This paper introduces a single, unified way to measure the quantum computational resource known as “magic” across three very different kinds of systems: spins, bosons and fermions. Quantum magic is the ingredient that makes a quantum state hard to simulate classically and is necessary for universal quantum computation. The authors build a measure called the magic Rényi entropy (MRE) that treats nonstabilizerness in spin systems and non-Gaussianity in bosonic or fermionic systems on the same footing.

The MRE is defined by taking several identical copies of a quantum state, applying a so-called convolution unitary that mixes those copies, keeping one output copy and discarding the rest, and then measuring how mixed that remaining copy becomes. In plain terms, the measure tracks how much purity is lost after this replica-mixing step. For spin systems with a local Hilbert-space dimension d, the MRE reduces to the previously used stabilizer Rényi entropy (SRE) when the number of copies is chosen appropriately (specifically when the replica number n is coprime with d). For bosons and fermions the same construction quantifies non-Gaussianity. The authors also show that the MRE satisfies standard properties expected of a resource measure, such as additivity and certain monotonicity rules under allowed operations.

The paper goes further by embedding the MRE into field theory. In one-dimensional critical systems—those described at low energy by a (1+1)-dimensional conformal field theory (CFT)—the authors rewrite the MRE as a partition function in a replicated Euclidean path integral. The replica-mixing operation becomes a boundary condition in that replicated theory. Under renormalization, that boundary condition flows to an infrared conformal boundary, and the MRE picks up a universal, size-independent contribution set by the Affleck–Ludwig boundary entropy (also called the g-factor). In other words, a universal number that characterizes the edge of the replicated system determines part of the MRE at criticality.