Hybrid Magic Entropy in Quantum Systems

• Hybrid Magic Entropy is a framework that merges nonstabilizerness (magic) with various entropy measures to characterize quantum states and resources.
• It integrates pure and mixed state analyses by combining stabilizer Rényi entropy, entanglement entropy, and hybrid phase-space distributions across spin, bosonic, and fermionic systems.
• Applications in random states, monitored circuits, and many-body systems demonstrate its effectiveness for resource certification, noise resilience, and tracking phase transitions.

Hybrid Magic Entropy denotes a family of entropic constructions that couple nonstabilizerness, or “magic,” to entropy, entanglement, or hybrid phase-space structure. The phrase does not refer to a single standardized invariant. In current quantum-information usage, it can denote a composite treatment of stabilizer Rényi entropy and entanglement entropy in random states, an entropy-corrected mixed-state witness such as $\mathcal W_\alpha(\rho)=M_\alpha(\rho)-2S_2(\rho)$, or a phase-space entropy for systems combining spin, bosonic, or fermionic degrees of freedom (Szombathy et al., 20 Jan 2025, Haug et al., 25 Apr 2025, Crew et al., 8 Aug 2025, Sarkis et al., 5 Sep 2025). This suggests that the term is best understood as an umbrella label for entropic formalisms that quantify magic together with statistical structure, mixedness, locality, or subsystem composition.

1. Foundational definitions

For pure $N$-qubit states, the most common starting point is the stabilizer Rényi entropy of order $2$. With Hilbert-space dimension $d=2^N$ and unsigned Pauli group $\mathcal P_N\equiv\{1,X,Y,Z\}^N$, one defines

$$\Xi_\psi(\sigma)=\frac{1}{d}\,\langle\psi|\sigma|\psi\rangle^2,$$

and

$$M_2(|\psi\rangle)\equiv -\log_2\sum_{\sigma\in\mathcal P_N}\Xi_\psi^2(\sigma)-\log_2(d).$$

This normalization ensures that $M_2=0$ for stabilizer states and that $M_2$ is additive for product states. The same framework also gives Clifford invariance, faithfulness, and the upper bound

$$M_2(|\psi\rangle)\le \log_2\!\left(\frac{2^N+1}{2}\right).$$

In the Haar-random setting, the paper further quotes

$N$0

as a lower bound on the average (Szombathy et al., 20 Jan 2025).

For mixed states, the central object is no longer the pure-state Pauli distribution alone but an entropy-corrected quantity. Defining

$N$1

one introduces the witness

$N$2

For pure states $N$3, $N$4, so the mixed-state witness reduces to the usual pure-state stabilizer Rényi entropy. This construction makes entropy explicit: mixedness decreases the witness through the $N$5 term (Haug et al., 25 Apr 2025).

In hybrid spin-boson settings, the definition moves from Pauli spectra to hybrid phase space. For a joint state $N$6 on bosons and spins, the hybrid Weyl function is

$N$7

with normalized phase-space density

$N$8

Its $N$9-Rényi entropy $2$0 is then shifted to define a hybrid magic entropy $2$1 that vanishes on the corresponding free sets, namely stabilizer states for spins and Gaussian states for bosons (Crew et al., 8 Aug 2025). The literature is therefore structurally unified by a common pattern: build a positive distribution from phase-space or Pauli data, take a Rényi-type entropy, and shift or combine it so that free states have zero magic.

2. Random states, entanglement, and composite magic–entropy functionals

For Haar-random pure states, the joint distribution of magic and entanglement provides one of the clearest meanings of Hybrid Magic Entropy. In the half-chain bipartition used in the random-state analysis, the von Neumann entropy is

$2$2

while magic is quantified by $2$3. Numerically, the joint distribution $2$4 becomes exponentially localized around

$2$5

with marginal widths

$2$6

and covariance

$2$7

Equivalently,

$2$8

Magic and entanglement are therefore both typically large, but their fluctuations become exponentially uncorrelated (Szombathy et al., 20 Jan 2025).

This asymptotic decoupling is the main reason the random-state literature motivates hybrid functionals. Although exponentially many states with $2$9 and $d=2^N$0 exist, they form an exponentially small fraction of Hilbert space. Typical Haar-like states instead simultaneously exhibit large magic and near-Page entanglement. The same work notes that product states produced by $d=2^N$1 gates can reach only

$d=2^N$2

which is substantially below the many-body upper bound, underscoring that entanglement facilitates higher many-body magic (Szombathy et al., 20 Jan 2025).

A possible hybrid metric, explicitly described as not proposed by the paper but consistent with its findings, is

$d=2^N$3

Using the reported scalings,

$d=2^N$4

and

$d=2^N$5

Because the covariance term is exponentially negligible, the variance is dominated by the marginal variances. This suggests that in Haar-like ensembles a composite magic–entanglement functional inherits sharp concentration from the separate concentration of $d=2^N$6 and $d=2^N$7 (Szombathy et al., 20 Jan 2025).

