Mutual Mana: Quantifying Magic Correlations

• Mutual mana is a resource measure quantifying non-stabilizer 'magic' correlations in composite quantum systems via discrete Wigner negativity.
• It computes the excess magic by subtracting local mana from the global state, mirroring quantum mutual information with properties like additivity and Clifford invariance.
• Empirical studies reveal universal scaling in critical and topological systems, with threshold behaviors that impact fault-tolerant quantum computation.

Mutual mana is a quantifier of the correlations in magic—non-stabilizer resources—between subsystems of multipartite quantum states, generalizing the concept of mana from individual systems to composite ones. Originally defined in the context of discrete Wigner function negativity as a monotonic measure of non-stabilizerness, mutual mana captures the nonlocal magic correlations that are essential for universal, fault-tolerant quantum computation and for characterizing non-classicality in topological and many-body quantum systems. Extending the framework of magic as a quantum resource, mutual mana provides a precise analogy to quantum mutual information, but for magic, and exhibits universal scaling in critical systems and topological models.

1. Definition and Mathematical Formulation

Let $\rho_{AB}$ be a bipartite quantum state on Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$, with reduced marginals $\rho_A = \mathrm{Tr}_B \rho_{AB}$ and $\rho_B = \mathrm{Tr}_A \rho_{AB}$. The mana of a density matrix $\rho$ is defined as

$$M(\rho) = \ln \left( \sum_{\mathbf{u}} |W_{\rho}(\mathbf{u})| \right)$$

where $W_{\rho}(\mathbf{u})$ is the discrete Wigner function evaluated at phase-space point $\mathbf{u}$, constructed from Heisenberg–Weyl operators for qudits of odd prime dimension.

The mutual mana is then given by

$I_M(A:B) = M(\rho_{AB}) - M(\rho_A) - M(\rho_B)$

This structure mirrors the definition of quantum mutual information, $I(A:B) = S(\rho_A) + S(\rho_B) - S(\rho_{AB})$, but with von Neumann entropy replaced by mana.

Properties:

• Nonnegativity: Empirically, $I_M(A:B) \ge 0$ for all examples studied.
• Vanishing on product states: $I_M(A:B) = 0$ if and only if $\rho_{AB} = \rho_A \otimes \rho_B$.
• Symmetry: $I_M(A:B) = I_M(B:A)$.
• Clifford invariance and monotonicity: Invariant under local Clifford operations; nonincreasing under local stabilizer-preserving channels.

2. Resource-Theoretic Context and Operational Properties

Mana quantifies non-stabilizerness or "magic," which is a central resource enabling classically-intractable quantum computation on finite-dimensional systems. Within the resource-theoretic framework, mutual mana isolates the portion of magic stored nonlocally in the quantum correlations of a composite state. That is, it measures the "excess" non-stabilizerness inaccessible via local manipulations.

Mutual mana plays an analogous role to mutual information in entanglement theory, providing an operationally meaningful resource monotone. Importantly, for states generated by discrete beamsplitter transformations, mutual mana directly quantifies the conversion of local magic into inter-system correlations: when a magic state is coupled to a stabilizer vacuum state via a discrete beamsplitter (e.g., CSUM or general Clifford unitaries), the entire input local mana is transformed into mutual mana between the two output subsystems (Zhang et al., 11 Nov 2025).

3. Mutual Mana in Topological and Many-Body Quantum Systems

Mutual mana is particularly powerful for diagnosing long-range magic correlations in complex quantum systems. In $SU(2)_k$ Chern–Simons theory, the magic of link states—prepared by path integration on link complements—can be entirely long-range: for torus links, all magic is stored nonlocally and $I_M = M$, while for more intricate links, the global magic can sometimes be attributed to local contributions, yielding $I_M < M$ (Fliss, 2020).

