OSQMBS: Open-System Quantum Many-Body Scars

• OSQMBS are nonergodic open-system structures that extend the conventional scar paradigm beyond closed Hamiltonians by exhibiting coexistence with thermal sectors.
• They are analyzed through several frameworks including effective constrained dynamics, Lindbladian embeddings into decoherence-free subspaces, and commutant-algebra methods.
• Revised ETH diagnostics and experimental studies on superconducting platforms highlight the practical implications of OSQMBS in controlling coherent quantum states.

Open-System Quantum Many-Body Scars (OSQMBS) denotes a family of nonergodic structures in open, effectively open, or open-system-dual quantum many-body dynamics that extend the scar paradigm beyond closed Hermitian Hamiltonians. In recent arXiv literature, the term is used in several non-identical but related senses: an effective open-system reformulation of kinetically constrained scar dynamics that leads to “Grand Canonical-like Thermalization” on an energy–quasiparticle-number plane (Wang et al., 6 Mar 2026); exact Lindbladian embeddings of scar towers into decoherence-free subspaces with persistent oscillations for generic initial states (Wang et al., 2023); a commutant-algebra definition of stationary pure states coexisting with a typical infinite-temperature stationary state on the orthogonal complement, together with asymptotic variants (Gotta, 22 Sep 2025); and a detailed-balance duality in which special Lindbladian eigenoperators become exact scars of a closed bilayer Hermitian Hamiltonian, notably a thermofield double state (Teretenkov et al., 2023). Experimentally, related work on single scars in superconducting-qubit platforms is relevant because it develops non-revival diagnostics under controlled imperfections, even though it does not formulate a Lindblad scar theory (Larsen et al., 2024).

1. Conceptual scope and relation to closed-system scars

Quantum many-body scars in closed systems are atypical excited eigenstates embedded in otherwise thermal spectra, often associated with ETH violation, low entanglement, and, in tower constructions, coherent revivals from special initial states. In the open-system setting, the same phenomenology is not transferred in a unique way. One branch of the literature treats scars as exact or asymptotically long-lived structures of a Lindbladian itself; another uses an effective open-system representation to reinterpret constrained unitary dynamics; a third identifies special open-system eigenoperators whose doubled-space images are exact scar states of a Hermitian bilayer model (Gotta, 22 Sep 2025).

A recurring misconception is that OSQMBS must mean a conventional scar that merely survives weak decoherence. The current literature is broader and more structured. In one formulation, OSQMBS are exact stationary pure states selected by an unconventional strong-symmetry structure, with all orthogonal initial states converging to an infinite-temperature state on the complement (Gotta, 22 Sep 2025). In another, a scar tower is embedded into a decoherence-free subspace (DFS), so that the asymptotic dynamics is not stationary but coherently oscillatory inside the DFS (Wang et al., 2023). In yet another, the “open-system” content is an exact non-Markovian representation of closed constrained dynamics, from which a revised ETH and an emergent quasiparticle number arise (Wang et al., 6 Mar 2026).

A common motif is the coexistence of an atypical sector with a thermal or infinite-temperature sector. This suggests a unifying interpretation, although the literature does not yet provide a single universally accepted definition. What is shared is that the exceptional sector is isolated relative to a thermalizing bulk, but the mathematical realization differs: stationary density matrices, nondecaying Liouvillian eigenoperators, operator-space eigenmodes, or low-density sectors in a generalized thermodynamic landscape.

2. Effective open-system formulation in kinetically constrained scar models

A direct open-system reinterpretation of scarred constrained dynamics is developed for a one-dimensional spin chain of length $D$ with periodic boundary conditions and on-site spin $j\ge 1$ (Wang et al., 6 Mar 2026). The unconstrained dynamics is

