Local Operator Entanglement (LOE)

Updated 8 July 2026

LOE is a quantum-information measure that maps local operators to pure states in a doubled Hilbert space and quantifies their bipartite entanglement.
It distinguishes chaotic dynamics from integrable or localized regimes by exhibiting linear growth in chaotic systems versus logarithmic or bounded behavior in integrable cases.
LOE serves as a diagnostic tool for operator complexity, scrambling, and the efficiency of tensor network simulations in quantum many-body and field-theoretic contexts.

Local Operator Entanglement (LOE) is a quantum-information quantity attached to local operators. In quantum many-body dynamics, it measures the amount of quantum entanglement generated by the time evolution of an operator that was initially local, after mapping that operator to a pure state in a doubled Hilbert space and evaluating bipartite entanglement across a spatial cut (Bertini et al., 2019). In conformal field theory, the same term is also used for the increase in (Rényi) entanglement entropy of a state created by acting with a local operator on the vacuum, especially its late-time value for a half-space bipartition (Nozaki et al., 2014). Across these settings, LOE has been used as a quantifier of operator-space complexity, a diagnostic of quantum chaos, a probe of scrambling, and a criterion for classical simulability (Dowling et al., 30 Jan 2025).

1. Definition, operator-state mapping, and scope

In the many-body setting, LOE is defined through the operator-to-state mapping, also called vectorization. A time-evolved local operator is mapped to a pure state in a doubled Hilbert space, and the entanglement of that state across a bipartition $A:B$ is then evaluated by Rényi entropies or the von Neumann entropy (Bertini et al., 2019). A standard definition is

$E_A^{(\alpha)}(O)=\frac{1}{1-\alpha}\log\!\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_{B}|O\rrangle\!\llangle O|\right)^{\alpha}\right]\right),\qquad \alpha\ge 0,$

with $|O\rrangle=(O\otimes \mathbb{I})|\phi^+\rangle$ in the doubled Hilbert space (Dowling, 5 Mar 2026). In this formulation, LOE is precisely the entropy associated with the distribution of squared Schmidt coefficients from an operator Schmidt decomposition (Dowling, 5 Mar 2026).

This operator-space notion is explicitly local in its initial condition: one starts from an operator supported on a small spatial region, often a single site, and studies its Heisenberg evolution under a local circuit or Hamiltonian (Bertini et al., 2019). Several works distinguish this from the operator entanglement of the full time-evolution operator itself, sometimes called global operator entanglement; the distinction is explicit in studies of non-Hermitian dynamics (Barch et al., 2023).

A second established usage appears in quantum field theory and conformal field theory. There one considers an excited state

$|\Psi\rangle=\mathcal{N}^{-1}\mathcal{O}(x_0)|0\rangle,$

defines a reduced density matrix for a spatial subsystem, and studies the excess Rényi entropies relative to the vacuum (Nozaki, 2014). In that literature, LOE is defined by the increased amount of (Rényi) entanglement entropy at late time for an excited state defined by acting the local operator on the vacuum (Nozaki et al., 2014). This usage does not involve Heisenberg evolution of an operator in operator space; instead, it characterizes the entanglement generated by a local excitation in a many-body state. The common element is the entangling action of locality under dynamical propagation.

2. Growth laws in chaotic dynamics and dual-unitary circuits

A central empirical and analytical theme is that LOE discriminates between chaotic and integrable dynamics. For chaotic systems, local-operator entanglement is expected to grow linearly in time, while it is expected to grow at most logarithmically in the integrable case (Bertini et al., 2019). This distinction has been used as a dynamical chaos indicator in local quantum circuits and spin systems (Bertini et al., 2019, Dowling et al., 2023).

