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

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.