Hole degrees of freedom play a central role in the exact solution of quadratic (mean-field) systems. Although a variety of experiments have suggested the existence of bosonic holes, a consistent and complete theory has long been hindered by the ghost problems. Here, we resolve the ghost problem and establish a unified theoretical framework for bosonic holes by introducing the $\mathcal{CPT}$ theory and bosonic particle-hole (PH) transformation. The bosonic analogs of the ‘‘Fermi surface’’ and ‘‘Fermi level’’ are proposed. Furthermore, PH duality between Hermitian and non-Hermitian quadratic bosonic systems (QBSs) is revealed. In both distinct QBSs, the $\mathcal{C}$ parity is shown to label PH-conjugate eigenspaces. Building on this duality, we demonstrate PH Bogoliubov quasiparticles in $\mathcal{APT}$ symmetric Hamiltonians, investigate the dynamical generation of PH entanglement, and predict Hermitian PH Aharonov-Bohm interference in non-Hermitian QBSs.

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• Bosons
• Open quantum systems & decoherence
• PT-symmetric quantum mechanics
• Quantum entanglement
• Non-Hermitian systems