Lattice gauge theory · quantum information

Magic and
entanglementin a gauge theory

Two different resources make a quantum state hard to fake on a classical computer. A new study measures both in the ground state of a non-Abelian gauge theory, on lattices worth 300 qubits — and finds they come apart.

The main result Read the paper

F1 The picture to hold in mind

Entanglement entropy S and magic density M₂/L of the ground state, against the coupling g. Both start high at g = 0 and vanish at strong coupling — but not together. Entanglement falls steadily; magic holds a plateau, then drops after a crossover coupling g_⋆ ≈ 1.9. Schematic, after Fig. 1 of the paper.

Non-Abelian, with matter

SU(2) gauge fields coupled to staggered fermions in 1+1 dimensions — the first large-scale study of magic in a non-Abelian gauge theory that includes matter.

Gauss's law exact

A dressed-site basis enforces gauge invariance identically: only 6 of 36 local states survive, and every state kept is physical.

Large scale

Tensor-network ground states up to L = 100 sites — 300 qubits' worth of Hilbert space — at bond dimension χ = 100.

Two kinds of quantum

Why are quantum computers hard to simulate? The reflex answer is entanglement — the non-local correlations that have no classical counterpart. Entangle enough particles, the story goes, and no classical machine can keep up.

The story is wrong, or at least incomplete. The Gottesman–Knill theorem says that a large family of quantum circuits — Clifford circuits acting on stabilizer states — can be simulated efficiently on a laptop, no matter how much entanglement they generate. A highly entangled stabilizer state is classically easy. What makes quantum computation genuinely intractable is a second resource: the distance from that easy set, called non-stabilizerness or, in the standard bit of quantum-information whimsy, magic.

Magic can be measured. The stabilizer Rényi entropy (SRE) of a pure state $|\psi\rangle$ of $N$ qubits asks how the state spreads over all Pauli strings $P$:

$$M_{2} \;=\; -\log_{2}\!\left(\frac{1}{2^{N}} \sum_{P \in \mathcal{P}_{N}} \langle\psi|P|\psi\rangle^{4}\right)$$

(1)

For a stabilizer state the sum concentrates on a few Pauli strings and $M_2 = 0$; the more evenly the state smears over the Pauli group, the larger $M_2$ grows. Entanglement and magic capture different departures from classicality, and a state can be rich in one and poor in the other. Where magic lives in physically relevant theories tells us where a quantum computer might one day earn its keep — which is exactly the question for lattice gauge theories, the quantum-field-theory workhorses that future quantum simulators are being built for.

F2 Playground — one qubit of magic

polar θ 55°

azimuth φ 45°

⟨X⟩, ⟨Y⟩, ⟨Z⟩—Pauli expectations

Magic M₂0.000max ≈ 0.585

Stabilizer.

A single qubit state on the Bloch sphere, with its magic computed live from equation (1): M₂ = −log₂[(⟨X⟩⁴ + ⟨Y⟩⁴ + ⟨Z⟩⁴ + 1)/2]. The six stabilizer states — the poles and the four equatorial axes points — have M₂ = 0: classically easy. Everything else is magical. The maximum, M₂ = log₂(3) − 1 ≈ 0.585, is reached at the eight points where ⟨X⟩ = ±⟨Y⟩ = ±⟨Z⟩, the face-diagonal directions. Drag the sliders and watch the state leave and re-enter the easy set. Entanglement never appears here — one qubit has none — which is the point: magic is a different resource.

Magic has a shorter history than entanglement, and its history in gauge theories is shorter still.

1998
Gottesman; Aaronson & Gottesman
Clifford circuits with stabilizer inputs and Pauli measurements are efficiently classically simulable — entanglement alone is not enough.
the easy set is defined
2005
Bravyi & Kitaev
Magic states: feed a non-stabilizer state into a Clifford machine and universal fault-tolerant quantum computation follows. Magic becomes a resource, literally.
magic = fuel
2022
Leone, Oliviero, Hamma
The stabilizer Rényi entropy — a measure of magic that can actually be computed for many-body states, later via Pauli sampling of matrix product states.
magic becomes measurable
2023–25
Many groups
Magic mapped in spin chains and critical systems; first gauge theories — Abelian U(1), string breaking in the Schwinger model, discrete and pure non-Abelian theories on ladders.
gauge theories, but no non-Abelian matter
Now
Jha, Toga, Taher, Bakalov, Kemper
Magic and gauge-invariant entanglement across the full parameter space of SU(2) lattice gauge theory with dynamical matter, up to L = 100 sites.
non-Abelian + matter, at scale

