2-\mu +2 v_{\textrm{el}}}{\mu E_{R}^{2}}\right) . \end{aligned}$$

(A15)

In consequence, in the trusted phase noise situation, the total channel-added noise, the total detection-added noise and the total added noise can be written as

$$\begin{aligned} \chi _{\text {line}}^{T}&=\frac{1}{T}-1+\xi _{\text {tot}}^{T}, \end{aligned}$$

(A16)

$$\begin{aligned} \chi _{\text {het}}^{T}&=\frac{2-\mu -2 v_{\textrm{el}}}{\mu }+\xi _{\textrm{error}}^{T}, \end{aligned}$$

(A17)

$$\begin{aligned} \chi _{\text {tot}}^{T}&=\chi _{\text {line}}^{T}+\frac{\chi _{\text {het}}^{T}}{T}, \end{aligned}$$

(A18)

where $\xi _{\textrm{tot}}^{T}=\xi _{\textrm{tot}}-\xi _{\textrm{error}}^{T} / T$ is the real excess noise.

With the scheme from [25], the $V_{\textrm{drift}}$ and $V_{\textrm{channel}}$ can be viewed as going to 0, thus Eq. (A7) can be derived as

$$\begin{aligned} \xi _{\text {phase}} \approx V_{A} V_{\text {error}}=\xi _{\text {error}}=V_{A}\left( \frac{\chi +1}{E_{R}^{2}}\right) . \end{aligned}$$

(A19)

Appendix B: Coherent states in discretely modulated CV QKD

A coherent states can be generalized to one with N quantum states $\left| \alpha _{k}^{N}\right\rangle =\left| \alpha e^{i 2 k \pi / N}\right\rangle$, where $k \in \{0,1, \ldots , N-1\}$ and $\alpha$ is a positive number related to the modulation of the quantum state as $V_{A} = 2\alpha ^2$.

In the prepare-and-measure (PM) version of discretely modulated CV QKD, Alice first selects a random bit string ${\textbf{a}}=\left( a_{0}, a_{1}, \ldots , a_{2 L-1}\right)$ of length 2L, and the coherent states are subsequently encoded according to the successive pairs of bit strings ${\textbf{a}}$ with the form $\left| \alpha _{k}^{N}\right\rangle$, where $k_l = 2a_{2l} + a_{2l+1}$. Alice sends these modulated coherent states to remote Bob through a lossy and noisy quantum channel. When Bob receives these states, he can apply a heterodyne detector to measure each output mode. The mixture state that Bob receives can be expressed by the following form:

$$\begin{aligned} \rho _{N}=\frac{1}{N} \sum _{k=1}^{N}\left| \alpha _{k}^{N}\right\rangle \left\langle \alpha _{k}^{N}\right| . \end{aligned}$$

(B1)

Note that the discrete modulation strategy of quadrature-phase-shift keying (QPSK) requires four nonorthogonal coherent states so that we have $N=4$. After the measurement, Bob obtains a 2L string ${\textbf{c}}= \left( c_{0}, c_{1}, \ldots , c_{2L-1}\right) \in {\mathbb {R}}^{2L}$. This string can be transformed into a raw key of 2L bits ${\textbf{b}}=\left( b_{0}, b_{1}, \ldots , b_{2L-1}\right)$, given by [26]

$$\begin{aligned} \left( b_{2 l}, b_{2 l+1}\right) = {\left\{ \begin{array}{ll}(0,0) &{} \text {if}\; c_{2 l+1}<c_{2 l}, \quad c_{2 l+1} \geqslant -c_{2 l} \\ (0,1) &{} \text {if}\; c_{2 l+1} \geqslant c_{2 l}, c_{2 l+1}>-c_{2 l} \\ (1,0) &{} \text {if}\; c_{2 l+1}>c_{2 l}, c_{2 l+1} \leqslant -c_{2 l} \\ (1,1) &{} \text {if}\; c_{2 l+1} \leqslant c_{2 l}, c_{2 l+1}<-c_{2 l}\end{array}\right. } \end{aligned}$$

(B2)

Bob then broadcasts the absolute values of $c_{2l} \pm c_{2l+1}$ through a classical authenticated channel. This side information allows Alice and Bob to turn the information reconciliation problem into a well-studied channel coding problem for the binary-input additive white noise Gaussian channel. After several postprocessing steps such as parameter estimation, reconciliation, and privacy amplification, Alice and Bob can establish a correlated sequence of a random secure key.