Nonlocal Magic in Quantum Systems

• Nonlocal magic is defined as the irreducible nonstabilizerness remaining after optimal local unitary rotations, capturing genuine correlations across bipartitions.
• It employs magic monotones, such as the stabilizer Rényi entropy, to separate local magic from nonlocal contributions in quantum states, including Gaussian and many-body systems.
• The measure distinguishes nonlocal magic from entanglement and Bell nonlocality, offering insights into operator entanglement, simulation complexity, and experimental verification.

Nonlocal magic is a quantum-information notion of irreducible nonstabilizerness that remains after all removable local-basis structure has been optimized away. In its standard bipartite form, it is defined by minimizing a magic monotone—most commonly a stabilizer Rényi entropy—over local unitaries $U_A\otimes U_B$, thereby isolating the part of nonstabilizerness that is genuinely tied to correlations across a bipartition rather than stored locally in either subsystem (Qian et al., 10 Feb 2025, Ahmad et al., 19 Nov 2025). Subsequent work has extended this idea to operator dynamics, fermionic Gaussian states, many-body critical systems, scattering processes, holography, and experiment, while also clarifying that nonlocal magic is distinct from Bell nonlocality and from the older “magic game” terminology of pseudo-telepathy (Andreadakis et al., 12 Apr 2025, Collura et al., 29 Apr 2026, Arkhipov, 2012).

1. Definition and resource-theoretic meaning

For a bipartite pure state $|\Psi_{AB}\rangle$, a standard definition is

$$\mathcal{M}_{AB}(|\Psi_{AB}\rangle)=\min_{U_A\otimes U_B}M\!\left(U_A\otimes U_B|\Psi_{AB}\rangle\right),$$

where $M$ is a magic measure satisfying faithfulness and additivity; the literature then commonly specializes to the second stabilizer Rényi entropy (SRE) (Qian et al., 10 Feb 2025). In the two-qubit experimental literature, the same structure is written as

$$M_2^{\rm NL}(\ket\psi)=\min_{U_A,U_B} M_2\big((U_A\otimes U_B)\ket\psi\big),$$

with the associated decomposition

$$M_2^{\rm L}(\ket\psi)=M_2(\ket\psi)-M_2^{\rm NL}(\ket\psi),$$

so that total magic is split into local and non-local contributions (Ahmad et al., 19 Nov 2025).

This optimization removes all magic that can be erased by local basis changes. What remains is the irreducible nonstabilizerness genuinely tied to entanglement across the cut. The quantity is symmetric under $A\leftrightarrow B$, stable under local basis rotations, nonnegative when $M$ is faithful, and subadditive when $M$ is additive (Qian et al., 10 Feb 2025). It vanishes for product states and stabilizer states, but need not vanish for mixed separable states, because some mixed states cannot be rotated by local unitaries into mixed stabilizer form (Qian et al., 10 Feb 2025).

A central conceptual point is that nonlocal magic is neither identical to total magic nor identical to entanglement. The experimental literature states this explicitly: not every entangled state has non-local magic, and not every bit of magic in an entangled state is non-local (Ahmad et al., 19 Nov 2025). This distinction recurs in essentially every later development.

2. Exact bipartite results and spectral structure

For two-qubit pure states, nonlocal magic is analytically tractable. Any state can be brought to Schmidt form,

$$|\psi\rangle=\cos(\theta)|00\rangle+\sin(\theta)|11\rangle,$$

and the exact result is

$|\Psi_{AB}\rangle$0

so for two qubits the basis-independent non-local magic is completely determined by the entanglement spectrum (Qian et al., 10 Feb 2025). Equivalent expressions appear in terms of Schmidt eigenvalues or concurrence in scattering work, again showing that for pure two-qubit states the nonlocal resource is fully controlled by entanglement-spectrum data (Gargalionis et al., 4 Mar 2026).