Dual-unitary circuits provide a class of statistically solvable models in which LOE can be computed almost exactly. In a class of completely chaotic dual-unitary circuits, LOE grows linearly and the asymptotic slope can reach the maximal possible value,

$S^{(n)}(y,t)\sim v_n t+\mathrm{const.},\qquad v_n=\log d^2,$

with $d$ the local Hilbert-space dimension (Bertini et al., 2019). The same work reports a phase transition in the slope of LOE as circuit parameters are varied; for $n>1$ the slope is controlled by the modulus of the largest nontrivial eigenvalue $\lambda$ of a one-site map, whereas the von Neumann entropy always grows at the maximal rate (Bertini et al., 2019).

A complementary channel-level viewpoint is developed through operator space entangling power, a measure for the ability of a unitary channel to generate operator entanglement (Andreadakis et al., 2024). In dual-unitary circuits, the average growth of local operator entanglement exhibits two distinct regimes in relation to the operator space entangling power of the building-block gate: for small entangling power the first nontrivial LOE grows monotonically with it, while above a threshold the first burst of LOE saturates (Andreadakis et al., 2024). The same work links short-time operator-entanglement generation to the Gaussian scrambling rate of the bipartite out-of-time-order correlator (Andreadakis et al., 2024).

The relation between scrambling and LOE is precise but not identical. Rapid OTOC decay is a necessary but not sufficient condition for linear growth of the LOE entropy (Dowling et al., 2023). In completely chaotic dual-unitary circuits, OTOC decay and LOE growth can become asymptotically equivalent, but integrable models can display exponential OTOC decay together with only logarithmic LOE growth (Dowling et al., 2023). This separates scrambling from chaos at the operator level.

3. Integrability, localization, and exceptional dynamical regimes

The late-time scaling of LOE in integrable systems is model dependent. In the Rule 54 automaton, for certain operators the half-system LOE saturates to a value that is at most logarithmic in system size, with a rigorous bound for diagonal operators,

$S_{\rm vN}\leq S_0 \lesssim 8\log_2 L,$

and early-time growth logarithmic in time (Jacoby et al., 12 Mar 2025). The mechanism described there relies on the presence of only two quasiparticle types and only two possible values of the phase shift between quasiparticles (Jacoby et al., 12 Mar 2025). The same study presents a heuristic argument, supported by numerical evidence, that generic interacting integrable systems such as the Heisenberg spin chain have a saturated von Neumann LOE that is volume-law in system size, while operator Rényi entropies of order $\alpha>1$ are at most logarithmic in $E_A^{(\alpha)}(O)=\frac{1}{1-\alpha}\log\!\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_{B}|O\rrangle\!\llangle O|\right)^{\alpha}\right]\right),\qquad \alpha\ge 0,$0 (Jacoby et al., 12 Mar 2025).

A distinct exactly solvable regime occurs in circuits with ultralocal solitons. If a circuit has an ultralocal soliton moving to the left or right, then the entanglement of local operators initially supported on the corresponding sublattice saturates to a constant value; for qubit chains this value is bounded by $E_A^{(\alpha)}(O)=\frac{1}{1-\alpha}\log\!\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_{B}|O\rrangle\!\llangle O|\right)^{\alpha}\right]\right),\qquad \alpha\ge 0,$1 (Bertini et al., 2019). By contrast, operators on the opposite sublattice can have unbounded entanglement, and in chiral circuits numerical evidence indicates logarithmic growth in time (Bertini et al., 2019). These results hold irrespectively of integrability (Bertini et al., 2019).

Disorder provides another contrast. In the disordered Heisenberg model with strong disorder, many-body localization prevents the information from propagating and being delocalized, and operator mutual-information diagnostics retain memory of the initial operator instead of approaching Page-like values (Mascot et al., 2020). In the thermalizing Ising model, by contrast, the early-time evolution qualitatively follows an effective light cone picture and the late-time value is well described by Page’s value for a random pure state (Mascot et al., 2020).

