Dissipation-Robust Quasilocalized State

• Dissipation-robust quasilocalized states are engineered phenomena in open quantum systems where phase-sensitive, local dissipation stabilizes localized modes and topological edge features.
• They can manifest as mixed steady states, boundary dark modes, or finite-size bound states, emerging from mechanisms like Anderson localization, non-Hermitian dynamics, and quasi-local Lindblad control.
• This approach demonstrates that targeted dissipation can be harnessed to protect and control quantum states, offering pathways for robust experimental realizations and quantum state engineering.

“Dissipation-robust quasilocalized state” is not a single standardized term in the current literature. It denotes a family of open-system phenomena in which dissipation does not merely fail to erase localization-like structure, but instead selects, stabilizes, or protects it. Depending on context, the relevant object may be a mixed steady state concentrated on localized Anderson modes, an exponentially boundary-localized topological state, a dissipationless localized bound state of an exact Green function, a finite-size atom-photon dressed state with quasilocalized photonic support, or a pure target state stabilized by quasi-local Lindblad control, where “quasi-local” refers to the locality of the control operators rather than to spatial localization of the state itself (Yusipov et al., 2016, Peng et al., 2024, Lin et al., 2024, Qiu et al., 10 Mar 2026, Ticozzi et al., 2011).

1. Terminological scope and principal realizations

The literature uses closely related but nonidentical notions. In disordered single-particle systems, the relevant object is typically a mixed asymptotic density matrix concentrated on selected exponentially localized Anderson modes. In topological settings, the object is usually an exponentially boundary-localized edge state or Majorana-related ground-state sector. In exact Green-function treatments, the robust object is a dissipationless localized bound state associated with a real pole outside the decaying continuum. In control-theoretic work, by contrast, the central notion is a pure state made globally asymptotically stable by quasi-local Lindblad generators, which is a locality-constrained stabilization problem rather than a spatial-localization problem (Yusipov et al., 2016, Peng et al., 2024, Lin et al., 2024, Ticozzi et al., 2011, Ticozzi et al., 2013).

Setting	Localized object	Representative paper
Disordered tight-binding chain	Mixed steady state over Anderson modes	(Yusipov et al., 2016)
SSH / Kitaev topological systems	Boundary-localized edge states or Majorana sectors	(Peng et al., 2024)
Odd SSH / Chern insulator with loss	Exact boundary dark mode	(Yang et al., 2023)
Dissipative photonic graphene	Finite-size atom-photon QLS	(Qiu et al., 10 Mar 2026)
Quasi-local dissipative control	GAS pure target state under locality constraints	(Ticozzi et al., 2011, Ticozzi et al., 2013)

This plurality matters because several papers explicitly do not use the phrase “quasilocalized state.” The SSH–Kitaev preparation work studies topological boundary states rather than generic disorder-induced quasilocalization (Peng et al., 2024). The two-mode Green-function work studies “dissipationless localized bound states,” not metastable resonances (Lin et al., 2024). The quasi-local stabilization literature studies DQLS and QLS states, where the adjective “quasi-local” modifies the control architecture, not the spatial profile of the target (Ticozzi et al., 2011, Ticozzi et al., 2013). A precise encyclopedia treatment therefore has to separate spatial localization, boundary localization, Hilbert-space freezing, and quasi-local control.

2. Phase-selective local dissipation as a localization mechanism

A recurrent mechanism is local, phase-sensitive, non-Hermitian dissipation acting on bonds or neighborhoods. In the Anderson-chain setting, identical local jump operators act on pairs of sites as

$$V_k=(b_k^\dagger+e^{i\alpha}b_{k+l}^\dagger)(b_k-e^{-i\alpha}b_{k+l}),$$

with the uniform phase $\alpha$ serving as a spectral selector. For $l=1$, $\alpha=0$ favors lower-band-edge Anderson modes, $\alpha=\pi$ favors upper-band-edge modes, and $\alpha=\pi/2$ makes the jump operators Hermitian, yielding the fully mixed state $\varrho_\infty=1/N$ (Yusipov et al., 2016). The key point is that the dissipator is not a projector onto a chosen eigenstate; it rewards a local phase pattern, and localized eigenmodes are selected because they inherit discriminable phase structure.

