Entanglement Asymmetry in Quantum Subsystems

• Entanglement asymmetry is an entanglement-based measure that quantifies the entropy cost of erasing symmetry-breaking coherence in a subsystem's reduced density matrix.
• It is computed as the difference between the entanglement entropy before and after symmetry twirling, separating coherent charge-sector contributions from classical mixtures.
• The concept extends to finite groups, conformal field theory, Gaussian states, and non-equilibrium dynamics, offering insights into symmetry restoration and emergent phenomena.

Entanglement asymmetry (EA) is an entanglement-based measure of symmetry breaking defined at the level of a subsystem. For a bipartition $A\cup B$ and reduced density matrix $\rho_A=\mathrm{Tr}_B\ket{\Psi}\bra{\Psi}$, it compares $\rho_A$ with the state obtained by removing coherence between symmetry sectors of the subsystem charge. In its original formulation for a global $U(1)$ symmetry generated by $Q=Q_A+Q_B$, EA is

$$\Delta S_A=S(\rho_{A,Q})-S(\rho_A), \qquad \rho_{A,Q}=\sum_q \Pi_q \rho_A \Pi_q =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},$$

and is equivalently the relative entropy $S(\rho_A\Vert \rho_{A,Q})$. It is therefore nonnegative and vanishes iff the subsystem state is already symmetric, i.e. iff $[\rho_A,Q_A]=0$ (2207.14693). Since its introduction as a subsystem-sensitive probe of symmetry breaking in many-body systems, EA has been extended to finite groups, conformal field theory, gauge theory, higher-form symmetry, random states and circuits, fragmented systems, and Gaussian-state formalisms (Capizzi et al., 2023, Benini et al., 2024, Florio et al., 3 Nov 2025, Lamas et al., 4 Oct 2025, Gotta et al., 2 Mar 2026, Travaglino et al., 29 Apr 2026).

1. Definition, physical interpretation, and basic structures

For $U(1)$ symmetry, EA quantifies the entropy increase produced by dephasing the reduced density matrix in eigen-sectors of $Q_A$. Physically, it measures the amount of charge-sector coherence stored in $\rho_A=\mathrm{Tr}_B\ket{\Psi}\bra{\Psi}$0: if $\rho_A=\mathrm{Tr}_B\ket{\Psi}\bra{\Psi}$1 contains off-diagonal matrix elements between different subsystem charges, symmetry twirling removes them and increases the entropy; if $\rho_A=\mathrm{Tr}_B\ket{\Psi}\bra{\Psi}$2 is already block diagonal, the entropy is unchanged and EA vanishes (2207.14693).

The same logic extends beyond $\rho_A=\mathrm{Tr}_B\ket{\Psi}\bra{\Psi}$3. For a finite group $\rho_A=\mathrm{Tr}_B\ket{\Psi}\bra{\Psi}$4, the symmetry-restored reduced state is

$\rho_A=\mathrm{Tr}_B\ket{\Psi}\bra{\Psi}$5

and the corresponding Rényi and von Neumann asymmetries compare $\rho_A=\mathrm{Tr}_B\ket{\Psi}\bra{\Psi}$6 with $\rho_A=\mathrm{Tr}_B\ket{\Psi}\bra{\Psi}$7. In this formulation, EA remains a subsystem measure of how much the reduced state fails to be invariant under the symmetry action (Capizzi et al., 2023). For compact Lie groups, the discrete sum is replaced by Haar averaging, and the same relative-entropy interpretation persists (Fossati et al., 2024).

A useful distinction runs through the literature. Standard entanglement entropy measures bipartite quantum correlations, whereas EA measures the entropic cost of enforcing local symmetry on the reduced state. Symmetry-resolved entanglement typically assumes $\rho_A=\mathrm{Tr}_B\ket{\Psi}\bra{\Psi}$8, so the reduced state is already decomposable into charge sectors; EA addresses the opposite regime, in which that block structure fails precisely because the subsystem retains symmetry-breaking coherence (2207.14693). This makes EA especially suited to out-of-equilibrium states, anomalous symmetry breaking, and situations where local order parameters are incomplete or unavailable.

The Rényi generalization,

$\rho_A=\mathrm{Tr}_B\ket{\Psi}\bra{\Psi}$9

is central in replica and experimental settings. Integer-$\rho_A$0 Rényi EAs may be experimentally accessible using randomized measurement protocols with post-selection (2207.14693).

2. Replica, defects, charged moments, and Gaussian formulations

A major formal development is the representation of EA through charged moments. For $\rho_A$1,

$\rho_A$2

If $\rho_A$3, then $\rho_A$4, so $\rho_A$5 (2207.14693). This charged-moment framework underlies exact free-fermion calculations, path-integral derivations, and several asymptotic analyses.

