Security analysis of linear optics cloning machine-enhanced passive state preparation continuous-variable quantum key distribution

Abstract

In this paper, the security of linear optics cloning machine-enhanced passive state preparation continuous-variable quantum key distribution is investigated. The employment of passive state preparation eliminates the strict requirements on electro-optical modulators, while an appropriately tuned linear optics cloning machine can compensate the preparation noise introduced in passive state preparation and improve the system performance. The expression of the composable secret key rate in the finite-size regime of the proposed scheme is obtained. Simulation results demonstrate that the proposed scheme can effectively enhance the secret key rate and increase the maximum secure transmission distance.

Appendix: The detailed derivation of \(I\left( {A:B} \right)\) and \(S\left( {B:E} \right)\) in PSP-LOCM CVQKD

Given the covariance matrix $\gamma_{{AB\varepsilon_{{{\text{PE}}}} }}$ in Eq. (22), the classical mutual information $I\left( {A:B} \right)$ can be expressed as

$$ I\left( {A:B} \right) = \log_{2} \frac{{V_{B} }}{{V_{B|A} }} = \log_{2} \left( {\frac{{\sigma_{B}^{2} + 1}}{{\sigma_{B}^{2} - {{c_{AB} } \mathord{\left/ {\vphantom {{c_{AB} } {\left( {\sigma_{A}^{2} + 1} \right)}}} \right. \kern-0pt} {\left( {\sigma_{A}^{2} + 1} \right)}}}}} \right). $$

(26)

On the other hand, $S\left( {B:E} \right)$ can be given by

$$ S\left( {B:E} \right) = S\left( E \right) - S\left( {E|B} \right), $$

(27)

where $S\left( \bullet \right)$ denotes the von Neumann entropy, $S\left( { \bullet \left| B \right.} \right)$ denotes the conditional von Neumann entropy on Bob’s measurements, and $B$ indicates Bob's measurement results. On the one hand, to maximize the information available, Eve can purify the system $EAB_{1}$, it follows that $S\left( E \right) = S\left( {AB} \right)$. On the other hand, Bob also can purify the system AEFG, one can get $S\left( {E|B} \right) = S\left( {AFG|B} \right)$. The above equation can be rewritten as

$$ S_{{\varepsilon_{{{\text{PE}}}} }} \left( {B:E} \right) = S\left( {AB_{{\varepsilon_{{{\text{PE}}}} }} } \right) - S\left( {AFG|B_{{\varepsilon_{{{\text{PE}}}} }} } \right) = \sum\limits_{i = 1}^{2} {G\left( {\frac{{\lambda_{i} - 1}}{2}} \right)} - \sum\limits_{i = 3}^{5} {G\left( {\frac{{\lambda_{i} - 1}}{2}} \right)} . $$

(28)

It can be noted that $S\left( {AB} \right)$ is determined by the covariance matrix $\gamma_{AB}$ and $S\left( {AFG|B} \right)$ is determined by the covariance matrix $\gamma_{AFGB}$. $\gamma_{AB}$ is described by

$$ S\left( {AB} \right) = \sum\limits_{i = 1}^{2} {G\left( {\frac{{\lambda_{i} - 1}}{2}} \right)} , $$

(29)

where $G\left( x \right) = \left( {x + 1} \right)\log_{2} \left( {x + 1} \right) - x\log_{2} x$ is the von Neumann entropy of the thermal field state [3, 4] and $\lambda_{i}$ is the i-th symplectic eigenvalue. The expressions of $\lambda_{1 - 2}$ read

$$ \lambda_{1} = \sqrt {\frac{1}{2}\left[ {A + \sqrt {A^{2} - 4B} } \right]} ,\lambda_{2} = \sqrt {\frac{1}{2}\left[ {A - \sqrt {A^{2} - 4B} } \right]} , $$

(30)

where $A = \det \gamma_{A} + \det \gamma_{B} + \det \sigma_{AB}$ and $B = \det \gamma_{AB}$. For $\lambda_{3 - 5}$, it is first necessary to derive $\gamma_{AFG}^{{x_{B} }}$ which can be expressed as

$$ \gamma_{AFG}^{{x_{B} }} = \gamma_{AFG} - \sigma_{{^{AFGB} }}^{T} \left( {\gamma_{B} + I_{2} } \right)^{ - 1} \sigma_{AFGB} , $$

(31)

where $\left( \bullet \right)^{ - 1}$ represents the inverse of a matrix. $\gamma_{AFG}^{{x_{B} }}$ is the system after Bob's measurement, which is obtained by projecting the measurement of the whole system on the mode B₂. Besides, the whole system can be described by the covariance matrix $\gamma_{{AFGB_{2} }}$, whose expression is

$$ \gamma_{AFGB} = \Omega^{T} \left( {\gamma_{AB} \oplus \gamma_{{F_{0} G}} } \right)\Omega = \left( {\begin{array}{*{20}c} {\gamma_{AFG} } & {\sigma_{AFGB}^{T} } \\ {\sigma_{AFGB} } & {\gamma_{B} } \\ \end{array} } \right), $$