$H_0=\sum_{k=1}^{D}s_k^x .$

Kinetic constraints forbid neighboring sites from occupying

$$|x\rangle_{k,k+1}=|j\rangle_k\otimes |-j\rangle_{k+1},$$

and define the constrained Hilbert space through

$$\hat P=\prod_{k=1}^{D}\left(\mathbb I-|x\rangle_{k,k+1}\langle x|\right).$$

The constrained Hamiltonian is obtained by projection,

$\hat P H_0\hat P=H\hat P=\hat P H.$

The same dynamics can be represented by a non-Hermitian effective generator together with two compensating dissipative channels. The resulting equation is written as

$$\partial_t \rho = \mathcal{L}(\rho) \equiv -i H_N \rho + i \rho H_N^{\dagger} + \mathcal{J}(\rho),$$

with a jump term $\mathcal J$ that annihilates all states in the constrained subspace because $\pi_k\hat P=\hat P\pi_k=0$. Restricted to $\mathcal H$, the construction reproduces the original unitary dynamics exactly. In this sense, the dynamics is “open-system” only effectively: the constrained closed system is represented as if it were coupled to an auxiliary environment. One dissipation rate is negative, $j\ge 1$0, so the effective dynamics is explicitly non-Markovian and incorporates information backflow (Wang et al., 6 Mar 2026).

This representation motivates an emergent quasiparticle number. For $j\ge 1$1, the two jump probabilities are equal, $j\ge 1$2, and the information-exchange rate is proportional to $j\ge 1$3, where

$j\ge 1$4

The operator $j\ge 1$5 counts local $j\ge 1$6-patterns and measures how strongly the state participates in the virtual transitions implementing the blockade. It is neither exactly conserved nor universal; it is emergent and model-specific. For an eigenstate $j\ge 1$7, the quasiparticle number is

$j\ge 1$8

Numerically, scar states have anomalously low $j\ge 1$9, which is interpreted as reduced information exchange with the auxiliary environment (Wang et al., 6 Mar 2026).

The construction was studied numerically for $H_0=\sum_{k=1}^{D}s_k^x .$0, $H_0=\sum_{k=1}^{D}s_k^x .$1, where the model maps to the PXP model on a chain of doubled length. Two special initial states are emphasized: $H_0=\sum_{k=1}^{D}s_k^x .$2 The first exhibits strong revivals, while both have large overlap with scarred eigenstates (Wang et al., 6 Mar 2026).

3. Revised ETH, generalized density of states, and coherence diagnostics

Within the effective open-system description, the conventional ETH is replaced by a two-parameter, “grand canonical-like” version (Wang et al., 6 Mar 2026). The diagonal ETH ansatz becomes

$H_0=\sum_{k=1}^{D}s_k^x .$3

with $H_0=\sum_{k=1}^{D}s_k^x .$4 and $H_0=\sum_{k=1}^{D}s_k^x .$5. For an initial state $H_0=\sum_{k=1}^{D}s_k^x .$6, the long-time average is

$H_0=\sum_{k=1}^{D}s_k^x .$7

where $H_0=\sum_{k=1}^{D}s_k^x .$8 and $H_0=\sum_{k=1}^{D}s_k^x .$9. The associated effective ensemble is

$$|x\rangle_{k,k+1}=|j\rangle_k\otimes |-j\rangle_{k+1},$$0

Because $$|x\rangle_{k,k+1}=|j\rangle_k\otimes |-j\rangle_{k+1},$$1 is not conserved, this is not a fundamental thermodynamic grand canonical ensemble; it is an eigenstate-weighted effective ensemble over $$|x\rangle_{k,k+1}=|j\rangle_k\otimes |-j\rangle_{k+1},$$2. Nonetheless, it predicts long-time averages and reduced density matrices more accurately than the canonical ensemble for the scarred states studied numerically (Wang et al., 6 Mar 2026).

The off-diagonal ETH is generalized as

$$|x\rangle_{k,k+1}=|j\rangle_k\otimes |-j\rangle_{k+1},$$3

so the suppression factor is no longer a one-dimensional density of states $$|x\rangle_{k,k+1}=|j\rangle_k\otimes |-j\rangle_{k+1},$$4, but a generalized density of states on the $$|x\rangle_{k,k+1}=|j\rangle_k\otimes |-j\rangle_{k+1},$$5 plane. The smoothed DOS is defined by Gaussian kernel smoothing over eigenstate coordinates $$|x\rangle_{k,k+1}=|j\rangle_k\otimes |-j\rangle_{k+1},$$6. Low-density regions in this plane are predicted to support relatively large off-diagonal matrix elements, and therefore anomalously large temporal fluctuations.