In conformal field theory, EA can be formulated using replica manifolds with symmetry defects inserted along gluing surfaces. For coherent excited states in $\rho_A$6-symmetric CFTs, the leading perturbative asymmetry is controlled by a replicated two-point function, and the von Neumann EA at quadratic order equals the Fisher information metric around the vacuum reduced state. In holographic theories, the same perturbative quantity is identified with Hollands–Wald canonical energy in the entanglement wedge and is related to the bulk $\rho_A$7 charge contained there (Benini et al., 2024). In $\rho_A$8-dimensional critical systems, the defect perspective also yields a characteristic $\rho_A$9 correction to EA, governed by the scaling dimensions of non-topological defects (Fossati et al., 2024).

Free-fermion systems motivated another formal refinement. Standard twirling typically sends a Gaussian reduced state to a non-Gaussian one, which obstructs direct correlation-matrix evaluation. A Gaussian asymmetry measure was introduced by replacing exact symmetrization with a Gaussian symmetrization that removes the anomalous block $U(1)$0 of the Nambu correlation matrix while preserving the normal block $U(1)$1. The resulting quantity,

$U(1)$2

is the minimal relative entropy from $U(1)$3 to the manifold of symmetric Gaussian states and is exactly computable from correlation matrices (Travaglino et al., 29 Apr 2026). This suggests a precise separation between symmetry breaking intrinsic to Gaussian data and extra non-Gaussianity generated by exact sector projection.

3. Dynamical symmetry restoration, non-restoration, and Mpemba phenomena

The foundational dynamical setting is a global quench from an initially symmetry-breaking state evolved with a $U(1)$4-symmetric Hamiltonian. In the XX chain, EA decays to zero at late times, thereby diagnosing dynamical restoration of subsystem symmetry. The original analysis established two characteristic features: larger subsystems restore symmetry more slowly, and states with stronger initial symmetry breaking can restore it faster, producing a quantum Mpemba effect (2207.14693).

The XY-to-XX quench sharpened this picture. In the anisotropic XY ground state, the broken $U(1)$5 symmetry is tied to Cooper-pair correlations after Jordan–Wigner mapping, and the equilibrium EA obeys

$U(1)$6

with $U(1)$7, so EA acquires a direct interpretation in terms of integrated Cooper-pair density (Murciano et al., 2023). After the quench to the XX chain, local symmetry is restored, but the relaxation law depends discontinuously on whether the initial state is critical: noncritical initial states yield $U(1)$8-type restoration, whereas critical initial states relax as $U(1)$9, which the authors identify as new strong and weak forms of the quantum Mpemba effect (Murciano et al., 2023).

Discrete symmetries exhibit analogous but not identical behavior. For cyclic $Q=Q_A+Q_B$0 groups, EA is bounded by $Q=Q_A+Q_B$1, unlike the unbounded logarithmic growth possible for continuous $Q=Q_A+Q_B$2 (Ferro et al., 2023). In the XY spin chain with broken $Q=Q_A+Q_B$3 parity, quenches reveal plateau structures and late-time restoration under periodic boundary conditions, while open boundaries can retain nonzero asymmetry when a post-quench boundary mode survives (Ferro et al., 2023).

Explicitly non-symmetric dynamics reverses the logic. In a non-symmetric random quantum circuit, subsystem EA generated from an initially symmetric state exhibits a pronounced early-time overshoot but ultimately vanishes for subsystems smaller than half the system, consistent with scrambling to an effectively fully mixed reduced state. In a non-symmetric Hamiltonian quench, the same early-time overshoot is present, yet the late-time subsystem EA remains nonzero because the locally thermal state inherits the symmetry-breaking terms of the Hamiltonian (Yu et al., 23 Jan 2025). This contrast underscores that EA distinguishes symmetry restoration driven by scrambling from persistent symmetry breaking inherited from local Gibbs structure.

4. Extensions across phases, symmetries, and field-theoretic settings

The scope of EA has broadened substantially, and several extensions have distinct physical meanings.

Before the table, two broad patterns are worth isolating. First, in ordered phases with finite symmetry groups, EA asymptotically counts degenerate vacua. In the ordered phase of the $Q=Q_A+Q_B$4-dimensional Ising field theory, where a $Q=Q_A+Q_B$5 symmetry is spontaneously broken, the large-interval asymmetry approaches $Q=Q_A+Q_B$6, and a broader conjecture states

$Q=Q_A+Q_B$7

with $Q=Q_A+Q_B$8 the subgroup leaving the state invariant (Capizzi et al., 2023). Second, in gauge and topological settings EA becomes a probe of symmetry-breaking information that is not naturally localized in conventional order parameters. In the massless Schwinger model it diagnoses anomaly-induced chiral symmetry breaking, and in topologically ordered phases it can be generalized to higher-form symmetry breaking (Florio et al., 3 Nov 2025, Lamas et al., 4 Oct 2025).

