Magic Rényi Entropy in Quantum Resource Theory

• Magic Rényi Entropy is a Rényi-type quantifier defined via the Pauli expectation-value distribution, distinguishing stabilizer states by yielding zero magic.
• Recent work generalizes MRE through a unified replica-convolution framework that extends its application to bosonic, fermionic, and hybrid systems.
• Efficient computational methods, including XOR-FWHT and QMC algorithms, enable exact and scalable evaluations of second-order MRE in many-body quantum states.

Searching arXiv for papers on Magic Rényi Entropy / stabilizer Rényi entropy and related developments. arXiv_search(query="stabilizer Rényi entropy magic Rényi entropy", max_results=10, sort_by="submittedDate") arXiv_search(query="Magic Rényi Entropy stabilizer Rényi entropy many-body", max_results=10, sort_by="relevance") Magic Rényi Entropy (MRE) is a Rényi-type quantifier of nonstabilizerness, or “magic,” designed to measure quantum resources that lie beyond the stabilizer formalism. In the qubit literature, MRE is often used interchangeably with stabilizer Rényi entropy (SRE): it is the Rényi entropy of the Pauli-expectation-value distribution of a pure state, shifted so that stabilizer states have zero magic. In more recent work, the term has been generalized into a unified replica-convolution construction that treats stabilizer magic in spins and non-Gaussianity in bosons and fermions on the same footing. Across these formulations, the central operational idea is constant: free states remain extremally structured under the chosen phase-space or replica representation, whereas resourceful states develop a broadened distribution and a corresponding loss of purity after convolution (Leone et al., 2021, Matsuda et al., 6 Jul 2026).

1. Foundational definition and resource-theoretic properties

For an $n$-qubit pure state $|\psi\rangle$, the original SRE construction defines the Pauli-string probability distribution

$$\Xi_P(|\psi\rangle):=d^{-1}|\langle\psi|P|\psi\rangle|^2,\qquad d=2^n,$$

with $P\in\mathcal P_n$ and $\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)=1$. The order-$\alpha$ magic Rényi entropy is then

$$M_\alpha(|\psi\rangle):=\frac{1}{1-\alpha}\log\!\left(\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^\alpha\right)-\log d.$$

For $\alpha=2$, this becomes

$$M_2(|\psi\rangle)=-\log\!\left(d\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^2\right),$$

and equivalent quartic Pauli-sum formulas are used throughout later work (Leone et al., 2021, Szombathy et al., 20 Jan 2025).

The resource-theoretic structure is one of the main reasons for the measure’s adoption. The foundational results establish faithfulness, Clifford invariance, and additivity: $$M_\alpha(|\psi\rangle)=0 \iff |\psi\rangle \text{ is a stabilizer state},$$

$|\psi\rangle$0

and

$|\psi\rangle$1

For $|\psi\rangle$2, later papers emphasize that SRE is also monotone under stabilizer protocols for pure states, whereas for $|\psi\rangle$3 that monotonicity fails. This divides the family into a resource-monotone regime and a more diagnostic regime (Hoshino et al., 14 Jul 2025, Zhu et al., 2024).

The second-order case is especially prominent because it is the simplest nontrivial member of the family and admits explicit algebraic reorganizations. Different papers use slightly different normalizations, including base-$|\psi\rangle$4 logarithms and shifts by $|\psi\rangle$5 or $|\psi\rangle$6, but the common content is unchanged: $|\psi\rangle$7 is a fourth moment of Pauli expectation values, so large magic corresponds to a broad Pauli spectrum and low concentration on stabilizer-compatible strings (Huang et al., 31 Dec 2025).

The original formulation also situates MRE among other magic monotones. It proves the upper bounds $|\psi\rangle$8, where $|\psi\rangle$9 is the stabilizer nullity, and for $$\Xi_P(|\psi\rangle):=d^{-1}|\langle\psi|P|\psi\rangle|^2,\qquad d=2^n,$$0,

$$\Xi_P(|\psi\rangle):=d^{-1}|\langle\psi|P|\psi\rangle|^2,\qquad d=2^n,$$1

with $$\Xi_P(|\psi\rangle):=d^{-1}|\langle\psi|P|\psi\rangle|^2,\qquad d=2^n,$$2 the free robustness. This places MRE in the same resource-theoretic hierarchy as robustness- and decomposition-based measures while avoiding variational optimization (Leone et al., 2021).