3. Mixed states: entropy-corrected witnesses and bounded-entropy testing

In mixed-state resource theory, Hybrid Magic Entropy is most naturally realized by witnesses that combine a magic term with an explicit entropy penalty. The basic witness $d=2^N$8 is genuine in the sense that, for any $d=2^N$9, $\mathcal P_N\equiv\{1,X,Y,Z\}^N$0 implies that $\mathcal P_N\equiv\{1,X,Y,Z\}^N$1 is nonstabilizer, while mixed stabilizer states satisfy $\mathcal P_N\equiv\{1,X,Y,Z\}^N$2. The same framework gives rigorous bounds on standard monotones: $\mathcal P_N\equiv\{1,X,Y,Z\}^N$3 A filtered variant $\mathcal P_N\equiv\{1,X,Y,Z\}^N$4 is stated to be strictly more sensitive than $\mathcal P_N\equiv\{1,X,Y,Z\}^N$5 while having the same asymptotic scaling (Haug et al., 25 Apr 2025).

The mixed-state theory is algorithmic as well as formal. For odd $\mathcal P_N\equiv\{1,X,Y,Z\}^N$6, there is an efficient procedure to estimate $\mathcal P_N\equiv\{1,X,Y,Z\}^N$7 to additive precision $\mathcal P_N\equiv\{1,X,Y,Z\}^N$8 with failure probability $\mathcal P_N\equiv\{1,X,Y,Z\}^N$9 using

$$\Xi_\psi(\sigma)=\frac{1}{d}\,\langle\psi|\sigma|\psi\rangle^2,$$0

copies, $$\Xi_\psi(\sigma)=\frac{1}{d}\,\langle\psi|\sigma|\psi\rangle^2,$$1 circuit depth, and

$$\Xi_\psi(\sigma)=\frac{1}{d}\,\langle\psi|\sigma|\psi\rangle^2,$$2

classical time. In the bounded-entropy regime $$\Xi_\psi(\sigma)=\frac{1}{d}\,\langle\psi|\sigma|\psi\rangle^2,$$3, this leads to a poly$$\Xi_\psi(\sigma)=\frac{1}{d}\,\langle\psi|\sigma|\psi\rangle^2,$$4-copy property test that distinguishes states with $$\Xi_\psi(\sigma)=\frac{1}{d}\,\langle\psi|\sigma|\psi\rangle^2,$$5 from states with $$\Xi_\psi(\sigma)=\frac{1}{d}\,\langle\psi|\sigma|\psi\rangle^2,$$6 by estimating $$\Xi_\psi(\sigma)=\frac{1}{d}\,\langle\psi|\sigma|\psi\rangle^2,$$7 and thresholding $$\Xi_\psi(\sigma)=\frac{1}{d}\,\langle\psi|\sigma|\psi\rangle^2,$$8 (Haug et al., 25 Apr 2025).

The same entropy-sensitive formalism supports certification problems. For noisy product $$\Xi_\psi(\sigma)=\frac{1}{d}\,\langle\psi|\sigma|\psi\rangle^2,$$9-states subject to mixed unital Clifford noise,

$$M_2(|\psi\rangle)\equiv -\log_2\sum_{\sigma\in\mathcal P_N}\Xi_\psi^2(\sigma)-\log_2(d).$$0

the paper proves an efficient algorithm, again assuming $$M_2(|\psi\rangle)\equiv -\log_2\sum_{\sigma\in\mathcal P_N}\Xi_\psi^2(\sigma)-\log_2(d).$$1, to distinguish whether $$M_2(|\psi\rangle)\equiv -\log_2\sum_{\sigma\in\mathcal P_N}\Xi_\psi^2(\sigma)-\log_2(d).$$2 or $$M_2(|\psi\rangle)\equiv -\log_2\sum_{\sigma\in\mathcal P_N}\Xi_\psi^2(\sigma)-\log_2(d).$$3. In the noiseless product case,

$$M_2(|\psi\rangle)\equiv -\log_2\sum_{\sigma\in\mathcal P_N}\Xi_\psi^2(\sigma)-\log_2(d).$$4

so the third-moment estimator calibrates a lower bound on the number of injected non-Clifford resources. The mixed-state witness is also robust under global depolarizing noise. For

$$M_2(|\psi\rangle)\equiv -\log_2\sum_{\sigma\in\mathcal P_N}\Xi_\psi^2(\sigma)-\log_2(d).$$5

the filtered witness remains positive for typical flat-Pauli-spectrum pure states whenever $$M_2(|\psi\rangle)\equiv -\log_2\sum_{\sigma\in\mathcal P_N}\Xi_\psi^2(\sigma)-\log_2(d).$$6, which the paper describes as persistence of magic under exponentially strong noise (Haug et al., 25 Apr 2025).