Random permutation circuits sharpen the distinction between classical sensitivity and quantum-chaotic LOE growth. When the local dimension is $E_A^{(\alpha)}(O)=\frac{1}{1-\alpha}\log\!\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_{B}|O\rrangle\!\llangle O|\right)^{\alpha}\right]\right),\qquad \alpha\ge 0,$2, random permutations are Clifford and the LOE of any local operator is bounded by a constant (Bertini et al., 14 Aug 2025). When $E_A^{(\alpha)}(O)=\frac{1}{1-\alpha}\log\!\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_{B}|O\rrangle\!\llangle O|\right)^{\alpha}\right]\right),\qquad \alpha\ge 0,$3, LOE grows linearly in time; this is proved in the large-$E_A^{(\alpha)}(O)=\frac{1}{1-\alpha}\log\!\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_{B}|O\rrangle\!\llangle O|\right)^{\alpha}\right]\right),\qquad \alpha\ge 0,$4 limit, and numerical evidence is reported already for $E_A^{(\alpha)}(O)=\frac{1}{1-\alpha}\log\!\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_{B}|O\rrangle\!\llangle O|\right)^{\alpha}\right]\right),\qquad \alpha\ge 0,$5 (Bertini et al., 14 Aug 2025). The same work further shows that LOE can be defined also in the classical field and puts it forward as a universal indicator chaos, both quantum and classical (Bertini et al., 14 Aug 2025).

Dynamical setting	LOE behavior	Reported mechanism
Chaotic dual-unitary circuits	Linear in time; maximal slope possible	Completely chaotic transfer-matrix structure (Bertini et al., 2019)
Rule 54 automaton	Saturates at most logarithmically in $E_A^{(\alpha)}(O)=\frac{1}{1-\alpha}\log\!\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_{B}\|O\rrangle\!\llangle O\|\right)^{\alpha}\right]\right),\qquad \alpha\ge 0,$6	Two quasiparticle types; restricted phase shifts (Jacoby et al., 12 Mar 2025)
Generic interacting integrable models	Volume-law von Neumann saturation	Many quasiparticle types; rapidity-dependent dephasing (Jacoby et al., 12 Mar 2025)
Strongly disordered Heisenberg chain	Information remains localized	Many-body localization (Mascot et al., 2020)
Random permutation circuits, $E_A^{(\alpha)}(O)=\frac{1}{1-\alpha}\log\!\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_{B}\|O\rrangle\!\llangle O\|\right)^{\alpha}\right]\right),\qquad \alpha\ge 0,$7	Bounded	Clifford structure (Bertini et al., 14 Aug 2025)
Random permutation circuits, $E_A^{(\alpha)}(O)=\frac{1}{1-\alpha}\log\!\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_{B}\|O\rrangle\!\llangle O\|\right)^{\alpha}\right]\right),\qquad \alpha\ge 0,$8	Linear in time	Non-Clifford operator spreading (Bertini et al., 14 Aug 2025)

4. Late-time volume laws, Page behavior, and random dynamics

For chaotic systems, numerical studies had indicated that LOE displays a volume-law behavior at late times, scaling proportionally with the number of local degrees of freedom. An analytical derivation of this late-time volume law was given under three assumptions: a higher-order non-resonance condition for the Hamiltonian eigenenergies, the ETH ansatz for matrix elements of the initial local operator, and the replacement of Hamiltonian eigenstates with random states in the final expression for LOE (Correr et al., 26 Mar 2026). The resulting formula exhibits volume-law scaling, with the leading term proportional to $E_A^{(\alpha)}(O)=\frac{1}{1-\alpha}\log\!\left(\mathrm{Tr}\left[\left(\mathrm{Tr}_{B}|O\rrangle\!\llangle O|\right)^{\alpha}\right]\right),\qquad \alpha\ge 0,$9, the logarithm of the Hilbert-space dimension of the subsystem (Correr et al., 26 Mar 2026). For a chain of qubits with $|O\rrangle=(O\otimes \mathbb{I})|\phi^+\rangle$0 sites in the subsystem, this means $|O\rrangle=(O\otimes \mathbb{I})|\phi^+\rangle$1 (Correr et al., 26 Mar 2026).