Setting	Symmetry/context	Representative EA result
Ordered Ising field theory	Finite-group SSB	$Q=Q_A+Q_B$9 (Capizzi et al., 2023)
Massless Schwinger model	Chiral $$\Delta S_A=S(\rho_{A,Q})-S(\rho_A), \qquad \rho_{A,Q}=\sum_q \Pi_q \rho_A \Pi_q =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},$$0 anomaly in gauge theory	$$\Delta S_A=S(\rho_{A,Q})-S(\rho_A), \qquad \rho_{A,Q}=\sum_q \Pi_q \rho_A \Pi_q =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},$$1 at $$\Delta S_A=S(\rho_{A,Q})-S(\rho_A), \qquad \rho_{A,Q}=\sum_q \Pi_q \rho_A \Pi_q =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},$$2 (Florio et al., 3 Nov 2025)
High-temperature Schwinger regime	Finite-$$\Delta S_A=S(\rho_{A,Q})-S(\rho_A), \qquad \rho_{A,Q}=\sum_q \Pi_q \rho_A \Pi_q =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},$$3 anomaly probe	$$\Delta S_A=S(\rho_{A,Q})-S(\rho_A), \qquad \rho_{A,Q}=\sum_q \Pi_q \rho_A \Pi_q =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},$$4 (Florio et al., 3 Nov 2025)
Toric code / Abelian topological order	Broken 1-form symmetry	$$\Delta S_A=S(\rho_{A,Q})-S(\rho_A), \qquad \rho_{A,Q}=\sum_q \Pi_q \rho_A \Pi_q =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},$$5; for non-chiral Abelian order, $$\Delta S_A=S(\rho_{A,Q})-S(\rho_A), \qquad \rho_{A,Q}=\sum_q \Pi_q \rho_A \Pi_q =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},$$6 (Lamas et al., 4 Oct 2025)
Fragmented systems	Commutant-algebra asymmetry	Conventional EA is logarithmic, fragmented EA can be extensive (Gotta et al., 2 Mar 2026)
$$\Delta S_A=S(\rho_{A,Q})-S(\rho_A), \qquad \rho_{A,Q}=\sum_q \Pi_q \rho_A \Pi_q =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},$$7 WZW model	Non-Abelian $$\Delta S_A=S(\rho_{A,Q})-S(\rho_A), \qquad \rho_{A,Q}=\sum_q \Pi_q \rho_A \Pi_q =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},$$8 EA	Real-time EA shows quantum Mpemba effect for fundamental and adjoint primaries (Fujimura et al., 6 Sep 2025)

The gauge-theory case is especially notable because the symmetry breaking arises intrinsically through the anomaly rather than through an explicit symmetry-breaking term. In the massless Schwinger model, EA decreases only logarithmically with temperature in the regime $$\Delta S_A=S(\rho_{A,Q})-S(\rho_A), \qquad \rho_{A,Q}=\sum_q \Pi_q \rho_A \Pi_q =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}\,e^{-i\alpha Q_A}\rho_A e^{i\alpha Q_A},$$9, whereas the chiral condensate decays as $S(\rho_A\Vert \rho_{A,Q})$0, making EA parametrically more sensitive to finite-temperature chiral-symmetry breaking in that model (Florio et al., 3 Nov 2025).

For higher-form symmetry, EA depends on both the topology of the subregion and the topological class of the symmetry operator support. In the toric code, contractible regions have vanishing higher-form EA, but for a suitable non-contractible cylindrical region one finds $S(\rho_A\Vert \rho_{A,Q})$1, matching the universal topological constant for that geometry while remaining conceptually distinct from topological entanglement entropy (Lamas et al., 4 Oct 2025). In fragmented many-body systems, the generalization via commutant algebras yields a symmetrized state

$S(\rho_A\Vert \rho_{A,Q})$2

and the resulting asymmetry can scale extensively because the number of dynamically disconnected sectors can grow exponentially with system size (Gotta et al., 2 Mar 2026).

5. Random states, circuits, black holes, and relational quantum frameworks

Random-state results reveal a sharp subsystem-size transition. For Haar-random pure states with arbitrary compact, semi-simple Lie-group symmetry, the average EA vanishes in the thermodynamic limit for $S(\rho_A\Vert \rho_{A,Q})$3, jumps to a finite value at $S(\rho_A\Vert \rho_{A,Q})$4, and for $S(\rho_A\Vert \rho_{A,Q})$5 grows as $S(\rho_A\Vert \rho_{A,Q})$6; fluctuations vanish, so this is also the typical behavior (Russotto et al., 2024). The non-Abelian result generalizes the earlier $S(\rho_A\Vert \rho_{A,Q})$7 random-state asymmetry Page curve and fixes the logarithmic coefficient by the group dimension (Russotto et al., 2024).

