---
title: Mutual Magic Entropy in Quantum Systems
url: https://www.emergentmind.com/topics/mutual-magic-entropy
type: topic
---

# Mutual Magic Entropy in Quantum Systems

Searching arXiv for recent papers on mutual magic entropy and closely related stabilizer Rényi/magic-correlation measures.
Mutual magic entropy is an entropic measure of non-stabilizerness correlations in multipartite quantum systems. In the literature summarized here, it is defined by subtracting subsystem magic contributions from a total magic quantity so as to isolate the portion of quantum magic that is genuinely shared across subsystems rather than localized within them. Closely related constructions appear under several names and in several settings: bipartite stabilizer mutual information in monitored free fermion circuits [2507.10688], mutual magic entropy in hybrid spin-boson systems [2508.06018], mutual two-point stabilizer Rényi entropy in interacting fermions [2601.13314], and mutual von-Neumann or mutual \(2\)-stabilizer Rényi entropy in matrix product state methods [2504.07230]. Across these works, the central theme is that total magic is often extensive and therefore insensitive to structural distinctions, whereas mutual or nonlocal magic measures diagnose delocalization, criticality, and correlation structure.

## 1. Concept and scope

Magic refers to the non-stabilizerness of a quantum state, a resource associated with dynamics or states beyond efficient Clifford simulability. Several of the cited works emphasize that global magic measures can be volume-law even in settings where the physically relevant distinction concerns whether magic is local or delocalized. In monitored free fermion dynamics, for example, the total stabilizer Rényi entropy remains extensive as the system transitions from a critical phase to an area-law phase, while the structure of magic itself undergoes a delocalization phase transition [2507.10688]. This motivates a mutual construction that removes local contributions.

In the most direct formulations, mutual magic entropy is defined by analogy with mutual information: total magic minus marginal magic. In hybrid spin-boson systems, the measure captures the portion of quantum magic that is genuinely distributed—shared—between the spin and bosonic subsystems, beyond what each subsystem would contribute alone [2508.06018]. In fermionic lattice systems, the mutual two-point stabilizer Rényi entropy is introduced specifically to probe correlation-induced magic by subtracting the one-site contributions from the two-site quantity [2601.13314]. In matrix-product-state formulations, mutual stabilizer Rényi entropy is described as quantifying the non-stabilizer correlations between subsystems and as detecting nonlocal magic in correlations [2504.07230].

A recurring implication is that mutual magic entropy is not merely a smaller version of total magic. Rather, it is intended to isolate an intrinsically relational component of non-stabilizerness. This suggests an analogy to connected correlation functions: what remains after local backgrounds are removed.

## 2. Definitions and main variants

Several formally distinct but conceptually aligned definitions appear in the literature.

In monitored free fermion circuits, the relevant quantity is the stabilizer mutual information between regions \(A\) and \(\bar A\),
\[
I_{\alpha} = M_{\alpha}(\rho_A) + M_{\alpha}(\rho_{\bar{A}}) - M_{\alpha}(\rho_{A\cup\bar{A}}),
\]
with the paper noting that for \(\alpha \le 1\) this formula is used directly, while for \(\alpha \ge 2\) a sign flip is implemented for positivity [2507.10688]. The stated interpretation is that only magic that is delocalized survives this subtraction.

In hybrid spin-boson systems, mutual magic entropy of order \(\alpha\) is defined for a pure state \(|\psi\rangle\) as
\[
I_\alpha(|\psi\rangle) := M_\alpha(|\psi\rangle) - M^S_\alpha(\rho_s) - M^G_\alpha(\rho_b),
\]
where \(M_\alpha\) is the hybrid magic Rényi entropy of the total system, \(M^S_\alpha\) is the stabilizer Rényi entropy of the spin reduced state, and \(M^G_\alpha\) is the Gaussian Rényi entropy of the bosonic reduced state [2508.06018]. The definition explicitly merges non-stabilizerness for spins and non-Gaussianity for bosons within a common subtraction scheme.

