Progress towards practical device-independent quantum key distribution with spontaneous parametric down-conversion sources, on-off photodetectors, and entanglement swapping

Device-independent quantum key distribution (DIQKD) guarantees unconditional security of a secret key without making assumptions about the internal workings of the devices used for distribution. It does so using the loophole-free violation of a Bell's inequality. The primary challenge in realizing DIQKD in practice is the detection loophole problem that is inherent to photonic tests of Bell' s inequalities over lossy channels. We revisit the proposal of Curty and Moroder [Phys. Rev. A 84, 010304(R) (2011)] to use a linear optics-based entanglement-swapping relay (ESR) to counter this problem. We consider realistic models for the entanglement sources and photodetectors: more precisely, (a) polarization-entangled states based on pulsed spontaneous parametric down-conversion sources with infinitely higher-order multiphoton components and multimode spectral structure, and (b) on-off photodetectors with nonunit efficiencies and nonzero dark-count probabilities. We show that the ESR-based scheme is robust against the above imperfections and enables positive key rates at distances much larger than what is possible otherwise.

Figure 1

Setup for DIQKD with a conventional entanglement-swapping relay (ESR) node based on linear optics. A source ${\rho }_{A{A}^{\prime }}$ distributes polarization entanglement to receivers Alice and Bob. The distributed states are subject to losses in fiber coupling (both in the channel to Alice, as well as to Bob) and transmission (in the channel to Bob, who is situated far from the source), the respective efficiencies being ${\eta }_T$ and ${\eta }_C$. Bob employs an ESR node, which consists of another similar entanglement source ${\rho }_{B{B}^{\prime }}$, beam splitters (BS, 50:50 beam splitter; PBS, polarizing beam splitter), and heralding detectors $D_H$ and $D_V$ corresponding to horizontal and vertical polarizations, respectively. Upon each successful entanglement-swapping event, Alice and Bob perform polarization measurement with polarizer settings $X, Y$, respectively, and the outcomes are denoted as $a,b\in \left\{+1,-1\right\}$, respectively. The detectors are assumed to be imperfect, on-off photodetectors, with nonunit efficiencies and nonzero dark-count probabilities.

Figure 2

(a) A Sagnac loop source for generating polarization entanglement based on Type II SPDC, meaning it produces down-converted light in two orthogonal polarization modes. The SPDC is pumped simultaneously by a clockwise (CW) and a counterclockwise (CCW) pump. DM stands for a dichroic mirror, which reflects light of frequency $2\omega$, while being transparent to light of frequency $\omega$. DPBS stands for a dichroic polarizing beam splitter, which splits light of both frequencies $\omega$ and $2\omega$ into its $H$ and $V$ polarization components. DHWP stands for a dichroic half-wave plate at an angle ${45}^{\circ }$, so that it flips the polarization in the $H,V$ basis as $H\to V$ and $V\to H$. The geometry of the source makes sure that the pump is perfectly recycled, while the down-converted light is output through the topmost and the rightmost modes. (b) A schematic representation of the source. The CW pump generates squeezing in the modes $a_H$ and $b_V$, while the CCW pump generates squeezing in the other two modes.

Figure 3

The ESR-assisted Bell testing setup for DIQKD in greater detail. The Type II SPDC crystal placed in a Sagnac configuration as depicted in Fig. 2 is used to generate a pair of two-mode squeezed vacuua (TMSV) over polarization modes. The polarizer angle settings at Alice and Bob are denoted as ${\theta }_A$ and ${\theta }_B$, respectively. The detectors are assumed to be imperfect on-off photodetectors, with nonunit efficiencies and nonzero dark-count probabilities.

Figure 4

Maximal loophole-free violation of the CHSH inequality $S$ as a function of distance in the ESR-assisted DIQKD of Fig. 3. The sources are assumed to be monochromatic. The detection efficiencies at Alice and Bob are assumed to be ideal (i.e., the product of coupling and detector efficiencies ${\eta }_D={\eta }_C{\eta }_{det}=1$ and dark-count probability in the detectors $P_{DC}=0$). The quantities ${\eta }_{HD}={\eta }_C{\eta }_{hdet}$ and $P_{DCH}$ denote the detection efficiency and dark-count probability, for the heralding modes and detectors, respectively. Curves corresponding to various values of ${\eta }_{HD}$ are plotted for the cases (a) without ($P_{DCH}=0$) and (b) with dark counts ($P_{DCH}\ne 0$) in the heralding detectors. The common reference (black) curve in both (a) and (b) corresponds to the case where the ESR node is absent. In (a), the $S$ curves coincide, while in (b), from top (blue) to bottom (gray), the curves correspond to decreasing values of ${\eta }_{HD}$.

Figure 5

Maximal loophole-free violation $S$ as a function of distance for different spectral spreads in the sources for the ESR-assisted DIQKD scheme. $\lambda$ here denotes the largest Schmidt eigenvalue in the Schmidt decomposition of the joint spectral density. The curves from top (blue) to bottom (gray) correspond to decreasing values of $\lambda$. The detection efficiencies at Alice and Bob are assumed to be ideal (i.e., the product of coupling and detector efficiencies ${\eta }_D={\eta }_C{\eta }_{det}=1$ and dark-count probability in the detectors $P_{DC}=0$), while the detection efficiencies in the heralding modes are taken as ${\eta }_{HD}=0.2$ and the dark-count probability in the heralding detectors as $P_{DCH}={10}^{-5}$.

Figure 6

(a) Maximal loophole-free violation of the CHSH inequality $S$ and (b) a lower bound on the key rate $K$ (bits per channel use), as a function of distance. The sources are assumed to be monochromatic. All detection efficiencies are assumed to be of nonunity (${\eta }_D$ and ${\eta }_{HD}$ denoting the detection efficiencies at the end users and the heralding detectors, respectively), but free from dark count ($P_{DC}=P_{DCH}=0$). The curves from top (blue) to bottom (gray) correspond to decreasing values of detection efficiencies ${\eta }_D={\eta }_{HD}=\eta$.

Figure 7

Effect of spectral impurity (where the leading Schmidt eigenvalue $\lambda$ is 0.99) in the sources. (a) The maximal loophole-free violation of the CHSH inequality $S$ and (b) a lower bound on the key rate, $K$ (bits per channel use), are plotted as functions of distance. All detection efficiencies are assumed to be of nonunity (${\eta }_D$ and ${\eta }_{HD}$ denoting the detection efficiencies at the end users and the heralding detectors, respectively), but free from dark count ($P_{DC}=P_{DCH}=0$). The curves from top (blue) to bottom [gray in (a) and pink in (b)] correspond to decreasing values of detection efficiencies ${\eta }_D={\eta }_{HD}=\eta$.

Figure 8

Variation due to the multimode spectral structure in the sources. (a) The maximal loophole-free violation of the CHSH inequality $S$ and (b) a lower bound on the key rate, $K$ (bits per channel use), are plotted as functions of distance. All detection efficiencies are assumed to be nonunity (${\eta }_D={\eta }_{HD}=0.98$, where ${\eta }_D$ and ${\eta }_{HD}$ denote the detection efficiencies at the end users and the heralding detectors, respectively) and dark-count probability $P_{DC}={10}^{-5}$. The curves from top (blue) to bottom (gray) correspond to decreasing values of the largest Schmidt eigenvalue $\lambda$.