Random quantum circuits turn this static picture into a dynamical one. In random unitary circuits that do not preserve the relevant $S(\rho_A\Vert \rho_{A,Q})$8, the Rényi-2 EA of the whole system and of subsystems larger than half the system approaches its stationary value on a time scale independent of system size, while for smaller subsystems the EA is non-monotonic and relaxes on the scrambling time. For local circuits that time scales linearly with system size; for geometrically non-local circuits it scales logarithmically (Ares et al., 21 Jan 2025). This connects EA directly to the distinction between local equilibration and true scrambling.

A related information-theoretic interpretation appears in the Hayden–Preskill protocol. There the radiation subsystem can display vanishing Rényi-2 EA before a transition at

$S(\rho_A\Vert \rho_{A,Q})$9

which is interpreted as an emergent $[\rho_A,Q_A]=0$0 symmetry of the radiation. When the initial black hole is maximally mixed, the averaged Rényi-2 EA of the radiation remains zero throughout the entire radiation process (Chen et al., 2024). The underlying explanation is a decoupling inequality showing that, below threshold, the radiation is close to the maximally mixed state, which is automatically symmetric (Chen et al., 2024).

There is also a conceptually adjacent but distinct correspondence in internal quantum reference frames. For a globally invariant pure state of a reference frame $[\rho_A,Q_A]=0$1 and system $[\rho_A,Q_A]=0$2, the asymmetry of the conditional state on $[\rho_A,Q_A]=0$3,

$[\rho_A,Q_A]=0$4

is exactly equal to the Rényi-2 entanglement entropy of the invariant state on $[\rho_A,Q_A]=0$5 (Hamette et al., 2021). This is not the standard many-body EA defined by symmetry-twirled reduced density matrices, but it places asymmetry and entanglement in direct quantitative correspondence within relational quantum descriptions.

6. Relation to neighboring notions, diagnostic power, and limitations

EA is often compared with local order parameters, charge fluctuations, symmetry-resolved entanglement, and quantum Fisher information, but the literature emphasizes that these quantities are not interchangeable. In the $[\rho_A,Q_A]=0$6 XY chain, a local order parameter such as $[\rho_A,Q_A]=0$7 can cross zero transiently while EA remains positive, showing that the subsystem still fails to commute with parity even when the conventional order parameter momentarily vanishes (Ferro et al., 2023). In the dual XXZ local-quench problem, ordinary entanglement entropy remains $[\rho_A,Q_A]=0$8 in the scaling limit while EA grows as $[\rho_A,Q_A]=0$9, explicitly tying EA to coherent occupation of a growing number of magnetization sectors (Ferro, 17 Feb 2026).

This suggests a common misconception to avoid: EA is not merely another fluctuation measure. In Gaussian free-fermion settings, charge-variance differences and full-counting-statistics diagnostics can detect symmetry breaking, but they do not encode the full asymmetry structure captured by the entropy difference $U(1)$0 or its Gaussian analogue (Travaglino et al., 29 Apr 2026). In the dual XXZ setting, EA and QFI are related by the mixed-state inequality

$U(1)$1

which generalizes the pure-state variance/EA bound, yet the bound is not always quantitatively sharp enough to reconstruct QFI behavior (Ferro, 17 Feb 2026).

Several limitations recur across the literature. Direct evaluation of standard EA generally requires access to reduced density matrices, charged moments, or equivalent replica data (2207.14693). In free-fermion problems, exact twirling produces strongly non-Gaussian states, motivating Gaussian-restricted measures when correlation-matrix tractability is essential (Travaglino et al., 29 Apr 2026). In higher-form and topological contexts, EA depends on the topology of the subregion and on the support of the symmetry operator, and it is not strictly equivalent to topological entanglement entropy (Lamas et al., 4 Oct 2025). In gauge theory, exact results currently rely heavily on special solvable models such as the massless Schwinger model (Florio et al., 3 Nov 2025). These constraints suggest that future progress will likely hinge on better nonperturbative access to charged moments, extensions to higher-dimensional gauge theories and non-invertible symmetries, and dynamical formulations compatible with monitored, dissipative, or interacting integrable settings (Florio et al., 3 Nov 2025, Travaglino et al., 29 Apr 2026).

In this broader perspective, EA has become a unifying diagnostic of subsystem symmetry structure. Its central invariant content remains simple: it measures the entropy cost of erasing symmetry-breaking coherence from a reduced state. What changes across applications is the object being twirled—charge sectors, finite-group sectors, higher-form sectors, fragmented commutant sectors, or Gaussian manifolds—and the physical phenomenon that those coherences encode.