Quantum systems promise unprecedented performance with respect to classical counterparts for various applications involving sensing, communications, control, and computing. Bringing this promise to reality hinges upon the distinguishability between quantum states. This paper explores the distinguishability of a broad class of non-Gaussian states, namely the photon-varied Gaussian states (PVGSs), which can be generated with current technologies from Gaussian states of the electromagnetic field. First, we show that the problem of designing orthogonal PVGSs is tantamount to determining the algebraic varieties of generalized Hermite polynomials, thus bridging the fields of quantum mechanics and algebraic geometry. Then, we prove the existence of orthogonal PVGSs by showing that proper choices of the states' parameters lead to nonempty algebraic varieties. By exploiting the geometrical properties of such algebraic varieties, we also establish methodologies for designing pairs of orthogonal PVGSs. Finally, we explore the distinguishability of PVGSs undergoing noisy time evolution in the presence of decoherence and show that, in such conditions, the developed methodologies still allow for achieving the highest distinguishability. By bridging quantum mechanics and algebraic geometry, the findings of this paper provide insights into designing a new class of quantum states that are distinguishable.

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• Quantum communication
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