Free probability yields a complementary random-dynamics benchmark. For Haar random dynamics, the LOE can be computed analytically for all Rényi indices, and for traceless normalized operators it asymptotically reproduces the Page curve for random states in the doubled Hilbert space, with exponentially deviating corrections (Dowling et al., 4 May 2026). At half-chain bipartition, the averaged von Neumann LOE takes the Page form

$|O\rrangle=(O\otimes \mathbb{I})|\phi^+\rangle$2

and the leading term is operator independent for traceless operators (Dowling et al., 4 May 2026). In contrast to higher-order out-of-time-ordered correlators, the leading-order LOE is independent of the initial operator (Dowling et al., 4 May 2026).

This operator independence at leading order suggests a reduced sensitivity of late-time LOE to higher cumulant data of the initial operator, relative to OTOCs (Dowling et al., 4 May 2026). A closely related observation in finite spin chains is that, in the thermalizing Ising model, the late-time value of operator-information diagnostics is well described by Page’s value for a random pure state (Mascot et al., 2020). Taken together, these results place LOE among the quantities whose chaotic saturation is controlled by random-state statistics.

5. Simulability, scrambling diagnostics, and non-Clifford resources

LOE has direct algorithmic significance. It dictates the complexity of simulating Heisenberg evolution using tensor network methods and serves as a strong dynamical signature of quantum chaos (Dowling et al., 30 Jan 2025). A rigorous formulation of this connection states that volume-law scaling of $|O\rrangle=(O\otimes \mathbb{I})|\phi^+\rangle$3-Rényi LOE entropies for $|O\rrangle=(O\otimes \mathbb{I})|\phi^+\rangle$4 implies that the operator cannot be approximated efficiently as a matrix-product operator while faithfully reproducing all expectation values (Dowling, 5 Mar 2026). By contrast, if one restricts to correlations over a relevant sub-class of ensembles of states, then logarithmic scaling of the $|O\rrangle=(O\otimes \mathbb{I})|\phi^+\rangle$5 Rényi LOE entropies implies MPO simulability; the covered tasks include infinite temperature autocorrelation functions, out-of-time-ordered correlators, and average-case expectation values over ensembles of computational basis states (Dowling, 5 Mar 2026).

This distinction between $|O\rrangle=(O\otimes \mathbb{I})|\phi^+\rangle$6 and $|O\rrangle=(O\otimes \mathbb{I})|\phi^+\rangle$7 is structural. For $|O\rrangle=(O\otimes \mathbb{I})|\phi^+\rangle$8, logarithmic LOE scaling yields polynomial bond dimension bounds for the relevant observables, whereas volume-law LOE for $|O\rrangle=(O\otimes \mathbb{I})|\phi^+\rangle$9 excludes efficient worst-case simulation (Dowling, 5 Mar 2026). The result puts on firm footing the heuristic expectation that low operator entanglement implies efficient tensor-network representability (Dowling, 5 Mar 2026).

LOE is also linked to operator scrambling diagnostics. OTOCs constitute a probe for LOE, and there is a quantitative upper bound on averaged OTOCs in terms of the Rényi-2 LOE entropy (Dowling et al., 2023). Yet the same work shows that rapid OTOC decay is a necessary but not sufficient condition for linear, chaotic growth of LOE (Dowling et al., 2023). In non-Hermitian local systems, the contrast becomes sharper: OTOCs fail to capture information scrambling in a simple non-Hermitian transverse-field Ising model, while operator entanglement remains a robust measure of dynamical properties of interest and detects entanglement phase transitions and long-time chaos indicators (Barch et al., 2023).

