Phase-matching quantum key distribution protocol based on orbital angular momentum

. A phase-matching quantum key distribution (PM-QKD) based on orbital angular momentum (OAM) protocol is proposed in the paper, named OAM-PM-QKD protocol, where the OAM of photon is used as an information carrier to implement the PM-QKD protocol. Moreover, the key generation rate performance of OAM-PM-QKD inﬂuenced by atmospheric turbulence is analyzed. The results show that the proposed OAM-PM-QKD protocol is able to exceed the linear key-rate bound when transmission distance exceeds 230 km , and its key generation rate is slightly larger than the original PM-QKD protocol, and the maximum secure transmission distance is also longer.


Introduction
The transmission loss of photons is still a major challenge in the practical implementations of quantum key distribution (QKD) protocol. In 2018, Lucamarini et al. proposed a twin-field QKD (TF-QKD) protocol [1] to overcome the rate-distance limits of point to point QKD protocol without quantum repeaters, and subsequently a lot variant of TF-QKD was proposed, such as 'sending or not sending TF-QKD' protocol [2] and phase-matching quantum key distribution (PM-QKD) protocol [3].
At present, phase is often used as the information carrier for PM-QKD protocol. Like phase, orbital angular momentum (OAM) is also an information carrier of quantum state, and has high-dimensionality, orthogonality, and rotational symmetry properties [4,5]. There are infinite OAM eigenstates in a single photon in theory, thus transmitting more information per photon. OAM state is invariant under rotations about the propagation direction, which further makes the sender and receiver avoid the reference frame misalignment. Until now, OAM states has been widely used as the information carriers in QKD protocol. For example, Mirhosseini [6] verified the high-dimensional QKD experiments encoded by OAM and weakly coherent states. We proposed the theoretical analysis of MDI-QKD protocol based on OAM [7] and round-robin differential-phase-shift quantum key distribution (RRDPS-QKD) protocol based on OAM [8]. Meng et al. [9] improved TF-QKD with the photon OAM to overcome the reference frame dependence. However, the PM-QKD based on OAM states has been so far much less analyzed and reported. 2 Phase-matching quantum key distribution protocol based on orbital angular momentum Fig.1 shows the schematic diagram of the proposed OAM-PM-QKD protocol. Both Alice and Bob generate pulses with a laser and modulate the photons of pulses to be the OAM states by emitting the pulses to a spiral phase plate (SPP). The OAM state is related to the azimuth phase of the photon, and it could be given as where R(r) is the amplitude, r and θ are the radial and azimuthal index, respectively. l is the topological charge of the OAM, whose value is an arbitrary integer. i is an image unit i = √ (−1). The OAM states prepared by Alice and Bob are superposition states of two opposite * shenzg@126.com † zhaosm@njupt.edu.cn ‡ njwanglele@163.com § maoqp@163.com topological charges. Then, Alice and Bob send their pulses carrying OAM states to Charlie through free-space. In the protocol, Charlie is the only one who operates measurements. Then Alice and Bob could generate the key based on Charlies successful measurement results. The details of the proposed OAM-PM-QKD protocol are listed below.
Step 1. Alice first randomly generates a binary key k a and a random phase φ a , φ a ∈ [0, 2π). Then, Alice prepares two SPPs to generate OAM states e ilθ and obtains her state as based on k a and φ a . Similarly, Bob randomly generates a binary key k b and a random parameter φ b , φ b ∈ [0, 2π), and then he generates a state of OAM state | B >. That is, Step 2. Alice and Bob send their signal pulses carrying OAM information to the untrusted third party, Charlie, through a free-space quantum channel.
Step 3. Charlie performs a measurement and records the detection results. The signal pulses sent by Alice and Bob first enter into a 50:50 beam splitter (BS) and then enter into OAM sorters. By calculation, the left output of the BS should be Note that there is an reflection from A to C in BS, and each reflection changes the sign of the topological charge of OAM mode. With the same way, the right output of the BS should be Similarly, there is an reflection from B to D in BS.
Here, the OAM sorter is an efficient OAM separation device based on the coordinate transformation method, which is composed of two optical elements R 1 , R 2 and a lens to achieve coordinate transformation from Cartesian coordinate (x, y) to logarithmic polar coordinate, The first optical element R 1 is with the transforma- ) + x], and a and b are scaling constants [10]. This transformation would introduce some distortions, which can be corrected by the second optical element R 2 [10]. The corrected phase ϕ 2 (u, v) is expressed as − 2abπ f λ exp(− u a ) cos( v a ). With the OAM sorter, the OAM modes ℓ and −ℓ are separated. Here, only OAM mode ℓ in the detectors D 1 and OAM mode −ℓ in D 2 are detected.
Step 3. After Charlie announces the detection result (D 1 click or D 2 click), Alice and Bob declare their random phases φ a and φ b .
Step 4. Only if |φ a − φ b | = 0 or π, then Alice and Bob retain the result of this communication and generate raw key. When |φ a − φ b | = 0, if D 1 click, then k a = k b ; and if D 2 click, then k a =!k b ; On the contrary, when |φ a − φ b | = π, if D 1 click, then k a =!k b , and if D 2 click, then k a = k b .
Step 5. Alice and Bob repeat the above steps until enough raw keys are generated. Then Alice and Bob perform error correction and privacy amplification on the raw key to generate secure key.
The proposed OAM-PM-QKD is a more practical version of PM-QKD. Hence, the key generation rate of the proposed OAM-PM-QKD protocol could be given by the same formula of the PM-QKD [3].
where Q µ is the total gain of pulses; E µ is the total quantum bit error rate(QBER)of pulses; E X µ is the phase error rate representing the information leakage; is the binary Shannon entropy function; µ is the light intensity; f is the error correction efficiency.
We simulate the performance of OAM-PM-QKD protocol with the parameters given in Table 1.  The simulation results are shown in Fig.2. The results show the key generation rate of the proposed OAM-PM-QKD protocol versus transmission distance, and com-pares them with the original PM-QKD protocol under the same conditions without the influence of atmospheric turbulence. From Fig.2, we can find that OAM-PM-QKD is able to exceed the linear key-rate bound when transmission distance exceeds 230km, and its key generation rate is slightly larger than the original PM-QKD protocol, and the maximum secure transmission distance is also longer.

Conclusions
In this paper, We have proposed the PM-QKD protocol based on OAM, where the OAM of photon is used as an information carrier to implement the PM-QKD protocol. We also have analyzed the transmission characteristics of the protocol in atmospheric turbulent channels. Simulation results have shown that the proposed OAM-PM-QKD protocol is able to exceed the linear key-rate bound when transmission distance exceeds 230km, and its key generation rate is slightly larger than the original PM-QKD protocol, and the maximum secure transmission distance is also longer.