2. Variants, system classes, and unified formulations

A major development after the original qubit definition is the extension of MRE beyond stabilizer magic narrowly construed. In the qubit case, the free states are stabilizer states and MRE coincides with SRE. In bosonic and fermionic settings, the analogous free states are Gaussian states, so the same construction becomes a Rényi measure of non-Gaussianity. The unifying perspective is that a replica-mixing convolution preserves purity on free pure states and generates mixedness only when computationally relevant resource is present (Matsuda et al., 6 Jul 2026).

System	Free states	Representative formulation
Qubits/spins	Stabilizer states	Pauli-characteristic-function SRE
Bosons	Gaussian states	Phase-space / replica-convolution MRE
Fermions	Gaussian states	Replica-convolution MRE with fermionic characteristic functions
Hybrid spin-boson systems	Gaussian $$\Xi_P(|\psi\rangle):=d^{-1}|\langle\psi|P|\psi\rangle|^2,\qquad d=2^n,$$3 stabilizer products	Hybrid magic Rényi entropy

For qubits, the unified formulation shows that for odd replica number $$\Xi_P(|\psi\rangle):=d^{-1}|\langle\psi|P|\psi\rangle|^2,\qquad d=2^n,$$4, the MRE exactly coincides with the standard SRE. For qudits, the same identification holds when the replica number $$\Xi_P(|\psi\rangle):=d^{-1}|\langle\psi|P|\psi\rangle|^2,\qquad d=2^n,$$5 and the local dimension $$\Xi_P(|\psi\rangle):=d^{-1}|\langle\psi|P|\psi\rangle|^2,\qquad d=2^n,$$6 are coprime. The same framework also explains why a corresponding convolution cannot exist for even $$\Xi_P(|\psi\rangle):=d^{-1}|\langle\psi|P|\psi\rangle|^2,\qquad d=2^n,$$7 in qubits, or more generally when $$\Xi_P(|\psi\rangle):=d^{-1}|\langle\psi|P|\psi\rangle|^2,\qquad d=2^n,$$8 in qudits (Matsuda et al., 6 Jul 2026).

Bosonic and hybrid generalizations recast the problem in phase space. For a trace-class operator expanded in displacement operators $$\Xi_P(|\psi\rangle):=d^{-1}|\langle\psi|P|\psi\rangle|^2,\qquad d=2^n,$$9, one defines the Weyl function $P\in\mathcal P_n$0 and constructs a normalized phase-space distribution from $P\in\mathcal P_n$1. The resulting Rényi entropy $P\in\mathcal P_n$2 becomes the ingredient from which stabilizer, Gaussian, and hybrid magic entropies are built. In hybrid spin-boson systems, the phase space is $P\in\mathcal P_n$3, and the hybrid MRE vanishes precisely on product states of the form $P\in\mathcal P_n$4. A corresponding mutual magic entropy,

$P\in\mathcal P_n$5

quantifies cross-sector resource that is not localized in either subsystem separately (Crew et al., 8 Aug 2025).

This broader usage changes the scope of the term “Magic Rényi Entropy.” In the earlier qubit literature it denotes a stabilizer-resource monotone; in the later unified literature it denotes a common Rényi framework for stabilizer magic and non-Gaussianity across spins, bosons, and fermions. A plausible implication is that “MRE” now functions less as a single formula than as a family of closely related Rényi constructions adapted to different free-state manifolds (Matsuda et al., 6 Jul 2026).

3. Computability, exact algorithms, and numerical estimators

One of the main practical attractions of MRE is that it avoids the optimization over stabilizer decompositions required by many other magic measures. In the original qubit construction, one computes Pauli expectation values and evaluates a Rényi functional directly; no minimization over the stabilizer polytope is required. The same paper also proposed a randomized-measurement protocol for $P\in\mathcal P_n$6, based on sampling random Clifford unitaries, measuring in the computational basis, and estimating four-copy probability correlators (Leone et al., 2021).

The most explicit exact speedup currently available is the XOR-FWHT algorithm for the second-order SRE. A brute-force evaluation of the Pauli quartic sum from a length-$P\in\mathcal P_n$7 state vector scales as $P\in\mathcal P_n$8. By rewriting Pauli strings as $P\in\mathcal P_n$9 with $\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)=1$0, identifying an XOR-convolution structure, and applying fast Walsh-Hadamard transforms, the computation is reduced to $\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)=1$1 FWHTs of length $\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)=1$2. The resulting deterministic and exact runtime is

$\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)=1$3

with natural parallelism across the shift index. In this formulation,

$\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)=1$4

which is algebraically equivalent to the original quartic Pauli sum (Huang et al., 31 Dec 2025).