The theory: SU(2) on a line

The stage is a one-dimensional chain of $L$ sites. Fermionic matter ("quarks", in two colors) lives on the sites; SU(2) gauge fields live on the links between them. The Hamiltonian is the Kogut–Susskind one:

$$H \;=\; g^{2}\sum_{n,A}\big(E^{A}_{n,n+1}\big)^{2} \;+\; m\sum_{n}(-1)^{n}\,\psi^{\dagger}_{n}\psi_{n} \;+\; \sum_{n}\big(\psi^{\dagger}_{n}\,U_{n,n+1}\,\psi_{n+1} + \mathrm{h.c.}\big)$$

(2)

Reading guide. Three terms, three jobs. The first is the electric field energy on each link, weighted by the coupling $g^2$ — turning $g$ up makes flux expensive. The second is the mass term; the alternating sign is the staggered-fermion trick for putting particles and antiparticles on one lattice. The third hops a quark to the neighboring site, with the link operator $U$ along for the ride to keep everything gauge-covariant. Color indices are suppressed. In one spatial dimension there are no plaquettes, so no magnetic term.

Gauge theories come with a constraint, not just a Hamiltonian. Gauss's law demands that physical states are annihilated by every local generator: $G^{A}_{n}|\psi\rangle = 0$. On the lattice this becomes a beautifully concrete rule. Each site sees three spins — the SU(2) flux $j_L$ on its left half-link, its matter charge $j_M$, and the flux $j_R$ on its right half-link — and they must fuse to a color singlet:

$$j_{L}\,\otimes\, j_{M}\,\otimes\, j_{R}\;\supset\;\mathbf{0} \qquad\Longleftrightarrow\qquad |j_{L}-j_{M}|\le j_{R}\le j_{L}+j_{M} \;\;\text{and}\;\; j_{L}+j_{M}+j_{R}\in\mathbb{Z}$$

(3)

The gauge fields are truncated at $j_{\max} = \tfrac12$ — the hardcore gluon approximation: each half-link carries spin $0$ or $\tfrac12$ and nothing higher. A site with its two half-links then has $6 \times 3 \times 2 = 36$ naive states, but the singlet rule (3) cuts that to exactly 6. Fusing site and half-links into one composite object — a dressed site — builds Gauss's law into the basis itself: no constraint to police, no unphysical states to leak into.

F3 Playground — build a physical state

left flux j_L

matter n_M

right flux j_R

matter spin j_M0from occupation

triangle rule✓|j_L−j_M| ≤ j_R ≤ j_L+j_M

integer rule✓j_L+j_M+j_R ∈ ℤ

Physical: a gauge singlet.

One dressed site: left half-link, matter site, right half-link. Pick the three quantum numbers and the two conditions of (3) are checked live. An empty or doubly occupied site is a color singlet (j_M = 0), so its fluxes must match; a single quark is a color doublet (j_M = 1/2), so the flux must change by half a unit across it — the quark is a source of flux, which is Gauss's law doing its job. Exactly 6 of the 36 combinations pass.

The six physical states of one dressed site at j_max = 1/2, labeled |j_L, n_M, j_R⟩. Four have matched fluxes and singlet matter; in the two single-quark states the flux steps by 1/2 across the site.
State	Matter	j_M	Flux across site
\|0, 0, 0⟩	empty	0	0 → 0
\|1/2, 0, 1/2⟩	empty	0	1/2 → 1/2
\|0, 2, 0⟩	baryon (2 quarks)	0	0 → 0
\|1/2, 2, 1/2⟩	baryon (2 quarks)	0	1/2 → 1/2
\|0, 1, 1/2⟩	single quark	1/2	0 → 1/2
\|1/2, 1, 0⟩	single quark	1/2	1/2 → 0

Two knobs remain: the mass $m$ and the coupling $g$. The whole study is a map of quantum resources over the $(m, g)$ plane, computed with tensor networks (DMRG) — with three observables tracked throughout: the gauge-invariant entanglement entropy $S$ across a cut link, the magic density $M_2/L$, and the particle density $\rho$, which counts quark–antiquark (meson) and two-quark (baryon) fluctuations in the vacuum.

The critical point, where everything is maximal

Set both knobs to zero, $m = g = 0$. All that survives of (2) is hopping, and the theory flows to a conformal field theory (CFT) with central charge $c = 1$ — a gapless, scale-free critical state. Criticality is where quantum correlations are at their most extravagant, and there is an exact prediction for how the entanglement of a block of length $l$ grows with system size:

$$S(l) \;=\; \frac{c}{6}\, \ln\!\left[\frac{2L}{\pi}\sin\!\left(\frac{\pi l}{L}\right)\right] + \text{const}$$

(4)

This is a stringent test: fit the measured entropies to (4) and the slope hands you the central charge. The magic obeys its own scaling law at criticality,

$$M_{2} \;=\; \alpha L + \beta \ln L + \gamma$$

(5)

— an extensive bulk piece, a logarithmic correction, and a constant.

