Two-Sided Long-Range Magic

• Two-sided long-range magic is a quantum resource that quantifies nonstabilizerness stored in bipartite correlations beyond local contributions.
• It is measured by subtracting individual subsystem magic from the total magic, highlighting genuine nonlocality in both circuit and topological formulations.
• This concept underpins applications in quantum circuits, TQFTs, and many-body diagnostics, demonstrating resilience against finite-depth local unitary optimizations.

Two-sided long-range magic is a context-dependent notion in the modern theory of quantum nonstabilizerness. In its bipartite form, it denotes the portion of magic that is genuinely stored in correlations between two subsystems rather than already present on either side separately; in its circuit-theoretic form, it denotes magic that survives optimization over finite-depth local unitaries; and in its strongest current formulation, it denotes states that cannot be prepared by a Clifford circuit and a finite-depth local unitary in either order (Fliss, 2020, Tarabunga et al., 2023, Li, 26 Mar 2026). Across these formulations, the common theme is that nonstabilizerness can be intrinsically nonlocal, topological, and correlation-based, rather than a merely local defect superposed on otherwise stabilizer structure (López et al., 2024, Wei et al., 6 Mar 2025, Zhang et al., 14 May 2026).

1. Terminology and conceptual scope

In resource-theoretic language, magic means nonstabilizerness. For odd-prime local dimension, a stabilizer state is an eigenstate of Heisenberg–Weyl operators, the convex hull of stabilizer states is denoted $\mathrm{STAB}$, and a state outside $\mathrm{STAB}$ is a magic state (Fliss, 2020). This definition already separates magic from ordinary entanglement: entangled states may still be stabilizer states, and thus may have zero magic.

The phrase “two-sided long-range magic” is not a single universally standardized definition. In link-state and many-body work, the closest formal objects are subtraction-based bipartite quantities such as long-range mana or long-range stabilizer Rényi entropy, both designed to isolate the nonstabilizer resource contained in correlations across a bipartition (Fliss, 2020, Tarabunga et al., 2023). In circuit-complexity work, long-range magic is instead defined by nonremovability under finite-depth local unitaries, and the explicitly “two-sided” version strengthens this by forbidding preparation with a Clifford circuit and a finite-depth local unitary in either order (Zhang et al., 14 May 2026, Li, 26 Mar 2026).

A recurrent misconception is that long-range magic is equivalent to long-range entanglement. The literature rejects that identification. In $U(1)_k$ Chern–Simons theory, link states can possess long-range entanglement while remaining stabilizer states with vanishing mana, whereas in non-Abelian theories the same topological construction generically yields nonzero magic (Fliss, 2020). A plausible implication is that two-sided long-range magic should be understood as a refinement of nonseparability, not a synonym for it.

2. Bipartite correlation measures

The most direct formalizations are subtraction-based quantities modeled on mutual information: one computes a magic monotone on the joint state and subtracts the corresponding monotones on the marginals. What remains is interpreted as correlation-only magic.

Setting	Quantity	Interpretation
Link states	$L_k(\rho_L)=M_k(\rho_L)-M_k(\rho_A)-M_k(\rho_B)$	Long-range mana
Many-body SRE	$$L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B)$$	Connected nonlocal magic
Measurement-only circuits	$$I_{\mathcal M}(A)=\mathcal M(|\psi\rangle)-\mathcal M(\rho_A)-\mathcal M(\rho_{A^c})$$	Mutual magic across a cut
Dual-unitary XXZ	Same subtraction with $\tilde M_2$	Exact long-range SRE in a solvable circuit

For link states, the mana $M(\rho)$ is defined from the discrete Wigner function, is additive on tensor products, and satisfies subsystem concavity. The bipartite quantity $L_k(\rho_L)$ therefore vanishes on product states and measures the part of the total mana not already visible on either boundary separately. Because framing changes act by the modular $T$-matrix, the paper also defines a topological long-range mana $\mathrm{STAB}$0 by minimizing over the framing orbit; if $\mathrm{STAB}$1, the magic is “entirely long-range” in the sense that some minimizing framing has zero subsystem mana on both sides (Fliss, 2020).

