# Phase-encoded measurement device independent quantum key distribution with practical spontaneous parametric-down-conversion sources

## Abstract

Measurement-device-independent quantum key distribution (MDI-QKD) with weak coherent sources has been widely and meticulously analyzed. However, the analysis for MDI-QKD with spontaneous parametric-down-conversion sources (SPDCS) is incomplete. In this paper, we propose two passive decoy protocols suitable for parameter estimation in MDI-QKD using SPDCS. By accounting for practical parameters of SPDCS with thermal distribution, we present an investigation on the performances of MDI-QKD under the active three-intensity decoy protocol, the passive two-intensity decoy protocol and the modified passive three-intensity protocol respectively. Phase randomization, inherently prerequisite for decoy protocol, is taken into consideration for evaluating the overall quantum bit gain and quantum bit error rate. The numerical simulations show that the MDI-QKD using SPDCS with practical decoy protocols can be demonstrated comparable to the asymptotical case and has apparent superiority both in transmission distance and key generation rate compared to the one using weak coherent sources. Our results also indicate that the modified passive three-intensity decoy protocol can perform better than the active three-intensity decoy protocol in MDI-QKD using practical SPDCS.

###### pacs:

03.65.Ud, 03.65.Yz, 03.67.-a## I introduction

Quantum key distribution (QKD) is an art to make sure that the two legal communication parties can share the same key based on quantum mechanics. QKD has achieved great development both in theory and experiment (1); (2); (3); (4); (5); (6) since Bennett and Brassard presented the first QKD protocol, BB84 protocol (7). The ideal QKD has been theoretically proved to be unconditionally secure in different ways (8); (9); (10); (11); (14); (12); (13), no matter how great the computation power and storage space the eavesdropper owns. However, practical QKD system undoubtedly exists sorts of imperfections. In fact, unconditional security of QKD never implies there is no restriction on the security in real situation. It holds under three important assumptions (15), i.e., quantum mechanics are correct, authentication of classical communication is secure and all devices are believable. Some loopholes due to imperfections of setups will open windows for the Eve to acquire the secret key. Just because of this, some quantum hacking strategies have been subtly derived and successfully attacked the practical QKD system, such as time-shift attack (16), detector-blinding attack (17), wavelength-dependent attack (18), and so on (19); (20); (21); (22).

Some countermeasures have been proposed to overcome these attacks. One approach is to derive efficient mathematical model to characterize the properties of devices and take imperfect factors into account in security proof as comprehensively as possible (15). But from the angle of philosophy, it will be difficult to implement on a practical level. Another way is to employ the device-independent QKD (DI-QKD) (23); (24), in which the security is based on the violation of a Bell inequality (25) without knowing the specifications of the devices used. However, it inherently requires a high detection efficiency to overcome the security loophole, which is hardly to achieve for practical single photon detectors, leading directly to a extremely low key rate at practical distances (26). Thus, people begins to look for feasible schemes for QKD that are intermediate between standard (device-dependent) QKD and DI-QKD (27); (28); (29); (30).

Measurement-device independent QKD (MDI-QKD) scheme is recently proposed by Lo et al (29), in which Alice and Bob both send photons to an untrusted third party who can even be an eavesdropper, say Eve. Eve performs a partial Bell-state measurement and announces the results to Alice and Bob for distilling a secret key. In MDI-QKD, the detection system can be considered as an oracle with input and output trusted, but no matter what has happened within it. Thus, MDI-QKD can remove all detector side channels, of which the security is guaranteed by the entanglement swapping techniques and reverse Einstein-Podolsky-Rosen (EPR) schemes (30). Once MDI-QKD has been put forward, some modified schemes have been proposed (31); (32) and several experimental demonstrations have been performed (33); (34); (35); (36).