(32)

where $\Omega$ presents the beam splitter transformation, and contents $\Omega = I_{A} \oplus \Omega_{{{\text{BF}}_{0} }}^{{{\text{BS}}}} \oplus I_{G}$ with $I_{A} = I_{G} = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right)$. Moreover, $\Omega_{{{\text{BF}}_{0} }}^{{{\text{BS}}}}$ can be given by

$$ \Omega_{{{\text{BF}}_{0} }}^{{{\text{BS}}}} = \left( {\begin{array}{*{20}c} {\sqrt {T_{{{\text{LOCM}}}} } I_{2} } & {\sqrt {1 - T_{{{\text{LOCM}}}} } I_{2} } \\ { - \sqrt {1 - T_{{{\text{LOCM}}}} } I_{2} } & {\sqrt {T_{{{\text{LOCM}}}} } I_{2} } \\ \end{array} } \right). $$

(33)

The matrix of additional LOCM tunable noise effects is

$$ \gamma_{{F_{0} G}} = \left( {\begin{array}{*{20}c} {N_{{{\text{LOCM}}}} I_{2} } & {\sqrt {N_{{{\text{LOCM}}}}^{2} - 1} \sigma_{Z} } \\ {\sqrt {N_{{{\text{LOCM}}}}^{2} - 1} \sigma_{Z} } & {N_{{{\text{LOCM}}}} I_{2} } \\ \end{array} } \right). $$

(34)

Fortunately, after the above intricate calculations, $\gamma_{AFG}^{{x_{B} }}$ can be presented by the following equation

$$ \gamma_{AFG}^{{x_{B} }} = \left( {\begin{array}{*{20}c} {\gamma_{A} } & {\sigma_{AF} } & {\sigma_{AG} } \\ {\sigma_{AF}^{T} } & {\gamma_{F} } & {\sigma_{FG} } \\ {\sigma_{AG}^{T} } & {\sigma_{FG}^{T} } & {\gamma_{G} } \\ \end{array} } \right). $$

(35)

The elements in the matrix of Eq. (35) can be expressed as

$$ \begin{aligned} \gamma_{A} = & \left( {\begin{array}{*{20}c} V & 0 \\ 0 & V \\ \end{array} } \right),\gamma_{F} = \left( {\begin{array}{*{20}c} {1 + TT_{{{\text{LOCM}}}} \left( {V_{A} + \varepsilon } \right)} & 0 \\ 0 & {1 + TT_{{{\text{LOCM}}}} \left( {V_{A} + \varepsilon } \right)} \\ \end{array} } \right), \\ \gamma_{G} = & \left( {\begin{array}{*{20}c} {1 - T\left( {V_{A} + \varepsilon } \right)\left( {T_{{{\text{LOCM}}}} - 1} \right)} & 0 \\ 0 & {1 - T\left( {V_{A} + \varepsilon } \right)\left( {T_{{{\text{LOCM}}}} - 1} \right)} \\ \end{array} } \right), \\ \sigma_{AF} = & \left( {\begin{array}{*{20}c} {\sqrt {TT_{{{\text{LOCM}}}} \left( {V^{2} - 1} \right)} } & 0 \\ 0 & { - \sqrt {TT_{{{\text{LOCM}}}} \left( {V^{2} - 1} \right)} } \\ \end{array} } \right), \\ \sigma_{AG} = & \left( {\begin{array}{*{20}c} {\sqrt {T\left( {1 - T_{{{\text{LOCM}}}} } \right)\left( {V^{2} - 1} \right)} } & 0 \\ 0 & {\sqrt {T\left( {1 - T_{{{\text{LOCM}}}} } \right)\left( {V^{2} - 1} \right)} } \\ \end{array} } \right), \\ \sigma_{FG} = & \left( {\begin{array}{*{20}c} {T\left( {V_{A} + \varepsilon } \right)\sqrt {T_{{{\text{LOCM}}}} \left( {1 - T_{{{\text{LOCM}}}} } \right)} } & 0 \\ 0 & { - T\left( {V_{A} + \varepsilon } \right)\sqrt {T_{{{\text{LOCM}}}} \left( {1 - T_{{{\text{LOCM}}}} } \right)} } \\ \end{array} } \right). \\ \end{aligned} $$

(36)

Then, it is straightforward to calculate $\lambda_{3 - 5}$

$$ \lambda_{3} = \sqrt {\frac{1}{2}\left[ {C + \sqrt {C^{2} - 4D} } \right]} ,\lambda_{4} = \sqrt {\frac{1}{2}\left[ {C - \sqrt {C^{2} - 4D} } \right]} ,\lambda_{5} = 1, $$

(37)

where $C = \frac{{A\chi_{{{\text{LOCM}}}}^{2} + B + 1 + 2\chi_{{{\text{LOCM}}}} \left( {V\sqrt B + T\left( {V + \chi_{{{\text{line}}}} } \right) + 2T\sqrt {V^{2} - 1} } \right)}}{{\left[ {T\left( {V + \chi_{{{\text{tot}}}} } \right)} \right]^{2} }}$ and $D = \left[ {\frac{{V + \sqrt B \chi_{{{\text{LOCM}}}} }}{{T\left( {V + \chi_{{{\text{tot}}}} } \right)}}} \right]^{2}$. Hence, $S\left( {B:E} \right)$ can be derived from $S\left( {AFG|B} \right) = \sum\limits_{i = 3}^{5} {G\left( {\frac{{\lambda_{i} - 1}}{2}} \right)}$.