To make the off-diagonal analysis less observable-dependent, the paper introduces the cross coherence purity (CCP),

$$|x\rangle_{k,k+1}=|j\rangle_k\otimes |-j\rangle_{k+1},$$7

where $$|x\rangle_{k,k+1}=|j\rangle_k\otimes |-j\rangle_{k+1},$$8 is the reduced cross-density matrix on subsystem $$|x\rangle_{k,k+1}=|j\rangle_k\otimes |-j\rangle_{k+1},$$9. For $$\hat P=\prod_{k=1}^{D}\left(\mathbb I-|x\rangle_{k,k+1}\langle x|\right).$$0, CCP reduces to the usual subsystem purity. For $$\hat P=\prod_{k=1}^{D}\left(\mathbb I-|x\rangle_{k,k+1}\langle x|\right).$$1, it quantifies the coherence retained between the two eigenstates on $$\hat P=\prod_{k=1}^{D}\left(\mathbb I-|x\rangle_{k,k+1}\langle x|\right).$$2. A rigorous bound implies that CCP controls the typical size of all local off-diagonal matrix elements on $$\hat P=\prod_{k=1}^{D}\left(\mathbb I-|x\rangle_{k,k+1}\langle x|\right).$$3, and the authors infer

$$\hat P=\prod_{k=1}^{D}\left(\mathbb I-|x\rangle_{k,k+1}\langle x|\right).$$4

They stress that this is not an exact theorem, since the CCP inequality is only an upper bound. Numerically, however, the inverse averaged CCP tracks the generalized DOS closely once data are binned in the $$\hat P=\prod_{k=1}^{D}\left(\mathbb I-|x\rangle_{k,k+1}\langle x|\right).$$5 plane and restricted to fixed $$\hat P=\prod_{k=1}^{D}\left(\mathbb I-|x\rangle_{k,k+1}\langle x|\right).$$6 windows (Wang et al., 6 Mar 2026).

This leads to a revised fluctuation estimate,

$$\hat P=\prod_{k=1}^{D}\left(\mathbb I-|x\rangle_{k,k+1}\langle x|\right).$$7

and an alternative CCP-based estimate,

$$\hat P=\prod_{k=1}^{D}\left(\mathbb I-|x\rangle_{k,k+1}\langle x|\right).$$8

Across all translation-invariant initial states tested, both estimates scale proportionally with the observed long-time fluctuation amplitudes. The anomalous fluctuations and quasi-periodic dynamics of scarred states are therefore recast as consequences of low-DOS regions in the generalized $$\hat P=\prod_{k=1}^{D}\left(\mathbb I-|x\rangle_{k,k+1}\langle x|\right).$$9 landscape rather than as isolated failures of thermalization (Wang et al., 6 Mar 2026).

The same low-DOS picture is connected to the approximate spectrum-generating algebra (SGA),

$\hat P H_0\hat P=H\hat P=\hat P H.$0

with small ladder deviation for scar states quantified by parameters $\hat P H_0\hat P=H\hat P=\hat P H.$1. Numerically, scar states have much smaller $\hat P H_0\hat P=H\hat P=\hat P H.$2 than thermal states, and the enhancement is attributed to a low generalized DOS that amplifies the relevant denominator $\hat P H_0\hat P=H\hat P=\hat P H.$3. In this interpretation, the SGA is not a competing explanation but a structure amplified in low-DOS sectors (Wang et al., 6 Mar 2026).

4. Lindbladian embedding into decoherence-free scar sectors

A distinct OSQMBS program constructs Lindblad dynamics whose asymptotic nondecaying sector is exactly the scar sector of a many-body Hamiltonian (Wang et al., 2023). The starting point is the Lindblad equation

$\hat P H_0\hat P=H\hat P=\hat P H.$4

If a scar tower $\hat P H_0\hat P=H\hat P=\hat P H.$5 satisfies

$\hat P H_0\hat P=H\hat P=\hat P H.$6

then the Liouvillian contains exact nondecaying eigenoperators

$\hat P H_0\hat P=H\hat P=\hat P H.$7

When the tower is equally spaced, the Liouvillian inherits a uniformly spaced set of purely imaginary eigenvalues, so observables display persistent periodic oscillations for generic initial states, including mixed states, after transient relaxation into the DFS (Wang et al., 2023).

