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arXiv:2508.06018 · Reports on Progress in Physics

Magic entropy in hybrid spin-boson systems

Samuel Crew, Ying-Lin Li, Heng-Hsi Li, Po-Yao Chang

Published Feb 1, 2026• 1 citation
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Abstract

We introduce entropic measures to quantify non-classical resource in hybrid spin-boson systems. We discuss the stabiliser Rényi entropy in the framework of phase space quantisation and define an analogous hybrid magic entropy and a mutual magic entropy that capture the distribution of quantum magic across spin and bosonic subsystems. We use these entropic measures to demonstrate two key phenomena: the detection of the superradiant phase transition in the Dicke model and the quantum dynamics of magic in the Jaynes-Cummings model. We develop a Monte Carlo numerical scheme to practically evaluate these entropic measures in interacting many-body systems.

DOI
10.1088/1361-6633/ae413c

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We introduce entropic measures to quantify non-classical resource in hybrid spin-boson systems. We discuss the stabiliser Rényi entropy in the framework of phase space quantisation and define an analogous hybrid magic entropy and a mutual magic entropy that capture the distribution of quantum magic across spin and bosonic subsystems. We use these entropic measures to demonstrate two key phenomena: the detection of the superradiant phase transition in the Dicke model and the quantum dynamics of magic in the Jaynes-Cummings model. We develop a Monte Carlo numerical scheme to practically evaluate these entropic measures in interacting many-body systems.
Characterizing a quantum state through the lens of quantum resources provides an information-theoretic perspective on many-body systems. While quantum entanglement serves as the paradigmatic example of a quantum resource, recent studies have shown that quantum magic, a resource for universal quantum computation, captures aspects of many-body states complementary to those described by entanglement. For instance, in spin systems, conformal field theory (CFT) analysis of the stabilizer Rényi entropy has revealed universal features of nonstabilizerness qualitatively distinct from entanglement. In bosonic and fermionic systems, however, a comparable formulation for their computational resource, non-Gaussianity, has yet to be established. In this work, we introduce a unified measure, the magic Rényi entropy (MRE), to quantify computational resources in spins, bosons, and fermions on an equal footing. We show that the MRE is a resource monotone under stabilizer and Gaussian protocols involving measurements and feedforward operations. The MRE reveals common universal aspects of nonstabilizerness and non-Gaussianity in critical many-body states. In particular, our CFT analysis shows that the universal contribution to the MRE appears as the size-independent term determined by the Affleck-Ludwig boundary entropy. We find that non-Gaussianity can continuously renormalize this universal contribution or drive a boundary transition through bulk-induced boundary renormalization-group flows. As a concrete example, we present a CFT analysis of non-Gaussianity in interacting spinless fermions described by the Tomonaga-Luttinger liquid, showing boundary transitions at the Luttinger parameters K=1/3K=1/3K=1/3 and K=3K=3K=3. Our field-theoretical predictions are confirmed by numerical calculations. These results provide a unified field-theoretical understanding of many-body magic across spins, bosons, and fermions.
Abstract Spin systems interacting with boson environments are ubiquitous in nature. To what extent quantum states in such systems depart from classicality is becoming an increasingly important issue. Here, we study nonclassicality in systems involving the interaction between a spin and a boson mode. We introduce a simple and effective method for quantifying spin–boson nonclassicality in terms of the entropy excess between the classical and the quantum Tsallis entropy. The method can be naturally extended to more general systems. Fundamental properties of the nonclassicality quantifier are revealed, which render it reasonable for spin–boson states. Basic features are illustrated by a variety of typical spin–boson states. As applications, we further investigate spin–boson nonclassicality in the Dicke model, which sheds light on the normal-superradiant phase transition and quantum metrology.
Quantum magic, or nonstabilizerness, provides a crucial characterization of quantum systems, regarding the classical simulability with stabilizer states. In this work, we propose an alternative and efficient algorithm for computing stabilizer R\'enyi entropy, one of the measures for quantum magic, in spin systems with sign-problem free Hamiltonians. This algorithm is based on the quantum Monte Carlo simulation of the path integral of the work between two partition function ensembles and it applies to all spatial dimensions and temperatures. We demonstrate this algorithm on the one- and two-dimensional transverse field Ising model at both finite and zero temperatures and show the quantitative agreements with tensor-network based algorithms. We analyze the computational cost and provide analytical and numerical evidences for it to be polynomial in system size. This work also suggests a unifying framework for calculating various types of entropy quantities including entanglement R\'enyi entropy and entanglement R\'enyi negativity.
The advent of quantum technologies brought forward much attention to the theoretical characterization of the computational resources they provide. A method to quantify quantum resources is to use a class of functions called magic monotones and stabilizer entropies, which are, however, notoriously hard and impractical to evaluate for large system sizes. In recent studies, a fundamental connection between information scrambling, the magic monotone mana and 2-Renyi stabilizer entropy was established. This connection simplified magic monotone calculation, but this class of methods still suffers from exponential scaling with respect to the number of qubits. In this work, we establish a way to sample an out-of-time-order correlator that approximates magic monotones and 2-Renyi stabilizer entropy. We numerically show the relation of these sampled correlators to different non-stabilizerness measures for both qubit and qutrit systems and provide an analytical relation to 2-Renyi stabilizer entropy. Furthermore, we put forward and simulate a protocol to measure the monotonic behaviour of magic for the time evolution of local Hamiltonians.
Entanglement and magic are fundamental resources that capture the complexity of quantum many-body systems. Non-local magic isolates the irreducible nonstabilizerness intrinsically tied to entanglement. However, evaluating this quantity generally requires a prohibitive minimization over the full Hilbert space, making it computationally inaccessible beyond a few qubits. Here, we overcome this bottleneck by establishing a closed-form expression for the non-local stabilizer entropies of fermionic Gaussian states over local Gaussian unitaries, which we prove at Rényi index α=2α=2α=2 for arbitrary subsystem size, and which can be evaluated in polynomial time directly from the eigenvalues of the reduced Majorana covariance matrix. We apply this framework to characterize fermionic non-local magic across diverse physical regimes: we derive an exact Page-like curve for typical random states, reveal logarithmic scaling at the quantum critical point of the XY model, and establish a quasiparticle picture for magic generation during out-of-equilibrium quantum quenches. Crucially, because our result relies solely on two-point correlation functions, it provides a scalable route for the experimental estimation of fermionic non-local magic in large-scale quantum processors via fermionic shadow tomography.
Quantum entanglement and quantum magic are two distinct fundamental resources that enable quantum systems to exhibit complex phenomena beyond the capabilities of classical computer simulations. While quantum entanglement has been extensively used to characterize both equilibrium and dynamical phases, the study of quantum magic, typically quantified by the stabilizer Rényi entropy (SRE), remains largely limited to numerical simulations of moderate system sizes. In this Letter, we establish a general framework for analyzing the SRE in solvable Sachdev-Ye-Kitaev models in the large-N limit, which enables the application of the saddle-point approximation. Applying this method to the Maldacena-Qi coupled Sachdev-Ye-Kitaev model, we identify a series of first-order transitions of the SRE as the temperature is tuned. In particular, we uncover an intrinsic transition of the SRE that cannot be detected through thermodynamic quantities. We also discuss the theoretical understanding of the SRE in both the high-temperature and low-temperature limits. Our results pave the way for studying the SRE in strongly correlated fermionic systems in the thermodynamic limit and suggest a new class of transitions for which the SRE serves as an order parameter.