F4 The CFT point, m = g = 0

system size L 60

cut position l L/2

S(l) from eq. (4)—c = 1, fitted const

central charge (measured)0.990(5)expected: 1

M₂ from eq. (5)—α=0.29, β=−0.69, γ=0.88

Left: the Calabrese–Cardy arc (4) for the chosen L, with the cut marker riding it; entanglement is largest for a cut in the middle and grows logarithmically with L. Right: the magic scaling law (5) with the paper's fitted coefficients α = 0.29(1), β = −0.69(22), γ = 0.88(59) — magic is extensive: every site contributes about 0.29 bits. Curves are drawn from the fitted formulas; the underlying data reaches L = 100.

The checks all pass, which is worth pausing on.

The central charge comes out right. Fitting (4) at the central cut across system sizes gives c = 0.990(5), against the exact c = 1 — even though the gauge fields are truncated to spin 1/2. The hardcore-gluon theory sits in the same universality class.
The critical state is the most magical state. Both S and M₂/L are maximized at m = g = 0. Critical ground states are the hardest states this theory owns, by both measures at once.
The opposite corner is free. As g → ∞ the ground state becomes a product stabilizer state: S = 0 and M₂ = 0 exactly. The two ends of the dial are settled analytically; the interest is everything in between.

The main result: magic and entanglement come apart

Now fix a small mass, $m = 0.2$, and turn up the coupling $g$, walking the theory from its critical point toward strong coupling — toward confinement, where flux is so expensive that the vacuum empties out. Both resources must die on this journey. The discovery is how they die.

Entanglement dies the way you would expect: monotonically, tracking the particle density, as the gap opens and correlations shorten. Magic does something else. It holds an plateau through the intermediate regime — the wavefunction stays firmly non-stabilizer even as its entanglement drains away — and only then drops, past a crossover coupling $g_{\star}$.

The finding

The coupling where magic begins its sharp decline coincides with the coupling where the entanglement entropy is changing fastest: $g_{\star} \approx 1.9$, read off from the extrema of $dM_2/dg$ and $dS/dg$ alike, and echoed by the particle density. Confinement suppresses quantum correlations and quantum computational complexity at different rates, but their milestones line up at one coupling.

F5 Walk the coupling — three observables, one crossover

coupling g 1.00

Entanglement S—falls monotonically

Magic M₂/L—plateau, then drop

Particle density ρ—vacuum fluctuations

|dS/dg|—maximal at g⋆

The three observables at m = 0.2 against the coupling, with the crossover g⋆ ≈ 1.9 marked. Drag the slider: left of g⋆ the state is magic-rich — entanglement is already falling but M₂/L barely moves. At g⋆ the entanglement is falling at its fastest, and magic begins its decline. Far right, the confined vacuum approaches a product stabilizer state and everything vanishes. Schematic reconstruction of Fig. 5 of the paper: the curves follow the behavior and values described there (S from ≈1.3, M₂/L ≈ 0.3 plateau, ρ from ≈0.4), not the raw data. The paper's version shows L = 20…100 collapsing onto these curves — no drift with volume, the signature of a crossover rather than a phase transition.

The interpretation offered is that the sustained plateau reflects the structure of the confinement crossover itself: even as the correlation length shortens, the vacuum remains a genuinely non-classical superposition of meson and baryon fluctuations. Only when the electric term crushes those fluctuations entirely does the state finally become classically easy. Magic, in other words, outlives entanglement.

The whole phase plane

One slice is suggestive; the paper maps the entire $(m, g)$ plane at $L = 70$. The picture is consistent everywhere: magic is maximal at the critical corner $m = g = 0$ and decays outward — at different rates on either side of $g_{\star}$ — while entanglement decreases monotonically with $g$, fastest near the same $g_{\star} \approx 1.9$. The crossover is not an accident of one parameter choice; it is a feature of the theory's resource landscape.

F6 Resource maps over (m, g)

at (m, g)—hover / tap the map

value—M₂/L

Schematic reconstruction of Fig. 6 of the paper: the observable over the (m, g) plane at L = 70, darkest where largest. The dashed white lines are slices of constant m/g — the paths one walks toward the continuum limit — and the vertical dashed line marks g⋆. Both maps peak at the critical corner and fade toward strong coupling; the magic map fades with a visible change of rate across g⋆, the entanglement map smoothly. In the paper the colors are drawn from data at the marked grid of simulated points, with χ = 100.