Note the fact that the sources of MDI-QKD should be trusted, a complete characterization of the source is indispensable. The weak coherent source is commonly used as a replacement of the perfect single photon source. However, like the standard QKD, the MDI-QKD with weak coherent sources is also vulnerable to the photon-number-splitting attack due to the multi-photon fraction (37). Fortunately, the decoy-state method (38); (39); (40) can be sufficiently applied to against this attack. More importantly, it can be used as a tool to efficiently estimate the contribution of the single-photon pulse. In Lo et al.’s seminal paper (29), the ideal infinite decoy-state protocol is conducted to evaluate the contribution of the single-photon pulse. But their result is almost impossible to be realized in real situations because of the limited source. Later, some practical decoy-state protocols and their improvements have been proposed by many researchers combining such as finite-size effect, basis dependent imperfection or quantum memories (41); (42); (43); (44); (45); (46). But most of these results are based on weak coherent source (WCS) except for the three-intensity decoy-state method proposed by Wang (44). For another candidate of photon sources within reach of current technology, i.e., the conditional generation of single photons based on parametric down-conversion (47); (48); (49), the performance of decoy-state MDI-QKD remains an issue of common concern.

Recently, we notice that Wang et al. (50) derived a formula for estimating the single-photon contribution for the MDI-QKD with Poisson distributed heralded single photon source (HSPS). However, in this paper, we will analyze the case when the spontaneous parametric-down-conversion sources (SPDCS) under a practical photon number distribution (PND) are used in MDI-QKD. Here, the PND is determined by a SPDCS with a threshold detector. Just simply considering the PND when threshold detector is triggered, we can apply the active three-intensity decoy-state protocol to estimate the single-photon’s contribution. While also taking the non-triggered PND into account, the passive decoy state protocol originally proposed by Adachi et al. is conducted (51). Note that full phase randomization of each individual pulse is a crucial assumption in security proofs of decoy state QKD and active phase randomization is implemented to protect against attacks on imperfect sources in recent experimental demonstration of MDI-QKD(36). Thus, differing from the analysis of Wang et al. (50), our evaluations for the overall gain and quantum bit error rate (QBER) are based on full phase randomization, which is internally demanded for decoy-state method. So our formulas for the overall gain and QBER are more stringent.

In this paper, inspired from the passive decoy-state method presented by Adachi et al.(51), we raise two passive decoy protocols, the passive two-intensity decoy protocol and the modified passive three-intensity decoy protocol, which can be applied in MDI-QKD using SPDCS with threshold detectors. Taking the phase-encoded scheme proposed by Ma et al. for an example(32), we analyze the performances of MDI-QKD with practical SPDCS when different decoy protocols are conducted. Numerical simulations show that the MDI-QKD using SPDCS has apparent superiority both in transmission distance and key generation rate compared to the one using weak coherent sources. Moreover, both of the modified passive three-intensity decoy protocol and active three-intensity decoy protocol(44) can be demonstrated comparable to the asymptotical case when active infinite decoy states are used. Our results also indicate the modified passive three-intensity decoy protocol can perform better than the active three-intensity decoy protocol in MDI-QKD using SPDCS with a practical photon number distribution. The rest of paper is organized as follows. In Sec. II, we shall briefly review the SPDCS with practical photon number distribution. The formulas for calculating the single-photon yield and error rate using active three-intensity decoy protocol are introduced in Sec. III. We derive the formulas for the passive two-intensity decoy protocol and modified passive three-intensity decoy protocol respectively in Sec. IV and Sec. V. We perform numerical simulations in Sec. VI, followed by a statistical fluctuation analysis in Sec. VII. We conclude the paper in Sec. VIII.