For many-body stabilizer Rényi entropies, the analogous quantity is

$\mathrm{STAB}$2

with $\mathrm{STAB}$3 and $\mathrm{STAB}$4 chosen as separated subsystems. This quantity is explicitly described as the connected component of magic and can be rewritten as

$\mathrm{STAB}$5

where $\mathrm{STAB}$6 is the Rényi-2 mutual information and $\mathrm{STAB}$7 is a Pauli-moment correction (Tarabunga et al., 2023). Unlike mutual information, however, $\mathrm{STAB}$8 is not guaranteed to be positive, because stabilizer Rényi entropies do not satisfy subadditivity in general. This sign issue is one of the sharpest formal differences between entanglement-based and magic-based bipartite diagnostics.

In measurement-only circuits, mutual magic is defined by the same subtraction principle but admits an especially concrete combinatorial interpretation: in the rotated-Bell-cluster description, $\mathrm{STAB}$9 is exactly the sum of the magic of clusters that have support in both $U(1)_k$0 and $U(1)_k$1 (Tarabunga et al., 2024). This makes the “two-sided” content literal: only cut-crossing nonstabilizer clusters contribute.

3. Link states and topological TQFT realizations

The most explicit topological realization of two-sided long-range magic appears in "Knots, links, and long-range magic" (Fliss, 2020). Given an $U(1)_k$2-component link

$U(1)_k$3

one removes a tubular neighborhood $U(1)_k$4, forming the link complement $U(1)_k$5. By the axioms of TQFT, Euclidean path integration on $U(1)_k$6 prepares a state

$U(1)_k$7

For $U(1)_k$8, $U(1)_k$9, and the wavefunction amplitudes are colored Jones invariants: $L_k(\rho_L)=M_k(\rho_L)-M_k(\rho_A)-M_k(\rho_B)$0

In the Abelian theory $L_k(\rho_L)=M_k(\rho_L)-M_k(\rho_A)-M_k(\rho_B)$1, all link states are stabilizer states, specifically weighted graph states, so their mana vanishes in all framings and in both computational bases. The nontrivial behavior begins in non-Abelian $L_k(\rho_L)=M_k(\rho_L)-M_k(\rho_A)-M_k(\rho_B)$2, where knot and link states are generically magical. For a two-component link, the state is pure on $L_k(\rho_L)=M_k(\rho_L)-M_k(\rho_A)-M_k(\rho_B)$3, and the long-range mana

$L_k(\rho_L)=M_k(\rho_L)-M_k(\rho_A)-M_k(\rho_B)$4

isolates the nonstabilizerness stored across the two boundaries.

The canonical example is the Hopf link L2a1. Its global state is magical, yet each reduced density matrix is maximally mixed,

$L_k(\rho_L)=M_k(\rho_L)-M_k(\rho_A)-M_k(\rho_B)$5

so the reduced states have zero mana in every framing and basis. Consequently,

$L_k(\rho_L)=M_k(\rho_L)-M_k(\rho_A)-M_k(\rho_B)$6

At $L_k(\rho_L)=M_k(\rho_L)-M_k(\rho_A)-M_k(\rho_B)$7, the topological mana is approximately $L_k(\rho_L)=M_k(\rho_L)-M_k(\rho_A)-M_k(\rho_B)$8 in the representation basis and $L_k(\rho_L)=M_k(\rho_L)-M_k(\rho_A)-M_k(\rho_B)$9 in the Verlinde basis. This is the cleanest known example of a two-sided state with zero local magic but nonzero global magic.

Torus links furnish an analytic family with the same qualitative structure. In the Verlinde basis their states acquire a GHZ-like form, and every reduced state on a proper subset of components is diagonal in that basis and therefore a classical mixture of computational-basis states. In that framing and basis, the reduced states have zero mana and

$$L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B)$$0

Numerically, at $$L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B)$$1, this equality persists after topological minimization in both the representation and Verlinde bases. For the tabulated two-link sample at $$L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B)$$2, only six of $$L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B)$$3 links have vanishing $$L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B)$$4, and the paper concludes that for a majority of link states the magic is entirely long-range.

4. Dynamical, critical, and measurement-induced manifestations

In extended many-body systems, long-range magic is used as an infrared-sensitive diagnostic precisely because full-state magic contains large local and UV-sensitive contributions. "Many-body magic via Pauli-Markov chains" introduces a Monte Carlo method that samples Pauli strings directly on a disconnected union $$L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B)$$5, making it possible to estimate the long-range quantity without separately resolving three extensive subsystem entropies (Tarabunga et al., 2023). In one-dimensional systems, the long-range magic $$L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B)$$6 peaks at the critical point of the transverse-field Ising chain $$L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B)$$7, and at criticality it scales as

$$L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B)$$8

In the spin-1 XXZ chain, it exhibits clear extrema at both the Ising and Gaussian transitions, even though the full-state magic density does not sharply identify them. The same paper emphasizes that this subtraction-based quantity is not guaranteed to be positive.