In interacting fermion systems, the mutual two-point SRE is defined as
\[
\tilde{\mathcal{M}}^{(\alpha)}_{i,j}(\rho)
=
\mathcal{M}^{(\alpha)}_{i,j}(\rho)
-
\mathcal{M}^{(\alpha)}_{i}(\rho)
-
\mathcal{M}^{(\alpha)}_{j}(\rho),
\]
where \(\mathcal{M}^{(\alpha)}_{i,j}(\rho)=M_\alpha(\rho_{i,j})\) is the two-site SRE [2601.13314]. The construction is explicitly local in real space and designed for computational accessibility.

In matrix product state methods, mutual SRE is written for two subsystems \(A\) and \(B\) as
\[
\mathcal{I}_\alpha (\rho)
=
\tilde{M}_\alpha(\rho)
-
\tilde{M}_\alpha(\rho_A)
-
\tilde{M}_\alpha(\rho_B),
\]
with both mutual von-Neumann SRE and mutual \(2\)-SRE discussed [2504.07230]. The same source also introduces an alternative \(\alpha=1\) construction based on a different Pauli-string distribution \(q_\rho(P)\), yielding \(\mathcal{I}_1^{[q]}\) [2504.07230].

For translation-invariant matrix product states, the mutual \(2\)-SRE is defined as
\[
L(\rho_{AB}) = \tilde{M}_2(\rho_{AB}) - \tilde{M}_2(\rho_A) - \tilde{M}_2(\rho_B),
\]
with \(\tilde M_2(\rho)=M_2(\rho)-S_2(\rho)\) [2508.03534]. This is interpreted as the magic encoded in correlations between \(A\) and \(B\).

These definitions differ in substrate—qubits, fermions, hybrid systems, or operator space—but preserve a common architecture: a mutualization of a magic monotone or magic proxy.

## 3. Relation to stabilizer Rényi entropy and other magic measures

Most mutual magic constructions discussed here are built on stabilizer Rényi entropy. In the free-fermion study, magic is quantified using the Stabilizer Rényi Entropy, computed numerically via a perfect sampling algorithm, and the mutual quantity is introduced because the total SRE is extensive even in trivial product states [2507.10688]. In interacting fermions, the two-point SRE is presented as a robust, computationally accessible probe for detecting magic in diverse fermionic phases, and its mutual counterpart removes one-site background magic [2601.13314]. In matrix product states, both mutual von-Neumann SRE and mutual \(2\)-SRE are treated as efficient correlation-sensitive measures [2504.07230].

The hybrid spin-boson formulation broadens the framework by combining spin stabilizer Rényi entropy with bosonic Gaussian Rényi entropy [2508.06018]. Here mutual magic entropy quantifies shared resource content across physically distinct sectors. This suggests that the mutual construction is portable across resource theories provided an additive subsystem-resolved magic measure is available.

A different but related direction is mutual mana, defined for bipartite states as
\[
\mathcal{M}_{\rm mana}(\rho_{ab})
=
\mathrm{Mana}(\rho_{ab})
-
\mathrm{Mana}(\rho_a)
-
\mathrm{Mana}(\rho_b),
\]
with mana based on discrete Wigner-function negativity [2511.08004]. That work presents mutual mana as a measure of magic correlations in close analogy with quantum mutual information. Although this is not mutual magic entropy in the strict Rényi-entropic sense, it belongs to the same family of mutualized magic-correlation measures.

The relation between these constructions is methodological rather than identity-based. Stabilizer-entropy mutual measures probe spread in the Pauli or Majorana basis, whereas mutual mana probes correlation in Wigner negativity. A plausible implication is that “mutual magic entropy” names a broader design principle: subtract local magic resources to isolate genuinely shared non-stabilizer structure.