Why it matters, and what still needs doing

The practical stakes are about where quantum computers will pay off. Simulating lattice gauge theories — especially non-Abelian ones with matter, where sign problems and real-time dynamics defeat classical Monte Carlo — is a headline application for quantum hardware. But a quantum computer only earns an advantage where the states involved are classically hard, and this work draws that map for a non-Abelian theory: hardness of both kinds concentrated near criticality and the continuum limit, a magic-rich intermediate regime, and a classically easy strong-coupling region. For fault-tolerant machines the point is sharper still: in error-corrected architectures Clifford operations are cheap and it is magic, injected through costly distilled states, that sets the budget. $M_2/L \approx 0.3$ bits per site is a statement about the resources a simulation of this theory will consume.

There is also a physics payoff. In this 1+1-dimensional model there is no deconfinement transition — the theory confines for all $g > 0$, and nothing here is a phase transition, only a crossover. But the demonstrated sensitivity of $dS/dg$ and of magic to the confinement structure suggests both as probes of phase diagrams in theories that do have transitions, such as SU(2) in 2+1 dimensions.

Honesty about scope, in the paper and here.

The truncation. Everything is at $j_{\max} = \tfrac12$, the hardcore gluon. The correct central charge is encouraging and preliminary checks suggest corrections are modest, but a systematic study at larger $j_{\max}$ is explicitly left to future work.
Finite bond dimension. Ground states are matrix product states at $\chi = 100$; SRE is estimated by Pauli sampling. Both are controlled but not exact.
This page's figures. F1, F5 and F6 are schematic reconstructions drawn from the functional forms and values reported in the paper, built for legibility — the quantitative record is the paper's own figures.

Next steps we point out: pushing $j_{\max}$ up, SU(3) in 1+1 dimensions for a step toward QCD, higher dimensions where tensor networks hit the entanglement barrier and neural quantum states may take over, and better sampling algorithms for magic itself.

Read it yourself

Magic and entanglement in 1+1-dimensional SU(2) lattice gauge theory — Raghav G. Jha, Goksu C. Toga, Jaber I. Taher, Bojko N. Bakalov, and Alexander F. Kemper (North Carolina State University). arXiv:2606.09971 [quant-ph]. Appendices A–C carry the representation theory, the dressed-site Hamiltonian, and the observables.

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Selected references

The load-bearing citations, from the paper's bibliography.

KS75
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Got98
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AG04
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BK05
S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev. A 71 (2005) 022316. arXiv:quant-ph/0403025. Magic states.
LOH22
L. Leone, S. F. E. Oliviero and A. Hamma, Stabilizer Rényi entropy. Phys. Rev. Lett. 128 (2022) 050402. arXiv:2106.12587. The measure used throughout.
Don12
W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory. Phys. Rev. D 85 (2012) 085004. arXiv:1109.0036. The gauge-invariant entanglement entropy.
CC09
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory. J. Phys. A 42 (2009) 504005. arXiv:0905.4013. Equation (4).
WCS21
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CMRS24
G. Calajò, G. Magnifico, C. Edmunds, M. Ringbauer, S. Montangero and P. Silvi, Digital quantum simulation of a (1+1)D SU(2) lattice gauge theory with ion qudits. PRX Quantum 5 (2024) 040309. arXiv:2402.07987. The dressed-site basis and the hardcore gluon.
FTF+25
P. R. N. Falcão, P. S. Tarabunga, M. Frau, E. Tirrito, J. Zakrzewski and M. Dalmonte, Nonstabilizerness in U(1) lattice gauge theory. Phys. Rev. B 111 (2025) L081102. arXiv:2409.01789. The Abelian predecessor.
THP23
P. S. Tarabunga, E. Tirrito, T. Chanda and M. Dalmonte, Many-body magic via Pauli–Markov chains — from criticality to gauge theories. PRX Quantum 4 (2023) 040317. arXiv:2305.18541.
CMS24
G. Cataldi, G. Magnifico, P. Silvi and S. Montangero, Simulating (2+1)D SU(2) Yang–Mills lattice gauge theory at finite density with tensor networks. Phys. Rev. Res. 6 (2024) 033057. arXiv:2307.09396. Where a real transition awaits these probes.
JTT+26
R. G. Jha, G. C. Toga, J. I. Taher, B. N. Bakalov and A. F. Kemper, Magic and entanglement in 1+1-dimensional SU(2) lattice gauge theory (2026). arXiv:2606.09971. The paper this page explains.