Phase-space representation of classical simulability for arbitrary bosonic encodings
Phase-space representations can be used to reveal resources for quantum computational advantage. In odd-dimensional qudit systems, states and operations admitting non-negative quasi-probability representations are efficiently classically simulable, and analogous conclusions hold for bosonic systems, usually represented in quadrature phase space. However, when quantum information is encoded in bosonic systems, the link between physical phase-space representations and computational resources becomes nontrivial: large negativities may arise in the physical space even when the encoded model remains efficiently simulable. We establish a general framework connecting physical phase spaces of bosonic systems to the effective computational phase spaces associated with arbitrary encodings. We show that bosonic Hilbert spaces with finitely many modes can always be exactly encoded on a quantum computer, without truncation or approximations, and that for any encoding there exists an adapted physical phase-space representation in which all provably classically simulable states and protocols are non-negative. Conversely, negativity can always persist in physical representations for arbitrary encodings. Our results formalize the operational meaning of phase-space negativity in bosonic quantum information and its relation with computational resources.