A more general structural theorem was given for bipartite nonlocal magic resource (BNMR). For a pure state

$|\Psi_{AB}\rangle$1

canonical encoding compresses the problem into a minimal Schmidt-support core of size $|\Psi_{AB}\rangle$2 qubits per side, and the resulting BNMR depends only on the nonzero Schmidt spectrum,

$|\Psi_{AB}\rangle$3

This extends invariance from local unitaries to local isometries (Huang et al., 23 Jun 2026). The same work identifies the zero set exactly: $|\Psi_{AB}\rangle$4 meaning $|\Psi_{AB}\rangle$5 and $|\Psi_{AB}\rangle$6 for some $|\Psi_{AB}\rangle$7 (Huang et al., 23 Jun 2026). Near such zeros,

$|\Psi_{AB}\rangle$8

for $|\Psi_{AB}\rangle$9, so the resource turns on quadratically under spectral perturbations (Huang et al., 23 Jun 2026).

This spectrum-only viewpoint also yields closed forms beyond qubits. For Schmidt rank $$\mathcal{M}_{AB}(|\Psi_{AB}\rangle)=\min_{U_A\otimes U_B}M\!\left(U_A\otimes U_B|\Psi_{AB}\rangle\right),$$0, the canonical core reduces to a single entangled qubit pair and

$$\mathcal{M}_{AB}(|\Psi_{AB}\rangle)=\min_{U_A\otimes U_B}M\!\left(U_A\otimes U_B|\Psi_{AB}\rangle\right),$$1

while for generalized GHZ states the hierarchy

$$\mathcal{M}_{AB}(|\Psi_{AB}\rangle)=\min_{U_A\otimes U_B}M\!\left(U_A\otimes U_B|\Psi_{AB}\rangle\right),$$2

collapses to equality (Huang et al., 23 Jun 2026). For higher local dimensions, analytic formulae were conjectured for prime $$\mathcal{M}_{AB}(|\Psi_{AB}\rangle)=\min_{U_A\otimes U_B}M\!\left(U_A\otimes U_B|\Psi_{AB}\rangle\right),$$3, based on “Schmidt attainment,” the hypothesis that the minimum over $$\mathcal{M}_{AB}(|\Psi_{AB}\rangle)=\min_{U_A\otimes U_B}M\!\left(U_A\otimes U_B|\Psi_{AB}\rangle\right),$$4 is achieved by a Schmidt-aligned state. The qutrit case yields a maximum non-local magic $$\mathcal{M}_{AB}(|\Psi_{AB}\rangle)=\min_{U_A\otimes U_B}M\!\left(U_A\otimes U_B|\Psi_{AB}\rangle\right),$$5, the ququint case numerically supports $$\mathcal{M}_{AB}(|\Psi_{AB}\rangle)=\min_{U_A\otimes U_B}M\!\left(U_A\otimes U_B|\Psi_{AB}\rangle\right),$$6, and the same constructions fail to be globally exact in composite dimension $$\mathcal{M}_{AB}(|\Psi_{AB}\rangle)=\min_{U_A\otimes U_B}M\!\left(U_A\otimes U_B|\Psi_{AB}\rangle\right),$$7 (Busoni et al., 10 Mar 2026). That work also shows that qubit-specific relations between non-local magic and simple entanglement diagnostics do not extend cleanly to qutrits and higher dimensions (Busoni et al., 10 Mar 2026).

3. Gaussian and free-fermion formulations

A major simplification occurs for pure fermionic Gaussian states. In the Majorana formalism, the state is characterized by a covariance matrix $$\mathcal{M}_{AB}(|\Psi_{AB}\rangle)=\min_{U_A\otimes U_B}M\!\left(U_A\otimes U_B|\Psi_{AB}\rangle\right),$$8, and local Gaussian unitaries bring any bipartite pure Gaussian state into a canonical product of independent entangled mode pairs (Iannotti et al., 29 Apr 2026, Collura et al., 29 Apr 2026). In this setting nonlocal magic over local Gaussian unitaries becomes a closed-form spectral functional of the reduced covariance matrix.

One formulation gives, for a subsystem $$\mathcal{M}_{AB}(|\Psi_{AB}\rangle)=\min_{U_A\otimes U_B}M\!\left(U_A\otimes U_B|\Psi_{AB}\rangle\right),$$9 of size $M$0,

$M$1

where $M$2 are the positive eigenvalues of the reduced Majorana covariance matrix $M$3 (Iannotti et al., 29 Apr 2026). An equivalent Gaussian-orbit expression uses the singular values $M$4 of the subsystem covariance block: $M$5 This identifies intermediate entanglement-spectrum modes $M$6 as the carriers of nonlocal nonstabilizerness, while both $M$7 and $M$8 contribute zero (Collura et al., 29 Apr 2026).