## Ii practical spontaneous parametric-down-conversion source

As to our knowledge, spontaneous parametric-down-conversion (SPDC) processes are widely used as the sources in QKD such as the triggered or heralded single-photon source(51); (52), and the entanglement source(53); (55). Up to now, there exist some promising implementation demonstrations of QKD based on SPDC processes(53); (54); (55). In this paper, as is shown in Fig.1, we consider the case when SPDC processes are used as triggered single-photon sources in phase-encoded MDI-QKD scheme proposed by Ma et al.(32)(other experimental schemes like the original polarization-based protocol(29) and the phase-encoded protocol proposed by Tamaki et.al(31) are also feasible). We remark that the security of scheme shown in Fig.1 relies on detectors and to be trusted, which maybe introduce side channel attack if the laboratories are untrusted(56). However, this type of attacks can be avoided if Alice and Bob have secret area respectively.

The two identical sources with sub-Poissonian distribution are emitted from non-degenerate SPDC process(47). This type of SPDC processes creates the two-mode state

(1) |

Suppose the intensity of the source to be , then the above description simplifies to

(2) |

When Alice(Bob) monitors one mode of her(his) SPDC source with a practical threshold detector described by detection efficiency () and dark count rate (), Alice’s(Bob’s) other mode changes to a mixed state under the condition that the detector has been successfully triggered. This state can be described as(54); (57)

(3) |

(4) |

where () is the mean photon number of Alice’s (Bob’s) one mode before triggering, () denotes the post-selection probability, and () is the correlation rate of photon pairs for Alice’s(Bob’s) source with intensity of (), i.e. the probability that one can predict the existence of a heralded photon given a triggered one. After phase randomization(integrating over ), Alice and Bob’s heralded mixed state shown as Eq.(3)and Eq.(4) can be described by the following photon number distribution

(5) |

where () denotes the case of triggered state output from Alice(Bob).

## Iii active three-intensity decoy-state protocol

Once the two-pulse state from Alice and Bob’s SPDCS arrives at the relay(controlled by an untrusted eavesdropper Eve)for detection, Eve will announce whether the two-pulse state has caused a successful Bell state measurement according to the detection results. Those states corresponding to successful measurement events at the relay will be post selected as the sifted keys and be input to the postprocessing procedure. The most key point is how to estimate the single-photon contributions and errors based on practical experiments. Luckily, the three-intensity decoy-state protocol for MDI-QKD recently proposed by Wang(44) can be introduced for this purpose. In this protocol, Alice(Bob) need to actively and randomly modulate her(his) sources to prepare different states with intensity of 0, and (0, and ). Here, () is the intensity of signal state, 0 is the intensity of vacuum states, ()is the intensity of decoy state and (). According to Wang’s theory, if holds true, the low bound of successful single-photon counting rate in the basis and upper bound of single-photon error rate in the basis can be estimated by(44)

(6) |

(7) |

Here, () denotes the overall gain when the intensity of Alice’s source is () and that of Bob’s source is (), , , , , and are the overall gain under different sources where represents that the intensity of the corresponding source is vacuum; is the yield of a background noise; , and is the overall gain in the basis under different sources ; , and is the overall quantum bit error rate of events in the basis under different sources ; is the error rate of a random noise.

Thus, the key rate for the active three intensity decoy-state protocol can be given by(44)

(8) |

where denotes the binary Shannon entropy function, and are the overall gain and error rate in the basis under source of . is the error correction efficiency.

## Iv passive two-intensity decoy-state protocol

Active decoy-state protocol typically needs a variable optical attenuator (VOA) in signal sender’s side to independently and randomly vary the intensity of each signal state. However, in some scenarios, if the VOA is not correctly designed, it may occur that some physical parameters of the sending pulses rely on the particular setting selected (58), which could bring threat to the security of the active schemes. Therefore passive preparation of intensity might be practically desirable to some extent (59). In (51), Adachi et al. present an efficient passive decoy-state proposal for the QKD based on SPDCS, where only a simple threshold detector is used. Later, Ma and Lo (56) combined the results of (51) and (59) to the most common case. In 2009, Curty et al. (60) generalized the passive decoy idea and proposed a new scheme for the case when weak coherent source is used. Here, inspiring from the idea proposed by Adachi et al. (51), we shall apply the passive decoy protocol to the MDI-QKD using SPDCS. The experimental setup, as is shown in Fig.1, is kept unchanged as the active three intensity decoy-state protocol, except keeping the detectors in the relay to work no matter whether there is a trigger signal or not in Alice and Bob’s threshold detectors. Therefore, the non-triggered events, acting as the role of decoy states, can be used to estimate the single-photon contribution. In this case, the signal -photon events can be divided into two parts, the triggered events with probability of and the non-triggered events with probability of

