Practical Quantum Key Distribution with Non-Phase-Randomized Coherent States

Quantum key distribution (QKD) based on coherent states is well known for its implementation simplicity, but it suffers from loss-dependent attacks based on optimal unambiguous state discrimination. Crucially, previous research has suggested that coherent-state QKD is limited to short distances, typically below 100 km, assuming standard optical fiber loss and system parameters. In this work, we propose a six-coherent-state phase-encoding QKD protocol that is able to tolerate the total loss of up to 38 dB, assuming realistic system parameters, and up to 56 dB loss, assuming zero noise. The security of the protocol is calculated using a recently developed security proof technique based on semidefinite programming, which assumes only the inner-product information of the encoded coherent states, the expected statistics, and that the measurement is basis independent. Our results thus suggest that coherent-state QKD could be a promising candidate for high-speed provably secure QKD.

Figure 1

Schematic of the six-coherent-state QKD protocol. The intensity modulator (IM) and phase modulator 1 (PM1) are positioned at Alice’s site to prepare the intended coherent states, $|{\psi }_z⟩$. As shown in the latter, the optimization suggests that all the reference pulses are the same; thus, IM and PM1 are only added in the “S” path to modulate the intensities of signal pulses. Phase modulator 2 (PM2) at Bob’s site is used to randomly rotate the phase of the signal signal by 0 or $\pi$. To avoid reducing the intensities of the coherent states, we can either use the polarization multiplexing method, as shown in the figure, or deploy active optical switches.

Figure 2

Distribution of coherent states in phase space. The key basis represents the real part of the space. Here, the intensities and phases $\{{\theta }_1,{\theta }_2\}$ are given by the optimization: in fact, the optimization suggests that the optimal encoding scheme is given by ${\mu }_1={\mu }_2={\mu }_3$ and ${\theta }_1={\theta }_2=\pi /2$.

Figure 3

The Gram matrix under six quantum states and the operator set of $\mathcal{S}=\{\mathbb{I},B_0^0,B_1^0,B^{\mathrm{\varnothing }}\}$. We assume that $B_0^{\mathrm{\varnothing }}=B_1^{\mathrm{\varnothing }}=B^{\mathrm{\varnothing }}$. Actually, this matrix is equivalent to the Gram matrix formed by the set $\{B_0^0,B_0^1,B_1^0,B_1^1,B^{\mathrm{\varnothing }}\}$, since the correlations related to $B_y^1$ can be represented by $\mathbb{I}-B_y^0-B^{\mathrm{\varnothing }}$. The reason we consider $\mathcal{S}$ is that it can reduce the matrix dimension.

Figure 4

The sub-block of the Gram matrix.

Figure 5

Statistical model considering dark count and misalignment error, where $y\in \{0,1\}$ represents the basis choice; $C_0$ and $C_1$ ($N_0$ and $N_1$) denote the event that detection $D_0$ and $D_1$ click (do not click), respectively. The notation $\tilde{b}\in \{\tilde{0},\tilde{1},\tilde{\mathrm{\varnothing }}\}$ represents the outcome without considering the misalignment error, while $b\in \{0,1,\mathrm{\varnothing }\}$ denotes the outcome considering the misalignment error.

Figure 6

For the key rate simulation, we assume a detector dark count rate of $p_{\mathrm{DC}}={10}^{-7}$ and a misalignment error rate of $\epsilon =2\mathrm{\%}$. The zero errors mean that the dark count rate and misalignment error are set to zero. The comparison is based on the same error model and same experiment parameters. The protocol of BB84 plus decoy technology we consider here comes from Ref. [34]. The one-decoy scenario of Ref. [34] and our six-coherent-state protocol both have two key states and four test states. Also, we have fewer test states compared to the two or infinite decoy states BB84 protocol. The optimization is carried out with the matlab package yalmip [38] and the SDP solver sedumi [39].