The engineered jumps are taken as

$\hat P H_0\hat P=H\hat P=\hat P H.$8

where $\hat P H_0\hat P=H\hat P=\hat P H.$9 are local projectors annihilating the whole scar tower and $$\partial_t \rho = \mathcal{L}(\rho) \equiv -i H_N \rho + i \rho H_N^{\dagger} + \mathcal{J}(\rho),$$0 are local operators chosen so that the dark space is not accidentally enlarged. The Hamiltonian is essential: it preserves the desired scar subspace while rotating unwanted states out of the null space of the dissipators, after which those unwanted states decay. This mechanism differs from ordinary dark-state engineering, where the focus is usually on a stationary dark state rather than a coherently oscillating dark tower.

The tower structure is encoded by a restricted spectrum-generating algebra on the scar subspace $$\partial_t \rho = \mathcal{L}(\rho) \equiv -i H_N \rho + i \rho H_N^{\dagger} + \mathcal{J}(\rho),$$1,

$$\partial_t \rho = \mathcal{L}(\rho) \equiv -i H_N \rho + i \rho H_N^{\dagger} + \mathcal{J}(\rho),$$2

To find projectors annihilating the whole tower, the paper compresses the tower into

$$\partial_t \rho = \mathcal{L}(\rho) \equiv -i H_N \rho + i \rho H_N^{\dagger} + \mathcal{J}(\rho),$$3

and then uses an MPS-based construction to determine local annihilating projectors. This “compressed-MPS” procedure is the technical backbone of the general framework (Wang et al., 2023).

The paper demonstrates the construction in four representative models: a Dicke-state toy model, a spin-1 $$\partial_t \rho = \mathcal{L}(\rho) \equiv -i H_N \rho + i \rho H_N^{\dagger} + \mathcal{J}(\rho),$$4 model, the AKLT model, and a domain-wall preserving spin-$$\partial_t \rho = \mathcal{L}(\rho) \equiv -i H_N \rho + i \rho H_N^{\dagger} + \mathcal{J}(\rho),$$5 model. These examples illustrate three different situations: exact coincidence between the dissipator null space and the scar subspace; a larger null space that is filtered by the Hamiltonian; and the need for additional higher-locality projectors to remove unwanted exact dark states. The AKLT example is especially important because it shows that two-local projectors can leave irrational unwanted modes, requiring three-local refinement (Wang et al., 2023).

A practically relevant extension is dissipative preparation. If a strong symmetry $$\partial_t \rho = \mathcal{L}(\rho) \equiv -i H_N \rho + i \rho H_N^{\dagger} + \mathcal{J}(\rho),$$6 commutes with both $$\partial_t \rho = \mathcal{L}(\rho) \equiv -i H_N \rho + i \rho H_N^{\dagger} + \mathcal{J}(\rho),$$7 and all $$\partial_t \rho = \mathcal{L}(\rho) \equiv -i H_N \rho + i \rho H_N^{\dagger} + \mathcal{J}(\rho),$$8, and distinct scar states carry distinct eigenvalues of $$\partial_t \rho = \mathcal{L}(\rho) \equiv -i H_N \rho + i \rho H_N^{\dagger} + \mathcal{J}(\rho),$$9, then each symmetry sector evolves independently, and a simple product state in the appropriate sector relaxes to the target scar state. The paper further proposes a digital quantum simulation protocol with ancilla qubits that are repeatedly reset. A Hamiltonian step, a system–ancilla coupling

$\mathcal J$0

and an ancilla reset reproduce the Lindblad evolution to order $\mathcal J$1. Superconducting qubits, Rydberg arrays, and trapped ions are identified as candidate platforms (Wang et al., 2023).