In the SSH and Kitaev settings, the bond-dissipative jump operators are number conserving and phase selective,

$$D_j=\sqrt{\Gamma_j}(c_{j+\ell}^\dagger+a\,c_j^\dagger)(c_{j+\ell}-a\,c_j),$$

with $a=\pm1$. Here dissipation does not create or destroy particles; it changes the relative phase between amplitudes on two sites. In the SSH chain, $a=-1$ favors the out-of-phase same-sublattice pattern encoded in the topological edge states, whereas in the Kitaev chain $\alpha$0 favors the ground state and $\alpha$1 favors the highest excited state. The effect depends crucially on placing the dissipation near the boundary, with exponentially decaying $\alpha$2 in SSH and end-bond-only dissipation in the Kitaev chain (Peng et al., 2024).

A third realization appears in clean lattices with aperiodic dissipation. There the bond operators are

$\alpha$3

with

$\alpha$4

The Hamiltonian itself is a clean nearest-neighbor tight-binding chain with only extended plane-wave eigenstates. Localization is induced by the dissipator, not inherited from the Hamiltonian. The paper reports that incommensurate modulation, $\alpha$5, is markedly more efficient than the commensurate case $\alpha$6, and that slow modulation $\alpha$7 supports localization while fast modulation $\alpha$8 destroys it (Roy et al., 18 Jun 2025).

Across these constructions, the concrete commonality is phase matching. Dissipation is engineered so that the target localized sector is closest to a dark state or least strongly damped state, while competing extended or bulk states are penalized. This suggests that dissipation-robust quasilocalization is often an interference-engineering phenomenon rather than a purely energetic cooling mechanism.

3. Mixed steady states, attractors, and localization diagnostics

In much of the literature the stabilized object is not a pure localized eigenstate but a mixed steady state with structured support in an appropriate basis. In the open Anderson model, the asymptotic state is nearly diagonal in the Anderson basis and concentrated on localized eigenmodes from selected spectral regions. In real space this can appear as several localized “hot spots,” because the steady state is a weighted mixture of a few localized modes rather than a single pure orbital. Quantum trajectories reveal intermittent dynamics with long-time sticking near selected localized modes interrupted by jumps between them (Yusipov et al., 2016).

The disordered XXZ chain with a delocalizing dissipator gives a more many-body version of the same theme. The steady state is mixed, and localization is diagnosed operationally. The main one-particle indicator is the weighted inverse participation ratio of natural orbitals,

$\alpha$9

For large disorder, $l=1$0 becomes large in two distinct regimes, weak interaction and strong interaction, whereas around $l=1$1 delocalized orbitals remain important. In the Hamiltonian eigenbasis, the occupations at $l=1$2 decrease approximately exponentially, while at $l=1$3 this decay is less pronounced and at $l=1$4 the profile becomes much flatter. Stronger dissipation $l=1$5 increases coherence and reduces both $l=1$6 and the concatenated inverse participation ratio $l=1$7 (Xu et al., 2021).

The aperiodic-dissipation work uses a different diagnostic triad: relative entropy of coherence $l=1$8, purity, and participation ratio

$l=1$9

For the incommensurate case at $\alpha=0$0, the slow-modulation point $\alpha=0$1 gives

$\alpha=0$2

whereas $\alpha=0$3 gives

$\alpha=0$4

The same paper studies $\alpha=0$5 and reports finite $\alpha=0$6, nearly unchanged purity, and stable PR at $\alpha=0$7, which it interprets as persistence in the thermodynamic limit (Roy et al., 18 Jun 2025).

A plausible implication is that “quasilocalized” in open systems often means localization of an attractor density matrix rather than strict localization of a single eigenvector. The appropriate basis may be the Anderson basis, the Hamiltonian eigenbasis, the site basis, the natural-orbital basis, or the Liouvillian rapidity basis, depending on which sector the dissipation actually organizes.

4. Boundary-localized topological states and Majorana sectors

A major branch of the subject concerns exponentially boundary-localized topological states rather than disorder-driven quasilocalization. In the SSH and Kitaev examples, boundary-focused bond dissipation prepares the boundary sector of the parent Hamiltonian. In the SSH chain, for $\alpha=0$8, $\alpha=0$9, and $\alpha=\pi$0, the total edge-state occupation is reported as $\alpha=\pi$1, with finite-size scaling

$\alpha=\pi$2

In the Kitaev chain, end-bond dissipation with $\alpha=\pi$3 yields $\alpha=\pi$4 for the ground state in both parity sectors, with

$\alpha=\pi$5

After dissipation is switched off, the diagonal occupations in the parent-Hamiltonian basis remain unchanged under unitary evolution, so the prepared edge-state or ground-state content survives (Peng et al., 2024).