The dual-unitary XXZ model gives an exact solvable instance of long-range SRE (López et al., 2024). For the partition

$$L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B)$$9

the reduced states on $$I_{\mathcal M}(A)=\mathcal M(|\psi\rangle)-\mathcal M(\rho_A)-\mathcal M(\rho_{A^c})$$0 and $$I_{\mathcal M}(A)=\mathcal M(|\psi\rangle)-\mathcal M(\rho_A)-\mathcal M(\rho_{A^c})$$1 are maximally mixed,

$$I_{\mathcal M}(A)=\mathcal M(|\psi\rangle)-\mathcal M(\rho_A)-\mathcal M(\rho_{A^c})$$2

so the long-range SRE equals the joint reduced-state magic: $$I_{\mathcal M}(A)=\mathcal M(|\psi\rangle)-\mathcal M(\rho_A)-\mathcal M(\rho_{A^c})$$3 In this exactly solved family, long-range magic vanishes when the light cones do not overlap, and for $$I_{\mathcal M}(A)=\mathcal M(|\psi\rangle)-\mathcal M(\rho_A)-\mathcal M(\rho_{A^c})$$4 it approaches the asymptotic value $$I_{\mathcal M}(A)=\mathcal M(|\psi\rangle)-\mathcal M(\rho_A)-\mathcal M(\rho_{A^c})$$5. This provides a controlled example in which all nonlocal magic is generated precisely at the light-cone scale.

Measurement-only circuits reveal a complementary phenomenon (Tarabunga et al., 2024). There, fixed-density non-Clifford measurements can produce extensive total magic in both phases of the transition, so the total amount of nonstabilizerness is comparatively featureless. By contrast, the mutual magic

$$I_{\mathcal M}(A)=\mathcal M(|\psi\rangle)-\mathcal M(\rho_A)-\mathcal M(\rho_{A^c})$$6

shows critical behavior analogous to entanglement. In $$I_{\mathcal M}(A)=\mathcal M(|\psi\rangle)-\mathcal M(\rho_A)-\mathcal M(\rho_{A^c})$$7D, at $$I_{\mathcal M}(A)=\mathcal M(|\psi\rangle)-\mathcal M(\rho_A)-\mathcal M(\rho_{A^c})$$8,

$$I_{\mathcal M}(A)=\mathcal M(|\psi\rangle)-\mathcal M(\rho_A)-\mathcal M(\rho_{A^c})$$9

and the time-dependent critical growth satisfies

$\tilde M_2$0

In $\tilde M_2$1D, at $\tilde M_2$2, the critical mutual magic obeys area-law scaling. The same work also defines a topological magic

$\tilde M_2$3

which is nonzero for $\tilde M_2$4 and vanishes for $\tilde M_2$5 in the studied $\tilde M_2$6D geometry.

5. Circuit-hierarchy, code, and topological-order formulations

A more basis-independent notion of long-range magic minimizes over shallow local basis changes. "Extensive long-range magic in non-Abelian topological orders" defines, for depth $\tilde M_2$7,

$\tilde M_2$8

where $\tilde M_2$9 may be the log-stabilizer fidelity, relative entropy of magic, max-relative entropy of magic, generalized log-robustness, or log-robustness (Zhang et al., 14 May 2026). In this language, short-range magic consists of states obtainable from stabilizer states by constant-depth local unitaries, whereas long-range magic is the part that survives such optimization. The paper proves extensive lower bounds for non-Abelian string-net ground states and low-energy states, shows that stabilizer states even up to constant-depth local unitaries cannot realize non-Abelian string-net ground states, and derives an Abelian compatibility condition: stabilizer-realizable Abelian orders must have mutual braiding phases that are $M(\rho)$0-th roots of unity for on-site qudit dimension $M(\rho)$1.