These formulae make the quantity polynomial-time computable and directly accessible from two-point correlators. For Haar-random Gaussian states, the average nonlocal magic is extensive, with a thermodynamic-limit density $M$9 that is maximal at the symmetric cut and satisfies

$$M_2^{\rm NL}(\ket\psi)=\min_{U_A,U_B} M_2\big((U_A\otimes U_B)\ket\psi\big),$$0

for the $$M_2^{\rm NL}(\ket\psi)=\min_{U_A,U_B} M_2\big((U_A\otimes U_B)\ket\psi\big),$$1 case (Collura et al., 29 Apr 2026). In ground states, nonlocal magic is suppressed deep in both trivial and topological phases and peaks near critical points in the Kitaev chain, with logarithmic scaling at criticality (Collura et al., 29 Apr 2026). In dynamics, random Gaussian circuits exhibit diffusive growth,

$$M_2^{\rm NL}(\ket\psi)=\min_{U_A,U_B} M_2\big((U_A\otimes U_B)\ket\psi\big),$$2

while XY-chain quenches show a sharp separation: at the $$M_2^{\rm NL}(\ket\psi)=\min_{U_A,U_B} M_2\big((U_A\otimes U_B)\ket\psi\big),$$3-symmetric XX point,

$$M_2^{\rm NL}(\ket\psi)=\min_{U_A,U_B} M_2\big((U_A\otimes U_B)\ket\psi\big),$$4

whereas for $$M_2^{\rm NL}(\ket\psi)=\min_{U_A,U_B} M_2\big((U_A\otimes U_B)\ket\psi\big),$$5 away from the special $$M_2^{\rm NL}(\ket\psi)=\min_{U_A,U_B} M_2\big((U_A\otimes U_B)\ket\psi\big),$$6 limit,

$$M_2^{\rm NL}(\ket\psi)=\min_{U_A,U_B} M_2\big((U_A\otimes U_B)\ket\psi\big),$$7

(Collura et al., 29 Apr 2026). This is one of the clearest demonstrations that nonlocal magic and entanglement can have parametrically different dynamics.

4. Unitary generation, operator entanglement, and simulation theory

For unitary dynamics, an exact operator-space formulation was established in “An Exact Link between Nonlocal Magic and Operator Entanglement” (Andreadakis et al., 12 Apr 2025). The central theorem states that a unitary map generates nonlocal magic if and only if it generates operator entanglement on Pauli strings. On that basis the paper introduces an average measure of a unitary’s Pauli-entangling power as a proxy for nonlocal magic generation, derives analytical formulae, and studies its typical value and upper bounds in terms of the nonstabilizerness properties of the evolution (Andreadakis et al., 12 Apr 2025).

This operator viewpoint was developed further in noisy Clifford encoding–decoding circuits. There the relevant diagnostic is the average Pauli-entangling power (APEP),

$$M_2^{\rm NL}(\ket\psi)=\min_{U_A,U_B} M_2\big((U_A\otimes U_B)\ket\psi\big),$$8

which vanishes for Clifford unitaries and is positive only when some evolved Pauli strings become non-Pauli and operator-entangled across a bipartition (Dallas et al., 24 Sep 2025). In the thermodynamic limit, local coherent noise injected on finitely many qubits produces finite, system-size-independent scrambling and nonlocal magic generation—described as a butterfly effect—whereas depolarizing noise gives

$$M_2^{\rm NL}(\ket\psi)=\min_{U_A,U_B} M_2\big((U_A\otimes U_B)\ket\psi\big),$$9

in the Clifford-averaged setting (Dallas et al., 24 Sep 2025).

In Haar-random circuit simulations, the same resource appears with an unexpectedly mild tensor-network cost. Using the SRE as the magic measure, one finds

$$M_2^{\rm L}(\ket\psi)=M_2(\ket\psi)-M_2^{\rm NL}(\ket\psi),$$0

so for fixed error

$$M_2^{\rm L}(\ket\psi)=M_2(\ket\psi)-M_2^{\rm NL}(\ket\psi),$$1

By contrast, faithful entanglement simulation near the chain center requires

$$M_2^{\rm L}(\ket\psi)=M_2(\ket\psi)-M_2^{\rm NL}(\ket\psi),$$2

The paper interprets this as an information separation between nonlocal magic and extra entanglement, and states that it is inappropriate to regard entanglement as the driving force behind the growth and spreading of nonlocal magic (Huang et al., 30 Sep 2025). A plausible implication is that nonlocal magic can be the more natural simulation target in regimes where entanglement is prohibitively costly.