A more stringent realization is dissipative boundary-state preparation by sublattice-selective loss. In the odd SSH chain, the zero-energy boundary mode

$\alpha=\pi$6

vanishes exactly on the lossy sublattice, so its Liouvillian rapidity is exactly $\alpha=\pi$7. Under ideal assumptions this gives an exact zero-rapidity boundary steady mode with infinite lifetime, while all bulk modes have strictly positive decay rates. In even SSH chains the surviving boundary mode is only exponentially long lived, with Liouvillian gap

$\alpha=\pi$8

but the same boundary-filtering principle remains operative. The construction extends to a momentum-resolved two-dimensional Chern-insulator geometry, again with loss on the nodal sublattice (Yang et al., 2023).

The dissipative atomic-wire construction realizes the same topological logic in a purely Lindbladian setting. The jump operators

$\alpha=\pi$9

prepare a $\alpha=\pi/2$0-wave-paired bulk dark state and leave a nonlocal decoherence-free Majorana subspace at the two ends of a finite wire. The bulk is separated from this edge sector by a dissipative gap; at the ideal point the damping spectrum is flat, $\alpha=\pi/2$1, so the minimal damping rate is $\alpha=\pi/2$2. Under deformations the edge modes remain exponentially localized, with localization length

$\alpha=\pi/2$3

and the topological protection is encoded by a winding number defined from the steady-state density matrix rather than from a Hamiltonian ground state (Diehl et al., 2011).

The experimental non-Hermitian photonic realization is different again: the boundary object is a finite-lifetime topological interface state rather than an exact dark state. In a one-dimensional 4-site-unit-cell lattice with uniform hopping and patterned losses, the non-Hermitian invariant is

$\alpha=\pi/2$4

The observed interface mode sits at $\alpha=\pi/2$5 and remains exponentially localized while $\alpha=\pi/2$6. Its absorption is smaller than that of a trivial isolated-waveguide configuration for $\alpha=\pi/2$7, but the mode delocalizes as hopping increases toward the exceptional-point regime near $\alpha=\pi/2$8 (Wetter et al., 2023). In this sense the literature contains both exact dissipation-protected boundary dark modes and quasilocalized topological resonances with finite attenuation.

5. Bound states, soft loss, and finite-size quasilocalization

Not all dissipation-robust localized states are steady-state density-matrix attractors. In exact Green-function treatments, the robust object may instead be a nondecaying pole. For the two-mode bosonic open system, the localized bound-state condition is

$\alpha=\pi/2$9

For symmetric coupling, the emergence thresholds are

$\varrho_\infty=1/N$0

The paper distinguishes three regimes: no bound state, one localized bound state, and two localized bound states. In long time this means, respectively, complete decay, nonzero asymptotic stationary amplitude/coherence, and persistent dissipationless oscillations with

$\varrho_\infty=1/N$1

This is a true localized bound-state mechanism, not merely a long-lived resonance (Lin et al., 2024).

The driven-dissipative spin-boson problem realizes a different kind of localization. Here “localization” means freezing of spin dynamics, not spatial confinement. With soft Markovian loss $\varrho_\infty=1/N$2, the bath kernel retains the long-time form

$\varrho_\infty=1/N$3

and the variational theory yields

$\varrho_\infty=1/N$4

so the transition occurs at

$\varrho_\infty=1/N$5

At this point $\varrho_\infty=1/N$6, the spin freezes, and the steady state becomes pure. The paper is explicit that the resulting pure state is not simply a trivial dark state of the original Liouvillian (Kamar et al., 28 May 2025).