"Long-range nonstabilizerness from quantum codes, orders, and correlations" recasts the same theme in asymptotic family language (Wei et al., 6 Mar 2025). A family has long-range magic if any constant-depth local circuit leaves it magical for sufficiently large system size, and it has short-range magic if some such circuit maps it to stabilizer states. The robust variant, strong LRM, requires

$M(\rho)$2

for all constant-depth local $M(\rho)$3 and all stabilizer families $M(\rho)$4. The paper proves that encoded logical states in a $M(\rho)$5-dimensional topological stabilizer code family exhibit LRM whenever the logical state lies outside the $M(\rho)$6-orbit of $M(\rho)$7; in $M(\rho)$8D, this implies that $M(\rho)$9 in the toric code has LRM. It also proves that a topological order cannot be realized by any topological stabilizer code if and only if, for any local Hamiltonian realization of that order, all ground states exhibit LRM. If the ground-space dimension is not a power of $L_k(\rho_L)$0, all ground states exhibit strong LRM.

The strongest current version of the term appears in "Explicit States with Two-sided Long-Range Magic" (Li, 26 Mar 2026). There, a state has two-sided long-range magic if it is not preparable by

$L_k(\rho_L)$1

thus placing it outside the first level of the magic hierarchy. The primary explicit example is the ZX-cat or “magical cat” state

$L_k(\rho_L)$2

The paper proves that this state cannot be prepared by either ordering within a constant approximation error. It also proves the same type of obstruction for ground states of certain quantum double and string-net models that admit no topologically transversal gates. By contrast, the related state

$L_k(\rho_L)$3

can be prepared by $L_k(\rho_L)$4, showing that exclusion from level $L_k(\rho_L)$5 is a precise hierarchy statement rather than a blanket impossibility.

6. Spatial diagnostics, finite bipartite analogs, and distinctions

The spatial organization of magic under dynamics can itself be two-sided. "Magic spreading under unitary Clifford dynamics" studies a state with a single injected unit of magic, $L_k(\rho_L)$6, and introduces the bipartite magic gauge, a canonical $L_k(\rho_L)$7 stabilizer-code representation adapted to a cut $L_k(\rho_L)$8 (Bejan et al., 26 Nov 2025). The gauge decomposes the state into stabilizers reducible to $L_k(\rho_L)$9, reducible to $T$0, or irreducibly supported on both sides. A central lemma shows that, for one logical qubit, either all three logicals are reducible to $T$1, or all to $T$2, or exactly one logical can be one-sided while the other two are supported on both $T$3 and $T$4. This yields a trichotomy for subsystem magic: $T$5 The intermediate value $T$6 is the paper’s explicit signature of partial, genuinely bipartite magic. The same work defines the linear magic length $T$7 and the full linear extent of magic $T$8; at early times they grow ballistically,

$T$9

and at late times the magic delocalizes so that $\mathrm{STAB}$00 and $\mathrm{STAB}$01.

At the two-qubit level, the closest analogous notion is non-local magic minimized over local unitaries on the two parties (Robin et al., 27 Oct 2025). In the scattering setting, this non-local magic is basis-independent, is supported by entanglement, and for the studied two-qubit pure states is empirically related to anti-flatness by

$\mathrm{STAB}$02

The paper finds that in low-energy nucleon–nucleon scattering below $\mathrm{STAB}$03 MeV only about one third of the generated magic is non-local, whereas in several Møller-scattering sectors the outgoing magic is entirely non-local. This is not long-range in the many-body sense, but it supplies a finite-system analogue of irreducibly two-sided magic.

"Maximal Magic for Two-qubit States" adds a complementary point (Liu et al., 24 Feb 2025). For two-qubit pure states, strong numerical evidence indicates

$\mathrm{STAB}$04

achieved by $\mathrm{STAB}$05 states that are exactly Weyl–Heisenberg MUB fiducials. Their concurrence takes only the values $\mathrm{STAB}$06 and $\mathrm{STAB}$07, and none is maximally entangled. A plausible implication is that even in the smallest bipartite setting, maximal nonstabilizerness is neither purely local nor reducible to maximal entanglement.

Taken together, these developments establish that “two-sided long-range magic” is not a single invariant but a family of closely related ideas. Depending on context, it may mean correlation-only magic across a bipartition, magic that survives all constant-depth local basis changes, or the stronger hierarchy obstruction forbidding Clifford and finite-depth local layers in either order. What remains constant across the literature is the central claim: nonstabilizerness can be stored nonlocally, protected topologically, transported ballistically, and detected by structures that have no purely entanglement-theoretic equivalent.