Nonlinear improvement of measurement-device-independent quantum key distribution using multimode quantum memory

Abstract

This paper proposes a quantum key distribution (QKD) scheme for measurement-device-independent QKD (MDI-QKD) utilizing quantum memory (QM), which is based on two distinct functions of QM: on-demand storage and multimode storage. We demonstrate a nonlinear increase in the secure key rate due to the utilization of QM. In the protocol incorporating on-demand storage, it is acknowledged that the secure key rate is scaled by $R=O(\sqrt{\eta _{{\text {ch}}}})$ as $\eta _{{\text {ch}}}$, while as an alternative approach, we reveal that the improvement is $O(m_{\text {s}}^2)$, with $m_{\text {s}}$ being the number of modes in frequency (spatial) multiplexing in the scheme incorporating multimode storage. We adopt an atomic frequency comb as a QM that incorporates the two functions and propose an architecture based on MDI-QKD to attain experimental feasibility. This scheme can be extended to quantum repeaters, and even for a single quantum-repeater node, there is a nonlinear enhancement and an experimental incentive to increase the number of modes.

Appendices

Appendix A: Calculation of key rates dependent on $T_2$

If QM with on-demand storage is utilized, $Y_{11}^{{\text {QM}}}$ can be described as follows [19, 36]:

$$\begin{aligned} Y_{11}^{{\text {QM}}}&= \frac{Y_{11}(\eta _{{\text {early}}},\eta _{{\text {late}}})}{N_{\text {L}}(\eta _{\mu A},\eta _{\nu B})}\cdot \frac{\eta _{1A}\eta _{1B}}{\eta _{\mu A}\eta _{\nu B}}\mu \nu e^{-\mu -\nu }, \end{aligned}$$

(8)

where $\eta _{1K}, \eta _{\mu (\nu ) K}\quad (K=A,B)$ represent the readout efficiency from QM in single-photon states and average photon number $\mu (\nu )$, respectively. Moreover, $Y_{11}$ is given as follows [51]:

$$\begin{aligned}&Y_{11}(\eta _A,\eta _B) \nonumber&\\&= (1-p_{{\text {dc}}})^2 \times \left[ \frac{\eta _{A}\eta _{B}}{2}+(2\eta _{A}+2\eta _{B}-3\eta _{A}\eta _{B})p_{{\text {dc}}}+4(1-\eta _{A})(1-\eta _{B})p_{{\text {dc}}}^2\right] , \end{aligned}$$

(9)

where $N_{\text {L}}$ represents the average number of attempts to load both QMs. The average number of attempts to load both memories, i.e., $N_{\text {L}}(\eta _A,\eta _B)$, is approximated by $N_{\text {L}} = \frac{3-2\eta _A}{\eta _A(2-\eta _A)} \approx \frac{3}{2\eta _A}$ when $\eta _A = \eta _B \ll 1$ in the no-multimode scenario [32]. Furthermore, $\eta _{{\text {early}}}$ and $\eta _{{\text {late}}}$ represent the loading efficiencies of the QMs for photons arriving early and late, respectively [19]:

$$\begin{aligned} \eta _K^{{\text {QM}}}= \left\{ \begin{array}{l} \eta _{{\text {late}}} = \eta _{r0}\eta _{\text {d}},\quad \text {if QM K is late}, \\ \eta _{{\text {early}}} = \eta _r(t)\eta _{\text {d}},\quad \text {if QM K is early}, \end{array} \right. \end{aligned}$$

(10)

where t represents the storage time of the QM in the event that it is loaded early, which is given as $(t=|N_A-N_B|\tau )$. The calculation of the expected value for $\eta _{{\text {early}}}$ is as follows:

$$\begin{aligned}&\overline{\eta _{{\text {early}}}} \nonumber&\\&= \eta _{\text {d}}\eta _{r0}E\{\exp (-|N_A-N_B|\tau /T_2)\} \nonumber&\\&= \frac{\eta _{\text {d}}\eta _{r0}\eta _{A}\eta _{B}}{\eta _{A}+\eta _{B}-\eta _{A}\eta _{B}} \times \left[ \frac{1}{1-e^{-\frac{\tau }{T_2}}(1-\eta _A)}+\frac{1}{1-e^{-\frac{\tau }{T_2}}(1-\eta _B)}-1\right] . \end{aligned}$$

(11)

Appendix B: Calculation of key rates dependent on temporal multiplexing $m_{\text {t}}$

Fig. 8

Schematic of a QM used in protocols that incorporate both frequency (spatial) multiplexing and time multiplexing. Here, each QM stores $m_{\text {s}} m_{\text {t}}$ photons and performs a frequency/time bin shift in the center for BSMs

The multimode AFC protocol (frequency multiplexing/spatial multiplexing scheme) in Sect. 3 was considered as temporal multiplexing $m_{\text {t}}$ with $m_{\text {t}}=1$, but it can be extended to the case of temporal multiplexing ($m_{\text {t}}>1$). A schematic of the QM when temporal multiplexing is incorporated is depicted in Fig. 8. It is necessary to store $m_{\text {s}}m_{\text {t}}$ photons across $m_{\text {t}}$ slots and match them accordingly.

$$\begin{aligned} Y_{11}^{{\text {QM}}}&= \frac{\mu \nu e^{-\mu -\nu }Y_{11}(\eta _A^{{\text {mm}}},\eta _B^{{\text {mm}}})}{m_{\text {t}}} \end{aligned}$$

(12)

Here, $Y_{11}$ represents the success rate of the BSM, as described in Appendix 1. The success probability $\eta _K^{{\text {mm}}}$ of the frequency-multiplexed and time-multiplexed S-BSM is given as follows:

$$\begin{aligned}&\eta _K^{{\text {mm}}} = 1-(1-Y_{11}(\eta _{{\text {ch}}}(L_{\text {K}})\eta _{\text {d}},\eta _{{\text {ent}}}\eta _{\text {d}}))^{m_{\text {s}}m_{\text {t}}} \quad (K=A,B). \end{aligned}$$

(13)

Here, in contrast to frequency/spatial multiplexing, temporal multiplexing results in a linear increase in $O(m_{\text {t}})$, as $m_{\text {t}}$ appears in the denominator of the rate Eq. 6, reducing the number of middle BSM attempts per unit time.

Fig. 9

Protocol with temporal multiplexing. Simulations were performed with $m_{\text {s}}=1$ in frequency multiplexing and $m_{\text {t}}=1,10^2,10^4,10^6$ for the number of modes in temporal multiplexing

As illustrated in Fig. 9, as $m_{\text {t}}$ increases, the plateau region expands; however, the overall rate decreases linearly with an increase in the time required to execute BSM, which is $m_{\text {t}}\tau$. Additionally, optimizing the number of temporal modes $m_{\text {t}}$ with consideration of the distance results in rate scaling of $O(\sqrt{\eta _{{\text {ch}}}})$, as observed in the envelope.