## 4. Physical interpretation: local versus delocalized magic

A central claim across the cited works is that mutual magic measures distinguish local magic from nonlocal magic. The monitored free-fermion paper states that total SRE includes both local and non-local contributions and that the stabilizer mutual information singles out delocalized, non-local magic [2507.10688]. In the area-law phase, the interpretation given is that magic is locally concentrated and any non-stabilizerness can be removed by local operations, whereas in the critical phase magic is delocalized and long-range [2507.10688].

The hybrid spin-boson formulation makes the same distinction in subsystem language: if the state is a product state, \(I_\alpha(|\psi\rangle)=0\), and all magic is local to the subsystems; if the state is entangled or correlated in a magic way, \(I_\alpha(|\psi\rangle)>0\) [2508.06018]. In that work, mutual magic entropy measures whether the nonclassical resource is genuinely shared between spin and boson sectors.

The interacting-fermion two-point construction sharpens this interpretation. Because the one-site SREs are explicitly subtracted, the mutual two-point SRE probes nonlocal magic due to correlations rather than intrinsic one-site magic [2601.13314]. In translation-invariant MPS, the mutual \(2\)-SRE is described as the magic encoded in correlations between two sites, that is, “nonlocal” magic not removable by local finite-depth Clifford circuits [2508.03534].

In multipartite settings, the inclusion-exclusion functional
\[
\mathrm{M}_{\mathrm{nl}}^{(n)}(\rho_{[n]})
=
\sum_{\emptyset \neq S \subseteq [n]} (-1)^{n-|S|}\, M(\rho_S)
\]
extends the same idea to genuinely global contributions [2601.03076]. This is not named mutual magic entropy, but it is a natural higher-order generalization of mutualization: subsystem subtraction isolates what cannot be assigned to lower-order parts.

## 5. Criticality, phase transitions, and scaling laws

Mutual magic quantities have become useful diagnostics of phase structure because they can change even when total magic does not. In monitored free fermion dynamics, bipartite stabilizer mutual information exhibits the same scaling behavior as entanglement entropy: logarithmic scaling in the critical phase and a finite constant in the area-law phase [2507.10688]. The study reports that total SRE is always volume-law and hence not a useful probe of criticality, whereas BSMI reveals a phase transition coincident with the entanglement phase transition [2507.10688].

In hybrid spin-boson systems, mutual magic entropy is used to detect the superradiant phase transition in the Dicke model. The reported finding is that the mutual magic entropy peaks sharply at the critical point [2508.06018]. In the Jaynes-Cummings model following a quench, the same quantity shows oscillations reflecting periodic delocalization and relocalization of magic between subsystems [2508.06018].

In the one-dimensional spinless \(t\)-\(V\) model, the mutual two-point SRE peaks at the quantum phase transition, with the peak sharpening for increasing system size, and its peak location accurately tracks Berezinskii-Kosterlitz-Thouless critical scaling [2601.13314]. In the two-dimensional honeycomb lattice \(t\)-\(V\) model, the mutual SRE at maximum separation decays with system size as a power law at criticality, mirroring the scaling of squared density-density correlations and allowing extraction of the anomalous dimension consistent with Gross-Neveu-Ising theory [2601.13314].

In matrix product state studies of ground states, mutual SRE is reported to characterize the critical point of the transverse-field Ising model independently of the chosen local basis [2504.07230]. For translation-invariant MPS, two-site mutual SRE vanishes asymptotically in injective MPS, but its derivative with respect to the tuning parameter is sharply peaked at the critical point and grows logarithmically with the separation in the Ising model [2508.03534]. This suggests that the quantity itself and its derivatives may diagnose distinct aspects of critical structure.

## 6. Dynamics, computation, and operational issues

Several works emphasize that mutual magic entropy is only useful insofar as it can be computed at scale.