5. Formal Lindbladian theory: exact stationary OSQMBS and asymptotic OSQMBS

A later work undertakes the problem of formally introducing quantum many-body scarring for Markovian open systems governed by the Lindblad equation and proposes a precise definition of OSQMBS in terms of stationary pure states and thermal complements (Gotta, 22 Sep 2025). In that formulation, a pure state $\mathcal J$2 is an OSQMBS iff

$\mathcal J$3

and, for every initial state $\mathcal J$4 such that

$\mathcal J$5

one has

$\mathcal J$6

with $\mathcal J$7. The scar is therefore an exact stationary pure state, while every orthogonal initial state thermalizes to the infinite-temperature state on its orthogonal complement (Gotta, 22 Sep 2025).

The theory is built from a bond algebra $\mathcal J$8 generated by local Hamiltonian terms and Hermitian jump operators, and its commutant $\mathcal J$9. The Hilbert space decomposes as

$\pi_k\hat P=\hat P\pi_k=0$0

OSQMBS arise when the commutant has an unconventional strong-symmetry structure: a one-dimensional scar block plus a single irreducible thermal block on the orthogonal complement. In the single-scar case, the stationary manifold is

$\pi_k\hat P=\hat P\pi_k=0$1

with $\pi_k\hat P=\hat P\pi_k=0$2. This shifts the scar notion away from revival physics and toward the stationary-state manifold of the Liouvillian (Gotta, 22 Sep 2025).

The framework also treats towers of scars. Depending on the commutant structure, the scar manifold can consist of multiple one-dimensional sectors or a higher-dimensional scar block. The paper studies benchmark spin-$\pi_k\hat P=\hat P\pi_k=0$3 and spin-1 constructions in which the atypical stationary states are ferromagnets or closed-system scar states adapted to the open-system commutant structure.

A further technical contribution concerns coherences between a scar sector and the thermal sector. For $\pi_k\hat P=\hat P\pi_k=0$4 and $\pi_k\hat P=\hat P\pi_k=0$5, the off-diagonal components evolve under an effective non-Hermitian operator

$\pi_k\hat P=\hat P\pi_k=0$6

and decay exponentially when $\pi_k\hat P=\hat P\pi_k=0$7 is gapped on the thermal block. For dephasing examples with isolated ferromagnetic scars, the paper derives explicit lower bounds on the decay rate of scar–thermal coherences (Gotta, 22 Sep 2025).

The same work introduces asymptotic OSQMBS (AOSQMBS): states that are not exact stationary states, do relax to the typical infinite-temperature state, but display anomalously large relaxation times. The proposed diagnostic is fidelity scaling. For one class of Lindbladians,

$\pi_k\hat P=\hat P\pi_k=0$8

while for another,

$\pi_k\hat P=\hat P\pi_k=0$9

These Ansätze are motivated by short-time expansions and are supported numerically for spin-1 tower models at modest system sizes. The paper emphasizes that the strongest exact results concern stationary-state structure and coherence sectors, whereas AOSQMBS are characterized operationally through scaling and finite-size benchmarks rather than by a single formal definition of the same precision as the exact OSQMBS definition (Gotta, 22 Sep 2025).

6. Detailed-balance duality and thermofield double scars

A complementary route to OSQMBS is based on an exact duality between open many-body systems satisfying detailed balance and closed bilayer systems with a self-adjoint Hamiltonian (Teretenkov et al., 2023). The open-system dynamics is written in the Heisenberg picture as

$\mathcal H$0

with Lindblad operators $\mathcal H$1 obeying

$\mathcal H$2

After a temperature-dependent similarity transformation and a Wick rotation $\mathcal H$3, the vectorized generator becomes a Hermitian bilayer Hamiltonian $\mathcal H$4 acting on $\mathcal H$5 (Teretenkov et al., 2023).

The operator-to-vector map is

$\mathcal H$6

Under this map, the identity operator is sent to the thermofield double (TFD) state,

$\mathcal H$7

and because $\mathcal H$8, one has

$\mathcal H$9

The TFD is thus an exact bulk eigenstate of the dual Hermitian bilayer Hamiltonian and is identified as a quantum many-body scar (Teretenkov et al., 2023).

