Bipartite Magic Gauge Framework

• Bipartite magic gauge is a framework that quantifies non-stabilizerness in bipartite quantum systems through explicit stabilizer decompositions and Clifford circuits.
• It enables efficient diagnosis of both local and nonlocal quantum magic using measures like reduced magic outcomes, BSMI, and Pauli–Markov chain estimators.
• Its cross-disciplinary applications range from optimizing quantum error correction and probing phase transitions to explaining emergent gauge fields in twisted bilayer graphene and graph labelings.

A bipartite magic gauge is a framework or construction that enables the explicit quantification, localization, or diagnosis of "magic"—non-stabilizerness or resource nonclassicality—in quantum systems with a bipartite structure. Across separate research areas, it provides both a quantitative measure and a mechanism for tracing the origin and flow of magic, especially in the context of either quantum error correction, quantum phase transitions, or emergent gauge structures in condensed matter systems.

1. Quantum Information: Bipartite Magic Gauge in Clifford Dynamics

The bipartite magic gauge (BMG) in stabilizer formalism is a canonical representation of pure quantum states encoding one logical qubit under bipartitions of a system. It reveals which logical operators—and thus which "magic"—are supported strictly in a subsystem $A$, its complement $B$, or jointly. The formal structure, as proven in Theorem 1, is that for any bipartition $A\cup B$ of the $L$-qubit system, one can choose stabilizer generators $\{a_i\}$ supported only on $A$, $\{b_j\}$ only on $B$, and $\{h_k\}$ that are nonlocal. This decomposition is achieved constructively via Clifford Normal Forms and Gaussian elimination over $GF(2)$, resulting in a polynomial-time ($O(L^3)$) algorithm for identifying the magic content in any contiguous region.

With respect to local magic quantification, BMG classifies subsystem $A$ into five "reducibility" patterns, corresponding to full, partial, or absent local magic—these are determined by the supports of the logical Pauli operators $X, Y, Z$. Explicitly, the reduced magic outcome (e.g., $\mathcal{M}_2(\rho_A)$) is assigned based on these patterns:

• Full magic: all logicals localized in $A$: $\mathcal{M}_2(\rho_A)=\log_2(4/3)$.
• Half magic: one logical in $A$, another in $B$, one nonlocal: $\mathcal{M}_2(\rho_A)=\log_2(6/5)$.
• No magic: logicals local to $B$ or nonlocal to $A$ alone: $\mathcal{M}_2(\rho_A)=0$.

This formalism yields precise operational length scales: the Linear Magic Length (LML), the minimal size of a symmetric interval supporting a full unit of magic, and the Full Linear Extent of Magic (FLEOM), the maximal span across which all magic is delocalized. Under Clifford circuit dynamics, both scales exhibit early ballistic growth governed by entanglement velocity. Thus, the BMG is essential for mapping the spatiotemporal structure of magic and enables efficient identification of quantum resource localization and transport in many-body dynamics (Bejan et al., 26 Nov 2025).

2. Measurement of Bipartite Magic in Many-Body Systems

The concept of a bipartite magic gauge extends to explicit diagnostic tools for probing non-stabilizer correlations in quantum many-body systems. Multiple, mathematically precise gauge constructions exist:

• Bipartite stabilizer mutual information (BSMI), defined as $$I_S(A:B) = M_1(\rho_A) + M_1(\rho_B) - M_1(\rho_{A \cup B})$$, where $M_1$ is the stabilizer Rényi entropy of order one, captures genuinely nonlocal magic. $I_S(A:B)>0$ precisely when magic cannot be generated by Clifford circuits restricted to $A$ or $B$ alone. In monitored fermion dynamics, this measure tracks a phase transition: $I_S \sim \ln L$ in the critical phase and saturates in the area-law phase (Wang et al., 14 Jul 2025).
• Pauli–Markov chain estimators of long-range magic, such as $$L(\rho_{AB}) = \tilde{M}_2(\rho_{AB}) - \tilde{M}_2(\rho_A) - \tilde{M}_2(\rho_B)$$ (where $\tilde{M}_2$ is the mixed-state 2-Rényi SRE), enable efficient extraction of nonlocal contributions even in large lattice geometries, using Metropolis sampling and Tree Tensor Networks. This method is sensitive to both conformal quantum critical scaling and gauge-theory confinement transitions (Tarabunga et al., 2023).