These constructions extend to many-body numerics. For an $$M_2(|\psi\rangle)\equiv -\log_2\sum_{\sigma\in\mathcal P_N}\Xi_\psi^2(\sigma)-\log_2(d).$$7-qubit subsystem $$M_2(|\psi\rangle)\equiv -\log_2\sum_{\sigma\in\mathcal P_N}\Xi_\psi^2(\sigma)-\log_2(d).$$8 of an MPS with bond dimension $$M_2(|\psi\rangle)\equiv -\log_2\sum_{\sigma\in\mathcal P_N}\Xi_\psi^2(\sigma)-\log_2(d).$$9, $M_2=0$0 can be computed to additive precision $M_2=0$1 in

$M_2=0$2

time, and $M_2=0$3 exactly for integer $M_2=0$4 in

$M_2=0$5

time. Applied to the transverse-field Ising chain, the witness grows roughly linearly with subsystem size near criticality and can be extensive despite entanglement. The same paper further states a cryptographic consequence: to mimic high-magic states with as little magic as possible, one requires an extensive amount of entropy, so entropy becomes a necessary resource to hide magic from eavesdroppers (Haug et al., 25 Apr 2025).

4. Monitored circuits, free fermions, and dynamical separation

Hybrid monitored circuits supply a second major arena in which Hybrid Magic Entropy acquires a concrete meaning. In a one-dimensional brickwork circuit of random two-qubit Clifford gates, stochastic $M_2=0$6-gate injection, and stochastic $M_2=0$7-basis measurements, entanglement and magic undergo distinct measurement-induced transitions. For the main case $M_2=0$8 with $M_2=0$9, the entanglement threshold remains near

$M_2$0

while the magic transition, measured by stabilizer $M_2$1-Rényi entropy, occurs at the larger value

$M_2$2

This yields an intermediate regime in which entanglement is area law but magic remains sub-extensive. The paper interprets the separation by noting that entanglement across a cut is constrained by the number of two-site gates crossing that cut, whereas magic can be created locally by single-qubit $M_2$3 gates and then protected by entangling Clifford dynamics (Fux et al., 2023).

In monitored free-fermion circuits, the structure is different but related. There the total stabilizer Rényi entropy remains extensive in both the critical and area-law entanglement phases. The phase-sensitive quantity is instead the bipartite stabilizer mutual information, which scales logarithmically in the critical phase and saturates to a finite constant in the area-law phase. For projective measurements, the transition is reported within

$M_2$4

The dynamics are also anomalously slow: in the purely unitary case,

$M_2$5

so the saturation time scales as $M_2$6, while in the monitored critical phase the collapse is controlled by $M_2$7 and the early-time form is

$M_2$8

The paper’s conclusion is that total magic is dominated by local contributions, whereas the nonlocal structure of magic tracks the entanglement critical point (Wang et al., 14 Jul 2025).

A broader dynamical comparison comes from ergodic Floquet and Hamiltonian systems. Using participation entropy and stabilizer entropy as paired diagnostics, Floquet dynamics exhibits exponential relaxation with size-independent rates,

$M_2$9

and saturation times

$$M_2(|\psi\rangle)\le \log_2\!\left(\frac{2^N+1}{2}\right).$$0

By contrast, the mixed-fields Ising Hamiltonian shows power-law relaxation with exponents

$$M_2(|\psi\rangle)\le \log_2\!\left(\frac{2^N+1}{2}\right).$$1

sub-Haar stationary values, and

$$M_2(|\psi\rangle)\le \log_2\!\left(\frac{2^N+1}{2}\right).$$2

The same work proposes a composite diagnostic

$$M_2(|\psi\rangle)\le \log_2\!\left(\frac{2^N+1}{2}\right).$$3

with corresponding interpolation between Floquet-like $$M_2(|\psi\rangle)\le \log_2\!\left(\frac{2^N+1}{2}\right).$$4 saturation and Hamiltonian $$M_2(|\psi\rangle)\le \log_2\!\left(\frac{2^N+1}{2}\right).$$5 saturation (Tirrito et al., 2024).

