Sufficiency of quantum non-Gaussianity for discrete-variable quantum key distribution over noisy channels

Quantum key distribution can be enhanced and extended if nonclassical single-photon states of light are used. We study a connection between the security of quantum key distribution and quantum non-Gaussianity of light arriving at the receiver's detection system after the propagation through a noisy quantum channel, being under full control of an eavesdropper performing general collective attacks. We show that while quantum nonclassicality exhibited by the light arriving at the receiver's station is a necessary indication of the security of the discrete-variable protocols, quantum non-Gaussianity can be a sufficient indication of their security. Therefore, checking for non-Gaussianity of this light by performing standard autocorrelation function measurement can be used for prior verification of the usability of prepare-and-measure schemes. It can play a similar role to the prior verification of the quantum correlations sufficient to violate Bell inequalities for entanglement-based protocols.

Figure 1

A setup for the detection of nonclassicality and non-Gaussianity of light arriving at Bob's station, applied for all of the DV QKD models introduced in Sec. 3.

Figure 2

The model for realistic DV QKD scheme with signal coupling to the channel noise in the form of the thermal bath.

Figure 3

Comparison between the lower bounds for the security of BB84 protocol realized using the scheme presented in Fig. 2 for (a) $p=1$ and (b) $p=0.01$ for two different values of the parameter $e:e=0$ (solid lines) and $e=0.05$ (dot-dashed lines), calculated numerically for the cases of $d=0$ (red, uppermost lines), $d={10}^{-5}$ (green, middlemost lines) and $d={10}^{-3}$ (blue, lowermost lines). The calculations were performed using the formula (3), with $Q$ given by expression (12). With black (upper) and brown (lower) dashed lines respectively we denoted the corresponding nonclassicality and non-Gaussianity criteria for the light arriving at Bob's detection system, calculated according to Eqs. (4) and (5), with the quantities $P_S$ and $P_C$ given by expressions (13) and (14).

Figure 4

The model for realistic DV QKD scheme with noise generated by an external source of light coupling to the signal before the quantum channel.

Figure 5

Comparison between the lower bounds for the security of BB84 protocol realized using the scheme presented in Fig. 4 and nonclassicality and non-Gaussianity criteria for light arriving at Bob's side in this case, found in the situation when the source of noise has thermal statistics of the number of photons emitted per pulse. The calculations of QKD security were performed using formula (3), with $Q$ given by expression (22). The nonclassicality and non-Gaussianity criteria were calculated according to Eqs. (4) and (5), with the quantities $P_S$ and $P_C$ given by expressions (24) and (25) respectively. The styles and colors of all the lines are exactly the same as in Fig. 3.

Figure 6

The change of Alice's setup which should be made in the QKD schemes illustrated in Figs. 2 and 4 in order to perform their security analyses in the case when the key is being generated using realistic single-photon source based on SPDC process.

Figure 7

Comparison between the numerically calculated lower bounds for the security of BB84 protocol realized using the scheme pictured in Fig. 2 with the modification presented in Fig. 6 and nonclassicality and non-Gaussianity criteria for light arriving at Bob's side in this particular case. The three panels illustrate the plots made for the following values of the parameter $\nu$: (a) $\nu ={10}^{-1}$, (b) $\nu ={10}^{-2}$, (c) $\nu ={10}^{-4}$. The calculations of QKD security were performed using the formula (2), with $y$ and $Q$ given by expressions (36) and (39) respectively. The nonclassicality and non-Gaussianity criteria were calculated according to Eqs. (4) and (5), with the quantities $P_S$ and $P_C$ given by expressions (40) and (41) respectively. The styles and colors of all the lines are exactly the same as in Fig. 3.

Figure 8

Comparison between the numerically calculated nonclassicality (dashed black, upper line), and non-Gaussianity (dashed brown, lower line) criteria and the lower bounds for the security of BB84 protocol (solid lines) realized by using DV QKD scheme presented in Fig. 2 for $p=0.5$ and $e=0.05$ in the cases of $d=0$ (red, uppermost line), $d={10}^{-5}$ (green, middlemost line), and $d={10}^{-3}$ (blue, lowermost line), with the analogous analytical criteria (corresponding dotted lines) calculated for $T\to 0$ and $d=0$, using formulas (44), (52), and (53). Additionally dot-dashed green (right) and blue (left) lines denote minimal secure values of $T$, analytically calculated for the cases of $d={10}^{-5}$ and $d={10}^{-3}$ respectively, using formula (67). The numerical calculations regarding QKD security, nonclassicality, and non-Gaussianity criteria were performed using exactly the same formulas as in the case presented in Fig. 3.

Figure 9

Comparison between the lower bounds for the security of BB84 protocol (solid lines) realized using the scheme presented in Fig. 2 and the respective criteria for non-Gaussianity of light reaching Bob's detection system (dotted lines), calculated numerically for $p=1$ and $e=0$ for the case when his autocorrelation function measurement is performed with the same noisy and untrusty single-photon detectors that are used for him for the key generation process with the probability of registering a dark count $d=0$ (red, uppermost lines), $d={10}^{-5}$ (green, middlemost lines), and $d={10}^{-3}$ (blue, lowermost lines). The corresponding criterion for the nonclassicality of light arriving at Bob's side (independent of the value of $d$) is drawn with black, dashed line.

Figure 10

Comparison between the lower bounds for the security of BB84 protocol realized with the use of the scheme presented in Fig. 4 and the corresponding non-Gaussianity criteria (brown lines, lowermost when $T\ll 1$) for the light arriving at Bob's side, found numerically for the cases when the source of noise has thermal (solid lines) or Poisson (dashed lines) statistics of the number of photons emitted per pulse. All the calculations were performed for $p=1$ and $e=0$. The QKD security criteria were found for two different values of the probability of registering a dark count in Bob's detectors per gate: $d=0$ (red lines, uppermost when $T\ll 1$) and $d={10}^{-3}$ (blue lines, middlemost when $T\ll 1$).