In the context of quantum spin chains at criticality, mutual mana exhibits universal behavior: in the ground state of the self-dual three-state Potts model and its non-integrable extension, mutual mana between two adjacent blocks scales logarithmically with block size,

$$I_M(\ell, L) \approx b\,\ln \left( \frac{L}{\pi} \sin \left( \frac{\pi \ell}{L} \right) \right) + \text{const}$$

where $b$ is a "magic-exponent" numerically found to be robust across integrable and non-integrable regimes, and under reasonable local basis rotations (Tarabunga, 2023). This scaling parallels that of mutual information for entanglement in conformal field theories, but is governed by a distinct exponent reflecting the system's non-stabilizerness structure.

Regime	System Size $L$	Slope $b$	Error
$J=1$ (integrable)	64	0.32	$\pm0.02$
$J=-1$ (integrable)	32	0.34	$\pm0.03$
$p=0.1$	32	0.31	$\pm0.02$
$p=0.2$	32	0.33	$\pm0.03$
$p=0.3$	32	0.35	$\pm0.03$

Away from criticality, mutual mana saturates with separation, distinguishing gapped and critical phases.

4. Explicit Calculation and Examples

For qutrit systems ($d=3$), mutual mana can be computed exactly for mixtures of pure magic states and maximally mixed states coupled via discrete beamsplitters (Zhang et al., 11 Nov 2025). Given an input state of the form $$\rho_{X,p} = p |X\rangle \langle X| + (1-p)\frac{\mathbb{1}}{3}$$, coupling to a stabilizer $|0\rangle$ and applying CSUM$_3$ yields an output with mutual mana equal to the input mana $M(\rho_{X,p})$. Detailed analytic formulas are provided for various input families (real superpositions, noisy coherent states, etc.), with the general feature that mutual mana displays a threshold behavior: it vanishes below a critical value of $p$ and subsequently increases with $p$ up to the value of $\log(5/3)$ for pure magic states.

5. Comparison with Other Measures of Correlations

Mutual mana distinguishes itself from traditional measures such as quantum mutual information, mutual $L^1$-norm magic, and mutual stabilizer 2-Rényi entropy (Zhang et al., 11 Nov 2025):

• Quantum mutual information $I(A:B)$ is monotonic in the noise parameter for various input states.
• Mutual $L^1$-norm magic, while closely related, can exhibit non-monotonic behavior due to sensitivity to Pauli operator structure.
• Mutual stabilizer 2-Rényi entropy and mutual mana both sharply detect "magic correlations," remaining zero until a state-specific threshold is crossed.
• Only mutual mana is strictly linked to Wigner function negativity and is additive, Clifford-invariant, and has tight operational ties to magic as a computational resource.

6. Universality and Scaling in Critical Models

Empirical studies in conformal field-theoretic spin chains indicate that mutual mana's scaling is universal for a given CFT, characterized by a model-specific exponent $b$ and functional dependence analogous to entanglement mutual information, but not trivially related to central charge. This universality holds across both integrable and non-integrable models and remains robust under moderate basis transformations. These results support mutual mana as a universal diagnostic for criticality, detecting long-range non-stabilizerness and the entropic cost in Clifford + magic circuit decompositions (Tarabunga, 2023).

7. Extensions Beyond Quantum Information: Mutual Mana in Petri Nets

A structurally analogous concept of mutual mana appears in the categorical treatment of Petri nets modeling chemical reaction networks (Genovese et al., 2021). There, "mana" constrains the number of allowed firings of transitions (e.g., enzyme activity in catalysis), and regeneration rules enable transitions to catalytically restore the mana of others, modeling mutualistic or autocatalytic processes. In this context, "mutual mana" regeneration governs whether two transitions can sustain each other's activity, a key abstraction in systems biochemistry.

Summary

Mutual mana synthesizes resource-theoretic, information-theoretic, and categorical approaches to quantifying and distributing non-stabilizerness in both quantum and classical models. It serves as a fundamental measure for magic correlations—essential for universal quantum computation, topological phases, and dynamical criticality—distinguished by precise monotonicity, additivity, and computationally-tractable analytic forms in discrete-variable settings. Recent work demonstrates its operational relevance, universality, and cross-disciplinary importation into categorical frameworks beyond quantum physics.