Several many-body algorithms reformulate MRE as a partition-function observable. A non-equilibrium QMC method for sign-problem-free spin Hamiltonians rewrites the Pauli sum as a ratio of replicated partition functions $\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)=1$5, interpolates with a parameter $\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)=1$6, and evaluates the resulting free-energy difference using a Jarzynski-type work estimator. The paper gives analytical and numerical evidence that the cost is polynomial in system size, with finite-temperature time and memory cost per sample scaling as $\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)=1$7, and reports an empirical signal-to-noise scaling $\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)=1$8 with $\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)=1$9 (Liu et al., 2024).

A complementary SSE-QMC construction evaluates integer-$\alpha$0 SRE by interpreting it as a ratio of generalized partition functions and then sampling reduced Pauli strings in a reduced configuration space. Its central technical point is that the sign problem from $\alpha$1 and $\alpha$2 operators is removed by grouping Pauli strings into reduced strings that retain only whether each local factor is diagonal or off-diagonal and by restricting to valid replica-parity configurations. This makes integer-$\alpha$3 SRE accessible in large 1D and 2D systems and is used to resolve singularities and volume-law corrections near criticality (Ding et al., 21 Jan 2025).

Tensor-network and local-proxy approaches provide additional computational routes. For translation-invariant MPS, the 2-SRE density in the thermodynamic limit is controlled by the dominant eigenvalue of a transfer operator,

$\alpha$4

and the bond-DMRG algorithm constructs an MPO for $\alpha$5 with bond dimension no more than $\alpha$6. For interacting fermions, global SRE remains difficult, but the two-point SRE $\alpha$7 and its mutual counterpart can be estimated from local observables and Monte Carlo-friendly Majorana/Pfaffian formulas, making them practical witnesses of local magic in DQMC (Liu et al., 5 Aug 2025, Fang et al., 19 Jan 2026).

4. Random states, typicality, and local spreading

For Haar-random $\alpha$8-qubit pure states, the joint distribution $\alpha$9 of magic and half-system entanglement becomes exponentially localized as $$M_\alpha(|\psi\rangle):=\frac{1}{1-\alpha}\log\!\left(\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^\alpha\right)-\log d.$$0 grows. The typical values converge to

$$M_\alpha(|\psi\rangle):=\frac{1}{1-\alpha}\log\!\left(\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^\alpha\right)-\log d.$$1

while the widths shrink as

$$M_\alpha(|\psi\rangle):=\frac{1}{1-\alpha}\log\!\left(\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^\alpha\right)-\log d.$$2

The covariance is even smaller,

$$M_\alpha(|\psi\rangle):=\frac{1}{1-\alpha}\log\!\left(\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^\alpha\right)-\log d.$$3

so magic and entanglement fluctuations become exponentially uncorrelated in the thermodynamic limit. This establishes that typical many-qubit states are simultaneously highly magical and nearly maximally entangled, but it does not identify the two resources with one another (Szombathy et al., 20 Jan 2025).

The same study makes the distinction concrete with counterexamples. The product state

$$M_\alpha(|\psi\rangle):=\frac{1}{1-\alpha}\log\!\left(\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^\alpha\right)-\log d.$$4

is unentangled but has substantial magic,

$$M_\alpha(|\psi\rangle):=\frac{1}{1-\alpha}\log\!\left(\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^\alpha\right)-\log d.$$5

Conversely, there are exponentially many stabilizer states with $$M_\alpha(|\psi\rangle):=\frac{1}{1-\alpha}\log\!\left(\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^\alpha\right)-\log d.$$6 and entanglement entropy close to the Haar value, yet they form an exponentially small fraction of Hilbert space relative to typical Haar-random states. This rules out the common misconception that large entanglement and large magic are interchangeable signatures of “quantumness” (Szombathy et al., 20 Jan 2025).

Random-circuit studies show that similar behavior emerges dynamically. For $$M_\alpha(|\psi\rangle):=\frac{1}{1-\alpha}\log\!\left(\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^\alpha\right)-\log d.$$7, a brick-wall random circuit built from alternating layers of two-qubit Haar-random gates rapidly converges to the Haar distribution of magic, and for depth $$M_\alpha(|\psi\rangle):=\frac{1}{1-\alpha}\log\!\left(\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^\alpha\right)-\log d.$$8 the two are nearly indistinguishable. This provides a circuit-based route to the typical high-magic regime (Szombathy et al., 20 Jan 2025).

