Reference-Frame-Independent Design of Phase-Matching Quantum Key Distribution

The recently proposed phase-matching quantum key distribution offers means to overcome the linear key rate–transmittance bound. Since the key information is encoded onto the phases of coherent states, the misalignment between the two remote reference frames would yield errors and significantly degrade the key generation rate from the ideal case. In this work, we propose a reference-frame-independent design of phase-matching quantum key distribution by introducing a high-dimensional key encoding space. With encoded phases spanning the unit circle, the error statistics at arbitrary fixed-phase-reference difference can be recovered and treated separately, from which the misalignment angle can be identified. By naturally extending the binary encoding symmetry and complementarity to high dimensions, we present a security proof of this high-dimensional phase-matching quantum key distribution and demonstrate with simulation that a 17-dimensional protocol is completely immune to any degree of fixed misalignment and robust to slow phase fluctuations. We expect the high-dimensional protocol to be a practical reference-frame-independent design for general phase-encoding schemes where high-dimensional encoding is relatively easy to implement.

Figure 1

Schematic diagram of the PM QKD protocol with $d$-dimensional encoding [11]. Alice prepares the coherent state ${|\sqrt{\mu /2}{e}^{i(2\pi /d){\kappa }_a}⟩}_A$, where ${\kappa }_a\in \{0,1,\dots ,d-1\}$. Similarly Bob prepares ${|\sqrt{\mu /2}{e}^{i(2\pi /d){\kappa }_b}⟩}_B$. They send the two coherent states to interfere at an untrusted measurement site. Ideally, if the phase difference $2\pi /d|{\kappa }_a-{\kappa }_b|=0$, the detector gives a single $L$ click. If $2\pi /d|{\kappa }_a-{\kappa }_b|=\pi$, the detector gives a single $R$ click.

Figure 2

Schematic diagram of the $d$-dimensional symmetric encoding QKD, where ${\kappa }_a$, ${\kappa }_b$ take values in $\{0,1,\dots ,d-1\}$. Alice and Bob share the bipartite state ${\rho }_{AB}$ and each applies a unitary operation $U(\kappa ):=U^{\kappa }$ according to their key values ${\kappa }_a$ and ${\kappa }_b$. The untrusted party Eve is supposed to announce the key difference $({\kappa }_a-{\kappa }_b)\mathrm{ mod }d$. The setup extends the binary symmetric encoding protocol in Ref. [15].

Figure 3

Schematic diagram of the entanglement-based $d$-dimensional PM QKD, where ${\rho }_{AB}$ is a bipartite state on two optical modes, and the encoding operation $U$ rotates the coherent state by $2\pi /d$. The optical mode is phase rotated by $2\pi k/d$ if the $k\mathrm{th}$ control dit is triggered. The encoded state ${\rho }_0$ is sent to the untrusted Eve for measurement, who is supposed to announce the key difference $({\kappa }_a-{\kappa }_b)\mathrm{ mod }d$, where ${\kappa }_a$ and ${\kappa }_b$ refer to the control dits triggered in $A^{\prime }{B}^{\prime }$. Alice and Bob distill secure keys from the qudit systems $A^{\prime }$ and $B^{\prime }$. The setup extends the binary entanglement-based symmetric encoding protocol in Ref. [15].

Figure 4

Encoding circles of low- and high-dimensional PM QKD against worst-case misalignment. In the low-dimensional case, both encoding phases are far away from the deviated phase locations, thus giving much uncertainty. Yet in the high-dimensional case, the deviated phases are closer to key phases, enabling the error correction to coordinate the phase shift.

Figure 5

Key rate of the 17-dimensional PM QKD at 100 km against fixed misalignment. The $\pi /17$ misalignment correlates the opposite key phases by $R$ clicks, and hence giving almost no effect on the key rate. The worst-case misalignment is reached at $\pi /34$, whose relative effect is negligible.

Figure 6

Rate-distance performance of two- and 17-dimensional PM QKD against various fixed-phase misalignment, in comparison with the repeaterless bound [6, 7]. The key-rate performance of the 17-PM under the worst-case $\pi /34$ misalignment is similar to that of the 2-PM with no misalignment. The key rate of 2-PM decreases gradually and cannot generate keys at the worst-case $\pi /2$ misalignment.

Figure 7

High-dimensional PM QKD under phase fluctuation. (a) Mutual information and privacy leakage against phase fluctuation range ${\varphi }_{\lim }$ for 2- and 17-PM, light intensity 0.2 and 0.03, respectively, communication distance 300 km. The fluctuation does not affect the privacy as a result of encoding symmetry. The mutual information of the 2-PM decreases faster than that of the 17-PM. (b) Mutual information and privacy leakage against light intensity $\mu$ for 2- and 17-PM, fluctuation range $\pi /3$, communication distance 300 km. The light intensity does not affect the mutual information. (c) Optimal light intensity $\mu$ against phase fluctuation range ${\varphi }_{\lim }$ for 2- and 17-PM, communication distance 300 km. The 2-PM light intensity decreases rapidly as the fluctuation increases in order to compensate its faster drop in mutual information. (d) Simulated key-rate performance: 17-PM under both $\pi /6$ fixed misalignment and $\pi /3$ range fluctuation (red line), 2-PM under the same scenario (blue dotted line), 2-PM under fluctuation only, no fixed misalignment (gray dotted line). The 17-PM is superior than the 2-PM under small phase fluctuation.

Figure 8

Schematic diagram of the entanglement-based $d$-dimensional PM QKD, where ${\rho }_{AB}$ is a bipartite state on two optical modes, and the encoding operation $U$ rotates the coherent state by $2\pi /d$. The optical mode is phase rotated by $2\pi k/d$ if the $k\mathrm{th}$ control dit is triggered. Same as Fig. 3 in the main text.