We use the Gaussian approximation describing photon count statistics for both the homodyne and the double-homodyne (heterodyne) measurements to study asymmetry effects arising from an 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. Additional noise sources are assumed to be represented by statistically independent components included in a single excess-noise parameter that enters the continuous-variable quantum key distribution (CVQKD) analysis. 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 the 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 semidefinite. The analytical results are used to perform analysis of the asymptotic security of the Gaussian-modulated CVQKD 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 squeezingdependent Holevo information whose extrema determine the lower and upper bounds for the secret fraction. For both types of measurements, the mutual information, the Holevo information, and the asymptotic secret fraction are sensitive to asymmetry effects, leading to degraded performance of the protocol.

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• Quantum communication
• Quantum communication, protocols & technology
• Quantum measurements