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

TL;DR

The paper presents a general framework to connect the physical phase-space structure of bosonic systems with their encoded computational representations across arbitrary dimensions, using a superselection-rule-compliant (SSRC) formalism. It introduces a spherical phase-space representation on $\mathbb{S}^2$, its planar continuous-variable (CV) limit, and a discrete toric phase space for qudit codes, establishing a one-to-one mapping between bosonic bases and qudit encodings. Key results show that the phase-reference choice fixes the planar origin and governs simulability, while the same encoded resources (SG vs SNG) manifest differently in planar spaces; in the CV limit, genuine quantum features can gradually vanish, linking physical simulability to the encoded code's phase-space properties. The framework extends naturally to bosonic error-correcting codes and clarifies how physical resources translate into discrete Wigner negativity signatures, providing a dimension-agnostic tool for designing and certifying bosonic quantum information processing architectures.

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.

Figures (1)

• Figure 1: (Color online) Intuitive illustration of the three phase space representations considered in this work. Right: SSRC bosonic states admit a spherical representation, such as the spherical Wigner function, which depends on the choice of a orientation (the physical phase reference). Coherent states, Fock states, and more generally particle-separable states fix a direction on the sphere, with the extremal eigenstates of $\hat{J}_{\vec{n}}$ occupying antipodal points along $\vec{n}$. The tangent plane at a chosen point on the sphere corresponds to the continuous variable (CV) limit ($\theta \ll 1$), and the point of tangency defines the vacuum of the planar CV phase space, given by its vicinity. Left: A qudit discrete phase space is constructed from generalized Pauli operators built from SSRC bosonic unitaries, providing the computational phase space description used throughout the work.