A further development places LOE in a resource-theoretic hierarchy with non-Clifford resources. LOE is always upper-bound by three distinct magic monotones: $|\Psi\rangle=\mathcal{N}^{-1}\mathcal{O}(x_0)|0\rangle,$ 0-count, unitary nullity, and operator stabilizer Rényi entropy (Dowling et al., 30 Jan 2025). In the average case for large random circuits, LOE and these magic monotones approximately coincide (Dowling et al., 30 Jan 2025). Related operator-space results establish that a unitary map generates nonlocal magic if and only if it generates operator entanglement on Pauli strings (Andreadakis et al., 12 Apr 2025). This suggests that LOE is not only a complexity diagnostic for tensor-network simulation, but also part of a broader operator-space characterization of nonclassical resources.

6. Quantum field theory, conformal field theory, and holographic LOE

In conformal field theory, LOE is formulated as excess entanglement of locally excited states. The basic replica-method quantity is

$|\Psi\rangle=\mathcal{N}^{-1}\mathcal{O}(x_0)|0\rangle,$ 1

with $|\Psi\rangle=\mathcal{N}^{-1}\mathcal{O}(x_0)|0\rangle,$ 2 the reduced density matrix of the excited state and $|\Psi\rangle=\mathcal{N}^{-1}\mathcal{O}(x_0)|0\rangle,$ 3 that of the vacuum (Nozaki, 2014). For free massless scalar theories and operators of the form $|\Psi\rangle=\mathcal{N}^{-1}\mathcal{O}(x_0)|0\rangle,$ 4, the late-time Rényi entanglement entropy is given by that of a classical binomial distribution, and the reduced density matrix is diagonal in the basis counting the number of quanta in the subsystem (Nozaki, 2014). The same work establishes a sum rule: for multiple, spatially separated single-species operators, the total Rényi entanglement entropy is the sum of the entropies of each excitation (Nozaki, 2014).

The earlier CFT formulation emphasizes late-time invariants of local operators. For a half-space subsystem, the late-time increase in entanglement entropy characterizes a given local operator from the viewpoint of quantum entanglement (Nozaki et al., 2014). In free scalar CFTs, specific local operators yield EPR-like values such as $|\Psi\rangle=\mathcal{N}^{-1}\mathcal{O}(x_0)|0\rangle,$ 5, while others give zero, in agreement with a heuristic picture of propagations of entangled particles (Nozaki et al., 2014). The physical interpretation is that the operator insertion emits an entangled pair of quasi-particles propagating at the speed of light in opposite directions, and the subsystem entropy records when the pair is split across the bipartition (Nozaki, 2014).

Large- $|\Psi\rangle=\mathcal{N}^{-1}\mathcal{O}(x_0)|0\rangle,$ 6 and holographic CFTs exhibit a different asymptotic structure. In large $|\Psi\rangle=\mathcal{N}^{-1}\mathcal{O}(x_0)|0\rangle,$ 7 CFTs, a naive large- $|\Psi\rangle=\mathcal{N}^{-1}\mathcal{O}(x_0)|0\rangle,$ 8 expansion can break down for the von Neumann entanglement entropy, while it does not for Rényi entropies (Caputa et al., 2014). For strongly coupled large $|\Psi\rangle=\mathcal{N}^{-1}\mathcal{O}(x_0)|0\rangle,$ 9 CFTs, the Rényi entanglement entropy of the excited state produced by a local operator grows logarithmically under time evolution, with coefficient proportional to the conformal dimension of the operator (Caputa et al., 2014). In two-dimensional holographic CFTs, the ordering of Euclidean and Lorentzian time evolutions after a local operator insertion is decisive: unitary time evolution induces late-time logarithmic growth of entanglement entropy, while non-unitary time evolution induces late-time constant behavior (Mao et al., 21 Dec 2025). For heavy primary insertions, the dual gravity description is a black brane with a spacetime-dependent horizon (Mao et al., 21 Dec 2025).

These CFT and holographic usages are not identical to the operator-space LOE of Heisenberg operators. A plausible implication is that the shared terminology tracks a common structural question: how a local operator, whether treated as an evolving observable or as a state-creating insertion, generates nonlocal entanglement under dynamical propagation.