Local spreading studies in random Clifford circuits reveal a different aspect of MRE: redistribution rather than creation. In a brickwork random Clifford chain initialized with a single local $$M_\alpha(|\psi\rangle):=\frac{1}{1-\alpha}\log\!\left(\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^\alpha\right)-\log d.$$9 excitation, the total pure-state magic is conserved by Clifford dynamics, but the sum of single-qubit SREs decays exponentially,

$\alpha=2$0

because magic becomes hidden in nonlocal correlations. After normalization,

$\alpha=2$1

the single-qubit SRE profile obeys an effective diffusion equation inside a ballistic light cone,

$\alpha=2$2

with width $\alpha=2$3. A restricted Clifford circuit retains non-ballistic spreading but changes the exponent to superdiffusive behavior, $\alpha=2$4 with $\alpha=2$5 (Maity et al., 11 Nov 2025).

5. Dynamics, chaos, and unitary-resource generation

MRE is not confined to state characterization; it has also been extended to quantify the nonstabilizerness-generating power of unitary dynamics. The amortized $\alpha=2$6-stabilizer Rényi entropy of an $\alpha=2$7-qubit unitary $\alpha=2$8 is defined as

$\alpha=2$9

with a strict version obtained by restricting the input to stabilizer states. This quantity is faithful on unitaries, subadditive under composition and tensor product, and therefore a resource monotone for quantum dynamics (Zhu et al., 2024).

A central conceptual result of the amortized theory is “nonstabilizerness dependence”: preexisting magic in the input can enhance the magic generated by a unitary. The paper demonstrates this explicitly for $$M_2(|\psi\rangle)=-\log\!\left(d\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^2\right),$$0, showing that the increase in $$M_2(|\psi\rangle)=-\log\!\left(d\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^2\right),$$1 on a suitable magic input exceeds the maximum increase achievable from any stabilizer input. This behavior distinguishes amortized SRE from amortized robustness of magic and amortized stabilizer extent, for which prior magic does not help (Zhu et al., 2024).

The same formalism yields gate-synthesis lower bounds. For the $$M_2(|\psi\rangle)=-\log\!\left(d\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^2\right),$$2 gate,

$$M_2(|\psi\rangle)=-\log\!\left(d\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^2\right),$$3

for $$M_2(|\psi\rangle)=-\log\!\left(d\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^2\right),$$4,

$$M_2(|\psi\rangle)=-\log\!\left(d\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^2\right),$$5

and more generally

$$M_2(|\psi\rangle)=-\log\!\left(d\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^2\right),$$6

which improves lower bounds on the $$M_2(|\psi\rangle)=-\log\!\left(d\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^2\right),$$7-count of $$M_2(|\psi\rangle)=-\log\!\left(d\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^2\right),$$8, $$M_2(|\psi\rangle)=-\log\!\left(d\sum_{P\in\mathcal P_n}\Xi_P(|\psi\rangle)^2\right),$$9, and one-dimensional Heisenberg time evolution in the examples analyzed (Zhu et al., 2024).

The earlier SRE literature had already connected magic generation to quantum chaos through out-of-time-order correlators. The linear nonstabilizing power of a unitary can be written in terms of averaged 2-OTOC and 8-OTOC expressions, and the conclusion drawn there is that maximal levels of nonstabilizerness are necessary for quantum chaos. In that sense, MRE is not merely a state-resource diagnostic but also a scrambling diagnostic (Leone et al., 2021).

A more direct chaos connection appears in thermofield-double dynamics. For chaotic all-to-all systems, the second-order SRE of a TFD state is proposed to obey

$$M_\alpha(|\psi\rangle)=0 \iff |\psi\rangle \text{ is a stabilizer state},$$0

until the stabilizer bound enforces saturation at $$M_\alpha(|\psi\rangle)=0 \iff |\psi\rangle \text{ is a stabilizer state},$$1. In the SYK model this is realized through an auxiliary-spin path integral with an emergent $$M_\alpha(|\psi\rangle)=0 \iff |\psi\rangle \text{ is a stabilizer state},$$2 symmetry: an early-time symmetric saddle tracks the spectral form factor, while a late-time symmetry-breaking saddle produces nearly maximal magic. The resulting saturation is interpreted as a first-order dynamical transition (Sun et al., 19 Jan 2026).