(9) |

(10) |

where or . If we consider the above photon number distribution shown in Eq.(9) and Eq.(10), the corresponding overall counting rates of triggered events and non-triggered events can be expressed as

(11) |

(12) |

where is the triggered gain when Alice sends -photon pulse and Bob sends -photon pulse, is the non-triggered gain when Alice sends -photon pulse and Bob sends -photon pulse, is the triggered overall gain when the intensity of Alice’s source is and that of Bob’s source is . The meaning of , and is similar to , and are the gains from background noises. Define where and , then . From a mathematical analysis, we can obtain that . Thus, it is obvious that for . Define , then for . Applying Eq.(11) and Eq.(12) leads to , we thus obtain the minimum value of as a function of the parameter :

(13) |

where and .

Note that the overall quantum bit error rates of triggered and non-triggered events are calculated by and , we can also get the upper bound of as

(14) |

where and is the error rate of a random noise.

In a similar way, we have another bound

(15) |

where .

Combining these two upper bounds, we have

(16) |

where .

Consequently, the key rate from the triggered events can be given by

(17) |

and the key rate from both the triggered and non-triggered events is given by

(18) |

Therefore, the final key rate of the passive one intensity decoy-state protocol is given by

(19) |

## V modified passive three-intensity decoy-state protocol

For the active three-intensity decoy-state protocol, the basic assumption is that the yield of decoy states is equal to that of signal states, i.e., for . And for the passive two-intensity decoy-state protocol, the assumption is that the yield of triggered states is equal to that of non-triggered states, i.e., for . Similarly, considering the case when active decoy states are used in passive decoy protocol, we can also assume that the yield of triggered decoy states is equal to that of non-triggered signal states, i.e., for . Based on this assumption, we improve the passive two-intensity decoy-state protocol and propose a new protocol called modified passive three-intensity decoy-state protocol.

In our new protocol, Alice(Bob) also need to actively and randomly modulate her(his) sources to prepare different states with intensity of 0, and (0, and ). Here, () is the intensity of signal states, 0 is the intensity of vacuum states, ()is the intensity of decoy states and (). The set of signal states is divided into two parts, one part is the set of triggered signal states and the other is the set of non-triggered signal states. And the set of decoy states is also divided into the part of triggered decoy states and the part of non-triggered decoy states. For convenience, we choose the non-triggered signal states and triggered decoy states to derive the yield of single photons. And we require that

(20) |

The convex forms of the overall triggered decoy events and non-triggered signal events can be expressed as

(21) |

(22) |

where , , and denote the triggered and non-triggered two-pulse state respectively when Alice sends pulse with intensity of and Bob sends vacuum pulse, and denote the triggered and non-triggered two-pulse state respectively when Alice sends vacuum pulse and Bob sends pulse with intensity of , and denote the triggered and non-triggered two-pulse state respectively when Alice and Bob both send vacuum pulse. Here, the state leads to the yield . Applying the same analysis as Wang’s theory(44), we can also derive a bound on based on Eq.(21) and Eq.(22). Under the condition that , and can be estimated by

(23) |

(24) |

where , , is the triggered gain when Alice sends pulses with intensity of and Bob sends pulses with intensity of , is the non-triggered gain when Alice sends pulses with intensity of and Bob sends pulses with intensity of , is the triggered overall gain when the intensity of Alice’s source is and that of Bob’s source is vacuum. The meaning of ,