5. Many-body, critical, and multipartite generalizations

In many-body systems, nonlocal magic is often studied through reduced bipartite states or inclusion–exclusion constructions. For the transverse-field Ising model at criticality, the two-point non-local nonstabilizerness $$M_2^{\rm L}(\ket\psi)=M_2(\ket\psi)-M_2^{\rm NL}(\ket\psi),$$3 decays algebraically,

$$M_2^{\rm L}(\ket\psi)=M_2(\ket\psi)-M_2^{\rm NL}(\ket\psi),$$4

while away from criticality it decays exponentially to a constant (Qian et al., 10 Feb 2025). In monitored Haar-random circuits at the critical measurement rate $$M_2^{\rm L}(\ket\psi)=M_2(\ket\psi)-M_2^{\rm NL}(\ket\psi),$$5, the averaged two-point quantity also decays as a power law and remains much smaller than the total magic (Qian et al., 10 Feb 2025).

The same work introduces measurement-induced nonlocal magic,

$$M_2^{\rm L}(\ket\psi)=M_2(\ket\psi)-M_2^{\rm NL}(\ket\psi),$$6

and reports “nonstabilizerness swapping,” analogous to entanglement swapping (Qian et al., 10 Feb 2025). In the critical monitored circuit, the decay exponent of post-measurement nonlocal magic is reported to be about $$M_2^{\rm L}(\ket\psi)=M_2(\ket\psi)-M_2^{\rm NL}(\ket\psi),$$7, whereas the pre-measurement mutual information decays with exponent about $$M_2^{\rm L}(\ket\psi)=M_2(\ket\psi)-M_2^{\rm NL}(\ket\psi),$$8; the post-measurement nonlocal resource therefore decays more slowly than any pre-measurement correlation inferred from the mutual-information bound (Qian et al., 10 Feb 2025).

A complementary construction is long-range magic,

$$M_2^{\rm L}(\ket\psi)=M_2(\ket\psi)-M_2^{\rm NL}(\ket\psi),$$9

which isolates the magic stored in correlations between distant subsystems (Tarabunga et al., 2023). The Pauli-Markov-chain method samples Pauli strings according to distributions built from $A\leftrightarrow B$0, and a Tree Tensor Network implementation reduces update costs to $A\leftrightarrow B$1 in system size (Tarabunga et al., 2023). In one dimension, long-range magic exhibits strong signatures of conformal criticality in Ising, Potts, and Gaussian models; in two-dimensional $A\leftrightarrow B$2 lattice gauge theories it identifies the confinement–deconfinement transition and displays critical scaling behavior at modest volumes (Tarabunga et al., 2023).

For genuinely multipartite structure, an inclusion–exclusion functional was introduced in fermionic systems: $A\leftrightarrow B$3 It vanishes whenever the state factorizes across any nontrivial partition, is invariant under local Clifford operations, and can be negative because it is an alternating sum rather than a sum of positive terms (Malvimat et al., 6 Jan 2026). In examples, the three-qubit GHZ state has

$A\leftrightarrow B$4

whereas the $A\leftrightarrow B$5 state gives

$A\leftrightarrow B$6

Applied to SYK, sparse SYK, mass-deformed SYK, and $A\leftrightarrow B$7 supersymmetric SYK, this functional resolves how nonstabilizerness is distributed across scales and reveals a pronounced disparity between thermal pure quantum states and thermal density matrices (Malvimat et al., 6 Jan 2026).

6. Scattering, holography, and experiment

In relativistic scattering, nonlocal magic provides a basis-independent refinement of earlier helicity-basis analyses. For gluon and graviton $A\leftrightarrow B$8 scattering, non-local magic is defined by minimizing the two-qubit SRE over local basis changes, and the helicity basis is found to coincide with the minimizing basis for many initial states, especially polarized product states (Gargalionis et al., 4 Mar 2026). The same work classifies all $A\leftrightarrow B$9 two-qubit stabilizer initial states into seven groups, and reports that for $M$0 out of $M$1 the local magic in the helicity basis exactly equals the basis-independent non-local magic (Gargalionis et al., 4 Mar 2026). This coincidence breaks in Yang–Mills theory deformed by the dimension-six operator $M$2, where helicity-basis local magic and non-local magic no longer agree (Gargalionis et al., 4 Mar 2026).