6. Criticality, topology, and limitations

Many-body applications have shown that MRE is sensitive to universal structure well beyond generic randomness. In the 1D and 2D transverse-field Ising model, 2-SRE exhibits singular derivatives at quantum critical points. In 1D, the 2-SRE density peaks at the critical point; in 2D, it does not, and instead peaks inside the ferromagnetic phase. The paper traces this to a competition between a $$M_\alpha(|\psi\rangle)=0 \iff |\psi\rangle \text{ is a stabilizer state},$$3-part tied to the Pauli characteristic function and a $$M_\alpha(|\psi\rangle)=0 \iff |\psi\rangle \text{ is a stabilizer state},$$4-part tied to ordinary thermodynamics. The subleading volume-law correction $$M_\alpha(|\psi\rangle)=0 \iff |\psi\rangle \text{ is a stabilizer state},$$5 shows discontinuous behavior near criticality and is proposed as a diagnostic of “nonlocal magic” (Ding et al., 21 Jan 2025).

Conformal-field-theoretic treatments sharpen this picture. In critical Ising chains with open boundaries, SRE acquires a universal logarithmic correction,

$$M_\alpha(|\psi\rangle)=0 \iff |\psi\rangle \text{ is a stabilizer state},$$6

whereas topological defects contribute an $$M_\alpha(|\psi\rangle)=0 \iff |\psi\rangle \text{ is a stabilizer state},$$7 constant determined by a defect-sector $$M_\alpha(|\psi\rangle)=0 \iff |\psi\rangle \text{ is a stabilizer state},$$8-factor. With multiple defects, these constants track fusion rules such as

$$M_\alpha(|\psi\rangle)=0 \iff |\psi\rangle \text{ is a stabilizer state},$$9

so SRE becomes an information-theoretic probe of noninvertible symmetry algebra (Hoshino et al., 14 Jul 2025).

The unified field-theory formulation extends the boundary-CFT interpretation beyond spins. In that framework, the universal contribution to the MRE has the form

$|\psi\rangle$00

with $|\psi\rangle$01 the Affleck-Ludwig boundary entropy of the infrared boundary induced by the replica convolution. Non-Gaussianity can either renormalize this $|\psi\rangle$02-factor continuously or trigger a boundary RG flow. For interacting spinless fermions in the Tomonaga-Luttinger liquid, this analysis predicts boundary transitions at

$|\psi\rangle$03

with numerical confirmation near the attractive-side transition (Matsuda et al., 6 Jul 2026).

Tensor-network and fermionic studies add further structure. In translation-invariant injective MPS, the non-local SRE density is bounded by a universal function of entanglement entropy, and the two-site mutual SRE vanishes asymptotically at large separation. In interacting fermion models, the mutual two-point SRE detects the Luttinger-liquid to charge-density-wave transition in 1D, captures the Gross-Neveu-Ising anomalous exponent $|\psi\rangle$04 on the honeycomb lattice, and reveals a short-range plateau associated with exclusion correlations in the $|\psi\rangle$05 Laughlin state (Liu et al., 5 Aug 2025, Fang et al., 19 Jan 2026).

The most significant limitation concerns mixed states. Early work proposed the extension

$|\psi\rangle$06

and established faithfulness on free resources, Clifford invariance, and additivity, with numerical evidence for monotonicity under partial trace (Leone et al., 2021). Later many-body work, however, reports that 2-SRE is not a reliable magic measure for mixed states such as Gibbs states: the mixed-state quantity can develop singularities at physically meaningless points and fail to diagnose thermal criticality cleanly. The present literature therefore supports a clear distinction between the well-established pure-state theory and the more delicate mixed-state case (Ding et al., 21 Jan 2025).

Taken together, these developments define MRE as a broad and technically versatile family of Rényi-based resource measures. In qubit systems it is a computable magic monotone built from Pauli statistics; in bosonic and fermionic systems it becomes a non-Gaussianity measure; in dynamics it quantifies magic generation; and in many-body theory it functions as a probe of criticality, defects, fusion algebra, and chaotic spectral structure. The unifying theme is not a single normalization or formula, but a common principle: computational resource is encoded in the Rényi structure of characteristic-function data, and that structure is often more accessible—analytically, numerically, and experimentally—than optimization-based alternatives (Leone et al., 2021, Matsuda et al., 6 Jul 2026).