5. Hybrid spin–boson and boson–fermion formalisms

In genuine hybrid quantum systems, Hybrid Magic Entropy becomes a phase-space quantity rather than a simple sum of subsystem entropies. For spin–boson systems, the construction uses the hybrid Weyl function

$$M_2(|\psi\rangle)\le \log_2\!\left(\frac{2^N+1}{2}\right).$$6

the normalized probability density

$$M_2(|\psi\rangle)\le \log_2\!\left(\frac{2^N+1}{2}\right).$$7

and the corresponding Rényi entropy $$M_2(|\psi\rangle)\le \log_2\!\left(\frac{2^N+1}{2}\right).$$8. The hybrid magic entropy is then defined by shifting $$M_2(|\psi\rangle)\le \log_2\!\left(\frac{2^N+1}{2}\right).$$9 so that product free states have zero resource value. For product states, additivity holds: $N$00 The associated mutual magic entropy is

$N$01

which vanishes on product free states and is positive in the perturbative Dicke-model regime studied in the paper (Crew et al., 8 Aug 2025).

This framework detects collective many-body phenomena. In the Dicke model

$N$02

the critical coupling is

$N$03

For $N$04 and $N$05, the spin magic entropy $N$06, the hybrid magic entropy $N$07, and the mutual magic entropy $N$08 all show a sharp peak or divergence-like trend at $N$09, while the bosonic Gaussian entropy exhibits a step-like transition. In the Jaynes–Cummings model, $N$10, $N$11, and $N$12 oscillate after quenches from Fock or coherent initial states, and the global state satisfies $N$13 at times $N$14 because it is then a product state (Crew et al., 8 Aug 2025).

A related boson–fermion program uses Grassmann phase space and a hybrid Wigner function $N$15. There the entropy-like quantity is

$N$16

built from a superspace $N$17 norm of the hybrid Wigner function. In product form it decomposes into a bosonic generalized mana and a fermionic stabilizer Rényi entropy. The paper uses this formalism for the Holstein polaron, where phonon–electron coupling enhances hybrid magic growth, and for the fermionic Jaynes–Cummings model, where the maximum hybrid magic for Fock initial states scales approximately as

$N$18

At the gate level it derives a closed form for the conditional displacement gate and finds saturation

$N$19

(Sarkis et al., 5 Sep 2025).

6. Generalizations, neighboring notions, and terminological boundaries

Several adjacent constructions broaden the meaning of Hybrid Magic Entropy. A convolution-based program defines “magic entropy” for qubits by

$N$20

for odd-prime qudits by

$N$21

and for mixed-dimensional systems by a tensor-product hybrid convolution channel

$N$22

leading to

$N$23

This formulation applies to states and, via Choi states, to gates, while preserving additivity, Clifford invariance, and experimental accessibility through swap tests (Bu et al., 2023).

Other extensions refine locality, multipartite structure, or low-order accessibility. The Heisenberg-picture operator stabilizer Rényi entropy defines an operator-space analogue with maximal value $N$24 and a Lieb–Robinson-type locality bound, making it suited to local dynamical magic generation (Dowling et al., 2024). The multipartite non-local magic functional

$N$25

isolates connected $N$26-body magic and can be positive, zero, or negative; this naturally suggests signed hybrid decompositions that separate local from genuinely global nonstabilizerness (Malvimat et al., 6 Jan 2026). For interacting fermions, the two-point stabilizer Rényi entropy

$N$27

provides a computable local proxy that captures the Luttinger-liquid–to–charge-density-wave transition, the Gross–Neveu–Ising critical exponent $N$28 on the honeycomb lattice, and short-range exclusion structure in the Laughlin state (Fang et al., 19 Jan 2026).

At a more abstract level, parameterized entropic magic quantifiers based on quantum $N$29 Jensen–Shannon divergences define

$N$30

and establish pure-state relations between the entropy-based and relative-entropy-based versions. These quantifiers are proposed as new tools for magic resource theory and can be efficiently computed in low-dimensional Hilbert spaces (Wang et al., 8 Apr 2026).

The term also admits two important disambiguations. First, “hybrid entropy” in the sense of Jizba–Arimitsu entropy combines Rényi and Tsallis axioms through escort averaging and $N$31-deformed composition, but the word “magic” does not appear there; it is a distinct generalized-entropy program rather than a nonstabilizerness measure (Çankaya et al., 2016). Second, in magic-angle twisted bilayer graphene, “magic” refers to the twist angle. The entropic phenomenology near $N$32 concerns a Pomeranchuk-like transition, a large entropy of about $N$33 per moiré unit cell, and a high-entropy correlated state with nearly-free magnetic moments. That usage is conceptually separate from stabilizer magic, even though the phrase “magic entropy” can appear superficially similar (Rozen et al., 2020).

Taken together, these strands show that Hybrid Magic Entropy is not a single formula but a research program. In one direction it means combining magic with entanglement or mixedness; in another it means defining entropic magic directly on hybrid phase spaces; in yet another it means isolating nonlocal, low-order, or operator-space components of magic. The common theme is the replacement of a bare binary distinction—stabilizer versus nonstabilizer—by a quantitative entropic landscape in which magic can be concentrated, witnessed, separated from entanglement, distributed across subsystems, or embedded in hybrid matter and hybrid architectures.