Heisenberg-Weyl bosonic phase spaces: emergence, constraints and quantum informational resources

Abstract

Phase space quasi-probability functions provide powerful representations of quantum states and operators, as well as criteria for assessing quantum computational resources. In discrete, odd-dimensional systems (qudits), protocols involving only non-negative phase space distributions can be efficiently classically simulated. For bosonic systems, defined in continuous variables, phase space negativities are likewise necessary to prevent efficient classical simulation of the underlying physical processes. However, when quantum information is encoded in bosonic systems, this connection becomes subtler: as negativity is only a necessary property for potential quantum advantage, encoding (i.e., physical) states may exhibit large negativities while still corresponding to architectures that remain classically simulable. Several frameworks have attempted to relate non-negativity of states and gates in the computational phase space to non-negativity of processes in the physical bosonic phase space, but a consistent correspondence remains elusive. Here, we introduce a general framework that connects the physical phase space structure of bosonic systems to their encoded computational representations across arbitrary dimensions and encodings. This framework highlights the key role of the reference frame-equivalently, the choice of vacuum-in defining the computational basis and linking its phase space simulability properties to those of the physical system. Finally, we provide computational and physical interpretations of the planar (quadrature-like) phase space limit, where genuinely quantum features may gradually vanish, yielding classically simulable behavior.