For two-particle scattering more generally, anti-flatness provides a compact proxy. In low-energy nucleon–nucleon and high-energy Møller scattering, the relation

$M$3

was verified for two-qubit pure states, with

$M$4

(Robin et al., 27 Oct 2025). The attraction of anti-flatness is experimental: it can be determined from one final-state particle and does not require spin correlations (Robin et al., 27 Oct 2025).

In holography, nonlocal magic is tied to entanglement-spectrum anti-flatness and to geometry. “Gravitational back-reaction is magical” proves that non-local magic is lower bounded by the non-flatness of entanglement spectrum and upper bounded by the amount of entanglement, and in holographic CFTs it vanishes if and only if there is no gravitational back-reaction (Cao et al., 2024). The same work states that non-local magic is approximately equal to the rate of change of the minimal surface area in response to the change of cosmic brane tension in the bulk (Cao et al., 2024). In holographic Schwinger pair creation, the refined Rényi slope gives

$M$5

which is strictly positive for $M$6 and vanishes for $M$7; the paper uses the criterion

$M$8

to conclude that Schwinger pair creation dynamically generates nonlocal magic for $M$9 (Grieninger, 5 May 2026).

Experimentally, the first direct demonstration was reported on a superconducting processor. Using a Contralto-D QPU, the experiment isolated a two-qubit register formed by D3 and C4, implemented both local-erasure and reduced-density-matrix-purity protocols, and found agreement between the two routes within statistical error and without free parameters in the noise model (Ahmad et al., 19 Nov 2025). For pure two-qubit states, the direct reconstruction formula is

$M$0

with $M$1 (Ahmad et al., 19 Nov 2025). The paper presents LM, M, and NLM state families, thereby separating local-only, mixed local-plus-nonlocal, and purely non-local magic on hardware (Ahmad et al., 19 Nov 2025).

7. Terminological boundaries and historical usage

A persistent source of confusion is that “nonlocal magic” in the resource-theory sense is not Bell nonlocality. The standard modern usage concerns nonstabilizerness that cannot be erased by local unitaries or local basis changes (Qian et al., 10 Feb 2025, Ahmad et al., 19 Nov 2025). By contrast, the older literature on “quantum magic games” uses “magic” in the Mermin–Peres sense of parity-constraint pseudo-telepathy rather than as a magic monotone.

In that earlier sense, an arrangement is “magic” if it admits a quantum realization with odd parity, and the central theorem is

$M$2

Every such magic game can be won with certainty using only three Bell pairs, and every magic arrangement has a realization using operators from the three-qubit Pauli group (Arkhipov, 2012). This is a theorem about nonlocal games, graph planarity, and perfect quantum strategies, not about local-unitary optimization of nonstabilizerness.

The bridge between the two terminologies is indirect but nontrivial. Recent work on shallow-circuit complexity constructs nonlocal games from linear binary constraint systems whose perfect quantum strategies exist but cannot be achieved by Clifford/stabilizer resources; for even $M$3, the game family has a perfect quantum strategy but no perfect Clifford or classical strategy, and this is then converted into an unconditional constant-depth separation between generic shallow quantum circuits and magic-free shallow circuits (Zhang et al., 2024). Separately, tomography-based Bell inequalities built only from Pauli measurements can sometimes witness quantum magic when the observed Bell value exceeds the maximal stabilizer value for the same operator family (Cieslinski et al., 31 Dec 2025). These connections show that Bell nonlocality, pseudo-telepathy, and nonlocal magic in the resource-theory sense are related through stabilizer structure, but they are not interchangeable notions.

The resulting picture is precise. Nonlocal magic, in the modern sense, is a basis-independent measure of irreducible nonstabilizerness bound to correlations. It can vanish for highly entangled stabilizer states, can be extensive yet spectrally constrained, admits exact formulas in several important settings, and has operational implications for simulation complexity, dynamics, scattering, holography, and quantum hardware (Huang et al., 23 Jun 2026, Andreadakis et al., 12 Apr 2025, Ahmad et al., 19 Nov 2025).