Randomness quantification of coherent detection

Continuous-variable quantum cryptographic systems, including random number generation and key distribution, are often based on coherent detection. The essence of the security analysis lies in the randomness quantification. Previous analyses employ a semiquantum picture, where the strong local oscillator limit is assumed. Here we investigate the randomness of homodyne detection in a full quantum scenario by accounting for the shot noise in the local oscillator, which requires us to develop randomness measures in the infinite-dimensional scenario. Similar to the finite-dimensional case, our introduced measure of randomness corresponds to the relative entropy of coherence defined for an infinite-dimensional system. Our results are applicable to general coherent detection systems, in which the local oscillator is inevitably of finite power. As an application example, we employ the analysis method to a practical vacuum-fluctuation quantum random number generator and explore the limits of generation rate given a continuous-wave laser.

Figure 1

(a) Schematic diagram of a shot-noise-driven QRNG. A homodyne receiver measures a certain quadrature $x\left(\varphi \right)$ of the vacuum state, which is controlled by the phase modulator. (b) Equivalent setting of (a). The two input coherent states $\left|{\alpha }_{\mathrm{LO}}{e}^{i\varphi }/\sqrt{2}\right.〉$ have the same phase, but their intensities are independent. The output is proportional to the photon-number difference measured by the two detectors. Here the randomness originates from the shot-noise effect. (c) Phase space presentation of classical modeled homodyne detection measuring $x\left(\varphi \right)$ quadrature of a vacuum state. (d) Generalized flow chart of a shot-noise-driven QRNG. The whole process can be divided into a quantum phase performing Fock basis measurement on certain input states and a classical phase performing a postprocessing on the Fock basis measurement outcomes. LO: local oscillator; PM: phase modulator; BS: beam splitter; $D_{0,1}$: photo detector; ADC: analog-to-digital converter.

Figure 2

Comparison of a Skellam distribution, given in Eq. (3), and a Gaussian distribution, with the same mean 0 and variance $2\mu =100$.

Figure 3

Dependence of the randomness before and after the postprocessing of subtraction on the intensity of LO. In the legend “$R_1$: after postprocessing” refers to Eq. (14) and “$2R_0$: before postprocessing” refers to Eq. (13). The dashed line and dot-dashed line are the practical upper and lower bound of randomness per sample, respectively, given in Sec. 3.

Figure 4

The lower bound of random number generation rate with different resolutions of the ADC and different local oscillator intensities.