Asymmetry effects in homodyne and heterodyne measurements: Positive operator-valued measures and asymptotic security of Gaussian continuous variable quantum key distribution

TL;DR

This work analyzes how asymmetry from unbalanced beam splitters and imperfect photodetectors affects Gaussian CV measurements and the security of Gaussian-modulated CV-QKD in the untrusted-noise model. It develops a POVM framework based on Gaussian approximations to photocount statistics, showing that noisy homodyne measurements correspond to an additive-noise channel acting on ideal quadrature measurements, while double-homodyne measurements require a non-unique representation that involves squeezing. The authors derive explicit expressions for mutual information and Holevo information, including a squeezing-parameter optimization for heterodyne-like measurements, and demonstrate that asymmetry degrades the asymptotic secret fraction and channel length. These results highlight device-induced vulnerabilities in CV-QKD and provide a formal basis for optimizing measurement settings to mitigate security loss under realistic imperfections, with future work needed for trusted-noise scenarios.

Abstract

We use the Gaussian approximation describing photocount statistics for both the homodyne and the double homodyne (heterodyne) measurements to study asymmetry effects arising from imbalance of the beam splitters and variations in quantum efficiencies of the photodetectors. After computing the $Q$ symbols of the positive operator-valued measures (POVMs) of noisy measurements that take into account the asymmetry effects, the operator representations for the POVMs are obtained in the form that assumes applying the additive noise quantum channel to the POVMs of noiseless (ideal) measurements. For double homodyne detection, it was found that the noiseless measurements should generally be expressed in terms of the projectors onto squeezed-states and the corresponding squeezed-state operator representation of POVM along with the measurement noise channel depend on the squeezing parameter that lies in the interval dictated by the condition for the excess noise covariance matrix to be positive semi-definite. The analytical results are used to perform analysis of the asymptotic security of the Gaussian-modulated continuous variable quantum key distribution (CV-QKD) protocol in the untrusted-noise scenario where the measurement noise is assumed to be accessible to an adversary. The inherent non-uniqueness of the operator representation for the double-homodyne POVM manifests itself in the squeezing dependent Holevo information that needs to be additionally optimized. For both types of the measurements, the mutual information, the Holevo information and the asymptotic secret fraction are sensitive to asymmetry effects leading to degraded performance of the protocol.

Figures (14)

• Figure 1: Scheme of a homodyne receiver: S is the source of the signal mode with the annihilation operator $\hat{a}$, LO is the source of the reference mode (local oscillator) with the annihilation operator $\hat{a}_{L}$, and BS is the beam splitter with the amplitude transmission and reflection coefficients $t=\cos\theta$ and $r=\sin\theta$, respectively; photodetectors $D_1$ and $D_2$ have quantum efficiencies $\eta_{1}$ and $\eta_{2}$, and $\mu\equiv m_1-m_2$ is the photon count difference.
• Figure 2: Exact (circle dots) and approximate (solid lines with markers) statistical distributions of photon count difference for the signal mode prepared in (a) the coherent state and in (b) the single photon Fock state computed for for different efficiencies at $\left|\alpha_L\right|=5$ and balanced beam splitter.
• Figure 3: Scheme of an eight port double homodyne receiver. S is the source of the signal mode $\hat{a}$; LO is the source of the reference mode $\hat{a}_{L}$; BS$_S$ (BS$_L$) is the signal mode (local oscillator) beam splitter; $\frac{\pi}{2}$ is the quarter wave phase shifter; BS$_i$ is the beam splitter of $i$th homodyne; $D_{1,2}^{(i)}$ are the photodetectors of the $i$th homodyne; and $\mu_i=m_1^{(i)}-m_2^{(i)}$ is the photon count difference registered by the detectors of $i$th homodyne.
• Figure 4: Double homodyne statistical distribution of photocount differences computed from Eq. \ref{['eq:P_mu1m2']} for various detector efficiencies at $\alpha=0.5$ and $\alpha_L=5$. All the beam splitters are taken to be balanced.
• Figure 5: Geometry in the $\delta$-$r$ (principal variance-squeezing parameter) plane. Noise variances \ref{['eq:sigma_N-sq-2']} are positive when the principal variance $\delta_1$ ($\delta_2$) is above $e^{2 r}$ ($e^{-2r}$), so that the squeezing parameter $r$ is ranged between $r_1$ and $r_2$. Solid grey lines represent the graphs of the exponents: $e^{2r}$ and $e^{-2r}$. In the case, where $\delta_1=q$ and $\delta_2=q^{-1}$, the interval is reduced to the point $r_1=r_2=\rho$.