These constructions further generalize robustness-based quantifiers, such as mutual robustness of magic $\mathcal{R}_{\mathrm{mut}}(A,B)$, which measures the resource-theoretic quantum correlation beyond entanglement, decaying algebraically at criticality and remaining robust up to a nonzero effective temperature (Timsina et al., 17 Jul 2025).

3. Magic Gauge and Emergent Non-Abelian Structure in Twisted Bilayer Graphene

In condensed matter systems—specifically twisted bilayer graphene (TBG)—the term "bipartite magic gauge" denotes a structural reduction that exposes a non-Abelian gauge field driving magic-angle flat bands. The continuum chiral Hamiltonian of TBG, when squared, decouples the two bipartite sublattices (A,B), yielding an effective $2\times2$ operator that acts on a triangular lattice. The emergent off-diagonal operator $A(\mathbf r)$ splits into a non-Abelian gradient piece, encoding interlayer currents, and an Abelian-like term. The "magic" in this context refers to the exact cancellation between positive kinetic and confinement energies and the negative (gauge-field) energy at specific twist angles—the magic angles. This cancellation creates flat electronic bands, corresponding to zero-frequency floppy-mode analogs in rigidity theory (Navarro-Labastida et al., 2021, Naumis et al., 2021).

The gauge-coupled triangular lattice, derived from the original bipartite honeycomb, thus acts as a "bipartite magic gauge": its non-Abelian structure and quantization condition (via Wilson-loop phases) determine the existence and stability of flat bands, which are essential for the correlated electron phenomena in TBG.

4. Bipartite Magic Gauge in Graph Theory: Labelings

Within combinatorial graph theory, bipartite magic gauges manifest as explicit constraints on labeling. For example, in the classification of edge-magic labelings of bipartite graphs, there are only four possible values of $b$ for which a connected bipartite graph admits a $b$-edge consecutive magic total labeling: $b \in \{0, |X|, |Y|, |X|+|Y|\}$, where $X,Y$ are the two color classes (Kang et al., 2014). For complete bipartite graphs, a $\Gamma$-distance magic labeling exists if and only if $m+n\not\equiv 2 \pmod 4$, with explicit constructions possible through partitioning of the Abelian group $\Gamma$ (Cichacz, 2013).

These results, though unrelated to quantum resource theory, showcase gauge-like constraints on combinatorial assignments in bipartite structures, providing full classification and explicit construction algorithms.

5. Physical and Operational Implications

The bipartite magic gauge, across all contexts, provides not only a measure of quantum resource but also operational protocols:

• In quantum error correction, BMG reveals extractable logical magic from a subsystem via Clifford circuits.
• In quantum many-body systems, BSMI and Markov-chain-based bipartite gauges distinguish genuinely non-Stabilizer correlations and detect universal phase transitions.
• In TBG, the gauge structure encodes the physical mechanism driving flat bands and the associated correlated states.

Furthermore, in field theory, an interplay between bipartite entanglement and magic underlies the emergence of gauge invariance: imposing maximal entanglement and minimal non-Cliffordness uniquely selects the physical gauge-invariant interaction from among possible symmetry-breaking quartic couplings (Núñez et al., 6 Nov 2025). This suggests a variational principle where fundamental interactions are extremals of entanglement–magic resource trade-offs.

6. Computational and Experimental Approaches

Algorithmic efficiency is a hallmark:

• Clifford normal forms and stabilizer tableaux enable $O(L^3)$ diagnosis of magic support in logical codes (Bejan et al., 26 Nov 2025).
• Tree tensor networks and Markov chain Monte Carlo make bipartite magic estimators scalable to hundreds of sites (Tarabunga et al., 2023).
• In quantum simulation and tomography, mutual robustness and related bipartite gauges can be reconstructed via QMC and column-generation methods, with direct physical interpretation (Timsina et al., 17 Jul 2025).

Experimental implementation, especially in lattice gauge theories and hybrid Rabi models, only requires Pauli observable measurements and enables mapping of magic phase diagrams or state preparation in optimal magic regimes (Campos-Uscanga et al., 1 Aug 2025).

7. Significance and Unification

The bipartite magic gauge formalism unifies otherwise distinct frameworks:

• In quantum information, it connects resource theory of non-Cliffordness, error correction, and logical encoding.
• In condensed matter, it links gauge-theoretic perspectives, topological frustration, and magic-induced flat bands.
• In graph theory, it organizes labeling assignments under combinatorial gauge-like symmetries.

This cross-disciplinary adoption underscores the centrality of bipartite magic gauges as canonical tools for quantifying, controlling, and exploiting nontrivial quantum structure in complex systems.