The entanglement structure is a central feature. Tracing out one layer yields a Gibbs state,

$j\ge 1$00

so the interlayer entanglement entropy is

$j\ge 1$01

At $j\ge 1$02, the TFD reduces to the maximally entangled EPR scar; at $j\ge 1$03, the interlayer entanglement is minimized or vanishes for a unique ground state. This gives an exact scar with temperature-tunable entanglement, a feature that is unusual in the conventional scar literature (Teretenkov et al., 2023).

More generally, explicit Lindbladian eigenoperators $j\ge 1$04 satisfying

$j\ge 1$05

have expectation values

$j\ge 1$06

for arbitrary initial states. Under the duality, these eigenoperators generate additional exact scar states or scar towers of the bilayer Hamiltonian. The paper provides broad model classes, including dissipative $j\ge 1$07-type spin models, dephasing-like $j\ge 1$08 dissipators, bond dissipators, and generalized Hubbard models (Teretenkov et al., 2023).

This duality-based construction is foundational for OSQMBS, but it comes with a clear caveat: the scar lives in the doubled Hilbert space of the dual bilayer system, not in the original single-layer open system. The open-system analogue is therefore most naturally described as an “operator scar,” while the scar nomenclature is literal on the Hermitian bilayer side.

7. Experimental diagnostics, misconceptions, and current limitations

Experimental work on single scars in fixed-frequency, fixed-coupling superconducting transmons is closely related to OSQMBS because it develops diagnostics adapted to imperfect, noisy, and stroboscopic settings, even though it is not a formal open-system theory (Larsen et al., 2024). The proposal implements single-scar parent Hamiltonians using trotterized cross-resonance interactions and studies a fully $j\ge 1$09-polarized product scar and a cluster-state scar. Because a single scar does not produce tower-like revival phenomenology, the paper proposes alternative signatures: persistence of nonthermal local observables, persistence of low entanglement, comparison with locally deformed nearby states, sensitivity to controlled coefficient randomization,

$j\ge 1$10

and sensitivity to Trotter resolution. Exact Floquet analysis for $j\ge 1$11 finds one mode with overlap $j\ge 1$12 with the $j\ge 1$13-polarized scar and $j\ge 1$14 with the cluster scar; under controlled protocol distortion the scar remains noticeably more stable than its deformed counterpart up to about $j\ge 1$15. The paper is explicit that this is not Lindblad decoherence, but a controlled coherent distortion of the protocol (Larsen et al., 2024).

The present OSQMBS literature contains several nontrivial limitations. In the effective constrained-dynamics approach, the quasiparticle number $j\ge 1$16 is emergent, model-dependent, and not conserved, while the open-system representation is non-Markovian because one jump rate is negative; most numerics are for $j\ge 1$17, $j\ge 1$18 (Wang et al., 6 Mar 2026). In the commutant-based formal theory, the strongest results assume Hermitian jump operators and focus on exact stationary-state structure rather than a full Liouvillian spectral theory; the AOSQMBS evidence is finite-size (Gotta, 22 Sep 2025). In the DFS-embedding program, local projectors annihilating the whole tower must be found model by model, unwanted dark states can persist, and higher-locality corrections may be needed, as in the AKLT construction (Wang et al., 2023). In the duality framework, Hermiticity of the bilayer Hamiltonian requires detailed balance, and the scar is directly realized only in the doubled space (Teretenkov et al., 2023).

These caveats also delimit the main conceptual controversies. One is terminological: whether OSQMBS should refer only to intrinsic Lindbladian stationary or nondecaying modes, or also to effective open-system reinterpretations of closed constrained dynamics. Another concerns universality: current constructions are exact and illuminating, but strongly model-specific. A further open issue is extension to genuinely dissipative scar systems in which environment-induced decoherence competes with scar coherence rather than merely protecting, dualizing, or reparametrizing it. The literature repeatedly points to this unresolved direction, along with scaling to larger systems, relaxation beyond Hermitian jumps, and the status of quasiparticle-number constructions beyond constrained models (Wang et al., 6 Mar 2026).