A direct use of the phrase “dissipation-robust quasilocalized state” appears in dissipative photonic graphene. There the finite-size QLS is an atom-photon dressed state

$\varrho_\infty=1/N$7

with support entirely on the lossless $\varrho_\infty=1/N$8 sublattice. Its residue is

$\varrho_\infty=1/N$9

so the finite-size stabilized population is dissipation independent, while in the thermodynamic limit $$D_j=\sqrt{\Gamma_j}(c_{j+\ell}^\dagger+a\,c_j^\dagger)(c_{j+\ell}-a\,c_j),$$0. The same zero-energy singular structure gives logarithmic relaxation,

$$D_j=\sqrt{\Gamma_j}(c_{j+\ell}^\dagger+a\,c_j^\dagger)(c_{j+\ell}-a\,c_j),$$1

and a pronounced quantum Zeno effect. In the two-emitter problem, this QLS cooperates with the atomic dark state

$$D_j=\sqrt{\Gamma_j}(c_{j+\ell}^\dagger+a\,c_j^\dagger)(c_{j+\ell}-a\,c_j),$$2

to yield decoherence-free coherent exchange (Qiu et al., 10 Mar 2026).

By contrast, a single localized squeezed reservoir in a bosonic lattice does not produce an exact asymptotic quasilocalized steady state. What it does produce is a long-lived prethermalized or quasi-stationary intermediate state confined inside a ballistic light cone, together with weakly damped modes and quantum-Zeno suppression at strong loss. The relevant object there is metastable and intermediate-time, not the final steady state (Yanay et al., 2020).

6. Conceptual boundaries, common misconceptions, and limiting cases

The first conceptual boundary is between quasilocalized and quasi-local. In the stabilization literature, a pure state $$D_j=\sqrt{\Gamma_j}(c_{j+\ell}^\dagger+a\,c_j^\dagger)(c_{j+\ell}-a\,c_j),$$3 is DQLS if and only if

$$D_j=\sqrt{\Gamma_j}(c_{j+\ell}^\dagger+a\,c_j^\dagger)(c_{j+\ell}-a\,c_j),$$4

This criterion characterizes when quasi-local dissipators can make the target the unique globally asymptotically stable steady state. It is a statement about locality-constrained control, invariant supports, and parent-Hamiltonian structure, not about spatial localization of the target itself (Ticozzi et al., 2011). The later extension shows that when DQLS fails one may still achieve QLS with quasi-local Hamiltonian assistance or conditional DQLS relative to an initialization subspace; GHZ states are not scalably stabilizable from arbitrary initial states under bounded-size quasi-local resources, whereas W states admit two-body Hamiltonian-assisted QLS and scalable conditional DQLS on $$D_j=\sqrt{\Gamma_j}(c_{j+\ell}^\dagger+a\,c_j^\dagger)(c_{j+\ell}-a\,c_j),$$5 (Ticozzi et al., 2013).

A second misconception is that openness generically preserves localization. The papers do not support that claim. In the Anderson-chain work, Hermitian jump operators drive the system to the fully mixed state $$D_j=\sqrt{\Gamma_j}(c_{j+\ell}^\dagger+a\,c_j^\dagger)(c_{j+\ell}-a\,c_j),$$6, which is completely delocalized (Yusipov et al., 2016). In the SSH boundary-preparation work, homogeneous dissipation fails to select edge states; strong boundary concentration and correct phase structure are essential (Peng et al., 2024). The relevant robustness is therefore highly structured and conditional: it arises from engineered jump operators, dissipative gaps, sublattice nodes, soft infrared loss, or exact spectral zeros, not from generic decoherence channels.

A third boundary is with closed-system quasilocalization. Confinement-induced quasilocalized dynamics in the longitudinal-field Ising chain is a disorder-free unitary phenomenon generated by confinement of excitations, Stark localization, suppressed entanglement growth, and exponentially long string-breaking times; it is highly relevant conceptually, but it is not a dissipative realization (Lerose et al., 2019). Likewise, quasilocalized states of self stress in packing-derived networks are conservative mechanical null-space objects characterized by the length scale

$$D_j=\sqrt{\Gamma_j}(c_{j+\ell}^\dagger+a\,c_j^\dagger)(c_{j+\ell}-a\,c_j),$$7

with no explicit damping or Lindblad dynamics (Lerner, 2017). These works delimit, rather than instantiate, the dissipative concept.

Taken together, the literature supports a precise but plural conclusion. A dissipation-robust quasilocalized state may be a mixed steady state concentrated on localized modes, an exact boundary dark mode, a finite-lifetime topological interface resonance, a Green-function bound state, a finite-size dressed state with logarithmically vanishing residue, or a quasi-locally stabilized pure target. What unifies these objects is not a single formal definition but a common structural theme: dissipation is engineered so that the localized or quasi-confined sector is the attractor, the dark sector, the least-damped sector, or the zero-pole sector of the open-system dynamics.