In monitored free fermion circuits, SRE is computed via a perfect sampling method over Majorana strings, with marginal probabilities obtained from submatrices of the covariance matrix [2507.10688]. The same work studies dynamics and finds that while total SRE becomes extensive in \(O(1)\) time, relaxation in the critical phase is parametrically longer than in generic random circuits, with relaxation time growing linearly with system size [2507.10688]. Because BSMI grows logarithmically in time at criticality, it tracks the slow buildup of nonlocal magic structure rather than the rapid saturation of total magic.

In hybrid spin-boson systems, the authors develop a Monte Carlo numerical scheme, specifically a Metropolis-Hastings sampling technique, to estimate the hybrid and mutual magic entropies in high-dimensional spin-plus-phase-space settings [2508.06018]. The computation jointly samples bosonic phase-space variables and spin Pauli data.

In matrix product states, mutual von-Neumann SRE and magic capacity are stated to be computable in time \(O(N\chi^3)\) for MPS of bond dimension \(\chi\) [2504.07230]. The same work introduces improved Monte-Carlo algorithms for mutual \(2\)-SRE and improved statevector methods for Bell sampling and SREs [2504.07230]. For translation-invariant MPS, a numerically stable bond-DMRG algorithm is introduced for SRE density, and the analysis proves that two-site mutual SRE vanishes asymptotically in injective MPS [2508.03534].

In interacting fermionic systems, the computational advantage comes from restricting to two-site reduced density matrices and using tomography plus Monte Carlo or determinant-quantum-Monte-Carlo-compatible estimators [2601.13314]. This is presented as a way around the intractability of global SRE for large interacting systems.

A common methodological lesson is that mutualization can improve interpretability without necessarily improving computational difficulty. Many of the advances therefore pair a mutual magic definition with a specially tailored sampling or tensor-network method.

## 7. Extensions, comparisons, and limitations

Mutual magic entropy is part of a broader family of “magic correlation” measures, and the literature also records important limitations.

A direct comparison appears in the mutual mana study, which compares mutual mana with quantum mutual information, mutual \(L^1\)-norm magic, and mutual stabilizer \(2\)-Rényi entropy for qutrit beamsplitter outputs [2511.08004]. That work notes that mutual mana can vanish for certain entangled outputs, highlighting that it captures only magic correlations, not all correlations [2511.08004]. This same caution applies more generally: mutual magic entropy is not an entanglement measure and need not track all correlated structure.

The MPS study stresses that mutual SRE is robust to changes in local basis and free of ultraviolet divergences, unlike bare SRE [2504.07230]. By contrast, the study of many-body SRE via reduced Pauli-string sampling reports that \(2\)-SRE fails to characterize magic in mixed states, yielding nonphysical results, and cautions that SRE is best used for pure states when interpreted as a bona fide magic monotone [2501.12146]. In translation-invariant MPS, the mutual \(2\)-SRE is defined through the shifted quantity \(\tilde M_2(\rho)=M_2(\rho)-S_2(\rho)\), reflecting exactly this mixed-state subtlety [2508.03534].

Higher-order generalizations also complicate interpretation. The multipartite non-local magic functional can be negative, with the negative sign interpreted as redundancy rather than synergy of lower-order magics [2601.03076]. This indicates that beyond the bipartite case, inclusion-exclusion-based magic decompositions may not behave like ordinary positive mutual informations.

The operator-space literature offers another extension. “Operator stabilizer entropy” is introduced as the Heisenberg-picture analogue of state SRE, and the paper notes that mutual-type entropies constructed from operator reductions are possible and natural, although not explicitly defined there [2408.16047]. This suggests a plausible continuation of mutual magic entropy into operator growth, locality, and Lieb-Robinson-constrained magic spreading.

Taken together, these works present mutual magic entropy not as a single universally fixed formula, but as a coherent class of subtraction-based measures designed to isolate shared non-stabilizer structure. Their principal significance lies in revealing phase transitions, long-range resource structure, and the distinction between having a large amount of magic and having magic that is genuinely distributed across a many-body system [2507.10688].

Source: https://www.emergentmind.com/topics/mutual-magic-entropy